The genesis of fluid mechanics, 1640-1780

The Genesis of Fluid Mechanics, 1640–1780

STUDIES IN HISTORY AND PHILOSOPHY OF SCIENCE VOLUME 22

General Editor: S. GAUKROGER, University of Sydney Editorial Advisory Board: RACHEL ANKENY, University of Adelaide STEVEN FRENCH, University of Leeds DAVID PAPINEAU, King’s College London NICHOLAS RASMUSSEN, University of New South Wales JOHN SCHUSTER, University of New South Wales RICHARD YEO, Griffith University

THE GENESIS OF FLUID MECHANICS 1640–1780 By JULIÁN SIMÓN CALERO

Library of Congress Control Number: 2008922901

ISBN 978-1-4020-6413-5 (HB) ISBN 978-1-4020-6414-2 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Original title: La génesis de la Mecánica de los Fluidos (1640–1780) UNED, Madrid, 1996

Translation: Veronica H. A. Watson

Printed on acid-free paper

All Rights Reserved © 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Acknowledgments

In my acknowledgements to the English edition, I would like to mention three persons who helped me with the original Spanish edition: Javier Simón Calero, my brother, for his comments; Eloy Rada García, for his efforts in publishing the work; and very specially to Manuel Sellés García, who was a decisive influence in the gestation of the entire work, in particular, he supervised with an eagle eye the more technical aspects of the book. In this new edition, I would like to include three new persons. The first is Veronica Watson, generous author of the translation of the Spanish text; she reworked all the quotations, starting from their original source languages. I acknowledge that my text was difficult, sometimes baroque and sometimes highly condensed; the result improves the original text considerably, in many occasions sweetening it and rendering it much clearer. Veronica took advantage of the different nuances of tone between the two languages, obliging me to rethink numerous paragraphs. The second person is Stephen Gaukroger, who was the first person to believe that the work was worth translating into English; he advised me on publishers, and in the end it was he himself who has edited the book. Stephen completed Veronica’s work, eliminating academic pedantry from the text, giving it an additional vivacity. The third person is Larrie Ferreiro, who, from the moment he knew it in its original version, has been its best and most assiduous reader to judge from his public comments, and to whom I am very grateful for his favourable views. For this new edition, he analysed the text in great detail, suggested improvements and made me complement and clear up obscure points. These three persons made me work considerably more, each in his own field. Although it is unnecessary, I cannot resist saying that I did it with great enjoyment. My profound gratitude to all, although I think that truly the ones who should be most grateful are the readers, as once the book arrives in their hands, my task concludes, and they become the proprietors of the text.

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Contents

ACKNOWLEDGMENTS ................................................................................................v ABBREVIATIONS ..........................................................................................................xi NOTE ............................................................................................................................. xiii INTRODUCTION.............................................................................................................1 PROLEGOMENON .............................................................................................................1 THE GENESIS OF FLUID DYNAMICS: A SUMMARY .............................................................5 PART I THE PROBLEM OF RESISTANCE............................................................................45 CHAPTER 1: THE FORERUNNERS OF IMPACT THEORY: HUYGENS AND MARIOTTE......................................................................................55 MEASUREMENT OF RESISTANCE ....................................................................................55 HUYGENS AND HIS EXPERIMENTS .................................................................................57 MARIOTTE AND THE THEORY OF JETS ............................................................................64 EXPERIMENT OF A JET AGAINST A PLATE.......................................................................71 CHAPTER 2: IMPACT THEORY: FORMULATION AND FORMALISATION ..............................................................................................73 THE CONTRIBUTION OF NEWTON ..................................................................................73 THE CONCEPT OF FLUIDS IN THE PRINCIPIA ...................................................................76 MOTION OF PENDULUMS ...............................................................................................81 RESISTANCE IN AERIFORM FLUIDS.................................................................................89 AQUIFORM FLUIDS (I)....................................................................................................95 AQUIFORM FLUIDS (II) ..................................................................................................99 EXPERIMENTAL RESULTS ............................................................................................107 FINAL COMMENTS ON THE PRINCIPIA ..........................................................................111 THE FORMALISATION OF IMPACT THEORY ...................................................................113 THE APPROACH OF JAKOB BERNOULLI ........................................................................114 THE SOLID OF MINIMUM RESISTANCE ..........................................................................122 THE SITUATION IN 1714...............................................................................................123

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CONTENTS

CHAPTER 3: THE EVOLUTION OF THE PROBLEM OF RESISTANCE ........................................................................................................125 PROBLEMS OF IMPACT THEORY ...................................................................................125 JOHANN BERNOULLI AND THE ‘COMMUNICATION OF MOTION’...................................128 THEORIES AND EXPERIMENTS OF DANIEL BERNOULLI ................................................132 JET AGAINST A PLATE: DANIEL BERNOULLI’S CLARIFICATIONS..................................137 THE NAVAL WORK OF PIERRE BOUGUER .....................................................................143 EULER’S SCIENTIA NAVALIS ..........................................................................................145 ROBINS’ NEW PRINCIPLES OF GUNNERY ......................................................................149 EULER’S STREAMLINE MODEL .....................................................................................158 D’ALEMBERT’S PARADOX ...........................................................................................167 OTHER WORKS OF EULER ............................................................................................175 THE THEORY OF JORGE JUAN ......................................................................................182 CHAPTER 4:

EXPERIMENTS ON RESISTANCE..............................................185

THE EXPERIMENTERS ..................................................................................................185 THE EXPERIMENTS OF BORDA .....................................................................................187 SQUARE PLATES ..........................................................................................................190 PRISMS ........................................................................................................................192 PYRAMIDS ...................................................................................................................192 SPHERES AND HEMISPHERES........................................................................................193 TOWING OF FLOATING CUBES ......................................................................................193 ROTATING ARM IN WATER ...........................................................................................194 CHAPMAN’S EXPERIMENTS..........................................................................................197 THE EXPERIMENTS OF BOSSUT ....................................................................................202 PROPORTIONALITY WITH THE SQUARE OF THE VELOCITY ...........................................207 PROPORTIONALITY WITH THE SURFACE.......................................................................208 PROPORTIONALITY WITH THE SQUARE OF THE SINE ....................................................209 CHAPTER 5:

FLUID-DRIVEN MACHINES AND NAVAL THEORIES .........215

FLUID MACHINES .........................................................................................................215 THE FUNCTIONING PRINCIPLES OF THE MACHINES ......................................................217 PARENT’S STUDIES ......................................................................................................221 PITOT’S WORKS ...........................................................................................................223 THE STUDIES OF BOSSUT AND OTHERS ........................................................................226 SMEATON’S EXPERIMENTS ..........................................................................................228 WINDMILLS IN EULER .................................................................................................237 NAVAL APPLICATIONS .................................................................................................242 THE FORCES IN THE HULL ............................................................................................246 FORCE IN SAILS............................................................................................................254

CONTENTS

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PART II THE PROBLEM OF DISCHARGE ...........................................................................267 CHAPTER 6: DISCHARGE FROM VESSELS AND TANKS ............................271 TORRICELLI’S LAW .....................................................................................................271 THE WORK OF THE PARIS ACADEMY ...........................................................................273 DOMINICO GUGLIELMINI .............................................................................................275 MARIOTTE’S EXPERIMENTS .........................................................................................278 DISCHARGE IN NEWTON ..............................................................................................282 GIOVANNI POLENI .......................................................................................................283 CHAPTER 7:

THE HYDRODYNAMICA AND THE HYDRAULICA .................293

DANIEL AND JOHANN BERNOULLI ..............................................................................293 THE BASIC PRINCIPLES OF THE HYDRODYNAMICA ........................................................298 MOTION THROUGH TUBES IN THE HYDRODYNAMICA ....................................................304 HYDRAULICO-STATICS ................................................................................................314 THE DISCHARGE OF ELASTIC FLUIDS ...........................................................................320 JOHANN BERNOULLI’S HYDRAULICA ...........................................................................326 D’ALEMBERT’S ACCOUNT OF MOTION IN TUBES .........................................................338 BORDA’S WORKS .........................................................................................................346 BY WAY OF SUMMARY…. ...........................................................................................352 CHAPTER 8: THEORETICAL CONSTRUCTIONS (I): CLAIRAUT AND D’ALEMBERT ...................................................................................................355 THE GRAND THEORISATION .........................................................................................355 THE SHAPE OF THE EARTH ...........................................................................................358 CLAIRAUT’S EQUILIBRIUM CONDITIONS ......................................................................360 GENERALISATIONS ......................................................................................................368 D’ALEMBERT ..............................................................................................................371 PRINCIPLES OF DYNAMICS AND HYDROSTATICS ..........................................................374 BODIES IN FLOWING CURRENTS ...................................................................................378 SOLUTION USING LATERAL FORCES .............................................................................381 SOLUTION BY CONSTANCY OF VOLUME.......................................................................386 BODIES MOVING IN A STATIONARY FLUID ...................................................................395 CONCLUSION: D’ALEMBERT’S CONTRIBUTIONS .........................................................399 CHAPTER 9:

THEORETICAL CONSTRUCTIONS (II): EULER....................401

LEONHARD EULER ......................................................................................................401 PRINCIPIA MOTUS FLUIDORUM ....................................................................................402 GENERAL PRINCIPLES OF THE STATE OF EQUILIBRIUM OF THE FLUIDS ........................419 GENERAL PRINCIPLES OF THE MOTION OF FLUIDS .......................................................426

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CONTENTS SEQUEL TO THE RESEARCHES ON THE MOTIONS OF FLUIDS .........................................435 EULER’S CONTRIBUTION .............................................................................................444 LAGRANGE’S PAPER ON FLUIDS ...................................................................................445

CHAPTER 10: APPLICATION OF FLUID MECHANICS TO PUMPS AND TURBINES ..........................................................................................................453 THE HYDRAULIC PUMP ................................................................................................454 PITOT’S THEORY OF PUMPS..........................................................................................456 PUMPS IN EULER’S WORKS ..........................................................................................462 BORDA’S STUDIES .......................................................................................................466 DANIEL BERNOULLI ON JET PROPULSION ....................................................................469 SEGNER’S TURBINE .....................................................................................................472 EULER’S ANALYSES .....................................................................................................475 PRESSURE IN THE TUBE................................................................................................477 TURNING TORQUE .......................................................................................................480 T ANK HEIGHT ..............................................................................................................481 SEGNER-EULER TURBINE ............................................................................................481 PITOT’S TUBE ..............................................................................................................490 RECAPITULATION: HYDRAULIC MACHINES..................................................................492 APPENDIX ....................................................................................................................493 UNITS OF MEASURE .....................................................................................................493 BIBLIOGRAPHY .........................................................................................................497 I PRIMARY SOURCES ..............................................................................................497 II. PERIODICAL PUBLICATIONS, COLLECTIONS AND COMPLETED WORKS..................505 III. SECONDARY SOURCES .........................................................................................506 DRAMATIS PERSONÆ..............................................................................................511 INDEX............................................................................................................................513

Abbreviations

Acta Erud.

Acta Eruditorum of Leipzig

Mém. Acad. Berlin

Histoire de l’Académie des Sciences et Belles Lettres of Berlin

Nouv. Mém. Acad. Berlin

Nouvelles Mémoires Histoire de l’Académie des Sciences et Belles Lettres of Berlin

Mém. Acad. Paris

Mémoires de l’Académie Royale des Sciences de Paris

Comm. acad. petrop.

Commentarii academiæ scientarum petropolitanæ of St. Petertsburg

Novi comm. acad petrop.

Novi commentarii academiæ scientarum petropolitanæ of St. Petertsburg

Phil. Trans.

Philosophical Transations of the Royal Society of London

Jour. Sav.

Journal de Sçavans

xi

Note

All measurements and sizes are expressed in the original units, followed by their conversion to the International System. The appendix lists the conversion factors used. Mathematical formulas and notation have been transcribed to present-day symbols, adding the constants required to render the formulas dimensionally consistent. This particularly affects the acceleration of gravity expressed by its customary symbol g, and consideration of density by its absolute value, that is to say mass/volume.

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Introduction

Prolegomenon In what follows we have set out to analyse the genesis of fluid mechanics as a modern scientific discipline. Two prior and interrelated questions must first be raised, however: what do we understand as the basic core of fluid mechanics, and when was this core developed? As seen today, fluid mechanics is a science of great complexity and diversity. Yet, the field derived from a common source, and evolved into a succession of disciplines, each one with an increasing degree of specialization. This hierarchy is headed by fluid dynamics and fluid statics, understood as applied to liquids and gases, which in turn gave rise to subsonic, transonic and supersonic disciplines. Although they all sprang from the same common theoretical basis, each discipline has specific problems determined by the nature of the fluid, the dominating phenomena, the motion undergone and the boundary conditions. Looking back, we find that the basic hypotheses that mathematically regulate fluid mechanics, the structure and formulation of its fundamental equations, and the problems of its applications were all forged in the second half of the seventeenth and first half of the eighteenth centuries, to the extent that the process can be considered in many respects complete by the end of the 1750s. Key concepts in fluid mechanics—such as turbulence, boundary layer, discontinuity surfaces, viscosity and thermodynamic processes—were introduced in the course of the nineteenth and twentieth centuries. We choose the decade of the 1750s as the jumping-off point for fluid mechanics, because a specific body of theory was built up by then, which (together with some experimental studies) provided the groundwork—despite a number of residual problems—for the modern scientific theory of fluids. This point was reached at the end of a century-long process in which the basic concepts gestated, were born, and evolved progressively with different theories and assumptions. It was only with the appearance of Isaac Newton’s Philosophiæ naturalis principia mathematica in 1687, that the scientific nature of the discipline made its first appearance, for this work analyses for the first time the dynamics of

1

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THE GENESIS OF FLUID MECHANICS, 1640–1780

fluids, basing itself on laws of nature of more general characteristics. Specifically, Newton introduces two basic premises: one concerning the definition of the constituents of the fluid, the other dealing with its behavior with respect to the general laws of dynamics as set out in the Principia. While it is true that before the appearance of this work, a large number of studies, some very worthy, existed on this subject, none of them provided the overall theoretical body of understanding supplied by the Principia. Nevertheless, despite a number of attempts over a period of 70 years, it was not until 1755 that Leonhard Euler was able to offer a definitive treatment. While this process opened with Newton and closed with Euler, there were considerable differences in their fundamental assumptions about the nature of fluids. For Newton, a fluid like air was an aggregate of particles that respond individually to the laws of mechanics, in such a way that the force generated by the impact of a current against an object consisted of the sum of the effects of each individual impact. For Euler, by contrast, a fluid was a continuum, ideally separated into elemental domains capable of supporting forces and internal pressures, whose space–time evolution is regulated by the laws of dynamics, expressed by a set of differential equations. These two radically different concepts were established in 1687 and 1755, respectively, and they define the termini of the genesis of fluid mechanics. Before 1685, the concepts of fluid mechanics were still based on a simplistic (to our eyes) view of the physical universe; but after 1755 those concepts were replaced by the construction we use today. The treatment of the development of fluid mechanics will deal with the dynamic aspects of fluids, liquids or gases subjected to forces, and limited by boundary conditions, as much as in their applications. Although static analyses are not dealt with as such, some aspects of fluid statics are looked at where this is needed to provide a more comprehensive picture. For example, in his 1755 work, Euler united static and dynamic accounts in the same theory, so in this case we deal explicitly to the former. On the other hand, we understand dynamics as the attempt to explain and reduce the motion of fluids in terms of the forces applied and the boundary conditions, and so do not deal with hydraulics, understood as the practical use of energy intrinsic to the movement of water. Although Newton is treated as the founder of the discipline, we have not neglected the contribution of several pre-Newtonian works, tracing the development of the discipline from Evangelista Torricelli’s work in 1644, through Christiaan Huygens and Edmé Mariotte. Among the post-Newtonians, the Bernoulli family (Jakob, Johann and Daniel), Benjamin Robins, Alexis Claude Clairaut, Jean Le Rond d’Alembert and Leonhard Euler play crucial roles, especially Euler’s 1755 work which brings the development to completion. In the 20 years after this, we witness a burst of activity in applied and experimental fields,

INTRODUCTION

3

in the work of Jean Charles Borda, Charles Bossut, Pierre Bouguer and Jorge Juan y Santacilia, so that the developments we are concerned with effectively come to a head only in the 1770s. Some clarifying points are worth developing before we begin. In particular, we need to say something about fluids from the point of view of theory, experiments and applications; we need to examine the distinction between hydraulics and hydrodynamics; and we need to raise the question of the relation between mechanics in general and fluid mechanics. Although it is common to think of science as a theoretical–experimental enterprise, problems arise when the question of applied science or of the application of science is introduced. For our part, we see a connection between both aspects, which we have called ‘applications’. We understand these to be the attempt to use available knowledge, be it theoretical or experimental, for modelling, explaining and reducing scientifically the behavior of machines, be they real or imaginary devices, but without these studies implying or intending their subsequent construction, which is a matter for technology.1 In a strict definition of science, one might be tempted to exclude these applications. But the advantages obtained in the theoretical analysis of any machine are a source of knowledge, obliging us to adjust the theories to a specific reality, something which is neither easy nor immediate. This is not the same as experiment, which has the great advantage of granting us freedom for creating an ideal space in order to render a specific effect. We treat the applications as forming part the science itself, on a different plane from experiments, but not constituting a separate group. Applications follow on from theory and experiments, and are continued by the actual construction of machines. This comes under the heading of technology, which is another subject, implying contributions of a different nature. Applications have another important aspect: they are the expression of a social need and through them society acts on science. They form another, additional nexus connecting the conceptual and social aspects of science. The relation between hydrodynamics and hydraulics comes under the ambit of interrelations between practical and theoretical worlds, if the former is understood as coming under the aegis of the theoretical world and the latter under the practical. We must remember that hydraulics flourished from antiquity as an 1

The problem of the technology is the production of technical entities, which is not exactly the same as engineering, although both partly coincide. The first step of the process starts by understanding the ideal object to be materialized. This is done according to the scientific knowledge available, which is sometimes very scanty or difficult to apply. In the second step this knowledge, together with materials provided by nature, allows these objects to be constructed, in a wide sense of the word, giving rise to the appearance of the technological sciences. In spite of the differences between science and technology, there is something both share: the scientific method. Therefore, the applications have to be treated scientifically.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

artisan activity, and remained so in the seventeenth and eighteenth centuries,2 its aim being to optimise the use, handling and distribution of water in watermills, waterwheels, canals, etc. It is natural that hydraulics posed problems which the sages of the time tried to study. However, hydraulics as then understood was not, nor did it become, a science in the modern definition. It remained a dignified artisan activity, which is what it had been since ancient times, with perhaps a few technological improvements. However, some confusion exists as to the terminology of the time, and until the term hydrodynamics appeared, the term hydraulics was often used in its place. D’Alembert tried to clarify the scope of both terms in the Encyclopédie Méthodique,3 but the word hydraulics continued to be used frequently in engineering circles. Nevertheless, while in its debt in some respects, hydrodynamics does not proceed from hydraulics. As regards mechanics, its evolution during those years deserves comment. The main aim of mechanics was the establishment of its general laws, allowing its problems to be reduced to mathematics. Several attempts were made to do this. One was Newton’s, with the establishment of the laws or axioms of motion, which introduced forces and the concept of mass. Newtonian mechanics assumed that these masses were points, or able to be reduced to points, and that the forces acted between these mass points. Previously, Huygens had considered the underlying principle to be the conservation of what he called live force (vis viva). This was the sum of the products of the masses multiplied by the square of the velocity of all the particles of the system. Conservation was limited to certain conditions, such as elastic collisions and processes under the action of gravity. In contrast to Newton’s laws, which deal with the interaction between individual particles, live force was extended to the entire fluid mass; which, together with the advance of infinitesimal calculus, made the live force theory very suitable for handling large groups of particles, the credit for this development going to Daniel Bernoulli. D’Alembert offered a different proposal, eliminating 2

The most important work on this subject Architecture Hydraulique de Bernard Forest de Belidor. The first edition appeared in 1737 and was published continuously until the nineteenth century. 3 In the volume III of the Encyclopédie Méthodique (1785), under the term Hydraulique it is said: ‘The part of Mechanics which contemplates the motion of fluids and which shows how water is channelled and how to raise them, as much for making the waters form a jet as for other uses. … The hydaulique deals not only with the water tubes and raising waters, but also the machines required to this end, but even more with the general laws of motion of fluid bodies. However, for quite a few years, mathematicians gave the name hydrodynamique to the general science of motion of fluids, and reserved the name hydraulique for the disciplines which, in particular, considered the motions of water: that is to say the art of channelling water, of raising it and handing it for the different requirements of daily life’. In the Hydrostatique talks of ‘the part of mechanics which considers the equilibrium of fluid bodies, together with the bodies which are immersed in them. … The hydrostatique is often confused with the hydraulique, as the subject matter is similar, and several authors hardly separate them at all’.

INTRODUCTION

5

the forces, and introducing conservation of the momentum in impact, along with a practical rule similar to virtual velocities. Subsequently, Maupertuis established the principle of minimum action. Whatever the principle was, the aim was the same: to obtain the laws of motion. In this connection, a little more needs to be said on the relation between fluid mechanics and mechanics understood in a general sense. If, as we have noted, hydraulics followed its own path, fluid mechanics emerged as part of mechanics.4 This is palpable in Newton, who dedicated Book II of his Principia to fluids, and the Book I to solids, but in both cases with the same title. This way of proceeding is also clear in Mariotte, who proposes some models of what we have called mechanics of jets. The mathematicians of the eighteenth century contributed greatly to mechanics, and they treated fluids as one of three types of bodies: solids, deformable bodies and fluids. The difficulty in treating fluids mathematically was enormous, and the help of differential analysis was absolutely decisive. Moreover, the process of mathematization in Euler and d’Alembert requires the definition of fluids as a continuum, in spite of the fact that they imagined them to be physically constituted by an aggregate of particles. It is interesting to note that in the second half of the 1700s, fluid mechanics, or rather the rational mechanics of fluids, was at the vanguard of theoretical mechanics. The genesis of fluid dynamics: a summary Let us begin with a brief summary of the entire process of the genesis and evolution of fluid mechanics in order to aid comprehension, and to place the rest of the work in context. It is a core thesis that this entire evolution took place along two main lines of activity: one dedicated to the effects that fluid current exercises upon a body immersed in it, and the other to dealing with how fluids discharge themselves through tubes or reservoirs. We have called the first ‘the problem of resistance’ and the second the ‘problem of discharge’. Almost all the authors of the time treated these subjects separately, and set them in one or other

4

In the Preface of the Principia, Newton says: ‘In this sense rational mechanics will be the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever. The ancients studied this part of mechanics in terms of the five powers that relate to the manual arts. … But since we are concerned with natural philosophy rather than manual arts, and are writing about natural rather than manual power, we concentrate on aspects of gravity, levity, elastic forces, resistance of fluids, and forces of this sort, whether attractive or impulsive. And therefore our present work sets forth mathematical principles of natural philosophy. For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces’. [p. 382]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

of these two contexts, sometimes even in the same book. Both enjoyed considerable independence, and a set of theories existed in each one, with its experiments and of course with its applications. Their methodology and evolution are not strictly comparable, as there are considerable differences between them, which makes them all the more interesting. Both respond to two sets of different preoccupations: in the case of resistance, to navigation problems and to machines driven by fluids such as mills and waterwheels; in the case of discharge, to water distributions and to jet reaction machines. The first was more practical and of more immediate interest; the second more theoretical, but conceptually superior, as the subsequent history bears out. In the main body of this work, we have separated both parts, dedicating the first part to the problem of resistance, and the second to that of discharge, and in each of these we have dealt with the theoretical, experimental and application aspects separately. This way of the presenting the facts emphasizes the coherence of each the two lines, but it breaks the general chronology of the work, so in this Introduction we shall describe the evolution of fluid mechanics as a whole, following a single chronological line, while making reference to the resistance or the discharge in each case, according to its type. We establish the starting point of the development, and more specifically of the line denominated ‘the discharge problem’ in Torricelli, who, in 1644, announced the law known today by his name, and which says that the efflux velocity of a liquid contained in a recepticle through an orifice is that which a heavy body would acquire when falling from a height equal to the depth at which the orifice is found.5 The law appeared in the work De motu gravium (On the motion of heavy bodies), which deals with the motion of projectiles, and was a result of his ballistic studies. He tells us that the clue that led him to this law is the commonly observed fact that when a reservoir discharges through a vertical tube acting as a spout, the water reaches a height very close to the level of the reservoir (Fig. I-1a). He observed that the jet ended up converted into droplets in its highest part, and he attributed this to its losing some height. It is interesting to note the empirical origin of his law, which is not so obvious, as the efflux of water is a different phenomenon from the fall of the heavy body that serves him as a reference. Should the orifice be in the lateral wall, he indicates that the trajectory followed would be a parabola, similar to the one a projectile would describe (Fig. I-1b). What Torricelli does, is to assimilate the water to a body in free fall. That is to say, he converts the liquids into an aggregate of solid bodies.

5

Mathematically, it is expressed as v e = 2 gh . That is, the efflux velocity is proportional to the square root of the depth.

INTRODUCTION

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The law, apart from being the first one to refer to fluids in motion,6 is of considerable interest for various reasons. First, it identifies fluids with solid bodies, in spite of their having such a different appearance and behaviour. Second, he proposes a mathematical relation for the first time. Third, he opens up an experimental field that would be a focal point for the next 50 years. In addition, it was the starting point of the discharge problem.

Fig. I-1. Torricelli springs

As regards the paternity of this law, Torricelli, who was in Italy when he wrote the aforementioned book, says that it was already previously known by Benedetto Castelli, his master. However, later authors, such as Daniel Bernoulli or Giovanni Poleni, say that some 3 years before (i.e., around 1641), Castelli had supposed that the outlet velocity was proportional to the depth, instead of its square root, as Torricelli had stated. It is difficult to know which account is correct; the fact is that the law was attributed to Torricelli by his contemporaries. The apparatus used for the experimental verification of this law was based on collecting water flowing out in the discharge during a certain measurable time interval, while the reservoir was kept full. Once the surface of the orifice was known, it was easy to calculate the outlet velocity of the water. A variant consisted in not filling the vessel during the process, and measuring the time it took to discharge completely, as by calculation this time was found to be double that in which an equal quantity of water would be discharged if the level of the vessel was maintained. Whichever procedure was used, the reality was that this apparently clear and simple experiment provided disparate results, a fact that caused enough headaches to stimulate an in-depth study of the phenomenon. Huygens and Mariotte in the newly founded Academy of Sciences of Paris, 6

The hydrostatics laws are older. The first one is due to Archimedes in the third century BC, and others were found by Simon Stevin (1548–1620) and Blaise Pascal, who was a contemporary of Torricelli.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

and Guglielmini and Poleni in Italy, are the only ones known to have published results, although many more must have tried. Huygens performed experiments in Paris in 1668, and his results confirmed Torricelli’s law as opposed to the supposition of Castelli.7 However, in the following year he went back on what he said, and questioned the previous results. What happened was that in the experiment less water was collected than predicted, and it would appear that the same thing happened in the various attempts made by him and other experimenters. We now know that the discharge process is not purely kinematic, as Torricelli’s law supposes, but dynamic, and all the water held in the recepticle intervenes in the phenomena; moreover, details such as the form of the vessel, the type of spout and the relation between the surface of the orifice and the surface of the reservoir play an important part. This was then unknown, and was only discovered with great difficulty. According to the sentiment of the times, these anomalies could be interpreted in two ways: either this velocity was not proportional to the square root of the depth; or the height, when considering the fall, was not that of the reservoir itself but a fraction of this. The experiments pointed towards this last alternative, which gave rise to the initiation of a process for revising the law, admitting the mathematical form of the square root, but changing the proportionality constant. Moreover, as the geometry of the apparatus influenced the results, the discrepancies among the measurements obtained by all and sundry were significant. Huygens’ work on motion in fluids was not just limited to discharge: he also devoted considerable effort to the effect undergone by bodies moving inside a fluid. Huygens, like Torricelli, was also a student of ballistics, and one of his preoccupations was to introduce the effect of the resistance of air into the motion of projectiles. If the trajectory followed by projectiles in the supposed absence of a resistant force was a parabola, when the resistance is included, the trajectory deviates from this geometric shape, and in addition, the determination of its path involves a considerable mathematical complication. Initially, it appears that Huygens estimated that the resistance encountered by the projectile was proportional to the velocity, but, after the experiments he made in 1669, he deduced that this was not so. Instead it was proportional to the square of the velocity. He introduced this law for the first time, and, with some nuances, it is still considered to hold. The ‘problem of resistance’ also started at this point: as we shall see, it is different in nature from that of the discharge, although it also originated in ballistics. An initial comparison between the two shows that the problem of resistance was dynamic in character, while discharge was rather more kinematic,

7

Cf. Jean-Baptiste du Hamel, Regiæ Scientarum Academia Historia Parisiis, 1698.

INTRODUCTION

9

even in spite of the fact that the concept of force was not yet clear.8 In order to measure the resistance, Huygens assimilated this to the weight of a column of fluid whose base was the frontal surface of the body in question. For the reference height of the column, he took the height from which the moving body would be dropped, in order for it to acquire a velocity equal to its motion; thus it was that the resistance was measured as being equivalent to one or two, or several times this height. We have called this height the ‘kinetic height’, and, as we shall see, it will play a fundamental role in the study of fluids. This height is the same as the depth in Torricelli’s law.9

Fig. I-2. Huygens experiments

Huygens carried out experiments on resistance following three different procedures as detailed in Fig. I-2. The first consisted in measuring the force produced by a jet against a plate, which was an extension of the discharge phenomenon. The second was to measure the towing force of a small ship in a pond; a method which has been perpetuated up to the present in the towing tanks that began to be built in the middle of the eighteenth century. The third, which also makes use of the motion of the object in a fixed medium, consisted of small sleds transporting a plate attached by counterweights, in such a way that the 8

To express it, they used terms such as impulse, impressions and thrusts and it would not be until Newton that the concept of force in a modern sense would enter physics, although it took some time to be admitted. However, the problem of force in Newton has to be dealt with cautious; see Westfall, Force in Newton’s Physics, especially Chapter 7. 9 Its mathematical expression was hc = v2/2g. Let us observe its identity with the Torricelli’s law.

10

THE GENESIS OF FLUID MECHANICS, 1640–1780

plate fell when the force produced by the moving air was superior to the value adjusted by the counterweights. As regards the last two methods, we can only say that his idea was clear, but that the capability of the instruments available to him was very rudimentary, although the methods were actually ahead of the times. Concerning the first of the three procedures, nowadays we know that the phenomenon of the impact of a jet against a plate differs from the resistance of a body submerged in a fluid. However, Huygens assimilated both in a single phenomenon, an idea that was a source of problems until Daniel Bernoulli separated them definitively in 1736. The jet experiment contributed to this identification, due to its closeness to the discharge. Diagrammatically, the matter was very simple: if the plate against which the jet strikes were to close the fluid outlet completely, the force on it would be the product of the static pressure times the outlet surface, i.e., the weight of a column of water whose height was the depth and area of the outlet. However, in the measurements made during the discharge, they found that the force was greater, and they could not find an explanation for this fact. Huygens did not confine his experiments to jets of water, but dealt also with discharges of air jets on plates, using an interesting apparatus that later served Mariotte as a standard. Huygens ended by giving specific values to the resistance of a plate moving in air or water, and the fact that his values were high, according to our present knowledge, does not detract one jot from his merit, given his rudimentary measuring apparatus. We have seen that Huygens approached the two major problems of fluids, resistance and discharge with different methods. A little later on we shall speak of Mariotte, who probably co-operated in the experiments with Huygens, and followed a very similar approach. However, for the sake of chronology, we shall go on to Domenico Guglielmini, who in 1683 carried out some experiments that are worth mentioning, and which were published in 1690 in his work Aquarum fluentium mensura (The measurement of the motion of the waters). Guglielmini is considered as belonging to the so-called Italian School of Hydraulics, a denomination that we owe to René Dugas,10 who traces its origins back to Galileo in the sixteenth century. The existence of the school was due to different factors, above all the Italian scientific flowering in the Renaissance, and the peculiar importance to Italy of solving practical problems of floods, lagoons and marshes. Guglielmini approached the problems of the motion of water in canals and its discharge from reservoirs. In order to study them he used an apparatus consisting of a barrow that discharged itself by successive regularly 10

Carlo Maccagni also calls in that way in his article ‘Galileo, Castelli, Torricelli and others. The Italian School of Hydrodynamics in the 16th and 17th centuries’. Hydraulics and Hydraulic Research. A Historical Review.

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spaced lateral orifices (Fig. I-3). The test procedure consisted in allowing the water to pass through an orifice, while maintaining the others sealed, and measuring the water flowing out in a specific time interval. Comparing the results obtained for the successive holes, he deduced the perfect proportionality of the outlet velocity with the square root of the height, as predicted by the Torricelli’s law. However, he did not base himself on the former measurements for the absolute velocities, but made a new reference measurement whose result he took as standard, and which curiously enough did not coincide with any of the previous ones. Using this standard value and the square root law, Guglielmini presented a table of velocities as a function of the height. If we apply Torricelli’s law to these, the height to be considered in the fall would be a quarter of the depth of water in the reservoir, a fact that Daniel Bernoulli made note of years later in his Hydrodynamica.

Fig. I-3. Guglielmini’s bucket

In these very same years, Mariotte carried out several studies and experiments which he published in his Traité de mouvement des eaux (Treatise of movement of the waters) which appeared in 1686, 2 years after his death. As the title of the work indicates, it deals with several problems: sources, winds, etc., and among these those of discharge and resistance are to be found. In general, Mariotte views his work as a practical treatise, thus including numerous measurements and application rules. In particular, his experiments on the outlet of fluids were fairly well systematized. He successively analysed the law of flows, and later he measured them, calculating the effect of the outlet area. Like Guglielmini, he not only found that the square root law was satisfied, but also that the height did not correspond to the depth, a fact that, as we have said, was repeated in almost all the experiments.

12

THE GENESIS OF FLUID MECHANICS, 1640–1780

He carried out several experiments regarding the measurement of resistance whose results were cited for a century. But apart from the experiments, Mariotte felt the need for a basic theory in order to handle the effects of the resistant forces. For this he assimilated these forces to the impact of what he called a ‘jet’ against a body, and he drew up five rules, defining this ‘mechanics of jets’. He imagined one jet as a cylinder of fluid moving like a solid; and given that he supposed the fluid to be formed by an infinity of small corpuscles, the jet would be a set of particles moving together. The behaviour of a jet before an impact against a plate would differ from that of a solid body, but in essence it would be the sum of the individual impacts against the obstacle. From this he deduces, as Huygens had likewise done, that the effect of the impact of a jet against an obstacle is proportional to the square of its velocity. The argument he brings to bear would become classic: if the fluid goes twice as fast, then double the number of particles will impact at the same time, and with double the effect of each one, so that the total effect will be four times greater. The interest of his theoretical proposals is complemented by the experiments. Even though his jets are a theorization of the phenomenon of the impact of the jet against a plate, he made the experiments with a plate totally submerged in a current, actually a river (Fig. I-4). Huygens, we remember, moved the models in fluids at rest. Mariotte did the opposite, these tests being the first of this type we know of. He interpreted the results in terms of his mechanics of jets, obtaining some standard values which he took as reference. If his tests are

Fig. I-4. Force on a plate in a flow

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interpreted under the form of the modern coefficient of resistance CD,11 we arrive at a value very close to 1, which is the equivalent of the weight of a column of liquid whose height is equal to what we call the kinetic height. Apart from the tests on the plate submerged in the current of a river, he also proposed another one for plates in air flows, although he presents the apparatus by itself without any reference whatsoever to a measurement. On comparing the values of resistance in the air and water, he established that the effects were proportional to the densities, another rule which also would persist. On the other hand, in the tests in water, this impinges on the plate perpendicularly, while in the case of air, it does so at an angle of 45°, thus breaking up the impinging force into two components. Mariotte interprets the phenomena as percussions, and in this sense this appraisal is the first appearance of the ‘impact theory’ which we shall presently consider. The question why Mariotte did not follow the experimental method of Huygens is not easy to answer; but we conjecture he was influenced by the fact that the results of the experiment of the jet against a plate were quite uncertain, because he was familiar with water mills and experimented with them on the river Seine, as he mentions in his Traité. To sum up, Mariotte’s contribution rightly merits the qualification of precursor that Dugas awards him. To recapitulate, during these years, on the one hand we find that experiments with discharges are only partially reconciled with their theoretical predictions, and on the other hand that there appears to be a proportionality of the resistance with the square of the impinging velocity, and with the density of the medium. Besides (and this is an important matter) the need was already felt to establish a theoretical basis enabling these phenomena to be explained, a need that Mariotte had anticipated. It is in this context that the first edition of Newton’s Principia saw the light. While Books I and III, respectively titled ‘The Motion of Bodies’ and ‘The System of the World’, are well known, Book II is not. It is also named ‘The Motion of Bodies’ although it deals with fluids, while the first deals with rigid bodies. The coincidence in the titles indicates that Newton put both solids and fluids under the same umbrella. However, Book I is structured as a closed set, characterised by an attempt to derive all motions from a few axioms or laws of motion. This is not the case in Book II, where he has to use several different additional hypotheses, which at the very least complement the others. In this respect, insufficiently justified suppositions abound, together with some small fudges and implausible constructions. In spite of this, Book II was of capital 11 The definition of this coefficient is CD = F/(½ρv2S), where F is the resistance force, ρ the fluid density, v the velocity and S the frontal surface. The meaning of this coefficient, as the times sense, was equivalent to the number of kinetic height the cylinder of equal weight that the resistance force would have. (cf. later Chapter 1, note 3).

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THE GENESIS OF FLUID MECHANICS, 1640–1780

importance throughout the eighteenth century, and almost all authors referred to it, some in order to follow it, others to refute it. Book II deals with several different problems and the most significant one refers to the motion of projectiles. He dedicates the first sections to the dynamics of projectiles in dense media, that is, to bodies whose movement is resisted by forces depending on the velocity, be it in simple proportion, squared or a combination of both. Assuming the existence of this resistance and its mathematical form, the resulting problem is similar in nature to that dealt with in Book I, of which these sections are but an extension. We deduce from this that Newton arrives at the problem of resistance due to his interest in ballistics, as Huygens also did, and unlike Mariotte, who arrived at it via hydraulic machines. Newton proposed a definition of a fluid as a body whose parts yield to any force impressed on it. However, he encounters difficulties when he tries to explain the different sources generating resistance. In terms of our present understanding, Newton considered the existence of a resistance produced by the forces of inertia, which he calls ‘inertia of matter’, and another resistance produced by viscous forces. His idea concerning the latter was a lot less clear, and he called it ‘lubricity, tenacity or fluidity’. Although he established the definition of viscosity as a shear force proportional to the variation of the velocity, the range of viscous phenomena and their mechanisms lay outside his horizon. This explains why he gave them various names according to the aspects they presented.

Fig. I-5. Pendulums in resistant media

Regarding resistance, and with the aim of determining its mathematical form by experiment, Newton proposed using pendulums oscillating in fluids (Fig. I-5). The idea was simple. The pendulum would maintain the amplitude of oscillation constant if there were no resistant forces. If these forces did exist, the amplitudes would be reduced with time, progressively diminishing until the pendulum came to rest, so that Newton could infer the mathematical laws of

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resistance from the observation and measurement of these diminished motions. Therefore, he introduced an extensive study of the motion of pendulums in resistant media, and arrived at the point of establishing a relation between the ratio of these reductions and the mathematical expression of the velocity in this law. Once the theory of the instrument had been established, he performed a set of experiments with pendulums of various sizes, making them oscillate in air, water and mercury. However, the conclusions he obtained did not indicate that the resistance was proportional to the square of the velocity, as was expected, and in consequence Newton tried to approach the problem by polynomials with different powers of velocity. What appeared more obvious to him was the proportionality of resistance to the cross section of the oscillating mass, and to a lesser degree to the density of the fluid. From the comments he makes, one can infer that the results of the experiment did not satisfy his expectations, but that in spite of these dubious results, he continued to think that the resistance was proportional to the square of velocity. After the work with the pendulums, which comes within the framework of experimental physics, Newton directly tackled the case of bodies in fluid currents or vice versa, that is, moving bodies in a fluid at rest. He dedicated the most memorable sections of Book II to this problem. He begins with a differentiation of behavior among liquids or uncompressible or non-elastic fluids, as opposed to air, which he qualified as being compressible or elastic. Although he considered both classes to be aggregates of particles, he established a difference between them: in the air the particles are separated from each other and repel each other due to forces which he called ‘centrifugal’. These are inversely proportional to the separation existing among the particles; for liquids all particles are in contact, and are only united by the forces of lubricity. That is to say, the particles in the air are easily individualized, while this is not true for those of a liquid. The theories in each case are different, we ought to say qualitatively different. But there is more; in the second edition of the Principia, published in 1713, Newton rewrote the section referring to liquids almost completely, although he maintained the same methodology as in the first edition, which also obliges us to treat the topic in two different moments. It is in the study of the motion of bodies in air that we consider fluid mechanics as becoming a truly modern scientific discipline, as it is there that Newton, in order to explain the phenomenon, introduces three basic hypothesis: one on the constitution of air, a second on how moving air and a body in its midst interact, and the third on the existence of a law regulating these interactions. The latter is the second law of dynamics. As we have said, he took a set of individual particles submitted to repelling forces for the air model, and for the interaction he adopted the mechanical model of impact. However, although the

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THE GENESIS OF FLUID MECHANICS, 1640–1780

theory can be constructed with these three elements, the difficulty of calculation was almost insurmountable, as the motion of one particle of air affected all others to a certain extent, given the repulsion among them. In order to solve this difficulty, he simplified the air model by eliminating these repulsive forces, thereby rendering the particles immobile. He called the fluid resulting from this operation a ‘rare medium’, which, although it did not have the properties that he attributed to air, would behave in a very similar way to air. In this new medium, the phenomenon would be modeled as the motion of all the particles, completely independent from each other, that would rebound elastically in the impact against the body, or would remain immobile if the impact was not elastic. In either case, they would transmit a certain momentum to the body, whose variation with time would be the resistant force.

Fig. I-6. Impact theory

This model, which we have called the ‘impact theory’, was already sketched by Mariotte, although it acquires its authentic meaning in Newton. He supposes (Fig. I-6) that there is a deflection in the impact, depending on the angle of incidence. The corollary of the supposition is the existence of shadow zones where the fluid cannot impact. As we shall see, this theory will remain in force in fluid dynamics for more than a century. It will be criticized and doubted, but because of the lack of a better theory it will prove essential due to its instrumental function. Newton applied this theory to a plate and a sphere. In the case of an elastic rebound, he obtained the values CD = 4 and CD = 2, respectively12 as resistance coefficients which were reduced by half when dealing with a non-elastic impact. 12 We quoted numerical values because the calculation of the actual magnitudes of these coefficients was one of the major milestones in the evolution of the problem of the resistance.

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In addition, he made a short incursion into the case of a solid of minimum resistance. When dealing with liquids, neither this model nor the impact theory was of use to him, as the particles are not only innumerable, but also they rub together, and when they impact with the object they transmit this action to the ones behind in a domino effect. Faced with this difficulty, Newton was forced to explain the phenomena using an alternative method. The model he used was the vertical discharge of fluid through an orifice, in whose current he had placed a resistant body (Fig. I-7). The method he followed in order to find the force acting on this body is based initially on finding the velocity of the discharging fluid as a function of the depth of the water in the reservoir. Afterwards, as a second step, he calculates the force upon the body also as a function of this height. Knowing both, and after eliminating the height parameter, the resistance is deduced as a function of the velocity. Obviously, this approach does not correspond with the impact theory, as there is no particle rebound. Likewise, we see that Newton made use of the discharge problem as an ancillary element, and although he used this methodology in the two first editions of the Principia, as we shall see, the solution he arrived at was different in each one.

a)

b)

Fig. I-7. Discharge with a body inside the current

In the first edition he offers an argument from which he concludes that the discharge velocity corresponds to a fall height that is half that of the depth; that is to say the contrary of Torricelli’s law. This was to some extent in tune with experimental results of the day, although it contradicted the experimental evidence of the vertical ascent of water in spring jets. The question was under the spotlight, and Newton must have convinced himself shortly afterwards of his error, as in 1690, in his personal copy of the Principia he made some notes from which we deduce that he had changed his mind. As regards the force exercised by the current on the sphere placed in it, he supposes it to be equal to the weight

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THE GENESIS OF FLUID MECHANICS, 1640–1780

of a column of water whose height was that of the reservoir and had the same cross section as the sphere. If the submerged body were to have another geometrical form, but the same cross section, the weight of the column would be the same; that is to say, the resistance for a liquid movement is independent of the shape of the body, which did not occur for a body in air. The result he obtains for a sphere in a liquid is that the resistance coefficient CD = 2, which is equal to that of the sphere in the air. Although Newton deals with the discharge quite briefly, we find it has two opposing aspects. On the one hand was Torricelli’s law, corroborated by the fact that vertical fountain discharges almost reach the level of the surface of the water in the reservoir. On the other hand, the results of the experiments indicated that when the reservoir is discharged by a lower orifice, and Torricelli’s formula is introduced, the height of the discharge will oscillate around half the depth of the reservoir. The situation was contradictory, and at first Newton opted for the latter solution, which was included in the first edition of the Principia. Later on, as we have said, with the correction he wrote in his personal copy, the new coefficient of resistance was reduced to CD = 1, that is, half the former, which furthermore partially agreed with the experimental values found by Mariotte. The solution to this contradiction will be given by Newton in the second edition. To recap, we must underline two aspects of Newton: the new approach to the formalisation of the impact theory using the ‘rare medium’ instead of air, and the use of discharge phenomenon as an instrumental element for resolving the question in the case of liquids, where impact theory, strictly speaking, is not used. There is therefore a crossing over of the two major lines of work, resistance and discharge, although this crossover is only partial. We now briefly leave Newton, but we will return to him when we consider the second edition of the Principia. In the two decades following Principia’s publication (1690–1710), a group of mathematicians—geometricians as they were known at the time—and another group of hydraulic mechanics entered the stage. The preoccupations of the former swung between what we could call pure applications, such as the discussions of the minimum resistance solid, and other more practical aspects such as applications to naval theory. Compared to them, the mechanics focused on the study of hydraulically driven machines, that is to say waterwheels with paddles. The characteristic common to the mathematicians was the acceptance and extension of the impact theory even to liquids, without anyone questioning of its physical reality. Among them Jakob Bernoulli stood out, having used differential calculus to analyse the effects of liquid flow on two-dimensional shapes of ships. His conclusions went partly unnoticed, but for many years they were not surpassed, as he obtained the maximum advantage from an analysis of this type.

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At the same time he extended his studies to sails, which, inflated by the wind, formed so-called sailing curves (velarias), related to the catenaries, which were also in fashion then. As regards the solid of minimum resistance, Newton offered a solution in the Principia, although almost without any justification. Its obscure nature caused Nicolas Fatio de Duillier, Guillaume Antoine de l’Hôpital and Johann Bernoulli to enter into the fray with new solutions in the final years of the century. All the solutions were purely mathematical; therefore we must consider the contributions they made to the study of fluids as being purely theoretical with little or no practical application. Proof of this is that they limited themselves to relative solutions between one shape and another, without including any values of the resistant forces. Analysts of hydraulic machines, like Philippe de la Hire and Antoine Parent, constitute the opposite extreme. In 1704 and 1705, they studied the behavior of fluid-driven machines. They based their experiments on impact theory, with a proportionality coefficient derived from Mariotte’s experiments, rounding it up to the nearest unit. To complement their reasoning, they introduced considerations concerning the mechanical efficiency of mechanical systems. The application of fluid mechanics to ships and to hydraulic machines responded to important social needs. It is not necessary to go into detail regarding the importance of the navy at this time, but by the eighteenth century, maritime power had become one of the defining elements of a nation’s military strength, and its most relevant exponent was the ship of the line. The ship of the line, at its most elemental, is a machine which balances the effects of the wind in the sails with that of the water on the hull. It is the epitome of the ‘fluid machine’. As regards other machines, it is nothing new to say that windmills and waterwheels had been almost the only sources of mechanical power from ancient times. The growing urbanization of the eighteenth century required a method of distributing water, that is to say, a source of driving power and a raising device, which were the waterwheel and the water-raising pump. Thus, we arrive at 1713, a year during which we find ourselves once again with Newton, and his second edition of the Principia. No new theory had appeared since the first edition, only, as we have seen, some mathematical and practical applications. Newton’s new contributions were important, as he practically rewrote the part referring to motion of liquids. We remember that, where we had left him, his findings contradicted Torricelli’s law, even when they complied with the discharge experiments. In order to deal with this, he modified the discharge process that had served as the basis of his method, introducing a theoretical construction, the cataract (Fig. I-8), with which he intended to reconcile both positions. In this type of discharge, a contraction of the stream is produced, which implies that the actual cross section of the outlets is smaller than the transit

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THE GENESIS OF FLUID MECHANICS, 1640–1780

orifice. Newton measured this contraction, and found that the contracted cross section was approximately half that of the outlet orifice. This allowed him to adjust things in order to say that the outlet velocity followed Torricelli’s law, but that due to this contraction the outgoing mass was reduced to half. Bringing the theory into line with the experiments in this way, he had to assume that the discharge took place as it was passing through a funnel of ice that became liquid without interfering with the falling water. The explanation was most unlikely, and was widely criticised. As regards the force exercised by the current on the body located in its midst, he placed another cataract on this body, concluding that the resulting force corresponded to their resistant coefficient of CD = 0.5, whether the body was a disk or a sphere, and that it was a quarter of what he had obtained in the first edition. To all this he added a new collection of experimental results, consisting in spheres falling into buckets of water, where he played with different sizes and densities, concluding that the theoretical predictions were verified. He even included among his experiments glass spheres dropped from the dome of St. Paul’s Cathedral. Further experiments were carried out by himself and others by Jean Théophile Desaguliers.

Fig. I-8. Newton’s cataract

The Essay d’une nouvelle théorie de la manœuvre des Vaisseaux (Essay on a new theory of ship manoeuvering) appeared in 1713. This treatise by Johann Bernoulli applied impact theory in order to calculate both the forces of water against the hull, and the wind in the sails. The work was the end result of a long debate between the French naval officer Bernard Renau d’Elizagaray, the Dutch scientist Christiaan Huygens and the Swiss scientist Jakob Bernoulli on the mechanics of naval ship maneuvering. For Bernoulli, the shape of the hull was

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responsible for the drift when the ship advanced, and he calculated it based on impact theory as applied to the position of the sail. Another author of the time was the Italian scientist and engineer Giovanni Poleni, and we mention two of his works in particular, the De motu aquæ mixto (On the motion of mixed waters) and the De castellis per quæ derivantur fluviorum aquæ habentibus latera convergentia (On constructions with convergent openings, through which the waters of rivers are discharged). In both works Poleni tackles the discharge problem. In the first, he makes measurements of some models of ‘mixed’ waters, which he means to include ‘live and dead’, which in turn recalls ‘live forces’ (vis viva) and ‘dead’ (i.e., static) forces. The method in evidence in De motu appears to follow a rather artisan line with not much future. In the second work, he goes back again to verify Torricelli’s law. In order to do this, he performs discharge experiments through orifices with spouts of varying shapes, simultaneously measuring the contraction of the outlet vein. He obtains confirmation of the proportionality of the efflux velocity with the square of the height, although he does not manage to explain the cause of the contractions. Poleni is possibly the last of the great experimenters of the Italian school. By 1720, it was common knowledge that discharge theory still suffered from a considerable conceptual weakness, while impact theory, supporting the problem of resistance, was well established by then. Nevertheless, the latter had a drawback: even though the model was accepted, it was not clear which values to use for the coefficients of proportionality. There were the values obtained by Newton, some theoretical as in the case of air, others justified experimentally such as for water, although these were almost double what Mariotte had measured. Nevertheless, it is not surprising that these were taken to be congruent, as the experiments made by Mariotte were made with flat plates, whereas Newton used spheres, and he had established that a sphere in air has half the resistance of a plate. Besides, the mathematicians had supposed that impact theory could be applied as much to air as to liquids, therefore both sets of results could be considered compatible. The decades of 1720 and 1730 were marked by the contributions of the Bernoullis, father Johann and son Daniel. Daniel contributed to the problems of discharge and resistance, and at the end of the decade he had sketched out the basic ideas of his Hydrodynamica, a work that would appear in 1738. Others made lesser contributions including Desaguliers, who went back to the idea of spring-driven pendulums oscillating in water and mercury. In 1724, we find Johann engaged in a work essentially about mechanics, ‘Discours sur le loix de la communication du mouvement’ (‘Discourse on the law of communication of motion’) which he presented for the Paris Academy’s annual Rouillé prize, and which contains a part dedicated to analysis of the

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THE GENESIS OF FLUID MECHANICS, 1640–1780

motion of solids in fluids. We note yet again that the contributions on fluids were frequently intertwined with those on solid mechanics. The work we quote is specifically Newtonian, and it extends the impact theory to bodies of revolution of any shape whatsoever, upon which he calculates the absolute forces, something he had not done in the Essay d’une nouvelle théorie de la manœuvre des vaisseaux. What can be considered as new is his application of impact theory to the problems of ballistics, which he outlined in the Essay, and complemented in the later work, ‘Problema Ballisticum’, at the same time making an incursion into the case of a pendulum in a resisting medium. In 1727, Daniel Bernoulli, who was living in St. Petersburg at the time, submitted the work ‘Disertatio de actione fluidorum in corpora solida et motu solidorum in fluidis’ (‘Dissertation on the effect of fluids in solid bodies and the motion of solid bodies in fluids’) on the analyses of bodies submerged in currents, to the Commentarii academiæ imperialis scientarum petropolitanæ (Memoirs of the St. Petersburg Academy), where he contributed a new theoretical construction in order to justify Newton’s results. This construction consisted of a plate inside a current flow, which he assumed was formed by elastic particles that impacted in successive layers on the plate, acting as a set of continuous percussions. The analysis of this model gave the resistance coefficient of value CD = 4, the same found by Newton in the motion in air. He compared this result with the experiment on the discharge of a vertical jet against a plate, which showed that CD =1. Faced with this dilemma, Daniel decided on the latter result, justifying this by experiment. Besides, when this corrected result was applied to a sphere using impact theory, it gave a CD = 0.5, which again coincided with the value found by Newton in the descent of spheres in water tanks. It goes without saying that any coincidence with Newton’s results was, at the time, a guarantee of acceptance. He ended the work with the application of impact theory to the ballistics of a cannonball fired vertically, in which he comments on the different flight times as a function of the projectile weight. His intentions are sound, but he lacks the data (the projectile velocity at the muzzle) needed to understand the phenomenon. In the following decade, the British artillery engineer Benjamin Robins will come back to this problem. There is another work of Daniel in the same 1727 volume of the Commentarii petropolitanæ, which is dedicated to the motion of water through ducts. The paper, entitled ‘Theoria nova de motu aquarum per canales quoscunque fluentium’ (‘New theory about the water motion flowing through whatever channels’) confirms the separation of the phenomena of resistance and discharge. This monograph marks a change in the treatment of movement through ducts, and it already contains ideas that appear later in the Hydrodynamica. The fundamental

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Fig. I-9. Motion through ducts

basis of the new approach consists of two principles: the conservation of the live forces and motion in parallel sections (Fig. I-9). For the first, he supposes that the set of particles comprising the liquid collide with each other and with the walls, in an elastic manner, therefore justifying the conservation of the total live force. However, in order to study motion of fluids in ducts, a second hypothesis is required: that the velocity be the same in all points of any section perpendicular to the axis, thus causing this section to advance like a plane. Given that the velocity is inversely proportional to the section, when this is known, the problem is reduced to a one-dimensional case, and the calculation of the total live force of fluid volume is reduced to a simple integral. In the specific case when it is applied to a reservoir in a vertical position with an exit orifice in the bottom part, the outlet velocity would be calculated by assuming that the live force of the whole reservoir is kept constant. The entire mass of the fluid would intervene in the process, and the final formula arrived at is no longer Torricelli’s, as it coincides with his only in the case where the cross section of the outlet orifice is negligible compared with the cross section of the reservoir and the upper surface of the water is not very close to the bottom. Although these ideas are greatly perfected in the Hydrodynamica, we can appreciate that there is a qualitative leap with respect to the previous approaches, which may be resumed by considering the motion of the fluid mass as a global phenomenon, not individualized in particles. Daniel presents another work dealing with discharges: ‘Experimenta coram societate instituta in confirmationem theoriae pressionum quaas latera canalis ab aqua tranfluente sustinet’ (‘Experiments performed before the Academy in order to confirm the lateral pressure in fluid currents’) which appeared in the 1729 volume of Commentari petropolitanæ. This is the first attempt to relate pressure

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THE GENESIS OF FLUID MECHANICS, 1640–1780

and velocity, a prelude to the formulation of the theorem later named for Bernoulli, and whose emblematic character all that work in the abstruse world of fluid mechanics knows.

Fig. I-10. Daniel Bernoulli’s experiment

At this point, a few additional comments on pressure might be helpful. The concept of pressure as a force that the fluid exercises on the walls belongs to the world of hydrostatics. Pascal extended it to the entire fluid by a very well-known principle taking his name. However, up to the date which concerns us, the concept of pressure remained as a property linked to hydrostatics, independent of motion, until Daniel Bernoulli demonstrated experimentally that this supposition was not true, and that the pressure on the walls of a discharge tube (for the moment only on the walls) depends on the velocity of the water in the duct (Fig. I-10). To this end, Daniel used the water height manometer, the apparatus shown in the figure, which consists of a vertical tube through which the water ascends according to the pressure in its base. If the outlet orifice of the discharge is blocked, and therefore there is no movement, the level of water in the tube reaches the height of the fluid in the reservoir. However, when circulation is allowed, the pressure will be lowered, which is evident from the drop in the tube level, and it will go down as the outlet flow increases. We repeat, the experiment has several remarkable points: it shows the relation between pressure and velocity, contributes a measuring instrument such as the water manometer, and presents an experimental result which will promote the theoretical development required to explain it. All these explanations are given in the Hydrodynamica, a work which was almost complete when Daniel left St. Petersburg in 1733, although its publication was delayed until 1738. But before we examine the Hydrodynamica, let us recall once again the case of the impact of the jet against the plate. This phenomenon had been mistakenly interpreted as being equivalent to the effect of

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the current on a body submerged in the same current, although this equivalence was strictly true according to the model of the impact theory. Daniel undid this supposition, identifying it as a different phenomenon (Fig. I-11) in which there is no rebound of the fluid, but where it is deflected towards the sides. The application of Newtonian mechanics, already well developed as a mathematical instrument, allowed him to estimate the expected forces, which he confirmed experimentally and quite precisely with a well-designed apparatus. The results appeared in the 1736 volume of the Commentarii petropolitanae. Three years later, another announcement appeared in the Commentarii with a new set of experiments, this time made by Georg Wolffang Krafft.

Fig. I-11. Jet against flat plate

In the decade of the 1730s, the figure of Henri Pitot also appears. Although he was principally a mathematician, he dedicated a good part of his activity to the analysis of machines. Among his contributions we find the work La théorie de la manœuvre des vaisseax reduite en pratique (The theory of the manoeuvering of ships reduced to practise), which is an extension with more practical applications of the work with the same title by Johann Bernoulli; and three monographs on water pumps in the Mémoires de l’Academie Royal des Sciences of Paris in which he studied the application of the theory of live forces. He is especially remembered for an instrument for measuring the velocity of a current, which is described in the 1732 volume of these Memoirs, and which is still known as Pitot’s tube. Apart from this invention, which continues to be in use, he made no theoretical or experimental contributions. However, his work concerning the application of fluid mechanics theory was a different matter: although water pumps were known from ancient times, until Pitot came on the scene there were no studies explaining how they functioned.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

We return to Bernoulli père et fils, Johann and Daniel. For unknown reasons, although it seems to have been due to fit of jealousy, the appearance of Hydrodynamica by Daniel aroused the ire of his father Johann. He prepared the Hydraulica, which came to light in its complete form in 1742, although according to Johann, it was written in 1732, that is, before his son had delivered his version. Today it has been shown that this was not true, and that the father plagiarized his son regarding the theory that bears his name. Nevertheless, Johann’s work is of higher quality, as it tackles the problem in a more general manner, and has a more solid Newtonian basis than the works concerning the conservation of live forces. As Daniel relates, the title of Hydrodynamica was dictated by a combination of the two branches of fluid mechanics: hydrostatics and hydraulics, which were separate up to then and which, he argued, should go together. Nevertheless, we note that in the text Daniel uses the word ‘hydraulic-static’. The subject matter of the work is very wide, and touches on the entire spectrum of the fluid problems of era. The contents can be divided into several areas, with differences of form and content between them, such as fluid discharge, oscillations of fluids in tubes, the passage of liquid between reservoirs and ‘hydraulic-statics’. To all these types of motions, which are due only to the force of gravity, he adds those produced by external forces like pumps, as well as the forces generated by the reaction of jets and the motion of liquids in reservoirs that are themselves in motion. All these areas deal with liquids, except one dedicated to a mass of air enclosed in a reservoir, which he analyses introducing a version of what we would nowadays call the kinetic theory of gases. The Hydrodynamica is a wideranging, dense and complex book whose premises are the principles of conservation of live forces and of motion through parallel sections, which Daniel presents in a more elaborate form in this work. In this respect, he introduces what he calls ‘potential ascent’ and real descent instead of the live forces. His definition of the first term is the height to which the moving body would ascend if its velocity were to be directed instantaneously upwards, while the real descent would be the amount by which the centre of gravity of the system descends in reality. That the loss of one must be the gain of the other, is easy to demonstrate, and can be seen intuitively. The application of this theory enables discharge problems to be solved. But Daniel is not content with theoretical solutions, proposing after each chapter a series of experiments to illustrate the solutions. When we turn to hydraulics-statics, which is the best-known and most important part of the Hydrodynamica, we find a change in the treatment that even affects the terminology, leading us to think that this part was written by Daniel later on. The problem posed by it is to determine the pressure in the discharge

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Fig. I-12. Bernoulli’s discharge

tube of a reservoir measured with a manometer (Fig. I-12). If the outflow were zero, then its pressure would be equal to the pressure existing at the bottom of the tank, but when an outlet current exists, the pressure diminishes. The procedure for evaluating this variation supposes that the water in the outflow tube is retained by the perforated plug, in such a way that if plug were to disappear, the water would flow freely to the exterior and thus the retention pressure would disappear. Therefore, the greater the flow rate (which implies greater velocity) the less will be the pressure. He bases the calculations for determining the pressure reduction on the conservation of the live forces of the entire fluid. The importance of the discovery lay in the existence of a relation between pressure and velocity, a fact that would be a crucial point in the evolution of fluid mechanics. Likewise experimental checking verified the predictions he made. In his studies he also contemplated cases which implied a loss of the live force, such as discharges among various reservoirs containing liquids (Fig. I-13). He has trouble with the quantitative determination of the losses, not an easy problem to solve. In this respect, these calculations should be taken as a set of good intentions, no more. He extended his principles to include even the discharge of a gas through an orifice. In this case he cannot speak of heights, as the air will tend to fill the entire recipient, but he imagines a fluid with the same density and an equivalent height, to which he applies the theory. The procedure may appear somewhat artificial, but Leonhard Euler and Jorge Juan would use it years later. Finally, it is worth noting the application of the discharge phenomenon as the driving element of a ship. The solution does not appear to be very practical, due to the low speed at which the jet comes out, but it was a forerunner of modern jet propulsion.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. I-13. Live force loss in discharges

The importance of Hydrodynamica was considerable. It covered almost the entire subject of fluid mechanics known to that date, where Daniel used the ‘potential ascent’/‘real descent’ (a derivation of the live forces conservation principle) as a method for resolving an extensive group of problems which we have grouped under the rubric of discharge and that he exploited it as far as he could. Regarding the Hydraulica of Johann, the basic contribution was twofold: the concept of internal pressure (although it would be better to say internal forces), and the division of the fluid into differential elements. Johann Bernoulli was a great mathematician with wide knowledge of differential calculus, which he used when treating the mechanics of solid bodies and flexible cords, precursors of the internal division of the fluid. At this time, pressure was considered as an external force to the fluid mass. We have seen how Daniel linked it to the velocity, but although he dimly saw the separation of the fluid into parts, he did not arrive at the heart of the question. With Johann the problem is completed. Whether it be in equilibrium or in motion, the fluid can be divided by imaginary internal surfaces, and we can substitute, for any part of the fluid, the forces it exercises on that surface. This enables us to isolate any element of the fluid enclosed by a surface, with the condition that the forces that the rest of it exercises on this element be located upon its surface. These forces, together with those of inertial origin affecting the isolated mass, will determine the evolution in time and space of the element, following the Newtonian equations of dynamics. It is easy to see that the method differs greatly from that of his son Daniel.

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The Hydraulica, which is a short work compared to the Hydrodynamica, starts with the movement through a duct with an abrupt change of section (Fig. I-14). In order to eliminate the discontinuity, Johann invented a whirlpool located in the narrowing of the duct, a most unlikely occurrence, but one, which enabled him to convert the jump into a continuous process of area reduction. When the fluid passes through this type of funnel it accelerates, and a force is required in order to produce this acceleration. In modern terms, this force is the difference of upstream and downstream pressures multiplied by the respective sections. If the system turns vertically, the accelerating force will be gravity, and thus the discharge problem posed by his son is solved.

Fig. I-14. Fluid motion through a narrowing duct

The whirlpool disappears when he analyses the motion through ducts with continuous variation (Fig. I-15). The local acceleration is determined by the section change and by the acceleration in a reference section. When this fluid is divided into layers, and the Newtonians laws are applied, he ends up with an

Fig. I-15. Motion in continuous tubes

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THE GENESIS OF FLUID MECHANICS, 1640–1780

equation whose formula is very similar to how Bernoulli’s theorem is understood nowadays. After his brilliant deductions, he allows himself the luxury of repeating the equations, using the method of conservation of the live forces. The results, logically, are the same. What we have said makes the difference between the two approaches quite clear. The merit of the idea and the initial outline are due to Daniel, its generalisation and improvement to Johann. Furthermore, this new picture will open up the way for the theoretical fundaments of hydrodynamics, and in a little over 10 years the culminating point will be reached. Nevertheless, for the sake of both their contributions, we think that instead of saying ‘Bernoulli’s theorem’ we ought to refer to it as the ‘Bernoullis’ theorem’. The years between 1742 and 1755 were full of frenetic activity, and very probably were the most brilliant in the entire history of the development of fluid mechanics. Great theoretical works appeared, as well as numerous works on applications, as much in the line of resistance as in that of discharge, and impact theory was eliminated from the theoretical field. We say ‘from the theoretical field’ because the options substituted for it did not solve the problems, and in practice this concept continued to be used. In this connection, it is curious to note in an author like Euler the difference between his theoretical equations and his analysis of the practical applications. The brilliance of the former contrasts with the clumsiness of the impact theory which he has to use in the latter, in spite of the corrections he introduces in order to refine it. As we have already said, one of the most pressing fields of application was that of naval mechanics. At the very beginning of 1740, two men, Pierre Bouguer and Euler, wrote their respective naval treatises: Traité du navire, de sa construccition et de ses mouvements (Treatise of the ship, its construction and its movements) and the Scientia navalis seu tractatus de construendis ac dirigendis navibus (Naval Science or treatise on the construction and steering of ships) which appeared in 1746 and 1749, respectively. Both Bouguer and Euler had previously dealt with naval questions in 1727, competing for a Paris Academy of Sciences Prize on masting of ships, which Bouguer had won. Bouguer later traveled to the Viceroyalty of Peru from 1735 to 1744 with the Geodesic Mission, aimed at measuring an arc of the meridian at the equator, in which the Spaniards Jorge Juan y Santacilia and Antonio de Ulloa also participated. Euler’s travels only took him from St. Petersburg to Berlin. These two naval treatises analyzed the ship as a floating, oscillating solid, therefore having aspects of stability while afloat, while being driven by the wind and resisted by the water. Although the stability analyses were linked to hydrostatics, the other two are fluid problems derived from impact theory: the resistance of the hull, and the force and shape of the sails. Euler made a theoretical incursion

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into the subject in order to obtain the basic coefficients of resistance, which was followed by an extensive mathematical analysis in applying them to various shapes of ships, together with a process for optimizing these shapes. Bouguer, who was more practical, took real ships as his starting point, and adopted measured values for the resistance coefficient. He also tried, without even attempting to offer any justification, to introduce the effect of the stern on the resistance, as it had been concluded from the strict application of impact theory that the stern produced no effect at all, being in the shadow of the current. In 1742, Benjamin Robins appeared on the scene, questioning Newton’s theory for the very first time in his New Principles of Gunnery. In passing, we note that ballistics is our eternal travelling companion of fluid mechanics. We recall that the values used as resistance coefficients for balls and plates in water were CD = 0.5 and CD = 1. These coincided in a certain way with the theoretical predictions, though it was puzzling that they quadrupled for air. Robins alleged that Newton’s theory was not completely applicable, and proposed separating the motions according to the capacity of the fluid to fill the vacuum that the body left behind itself. In this respect, he said that there were two types of motion: some, in which the fluid always occupies the space left by the moving body; others, in which this space is only partially filled, leaving some vacuum behind. (Fig. I-16). The former takes place in water and the latter in air, although in air he considers that it depends on the body’s velocity. If this is slow, the air will have time to fill the space, and it will be equivalent to the first type. By contrast, if the motion were very fast, there would not be time to occupy the space, and a vacuum would be left behind that would increase the resistance. In the first type, the resistance is given by Newton’s theory, while in the second he estimates that the resistance is tripled with respect to the same theory. The intermediate situations also have intermediate values. Robins performed tests with cannon balls, where he measured velocities in various points of the trajectory with the help of a ballistic pendulum, and obtained some experimental results quite compatible with his hypotheses. They were the first supersonic experiments made in history, even though he did not know it at the time.

Fig. I-16. Robins’ motions

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Robin’s New Principles of Gunnery had a continuation in its German translation by Euler, published in 1745. The translation carried an extensive commentary, so much so that in the part that interests us, the notes take up more space than Robins’ original text. Euler followed Robin’s ideas in his comments, but he went much further in his arguments. Following this line, he calculated the velocity at which the body ought to move in order to begin to form the vacuum, and, with the aid of his mathematical skillfulness, he carefully analyzed the experimental results presented by Robins. However, it is of some importance that in the commentaries Euler supposed that the fluid did not collide with the body, but went round it in a curvilinear fashion (Fig. I-17). With the exception of the analysis made by Daniel Bernoulli for the impact of a jet against a plate, this was the first time that the impact model was substituted by deviations following streamlines: a more realistic model but much more difficult to process mathematically. Basing himself on this new model, Euler made calculations whose purpose was to determine the value of resistance, although he followed the assumption that the pressure was proportional to the normal component of the velocity, which in turn would be a function of the separation of the streamlines. With these hypotheses he arrived at the surprising result that the resistance of the body was null. It was the first appearance of this contradiction between theory and reality.

Fig. I-17. Euler’s streamlined model

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Two years before, in 1743, a young mathematician, Jean Le Rond d’Alembert, published the Traité de Dynamique (Treatise of Dynamics). At the end of it was a small part dedicated to fluids. His basic thesis was that forces are strange entities, and that mechanics cannot base itself upon them, but must instead be based on phenomena with a physical content, such as impacts and the conservation of the momentum. His application principle, which is more of a rule, consists in supposing that if a system of linked masses, with given links and external restrictions, is driven with a certain velocity at each mass, the system will behave in a certain manner, producing a velocity in each mass. If the differences between the velocities produced, and those introduced at the beginning, were given as the initial system velocities, the system would not move: both introduced and produced velocities would destroy themselves. It is worth noting that d’Alembert is the most philosophical of the mechanicians of the eighteenth century: the notion of force certainly had many obscure points, and it is not surprising that he tried to escape from this idea. In the following year another work of his appeared: the Traité de l’équilibre et du mouvement des fluides (Treatise on the balance and motion of fluids) in which he studied the resistance of a body in a fluid and the discharges. An interesting point of this work is that when he tries to submit fluids to the laws of mechanics, he finds himself with the already known difficulty that the multitude of corpuscles makes the study almost impossible. Therefore, in order to solve this problem, he uses a hypothesis he calls the ‘principles of experience’, and which will substitute for those of mechanics, which he considers to be metaphysical in nature and located beyond experience. He takes two principles of experience; one for hydrostatics, where the pressure applied in a point of a vessel full of fluid is distributed uniformly in all directions, and another for hydrodynamics, where in the vertical discharge of a vessel, the horizontal layers continue being parallel planes in the discharge motion. For fluid resistance he applied impact theory, in which the fluid, comprised of tiny particles, collides and transfers its total momentum to the body. He presents formulas that, with some exceptions, are derived from his theoretical conceptions, and are similar in nature to those encountered in other authors. However, he does make a new contribution to the study of elastic fluids, although his hypotheses in this respect are not very consistent. Concerning the discharge, he bases it on the principle of conservation of the live forces, which he demonstrates in turn, by starting out from his general principle of mechanics. Given his aversion to the forces, his criticism of the Hydraulica and his acceptance of the Hydrodynamica is understandable.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

In 1743, the Théorie de la figure de la Terre, tirée des principes de l’hydrostatique (Theory of the shape of the Earth, obtained from the principles of hydrostatics) by Alexis Claude Clairaut appears. Together with d’Alembert and Euler, these three were the authors of what can be considered the ‘grand theorisation’ of fluid mechanics, who together raised the discipline to its zenith during the eighteenth century. Clairaut’s work dealt with the question of the shape adopted by the Earth due to the combination of gravitational forces and its rotation. The root of the question resided in the nature of the gravitational forces, a subject that had two different and opposing conceptions: the Newtonian and the Cartesian. The scientific community of the time estimated that if the shape of the Earth was known, one could deduce which hypothesis was valid. Therefore, the measurement of the Earth’s shape was a sort of crucial experiment that the Academy of Paris tried to solve by promoting two expeditions, one to Lapland (modern Finland) and the other to the Viceroyalty of Peru (modern Ecuador), with the aim of measuring arcs of the meridian. The mutual comparison of these, together with the measurements already made in France, would enable the sought after shape to be determined. Clairaut was one of the participants of the Lapland expedition. The roots of the theoretical problem lay in fluid mechanics, as everyone supposed that the Earth behaved like a fluid in rotation submitted to gravitational forces. The solution in the case where there was no rotation was quite simple: a sphere. However, the determination of this shape when rotation existed made the addition of another condition mandatory. In this respect, Newton had supposed two imaginary canals that were to be found in the centre of the Earth, one originating in the Pole and the other in the Equator, and he postulated that the pressure must be the same at the meeting point. As the channel coming from the equator was alleviated by the centrifugal force, its length would be greater. On another tack at about the same time, Huygens suggested that for any imaginary channel located on the surface, the fluid filling it would have to be at rest. The conditions of one and the other were both necessary, and it appeared that both should always be satisfied jointly. Nevertheless, Bouguer presented certain models in which the shape he obtained for the Earth was different, according to which condition was used. This made Clairaut ask himself if a more general relation existed between the forces and the fluid in the equilibrium condition, and he began his work from this line of argument. His starting point was the idea was that the fluid contained in the two imaginary canals joining two points on the surface had to be balanced

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Fig. I-18. Clairaut’s channels

(Fig. I-18), a condition that included those of Newton and Huygens. Following this idea, and by resorting to closing the canals internally, he arrived at the conclusion that if an equilibrium existed for a solid mass at rest, it would also exist if the mass were to rotate, only that the resulting figure would not be the same. This indicated that the existence of an equilibrium condition was independent of rotation, and depended only on the nature of the forces existing in the interior of the fluid mass. These forces, which we would nowadays call a force field, are those attempting to move the fluid enclosed in any channel. For his closedcanals hypotheses, it is obvious that the total forces on any canal passing through two fixed points would only be a function of the position of these two points, and not a function of the shape of the canal. Therefore, with the help of mathematical analyses, he arrived at the conclusion that the condition required for the equilibrium was expressed by a differential equation relating the components of the force field, and which did not depend on the nature of the fluid. The importance of Clairaut’s findings was precisely this: the transfer of the condition of equilibrium of the fluid to a mathematical relation between the components of the force field. Once this conclusion was obtained for a field with axial symmetry, he generalised it to include three-dimensional fields, obtaining a similar, although rather more complex result. A few years later in 1749, d’Alembert applied for the prize awarded by the Academy of Sciences of Berlin on the resistance of fluids. The jury decided not to award the prize to anyone, as they considered that none of the works presented showed experimental evidence of the theories they defended. The decision profoundly irritated d’Alembert, and even more so because Euler was among the members of the jury, a fact which caused the already poor relations between them to deteriorate further. The work was published in 1752 with the

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THE GENESIS OF FLUID MECHANICS, 1640–1780

title Essai d’une nouvelle théorie de la résistance des fluides (Essay on a new theory of the resistance of fluids), constituting a second milestone in the theorisation process. D’Alembert based his theory on his principle of cancellation of velocities, with which he made some basic contributions that served for the rest of the work. One of the contributions is the generalisation of the results obtained by Clairaut for fluids of variable density, and the other is a variant of Bernoulli’s theorem. It should be noted that d’Alembert did not cite Clairaut’s work, possibly because their personal relations were, it appears, not very cordial.

Fig. I-19. Fluid field

The most significant thing in this work is the treatment of the structure of motion in a fluid current around a body, although he limits himself to the twodimensional and axisymmetric cases. Here, there is nothing that remotely resembles the impact theory. He supposes the fluids to be animated by a uniform movement upstream, which starts to deviate as it approaches the body, forming streamlines surrounding it and outlining it, to return later to the uniform condition downstream (Fig. I-19). The change is continuous and gradual, and demonstrates that the structure of the fluid field, that is to say the geometrical form of the current lines, is independent of the speed, and depends only on the shape of the body. As a consequence of this, the velocity at a specific point will be proportional to the velocity upstream, a fact that allows him to make the motion dimensionless. On the other hand, the effect of the fluid on the body will be that produced by the pressure of the channel adjacent to the body, and not by an

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impact. In order to obtain the fluid field he starts from two premises: continuity and dynamics, and as a result arrives at the fact that the overall problem is reduced to only two equations in partial derivatives between the axial and lateral components of the velocity. The fluid problem had been reduced to a mathematical problem. The difficulty, however, lay in how to solve these equations. In order to do so, d’Alembert makes an impressive leap: the conversion of these equations into simpler ones by means of a complex variable transformation. This has been one of the most important and productive resources of fluid mechanics, but solving the transformed equations was not easy either, a task he attempted by using developments in series. The first step was to determine the fluid field, at least in theory, so that velocities were found at each point, and in particular on the surface of the body in question; then, it was necessary to determine the force on the body, which he called ‘pressure’. This he obtained as an integration of the local pressures. There is a rather interesting point in these analyses: d’Alembert’s attempts to demonstrate that the case of the moving fluid with the body at rest was equal to its opposite case, the moving body with the fluid at rest. Such equivalence had been justified by Newton, but d’Alembert demonstrated it by showing the nullity of a mathematical term. As a consequence of the theory, when performing the calculation of the resulting force, he arrived at the result that the total resistance on the body is zero. This conclusion, known today as d’Alembert’s paradox, is totally contrary to experience. It would be a source of problems, and would remain without any explanation for more than 150 years. D’Alembert tried to find an outcome to this paradox, by giving the theory less validity near the bow and stern. Years later, in his Opuscules mathématiques, he proposed the problem to mathematicians once again, as a paradox to be resolved. We recall that Euler arrived at a similar conclusion, but starting from very different hypotheses. In spite of the fact that the Essai was very difficult reading, it completely changed the theory of fluids and as such should have been recognized for its ground-breaking nature. It was, but only for 3 short years; in 1755, Euler published three memoirs that greatly surpassed everything done by his predecessors. Despite of the higher quality of Euler’s work, however, he owed a lot to d’Alembert, whose work he knew as a member of the jury, and to Johann Bernoulli. The three Memoirs appeared in the 1755 Mémoires de l’Académie Royal of Berlin, although his ideas had already been outlined in another work, Principia motus fluidorum (Principles of fluid motions), read in this Academy in 1752 but not published until 1760. In this work, Euler treats a fluid as a continuous entity, just as d’Alembert, Clairaut and Johann Bernoulli had done, but the difference is that he justifies the fluidity conditions. Strictly speaking, this hypothesis is the

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THE GENESIS OF FLUID MECHANICS, 1640–1780

only one that allows differential analysis to be applied to the fluid medium. However, this does not mean to say that he believed that this was the reality, as years later Euler talked once again about the corpuscular nature of air. It was therefore a working hypothesis, as it only allowed differential analysis to be applied. As was customary at the time, the fluidity condition he used in the Principia motus fluidorum is the impenetrability and contiguity of the particles of an uncompressible fluid, and this condition will be the one determining which motions are possible and which are not. The analysis is based on this condition and on Newtonian dynamics, and leads him to two equations, coming close to d’Alembert, in a manner of speaking. One equation refers to continuity, and the other to momentum. The first is the mathematical expression of the constancy of the volume of a fluid element evolving with time, and it is a kinematic condition. As regards the second, this is the application of Newtonian equations to the same element (Fig. I-20), with the forces acting upon it: namely those of pressure on the walls plus the mass forces, which leads him to another group of equations. The mathematical development takes him to the irrotationality condition and the introduction of the velocity potential. We can appreciate Euler’s genius in this work in the clarity of concept and exposition, as well as the ease with which he handles the mathematics, of which he was an absolute master. These abilities, which are present in all his work to a greater or lesser degree, allowed Euler to surpass his contemporaries, and to take the ideas of others much further than they themselves could have imagined. The limitations of the Principia motus fluidorum were the incompressibility and the irrotationality. He would lift both restrictions in his following works.

Fig. I-20. Element of fluid

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The three foundational monographs appeared in the same volume of the Memoirs of the Berlin Academy, and corresponded to a general approach in Euler’s ideas on fluids. The first was dedicated to hydrostatics and the other two to dynamics. As a condition of the existence of equilibrium, took the constancy of the pressure on the external surface of a fluid at rest, to a certain extent coming close to d’Alembert’s approach. In addition, he argued against the corpuscular hypothesis, maintaining that this would not allow the stability of the fluid to be maintained. In the treatment of motion he declared clearly that he would try to investigate the fluid motion evolution as a result of the application of forces starting from initial conditions. The resulting equations are the same ones: continuity and momentum. The difference with respect to the Principia motus fluidorum is that he derived the first from the constancy of the mass enclosed in a controlled domain, instead of the volume. The mathematical development which he carried out has remained almost invariable since then; so much so that these equations are known today as ‘Euler’s equations’. He extended the application of the mechanics to both compressible and incompressible fluids. For the former he added an additional equation that we nowadays call the equation of state of an ideal gas, which links the relation between the gas temperature, density and pressure. The resulting equations give one for continuity, and the three for the momentum, which in reality represent the three components of a vectorial equation, all of them in partial derivatives. These equations, which are not without a certain formal beauty, have only one inconvenience: the lack of mathematical algorithms permitting their application, a difficulty which has persisted until the recent past. This circumstance, noted by Euler on more than one occasion, leads him to try to apply them to the simpler cases. Some of these particular cases correspond to situations already studied by other authors, only now they are deduced much more quickly and clearly. Euler’s great virtue was to have clarified the initial basic concepts: the use of the Newtonian equation and his mastery of differential analysis. The equations represented a moment of mathematical beauty and elegance unsurpassed in the century; but, as Euler himself recognised, mathematical analysis was not sufficiently advanced to enable all the consequences to be extracted from the equations. Certainly the equations were left there. The only person in the eighteenth century to dare to return to this subject was Giuseppe Lodovico Lagrangia (better known as Joseph Louis Lagrange), in 1781. These works were not Euler’s only contribution to the question of fluids during these years. In 1752 and 1756, he produced several applied studies on water jet turbines, for which he developed an application of his general theories

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to motion in tubes and windmills. In this respect it is interesting to compare a number of Euler’s works written in the same years, and to see how in some he reaches the peak of abstraction, while in others he is obliged to return to the impact theory, in which he does not believe and tries to correct. As we have already said, the line of work concerning discharge gave more results and facilitated the evolution of fluid dynamics, while that of impact, which was perhaps more necessary in practice, was almost a dead end. From 1760 onward, there were no further theoretical advances during the eighteenth century, with most of the later efforts focused on experimentation, since there developed a general belief that the only way to obtain useful practical data was through measurement. Jean Charles de Borda, better known for his experimental work, returned to the discharge problem in 1766, stating that theoretical equations were of no use to him. He used the conservation of live forces, but in spite of some contributions made by him, his works marked an advance only with respect to Daniel Bernoulli. But in the experimental field, Borda’s apparatus, measurement methods, and theory of experiments had undergone notable improvement since the Bernoullis. This was noticeable in his works on the resistance of bodies in air and water. Borda’s methods followed those of Robins: he used rotating arms, turned by weights, to propel his models in fluid streams (both in air and in water). The objects used in his experiments were geometrically simple, such as spheres, plates and cones; therefore, his results could be reduced to the simple formulas. These results demolished many of the existing ideas about the relation between the geometric shape and the resistance coefficient. Later on, in the following decade, the experiments were focused on specific objects such as ships’ hulls, which, as we have said, was one of the areas that prompted the greatest institutional and political interest. Following this line, and by mandate of the French government, Charles Bossut and others carried out several set of experiments in basins, whose results published in 1775 and 1778. The methodology consisted in towing the model by a cable attached to falling weights, as Huygens had outlined. His models were geometric shapes resembling ships hulls, although with parametric variations in the dimensions. There was also another experimenter in this decade who followed the same methods, the Swedish naval constructor Fredrik Henrik af Chapman. It is interesting that the greatest efforts in the naval application of fluid mechanics, at both experimental and application level, did not arise in Britain until years later, even though Britain possessed the greatest fleet of the times, and had an empirical tradition in science. It was the French Bouguer, the Swiss Euler, the Spaniard Jorge Juan, and the Swede Chapman who wrote naval treatises.

INTRODUCTION

41

These same naval treatises, which served to close the period we are analysing, constituted the expression of remarkable political interest in the science of fluids. We said that in the 1740s the naval treatises of Euler and Bouguer appeared. The latter was widely commented on at the time, while Euler’s must not have been well understood in the environment of shipbuilders and budding naval engineers, possibly due to its demanding mathematical level and the fact that it was written in Latin. Perhaps for this very reason, Euler produced a simplified French version in 1773. If in the former version he rose to high abstractions in the mathematical forms of the ship, he now simplified his treatments in the opposite sense, even to excess. Two years before, in 1771, the Examen marítimo (Maritime Examination) of Jorge Juan appeared. This was a treatise of naval theory and construction that can be considered as one of the best of the century. The work began with a large section on applied mechanics and the theory of the resistance of fluids, in which he rejects Euler’s theories and those of others, proposing a new theory on the effect of fluid current on a body moving in it. Although this theory turned out to be incorrect, it did however have a justification: the results of impact theory were notably erroneous, and Jorge Juan sought something to substitute for them.

PART I The Problem of Resistance

The Problem of Resistance

As we have explained in the Introduction, ‘the problem of resistance’ is understood to be the set of studies carried out with the aim of determining the ‘forces’ generated on a body moving in a fluid; or conversely, the forces produced on a stationary one in a current of fluid.1 The interest in this problem is explained in part by its obvious practical utility, since ships, mills and other machines are affected by these ‘forces’, and all of them perform their functions through the same ‘forces’. Therefore, it is no wonder that this problem gave rise to a line of speculation and research that began in the middle of the seventeenth century, and continued with increasing importance up to the present day. We note that the term ‘forces’ has been placed in inverted commas to emphasize its use and meaning, because although nowadays it is a familiar and well-understood concept, this was not so in the seventeenth and eighteenth centuries when its use was tinged with ambiguity. The word ‘force’ was applied with various nuances to actions implying production or alteration of motion; whereas the word ‘force’, as we understand nowadays, was designated in those days by different synonyms.2 In the specific application of the word ‘force’ to the phenomenon of the motion of fluid, the authors of the time coined the term ‘resistance’, by which they wished to express opposition to motion. This meaning has continued to the present day, and we will we shall use it in this sense from here on. Before looking at the development of theories explaining resistance, however, it is worth noting how the phenomenon of the generation of forces in a moving body in a fluid is considered nowadays. This explanation will also serve as a reference in our investigations. In this respect, we will use a simple model consisting of a streamlined body (Fig. P-1) immersed in a uniform flow, which will assumed to be an ideal fluid in a subsonic incompressible regime with

1

These two situations are dynamically equivalent. Nevertheless, more than one physicist questioned the foundation of such a principle. Newton bases it on the fact that the action of the medium on the body is identical to that of the body on the medium as consequence of his laws [Book 2, Prop. 34]. D’Alembert takes them as equivalent by virtue of a proof. 2 The topic is interesting and complex, since it is in the seventeenth and eighteenth centuries that the concept of force was defined. A most interesting work on this subject is Force in Newton’s Physics by Richard S. Westfall. As regards its usage the terms live forces, dead force, accelerating forces, and many more were used.

45

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THE GENESIS OF FLUID MECHANICS, 1640–1780

absence of turbulence. In this flow, the velocity field upstream may be represented by a set of parallel streamlines, with constant velocity in magnitude and direction, and with constant pressure as well. When these streamlines approach the body they will progressively separate, as they go around the body. This implies that both the amplitude and the direction of velocity will change, thus modifying the pressure field, to return downstream to a configuration similar to the initial one. The total force on the body will be the result of the sum of the pressure forces plus the viscosity forces, both of which act over the surface of the body; the first acting normally to the body and the second ones tangentially to it. The components of these forces are called resistance and lift, following the incident velocity and its perpendicular, respectively. The lift would not exist if the configuration was symmetrical. The forcegenerating process can be modeled by assuming that the presence of the body alters the velocities’ field, and these in turn alter the field of pressures, which are the ultimate cause of the force. The deciding factor in the phenomenon is therefore the geometrical shape of the body and its position with respect to the current flow.

Boundary layer Separation point Turbulent wake

Fig. P-1. Body in a current flow

In order to achieve an analytical solution to this problem, the equations of momentum, continuity and energy must be solved. These comprise a set of nonlinear partial differential equations, together with the boundary conditions of the body. A simpler mathematical solution of the problem is obtained by eliminating the terms related to the viscosity, which makes treatment of the equations somewhat easier. This set of simplified equations is known as ‘Euler’s equations’, since Leonhard Euler was the first to establish them. The absence of viscosity would imply that the current slips over the entire body surface, from bow to stern, at a velocity always tangent to the surface. Nevertheless, in reality the viscosity does not allow this ideal ‘slipping’, but instead causes the velocity to

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be zero at the surface itself. As a result, a thin layer will appear around the body, called ‘boundary layer’. Outside this layer everything occurs as if viscosity did not exist, while its effects are only seen inside this layer. In the outer limit of the boundary layer the velocity will fulfill the tangency condition, whereas in contact with the body the velocity will be zero. In other words, the boundary layer allows viscous phenomena to be introduced in a relatively simple way, rendering the imaginary assumption of eliminating the viscosity to be of real practical interest. In fact, almost all theoretical aerodynamics was developed on the assumption that air is an ideal fluid without viscosity. However, not everything is so easy, since viscosity has other effects. The boundary layer begins at the leading edge and its thickness grows progressively, but experience shows that it does not remain adhered to the entire length of the body, but separates from it at a point where a turbulent wake is originated. Before the separation point, the pressure in the outer limit of the boundary layer retains the same value in all points of the line perpendicular to the body surface; but does not do so after the separation. Therefore, in the adhered boundary layer zone, the pressure forces are the same as if viscosity did not exist, but this does not hold after the separation point. Thus, we observe that the effect of viscosity is double: in the front part it does not modify the field of pressures, but only produces an additional drag, called friction, which is generated inside the boundary layer; whereas behind the separation point it profoundly alters the field of pressures, which gives rise to a resistance called form drag, because it is due to the form of body. A further complication to the phenomena of resistance is that the boundary layer has two regimes: laminar and turbulent, characterized by the behavior of the fluid particles inside it. At the bow it always starts as laminar (the pressure field having well-defined layers), changing at a transition point to turbulent (the layers mixing together) which also influences the fluid field around the body. This brief review provides some idea of the complexity of the subject. With the help of modern computational sciences, the discipline called ‘computational fluid dynamics’ has emerged. Its aim is to apply numerical analysis and computation techniques in order to obtain computing codes to be used in huge computers. These machines solve Euler’s equations for bodies whose forms are not very complex, and the present frontier lies in solving the Navier–Stokes equations, with include the viscosity. But the computational advances have not been able to eliminate the need for experiment in aerodynamic tunnels or hydrodynamic channels, even in the simplest cases. In the last third of the seventeenth century, the first attempts to explain resistance followed very different paths. The basic equations of movement were developed in the middle of the eighteenth century, but deficiencies in the ability

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to carry out for mathematical calculation, and lack of knowledge of viscosity, made it impossible to advance in this direction. In this period fluids were conceived as being composed of particles, and in consequence it was assumed that resistance took place as result of the collisions of these particles against the body. A basic theory arose as an explanatory hypothesis of this paradigm, which has been termed ‘impact theory’. It was established between the end of the seventeenth century and the beginning of the eighteenth, and was exploited, applied, discussed and refuted during the middle and later part of the eighteenth century. This theory was based on two hypotheses: the first supposed that the fluid was formed by an individualized and independent set of particles; the second assumed that the resistance was the sum of the mechanical effects of all the impacts. A fluid current consisted, therefore, in the uniform parallel motion of a endless number of material particles, which struck against the external surface of any body in the current, rebounding more or less according to its degree of elasticity. Each individual effect would be a local force perpendicular to the surface of the body, and would depend only on the normal component of the velocity at the point of impact.

F v

Fig. P-2. Impact of the fluid flow on the plate

This phenomenon is depicted in Fig. P-2, which illustrates a flow with velocity v that impinges on a flat surface S, inclined with respect to the current at an angle of incidence α, so that when the flow rebounds mirror-like against the plate it produces a force F. The magnitude of this force is easily computable according to the concepts of Newtonian mechanics, and it will be equal to the variation of the momentum between the incident fluid and the rebounding fluid,

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which, after the appropriate calculations,3 turns out to be F = 2ρSv²sin²α. In other words, the resistance force is proportional to the surface, to the square of the velocity, and to the square of the sine of the angle of incidence, a proposition that we will see repeated in one way or another by almost every author during the seventeenth and early eighteenth centuries. This formula can be expressed as F = 2ρSvn², where vn is the normal velocity in the incident point, that is to say, the resistance is proportional to the square of the normal velocity. But it can also be written in the following form: F = 4Ssin²a (½ρv²), in which the term ½ρv² is known today as dynamic pressure; thus, the force can be read as 4Ssin²a times this dynamic pressure. If the incidence were perpendicular to the body, the formula would indicate 4S times the dynamic pressure, or a resistance coefficient of CD = 4. If the collision were inelastic and there were no rebound at all, the forces would be half of the previous ones, which would lead to CD = 2. Throughout the development that follows, we will see that these values will be repeated in one form or another. The first to conjecture that resistance was generated as a set of collisions were Christiaan Huygens and Edmé Mariotte. In 1669, Huygens, who was concerned with the effects of resistance in the ballistics of projectiles, performed a series of experiments of several types centered on the topic of the resistance. In these he spoke about the ‘impressions’ caused by the fluid. Mariotte, who demonstrated his ideas and experiments in his Traité du mouvement des eaux, published posthumously in 1686, spoke of the impacts of jets on bodies. We can consider both as forerunners, but the person who really established impact theory was Isaac Newton, as he presented it in Book II of his Principia (1687). Newton accepted the hypothesis of the impacts only in case where the fluid was air, as he supposed air to be constituted by physically separated particles, whereas in case of a liquid it was no longer possible to assume individual collisions, because the particles were in contact, and so would interfere with one another. For that reason, he provided a rigorous demonstration of the theory of

3

The calculation begins determining the volume of fluid per unit of time that will impact to this surface and that will be Svsinα. In the corpuscular hypothesis, if each particle has a mass µ and it is assumed there are N particles for unit of volume, then the number of shocks for unit of time will be NSvsinα. The force that every particle generates in the impact will be proportional to the variation of momentum of each one. Assuming an elasticity coefficient in the rebounds designated as k, the velocity variation will be expressed as ∆v = kvn = kvsinα, where 1 < k < 2, according to have been mentioned before. Therefore, the total resultant force, sum of all the individual shocks for unit of time, will be expressed as the product of the previous magnitudes, this is: kµNSv²sin²α. Now then, µN it is the mass for unit of volume of the fluid, that is the density ρ, therefore the expression for the resistance results in F = kρSv² sin²α, equivalent to the one given with k = 2.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

impact on the basis of his dynamical conceptions only for air, whereas for liquids he presented less rigorous arguments, which he juggles to make coincide (more or less) with the results of his own experiments. Though the merit of the justification of the theory was due to Newton, its formalization—with the introduction of the differential analysis as a tool—was due to other excellent mathematicians, in particular Jakob and Johann Bernoulli, Guillaume Antoine de l’Hôpital and Nicolas Fatio de Duillier. Between 1690 and 1710, these mathematicians developed all the potential of the theory with the aid of differential analysis, developing it into close to its final form. Unlike Newton, Huygens and Mariotte, for them the problem was almost exclusively mathematical, and they ignored, or at least forgot about, its physical origin. In this respect they did not recognize any difference between air and water, nor did they calculate forces in an absolute sense, but only the relative effects. Thus by 1720, we have a formalized theory for use with the aid of differential analysis, justified theoretically for motion in air, and a set of experimental results for the movement of plates and spheres in the water, although there were doubts concerning the proportionality constant between the forces and the dynamic pressure. Impact theory was established to explain resistance, but it was also applicable to the form of ships (based on the optimal form for minimization of resistance), to the sails and to some hydraulic machines, such rudders and oars. Nevertheless, the theory had some grey areas. On the one hand it had only been demonstrated theoretically for air, while the only experimental results available were for water. The attempts to extend the predictions for air to liquids stumbled over the fact that the theoretical forces were two or four times greater than the empirical data, according to the hypotheses, and this was difficult to account for. A second problem was that impacts of particles postulated by this theory only produced effects in the front part of the bodies, leaving the rear parts devoid of any impacts, whereas in normal observation the fluid fills the rear part of any body moving in it. A third obstacle was the reality of the collisions. This set of problems gave rise to a series of variants and explanations that shaped the evolution of the theory over the next 30 years. The search for a theoretical justification for impacts in liquids reached the phase of imagining a physical construction with a liquid in movement, in which the laws of mechanics were applicable, thus obtaining the value of the proportionality constant for resistance. The first person to attempt it was Daniel Bernoulli in a report that appeared in the Commentarii petropolitanæ in 1727, where he proposed an imaginary experiment with a liquid, which in essence was equivalent to the impacts in the air, and so too the resultant coefficient of resistance. Faced with a notable disagreement with experiments Daniel took the commonly used coefficient values for liquids, with the sole justification of not

THE PROBLEM OF RESISTANCE

51

diverging from experiments. Euler followed the same process in his Scientia navalis (Naval Science), written from 1736 to 1740, but not published until 1749. He proposed two different explanatory models, and in face of the fact that the result for one was the double that for the other, he chose the one that came nearest to reality, without justifying why the other one did not serve. The fact is that both Daniel Bernoulli and Euler ended up taking the theory the impact with a proportionality constant, justified only by experiment. By contrast, Pierre Bouguer, scientist and author of Traité du navire (Treatise on the Ship), published in 1746, took the result given by the measurements directly, without any need for theoretical justification. The second problem was the impact theory’s inability to explain the forces in the rear part of a body. To correct this, the authors admitted the validity of the impact theory of the frontal part, but added additional hypotheses for the rear part which gave rise to ‘mixed’ theories. Bouguer makes very clear the effect that the shape of the stern has on the progress of a ship, and he establishes a hypothesis which attempts to explain how the water enters the hollow left behind in the ship’s wake. At the same time, independently, Benjamin Robins published in 1742 his New Principles of Gunnery, a work on artillery in which one of the problems treated was the resistance of projectiles. In somewhat similar fashion, Robins supposes that a spherically shaped bullet when advancing leaves behind a vacuum that the surrounding air tries to fill. At a very low velocity the behavior in air is similar to that given by Newton for a sphere in a liquid, but as the velocity increases so does the rear vacuum, with the resulting increase in resistance. Robins establishes the dependence of the resistance coefficient on the velocity, which he confirms with a series of experiments consisting of balls fired by cannons. Some years later in 1756, Euler analyzes the effect of the wind in the sail of a windmill, for which he also assumes the existence of a partial vacuum at the back of the sail, and advances several types of mathematical laws for the pressure in this place. These three comprise the more significant examples of the ‘mixed’ theories. The New Principles of Gunnery had an interesting history. There was significant interest in the topic, since all armies were interested in artillery, and Euler was entrusted to translate this work into German, which he did in 1745, but with the addition of some very extensive commentaries regarding resistance. Euler carried forward Robins’ suggestions to a point that the latter could hardly have imagined, and in addition, as was usual in him, he completed the mathematical analyses, organising the experimental information with great rigor, and questioned the existence of impacts for the first time. On the contrary, he surmised that the fluid flowed around the body, introducing the ‘stream-line’ model with which he came closer to present-day conceptions. Nevertheless, in this work and

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in a later one, the difficulties in finding the pressures in these new models were so great, that he had to admit that although the fluid followed the streamlines, he had to obtain the forces in each point using the formulae of impact theory. This is to say, it is a ‘hybrid’ theory. Strictly speaking, the first person to make clear the inadequacy of impacts was Daniel Bernoulli, who demonstrated theoretically and practically that the phenomenon of the impact of a jet against a plate could not be studied by means of impact theory, as the jet curved, forming the shape of a bow, on reaching the plate. Thus, we observe that the situation of impact theory becomes rather curious: nobody believed in it, as several authors explicitly state, but as they do not have an alternative theory, they continue to use it, though with distrust and protests. With impact theory practically exhausted, a key work appeared: d’Alembert’s Essai d'une nouvelle théorie of the résistance des fluides, presented for the prize of the Academy of Berlin in 1749, and published in 1752. This work is a result of the ideas on fluid dynamics coming from the discharge problem. In it d’Alembert deduces for the first time what are nowadays known as general equations of the motion of fluids, continuity and momentum equations. But their mathematical solution was almost impossible. Besides, to his utmost desperation, he found that the resistance suffered by a body in a stream flow was zero, contrary to all practical evidence. He qualifies this fact as a paradox (even today it continues to be known as ‘d’Alembert’s paradox’), and it held fluid mechanics in check for 150 years. In spite of the attribution of the paradox to this author, Euler in his first streamlined model had already arrived at a similar result, but with a hybrid theory, as we have seen. The paradox arises due to viscous phenomena, because if the viscosity did not exist, the fluid flow would remain adhered to the body without any loss of energy and the resistance would be zero. The boundary-layer separation, with the ensuing alteration of the pressures field, is the principal factor responsible for resistance and today, a good part of the aerodynamics of aircraft wings or fuselages focuses on moving the separation point back as far as possible in order to diminish the resistance. According to Roger Hahn,4 this contradiction between theory and practice was the reason for the divorce between these two branches. This appraisal seems too excessive, but it is true that in the second half of the eighteenth century, experimentation was promoted, and the theoretical inability to predict the resistance had some influence on it. Finally, we must turn to the theory of Jorge Juan y Santacilia, who presented it in his Examen marítimo, where he applied it to ships in 1771. For him gravity was the reason for the resistance, therefore the local force at a point is a function of the local conditions, as in impact theory, but it is also a function of 4

Cf. L’hydrodynamique au XVIII e siècle.

THE PROBLEM OF RESISTANCE

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the depth. The curious thing about this conception is that it results in finding the resistance to be a function of the velocity and not of its square, a fact that Juan repeatedly stresses. We have summarized the evolution of the problem of resistance and how the impact theory led to a blind alley. In the following chapters we will give a more detailed breakdown of all this. First of all, let us look at the work of earlier scientists.

Chapter 1 The Forerunners of Impact Theory: Huygens and Mariotte

We award the title of forerunners to Christiaan Huygens and Edmé Mariotte, two men closely linked to Academy of Sciences of Paris in the first years of its progress, and with whom impact theory began. The fact that Huygens at this time was paying special attention to problems of motion of bodies in dense media, especially projectiles, contributed to this. Mariotte was likewise concerned with hydraulic phenomena in general, and the forces on the blades of a hydraulic wheel in particular. Both pursued their preoccupations in a wider context, where experiments and solutions with practical applications converge. Measurement of resistance Of some importance here is the question of how the resistance of a body in a current was measured in the seventeenth and eighteenth centuries. As we have already said, the concept of force did not appear clearly in physics until the advent of Newton, and its diffusion throughout Europe was not straightforward. What was really in common usage was the notion of weight, and as a result we should not find it strange that resistance was equated to weight, and specifically to that of a cylindrical column of fluid whose base was equal to the cross section of the body, with a certain height H. In this procedure the resistance is reduced to the height of this column, as is shown in Fig. 1-1. There is an additional point to be made: instead of taking this height as an absolute measurement, a relative measurement was used which took as its standard of reference a height equal to that from which one had to let a heavy body fall, such that it attained the same velocity as the fluid. In these conditions, the resistance was said to be equal to one, two, four or x-times this reference. If the velocity of the motion was v, this standard height would be h = v2/2g, a magnitude which we today call the ‘kinetic

55

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THE GENESIS OF FLUID MECHANICS, 1640–1780

height’,1 given that it associates a height with the velocity. Nowadays, the measurement of resistance is represented by a dimensionless coefficient called the ‘resistance coefficient’ and designated CD, or sometimes Cx, and defined as2:

CD =

D 1 ρ v2S 2

[1.1]

Where D is the resistance, ρ the fluid density and S the frontal cross section of the body. If we relate this coefficient to the old measurement according to x-times, CD = x results. That is to say, what we now call the resistance coefficient coincides with what was then expressed as being so many times the weight of the cylinder whose base was the frontal cross section and whose height was the kinetic distance described above.3 In what follows, we will make almost exclusive use of CD, because this makes presentation and understanding easier. V

S

S

H

D D

Fig. 1-1. Measurement of resistance

1

This height is also known as ‘sub-limited’, resulting from the Italian translation of ‘sublimatta’, which in his turn comes from the Latin ‘sublimitata’. We think the ‘kinetic’ denomination is much clearer and explanatory. 2 In the mechanics of fluids it is very common to work with non-dimensional parameters. This has the advantage that these parameters are constant, or almost so, within a broad margin of variables, which simplified the applications and experimental measurements. On the other hand, some of the effects to be considered in the motions of fluid are negligible, according to the values of these coefficients, which enable us to ignore, or only consider, part of the terms. As an example of nondimensional parameters, we mention Reynolds, Mach, Froude, Nuldset, Prandtl, Stanton numbers, among others, as the commonest ones. 3 Another way to see it is that the pressure in the base of a cylinder, with the kinetic height as the generatrix, is the dynamic pressure. The proof of the equivalence is easy. The weight of the reference column is W = ρgh, and the resistance D = xW. Then, Xw = ½ρv2SCD, resulting in CD = x.

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The expression of resistance as x-times was of great importance, as there are various concurring circumstances for which this value was 1. On the one hand, we have Torricelli’s law which represents the outlet velocity of a liquid through an orifice in a reservoir whose depth h was precisely v2 = 2gh, coinciding with the kinetic height. Furthermore, the experiments attempting to measure the force of a jet against a plate pointed towards values close to 1. Finally, for a body that accelerates to a velocity v, if suddenly it were pointed upwards, the body would ascend to a height equal to the kinetic height. This property was used to deal with the conservation of live forces. Huygens and his experiments Christiaan Huygens lived in Paris almost uninterruptedly from 1666 to 1681, and had been a member of the Academy of Sciences since its foundation in 1666. In 1668, he began the study of the fall of bodies and the movement of projectiles in dense media. For this he introduced the concept of the effect of the resistance of the medium. Although his first hypothesis was that the resistance was proportional to the velocity, one year later, after various experiments, he substituted the simple ratio for that of the square. His personal notes on these experiments have come down to us, and they are currently published in his Œuvres Complètes,4 as well as the references quoted in the Regiæ Scientarum Academiæ Historia Parsiis (History of the Royal Academy of Sciences of Paris), (1698)5 of Jean-Baptiste du Hamel, which relates experiments made in 1669. It is important to note the fact that Huygens carried out two classes of experiments on resistance. On the one hand he tried to direct measurements of the resistance force which opposes the movement of cubes dragged through water, or that of air against plates. But on the other hand, he also performed measurements on the impact effect of jets of water or air against plates. On the hypothesis of impact theory, this latter phenomenon is equivalent to the resistance of a moving plate immersed in fluid, as both are equally reduced to a set of mechanical shocks of fluid particles against a body. The convergence of these two phenomena was maintained during a quite a long time, giving rise to numerous problems.

4

Edited by the Societé Hollandaise des Sciences. The volume of interest, Volume XIX, was published in 1937. 5 Cf. Section III, Chapter IV, under the epigraph ‘De Hydrostaticis’.

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According to what du Hamel tells us, Huygens had co-operated with, or had been present in various attempts to verify the Torricelli’s law,6 one of the activities promoted by the Academy during those years, and is which included others as Picard, Castelli, Roberval and Mariotte. The test apparatus and experimental means that they used also served, with few modifications, for measuring the force of a jet against a plate. Du Hamel wrote about this experiment, and after describing the parameters he states ‘This proof concerning the water and air driving force was continued and more extensively developed in the month of 7 April of 1669 and the following months.’ He does not indicate the authors of the experiments, although we suppose they must have been Mariotte and Huygens. Such experiments made in the Academy must have been the first of this kind, as years later Daniel Bernoulli affirmed: As far as I know, the first experiments made to determine the force of a water stream impacting on a flat plane were set up under the auspices of the Paris Academy of Science in 1679.8 This appears in the history of the aforementioned Academy, written by du Hamel. These [experiments] were followed by countless others. 9

For his part, Huygens describes these experiments in his notes, where a clear parallel exists between his reports and du Hamel’s, although Huygens contributes more details, and adds some sketches of the apparatus that du Hamel only mentions in a simple description [p. 120].10 Figure 1-2 shows the two drawings made by a Huygens. One can appreciate that they are water reservoirs with an outlet orifice, in the bottom in one case, and in the side in the other, from which the jet of water spouts, impinging on a plate fitted or articulated to the arm of a pair of scales. The force produced in this ‘impression’ was measured by means of a mass on the other pan of the scale. According to the text, the apparatus used by Huygens followed the first drawing, although it says that it could be done using the second. The reservoir was a cylinder of 3 pieds (927 mm) in height, and of some 6 pouces (162 mm) in

6

This established that the outlet velocity of a liquid through an orifice in a vessel was equal to the velocity a body would reach when falling from a height equal to the distance between the free surface of the liquid and the outlet orifice. That is to say, the kinetic height of the outlet velocity was equal to the depth at which the orifice was allocated. About the same cf. supra 6.1. 7 Cf. Regiæ Scientarum, §. III, p. 48. 8 Bernoulli mis-dated the 1669 experiments. 9 ‘Disertationes de legibus mechanicis nondum descriptis’, Second Part, §.1, Comm. petrop. Vol. VIII, 1736 (1741). 10 We quote the pages of the Œuvres Complètes, Vol. XIX between brackets.

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Fig. 1-2. Huygens apparatus for measuring water discharge

diameter, with a circular orifice in the bottom of 4 lignes (9.02 mm) from which the water flowed out. This orifice was closed by a plate P around 1 pouce in size (27 mm) located at the end of a lever articulated to one arm of the scale. The reservoir was filled with water up to a certain height, and in order to counteract the effect of the jet of water against the plate, the scale was loaded in such a way as ‘not to be either too near or too far from the opening of the cylinder’ [p. 123]. The situation was difficult to adjust, and as he confesses in his manuscript he was forced to carry out successive and repeated attempts. Specifically, he cites the case in which a height of the water did not exceed 35 pouces (947 mm) and for which ‘it was found that the weight as two onces, and at least three gross (73 g) or at most four gross and a half (78 g)’ [p. 123]. The difference between one method and the others is 5 g. 800

Force (mN)

600

Experiment Theory

400

200

0 200

400

600 Heigth (mm)

800

Fig. 1-3. Forces on the plate

1000

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THE GENESIS OF FLUID MECHANICS, 1640–1780

With these experiments, Huygens’ aim was to compare the force measured with the weight of a cylinder whose base was equal to the orifice exit area and the height of the reservoir. The outlet velocity of water was introduced directly, as it depends on the height of the fall. Thus, ‘the force or imprint of the water, whose velocity is known, against any given surface can be determined, e.g., against the blades of a mill wheel’ [p. 124]. The way to obtain the velocity is via Torricelli’s law, which Huygens still accepted. Figure 1-3, in which the abscissa represents the heights of the reservoir and the ordinate the forces, was made with the four cases registered by du Hamel and the five by Huygens. The dotted line indicates the points in which the force would be equal to the weight of the column. A certain approximation of this line to the values measured can be appreciated, and this lead Huygens to say that ‘the imprint that the water coming out of an orifice in the bottom of a vessel makes, is equal to the absolute weight of a cylinder of water having the same orifice as its base, and the height is equal to the water in the vessel’ [p. 124]. M

Q P

F R

Fig. 1-4. Apparatus for measuring the force of air

But Huygens was not satisfied with the experiments with water, and in the same year he extended his experiments to air. The basis was similar: to produce a jet of air and project it against a plate, something rather more difficult to achieve. In order to do this he devised an apparatus whose design he presents in his manuscripts [p. 132] with the epigraph ‘in order to observe the wind force’, and which was reproduced years later in Machine et Inventions aprouveés par l’Académie Royale des Sciences depuis son établissement jusqu’a present; avec leur description. Desinées et publiées du consentement de l’Académie, par M. Gallon11 (Machines and Inventions approved by the Royal Academy of Sciences 11

This book was published in Paris in 1735, as it is quoted in the Œuvres Complètes, note in p. 128, from where the shown reproduction has been taken. There is a hand scheme of it in the page 132.

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since its foundation until the present, with their description). Both were used to construct the diagram shown in Fig. 1-4, where it is easier to understand how the experiment functions. The apparatus consisted of an air bell formed by a cylinder whose upper part was closed, and whose diameter was 8 pouces 7 lignes (232 mm). It was placed in a vessel with water in order to insulate the exterior from the interior. Two tubes were connected to it: the first was connected to a manual bellows F which was used to fill the air bell and the second, with a diameter of 2 pouces 5/12 lignes (5.45 mm), serving as a discharge outlet to the exterior. The outlet orifice of the latter was closed by a plate R which acted as a plug, and formed part of an articulated mechanism in Q, in such a way that the force that this plate exercised on the outlet nozzle varied, moving the mass P over the other arm of the set, thus enabling the force to be calculated. It is clear that as the bell descended an air current was produced through the discharge tube which struck the plate R. The experiment consisted of two phases. In the first a mass M is placed in the upper part of the bell, and this was filled with air using the bellows until the bell ascended to a determined height, which was 9 pouces (244 mm). Next, the free discharge of air was allowed unhindered, registering the time that the cylinder took to descend from this height. With this time, the volume of air expelled and the area of the outlet tube, Huygens obtained the ejection velocity of air from the nozzle. In the second phase he refilled the bell using the bellows, and measured the force on the plate using a similar method to that employed in the water, by moving the load P on the arm of the lever. He thus found three direct parameters: the mass M, the retention force and the discharge time plus an indirect one: the velocity. From these he concluded that ‘In both air and water, the imprints are found to be the square of the velocity’ [pp. 132–135]. 40

40

2000

2000

p/v2

20

20

10

10

0

0 0

2

4

6

8

1500

1500

F/S

1000

500

1000

500

0 0

500

Mass (kg)

1000

Pressure (Pa)

Fig. 1-5. Air discharge results

1500

0 2000

p/v2

30

Time (s)

Force (N)

Force

F/S (Pa)

Time

30

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THE GENESIS OF FLUID MECHANICS, 1640–1780

In order to understand his line of reasoning, we shall follow the graphs presented in Fig. 1-5 which show the values found by Huygens. In Fig. 1-5 (left) the mass that makes the cylinder descend is placed on the abscissa, and the measured retention force F and the discharge time vary with this. The ratio between the force and the mass can be easily seen. The other figure, Fig. 1-5 (right), shows the force over the plate divided by the area of the outlet orifice, and the quotient of pressure over the square of the velocity varies with the calculated internal pressure p. The proportionality between the first two and the constancy of the second are remarkable. After a laborious reduction [pp. 129–140], he ends by offering resistance values which he says are useful for constructing all kinds of watermills and windmills. In the case of water, a current of 1 pied/s (0.325 m/s) acting on a surface of 1 square pied (0.105 m2) produces a force of 44 ½ onces (13.26 N), which is the equivalent of a resistance coefficient CD = 2.39. In the case of wind, when it blows against a plane of one pied square with a velocity of 10 pieds (3.25 m/s) it makes an impression equal to 9 onces (2.7 N) which corresponds to CD = 3.96. The resistance coefficients in both cases turn out to be very high.12

Fig. 1-6. Cube dragged through water

Let us remember that the experiments referred to both water and air, dealing with the impact of a jet against a plate, where Huygens likens the ‘impressions’ to impacts. Nevertheless, he also deals with the measurement of resistance directly, that is to say of bodies in currents. In this respect, he presents a drawing of three interesting apparatuses. One depicts a wooden cube being dragged through a channel of water. The apparatus he used was later converted into a classic assembly: a weight which pulls the cube by a cable and pulley (Fig. 1-6). In the explanations and comments made by Huygens it is worth noting two points. First, note the equivalence between the movement of the box and that of the water, ‘because the effect of the pressure is equal whether the water goes against the surface of the body, or whether the body moves with a similar velocity 12

According to well established present day measurements, the CD of a square plate for Reynolds numbers larger than 1·103 is 1.14. Referring to Huygens former statement, he appears to have made a mathematical error, as translating the result 35 pouces gives a force of 23.64 onces (7.08 N) instead the 44.25 he states.

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in still water’ [p. 122]. Second, as a result of the measurements he deduces that there is proportionality between the ‘impressions’ and the square of the velocities: ‘thus, if the velocity of the water is double, the weight D must be quadruple, so that the impressions of water against the same surface are the squares of the velocities’ [p. 123]. This is the first time that this rule is established,13 and we recall that previously Huygens had supposed the resistance to be proportional to the velocity. Thus it may be that these experiments convinced him of the contrary. The other two apparatuses concern wind. One was a kind of weathercock, similar to that presented by Mariotte later on. Its conception is simple, designed to balance the force of an air jet on a surface using a weight on the arm of a lever [p. 137]. In some ways it was similar to the case of the water jet. The second was a type of sleigh, moved by a system of pulleys, upon which a vertical surface was placed and held in place by an ingenious counterweight mechanism [p. 138]. This worked in such a way that the vertical surface fell when the wind force exceeded a preset value. Knowing the velocity of the sleigh at the moment of its fall, he obtained a relation between this and the force. The original drawing of Huygens is shown (Fig. 1-7) in which we can see the aforementioned mechanism. The apparatus had two drums, the diameter of one was double that of the other, so that two sleighs were moved simultaneously, one at twice the velocity of the other.

Fig. 1-7. Dragging plates using a sleigh

Before leaving Huygens, we must underline his undeniable talent as an inventor of apparatus and mechanisms, among which his studies on clocks stand out. He was the first to establish the proportionality between the resistance and the square of the velocity, which came to be accepted almost as a law. He also proposed the hypothesis that the force exercised by impact upon a plate was equivalent to the weight of a column whose height was equal to the kinetic height. Moreover, his experiments would be a source of new novelties and would be repeated many times. 13 This is the opinion that we share, expressed by H. J. M. Bos in the Gillispie’s Dictionary of Scientific Biographies under the entry of Huygens.

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Mariotte and the theory of jets Edmé Mariotte played an outstanding role in the Academy of Paris, where he overlapped with Huygens, who, after Mariotte’s death, accused him of plagiarising some of his experiments. We shall not go into this affair, although it is certain that the apparatus used by Mariotte was similar to those of Huygens. However, although the ideas of Mariotte were never synthesized into a formal theory, he proposes an interesting set of rules on the effect of the ‘impacts’ of the jets of fluids upon the bodies. These effects have their basis in likening the jets to an aggregate of particles moving as a solid body, but when they collide, they do so as individual elements. Of the various works by Mariotte, of special note is his great synthesis, the Traité du mouvement des eaux14 (Treatise on the movement of water), a posthumous work (he died in 1689) which saw the light of day thanks to Gaston De La Hire in 1686, and which comes close to a manual of hydraulic engineering in use at that period. Nevertheless, some of his ideas on fluids had already been anticipated in the Traité de la percussion ou choc des corps (Treatise of percussion or impacts of bodies) of 1673. One of the parts of the first of these treatises is dedicated to the equilibrium of fluid bodies, and one of the ‘Discourses’, into which each part is divided, deals with the so-called balance by impacts. Mariotte considers fluids in motion, in particular (but not exclusively) water, to be somewhat similar to solid bodies, in order to thus establish some rules for fluid impacts similar to those accepted for solids. The underlying preoccupation is to express the phenomena mathematically, as his own words show: Air and water are used in machines in order to make them move by their impact. The equilibrium established between them and closed bodies against which they may strike, can be known through the following rules. [p. 188]15

And he continues with the five rules which will define what we have called ‘the mechanics of jets’.

14

The work comprises five parts, as follows: 1/ ‘Of the various properties of the fluid bodies, on the origin of the fountains, and the cause of winds’. 2/ ‘On the balance of the fluid bodies’. 3/ ‘On the measurements of flowing waters and out springs’. 4/ ‘On the height of jets’. 5/ ‘On the ducting of waters and resistance in tubes’. 15 The references between brackets will refer to the Traité du mouvement des eaux.

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Fig. 1-8. Fluid jet

In the first place, we ought to know what it is understood by ‘jet’ (jet d’eau). Mariotte explains it with the help of Fig. 1-8 [p. 188] in which he imagines a tank CD with an orifice in the bottom side from which water flows out horizontally, giving rise to the ‘jet’ AB. In order to give us a better idea of its composition and behaviour, he compares it with a wooden cylinder EF which also moves horizontally. He notes the fact that if both strike against a body, the wooden cylinder suffers the effect as a whole, as all its internal parts are joined together, while the case of the jet is different: [B]ecause as water is fluid and composed of an infinite number of small corpuscles which slide over one anther, as if they were tiny grains of sand, only the first moving towards B can exercise the first force on the bodies they encounter, and they rebound or separate before the others like them at d can impact in their turn.

That is to say, given the type of union existing between its particles, the jet does not behave like a solid but like a conglomerate. As a consequence of this constitution, the particles jump aside after impact, influencing those following them. In view of these explanations, we can see he understood by the term ‘jet’ a column of water which moves as a whole and following its axial direction, and which is defined by its cross section and length. Its materialisation would be the result of the discharge of a vertical tube filled with water. The liquid at its exit point, and in following its path, would behave kinetically as if it were a solid, but it does not do so when it strikes against a surface. It is very clear that in this mechanics of impacts the problem of resistance is identified with a physical impact.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Mariotte based his ‘mechanics of jets’ on five rules. I. ‘Jets of water do not collide due to the effort of all their parts as closed bodies do.’ [p. 188] This rule corresponds with the definition of a jet. For a closed (fermé) body he means the ‘solids’. II. ‘The water which springs from the bottom part of a tank through a round aperture, remains in equilibrium due to its impact with a weight of a cylinder of water whose base is this very aperture, and whose height is that measured from the centre of the aperture to the upper surface of the water.’ [p. 196] This proposition is understood with the help of the experiment he proposes, as it similar to those of Huygens, and will be frequently repeated by other scientists. In Fig. 1-9 (left), the water contained in the tube MN forms a jet which is balanced by a weight Q equal to that of the column. What Huygens said is applicable to this experiment. However, although the rules only mention water, Mariotte also extends them to air, and he uses an apparatus similar in style to that used by Huygens for this purpose, the difference being the air bell is replaced by a cylinder whose walls are formed by a bellows, Fig. 1-9 (right). When compressed by the weight P, this accordion-like device expels the air through the orifice N which acts as water outlet in the other case. The sealing force is found to be

Fig. 1-9. Equilibrium at the fluid outlet

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proportional to the weight compressing the bellows, and the ratio of the cross section of the orifice at the base of the tank, i.e., the product of the internal pressure multiplied by the orifice section, as the first is determined by the quotient between the weight and the cross section of the tank. III. ‘Jets of equal length, coming out from the small holes made below various tubes filled with water from different heights are equilibrated by weights that in turn are in relation to the heights of the tubes.’ [p. 201] The rule is quite clear: if in the previous experiment tubes of different lengths were put in place, the weights would have the same relation. In the three preceding rules, no kinematic consideration has yet been introduced, thus before defining the following rule he warns us that according to the law of Galileo16 the velocity of the fall of the weights is related to the square roots of the heights. IV. ‘Jets of equal length and with different velocities on impact sustain weights which are in their turn related to the square of their velocities.’ The basic reasoning is the same as Huygens’: given that water is formed by tiny particles, ‘of necessity we arrive at the point that when they move two times more quickly, two times more collide at the same time, … and as they move twice as fast, they make twice the effort, and thus the two effects must cause a quadruple effect’ [p. 186]. Here, he also goes on to analyse the cases of discharges of air, using air bells very similar to those of Huygens. We underline the fact that he establishes that the effects of air and water are equivalent when the velocity of the water is 24 times greater, that is to say ‘the air is 576 times more rarefied than water’ [p. 187].17 V. ‘Jets of equal velocity and with different openings sustain on impact weights which are in turn related to the square of their diameters.’ [p. 212] To say same diameters here is the same as saying same surface area.

16

He refers to the law of the fallen heavy bodies fall, according to which, the square of the time of descent is to the height of the fall, expressed as s = ½gt2. 17 The actual ratio between their densities is 820.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

These rules, which he had already implied in the Traité de la percussion, demonstrate an attempt to look for an equivalence between solid and fluid bodies. Specifically, he tries to reduce liquids to the same principles as solids, considering the first as an aggregate of particles, and his comparisons with grains of sand make this very clear. These explanations, together with the observation of a phenomenon of the jet shock against a plate, in which water jumps and splashes towards the sides, prefigured the impact effect, and therefore the resistance as a phenomenon of mechanical impact. Apart from the notable effort made by Mariotte upon the introduction of these rules, it is important to underline his experiments, which were quoted up to a century afterwards. For Mariotte, the resistance of a completely submerged body was equivalent to the impact of a jet of water against the same body. This idea is consistent with the hypothesis that the resistance is generated as an effect of the impacts of individual particles. In this respect, when he carried out the experiment he did not do so with jets, as Huygens had done before, but he mounted his apparatus with plates located inside a flowing current. At first it seems strange that he followed this indirect path in order to check his theories on the jet forces. However, from the mention that he makes of ‘mills and other machines’, and what he adds a few pages further on about ‘how the force of the mill wheels of the Seine can be calculated’ [p. 221], we can conclude that Mariotte paid particular attention to resolving this particular case, and thus he started out from the force on a flat plate, which is a basic element of the mill, and then adjusts it to his theory. This preoccupation is evident at other points of the Traité, as much for fluid currents as for windmills and ships’ sails. His experiments with plates in currents are the first to be historically registered, and although he must have carried out considerably more, he only quotes the results of two in his treatise, apart from another couple that are merely qualitative.

F

F v

Fig. 1-10. Measurement of the resistance in a water current

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As he does not present any figure [p. 214], we have constructed a diagram following his descriptions of the apparatus he must have used in Fig. 1-10. It was formed by a square plate whose sides measured 6 pouces (162 x 162 mm) submerged perpendicularly to the current at a depth of 2–3 pouces (27–54 mm) and fixed to a mobile bracket. The shock of the water is equilibrated by a weight located on the other arm of the bracket at a distance from the shaft equal to that of the center of the plate. In order to measure the velocity of the current he says that small bits of wood or blades of grass are dropped from a large boat anchored in the centre of the river, and using a pendulum of semi-seconds, he measured the time that these took to cover the distance of 15 pieds (4.87 m), between this boat and the other small one tied to it. The first measurement that he quotes corresponds to a velocity of 3 1/4 pieds/s (1.06 m/s) which required a force of 3 3/4 livres (18 N) for the balance. Reducing these values to the resistance coefficient, a value of CD = 1.22 was determined, close to the present values. He corroborates the quality of the demonstration by stating that ‘I have found the same force of water current in several other places in the river, and even in the aqueduct of Arcüeil’ [p. 216]. He made a second measurement near the banks of the river where the velocity turned out to be 1 1/4 pieds (0.406 m/s), which, in order to be a equilibrated, requires a force of 9 onces (2.70 N). On comparing these values with the previous ones he makes note of the good agreement between them when considering the proportionality of the forces with the squares of the velocities. If we calculate the resistance coefficient we obtain CD = 1.25 which is almost identical to the value of the previous case. Mariotte, when interpreting these results, does so by likening them to impacts, and thus generalizing them. In this respect he states that: [I]n order to know the forces of the water currents when they strike against the blades of the mill or any other machine, it is necessary to know their velocity and to compare it with that of the waters springing from the bottom of a tank. [p. 213]

Thus, for the first experiment he supposes a tank elevated to 12 pieds (3.9 m) and says that, according to the doctrine of Galileo, the velocity of a jet falling from the same would be 24 pieds (7.8 m/s) which is 7 1/2 times greater than the 3 1/4 of the current of the river.18 Therefore, the force of the jet coming out of the tank must be 7.52 = 56.25 times greater. The weight of a column 12 pieds high and a square base of 1/2 pieds is 210 livres (1,007 N), therefore the force on the blades will be 56.25 times less, that is to say, 3 3/4 livres, which is exactly the force found. 18

The calculation is not very precise, for the result is 26.9 pieds/s (8.74 m/s).

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Fig. 1-11. Apparatus for measuring the force of the wind

Mariotte tried to extend this type of experiments to air, and proposed and apparatus for this similar to one belonging to Huygens,19 in order to measure the force of the wind shock, although it did not give any measurement (Fig. 1-11). In contrast to other cases of jets, in this one the incidence of the fluid is not perpendicular to the plane, but forms an angle of 45 degrees with the plane of incidence. And in these circumstances he affirms ‘that the weight R will not have more than half its weight, … as has been more fully explained throughout the ‘Treatise of percussion’ [p. 224]. Let us say, that the resulting effects are proportional to the squares of the sine of the angles of incidence, as was established in the theory of mechanical percussions. In this case this angle is α = 45°, and sin α = 1/√2, whose square is 1/2, which justifies what has been said. It also quotes the calculation of the force on a sail of a ship. For this he supposes that the air is 576 times less dense than water, a result he had already obtained in expansion models based on his fourth rule. Let us now leave Mariotte. Regarding his contribution to the problem of resistance we must state two points. The first is that the attempt to look for an explanatory model led him to enunciate a set of rules, which, although they did not have the standing of a theory, were already a precursor of impact theory. The second is his experimental ability, and the values he supplied would prove useful for many years.

19

Cf. the hand drawing figure in the Œuvres, vol. XIX, p. 137.

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Experiment of a jet against a plate As we have said, the phenomenon of the impact of a jet against a plate was considered to be the equivalent of that of a plate moving in a fluid, and it gave rise to an experiment that was repeated many times. In the first instance, we have to say that in spite of its apparent simplicity, the phenomenon is not easy to explain, as several different factors intervene. Thus, the formation of the jet at the outlet of the tank depends greatly upon the geometry of the outlet nozzle and its adjustment to the tank. Supposing these elements are well designed in order to avoid perturbations in the current, and that the volume of the tank is very large when compared to the flow rate, the outlet velocity will be determined by Torricelli’s Law, and the volumetric consumption will be the product of this velocity multiplied by the area of the nozzle. In real cases this will only be the upper limit. As regards the force on the plate, there are two extreme cases: that the plate acts as a seal, or that it is quite separate. In the first, Fig. 1-12a, the force on the plate will be only that due to the hydrostatic pressure, that is F = Ps = ρghS, and besides, it will only act on the part in contact with the liquid.20 In the second case, when the plate is separated by a distance enabling distribution of the liquid without modifying the flow upstream in the nozzle, if there are no rebounds or splashing, a layer will form whose thickness will decrease as it separates from the central point, as in Fig. 1-12b. The force on the plate will be equal to the variation of the momentum of the flow, which goes from being an axial vector to a perpendicular distribution. The volumetric flow will be Sv, and the momentum the same as ρSv, thus the total force is F = ρSv2. If we substitute the velocity by the height with the aid of Torricelli’s theorem, this result can be expressed as F = 2ρghS. This is exactly double the value found in the first case. What happens is that the dynamic phenomenon is different from the static situation. In the latter circumstance, the pressure distribution is uniform, while in the dynamic discharge it is not so, as it decreases radially with the distance. On the supposition of an infinite plate, the result of integrating the distribution of forces will coincide with the previously calculated value. Furthermore, all these considerations are taken for an ideal environment without gravity, because if this existed it would modify the results.

20 It is assumed that the dimension of the orifice is very small compared with the height, allowing the pressure to be assumed as uniform.

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a)

b)

Fig. 1-12. Forces against a plate

The phenomena as described took some time to understand, and the experimenters took the static value as the upper limit. The person who cut through the Gordian knot of misunderstanding was Daniel Bernoulli, when he noted in the Hydrodynamica the difference between the start of the discharge and the stationary condition. Later on, in 1736, he explained the second phenomena clearly and performed experiments to confirm it.21 It is still doubtful what they really measured when they carried out this experiment. It must have been something in-between, as the plate was small, and being very close to the orifice it distorted the flow, which was also subject to the influence of gravity and other factors.

21 Cf. Comm. acad. petrop. 1736. See later Chapter 3, ‘Jet against a plate: Daniel Bernoulli’s clarifications’.

Chapter 2 Impact Theory: Formulation and Formalisation

The contribution of Newton Impact theory went through two decisive phases. The first was its formulation, due to Isaac Newton, who set it out in the Principia; the second, its formalization, by various mathematicians over a longer period of time. It was Newton certainly who gave the theory scientific status, as we currently understand this difficult and ambiguous expression. However, the preparation of the theory for use as a mathematical model, with all the resources of differential analysis, was due to other mathematicians. The importance of Newton’s Philosophiæ naturalis principia mathematica (Mathematical Principles of Natural Philosophy) extends from the dynamics of rigid bodies to that of fluid bodies. Of the three Books making up the Principia, Newton devoted Book I to solids and Book II to liquids, although he used the same title for both, ‘The Motion of Bodies’, which allows us to conjecture that he considered solids and fluids as being subject to the same laws, in spite of the peculiarities of each. He made this explicit in the preface, where he explained what he understands by mechanics: In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. … [I] consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore I offer this work as the mathematical principles of philosophy, … and to this end the general propositions in the first and second Books directed.1

1

The quotes of the Principia were taken from the third edition translated by Andrew Motte and corrected by Florian Cajori, because we believe that, from an historical point of view, it may be closer to Newton’s thinking, even though it would be easier for a twenty-first century reader to follow a more modern translation, like the one by I. Bernard Cohen and Anne Whitman. In these quotes, the number of the proposition and details are indicated between brackets, referring always to the Book II, except when indicated. The page number, when quoted, corresponds to the University of California Press, ed. 1962. The quotations from the first edition were translated with the help of Eloy Rada García.

73

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THE GENESIS OF FLUID MECHANICS, 1640–1780

But in spite of the concurrence of the objectives, the methodology in each one of the books is very different, and the same applies to the survival of the teachings contained therein. In Book I the developments are characterized by their deductive nature: they begin with the ‘Axioms or the Laws of Motion’, that constitute the cornerstone for the entire system; taking them as the starting point, the rest of the results are derived in successive steps. This does not happen in Book II, where the concepts and hypotheses are indeed introduced successively, but without the coherence and consistency of the other Book. This new way of doing things is probably due to the difficulty in treating the fluids, which he considers as aggregates of innumerable individual corpuscles interacting among themselves. Concerning this theme, Truesdell states that ‘Newton’s program of deriving all results rationally from the ‘axioms, or laws of motion’, while fairly successful for mass-points in Book I, broke down completely in Book II, where a fresh hypothesis starts up at every turn’.2 In spite of Truesdell’s opinion, we must consider that the Principia mark the definitive point of the birth of fluid mechanics as a science, as it provides a new and innovative attempt to submit fluids to a theoretical treatment explaining their behaviour as a function of their constitution, the conditions in which they move, and some general laws. It is quite true that Newton only does this for air and not for liquids, but this underlines precisely the coherence of his hypotheses, as these cannot be applied indiscriminately, but only to well-defined constituting models. What is more, he does not even apply them strictly to air, which he identifies as an elastic fluid, but to what he calls a ‘rare medium’ (medium rarum) which is a kind of imaginary fluid, about which he conjectures that its behavior, when faced with a body that moves in it, comes closer to that of real air. Newton’s hypotheses are no longer accepted, but the contribution of Book II is significant. Newton’s ‘failure’ is only to be expected, as the nature of fluids proved to be difficult and unfriendly over many more years, and even some mathematicians of the eighteenth century thought that even if fluids were bodies like the others, they required different laws to explain their behavior.3 The successive transformation of the discipline of fluid mechanics makes it evident that the Book II would be a reference work throughout the entire eighteenth century. The Principia had three editions in Newton’s lifetime. The first one was in 1687, the second in 1713 and the last in 1724. Even though overall few differences exist among them, this is not true as regards the movement of bodies in liquids. The criticisms made by Nicolas Fatio de Duillier and Roger Cotes, together with his own investigations, required almost a complete rewriting of the 2

Cf. ‘Rat. Fluid Mech-12(2)’, p. XIV. In this sense d’Alembert is expressed in the Prologue of the Traité de l'équilibre et du mouvement des fluides.

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propositions dealing with resistance of bodies in liquids, as well as requiring him to try to verify them experimentally.4 There are no significant differences with the third edition, except for the addition of some supplementary experiments. On analysing Book II, it can be appreciated that is does not form a coherent whole, bur rather it is a juxtaposition of various themes with clear differences among them. These major themes are: hydrostatics, resistance of moving bodies, propagation of perturbations and motions of vortices.5 But, even if we limit ourselves to resistance, our principal theme, we find two different approaches: the study of the motion of pendulums in resisting media, and the motion of projectiles. In the first, Newton devoted himself to investigating the mathematical form of the law of resistances in relation to velocity, body size, and fluid density, basing himself on experimental data. In the second, when treating the motion of projectiles he introduces the basic hypotheses that translate the impact theory into mathematics and confer upon fluid dynamics its nature as a science in the modern sense of the word: how the phenomenon is developed, what theories are applicable to it, and how to calculate the effects. Using the pendulums he measures and attempts to explain the results mathematically. In the case of projectiles he explains the phenomenon intrinsically, and later goes on to try to confirm it experimentally. There is no logical continuity between the two methodologies. We should emphasise that the approximation we have designated as scientific is only applicable to this air-like ‘rare medium’, though real air can be modeled upon it. In the case of liquids, the same hypotheses are not valid, and he looks for empirically explainable alternatives, though these cannot be reduced to more basic theories. Besides—and this is important—there is a difference in the treatment of liquids between the 1687 edition and that of 1713, and although, as we shall see, the difference is more quantitative than qualitative, it is however linked to certain conceptual developments. The consequence is that Newton’s contribution must be treated separately for each edition.

4

Such as it is indicated by Newton himself in the Preface to the Second Edition: ‘In the seventh section of the second Book the theory of resistances of fluids was more accurately investigated, and confirmed by new experiments’. Cotes, in the Editor Preface, also states it. 5 The first four sections can be understood as an extension of the First Book, because it treats the motion of a point mass subject to defined forces. The titles of the rest sections are the following: ‘The density and compression of fluids; hydrostatics’ (Section V). ‘The motion and resistance of pendulous bodies’ (Section VI). ‘The motion of fluids, and the resistance made to projected bodies’ (Section VII). ‘The motion propagated through fluids’ (Section VIII). ‘The circular motion of fluids’ (Section IX).

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The concept of fluids in the Principia Newton defines a fluid in these terms: ‘A fluid is any body whose parts yield to any force impressed on it, and, by yielding, are easily moved among themselves’ [Sec. V]. This suggests a phenomenology of fluids rather than its internal constitution.6 This definition covers both liquids and air. However, Newton distinguishes very clearly between one and the other. In the first case he nearly always refers to water, although he does sometimes mention oil, mercury, and some other liquids. As regards gaseous media, he deals exclusively with air, as at the time no other gas was known, although he occasionally mentions vapours.7 We shall go over the qualifications and qualities that Newton attributes to them throughout his Principia, chapter by chapter, in order to underline the nuances, variations and even the contradictions that exist. The definition of fluids is presented at the opening of the chapter ‘The density and compression of fluids; hydrostatics’ [Sec. V]. A little further on he talks of a ‘homogenous and unmoved fluid’ [Prop. XIV] in a contextual reference to liquids. For air he introduces fluids where ‘Let the density of any fluid be proportional to the compression’ [Prop. XXI]. Then, after establishing that he writes: ‘… as to our own air, this is certain from experiments, that its density is either accurately, or very nearly at least, as the compressing force’ [Prop. XXII, Sch.].8 He talks of a fluid constituted by particles that repel each other, so that ‘particles fleeing from each other, with forces that are inversely proportional to the distances of their centres, compose an elastic fluid, whose density is as the compression’ [Prop. XXIII]. The composition of air is inferred from the conjunction of these statements. However, then he adds:

6

As antecedents of the nature the fluids, we have the opinions of Mariotte and Hooke. The first, at the beginning of the Traité du mouvement des eaux (1686), says ‘Air and flame are fluid bodies; water, oil, mercury, and other liquors are fluid bodies and liquid; all liquid is fluid but not all fluid is liquid. I call liquid to that which being in enough quantity flows and extends under the air until its surface is placed at level; and as the air and the flame have not this property, I do not call them liquid but only fluids’. Robert Hooke, in the Micrographia (1665) in the explanations referred to the Observation VI, introduces what he considers to be the cause of the fluidity. He says: ‘for heat being nothing else but a very brisk and vehement agitation of the parts of a body … the parts of a body are thereby made so loose from one another, that they easily move any way, and become fluid ’ (p. 12). 7 The first recognized work that mentions other gases is the Vegetable staticks by Stephen Hales, published in 1727. Nevertheless, there are some experiments of Robert Boyle, followed by Robert Hooke and John Mayow, in which they managed to express the idea of the ‘aerial nitro’. 8 This is the Boyle law, also denominated frequently as Boyle–Mariotte.

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But whether elastic fluids do really consist of particles so repelling each other, is a physical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question. [Prop. XXIII, Sch.]

In the following chapter, that deals with ‘The motion and resistance of pendulous bodies’ he does not establish any difference between air and water. What is more, he treats all the fluids as equals, and, using experimental measurements obtained with damped pendulums, tries to confirm that the resistance is proportional to the densities. For this he uses the same pendulum that he makes oscillate in both air and water. This apart, there is a small detail to be pointed out: the reference to tenacity in fluids: [F]or the more tenacious fluids, of equal density, will undoubtedly resist more than those that are more liquid; as cold oil more warm, warm oil more than rain water, and water more than spirit of wine. [Prop. XXXI, Gen. Sch. p. 324]

On the basis of what he says here, this property, which he designates as tenacity, is what we currently call viscosity. In the chapter ‘The motion of fluids, and the resistance made to projected bodies’, a key point of the theory of resistance, there is a frequent repetition of the word ‘particle’ referring as much to air as to water. The reason is that he supposes that the resistance to motion is produced in both fluids as the result of the impacts of the individual particles, although the mechanism and the theoretical solution will be different in air and in water. As a preamble to the treatment of air, Newton begins by presenting a model of the system which he still does not identify as a fluid, formed by separate particles and with some type of force among them. This model helps him to theorise the effect that such a system would have on a body moving in it. Almost immediately he goes on to suppose that the internal forces between two particles will be inversely proportional to the distances between them, and a little further on he identifies this fluid as air. He had formerly assumed air to be a fluid of this type, as its experimental behavior indicates that its density is proportional to its compression [Prop. XXII, Sch.], and besides: If a fluid be composed of particles fleeing from each other, and the density be as the compression, the centrifugal forces [or forces of repulsion] of the particles will be inversely proportional to the distances of their centres. [Prop. XXIII].

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The identification is clear when, talking about these systems, he says ‘if those systems are two elastic fluids, like our air’ [Prop. XXXIII, Cor. I], i.e., there is an identification of elasticity as a distinctive property of air. However, in explaining the resistance, he transforms it further, converting it into what he calls a ‘rare medium’, which he supposes ‘consisting of equal particles freely disposed at equal distances from each other’ [Prop. XXXIV]. This ‘rare medium’ is the elastic fluid from which he has removed the repelling forces existing between the particles, and which he considers to be the equivalent of air with respect to the impact of a body at very high velocity. In brief, Newton has a clear model of air as a compound of particles repelled by forces inversely proportional to their distances. But, as the analysis of air on a moving body is very complex, he demonstrates that the effect of these repelling forces is negligible if the velocity of the body is high. On this model, therefore, the air is treated as equivalent to a rare medium. We note that the term ‘air’ disappears and is replaced by ‘medium’. This model of air is not valid for liquids. Nevertheless, in the first edition Newton studied the case where, due to very high compression, the particles came as close together as possible, a situation in which ‘they slide against each other as easily as if they were extremely lubricated, and if they collide they will rebound each other by the power of the aforementioned forces, just as if they were elastic’ [1st Ed. Prop. XXXIV]. It follows from this that everything continues to happen the same way as in a rare medium. But he suppressed this proposition in the second edition. Newton does not expound clearly in this section what is the constitution of the liquids. However, there are indirect references. In the Scholium to Proposition 35, which acts as a transition between the studies of aeriform fluids and liquids, and in which he gives us a suggestion as to what happens to a body moving in the latter, he shows that: But in continued mediums, as water, hot oil, and quicksilver, the globe as it passes through them does not immediately strike against all the particles of the fluid that generate the resistance made to it, but pressed only the particles that lie next to it, which press the particles beyond, which press other particles, and so on. [Prop. XXXV, Sch.]

This description coincides with the constitution that he presented in the Principia and this is reproduced here as Fig. 2-1. When treating liquids, Newton includes other properties and characteristics like friction or tenacity. Of these, when commenting on the resistance of a body in a canal he says ‘Some difference may arise from a greater or less friction; but

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Fig. 2-1. Liquid

in these Lemmas we suppose the bodies to be perfectly smooth, and the medium to be void of all tenacity and friction’; [Prop. XXXVII, Sch.], and concerning the same theme, talking about the different types of resistance, he adds: And though air, water, quicksilver, and the like fluids, by the division of their parts in infinitum, should be subtilized, and become mediums infinitely fluid, nevertheless, the resistance they would make to projected globes would be the same. For the resistance considered in the preceding Propositions arises from the inactivity [inertia] of the matter; and the inactivity [inertia] of the matter is essential to bodies, and always proportional to the quantity of matter. By the division of the parts of the fluid the resistance arising from the tenacity and friction of the parts may be indeed diminished; but the quantity of matter will not be at all diminished by this division; and if the quantity of matter be the same, its force of inactivity [inertia] will be the same. [Prop. XL, Sch. p. 366]

In this paragraph he seems to indicate that the degree of fluidity is related to the size of the corpuscles forming the fluid, and what is more, he places air and liquids in the same category.9 And then as a final quote, in the final Scholium of Section III, added to the third edition, and for the same reason as before, he says ‘the resistance of spherical bodies in fluids arises partly from the tenacity, partly from the attrition [friction], and partly from the density of the medium’ [Prop. XIV, Sch.].

9

A few lines further on, when he speaks of the interplanetary spaces, says that this ‘must be utterly void of any corporeal fluid, excepting, perhaps, some extremely rare vapours and the rays of light’. In the Opticks he insists ‘it is necessary to empty the heavens of all matter, except perhaps some very thin vapours, steams or effluvia, arising from the atmospheres of the Earth, planets and comets, and from such an exceedingly rare aethereal medium as we described above’, Ed. 1717, Book III, Part I, Query 28, p. 343.

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As a complement to these statements, consider the views that Newton states in the Opticks. Here he says ‘For the resisting power of fluid mediums arises partly from the attrition of the parts of the medium, and partly from the vis inertiæ of the matter’.10 He continues to affirm that in a spherical body, the part due to friction is proportional to the factor of diameter and the velocity, while that due to inertia is proportional to the square of this factor. Thus there are two classes of resistance: And by this difference the two sorts of resistance may be distinguished from one another in any medium; and these being distinguished, it will be found that almost all the resistance of bodies of a competent magnitude moving in air, water, quicksilver, and such like fluids with a competent velocity, arise from the vis inertiæ of the parts of the fluids.11

Of the two components the one due to friction or rubbing can be reduced by division, or according to his words, making them smoother and more slippery. By contrast, the other one, proportional to the density, cannot be reduced, as it is due to the inertia of matter. In the chapter ‘The motion propagated through fluids’ [Sec. VIII] he presents the drawing shown in Fig. 2-1, and he uses it to explain the propagation of the perturbations through the fluid. Although he does not mention the type of fluid, it seems he is referring to a liquid. However, a few pages further on, in the explanations on the propagation of an ‘elastic medium’ (medium elasticum) he suggests that we ‘Conceive the pulses to be propagated by successive condensations and rarefactions of the medium’ [Prop. XLII, Case 2]. Note that the term ‘medium’ is now employed much more extensively than ‘particle’. When he deals with ‘The circular motion of fluids’ [Sec. IX], he supposes that this type of motion is generated by means of the rotation of a cylinder or sphere, which drag the fluid to an irrotational vortex (Fig. 2-2). To produce this vortex each of the concentric cylinders makes an ‘impression’ upon the contiguous one. Concerning this type of motion, he establishes that ‘The resistance arising from the want of lubricity in the parts of a fluid, is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another’ [Sec. IX, Hypot.].12 Further on he affirms: ‘I suppose the fluid to consist of matter of uniform density and fluidity’ [Prop. LII, Sch. p. 392], and 10

Ibid. p. 339. Ibid. p. 340. 12 This law received the name of Newton’s and it is the definition of viscosity. Specifically, if the former is designated as µ, this one will be µ = F/(dv/ds). Even more, it is used to classified the fluid in Newtonian or non-Newtonian according to its behavior. 11

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he continues that ‘if the parts of the fluid are in any place denser or larger than in the others, the fluidity will be less at that place’ [ibid. p. 393], and in these conditions ‘I suppose the defect of the fluidity to be supplied by the smoothness or the softness of the parts or some other condition’ [ibid.] We see that there are three different, but related things: the ‘fluidity’; the ‘parts of the fluid’; and the ‘lubricity’.

Fig. 2-2. Vortex

In general terms, what Newton wanted to make clear is the existence of what we nowadays call viscosity, and which can be defined grosso modo as a resistance to flow.13 Thus, faced with a lack of a precise knowledge of the viscous phenomena; fluidity, tenacity and lubricity would be ways of referring to the viscosity. Certainly the viscosity phenomena are complex, and it is understandable that Newton addressed them in not very exact terms, following resemblances with other classes of phenomena. Finally, note that what appeared to be a very clear separation between the behavior of the liquids and the air in the generation of resistance [Sec. VII] ends by breaking down, as finally Newton performs experiments in which balloons are dropped in liquids and in the air, and he applies the same formulae to both. We shall return to this point. Motion of pendulums In the evolution of Newton’s thought, his first preoccupation was the laws governing the dependence of resistance with the velocity and density of the medium. In order to untangle these relations he used an oscillating pendulum in resisting media as an instrumental apparatus. He devoted considerable space to these 13 More exactly, the viscosity is measured as the shear force opposing the motion of two parallel layers, and the definition follows what Newton said in Section IX. Cf. previous note.

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aspects, and we begin by analysing these themes. As is evident from the Principia, his idea was that if a pendulum moves in a fluid, whether air or liquid, this medium will induce a resistance in the pendulum mass, and as a consequence, it will dampen the amplitude of the oscillations. The law of resistances can be inferred from the measurement of the dumping decrement. In line with this idea, he developed an entire mathematical apparatus with which he tried to determine both the mathematical form of this law and the specific values of the resistance at any given point.

mg Fig. 2-3. Simple pendulum

The pendulum theory was already known before Newton. This applied as much to the simple or circular pendulum as to the cycloidal one. In Book I [Prop. L-LII], he devotes some attention to them, referring to Christopher Wren and Christian Huygens as students of the same, and to the latter in particular as discoverer of the isochronal nature of the cycloidal pendulum.14 In Fig. 2-3 a simple pendulum is shown with the forces acting on its mass. The equation governing its motion is:

lϕ + g sin ϕ = 0

[2.1]

14 In a circular pendulum the oscillating mass swings a circumference arc, while in the cycloidal one it swings a cycloid. The main difference between both is that the period of the latter is independent of the in initial amplitude which does not occur in the former. However, for small oscillations the behavior of is similar for both (for 30° of initial amplitude the difference is 1.5%). This small error, together with the difficulty of constructing a cycloidal pendulum, limits this to theoretical constructions, therefore the circular ones are always used in practice, correcting their results.

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Resolving this differential equation the duration of an oscillation or period is obtained resulting in: l

T = 4K

[2.2]

g

Where K is the complete elliptic integral of the first kind 15 with sinϕ0 /2 as parameter, ϕ 0 being the initial amplitude, which means that the period depends on the amplitude. However, for small values of ϕ0 the value of K tends to π/2, so that the period will become: T = 4π

l

[2.3]

g

which is the commonly used formula. Newton bases his theoretical calculations on cycloidal pendulums, as these are pure oscillators, and the period is constant and independent of the initial amplitude. Nevertheless, when performing the experiments, he used circular pendulums, as it was difficult to construct cycloidal ones. Henceforth we shall suppose that the pendulums behave as pure oscillators, taking the arcs described as variable. Given that these are related to the angle by s = lϕ, the resulting equation will be:  s+

g

s=0

[2.4]

l

This formula is moreover in consonance with the methods and proceedings of Newton, who speaks more of described arcs than angles. In the case where the pendulum oscillates in a fluid, the previous formula has to be modified, as the resistant force, whose sense is contrary to the velocity, should be added to the recovery force. In order to resolve the problem Newton used a geometric construction (Fig. 2-4), in which the abscissa represents the amplitude of the oscillations. The ith starts in ai goes up to bi and returns to ai + 1, which will be a little less than ai. The relation of the decrement, ∆ai = ai + 1 – ai , together with ai are the keys to the form of resistance. Newton states:

15

The definition of the complete elliptic integral of the first kind is: π/2

K (ψ ) =

∫ 0

dθ 1 − sin 2 ψ sin 2 θ

For values of ψ close to zero this integral tends to π/2. The fact that the period depends of the initial amplitude is due to the recovery force is proportional to the sine of the shift angle, and not to the angle itself. That has as consequence the system is not a harmonically pure oscillator. For small angles the system behaves in this way, for in this cases is verified that sinϕ → ϕ.

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Y K bi

O

D

a i+1 a i

S

Fig. 2-4. Decrements and amplitudes [A]nd at each of its points D the perpendiculars DK be erected, which shall be to the length of the pendulum as the resistance of the body in corresponding points of the arcs is to force of gravity. [Prop. XXX]

That is, R(s) represents the resistance in point s, the height DK that he cites will be: DK = y ( s ) =

lR ( s )

[2.5]

g

His reasoning continues: I say, that the difference between the arc described in the whole descent and the arc described in the whole subsequent ascent multiplied by half the sum of the same arcs will be equal to the area Bka which all those perpendiculars take up. [Ibid.]

As the reduction of amplitudes is ∆ai, the area enclosed Σ = ∫yds, this theorem can be expressed as: l

∫ yds = g ∫ Rds =

ai + ai +1 2

∆ai

[2.6]

In this equation the term ∫Rds represents the energy dissipated in a cycle,16 therefore the physical significance of the previous mathematical expression indicates 16

Another way to reach the same conclusion would be to start with the energy of the system. We know that in a simple harmonic oscillator of the type analysed, d²s/dt ² + ks = 0, with an initial amplitude of s0, its total energy is E = ½ks0². Therefore, in the case in point, it will be: Ei =

l g 2 ai 2 l

This equation reflects the relation between initial amplitude and the total energy. If a small variation ∆E were produced in a cycle, its effects on the total energy will be:

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the existence of proportionality between the energetic dissipation and the decrement of the amplitude per cycle. Another theorem derived from this states: If the resistance … be augmented or diminished in a given ratio, the difference between the arcs described in the descent and the arc described in the subsequent ascent will be augmented or diminished in the same ratio. [Prop. XXXI]

The demonstration is an immediate consequence of [2.6]. The consequences are important, as the following corollaries make clear, which we resume: Hence if the resistance be as the velocity, the difference of the arcs in the same medium will be as the whole arc described; and conversely. If the resistance varies as the square of the velocity, that difference will vary as the square of the whole arc; and conversely. And generally, if the resistance varies as the third or any other power of the velocity, the difference will vary as the same power of the whole arc; and conversely. [Ibid. Cor. I–III]

That is, if the resistances follows the n-power law of the velocity, according to the standard R = k1vn, the decrements will do so as ∆a = k2an,17 conserving the exponent n. Therefore, the decrements can be deduced primarily from the measurements of the successive amplitudes, and from these the value of n can be inferred that will define the law of resistances. This is the basic theory of pendulums as an experimental apparatus, and with its help Newton performed an important experimental task that he included in the General Scholium of this Section.18 ∆E i =

g a i ∆a i l

In the case analyzed, the lost ∆E is precisely the dissipated energy ∫Dds, which substituted in the former, leads to the same expression given by Newton, with the exception that he takes the average of two successive amplitudes. Well now, both ways are but approximations, and are even more valid when D → 0, which in turn implies ai → ai + 1. 17 A mathematical justification of the statement is given below. From the solution of the oscillator, the amplitude and velocity are: s = a 0 cos ωt ; s = −a 0ω sin ωt = −ω a 02 − s 2 Therefore, the energy dissipated per cycle will be: a0

∫

∫

∫

1

∫

∆E = 4 Dds = 4k1 s n ds = 4k1ω n (a 02 − s 2 ) 2 ds = 4k1ω n a 0n +1 (1 −t 2 ) 2 dt 0

n

n

0

Which equal to ∆E = ω²a0∆a0, leads to the expression: ∆a 0 = − k1ω n − 2 Φ (n)a 0n Where Φ(n) is the value of the integral. The last formula clearly illustrates this dependence. 18 In the first edition this Scholium was at the end of the Section 7.

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The experimental apparatus consisted of a simple pendulum whose mass oscillated in a vessel containing the fluid, with a ruler for measuring the amplitude of the oscillations. The test procedure consisted in drawing back the mass a certain arc from the pendulum leaving it to oscillate freely. The average damping per oscillation was obtained from the initial amplitude and the number of oscillations required to reduce this amplitude to a determined fraction. He repeated this process for several initial amplitudes, thus obtaining a set of pairs {∆a, a} that, in accordance with the theoretical predictions, should follow a law of the type ∆a = kan .19 The constructive descriptions he offers of the apparatus, and circumstances of the experiments are fairly complete, although there are a few loose ends to be cleared up, such as those concerning the nature of the suspending threads or the surface finish of the spheres. In one of the justifications he offers on the discrepancies between the results and the predictions, he alleges that ‘But the greatest of the globes I used in these experiments was not perfectly spherical’ [Sec. VI, Sch. Gen. p. 321]. This throws some doubts and uncertainties on the materials used, although possibly their effect was not very meaningful in the final results. In total, he describes various series of experiments with six different pendulums oscillating in air, water and mercury.20 By way of example, and especially with a view to illustrating Newton’s experimental frame of mind, we shall describe the first of these in some detail, while we illustrate only the consequences of the remaining ones. This first experiment was one oscillating in air. The mass was a wooden sphere of 57 7/22 ounces (1,635 g) and 67/8 in. (175 mm) diameter,21 suspended from a thread of 10 ½ ft (3.3 m).22 In order to measure the oscillation he used a graduated ruler fixed to the wall at 10 1/12 ft (3.07 m) from the center of the oscillation.23 The test procedure consisted in counting the number of oscillations made until the initial amplitude was reduced by 1/8. The average reduction would be ∆a = (a0/8)/n1/8, a0 being the initial amplitude, and n1/8 the number of oscillations. In order to interpret the results better, in Fig. 2-5 the reductions are represented as a function of initial amplitudes for all the tests presented in the Principia, those marked as Exp. 1 corresponding to the ones that interest us now. 19 The easy way to extract the value of n from this set of pairs is using a logarithmical representation, as log(kan) = logk + nloga. As for each series the value of logk is constant, n will be the slope of the regression line. 20 There is a seventh case where he tries to find the resistance concerning a hypothetical ether, see later note 25. 21 According to the data the density of the wood used will be 0.58 g/cm3. 22 The period of this pendulum in a vacuum will be 3.59 s. 23 He does not say if the rule is straight or curved, we assume that it was curved because he speaks later of ‘arcs’. Further more, it is not possible to shift 64 in. laterally with the dimensions he gives.

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If the correlation was perfect, the points ought to be on a straight line whose slope is n = 2, parallel to the line drawn as continuous. We can see that is approximates this condition sufficiently for high amplitudes, but goes away in the small ones.24 300

Exp.4w

100

n=2

Exp.5w

Decrement (mm/osc)

Exp.1 Exp.3

10

1

Exp.2

Exp.4a

0,1

0,01 1

10

100

1000

3000

Amplitudes (mm)

Fig. 2-5. Experimental results with pendulums

After the first experiment, he substitutes the wooden sphere for one of lead, measuring 2 in. (50.8 mm) in diameter and having 24.5 ounces (744 g) of mass. He repeats the measurements, whose results correspond to those marked Exp. 2 in the previous graph, and which deviate even more from the n = 2. In connection with the poor correlation with the law of dependence for lower velocities, he says that: The resistance of the globe, when it moves very swiftly, varies as the squared of the velocity, nearly; and when it moves slowly, in a somewhat greater ratio. [Ibid. p. 316]

This lack of agreement leads him to try to make and adjustment with a polynomial of the type Av + Bv3/2 + Cv², for which he determines the numerical values of these three coefficients, after long and laborious calculations. As a third experiment in air he augments the diameter of the sphere until 18 ¾ in. (476 mm), with a thread length between centers of 122½ (3.11 m). By 2

The reason of this it is, according to his ideas, the parameter ∆a/a0 must be constant and in a logarithmic diagram, as used in Fig. 2-5, the slope will be equal to exponent, that is 2. 24

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comparing the results of this experiment and the previous one, he deduces the proportionality of the resistance with the squares of the diameters. He only presents one measurement, expressed as Exp. 3. He follows with some fairly prolific calculations tending to compare this last measurement with the results given by the first of the pendulums, concluding that they are consistent, so he adds: Therefore those parts of the resistances which are, when the globes are equal, as the squared of the velocities, are also, when the velocities are equal, as the squared of the diameters of the globes. [Ibid. p. 321]

After this he goes on to compare oscillation in air and in water. The mass is a lead sphere of 3 5/8 in. (92 mm) and 166 1/6 ounces (4.71 kg) with a thread of 134 3/8 in. (3.41 mm) between centers, that moves in a box of 4 × 1 × 1 ft (1.23 × 30 × 30 m) full of water. He thinks there must be a proportion between the resistances and the densities of air and water, and therefore carries out a series of measurements in the air and another in the aforementioned tank filled with water. His sums do not work out, for although the currently accepted ratio of the densities of water and air is 860:1, the one he obtains is 571:1. He goes back to make some fairly sui generis corrections, blames the discrepancies on the fact that the tank is small, and modifies the pendulum, giving it two spheres, one that submerges and another that does not, although without much success. The results are collected as Exp. 5. On the other hand, he finds that it does not follow the law of squares in the water either. We have represented the results as Exp. 4a and in Fig. 2-5, corresponding to air and water, respectively. He makes the sixth pendulum, made of iron, oscillate in mercury, and here he does find a good correlation with the comparative densities of mercury and water, although he does not present the results. He ends by stating that ‘it is clear enough that the resistance of bodies moving swiftly is very nearly proportional to the density of the fluids they move’ [ibid. p. 721]. Two points are worth noting about these experiments. First, they are not conclusive, either as regards the proportionality with the square of the velocity, or with the densities or with the square of the diameters. His attempt to seek confirmation leads him to a number of ad hoc estimations in reducing the data.25 25

He carried out another set of experiments because ‘Lastly, since it is the opinion of some that there is a certain ethereal medium extremely rare and subtle, which freely pervades the pores of all bodies; and from such a medium, so pervading the pores of the bodies, some resistance must needs arise; in order to try whether the resistance, which we experience in bodies in motion, be made upon their outward surfaces only, or whether the internal parts meet with a considerable resistance upon their surfaces, I thought of the following experiment, …’ (Prop. XXXI, Gen. Sch. p. 325). The pendulum he described consists in round pinewood box by a cord of 11 ft (3.38 m) which he separated initially by 6 ft (1.82 m), marking very exactly the returning point in each ofr the three

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Second, the experimental apparatus is not as simple as it looks at first glance. Apart from the technical difficulties of construction for the sphere, the cord, or the tank, the theory of the instrument is complex. Strictly speaking, additional factors should be taken into account, such as the motion induced in the fluid by the sphere, which is variable with respect to the number of oscillations; the existence of two coupled motions, as the point of union of the cord with the sphere is not the geometrical center of the latter, and the energy dissipation in the cord, which is neither a non-deformable nor perfect cord. Newton must have been disappointed in his hopes for the apparatus, and in the first edition he concluded with an interesting paragraph which he omitted in the second edition: The same method we used to find the resistance of spherical bodies in water and mercury can be used to find the resistance of bodies with other shapes; thus the different shapes of small models of ships can be compared in order to find at low cost the ship most apt for navigation. [Ed. 1st, Sec. VII, Prop. XL, Scholium generale]26

But the experiments with ships would still have to wait quite a few years. Resistance in aeriform fluids The claim that it was Newton who established the basis of fluid mechanics is justified by his introduction of three decisive hypotheses: one on the constitution of air, the second on the way in which air and a body interact, and the last on the laws regulating this. Certainly, in light of present knowledge, the hypotheses devised by Newton are far from the reality he wanted to represent. But the important thing is the existence of a method, which that was radically different from anything up till this point. Some of the ‘ingredients’ were not new with Newton, such as the supposition of the corpuscular nature of the fluid, and the

first oscillations. Afterward he filled the box with lead and other heavier metals with a weight 78 greater than the empty box. His intention was to determine if the ether influences the heavier metals, damping more than expected of the inner mass. He counts the oscillations until the first mark is reached, these result to be 77, while according to the weight they ought to be 78. Therefore, he played with the relation 77/78, deducing that: ‘The resistance of the empty box in its internal parts will be above than 5,000 times less than the resistance on its external surface’ [p. 326]. However, Newton does not compromise himself with these measures, because he says: ‘This experiment is related by memory, the papers being lost in which I had described it’ [ibid.]. His intentions are not clear. 26 This paragraph was included in the General Scholium that was placed at the end of Section 6 in the 1st edition. In the 2nd it was moved to Prop. 31, with some corrections and suppressions. This paragraph was one of them.

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identification of the resistance with a series of impacts, but the general theory was lacking. Let us go over these three hypotheses. The basic regulating law that applies is the second law of dynamics which he called Law II, stating: ‘The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed’ [Book I, Law II]. In an update of this law, and in accordance with the context of the work we should read ‘quantity of motion’ for ‘change of motion’, ‘applied impulse’ for ‘motive force impressed’ and ‘vector direction’ for ‘right line in which… is impressed’. Thus, the definition would be translated in the following equation: G G ∆ ( mv ) = k ∫ Fdt [2.7] Or rather, the variation in the quantity of motion is proportional to the impulse applied. We would also substitute, for a suitable system of units, the ‘proportionality’ for the ‘equality’, that is k = 1. There is another way to express this law in Newton, that says ‘The motive quantity of a centripetal force is the measure of the same, proportional to the motion which it generates in a given time’ [Def. VIII], that is, F = a∆(mv)/∆v, but we prefer the first as it adapts better to the impacts.27 For the air model he supposes that this elastic fluid is formed by particles at rest, repelling each other by means of forces inversely proportional to the distances between their respective centers. If a body is launched into this fluid at a certain velocity, it will strike some of the particles, creating resistance as follows: For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the system act on each other, partly from the collision and the reflections of the particles and the greater parts. [Prop. XXXIII]

That is to say, the resistance is the sum of two terms: one due to the forces between the particles, which he calls centrifugal or centripetal, and the other produced by the impacts. The last of these terms is proportional to the square of the velocities, as in each individual impact the force is proportional to the quantity of motion: that is the velocity, and moreover the number of impacts is also 27

About the evolution of the Newton’s ideas see Force in Newton Physics by R. Westfall. In the Chapter VIII titled ‘Newtonian Dynamics’, he says (p. 471) that Newton proposes two mutually incompatible definitions of forces See also ‘The Reception of Newton’s Second Law of Motion in the Eighteenth Century’ by T. L. Hankins, who says that Newton uses one or another according to how it is deals with shock forces or continuous action. In this respect we rather think that Newton sometimes understands the force in the current sense, as it is shown in Definition VIII, and other times he considers it as the impulse, as in Law II.

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proportional to the velocity. The result is the simple application of impact theory. The sum of the interactions among the particles is more difficult to calculate. For this, if a projectile had been launched in the fluid, he says that ‘in the same fluid a projected body that moves swiftly meets a resistance that is as the square of its velocity, nearly’ [ibid. Cor. II]. This term ‘nearly’ is due to this first centrifugal term which will disappear if internal forces do not exist, as he points out: ‘and therefore in a medium, whose parts when at a distance do not act with any force on one another, the resistance is as the square of the velocity, accurately’ [ibid.]. But as he believes that in reality there are always forces between the particles of the fluid, he adds: ‘the resistance of a body moving very swiftly in an elastic fluid is almost the same as if the parts of the fluid were destitute of their centrifugal forces, and did not fly from each other’ [ibid. Cor. III]. That is to say, Newton tries to eliminate the influence of the repelling forces, which he calls ‘centrifugal’, rendering them inappreciable for fast bodies, although he provides no indication of the degree of speed. The purpose of this is to make the mathematical calculation of the forces possible, and to do this he makes each particle independent by breaking the links between them, links due to the repelling forces. By the complete elimination of these forces the fluid converts into ‘… a rare medium, consisting of equal particles freely disposed at equal distances from each other’ [Prop. XXXIV]. We can now clearly see that this ‘rare medium’ is the previous elastic fluid from which all the forces have been eliminated, and therefore will neither be elastic nor fluid. This is the reason he substitutes the designation of fluid for that of rare medium.28 In this medium, as the resistance depends only on the impacts, the size of the fluid’s corpuscles is of no importance, as the quantity of motion received will be the same. He sets this out clearly in these terms: [S]wift bodies, moving through mediums of the same density, whose particles do not fly from each other, will strike against an equal quantity of matter in equal times, whether the particles of which the medium consists be more and smaller, or fewer and greater, and therefore impress on that matter an equal quantity of motion, and in return (by the third Law of Motion) suffer an equal reaction from the same, that is, are equally resisted; … For the resistance to projectiles moving with exceedingly great celerities is not diminished by the subtilty of the medium. [Prop. XXXIII, Cor. V] 28

Comparing with current ideas, it is true that the gases are composed by individual molecules, however, they are not still but in constant agitation, therefore they collide with each other but they will not repel each other as if they had springs. Now, in the atmosphere at altitude of 100 km (generally considered to be the edge of space) the air density is so low that the molecules have to run a long distance before colliding with another molecule. Therefore, faced with a moving body the gas behaves just as Newton forecast, with individual impacts between molecules.

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Of course this is also true for elastic fluids, when the motions occur at ‘great celerities’. For the impact model, the explanation adduced by Newton for this theory is simply mechanical. The particles of fluid, already individualised as components of the rare medium, impact on the body and transmit a quantity of motion to it that will be a function of the difference between the striking and the rebounding velocities. The total resistance will be the sum of all the quantities of motion transferred per unit of time, which in turn depends on the form, size and velocity of the body, as well as the number of particles per unit volume of air. In the application of impact theory, Newton distinguishes two types of reflections, those where ‘the globe and the particles of the medium be perfectly elastic, and are endued with the utmost force of reflection’ and those where ‘particles of the medium are infinitely hard and void of any reflecting force’ [Prop. XXXV, Sch.]. In the first situation, the particle will rebound after impact, while in the second one, it will remain immobile next to the body. In the first of these two cases, that of frontal impact, a particle will pass from an initial velocity vi = v to another final vf = –v. There is a complete change of direction, thus the variation of the momentum will be m(vi – vf ) = 2mv, m being the individual mass. The other case will end with a vf = 0, and consequently the variation will be half, that is mv. Before entering into the calculation of resistance, he states the equivalence between the cases of motion of a body in a fluid at rest, and a body at rest in a moving fluid. In justification, he says that: ‘the action of a medium upon a body is the same (by Cor. V of the Laws29) whether the body moves in a quiescent medium or whether the particles of the medium impinge with the same velocity upon the quiescent body’ [Prop. XXXIV]. We take note that he refers in his analysis to the laws of motion. With the exception of a digression on bodies of minimum resistance, Newton studies the resistance undergone by the spheres and cylinders moving in an axial direction. From a logical point of view, it seems more appropriate to begin by the analyzing the cylinders, as they are equivalent to a flat plane where the impacts are uniform in the entire surface, and then to go on to the spheres where the intensity of the impacts depends on the separation of the striking point with respect to the axis. Newton, however, follows the opposite path, which is much more round about, although conceptually equivalent.

29

This corollary states: ‘The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion.’

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He structures the process into the following steps: first, he demonstrates that the resistance generated on a sphere is half that of a cylinder. Second, he calculates the momentum transferred to the fluid by the cylinder in motion. Third, given that the relation of volumes between both bodies is known, he calculates the transference of the momentum to the sphere, and in consequence the force upon it. Although this method could be changed without altering the soundness of his reasoning, we have preferred to maintain his same way of reasoning. V.cos

v

V

R 2R 2R

a)

b)

Fig. 2-6. Sphere and cylinder

The demonstration that the resistance borne by a sphere is half that borne by a cylinder30 is based on the fact that the angle of reflection of a particle in a sphere becomes smaller as its goes away from the symmetrical axis, while at the flat face of the cylinder it is constant [Prop. XXXIV] (Fig. 2-6). In both cases the mass of the fluid impacting per unit of time is the same, but this does not happen with the axial projection of the reflected velocity. Newton calculates the equivalence between both cases by means of a geometrical construction, although nowadays it would be easier to use integral calculus.31 We note that only the frontal part produces resistance, as it is here where the fluid strikes directly, while the back part produces none. The next step is to calculate the momentum transferred by the cylinder to the fluid. Concerning this Newton states: 30

The cylinder he deals with is the one circumscribed in the sphere, as it is shown in the figure, that is, with a generatrix of 2R. As the volume of the sphere is 4πR3/3 and the volume of a cylinder with a generatrix 2R is 2πR3, the relation between both is 2/3. 31 The mass of fluid is ρπR²v. In the cylinder, If it rebounds from the cylinder, this mass is sent back with velocity –v, therefore, the variation of velocity is 2v and the momentum will be 2ρπR²v². For the sphere, a ring of radius Rdsinα and width Rdα, that is a surface of πR²sinαdα, will receive a mass ρπR²sinαcosαdα. This mass will impact with a normal velocity of vcosα and rebound with an angle 2α, so that horizontal projection of the change will be 2vcos²α. By grouping the total resistance arrived at is the result of the integral 4ρπv²R²sinαcos3αdα between 0 and π/2. The result is ρπR²v², exactly half of the resistance corresponding to a cylinder.

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THE GENESIS OF FLUID MECHANICS, 1640–1780 [A]nd let us suppose that the particles of the medium, on which the globe or cylinder falls, fly back with as great a force of reflection as possible, … and since the cylinder by falling perpendicularly on the particles, and reflecting them with the utmost force, communicates to them a velocity double to its own: it follows that the cylinder in moving forward uniformly half the length of its axis, will communicate a motion to the particles which is to the whole motion of the cylinder as the density of the medium is to the density of the cylinder. [Prop. XXXV, Case 1]

The fact that the ratio of densities of the body and of the fluid comes into the reasoning, somewhat obscures the text, but this is due to the fact that Newton deals with bodies with mass, and not with geometric forms. The momentum taken by the cylinder is 2πR3ρcv, and the time it takes in traveling along what he calls the semi-axis (which given the construction is a radius) has a value v/R, then, if the density of the fluid is ρa, the quantity of motion communicated will be 2ρaπR²v². Next, and as the ultimate step, he transfers this reasoning to the sphere, which on one hand, communicates only half the momentum of the cylinder, and on the other has only two thirds of its mass; thus the time taken to make the transfer will be that taken to travel along ‘the two thirds of its diameter’. Going on to the forces he adds: And therefore the globe meets a resistance, which is to the force by which its whole motion may be either taken away or generated in the time in which it describes two-third of its diameter moving uniformly forwards, as the density of the medium is to the density of the globe. [Ibid.]

The calculations, a continuation of the above, bring us to where the force is: D = ρπ R 2 v 2

[2.8]

That is to say, the force is proportional to the density of the fluid, to the frontal surface and to the square of the velocity. Clearly, for the cylinder the force will be double this. If instead of absolute forces, we use the dimensionless resistance coefficient,32 the values CD = 4 and CD = 2 are obtained for the cylinder and sphere, respectively. All this refers to the case where the reflections of the particles are perfectly elastic, if this were not so but ‘destitute of all elastic force, and therefore of all force of reflection’ [Prop. XXXV, Cor. I], the resistance would be half, that is to say CD = 2 and CD = 1, respectively. 32

That, we remind, is defined as CD = D/(½ρv²πR²).

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Up to now we have expounded on the more outstanding points of Newton’s theory for aeriform, non-continuous or elastic fluids, which did not undergo any substantial modifications in the second edition of the Principia. But before going on to the liquids, he introduces a Scholium in the Proposition XXXV which he reworked in the second edition, where he relates both types of fluids, about which we need to make a few comments. In the Scholium he resumes what he has said about the resistant force in spherical projectiles, and repeats the already mentioned value the force should have (CD = 2). Then, he again notes that the force is half when the particles lack the force of reflection (CD = 1). Just after he adds: ‘But in mediums, as water, hot oil, and quicksilver, the globe as it passes through them does not immediately strike against all the particles of the fluid that generate the resistance made to it, but presses only the particle that lie next to it, which press the particles beyond, which press other particles, and so on; and in these mediums the resistance is diminished one other half’ [Prop. XXXV, Sch.], that is, CD = 0.5. This is a harbinger of what is to come. He thus moves on to resistance in fluids. Aquiform fluids (I) Newton considers that the behavior of a moving body in a liquid follows different patterns than in air. By contrast with air, which is formed by separate individual particles without any interference between them, he supposes the liquids to be constituted by an aggregate of particles in contact with each other. Therefore, the impact against a body does not produce rebound, ‘but presses only the particles that lie next to it, which press the particles beyond, which press other particles, and so on’ [ibid.]. The mathematical treatment of this conglomeration would oblige him to take all these collisions into account, which is practically impossible. Because of this, he has to look for an alternative model or theory that sidesteps these difficulties, but enables him to calculate the forces.33 For this Newton approaches the problem indirectly, dividing it into two other independent ones that he resolves separately. The first consists in producing a current of liquid by means of discharging a reservoir through an orifice in its base (Fig. 2-7a), obtaining the velocity of this jet in function of the water level in the tank, that is v = v(h). In the second problem, he places the body whose resistance he wishes to determine in this jet (Fig. 2-7b), and finds the force produced in the body also as a function of the water level, that is D = D(h). Having obtained both, the velocity and the resistance as a function of the water level, he eliminates this one and thus finds the resistance as a function of the 33 The same will happen to d’Alembert when he attempts to apply his general principles to the Mechanics of Fluids.

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velocity. Throughout this construction there are neither impacts nor changes in the momentum. Now, although this idea is common to all the editions of the Principia, Newton employed a different procedure in each of the first two editions, both for analyzing the discharge and for calculating the force over the object located in the jet, and consequently he obtained different results. The differences between then were conceptual, which obliges us to treat Newton’s contribution as two separate ones.

h

h

v a)

b)

Fig. 2-7. Discharge and body

As regards the generation of the discharge, in the first edition Newton supposes a cylindrical glass full of water with an orifice in its bottom closed by a plug of area A. The bottom of the glass supports the entire weight of the water and to the plug corresponds the column whose base is the plug itself. When this is withdrawn he says that the weigh it sustains will cause the water to descend, and ‘that the motion of all the effluent water will be what the weight of the water placed perpendicularly upon the outlet can generate’ [Ed. 1st, Prop XXXVII]. That is to say, he imagines that only this column generates a momentum, whose mass is ρAh, and which will act as a cylinder or piston. For the calculation of the exit velocity of the fluid from the orifice, he hypothesizes that the momentum acquired by the cylinder in free fall after any given time must be equal to the quantity of motion that the water poured in this same time would have acquired. He does not explain the whys and wherefores of such a statement. He says: Therefore, F designates the outlet area, h the height of water perpendicular to the outlet, p its weight, Ah its quantity, s the space run in free descent in vacuum in any given time t, and v the velocity acquired during the fall in this time: the acquired momentum Ahv will be equal to the momentum of all the effluent water in this time. [Ed. 1st, Prop. XXXVII]34 34 In the original is ‘AfxV’, where ‘A’ is the height of the liquid, ‘F’ the section of the exit and ‘V’ the velocity. In strict sense the density is lacked as a factor of Ahv.

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We go on to translate his reasoning to modern formulas. At the end of a time t the falling velocity of the cylinder of water will be v = gt. On the other hand, if the effluent velocity is vs, during this time a volume of water ρAvst = ρAvsv/g would have flowed out, possessing a momentum ρAvs²v/g which, equalled to the quantity the cylinder would possess (which he quotes as being Ahv), leads to vs² = gh. This is equivalent to the velocity of the fall of a heavy body from a height equivalent to he = h/2, just half that required by Torricelli’s law. ‘Therefore if the motion in that place [the outlet] is directed upwards,’ he writes ‘the effluent water will ascend to a height of ½h’ [ibid.].

h

h

S

vs h gt

a)

b)

Fig. 2-8. Discharge

We have not found any justification for his arguments, as he identifies disparate phenomena. On one hand the fall of a cylinder of water (Fig. 2-8a) behaves like a solid (according to his reasoning) while on the other hand, (Fig. 2-8b) there is a continuous discharge. It would appear that he intended to justify the results of some experiments with these peculiar arguments. However, shortly after, he must have become convinced of the error of his ways, as in his own copies of the Principia he made handwritten corrections tending towards agreement with Torricelli’s law.35 In particular Fatio, in 1690, when referring to this point writes: ‘I could scarcely free our friend Newton from this mistake, and that only after making the experiment with the help of a vessel which I took care to prepare’.36 Continuing the thread of his argument, for the second part of the problem Newton proposes an imaginary apparatus that will enable him to find the force on a sphere located in the discharge current. The said apparatus (Fig. 2-9) consists of a reservoir ABCD that discharges into a lower vessel IKLM through an outlet EF, prolonged by another tube EFGH. The second vessel will be full, and 35 36

Cf. the edition of Koyré and Cohen, who record them in the E1a and E1i. Vide Notes in p. 777-ff. Quoted by Westfall, Never at Rest, p. 495.

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will spill over its edge IM, ‘in such a way that the water descends throughout the channel [EFGH] with a uniform motion’ [Ed. 1st, Prop. XXXVIII] and whose velocity can be calculated from what has already been set out. According to this, it will be equal to a fall from a height equal to half of SR. The function of the tube EFGH is to cause the water to descend at uniform velocity, as it is confined in the tube. In the center of the outlet tube SR he placed a small sphere P held in place by a thread PR, and he states that this sphere acts as a plug, and that the quantity of liquid flowing through will be less when the sphere is in place. ‘And therefore, the contrary action of the water in the sphere is equal to the force that may be generated or eliminated by the same motion, which is the weight of a water column perpendicular to the sphere, and whose altitude is RS’ [ibid.]. We note that he is supposing the resistance to be independent of the form of the sphere, and it is only a function of the straight section and the height of the reservoir.

Fig. 2-9. Apparatus for producing a uniform discharge

At this point he already has the velocity of the fluid depending on the height of the reservoir, and likewise the resistant force as a function of the same. The last step is to express the resistance as a function of the velocity, eliminating the height. He specifies the latter with a heading similar to that used for the sphere and the air, and which seems to please him: And therefore, the resistance of the sphere which advances uniformly in any highly fluid medium, … is equal to the force, that thrusting uniformly over an sphere of the same dimension as the sphere, and with the same density as the medium, would

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be able to generate the velocity of the sphere in the time taken by the sphere to describe two third of its diameter. [Ibid.]

Translated into a resistance coefficient the result is CD = 2. In the correction made by Newton in his own handwriting,37 he changed the two thirds for four thirds, giving us CD = 1. That is to say, he annulled his argument concerning the discharge and returned Torricelli’s law. In sum, in the first edition, the resistance coefficients for a movement in air depend on the shape of the moving body. However, when the movement takes place in water, the force is independent of the shape. Numerically, for a cylinder with a flat head, the coefficients in air and water result in CD = 4 and CD = 2, while for the sphere both were CD = 2, Aquiform fluids (II) In the second edition published a quarter of a century later, Newton completely reworked these propositions, although he retained the general approach of analyzing the resistance as a discharge rather than a force. Fatio had already refuted the solution given for the discharge in the first edition, by simply proposing the experiment of curving the water outlet tube upwards, so that the water came out as a jet, thus making it plain how the jet reached the level of the water in the reservoir, contrary to what Newton had said.38 That is to say, the outlet velocity of the fluid corresponded to that indicated Torricelli’s law, and not to half this value. In the correspondence with Cotes, the editor of the second edition of the Principia, it seems that Newton had already included the ‘cataract’ as a new form of discharge in 1710, contrary to the view of Cotes, who supported his own case with the experiments of Mariotte, which Newton said he did not know.39 Although this is not expressed in an explicit manner, probably Newton knew that in the discharge of a liquid Torricelli’s Law was correct, and he tried to reconcile the contrary experimental results by introducing the phenomena of the contraction of the stream. A little further on we shall see the new version of the discharge, which has remained associated with the term ‘cataract’, as well as the new mechanism of generating force upon a body in its midst.

37

In the E1a, by Koyré and Cohen, op. cit. Westfall says that Fatio did it in the early 1690s. Op. cit. p. 708. 39 About that, see Westfall, op. cit. pp. 708–710. 38

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ice

I h0

A

H

A

B

ice

H

B

h

E

G

E

F

a)

ice

G

F

b)

Fig. 2-10. Cataract (1)

The new discharge, which continues to be imaginary, is studied by Newton in a more complex form in three steps. Firstly, he supposes an empty vertical vessel, over which a cylinder of ice of equal diameter descends uniformly. This mass liquefies instantly when it reaches the upper edge AB of the glass (Fig. 2-10a), giving as a result a uniform flow of water starting from this plane. The purpose of this assembly is to produce an inlet current with an initial velocity v0 that would be equivalent to a fall from a height IH. Upon falling into the vessel, this current would increase its velocity due to the action of gravity, and consequently would also reduce its diameter, which would be EF on arriving at the outlet section. At the same time, the velocity of the liquid in this section EF would be equal to a fall from a height IG. He says: Therefore, by Galileo’s Theorems, IG will be to IH as the square of the velocity of the water that runs out at the hole to the velocity of the water in the circle AB, that is, as the square of the ratio of the circle AB to the circle EF; those circles being inversely as the velocities of the water which in the same time and in equal quantities passes through each of them, and completely fills them both. [Prop. XXXVI]

That is to say, the squares of the velocities acquired in fall are proportional to the heights, but he does not say whether the water flowing out will follow the Torricelli’s Law or not. He gives the name ‘cataract’40 to the figure formed by the water when it descends inside the vessel, due to the effect of gravity and to

40 Although Newton does not do it, the mathematical demonstration to find out the contour of the cataract is very easy. By the continuity equation the mass flow has to be constant, as Q = πR2v0 = πr2v. As the velocity follows the law v2 = 2g(z + zo), it results r = R/√(z0/(z + z0)), where r is the radius of the stream. This equation is independent of the assumption of the Torricelli’s law.

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its being a continuous fluid. Internal forces also intervene in its formation, as he argues: We suppose, indeed, that the parts of the water cohere a little, that by their cohesion they may in falling approach to each other with motions parallel to the horizon in order to form one single cataract, and to prevent their being divided into several; but the motion parallel to the horizon arising from this cohesion does not come under our present consideration. [Ibid.]

With these words he tries to justify what we would nowadays call the laminarity of the flow. Even in spite of the fact that the generation of this cataract is forced, it is not very different from what can be seen at first glance in the exit of some jets form the orifices of taps or tubes. As his second step, he supposes the empty cavity of the glass, between the cataract and the walls, to be filled with ice (Fig. 2-10b): [S]o that the water may pass through the ice as through a funnel. Then if the water pass very near to the ice only, without touching it; or, which is the same thing, if by reason of the perfect smoothness of the surface of the ice, the water, though touching it, glides over it with the utmost freedom, and without the least resistance; the water will run through the hole EF with the same velocity as before. [Ibid. Case 1]

That is to say, for the moment the ice is something completely superfluous. ice

I

I h0

A

H

A

B

ice

H

B

h V E

G

E`'

G'

F F'

S E

T W G

d

F d

a)

b)

Fig. 2-11. Cataract (2)

For the third step he states: ‘Let now the ice in the vessel dissolve into water; but the efflux of the water will remain, as to its velocity, the same as before’ [ibid.]. Let us admit the liquefaction as an imaginary phenomenon (Fig. 2-11a). However, it is difficult to believe that everything would continue the same, as

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naturally the falling water would mix with the water resulting from the liquefaction, and this would alter the phenomena. To this he replies: It [the velocity] will not be less, because the ice now dissolved will endeavor to descend; it will not greater, because the ice, now become water, cannot descend without hindering the descent of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water. [Ibid.]

Nevertheless, he admits that not everything would continue the same, as: For now the particles of water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with oblique motions; and in tending downwards they meet in a stream whose diameter is little smaller below the hole than in the hole itself. [Ibid.]

This is to say, with the ice liquefied, a contraction of the stream will exist once it emerges at the exterior, a phenomena never before described, and whose magnitude he will determine experimentally. The same Fig. 2-11a illustrates the process. Following the thread of his explanations, after its exit through EF, the fluid stream starts to contract in such a way that at a certain distance below EE the diameter will be E′ F′ , maintaining itself constant from this point on. This cannot be possible if the discharge is vertical, as the fluid would continue to accelerate due to the action of gravity and it would contract progressively. Newton refers to the horizontal outlet, thus this explanation is an idealisation. In fact, in order to determine the value of this contraction experimentally he perforated a tank: And that the stream of running water might not be accelerated in falling, and by that acceleration become narrower, I fixed this plate not to the bottom, but to the side of the vessel, so as to make the water go out in the direction of a line parallel to the horizon. [Ibid.]

He does not mention the size of the hole, but the diameter must have been 25/40 in. (15.9 mm), and he observed that at a distance of almost half an inch (12.7 mm) the stream had contracted to a diameter of 21/40 in. (13.3 mm), a value that he says was measured with the same precision. From this he deduces that the narrowing is ‘in the ratio of 25 × 25 to 21 × 21 or very nearly 17 to 12, that is, roughly as the square root of the ratio of 2 to 1’ [ibid.].41

41

About this experiment see later Chapter 6, ‘Discharge in Newton’.

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Another observation follows these, one which we consider to be of major importance, as it constitutes the agreement between the results of the experiments and Torricelli’s Law. Now it is certain from experiments, that the quantity of water running out in a given time through a circular hole made in the bottom of a vessel is equal to the quantity, which, flowing freely with the aforesaid velocity, would run out in the same time through another circular hole, whose diameter is to the diameter of the former as 21 to 25. [Ibid.]

Newton subsequently explains the phenomena, which acts as a recapitulation. The water that exits from Ef contracts (Fig. 2-11a), so that at a distance EE’, approximately equal to EF, the diameter is 21/25 less than that of the outlet. Consequently the outlet surface will be (21/25)² = 0.7056, a value very close to 1/√2, and which will also be the ratio the velocity will maintain in G and J, respectively. Now, he argues that the velocity in G’’ will the same ‘that a heavy body can acquire very nearly in falling and describing by its fall a space equal to half the height of the water standing in the vessel [ibid. p. 736], which expressed mathematically would be vG² = 2g(zIG/2). Due to the contraction in the stream, the velocity in G’ would be vJ² = 2vG², then following the same formula the height equivalent to the fall would be zIJ ≈ zIG, ‘very nearly the velocity that a heavy body can acquire in falling and describing by its fall a space equal to the whole height of the water standing in the vessel’ [ibid.]. That is to say, the final practical result is that the Torricelli’s Law is satisfied, but the quantity of fluid flowing corresponds to an orifice whose area would be half. But Newton does not want anything outside the vessel, thus he transfers the contraction inside it, (Fig. 2-11b) so as ‘to make the solution more simple and mathematical’ [ibid.]. He now introduces a new fictitious orifice: ST in an approximated ratio 25/21 to EF, and at a distance ZG equal to the diameter of the former. He concludes that ‘the velocity of both falling streams will be in the hole EF, the same that which a body will acquire by falling freely from the whole height IG’ [ibid.]. The cataract was not only the origin of a long lasting argument with Cotes, but also gave rise to abundant criticism, which is not at all surprising. For some it was artificial, and for others unlikely, and Johann Bernoulli demonstrated its impossibility.42 Westfall says that in spite of the agreement with the experiments, ‘few would care to dignify it with the title of scientific investigation’.43 42

In the Hydraulica, Part II, Sec. LX. He finished by saying: ‘Therefore the Newtonian explanation, since it is adverse to the laws of the hydrostatics, can not stand.’ 43 Never at Rest. p. 712.

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We have seen that theoretical laws, imaginary constructs and experimental intervene in his conception of things, but all the same, although it be artificial not to say inexact, we must consider that Newton was disposed to admit an almost improbable construction rather than break with those which seemed to him more natural, like the fall of weights. ice h0

A

H

B h

E

PQ

F

ice

Fig. 2-12. Cataract with disc

Once the outlet current was defined, Newton placed the resistant object in it. But contrary to the first edition, now the object is a disc not a sphere, although later he will identify both types of bodies. He situates the small disc PQ in the flow (Fig. 2-12), and in order to maintain the same explanation as in the case of the discharge, he supposes that a pinnacle of ice forms upon it, and together with the ice formed on the sides it will drive the cataract. The rest of his reasoning is the same as before mutatis mutandi. The resistance will be the weight of this pinnacle, although the calculation he makes lacks meticulousness, and is more or less like a simple estimation. He says its volume will be greater than that of a cone having the same base and height as the vessel; but its volume will be less than the semi-spheroid44 with the same base and height. The volume of the latter is two thirds the cylinder with the same base and height, and that of the cone is one third. In conclusion, he calculates that the pinnacle will have a geometric shape between a cone and a spheroid, as a ‘consequence of this’ he takes the arithmetic average between both.45 To be precise, he writes that if the circle is very small:

44 45

He understands for spheroid a paraboloid of revolution. The arithmetic average between 1/3 and 2/3 is 1/2.

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The weight of the water which the little circle PQ sustains, when it is very small, is very nearly equal to the weight of a cylinder of water whose base is that little circle, and its altitude ½GH; for this weight is an arithmetical mean between the weights of the cone and the hemispheroid above mentioned. [Prop. XXXVI, Cor. IX]

He realises that if the disc should block the orifice, it would have to support the weight of the entire cylinder. Thus, somewhat cautiously, he advances the following formula for the weight: 1

2 GH

EF 2 EF 2 − 1 2 PQ 2

[2.9]

That at the limit, when PQ → 0, it value is ½. That is to say, a cylinder with the same base and half the height. It is quite evident that the arguments are not very sound, oscillating as they do between precise and approximate values. Nor is the reasoning very intuitive, jumping between qualitative and quantitative arguments, the whole appearing quite ad hoc. With this he completes the process, as he has both the velocity and the force supported according to the height of the glass. Changing these values to the resistance coefficient results in a value of CD = 0.5. Comparing it with the solution of the first edition, in which he places a sphere supporting the weight of a cylinder whose height was the same as the vessel, now it is a disc that supports half of this cylinder. In order to pass the disc to the cylinders and spheres he imagines bodies with these forms, but with equal cross sections, located in the center of a channel with a finite width. In this respect he announces two main themes. The first is: If a cylinder moves uniformly forwards in the direction of its length, the resistance made thereto is not at all changed by augmenting or diminishing that length; and is therefore the same with the resistance of a circle, described with the same diameter, and moving forwards with the same velocity in the direction of a right line perpendicular to its plane. [Lemma IV]

The other, If a cylinder, a sphere, and a spheroid, of equal breadths be placed successively in the middle of a cylindrical canal, so that that their axes may coincide with the axis of the canal, these bodies will equally hinder the passage of the water through the canal. [Lemma V]

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The conclusion follows immediately: all have the same resistance, that is, ‘in a ratio compounded of the square of the ratio of the velocities and the square of the ratio of the diameters, and the ratio of the density of the mediums’ [Pop. XXXVII, Cor. I]. And if this is true in a narrow channel it will also be true in an infinite one. The independence of the form is striking, and even more so when does not occur in the aeriform fluids. Perhaps because of this he subsequently ask the question about the resistance of a faired figure similar to that represented in Fig. 2-13 [ibid. Sch.]. After some weak arguments, he ends without providing a conclusive value, and makes a comparison with the cylinder saying that ‘And the resistance must be to this force in the ratio 2 to 3, at least’ [ibid.].

Fig. 2-13. Fairing body

For a sphere, he expresses its resistance with an explanation of structure similar to that given for gases, ending that the momentum would be braked by a force equivalent to that acting in the time taken to travel the 8/3 of its diameter, which takes us to the known CD = 0.5 [Prop. 38]. This value is four times less that the one he gave in the first edition of the Principia. To sum up, the resistance coefficient in motions in rare media depends on the form of the head or bow of the body, with the value CD = 4, when the body is flat and CD = 2 when it is semi-spherical. Compared to this, when the motion is in a liquid, there is no dependence upon the form of the bow, and the value is always CD = 0.5. 46 Before finishing with the theoretical aspects, let us compare what he says about the discharge in this second edition with reference to the first. In the first he said that the force causing the jet to descend was equal to the weight of the column of liquid whose base was the cross section of the orifice. Now he corrects this point, and referring to the same constructions, he deduces that this force ‘is equal to the weight of a cylindrical column of water whose base is the 46

We disagree with Dugas, who in the Chapter 9 of his Histoire de la Mécanique states that the resistance of a cylinder of air is ‘equal to the weight of a cylinder of fluid with the same base and whose height is double that from which it has to drop to acquire the velocity of the moving body’ [p. 310]. Although Newton does not express himself in this way, the resultant value will be CD = 2, instead of CD = 4. For the sphere in air Dugas gives half the correct value. For the case of water he does take the correct value, that is CD = 0.5.

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hole EF, and its altitude 2GI or 2CK’ [Prop. XXXVI, Cor. II], which is double the previous one. In his new argument he states that ‘the effluent water, in the time it becomes equal to this column, may acquire, by falling by its own weight from the height GI, a velocity equal to that with which it runs out’ [ibid.]. Torricelli’s Law exactly. There is a nuance that differentiates the reasoning in both editions: the first is static, while the second is dynamic.47 Experimental results Newton follows up his theory by providing the results of a considerable amount of experimental work. The foundation of his experiments is to a sphere of known characteristics fall in a fluid, recording the time taken to travel a certain distance, and thus infer the value of the resistance.

water s

D A W

h

Fig. 2-14. Falling vessel

He analyses the theoretical basis of the apparatus in detail, providing a set of auxiliary tables for this purpose which later serve to help him in the calculations [Prop. 40]. Although he says nothing about how he obtained these tables the basic point is simple. At a point in its fall (Fig. 2-14) the sphere will be subjected to three forces: gravity, the ascending or hydrostatic force, and the resistance. If its radius is R, its density ρe, that of the water ρw, and the velocity v, its mathematical expressions are:

47

This was noticed by Daniel Bernoulli in his Hydrodynamica, Sec. I, §.2, where he considered both as valid. One for the initiation of the motion, the other for its continuation; i.e., the transient phase of the discharge was initiated according to the first model, and the dynamic evolution of the motion lead to the stationary condition, which is the second model.

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W=

4 3

π R3 ρe g ;

A=

4 3

π R3 ρw g ;

D=

1 2

ρ w v 2π R 2CD

[2.10]

The resulting W-A-D must be equal to the product of the mass of the sphere due to the acceleration, from which the equation of motion is obtained:

ρ e s +

ρ wCD 8R

s 2 − ( ρ e − ρ w ) g = 0

[2.11]

This equation merits comment before giving its solution. When reflecting on the phenomena of the fall of the sphere, it is easy to understand that when it is released it starts to accelerate, and consequently as the velocity increases so does the resistance, in such a way that the total force of the fall will diminish. Therefore, the acceleration will also diminish, tending asymptotically to zero, a condition in which the velocity will be constant, and which receives the denomination of limit velocity. The value of this is very easy to determine from the last equation, as in the limit condition d²s/dt² = 0, from which gives us: 8 Rg ( ρ e − ρ w )

vl =

[2.12]

3CD ρ w

Returning to equation [2.11], and defining the parameter λ = 8Rρe /CD ρw , its solution is the following48:

v = vl tanh

vl t

λ

;

s = vl ln cosh

vl t

λ

[2.13]

If we eliminate the time from this two equations we are left with the relation between the velocity and space which is: v = vl 1 − e −2 s / λ

[2.14]

In this expression we can see that the parameter λ, which has the dimension of a length, is the parameter that regulates the approximation to the limit condition.49 48 To solve this equation the change ds/dt = v is made, reducing it to μdv = (vl²-v²)dt, which is solved by a single integral. To find the space s there is another simple integral. The symbols cosh and tanh are the hyperbolic cosine and tangent, respectively. 49 Precisely, for λ = s, results v = 0.9298.

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An example illustrating this variation is presented in Fig. 2-15, where the asymptotic character of the velocity is obvious. In the practical measurements the bigger λ is the slower the approximation to this condition will be, and it is even more mistaken to suppose that the velocity of the fall is constant. Newton was conscious of this, and justified that the influence was small.50

V Vl

t Fig. 2-15. Evolution of the velocity

Once the theory of the experiment has been described we pass on to the experiments themselves. In total he presented fourteen rounds of tests [Prop. XL, Sch.]. The first twelve deal with the fall of spheres in a vessel of water, while the last two are in air. In order to reduce the data, in all the tests he takes a resistance coefficient for the sphere of CD = 0.5, be it in water or air. This value 50

As a sample of the differences of the Newton’s presentation with respect to the present formulation, we transcribe part of the Proposition XL: ‘Let A be the weight of the globe in a vacuum, B its weight in the resisting medium, D the diameter of the globe, F a space that is to 4/3D as the density of the globe to the density of the medium, that is, as A to A-B, G the time in which the globe falling with the weight B without resistance describes the space F, and H the velocity which the body acquires by that fall. Then H will be the greatest velocity with which the globe can possible descend with the weight B in the resisting medium, by Cor. II, Prop. XXXVIII; and the resistance which the globe meets with, when descending with that velocity, will be equal to its weight B; and the resistance it meets with in any other velocity will be to the weight B as the square of the ratio of that velocity to the greatest velocity H, by Cor. I, Prop. XXXVIII. This is the resistance that arises from the inactivity [inertia] of matter of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated. Let the globe be let fall so that it may descend in the fluid by the weight B; and let P be the time of falling, and let that time be expressed in seconds, if the time G be given in seconds. Find the absolute number N agreeing to the logarithm 0.4342944819(2P/G), and let L be the logarithm of the number (N + 1)/N; and the velocity acquired in falling will be (N – 1)H/ (N + 1), and the height described will be 2PF/G-1.3862943611F + 4.605170186LF.’ Only a small clarification, where he says logarithm, antilogarithm must be understood.

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comes close to the measurements made in the laminar movement condition when compared with the data currently available. 51 The dimensions of the wooden vessel of the first three experiments were 25 × 25 × 290 cm, in the next four 22 × 22 × 472 cm and up to 462 cm deep for the rest. The diameter of the spheres is not cited in all the cases, although it seems that it was around 2 cm, and the spheres were made of wax with a lead core so as to obtain different densities. Newton noted the weight in water and in air, varying between 3 and 24 g, which is the equivalent of densities between 1 and 2, and he measured the time with a pendulum that beat semi-seconds.52 The test procedure consisted in letting the sphere fall and measuring the falling time. From the data of the sphere and the fluid, he calculated the space that should be travelled in this time, comparing it with the real space in the vessel. The times went from 4 s in the first of the tests to 30 in. the twelfth. In the latter the times measured oscillated between 30.5 and 32.5 s, and, from what he says, following the theory it should be 32.25 s. The results he obtains confirm his prediction, although there are cases with discrepancies that he justifies, be they due to the existence of bubbles or malformations, defective weights or other causes. Experiment 13 concerned falls of glass spheres in air, which he left drop from the dome of St. Paul’s church in London53 at a height of 220 ft (67 m). He carried out six drops. In each one a large sphere filled with air and another smaller one filled with mercury were released simultaneously. The air-filled ones had diameters between 5 and 5.2 in. (127–132 mm) and weighed between 483 and 642 grains (31–42 g), and the mercury-filled ones were 0.75–0.8 in. (19–20 mm) with weight 747–983 grains (48–64 g). These served him as reference.54 The time of descent measured for the mercury spheres were of 4 s,55 which were longer than the 3.7 s calculated by the law of falling weights. He attributes this 0.3 s difference to the release mechanism, and uses it to correct the falling times of the air ones. He found for these values between 7.7 and 8.2 s that corrected to 51 In the twentieth century the fluid dynamic resistance of a sphere was experimentally studied both in wind tunnels and hydrodynamic channels with a wealth of detail. We must note that in the motion of a sphere two different regimes are presented: the laminar and the turbulent. The parameter that regulates the existence of one or the other of these is the called Reynolds number, which is defined as Re = ρvD/μ and it indicates the ratio of the mass forces over the viscous ones. The step from one regime to another occurs at an approximated value of 5·105 for the Reynolds number and it is called transition. Now, for the laminar regime the resistance coefficient is approximately 0.5, while for the turbulent one it drops to values of the order of 0.2. All the experiments carried out by Newton were in the laminar regime. 52 This values lead to λ = 16 cm, which is very small respect to the 290 cm of the vessel depth. That means that the sphere reaches the limit velocity very soon. 53 He quotes June 1710. 54 The falling regime is laminar with a Reynolds of 68,000. 55 Newton uses seconds and third minutes in measuring time. Each second has 60 third minutes.

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8 and 8.5 s, respectively. For each of the six drops he calculated the distances which should have been travelled in the measured times according to his theory, and compared them with the 67 m height of the dome. He found that the differences varied between 1.65 and 3.27 m, which leads him to say that ‘the difference is inappreciable’ [ibid.]. Experiment 14 was not carried out by Newton but by Desaguliers,56 who dropped spheres at the same church, but at a height of 272 ft (some 84 m). The methodology is the same and the agreement also. These results lead him to assert: Our theory, therefore, exhibits rightly, within a very little, all the resistance that globes moving in air or in water meet with; which appears to be proportional to the densities of the fluids in globes of equal velocities and magnitudes. [Prop. XL, Exp. 14]

We have seen that he performed the tests on the fall of spheres in liquids and also in air, affirming the validity of his theory for both. However, the resistance coefficient for air was CD = 2 (and this is remarkable as we have already said), while he applies the value CD = 0.5, which is that corresponding to motion in liquids. This contradicts his own theory, and, as we have already warned, throws doubts and uncertainty on his conception of fluids. On the other hand Newton was lucky in that all the experiments he performed (or at least those he relates), the flow regimes of the spheres were laminar, which is to say that the resistance coefficient is in the order of 0.5. If the velocity had been greater, and if he had entered into a turbulent regime, this value would have been reduced to less than half.57 Final comments on the Principia In recapitulating briefly, the first consideration is the incoherence in the treatment of the two classes of fluids: air and liquids. In the studies on pendulums he makes no distinction between the two, although in this particular apparatus such differentiation is perhaps not important, as he was looking more for the mathematical form of the law, not its numerical value. There are qualifications, however: he tried to correlate the resistances in air and in water, and this is not viable in the case of the fall of the spheres, where he takes the same coefficient for air and water. With this, all the statements that he had made previously become rather questionable. 56 57

In June 1719. Cf. supra note 51.

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The second point is that the shape of the front or rear of the moving body is not considered either in the case of continuous fluids or in the discontinuous fluid, and his reasons are different in each case. However, he does affirm that ‘projectiles excite a motion in fluids as they pass through them, and this motion arise from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts’ [Prop. XL, Sch. p. 366]. In this connection, recall the young Newton who in the Questiones, when he spoke about violent motion said: ‘For you may observe in water that a thing moved in it does carry along with it the water behind it, as in a cone, or at least the water is moved from behind it with but a small force as you may observe by the motes in the water’.58 That is to say, the phenomena had already been observed, although the difficulty in treating it prevented it from being considered.59 As a third and last commentary we consider the different sources producing resistance. Everything said refers to the resistance that ‘arises from the inactivity [inertia] of the matter’ [ibid.], or to put it another way, it is derived from the variation in the momentum produced by bodies in the current. This is impossible to avoid, as it is essential to the nature of the bodies, and derives directly from mechanics. But Newton also detects another ‘the resistance arising from the tenacity and friction of the parts’ [ibid.], which can be diminished, either by reducing the size of the parts of the fluid, or if the bodies are extremely polished. He does not specify if the latter components are applicable to both liquids and gases, however, according to the text they are only applicable to the first. In the third edition (1724), he introduced a Scholium at the end of Section II in which he manifests that: The resistance of spherical bodies in fluids arises partly from the tenacity, partly from the attrition, and partly from the density of the medium. And that the part of the resistance which arises from the density of the fluid is, as I said, as the square of the velocity; the other part, which arises from the tenacity of the fluid, is uniform, or as the moment of the time. [Prop. XIV, Sch.]

In this remark he explains nothing about the resistance due to friction. We are tempted to suppose it to be proportional to the velocity, but the only possible justification would be what he asserts for the circular motions: ‘The resistance arising from the want of lubricity in the parts of a fluid, is, other things being 58

Cf. Questiones, p. 21 98r. We have followed the edition by J. E. McGuire and Martin Tamny, with the transcription given in p. 367. 59 It took a long time to understand the mechanism of wake formation and evolution. Theodore von Kármán was the first who established the ‘vortex streets’. By the way, he said that these vortices were portrayed in a picture of the Italian Renaissance. (Cf. Chap. 3, in Aerodynamics: Selected Topics in the Light of their Historical Development. Cornell University Press, New York, 1954).

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equal, proportional to the velocity with which the parts of the fluid are separated from one another’ [Sec. IX, Hypothesis]. Finally, we wish to stress the immense amount of work that Book II of the Principia represents, as much for the theories he puts into play as for the experimental work. Both may be arguable, but the Book attracted the interest of all the authors over at least a century. We repeat again our basic argument: with the Principia the fluids find themselves submitted to a theory. The formalisation of impact theory The formalisation of the impact theory follows the proposals made by Newton in the first edition of the Principia. We have seen how he formulated the phenomena with absolute precision for movements in air, although his mathematical approach leaned on geometrical constructions and the solutions were of an integral nature. The same happens with his formulation of the laws of dynamics. In the 1690s and the 1700s there were a number of mathematicians who transformed these approaches into a differential formulation applicable first to bodies with simple forms in order to extend them later to all bodies. Among these geometricians the brothers Bernoulli, Jakob and Johann, Nicolas Fatio de Duillier and Guillaume Antoine de l’Hôpital stand out. The formalisation to which we refer consists of the following propositions: (a) The surface of the body upon which the fluid impinges is divided into differential elements defined analytically. (b) The impinging fluid is reflected mirror-like in each one of the points of the body upon which it impinges. (c) The effects, or forces, are normal in each point to the square of the impact velocity, to the density of the medium, to the differential surface considered and are perpendicular to this surface. (d) The total force is the integral of the local forces extended to the entire surface in which the normal velocity is positive. As corollaries to the above we have: (a) The analytical expression of the elemental force as dFn = kρv²ndσ, where k is an adequate constant of proportionality. (b) There will be a shadow zone in which the normal velocity will be negative.

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vn

v

v

dF ds

Fig. 2-16. Elemental force

We note that this approach is essentially mathematical (Fig. 2-16). Newton was essentially a physicist, however, and as such, his theories also have physical content. By contrast, the figures mentioned were mathematicians, and their main interest lay in the formal resolution of the problem posed from specific theoretical bases. In this respect for them there is no difference between water and air: in both cases the mathematical process is equivalent to a local reflection of a current flow in a body. Neither do they concern themselves in principle with the absolute magnitude of the forces that appear, but they do make relative evaluations, comparing now the form of a body with the case of the plane, now the effects of air and water. What is more, when Johann Bernoulli wants to translate his results to the physical world he has to use the same Newtonian approach, and when it is his son Daniel who tries to do it, he has to propose an experiment. Another interesting fact is that, in line with what we have said, these men focused their enquiries towards specific problems: those related to naval theory, whose importance at this time is manifest, or to the solids of minimum resistance, which lend themselves to mathematical virtuosity. The approach of Jakob Bernoulli Jakob Bernoulli was the first of a long and productive family dynasty of great renown in the fields of mathematics and mechanics. Despite occasionally being overshadowed by other members of the family or by his contemporaries, his contribution to mechanics was very important, and as regards fluid dynamics his main contribution was in the mathematical analysis he devised in order to apply the effects of resistance to naval forms. Apart from this, he also considered the problems of the ‘sail’ curves, that is the curves formed by the sails when full of wind, although he did not employ the analytical method of impact theory for these, but rather he approximated them qualitatively to catenary curves. Jakob

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extended the impact theory to liquids, converting the problem of physics into a purely mathematical one, in which he used differential calculus with great mastery. His contribution arrived via the studies of naval applications. Two works are particularly significant. The first appeared in 1693, as ‘De resistencia figurarum in fluidis motarum’ (‘On the resistance of motion if the figure is in a fluid’),60 and the second, entitled ‘De Celeritate & Declinatione [Dérive] Navis’ (‘On the velocity and declination [leeway] of ships’), which it seems was not published in Jakob’s lifetime,61 but was included in his Opera in the part corresponding to the ‘Varia Posthuma’.62 The interest of both lies in the extension of the theory to curved forms, which he achieves with the help of differential calculus. In the first of them he begins with a reflection on the velocity achieved by a ship driven along by the force of the wind on a sail, for which he supposes a model ship in which both the bow and the sail are two flat rectangles of equal surface. In this ship the force of the wind upon the sail has a driving function, and must be equal to the resistance of the water on the hull. In spite of the differences, the two are both technically resistances. In this respect he states: [I]f the part of the bow surface immersed in the water is flat and equal to the sail’s subtended surface (i.e. the base of the circular segment which is described by the sail) and in addition the air and water gravity as 1 to 841,63… I deduce that the maximum ship velocity, whichever the mass, will be precisely one thirtieth of the wind velocity. [p. 563] 64

Assume the frontal surfaces of the sail and the submerged part of the hull are equal. If the velocity of the ship is v, the resistance will be such that kρwv², where ρw is the density of water. For the resistance of the sail, which is the driving force, this value would be kρa(w – v)², where ρa is the density of air and w the velocity of the wind. The difference v – w is used because the total velocity of the wind does not act on the sail, but only the relative velocity, which is the difference between the wind and the ship. Equalling the two expressions and after a short calculation he obtains:

60

In the Acta Erud., June 1693. In Opera, pp. 561–572. He died in 1705. 62 Opera (1744), Art. XIII, p. 1057. 63 The exact value at 15 C and sea level is 1:813. 64 The quotes between brackets refer to the Opera. 61

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w v

−1 =

ρw = 841 = 29 ρa

[2.15]

The number under the square root, 841, is the one he gave as the ratio of the densities, resulting v = w/30. In these arguments he employed impact theory applied to a plane surface, but, as he observes in the next line, the form of the bow of a ship is not usually plane. Although the bow surface, which is immersed in water, is not flat, but pointed or hooked as commonly occurs in order to cleave the water more easily, nevertheless it is agreed that here no other difficulty appears, except those … of defining how much more or less this or that figure moving in the fluid is resisted. [p. 563]

He therefore goes on to consider the cases of bows which are triangular, square, segments of a circle, semicircular, parabolas, hyperbolas and of any shape or form, giving in each case the multiplication factor of the bow with respect to the case of the plane. He makes no demonstration, but only expresses the results in a narrative form, and only accompanies this with a figure in which he compares the triangular case with a rectangle of equal base (Fig. 2-17). Examining this one can deduce: (a) That of the resistant force of the rectangular case (HQ) only its normal component (QI) acts on the triangular side. (b) Of this one in turn, the resistance is the component according to the axis (MN). The use of trigonometry leads to the fact that this resistance will be equal to the force (HQ) multiplied by the square of the sine of the angle in the vertex. I Q

H

M N

Fig. 2-17. Triangular bow

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For any shape whatsoever he says: ‘The resistances are regarded between them as the [ratio] of the base of the figure to the integral of the cubes of the elements of the base, divided by the squares of the elements of the curve’ [p. 568]. One can arrive to this conclusion taking Fig. 2-18 as starting point. We have taken this figure from his second work quoted, which represents a symmetrical bow, defined by an equation y(x), upon which a current impinges parallel to the axis of symmetry with a velocity v. Upon impact against a point, let it be P, the current breaks down into its normal and tangential components, represented as vn and vt. Of these two, only the normal one will produce resistance as would correspond to the case of a reflection or rebound, although in this physical model it is not explicitly mentioned. Y

A

vn v

P

vt

F

O

B

Fig. 2-18. Incident in a symmetrical bow

Besides, the magnitude of the resistance is proportional to the square of this velocity, that is kvn² and in the normal direction. Therefore, considering a differential element ds, the force upon this would be Df = kvn²ds. Now, at this point, if the angle between the normal velocity and the horizontal velocity is β, then vn = vcosβ, and dF = kv²cos²βds, which, as we have already said would be normal, and that when projecting itself over the OX axis it would give rise to the resistance dD = kv²cos3βds. As can be seen in the detail of the previous figure, for simple geometric considerations it holds that cosβ = dy/ds when y > 0, so that the previous equation will be expressed as dD = kv²dy3/ds², and the total c value would be: B

D = 2kv

2

dy 3

∫ ds O

2

[2.16]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The integral extends between the vertex and the end, thus the total resistance is twice this.65 If we were dealing with a flat base it would be ds = dy and the resistance would have the value D0 = 2akv². Therefore any type of shape would have a resistance of D = D0.Γ, where: B

Γ=

1 dy 3 a ∫O ds 2

=

1

B

( y ')3

a ∫O 1 + ( y ') 2

dx

[2.17]

Which is just what Jakob said.66 Although he did not develop it analytically at all, the equation presented here is directly inferred from what he did later in the second of his articles. In this second work, ‘De Celeritate & Declinatione [Dérive] Navis’ Jakob deals with the more general case, when the current does not follow the axis of symmetry but reaches the body with angle of incidence. In this circumstance, obviously, the resulting force will not follow the axis of symmetry but will go in another direction, which will not coincide either with the incident current. Therefore, the problem requires two elements to be calculated: the magnitude of the resistance force and its direction. He begins by analysing the behaviour of a rectangular ship; first when the wind follows the direction of the keel and the motion is symmetrical; and later when the wind strikes with an angle, so that the ship drifts to leeward. For this case he establishes what he calls the ‘Fundamental lemma’ [p. 1058], in three points: At maximum ship’s velocity, the water resistances are like the forces by which the ships are propelled [by the wind] towards those parts [of the ship] to which [the water] offer resistances. The resistances are produced by the ship’s sides [that are] opposing the waters, and [the resistances are as] the square of the ship velocity [acting] in those parts offering resistances. Indeed, the forces [depend on] the squares of the wind speeds acting on the ships, and of the reciprocal sinus of the angles between the wind and the direction in which [the ships] are resisted.

In the first paragraph he establishes the equality (in magnitude and direction) of the driving force and the resistance, in the direction of the latter, which is not 65

Also when y < 0 the sign of the cosine changes, that is, cosβ = –dy/ds, and consequently cos3β also. If this is not considered the final value is zero. 66 It is reminded that y’ = dy/dx.

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necessarily that of the keel. The first of both is that produced by the wind, which, as he says in the third point, should be proportional to the squares of the wind velocity and the sine of its direction. As regards the resistance, in the second paragraph he establishes that it depends on two factors: the square of the advance velocity and the velocity of the lateral parts of the ship according to the direction of the motion. In the mathematical development he finds that a rectangular ship runs faster when following the diagonal instead of the keel. Applying that to the ship already studied in the ‘De resistentia’ (equation 2.15), and assuming that the rectangle has a length to width ratio of 10:1, the ratio of velocities will be 401:400 [p. 1059]. After the rectangular shape, he analyses a ship of whatever shape, similar to the depicted in Fig. 2-19. v

Y vn P O

vt

A

T Fy

F Fx

X B

Fig. 2-19. Oblique incidence

The difference between the case of the oblique current and the symmetrical one is the dissymmetry of the velocities on both sides. According to Fig. 2-19, if the current attacks at an angle α,67 the normal and tangential velocities at point P, whose perpendicular forms the angle β with the axis, will be: vn = v cos( β + α ) ; vt = v sin( β + α )

[2.18]

The resistant force on a differential element will be dF = kv²cos²(β + α)ds in the direction of the perpendicular. The total force components Fx and Fy over the OX and OY axis are obtained by integration along the curve, and they are: 67 We use the modern notation, because his one is rather tricky, as corresponds to the early stages of infinitesimal calculus.

120

THE GENESIS OF FLUID MECHANICS, 1640–1780 T

Fx = ∫ dF cos β ; B

T

Fy = ∫ dF sin β

[2.19]

B

We have taken point T as the upper limit of the integral, which will be the tangent of the velocity with the body, and in which the normal velocity is zero. The zone between T and A is characterised mathematically by the fact that the normal component of the velocity is negative, that is, there is no shock. It is a ‘shadow’ zone, which he supposed produced no resistance whatsoever. The existence of this point, which may not exist, depends on the geometrical shape of the body and the value of the incidence of the current. One of the consequences of the existence of T is the ensuing complication in the calculations. Continuing with the forces, the magnitude and direction of the resulting force will be: Fy F = Fx2 + Fy2 ; [2.20] tan γ = Fx The direction γ of the resultant force is different from the incident angle α. Thus, if the reference axis is changed passing from the axis shown, called body axis, to others formed by the velocity and its perpendicular, which would be the velocity axis, a lateral force appears that is the cause of the so called drift of the ship, the determination of which was, in those days, subject to considerable controversy. The integrals [2.19] result in: Fx = kv 2 ∫ cos 2 ( β + α ) cos β ds

[2.21]

Fy = kv 2 ∫ cos 2 ( β + α ) sin β ds

[2.22]

In the case that T and A coincides, i.e., when there is symmetry, the previous results are detailed as: Fx = 2kv 2 (λn cos 2 α + 2λm sin 2 α )

[2.23]

Fy = 2kv 2 (λm sin 2 α cos α )

[2.24]

And for the direction: tan γ =

2λm sin α cos α

λn cos 2 α + λm sin α

[2.25]

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The parameters λn and λm appear in these expressions and are auxiliary integrals whose definitions are: B

λm = ∫ O

B

λn = ∫ O

dy 3 ds 2

=∫

dydx 2 ds

2

T

O

y '3 dx 1 + y '2 T

=∫ O

y ' dx 1 + y '2

[2.26]

[2.27]

Which are the expressions presented by Jakob in the aforementioned work. We underline that their formulation is very similar to present-day terminology, with perhaps the exception of the integration limits.68 Compared with Newton, one appreciates that Jakob presents an important refinement as regards calculus. However, and in line with mathematical preoccupations, he does not calculate the absolute forces but the relative ones instead. Moreover, and here there is an important difference, he only differentiates water from air by their respective densities, not because they generate resistance using different mechanisms. The treatment given by Jakob in this last work left the impact theory almost completely configured, except for the numerical values of the resistance coefficients; nevertheless, its appears that this work was unknown until the appearance of his Opera in 1744.69 68

We explain below the steps driving to these expressions. The development of the sub-integral functions are: cos 2 ( β + α ) cos β = cos 3 β cos 2 α − 2 cos 2 β sin β cos α sin α + sin 2 β cos β sin 2 α cos 2 ( β + α ) sin β = cos 2 β sin β cos α − 2 cos β sin 2 β cos α sin α + sin 3 β sin 2 α

Also, cosβ = ±dy/ds and senβ = dx/ds, corresponding the positive sign to the also positive values of y, and negative conversely. Introducing this values in the former sub-integral functions, omitting the ± in the signs, it will be: dy 2 dx dx 2 dy 2 dy 3 cos 2 α − 2 sin α cos α + sin α 3 ds 3 ds 3 ds dydx 2 dx 3 dy 2 dx cos 2 α − 2 sin α cos α + 3 sin 2 α 3 ds 3 ds ds

Now, when realizing the integrations, the terms in dy² will be cancelled by symmetry reasons, leaving only two terms in first integral and one in the second, as in accordance to the given results. 69 We have not found any reference to this one. Also, Bouguer seems to rediscover it in the Traité du navire [III.II.IV. §.I].

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The solid of minimum resistance The search for the figure of minimum resistance has a long tradition in fluid dynamics, even when treated purely as a mathematical problem. The matter is to find the external shape, determinable using a mathematical expression, for which the resistance of the body will be minimum. In general, an additional condition is required. To try to solve this problem one needs to take as starting point an expression that relates resistance to the geometric shape. By manipulating this law, plus the other conditions, one arrives at the shape fulfilling the required condition. We shall see that both Bouguer and Euler dealt with this problem in three-dimensional bodies, and in order to do this, they developed the method of variational calculus. However, we owe the first known solution to Newton, who set out the relation that the elements of the trunk of a cone would need to satisfy to possess this property [Principia, Book II, Prop. XXXIV, Sch.], and he followed this up with an extension to other curves, although his explanations are quite cryptic. But Newton expounded these problems and their solutions without demonstration of any kind. This meant that the solutions, because of their difficulty, were received with total incomprehension, and it seems that only Huygens was capable of reproducing Newton’s reasoning. Years later, around 1694, he returned to these themes, as some drafts, which appear to be clarifications for the second edition of the Principia, indicate, although later on they are not included in the edition.70 The somewhat obscure nature of these solutions gave rise to a number of works by Fatio de Duillier, the Marquis de l’Hôpital and Johann Bernoulli. In 1698 the first of these wrote the ‘Investigatio geometrica solidi rotundi, in quod minima fiat resistentia’ (‘Geometric investigation of round solids in which the resistance is minimum’), a summary of which was published in the Acta Eruditorum 1699 [p. 510]. Fatio emphasised the obscure nature of Newton’s solution, noting that ‘Dr. Newton resolved it, but he gave a construction in which the solution throws no light on the object sought’. In the same volume of the Acta, l’Hôpital presents the work ‘Facilis et expetita Methodus inveniendi Solidi Rotundi, in quod, secundem axem motun, minor fiat a reside fluido resistentia, quam in quodvis aliud ejusdem Longitudinid & Latitudinis’ (‘The easiest and most expedite method to find the solid of revolution in which the resistance to the fluid, according to the motion axis, becomes smallest, and what are the longitude and the latitude [coordinates x and y]’), where, after corroborating what Fatio stated, he expound his own method of finding the solution. 70

Cf. ‘The solid of revolution of least resistance to motion in a uniform fluid’, in The Mathematical Papers of Isaac Newton, Vol. VI (1684–1691), Chap. 2, §.1. Ed. D. T Whiteside, Cambridge University Press, Cambridge.

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Johann Bernoulli, the third quoted, presents his work in the following volume of the Acta (1700) under the title ‘De solido rotundo miniae resistentia, addensa iis quae de eadem material habentur in actis an. Super. Mens Novemb’ (‘On the revolution solid of minimum resistance. An addenda to what was stated concerning this subject in the acts of the month of November’). The work it claims to be a continuation of is an ‘excerpta ex literis’ for the volume of 1699. We shall not go into the peculiarities of each solution we repeat that they are rather more mathematical problems. However, all have a common base: to divide the contour of the body into a polygon whose number of sides tends to infinite, and what is of greater significance for the investigation that follows, they manage to determine that the contribution of each element is the form Δy3/Δs², an expression equivalent to that given by Jakob Bernoulli. The situation in 1714 We argue that the theory of impact was fully consolidated in the publication of the Essay d’une nouvelle théorie de la manœuvre des Vaissseaux (Essay of a New Theory of Ship Manoeuvering) in 1714, for it here with this that all the elements of the impact theory became established. These elements can be summarized in the following points: (a) The effects produced when a moving body is displaced in a fluid, or viceversa, are proportional to the density of the fluid, the square of the velocity and the frontal cross section of the body. This point is generally admitted. (b) These effects are also proportional to the square of the sine of the angle of incidence, and in the direction normal to the surface. This proportionality together with that of the velocity is equivalent to say that it is also thus with respect to the square of the normal velocity. Newton only admits this premise for rarefied fluids. (c) There is no coincidence concerning as regards the value of the proportionality factor or the resistance coefficient. Newton deduces a value of CD = 4 for the rarefied fluids, while for the liquids he takes CD = 1 in the first edition of the Principia and CD = 0.5 in the second edition. De la Hire takes the value of CD = 1 and Mariotte an experimental value. The mathematicians do not define themselves, but consider the effects between the air and water as relative, but not absolute values.

Two divergent but simultaneous approaches begin from this date. On the one hand there is the exploitation of the theory, that is, its application to ships and hydraulic machinery. On the other, there is a questioning of the theory, that is, the doubts as to the model’s validity. These will be discussed in Chapter 5.

Chapter 3 The Evolution of the Problem of Resistance

The evolution of impact theory can be seen to progress in three phases. The first preliminary one consists of guesswork and poorly grounded solutions. The second phase establishes and formalizes the basic theory. The third throws doubts on its verisimilitude, which initiates a drive to introduce corrections to the basic theory. This process, whose general lines of thinking are common to many other fields of science, can be represented by an hourglass shape. The first phase, with a broad but ill-defined form due to the extent of ideas and concepts, narrows down to a select few mathematical formulas. It then opens out later into a range of applications and corrections due to real-life limitations of the theories. Following this scheme, we already have passed through the first two stages analysed in the previous chapters. We now come to the third stage: that of setting out the problems, along with solutions and corrections that were proposed which gave rise to a set of theories we have named ‘mixed’ and ‘hybrid’. The former interpret the effects in the front and rear parts of bodies using different assumptions; the latter calculate local effects using one theory which is superimposed on the determination of the fluid field obtained by another one. Problems of impact theory We have seen that in impact theory, one of the points that remained unresolved was the lack of agreement on the proportionality between the forces generated on a body and the dynamic pressure of the fluid, i.e., the value of the coefficient CD. As has already been noted, the reason for this was that the theoretical formalisation was derived from the assimilation of resistance to a set of impacts. This was considered to make sense physically only for the air, but not when dealing with liquids. Given this premise, it was possible to calculate this constant (CD) in the case of air by analysing the phenomenon as Newton had done, because air was assumed to be to the sum of individual particles. However, this procedure was not feasible for liquids, as liquids did not have discrete, non-interactive particles. The conjecture of using the same coefficients in both fluids was rejected by the experimental evidence, as the data obtained in experiments with water did not coincide with the theoretical findings for air, for which hardly any experimental data existed. 125

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Two strategies appeared in response. First, there were those more preoccupied by the applications, such as Pierre Bouguer, who, without further consideration, took for CD the value given by experiment for the flat plate. Others, with a more mathematical bent, such as Daniel Bernoulli and Leonhard Euler, tried to model the phenomenon of resistance in water in order to deduce this constant from it. Nevertheless, at the end, the results of the speculations continued to be divorced from experiment. Another factor which threw doubt on the validity of impact theory was its postulation of the absence of any effect on the rear part of a moving body, as particle impacts on this area cannot exist, since they are in the fluid ‘shadow’ of the body. In reality, the simple observation of the movement of any liquid around the body demonstrated that this is not the case, as all authors recognised. To remedy this fact, variations of the theory appeared intending to deal with the rear effect, such as those of authors like Robins, Euler and Bouguer. Euler commented on and complemented Robins’ studies as well as bringing his own ideas to bear, and Bouguer studied the optimum shape for the sterns of naval ships. All these variants have in common the assumption of the existence of impacts on the front part or bow, but they admit a completely different behaviour of fluids in the rear zones. We call these theories ‘mixed’, as they combine a resistance theory for one part of the body and another for the remainder. At the same time, whether it was a consequence of all these difficulties or simply due to common sense, doubts grew as regards the verisimilitude of the impacts. So much so that Euler came to say that ‘the resistance is represented as an effect of an impact … this representation is a chimera’.1 Although he considers this to be an approximation of reality, and truer in water than in the air, he ends up admitting that what really happens is that the fluid surrounding the obstacle follows streamlines instead of impacting upon it. This is the ‘streamline’ model, more in accord with present-day ideas. This new model becomes more and more specific from the 1740s onwards, although its precursor was Daniel Bernoulli, who in his study of the effect of the jet against a plate (1727) said that there were no impacts, but rather that the fluid shifted laterally. The difficulty of the new model lies in the treatment of streamlines, which favours the appearance of ‘hybrid’ theories, which assume that the fluid surrounds the body, although the local effects continue to be estimated according to the postulates of impact theory. That is to say, the prognostics obtained by one theory are superimposed upon the other. Finally, apart from impact theories, be they pure, mixed, or hybrid, there was another somewhat singular one: that of Jorge Juan y Santacilia. He assumed 1

‘Recherches plus exactes sur l’effect des moulins à vent’, §.III. Mém. Acad. Berlin, 1756.

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127

that gravity is responsible for resistance, thus local force will be a function not only of velocity and density, but also of the depth of the point in question. In his theory, Juan deals with the effects of the stern and unifies the behaviour of fluids in static and dynamic conditions. However, local effects continue to form the basis of the formalisation of impact theory. To sum up, we distinguish several strands of impact theories: • Pure impact theory formalised as dealing exclusively with impacts, the overall effect over the entire body being the sum of the local effects. In their turn, local effects depend solely upon the geometric conditions at the point where they are generated. • Hybrid theories, although they assume that the fluid surrounds the body, adopt the same basic assumptions as the impact theories in order to evaluate local effects. • Mixed theories adopt the postulates of frontal impacts and introduce other hypotheses for the rear areas. • Gravity theory, although based on different assumptions, continues to consider local effects in the same way as the pure impact theory. As we have said, the resistance problem was sustained by impact theory. However, in the other line of research, which we have called the discharge problem, a set of equations was developed that represented the fluid field around any body. Thus the resistance value had to be obtained from the solution of such equations and the application of the results of that calculation, thereby solving the problem. But it was not so as, according to the findings of d’Alembert, this approach concluded that the total resistance over a body was zero, a fact which contradicted experience. That, using Truesdell’s words, led to ‘misunderstandings over the next 150 years’.2 The theories provided no help, and experimentation was in the early stages. Nevertheless, the need for modelling phenomena for use in applications was evident, and in consequence this made the impact theory inevitable. Thus we find ourselves in a somewhat ironic situation, which is by no means unique in the history of science, where the reality of the facts made it mandatory to use a theory in which no one believed. In short, impact theory continued to be the basis of the resistance problem. From here on, we shall develop these ideas in greater detail in an approximately chronological sequence. There is, however, a methodological difficulty. Bouguer and Euler completed their two naval treatises around 1740, but these were not published until 1746 and 1749, respectively. In the intervening years 2

Cf. Essays in the History of Mechanics. Chap. II, p. 122.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

the work of Robins appeared, and also the Euler’s commentaries on the same, which make clear how Euler’s ideas had evolved. Thus if a strictly chronological presentation is followed, Euler’s opinions appear outmoded. In order to avoid this we shall follow the sequence: Bouguer; Euler; Robins. We start with Johann Bernoulli, who applied pure impact theory to a body moving in the air. Next we look at his son Daniel, who studied liquids, and tried to justify resistance using an imaginary construct, although the theoretical result contradicted the experimental one. This is the same Daniel who a little later produced a definitive analysis of the phenomenon of the impact of a jet against a plate, thus giving rise to the streamline model. We continue with the naval works of Bouguer and Euler, arriving at Robins, who was the author of an alternative theory to Newton’s. It was later commented on by Euler, who as a consequence introduced streamline models for the first time. The chapter is complemented by other contributions of Euler and the Jorge Juan’s theory that takes gravity as the cause of the resistance. We insist that the solution given by d’Alembert, with its application of motion equations, did not derive from impact theory, although it makes reference to resistance. This will be treated in greater depth in Chapter 6, but a short note is included here to round out the issues. Johann Bernoulli and the ‘Communication of motion’ Earlier we found Johann Bernoulli in the process of formalising impact theory, and as we have seen, he made a considerable contribution to this with his studies on the optimisation of the shape of moving solids in fluids. As befits a mathematician, his reflections do not dwell for very long on the nature of phenomena, but turn towards the form of mathematical laws and their exploitation in differential analysis, which he himself helped to forge. In accord with this strategy, he did not enter into the calculation of the absolute value of the resistance produced over a given geometric shape, but limited himself to making a relative evaluation, taking a flat plate as a comparison standard. This is the procedure he follows in his Essay d’une nouvelle théorie de la manœuvre des vaisseaux (1714), a naval work whose origin was the refutation of the ideas of Bernard Renau d’Elizagaray on the leeway problem.3 In the Essay he affirms that: the corpuscular nature of fluids:

3

Renau had published his De la théorie de la manœuvre des vaisseaux in 1689, giving a solution to the leeway (direction of a ship pushed laterally by the wind) for first time. As consequence, a long drawn-out controversy took place about the problem, Huygens, Parent and Bernoulli intervening in it. An account in the time was given by Jorge Juan in the Preface of the Examen

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129

[I]s a truth known by everyone, and is easily demonstrated. When considering a fluid as a set of small balls whose motion is uniform and parallel, it is clearly seen 4 that each one of these balls impacts upon the surface.

We would like to emphasize the ‘every one’ mentioned in this proposition, as this at the very least indicates the existence of a certain consensus at the time. In the work quoted, Johann applied the impact theory to the sails and hull of ships, and, as he had not decided upon any value for the resistance coefficient, he calculated the relative effects of water upon the hull and of wind upon the sails, indicating that with the craft in movement both should be equal. Not much later, in 1724, he presented a work titled ‘Discours sur les loix de la communication du mouvement’ (‘Discourse on the laws the communication of the motion’) for the Rouillé Prize of the Paris Academy, in which he follows a clearly Newtonian approach, and where he attempts to obtain absolute values for the resisting force. In this respect he writes: However, I believe I must state that by means of this Théorie it will be easy to determine the absolute effects of the resistance of the medium composed of given molecules of a perfect elasticity, and separated one from another by small interstices.5

It is noteworthy that this definition of fluids follows very closely that of Newton’s ‘rare medium’. What is more, regarding the loss of velocity undergone by a moving body when it advances in a fluid of this type, he adds: The quantity of this loss depends upon the shape of the body in motion, its consistency, or the density it has with respect to the density of the fluid composed by the elastic molecules.6

As an application of these ideas he analyses the motion of a body with axial symmetry, which he calls ‘conoid’ and which is represented in Fig. 3-1. He establishes that if such a body were launched with an initial velocity v0, the space s which it travels until the velocity diminished to a value v would be given by the following formula:

s=

c Γ CD

ln 0

v0 v

[3.1]

Marítimo. Larrie Ferreiro presents a very detailed relation of the controversy in Ships and Science (Cambridge: MIT Press, 2007), Chapter 2, under ‘The Debate over the Dérive’. 4 Cf. Essay d’une nouvelle théorie de la manœuvre des vaisseaux. Chap. I, §.I. 5 Cf. ‘Discours sur les loix…’, Chap. XII, §.11. 6 Cf. op. cit. Chap. XII, §.12.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Where c is defined as the length of a cylinder of air with a base equal to the body and with the same weight.7 If the conoid has a base with a radius R, its mass is m and the density of the fluid ρ, will be c = m/πR2ρ. The resistance is determined by the product ΓCD0, being CD0 being the resistance coefficient for the flat plate whose value is taken as 4 and Γ a shape factor of the conoid having the value: Γ =

1 R2

lc

∫ 0

r dr 3 ds

2

[3.2]

The symbols correspond to those shown in Fig. 3-1. As an application of these formulas he affirms that if a spherical lead ball is launched into a fluid, it will reduce its initial velocity to half when it has followed some 3,700 diameters. If, instead of the sphere, a lead cube is launched, its velocity will be reduced by half at 2,770 times its length, and if it is a straight cone launched at the point it will be at 924, and at 462 if launched at the base. If we repeat Johann’s calculations we must correct his values to 4,278 diameter for the sphere, 1,069 for the cone by its point and 534 by the base.8 These values are greater than those found by Johann, and the only cause must be a lack of precision in the relation of the densities of lead and air.

7

Johann Bernoulli does not explain the source of equation [3.1], however the way leading us to it is not complicated. The first step would be to determine the conoid resistance using the impact theory, which will be: lc

D=

rdr 3 1 1 ρ v 2 ΓC D 0 = ρ v 2 C D 0 2π 2 2 ds 3 0

∫

The second step would be the one related to braking of the conoid once it is launched with the initial velocity v0. The general equation is D = mdv/dt, which is the second law of dynamics as we know it today, and which represents the speed variation with time. Nevertheless in those years it was not yet used in this form, but with the speed variation with space, that is: D = mvdv/ds. Introducing all this terms together the following differential equation is obtained:

dv ΓCD0 = ds v c

Whose solution is the aforesaid [3.1]. 8 Applying all the former formulae it is found that Γ = 1/4 for both the sphere and sharp cone, and Γ = 1 for the flat part. With regard to c = 4ρcR/3ρa for the sphere and ρcR/3ρa for the cone. With these values for the sphere the following expression is obtained: 2 ρc s = ln 2 2R 3 ρ a Which will be four times less for the first case of the cone, and eight times in the second case. The densities used are 11,340 kg/m3 for the lead and 1.225 kg/m3 for the air.

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131

r dr R

dx 0

lc

x

Fig. 3-1. Conoid

A complement to all the above appeared in the article ‘Problema ballisticum’9 which makes use of the same equations. He tried to find the trajectory of a moving body launched vertically in a uniformly dense medium, which he conceived ‘as though it consists of equal and elastic corpuscles disseminated through the space at equal intervals. The air would be probably of this nature’.10 The basic ideas are the same as those in the ‘Discours’, which he cites, and from which he takes the formula [3.1]. In this new work he applied the same equations to an iron ball, whose density he said was 7,000 times greater than that of air, and for which he found that the velocity diminished to half when it had followed 3,235 diameters. At this point he refers to the ‘Discours’, and records that there under the same circumstances he obtained 3,700 for lead. A simple check shows that he had difficulties with the densities. For the proportion of iron to air the value of 7,000 is fairly acceptable (with present-day data it is 6,367 for standard conditions). But, according to his calculations, for the densities of iron and lead the value should be in the proportion of 3,700/3,235 = 1.144, a figure quite distant from the actual value which is approximately 1.448. Johann completed other works on resistance, such as the ‘Oscillationibus penduli. In medio quod resistit in ratione simplici velocitate’11 (‘On pendulum oscillations. In a medium which resisted in a simple ratio of velocity’), in which 9

Opera omnia, Vol. IV, Nº 182, p. 354. It has not been possible to find the date of this work, even though it is subsequent to Discours, as it is mentioned in it. 10 ‘Problema ballisticum’, §.V. 11 Cf. Opera omnia, p. 374.

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he also follows Newton, however we will refrain from commenting on it here, given its predominantly mathematical concerns. As a final comment, we would like to emphasize that Johann carried out the purest possible application of impact theory, his capacity for analysis and the rigorous application of Newtonian dynamics being especially noteworthy. This is the neck of the hourglass we spoke about earlier, which synthesises all the prior imaginative work and is, at the same time, the point of expansion and diversification of the theory. Theories and experiments of Daniel Bernoulli Daniel Bernoulli, unlike his father Johann, searched for the means to justify the application of impact theory to liquids using experimental means, and thus to validate conclusions by experimental results. In this respect he stated that he intended ‘to establish nothing that was not confirmed by experience’.12 In order to achieve this in his ‘Disertatio de actione fluidorum in copora solida et motu solidorum in fluidis’ (‘Dissertation on the action of fluids on solid bodies and of motion of solids in fluids’) he proposes and evaluates an imaginary construction for finding the forces that appear on a plate submerged in a fluid stream. However, when the time comes to look for a confirming experiment, he has to make use of the force produced by the discharge of a reservoir through an orifice, which is not comparable to the phenomena of the force on the plate. Even if one took this force as being representative, as he does, however, it does not support his theses, but rather refutes them. Daniel takes these results as sound, as opposed to the theoretical ones. Let us have a look at all this. The imaginary construction described in the first paragraph of the ‘Disertatio’ which he subtitles ‘on the pressure of flowing waters’ is reproduced in Fig. 3-2. It consisted of a plate (bb′ ) submerged in a stream of fluid and retained by a cable. A mass M hangs from it by a pulley in which the equilibrium condition balances the force produced in the plate, and whose value will be in function of its size and the speed of the fluid. Daniel supposes that ‘the fluid consists of elastic particles that jump to all sides immediately after the impulse and make room for the subsequent particles’ [Part I, §.II].13 That is to say, the phenomenon consists of a succession of non-elastic impacts. In order to calculate the equilibrium state he imagines the fluid divided in ‘very thin and solid’ layers pp, oo, etc., δ in width, which strike the plate making it recede in a 12

Cf. ‘Disertatio de actione fluidorum in corpora solida et motu solidorum in fluidis’, Part I, §.II. Comm. acad. Petrop. Vol. II, 1727 (1729). 13 In this section the quotes between brackets refer to the ‘Disertatio’.

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v

133

q

o

p

o'

p' b' q'

b

m M

h

Fig. 3-2. Bernoulli experiment

‘uniformly delayed movement’ until it stops. This is the moment in which the particles disappear from the layer, allowing the plate to return to the initial position. Due to the pulley system at each impact of the layer the mass M rises to height h and then goes back down as the fluid disappears from the layer, repeating the cycle. He interprets the phenomenon as a continuous succession of small blows causing the plate to oscillate. v v0

t

0

t

0

t0

t

Fig. 3-3. Plate oscillations

In accordance with this, the mass ascends and descends from an initial velocity v0 down to zero and it recovers again up to v 0 as it is shown in Fig. 3-3. Bear in mind that he does not say that this final velocity is equal to that of the fluid, which we call v. If we term the deceleration, which he took as a constant, a, the time taken in decelerating from v0 to zero will be t0 = v0/a, which will be

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half a complete cycle, and the space traversed by the counterweight in the semicycle h = v02/2a. Regarding this he says that: But in the same time that [the body M] ascends and descends the space h, the body can run the space 4h if it advances uniformly with its initial or final velocity, which it evidently [also] has in position M. Thus the ratio between the fluid velocity and the initial velocity of the body is as l to 4h. [Part I, §.II]14

The first statement is clear, as if the velocity v0 was maintained in the time 2t0 it would travel the space h = 2v0t0 = 2v02/a = 4h. In the second part of the quote, he establishes that the ratio of the speed of the fluid to the initial speed of the plate is: v

δ

=

[3.3]

4h

v0

To which he adds that ‘the weight of the layer … is to weight M inversely as its velocity, or directly as 4h to δ’ [ibid.]. That is: v

=

v0

δ

=

4h

M

[3.4]

me

Where it is possible to obtain the mass M. Taking into consideration that me = ρSδ is the weight of a layer, the mass will have the following value: M =

ρ Sδ2 4h

[3.5]

To eliminate the terms h and δ from the above formula he introduces the time a particle takes to pass through in the layer, which will be t = δ/v, that is, the interval corresponding to a complete cycle comprising the ascent and descent of a weight, t/2 being therefore the time taken in each phase. This t/2, ‘that is the time the weight takes to fall h’ [ibid.], allows us to obtain the value of h as the space descended by the weight due to the action of gravity, that is, h = ½g(t/2)2 = gδ2/v2/8, which when introduced in the formula leads to the following formula for the mass M: 14 That is due to the fact that the real movement is ‘uniformly retarded’ [ibid.], i.e., at a constant acceleration. A body with an initial velocity v0 and deceleration a will travel a space sp = v02/2a until it stops, in a time tp = v0/a, whereas if it does not decelerate it will travel sp = v02/a. That is double. For the case in point, there are two cycles therefore it will be four times.

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135

M g = 2 ρ S v2

[3.6]

This Mg is precisely the resistance force, which we have designated as D. Applying the definition of the resistance coefficient CD, the value CD = 4 is found for this coefficient, coinciding with the value found by Newton for the air. This coincidence is not accidental, for Newton’s model and this one are basically equivalent. Having said all this, we must nevertheless note that this impact model is somewhat unusual, and the reasoning used to explain it somewhat tortuous. Nevertheless, the phenomenon makes more sense, if jumping from the intermediate step of proportionality, we identify the resistance force as the variation of the quantity of motion during the action time of the layer. This is: D=

∆ me ∆t

=

ρ Sδ v = 2 ρ S v2 (δ / 2) /v

[3.7]

After developing these results Daniel makes note of the fact that the results agree with a fragment of Prop XXXV of the Principia,15 dealing with the resistance of spheres in a fluid [Part I, §.III]. However some clarification is required, for it does not appear to be directly applicable. The proposition of Newton quoted corresponds to the movement in as far as this is considered a rare medium, while Daniel only speaks of ‘fluids comprised of elastic particles’ without specifying anything else, and without bearing in mind the difference of behaviour between liquids and gases. On the other hand, the Newton quote refers to a ball, as opposed to the flat plate analysed by Daniel, and for which he had found a value twice as large. After the theoretical analysis, Daniel went on to the experimental part. In this respect he states ‘in order to know experimentally if the determination of pressures in the fluids was true, I considered the fluid coming out of a perforated cylinder’ [Part 1, §.IV]. The apparatus proposed (Fig. 3-4, right) is similar to that employed by Huygens and Mariotte in the impacts of a jet against a plate, and of which he says ‘the experiment had already been made by others’ [ibid.]. According to the previously found theoretical results, the resulting force upon the plate

15

The fragment quoted by him is the following: ‘And therefore the globe meets with a resistance, which is to the force by which its whole motion may be either taken away or generated in the time in which it describes two thirds of its diameter by moving uniformly forwards, as the density of the medium is to the density of the globe.’ (Principia, Prop. XXXV, Case 1). For more details cf. previous Chapter 2, ‘Resistance in aeriform fluids’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 3-4. Experiment for the discharges of liquids

ought to be equal to four times the depth of the water in the reservoir. However, the result found by Daniel was that the force upon the lever is not four times but one: ‘I saw that these weights were exactly equal to each other, when however, one ought to be four times the other, in such a way that the supposed hypothesis could not take place’ [ibid.]. Faced with this contradiction, he proposes a variation which is likewise reproduced in Fig. 3-4 (left) and in which he assumes that the outlet’s jet impacts against a conical obstacle firmly attached to the cylinder, and leaving only a slot through which the fluid can flow, and in this respect he states: Its it is obvious to anyone that am cannot have pressure in any other way than by the cylinder of water orsn which is above (disregarding the small part onma), and likewise all parts of the cylinder are pressed down by the cylinder of water above them. [Part I, §.VI]

This argument is not very clear; it is more like a justification as he goes on to add that from this ‘paragraph it follows that the fluid pressure is four times less than that determined in the second paragraph’ [Part I, §.VII]. This second paragraph is where he carries out the theoretical development. As a consequence, he adopts a resistance coefficient of CD = 1 for the flat plate. If the impact is against a curved surface, then he adds the factor of proportionality with the square of the sine, and the formulas he arrives are equal to those given by a Jakob Bernoulli, so we shall not repeat them. For the case of the sphere the value found is half that of the flat plate, that is CD = 0.5, coinciding with Newton’s results for a liquid-like fluids.

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137

Daniel’s procedure is somewhat curious, as after presenting a model from which he extracts general results, he has to reject it since it does not agree with the experiments. This is praiseworthy, but he does not analyse in what respects the model fails, which would be the appropriate thing to do. Finally, it should be noted that the fourth part of the ‘Disertatio’, subtitled ‘About the moment of bodies launched upwards, where the experiments made with cannons by Dr. Gunther come up against the calculations’, presents some experiments made with cannon balls. In the fifth part, ‘Concerning the horizontal motion of bodies in perfectly fluid and resistance media’, he comments upon his father’s ‘Discours sur les loix de la communication du mouvement’. In the sixth and last, ‘That contains general comments about resistance in fluids’ he asks about the nature of resistance, which he estimates as having two causes: one, the inertia that produces forces proportional to the square of velocity; and the other, tenacity, which is simply proportional to the velocity. In this he follows Newton, as he himself recognises. [Part VI, §.2]16 Jet against a plate: Daniel Bernoulli’s clarifications For a fairly long time the phenomenon of the effect of a jet against a plate and that of the resistance of a submerged body had been considered equivalent. This assimilation has its origin in Huygens and Mariotte, and strictly according to impact theory it would be true, as in both cases we are dealing with impacts of the particles constituting a fluid against an object. Daniel Bernoulli undermined this identification and backed his case experimentally. However, we would like to emphasize expressly that it is in his analysis that the impact model is rejected for the first time, and where it is shown that moving particles follow trajectories which curve progressively. Consequently, with the new approach, the resulting force is obtained as a variation of the momentum of the fluid stream in the process of curving. Thus we need to know the velocity field, and it is obvious that the matter is important. The identification, which could be called confusion, between the two phenomena was made throughout the first third of the eighteenth century, and even Daniel Bernoulli, as we have just seen, made use of this idea when he proposed

16

Nevertheless, Newton does not say explicitly that a part of the resistance be proportional to velocity. Daniel invokes the theoretical assumption given in the Principia Prop. XI, referring to bodies that resist ‘partly in the ratio of the velocities, and partly as the square of the same ratio’. But these are the mathematical prolegomena made by Newton, and not the theory itself, as we have warned. Cf. previous Chapter 2, ‘The concept of fluids in the Principia’.

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the experiments to confirm his resistance theory. A few years later in the Hydrodynamica, on treating the equivalence between the force on the plate and the height of the water level he was to say: The majority not to say all have defended and still defend this opinion, because it agrees admirably with other experiments, mainly with those which are customarily performed with spheres moving in a resistant medium.17

He explicitly appears to place spheres and jets in the same group. However, in the same point, after referring to his previous works in the Commentarii petropolitanæ18 and after admitting that he himself took these results as being valid, he takes this back with the following words: In truth, after having weighed the matter with more attention, and having added new principles and at the same time carrying out a new type of experiments, I saw clearly that he common opinion about the impetus of a stream of water had to be changed.19

Note that he speaks of new principles. Almost immediately he explains the new model: Moreover, I wish it to be taken into account that here I only speak here of single jets which are received in their totality by planes, but not about the fluids surrounding bodies which produce impetus in them, such as the winds or rivers. Indeed, I say that these two types of impetus, which up to now authors have confused, must be perfectly distinguished between them, for reasons that I shall proceed to express briefly further on.20

The reasons he announces are perfectly developed in the memorandum: ‘De legibus quibus mechanicis, quas natura constanteer affectat, nondum descriptis, earumque usu hydrodynamico, pro determinanda vi venæ aquea contra planum incurrentis’21 (‘About certain as yet unwritten mechanical laws, to which nature constantly tends, and about their hydrodynamic use in order determine the force of the vein of water inciding on a plane’), and to which we will refer further on. Daniel establishes the divergence between the impact experiments and those of resisting bodies with an acknowledgement of Newton: 17

Cf. Hydrodynamica, Sec. XIII, §.15. It is refers to the ‘Disertatio de actione fludorum in corpora solida et motu solidorum in fluidis’ (1727). 19 Cf. Hydrodynamica, Sec. XIII, §.15. 20 Ibid. 21 Cf. Comm. acad. petrop. Vol. VIII, 1736 (1741). 18

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139

As far as I know, Newton was the first to investigate the force of waters using means other than experience; but however he did not investigate the force of the fluid stream as it flows from a vessel, rather he investigated the resistance of the fluid surrounding the body to which it resists. [Part 2ª, §.5]22

Daniel’s criticism is very sharp, and affects in equal measure the stream contraction given by Newton and the experiments described by du Hamel. In the latter case, he refers to one of the experiments presented in the Regiæ scientarum Academiæ Historia Parisiis, and after analysing it in some detail he ends up stating: ‘therefore it is evident that with this experiment the opinion of the authors is not only not confirmed, but rather it is rejected’ [Part 2ª, §.2]. The opinion he quotes was that the force was equal to the weight of a cylinder of water whose height was equal to its depth.

a)

b)

Fig. 3-5. Jet impact against a plate

He rejects the impact theory of Newton, in which Newton obtained a quadruple or double force according to whether the existence or absence of rebound was taken into consideration, along the same lines. In this second case he says, ‘however, this hypothesis can by no means be admitted, as it supposes that all particles impact immediately and directly upon a plane; as no one can conceive how this could take place differently to the way in which the particles are annihilated immediately after their impact’ [Part 2ª, §.5]. This provides him with the reason for saying ‘the phenomenon of which I speak consists precisely in that all the particles of water slide along the plane in the direction of the self-same plane’ [Part 2ª, §.6], as is shown in a Fig. 3-5a. We note that this is the first appearance of the streamline model. 22

The quotes between brackets refer to the ‘De legibus…’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

In order to calculate the force he used the general principles of Newtonian mechanics, matching the force in one direction with the variation of the momentum in the same direction [Part 2ª, §.6–8].23 If the area where the jet impacts is S and its velocity is v, then as the velocity changes 90° in direction, the resulting force in this direction will be F = Sρv2. This is the equivalent of a column whose height is double the kinetic height.

Fig. 3-6. Oblique incidence

However, Daniel points out that the former is valid when the size of the plane is much greater than the diameter of the stream, but in the contrary case this hypothesis would not hold, as in Fig. 3-5b. Something similar happens when the incidence is not perpendicular but oblique, as is shown in Fig. 3-6. In this case he comments: ‘I say then, that this case cannot be correctly estimated, as much due to the variation of the intermediate velocity bearing upon the case, … as to the unequal variation in the direction of the particles’ [Part 2ª, §.12]. This was true, as he was not able to determine what proportion of particles goes to each side and at what velocity they would go. That is, velocity distribution is the intermediate step in finding the forces. Assuming that the jet was prismatic, and that this proportion was known, it would be possible to deduce the force, something he does by introducing the distribution of the liquid between the two resulting branches as a parameter. To confirm this hypothesis he prepared an experiment, Fig. 3-7, that he says he carried out in his home in the presence of his father, his uncle Nicholas, and Emmanuel König. It consisted basically of a vessel with a horizontal tube as an orifice (in order to avoid the stream contraction) which impacted against the plate. The idea behind the apparatus is very similar to that of the one presented by Huygens, although it was constructed more carefully. In order to measure the 23 We have spoken about this experiment previously in Chapter 1, ‘Experiment of a jet against a plate’.

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141

131 (106.4)

M 19 (15.4) 92 (74.7) h

e

Fig. 3-7. Jet against plate experiment

outlet velocity of the water, he allowed the free fall of the jet and measured the point where it strikes the ground.24 As the trajectory of the jet is equivalent to a parabolic fall, the velocity will be given by v02 = e2g/2h. As a unit of measure he employed 1/400 Paris pieds (0.812 mm). In accordance with the measurements shown, the weight of a water column of diameter 19 units (15.43 mm) and height 131 (106.4 mm) was 8.937 grosses (34.17 g). Both h and e had the value 900 (730.8), from which was found that the kinetic height of the outlet speed was 225 units (182.7 mm), which represented a real velocity of 1.892 m/s. The force he measures is 17 grosses (65 g), slightly less than double the weight of the column (17.874 grosses), which was the expected theoretical result. Daniel was very satisfied with these results, and insisted on distinguishing the two phenomena. The last words of his article are: One thing and another contribute to the fact that the force of the current against the plane cannot be determined with accuracy, and thus the resistance suffered by bodies moving in an infinite fluid cannot be defined either, unless experiments are recurred to. [Part 2ª, §.14]

The state of Daniel’s knowledge did not permit him to assume the possibility that the velocity field could be determined by a non-experimental procedure. After the works of Daniel, yet another new experimental determination was made by George Wolffgang Krafft, which appeared in the Commentarii 24 For an exit velocity v0 the distance in this plane will be x = v0t, and the vertical drop y = ½gt2. Eliminating the time between both, y = gx2/2v02 is obtained.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

petropolitanæ, entitled ‘De vi venæ aquæ contra planum incurrentis experimenta’25 (‘Experiments on the force of a vein of water impacting against a plane’) in which he describes six experiments carried out in order to confirm the theories of Daniel Bernoulli, and performed between June 1736 and January of the following year. According to the description he gives, the experimental work was carried out with great care as regards both the apparatus and the taking of measurements. The apparatus was similar to that employed by Bernoulli, as was the methodology followed. The vessel had a diameter of 15/16 English feet (286 mm) and a height of 2 ft (610 mm). The units of measurement he employed were 1/2,000 English feet (0.1524 mm) for the lengths and 1/7680 Dutch ponds (0.064 g) for the weights. He used three orifices, all with an internal diameter of 89 units (13.6 mm): one long, another short one and another flush with the outlet plane. Of the first he says that the jutting part was of 218 units (33 mm), a detail he did not give for the second orifice. The height of the water was 578 mm. We present the results in the graphic form shown in Fig. 3-8. The abscissas show the kinetic heights of the outlet speed, and the ordinates show the relation between the measured force and the predicted force according to the kinetic height. From the figure it can be appreciated they are close to unity, which means a good agreement with the theory. In this regard, Krafft adds to his results of Daniel Bernoulli’s experiment telling that ‘it is assumed by the illustrious Bernoulli, and it is described in the highly praised dissertation upon the same’.26 1

Exp/theory rate

0,95

no indication

flush

0,9

no indication short orifice

0,85

0,8 370

long orifice 380

390

400

410

420

Kinetic heigth (mm)

Fig. 3-8. Results of Krafft’s experiment 25 26

In Volume XI, 1739 (1750). Op. cit. Experimentum VI.

430

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143

In each one of the three experiments he used a different orifice. In the fourth27 he only says that he maintained the reservoir constantly full, and makes no special mention of anything for the fifth experiment. In his observations, Krafft points out that the kinetic heights of the outlet are less than those of the reservoir, when in theory they ought to be the same, and that the forces measured are also less than expected. But Krafft does not explain the reasons for these discrepancies. Certainly they are fairly small. Here we leave the jets impacting on plates. Later, d’Alembert returned to this problem in his Essai d’une nouvelle théorie de la résistance des fluides,28 in which he applied his theories and developed a new approach in order to calculate the force. The essay makes several references to Daniel Bernoulli and to Krafft’s experiments. The last reference we have found on this theme in the eighteenth century belongs to Lagrange, who analysed the question in 1784, in an article titled ‘Sur la percussion des fluides’,29 although it brought nothing new to bear on the matter. The naval work of Pierre Bouguer Towards the middle of the eighteenth century, two of the most important naval works of that century were about to appear: Euler’s Scientia navalis (Naval Science) and Bouguer’s Traité du navire (Treatise of the ship). They were published in 1749 and 1746, respectively, both with more delay than their authors would have desired, as they confess in the prologues. Both are extensive, and their main object was the application of fluid mechanics to naval architecture. Although there are parallels between them, thre are also differences, the most outstanding being that Bouguer focused on the problem more as a naval engineer, while Euler centred on it as a mathematician. The ships of Euler are defined by equations, while those of Bouguer have frames and ribs. Bouguer devoted considerable activity to naval themes although he was not a seaman. Apart from the work already quoted, he had previously obtained the Prize of the Academy of Paris for his De la mature des vaisseaux (On the Masting of Vessels), and was the author of other books and monographs on this theme. As a mathematician, he paid special attention to calculating optimum shapes for the bow of ships, for which he used impact theory. As regards the resistance problem, Bouguer’s merit lies in the applications field, as he contributes no new theoretical development to the problem, either in the work we are concerned with, or in the other writings he dedicated to naval 27

Euler was present at this one. In Chapter VII. 29 Mem. Acad. Sci. Torino, Vol. 1. Also in the Œeuvres, Vol. 2, pp. 237–249. 28

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themes. He approaches the question logically, with the calculation of the resistance force of the ship, as well as the forces acting on sails, and he assumes the resistance to be proportional to the square of the velocity, stating that ‘when the same fluid impacts on the same surface at different velocities and with different obliquities, the impulses are the product of the square of the velocities and the square of the sine of the angle of incidence’ [I.II.§.I, p. 356].30 But he does not state the proportionality of the resistance with the surface, and concerning this a little later he adds: ‘the impressions of the same fluid are noticeably similar to the products of the square of the velocity and the square of the sine of the angle of incidence, multiplied by the extension of the plane receiving the impact’ [ibid.]. We emphasize the term ‘noticeably’ (sensiblement), which indicate the assumption of conceptual inexactitudes. An innovation, and an important one, is the value he uses for the resistance of a plate: One can take for example as an experimental principle that sea water upon impacting perpendicularly on a surface of one square foot with a velocity of a foot per second makes an impression more or less equal to one pound seven ounces [of Paris]. [Ibid. p. 357]

It is very clear that Bouguer did not intend to reduce the resistance to a column of water whose height is one, double or four times the kinetics, as all the previous authors did, but rather he took a magnitude directly obtained from experiment. The coefficient resulting from the values he quotes is fairly similar to that obtained by Mariotte in the Traité du mouvement des eaux. Apart from this eminently practical proceeding, another outstanding fact in Bouguer is that he considers that the rear parts, or the sterns, of the ships as being resistance generators, thus giving rise to a mixed theory. As we have explained, in impact theory the stern does not contribute to the resistance. In fact, all investigators from Newton onwards pointed out that this is not exactly true, because at the very least water tends to occupy the vacuum left by the ship as it advances. Bouguer in the Traité points out that ‘If the end part [stern] does not contribute as much as the front to the rapidity of the advance velocity, it is however certain that it does indeed make a contribution’ [V.VII], and in consequence he analyses sterns with some care. This was the first time that this problem was broached.

30

In this section the quotes between brackets refer to the Book III of the Traité du navire.

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145

y

x

ds

vn

vo

Fig. 3-9. Stern effect

The basis of his argument is that as the ship advances and leaves a space, the surrounding water, pushed by the water above it, tends to occupy the space. Thus this movement can be facilitated by giving a specific shape to the stern, and although the impact thus produced does not contribute much to increasing the velocity, at least it eliminates part of the resistance. Figure 3-9 represents the stern which advances towards the left, and the dotted line shows the position it occupied previously. In this process, the water rushes towards the gap at a velocity that Bouguer calls ‘absolute’, and which is the function of the depth, as the greater the volume of water it has on top the greater it will be. If we consider an element ds and the stern equation is y = y(x), the component of the ship’s movement would have to be subtracted from the absolute velocity, that is, v n = v0sinθ = v0dy/ds, given v0 –vn as the impact speed, which would reduce the resistance effect. In the case of a flat stern the speed effect would be eliminated, reducing the resistance relief which would disappear if the speed of the ship is greater than the ‘absolute’ [V.VII.§.II]. Referring to the latter, he says ‘it is noticeably proportional to the square roots of the depths, as it is taught by hydrostatics’ [V.VII.§.IV], which is to say that it would be the discharge speed defined by Torricellis Law.31 This model for the stern has nothing to do with the impact model, giving rise to a mixed theory. Euler’s Scientia navalis The Scientia navalis marks the beginning of Euler’s contributions to the resistance problem, and from the content of other articles and memoranda, the ideas presented in this work were derived from around 1740. Euler, just as Daniel 31

This reasoning shows a possible connection with the Jorge Juan’s theory, who also makes use of this speed.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Bernoulli had done, looked for the justification of the resistance in motions in liquids. He did this by analyzing the problem pursuing two different methods. He based the first on the conservation of the momentum, while in the second case the live forces are the ones to be conserved. The results obtained by one or the other differ by a factor of two, and, faced with the dilemma, he chooses the second option as it adjusts better to the experimental data, but he does not justify the failure or success of his initial hypothesis in either case. In both cases, the figure he bases the analysis on is a flat rectangular shape (Fig. 3-10) whose frontal surface is S and whose total mass is M. That is to say, it is a physical body and not a geometrical shape. As it is displaced, this moving body communicates part of its motion to the water in front of it, which in turn is displaced jointly with it. The calculation procedure consists in equalising the physical magnitude to be conserved between one instant considered as initial, and another subsequent instant. A

a

M dx b

B

Fig. 3-10. Moving body in a fluid

In a certain time interval the front of the moving body will pass from position AB to ab, displacing dx. After this advance, the displaced water Abab will move together with the body, with the result that the speed will be reduced.32 In this respect Euler states: In order to define the reduction in motion, an invocation of the rules of the communication of motion would be convenient, and precisely those referring to completely soft bodies, as the experiments show to a sufficient extent, at least in this case, that water lacks any kind of elasticity. [I.§.465]33

32

If we remember Newton, we can see this is exactly the case of the impact of an elastic fluid without rebound, Book II, Prop. XXXV, Case 2. 33 The quotes between brackets refer to the Scientia navalis.

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147

In the first of these two hypotheses he equates the momentum before and after, while in the second the total live force is the magnitude to be preserved. If the advance velocity is designated as v, the initial quantity of motion will be Mv.34 After travelling a distance of dx the speed will diminish to v – dv and the new mass in movement will be M + ρSdx, ρ being the water density. Then: M v = (M + ρ S dx)(v − dv)

[3.8]

Solving and ignoring the second order terms35 we arrive at the following expression: M dv = ρ S v dx

[3.9]

The next step is to look for a force (potentia) which produces the same effect upon M. According to the second law of dynamics, this force P, will be P = Mvdv/dx,36 and substituting it in equation [3.9] we end up with: P = ρ S v2

[3.10]

Which corresponds to the weight of a column double height of the kinetic one, that is: CD = 2. But Euler seems not to want to accept this value, since the experiments suggest otherwise, and he adds: Even now it much argued among the authors who have written about water resistance, upon whether the resistance is equivalent either to a double cylinder of water whose base be equal to the surface directly exposed, as we have found here, or to a single cylinder. [I.§.472]

And after some considerations in which he remarks on the non-elasticity of water, he goes on to treat the problem using the principle of conservation of live forces. The difference is now he conserves the live force, Mv2, instead of the quantity of motion. The resultant equation is: M v 2 = (M + ρ S dx)(v - dv )

34

2

The notation has been modernised, since Euler uses kinetic heights as speeds. That is ρSdxdv. 36 This equation is equivalent to P = Mdv/dt. 35

[3.11]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

That mutatis mutandi leads him to: P =

1 2

ρ S v2

[3.12]

That is just a half of equation [3.10], and in is contradiction with the other case as himself notes. But he continues: Thus this controversy does not worry us much, and in either case the proportions we are mainly concerned about stay equal. … As the experiments made on the resistance of bodies moving in water seem to incline towards the simple cylinder, thus agreeing more with the argument derived from the conservation of the live forces. [I.§.472]

As an aid in adopting these new values, he comments on Newton’s falling sphere experiments. Concerning these he says that, ‘they appear to show with sufficient clarity that the resistance should be shown just by the simple water cylinder, whose height is equal to the simple height generated by the velocity’ [I.§.473]. This is not exactly true. Newton does not use heights to express resistances, but of forces capable of slowing the motion, and if we make the calculation the resulting height for the Newton’s experiments turns out to be a half. As it is not likely that Euler misinterpreted Newton, what possibly happened is that he took the results of Newton’s experiments, but did not accept the interpretation that Newton put on them. Let us remember that Newton assumed that the resistance of a cylinder moving axially in a liquid was independent of the shape of the front part or head.37 This was not admitted by Euler, for whom if the head is hemispherical, the resistance is halved. Then if Newton measured a coefficient of 0.5 experimentally for the sphere, it can be inferred that he would have obtained a value of 1 for a flat surface if he had carried out the appropriate experiment. We believe that Euler’s words should be interpreted in this sense, and he corroborates this when he affirms that ‘the resistance undergone by the flat surface impinging directly in the water shall be measured by the weight of the water cylinder whose base is equal to the surface and whose height is equal to the same height due to the velocity’ [I.§.473]. In any case Euler’s choice is quite surprising, as it is based solely on some doubtful experimental results. One gets the impression that he tried to save the phenomena by any means. It is also worth mentioning the supposition that water is swept along by the body, is nor realistic, because only with looking at a channel in which there is a moving body it can be seen that this is not so. 37

Cf. Principia, Book 2, Prop. 37, Lemma 5.

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Finally, when the fluid incidence is not perpendicular, he follows the other treatise writers [I.§.474], supposing the proportionality to be the square of the sine of the angle of incidence. If the shape of the body is curved, he divides it into differential elements which he later integrates [I.§.493]. Likewise, he also agrees with Jakob Bernoulli in the estimation of the integration limits in the sense of supposing that resistance is only produced in those parts which are in front of those points in which the current is tangential to the curve [I.§.495]. As a last comment concerning the Scientia navalis, we emphasise that this work is also memorable for its contributions to general mechanics. By way of example, we note that it gives the first definition of the principal axis of inertia, and provides the breakdown of oscillating motions into components according to these axis. Robins’ New Principles of Gunnery In London in 1742 Benjamin Robins’ book New Principles of Gunnery appeared. It hardly needs mentioning that a basic question of gunnery is aiming, which, in the form of predictions, is translated into the so-called firing tables, i.e., the gun elevation required for the projectile to reach the desired target. The drawing up of these tables is based on the calculation of the projectile trajectory, which is moved in turn by the forces of gravity and the aerodynamic resistance. It is in this last link in the chain where Robins makes an important contribution not only to gunnery, but also to fluid mechanics. His theoretical analyses were a criticism of the ideas of Newton, for which he substituted his own theory. Furthermore, this work was complemented by experimental measurements of the real velocity of projectiles, for which he used the ballistic pendulum, an instrument of his own invention.38 As an indication of the importance that Robins placed resistance, on the front cover of his work, after the title, there is a note that reads: ‘containing:

38 W. Johnson, Robins’ scholar, in ‘Benjamin Robins, FRS (1707–1751): New details of his life’, Notes Rec. R. Soc. Lond, 46 (2), 1992, pp. 235–252, says the pendulum was diagrammatised by the French in 1707. Without going into this point, Robins makes a very precise description of it in his book [Chap. I, Prop. VIII]. Basically, it consists in a heavy mass that can oscillate over a horizontal axis, and over which a projectile of known mass is fired. The projectile will embed itself, and the quantity of motion will be transferred to the pendulum, which will move ascending its centre of gravity according to this amount. The displacement is recorded by means of tapes that are untied and lengthened to the maximum pendulum deviation. The bullet’s velocity can be determined using these deviations, the system geometry, its mass and the projectile mass. See also ‘Musket and pendulums: Benjamin Robins, Leonhard Euler and the Ballistics Revolution’ by Brett D. Steele in Technology and Culture, Vol. 35, nº 2, 1994, although that work is focused on the ballistic revolution. A nice picture of a pendulum is depicted in the Gunnery (p. 25).

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The Determination of the Force of Gunpowder, and an Investigation of the Difference in the Resisting Power of Air to Swift and Slow Motions’. The work cited was translated into German by Euler in 1745,39 who also commented on it extensively, more as a mathematician than as a gunner, and thus produced what was practically a new text inserted into the original. To quote Truesdell: ‘the annotations which he added to his translation of Robins’ little collection of rules, experiments, and guesses transformed it into the first scientific work on gunnery’.40 In contrast with this reading, which seems to belittle Robin’s work, we wish to emphasise that the Robins’ work has considerable merit in itself, as he questioned the reality of the corpuscular fluids of Newton, introduced the stern effect, and discussed the ratio of the resistance to the square of the velocity. These points were accepted and commented on in depth by Euler, who in turn, added new ideas which did not follow the original texts of Robins exactly. The most outstanding of these was the introduction of the streamline resistance model as opposed to the impact one, which he presented for the first time for a body, although we must not forget the precedence of Daniel Bernoulli in his analysis of the jet against a plate. New Principles of Gunnery is an extensive work, although we shall centre on the second chapter, ‘Of the Resistance of the Air and of the Track described by the Flight of Shot and Shells’. A few lines from the beginning Robins, states41: That the greatest part of authors have established it [the resistance] as a certain rule, that, whilst the same body moves in the same medium, it is always resisted in the duplicate proportion of its velocity; This rule, though excessively erroneous, (as we shall hereafter shew) when taken in a general sense, is yet undoubtedly very near the truth, when confined within certain limits. [Chap. II, p. 66]

By this he wishes to make clear that the resistance coefficient varies slightly with the velocity. Thus, for small variations in velocity the resistance can be considered constant. Robins distinguishes two types of fluids, which do not coincide with those of Newton, although a certain relation exists between both types. According to his own words:

39

The title of the traslation is Neue Grundsätze der Artillerie, aus dem Englischen des Herrn Benjamin Robins übersetzt und mit vielen Anmerkugen versehen, Berlin 1745. 40 Cf. ‘Rat. Fluid Mech-12(2)’, p. XXXVIII. 41 The calling in brackets will correspond to the New Principles of Gunnery, that are frequently quoted merely as Gunnery.

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In order to conceive the resistance of fluids to a body moving in them, it is necessary to distinguish between those fluids which, being compressed by some incumbent weight, perpetually close up the space left by the body in motion, without permitting for an instant any vacuity to remain behind it; and those fluids in which (they being not sufficiently compressed) the space left behind the moving body remains for some time empty. [Chap II, Prop. I, p. 67]

That is to say, whether an empty space remains or not behind a moving body is decisive for the law of motion as Robins conceives it. Fluids exist in which this effect is never produced, and these would be totally compressed fluids, and others exist in which the effect could take place, although it would depend upon the velocity of the moving body. He does not say clearly which ones belong to one type or the other, although he states that ‘the air partakes of both these affections, according to the different velocities of the projected body’ [ibid.]. Robins recalls the two types of fluids surmised by Newton, as well as the resistance value of a cylinder and a sphere to both types of liquid. One type, aquiform, is incompressible, and its molecules are always in contact, and the other, the aeriform, is compressible and its particles separated among themselves. In the former, resistance is generated by individual impacts, while in the other the action upon one particle is transmitted to the rest. Regarding this he states that while the hypothesis of discontinuous fluids: But though the hypothesis of a fluid, thus constituted, [the discontinuous one] be of great use in explaining the nature of resistances; yet, in reality, no such fluids does exist within our knowledge. All the fluids with which we are conversant are to be formed, that their particles either lie contiguous to each other, or at least they act on each other in the same manner, as if they did. [Ibid. p. 69]

Robins partly agrees with the Newtonian mechanism, but he rejects the existence of discontinuous fluids; all fluids are continuous, even when some never leave a vacuum behind them and others sometimes do so. In the case of compressed fluids, which were those which never leave a vacuum, he is of the opinion that the shape of the body has little influence on the magnitude of the resistance: [F]or the principal resistance in compressed fluids arises from the greater or lesser facility, with which the fluid, impelled by the fore-part of the body, can circulate towards its hindmost part; and this being little, if at all, affected by the form of the moving body. [Ibid. p. 70]

That is to say, Robins agrees that his compressed model behaves like Newton’s aquiform type which he corroborates further on.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

But as for the other type, which can leave a vacuum, he establishes a second division according to whether the motion is slow or fast, arguing that when the velocity of the object is high, the molecules do not have time to rush behind the moving body thereby creating a vacuum which would increase the resistance, as: [T]he fluid cannot instantaneously press in behind it, and fill the deserted space; for when this happens, the body will be deprived of the pressure of the fluid behind it, which in some measure balanced its resistance. [Ibid. p. 71]

This type of behaviour would be similar to the air-like fluids of Newton, as in these all the resistance is generated in the front part (Fig. 3-11a). Besides, in this case the influence of the shape of the body is considered to be significant. On the other hand, when the speed is slow and if there is time to fill the vacuum, then the behaviour will be similar to the case of the compressed fluids. Obviously there is an intermediate situation with the partial vacuum that we have tried to illustrate in Fig. 3-11b.

a)

b) Fig. 3-11. Robins motions

Robins again recalls that according to Newton’s results, the resistance suffered by a cylinder which moves along its axis in a rare medium is four times that of a continuous medium, a result which he accepts for his rapidly moving cylinder. Concerning this he states: And therefore, as we before observed, since the resistance of a discontinuous fluid to a cylinder, moving in the direction of its axis, was four times greater than the resistance of a fluid sufficiently compressed of the same density, it follows, that the resistance of a fluid, when a vacuum is left behind the moving body, may be near four times greater than that of the same fluid, when no such vacuity is formed. [Ibid. p. 72]

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In the case of the sphere, Newton had said that the resistance was one-half the cylinder in a rare medium, and equal in an aquiform medium. On comparing it to the cylinder Robins says: In a globe the difference will not be so great, because, on account of its oblique surface, its resistance in a discontinuous medium is but about twice as much as in one fully compressed; for its oblique surface diminishes its resistance in one case and not in the other: however, as the compression of the medium, even when a vacuity is left behind the moving body, may yet confine the oblique motion of the parts of the fluid, which are driven before the body, and its elastic fluid (as is our air) there will be some degree of condensation in those parts, it is highly probable, that the resistance of a globe, moving in a compressed fluid with a very great velocity, will be between that of a globe and of a cylinder, in a discontinuous medium. That is, (in proportion to its velocity) we may suppose it to be more than twice, and less than four times the resistance of the same globe, moving slowly through the same medium. Whence, perhaps, we shall not much err in supposing the globe in its swiftest motions to be resisted near three times more, in proportion to its velocity, than when it is slowest. [Ibid. p. 72]42

Our intention with this long quote is to give an idea of the line of argument followed by Robins, and we emphasise that ‘near three times’ for the resistance of the sphere is an intermediate value between the cylinder and the sphere under vacuum conditions. To sum up, Robins and Newton agree regarding the resistance of fluids which do not leave a vacuum behind them, either because they are very compressed or because the movement is very slow. In these cases the shape of the body has little or no influence, and the resistance coefficient is CD = 0.5. At the opposite extreme, if the advance is very rapid, that is to say with the total rear vacuum, then the resistance will be three times greater than this value for the sphere, and four times for the cylinder. For the intermediate cases the resistance will be between these extremes, i.e., between 1 and 3 for the sphere, and 1 and 4 for the cylinder. Now, if what is understood by low velocity is easy to define, the same cannot be said concerning the so-called rapid velocity. Robins surmised a maximum value without any theoretical justification, as we shall see when we come to his experiments. Euler however contributed a more precise definition.

42

Here we make a qualification that this is true if there are no rebounds, but in this case it will be eight times. Robins says: ‘ that neither the cylinder, nor the parts of the fluid, were elastic’ [Prop. I, p. 68].

THE GENESIS OF FLUID MECHANICS, 1640–1780 600

6

500

5

400

4

Fligth time (s)

Velocity (m/s)

154

300

Set 1 Set 2 Set 3

200

100

3

2

1

Pendulum 0 0

20

40

60

80

0 250

Distance (m)

Experiments Theory Lake 300

350

400

Range (m)

Fig. 3-12. Cannon firing results

Robins did not limit himself to theory, but also contributed a considerable amount of experimental work on the resistance of spheres in air [Chap. II, Prop. 2, p. 74-ff]. For this he used a cannon that fired a spherical lead bullet of 3/4 in. in diameter (19.02 mm), whose approximate mass was 1/12 lb (37.8 g),43 against a ballistic pendulum located at different distances from the muzzle. He tried to get the bullet resistance by measuring the deceleration over the trajectory. He performed three sets of shots, and previously he assured a muzzle velocity tolerance of 20 ft/s (6 m/s) from the result of various preliminary tests. For the first group, the pendulum was placed at 25, 75 and 125 ft (7.6, 22.9 and 38.1 m), in the second at 25 and 175 ft (53.3 m). For the third he reduced the power charge and placed the pendulum at 25 and 250 ft (76.2 m), getting velocities logically lower than the previous ones.44 For each case several shots were made (three or five), in order to obtain the average velocities. The results are shown in Fig. 3-12 (left). He also performed three firings over a lake,45 determining the range and flight time, assuming that the muzzle velocity of the projectile was 400 ft/s (122 m/s)46 presented in Fig. 3-12 (right). 43

The results obtained from this data show the density of the material to be 10.5 g/cm3, which correspond to lead whose density is 11.3 g/cm3. 44 In the first one he placed the pendulum at 25, 75 and 125 ft, obtaining 1,670, 1,550 and 1,425 ft/s for the velocity. In the second he only placed the pendulum at 25 and 175 ft, with velocities of 1,690 and 1,300 ft/s. In the third one, with less powder charge, he repeated the distance of 25 and moved the second to 250 ft, obtaining 1,180 and 950, respectively. He says that the firing times are the average of three shots in the first sets and of five in the second. 45 A possible reason for firing over the lake could be the flatness of the water surface and the easy determination of the impact time. 46 He measured 4.25, 4 and 5.5 s with ranges of 313, 319 and 373 yards, respectively.

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The data reduction made by Robins was not very detailed, as he obtained the deceleration as the difference in velocity between two positions.47 As we have already noted, he compared the results with the values given by Newton for the sphere in a liquid, which he took as the unit (for this CD = 0.5). From the first test group he concluded that the relation between the forces measured and thus calculated was 2.4 (equivalent to CD = 1.2). Robin says of this value that it was relatively close to 3, which was what he had estimated for the sphere in the previous Proposition.48 In the case of the second group, he says that the same parameter comes even closer to the value of 3. From these results, in particular those confirming the triple value for the sphere, he deduced that the vacuum had already formed behind the sphere for the velocity of 1,700 ft/s (518 m/s).49 Furthermore, he estimates this value as being the maximum possible ‘as no large shot was ever projected in practice with velocities exceeding that of 1,700 ft/s’ [Cap. II, Prop. III]. This was the reason why he reduced the quantity of gunpowder in order to achieve lesser velocities. Repeating the process using the data of the third set he obtained a relation of 1.57 (CD = 0.78).50 Finally for the shot over the lake, he estimated that according to the quantity of gunpowder used the exit velocity should be 400 ft/s (122 m/s). In accord with this, and with the resistance value for a slow body (CD = 0.5), the theoretical times upon impact were found to be smaller than those measured and thus the resistance was considerably greater.51 1/C D0 1/C D 1/C Dm A

C

V

B

D

Vm Fig. 3-13. Robins construction 47

Using the equation a = vdv/ds. We have repeated his calculations and find similar values. Properly that relation is 2.14 instead 2.4. 49 Euler, as we will see, showed in his commentaries that the vacuum would be produced at more than 1,348 ft/s (425 m/s). 50 For this case we have found 1.42. 51 The theoretical times were 3.2, 2.28 and 4 s for the ranges measured. See previous note 46. 48

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These results reassure him that the resistance is greater than that predicted by Newton, increasing from the value given by Newton for slow movements up to a factor of three times. In order to determine the resistance at velocities less than the maximum, which, as we have said, he took as 1,700 ft/s, he uses the geometrical constructions shown in Fig. 3-13. In it the AB is a segment corresponding to the maximum velocity (v m) and AC corresponds to the velocity (v) whose resistance he wished to know. Likewise he states that BD and CD are respectively proportional to the inverse of the maximum resistance (CDm) and the one he wants to find (CD), AD being the minimum (CD0). He gives no reason for justifying this construction, only that it fulfils the conditions at zero and maximum velocities; which will be when point C coincides with B or A, respectively. The following geometric relations are in accordance with the description of this construction:

AB v = ; AC vm

BD C D0 = ; AD C Dm

CD C = D0 AD CD

[3.13]

To these we have to add the obvious ones: AC + CD = AD ; AB + BD = AD

[3.14]

Once resolved, we end up with the following equation: CD =

CD

0

⎛ C ⎞ v 1 − ⎜1 − D ⎟ ⎝ C D ⎠ vm

[3.15]

0

m

It can be confirmed that, for v = 0 and v = vm the resistance coefficients are CD0 and CDm, respectively, these being the minimum and maximum. Robins did not arrive at this equation, which was deduced by Euler starting out from the given premises. Before passing on to analyse Euler’s commentaries to Robins’ Gunnery, we will make two additional observations. The first refers to the tests with cannons. As we indicated in the introduction, these were the first supersonic tests ever performed. The exit velocities in the first tests were 509 and 515 m/s, which correspond approximately to the Mach number M = 1.50,52 for which, according 52 The Mach number is defined as the relation between the fluid speed and local speed of sound. It is 340 m/s for a temperature of 15°C.

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to current data, the resistance coefficient of a sphere has the value of CD = 1.41; which is fairly close to the value found by Robins. The other point we wish to note does not derive from the New Principles of Gunnery, but is still related to gunnery. We refer to the experiments made on several occasions in 1746 by Robins before the Royal Society concerning the resistance of spheres at low velocity. He used an apparatus of his own invention, as is described in the Mathematical tracts of the late Benjamin and Robins, Esq (1761).53 Robins set out to demonstrate two propositions: ‘that the resistance of the air to a 12 lb iron bullet, moving with a velocity of 25 ft in a second, is not less than half an ounce avoirdupois’ and ‘that the resistance of the air, within certain limits is nearly in the duplicate proportion of the velocity of the resisted body’.

Fig. 3-14. Robins rotating apparatus

As can be seen in Fig. 3-14, the apparatus consisted of a rotating arm moved by a cable coiled around a central cylinder with a weight. The sphere, whose resistance was to be measured, was placed at the end of the arm. This system is an alternative procedure to the falling spheres used by Newton.54 The ball had a diameter of 4.5 in. (114 mm), the cylinder 2.06 (52.3 mm) and the arm up to the ball 49.5 (1,257) resulting in 51.75 (1,314) from the axis to the centre of the sphere.55

53 Edited by James Wilson. The work consists of two volumes. The chapter we are interested in is titled ‘An account of the experiments, relating to the resistance of the air, exhibited at different times before the Royal Society, in the year 1746’ (No. III, p. 200). 54 A third procedure would be the wind tunnel. 55 But it is not only Robins who had the idea of this apparatus. According to Smeaton citation (Cf. ‘An experimental Enquiry …’, Phil. Trans., 1759, in footnote p. 139) in same years another Englishman, Mr. Rouse of Harborough, presented another similar one, but neither knew the other. That

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THE GENESIS OF FLUID MECHANICS, 1640–1780

His operative procedure was to let the system accelerate, and then he measured the time taken by each turn. With a mass of 0.5 lb (0.227 kg), and after leaving it for three turns, it measured 27 3/4 s for the 10 remaining turns, 27 1/4 for the other 10, and 37 1/2 for 10 more. With 3.25 (1.508 kg), and after 10 revolutions, he measured 20 more as 21 1/2. He then substituted the sphere for a mass of lead with a mass of 1 lb (0.454 kg), and measured 20 revolutions at 19 s. As this time was similar to the previous one, he concludes that the mass of 1 lb will compensate the parasitic resistance, leaving 2.25 for the resistance. He carried out the reduction of revolutions to velocities, and the turning moment produced by the falling mass obtaining a resistance force of 0.72 ounces, thus confirming his proposition which says that a 12 lb cannon ball has a similar diameter to the sphere used. Certainly his calculations were made in a very rough and ready way. Taking his measurements, and repeating the calculations in a carefully way56 we would arrive at a force of 0.78 ounces (0.22 N) for a translation velocity of 7.68 m/s, which, transformed into a resistance coefficient, would result in C = 0.59. This value is close to present-day measurements. In order to check his second proposition he used masses of 2, 4 and 8 lb, i.e., in a squared proportion, measuring the time spent in 20, 30 and 40 revolutions, respectively. For the three cases he obtained 27.5 s, thus confirming his predictions. Euler’s streamline model Although the most significant of the contributions made by Euler in the translation of the Gunnery into German57 was the introduction of the streamline model, there are other aspects that should not be forgotten. The first of these is the new reduction of the data obtained by Robins. Euler repeated the process, but used more elaborate and exact methods, which in essence were based upon more complete approximations of the velocity as a function of the distance. In this way, using the data of the firings against ballistic pendulums for the relation between the experimental values in the first set of tests and taking Newton’s sphere as the unit, he obtained the values of 2.35 and 2.48 (CD = 1.17 and 1.24),

makes Smeaton comment: ‘this happens, that when two people think about the same subject matter, their experiments are similar’ (ibid.). 56 The only hypothesis to make is that resistance be proportional to square velocity. If the parasitic resistance for 19/20 s per revolution was compensated with a pound, for 21.5/20 it will be 0.78 lb, and the one to compensate the resistance D = 3.25 – 0.781 = 2.469. The Reynolds number is 5.38·105, which is very close to the transition. 57 The translation from German to English has been done with the help of Amparo Díez Martínez.

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according to the interval.58 For the second set he found the value of 2.59 (CD = 1.29), and in the third 1.549 (CD = 0.78). For the firings over the lake he studied the trajectory in great detail, and a value of 1.605 (CD = 0.81) can be deduced from his application. The treatment is far better than that of Robins. A second aspect refers to the formula relating the resistance coefficient with the velocity formula [3.15]: here he says the expression that can be deduced from the construction proposed by Robins is one of many possibilities that verify conditions at the ends. Against this formula he maintains that although the range of application has to be limited between v and vm, there is not much sense in the fact that for a higher speed the result tends to infinity.59 As an alternative, he is of the opinion that it would be better to represent the resistance by any equation of the type v2 + v2n/2g, that after a few additional considerations he estimated that n = 2. To obtain the value of g he used the data of the three sets of shots carried out by Robins and obtained 22,102; 18,176 and 30,781 Rhenish Fuß, respectively.60 For consideration of the range of the trajectory he estimated that the third one should be the most exact. Now Euler compared this figure with the ‘height of the natural column of air whose weight is equal to the elastic force of air’ [Prop. III, 2nd. Com. p. 311],61 that is, 28,845 ft, equivalent to 27,979 Fuß. Even though this result is smaller than the one obtained before (22,102), he thought that the truth is that the value of g conforms with the height of that column, that is, g = 27,979 Fuß (8,792 m). This could be a coincidence, but his method of proceeding is not unusual because by this that time this height played an important role, as we shall see. The commentaries of a conceptual nature are much more substantial and wide-ranging.62 They can be classified in two categories: one, in which Euler exploits the ideas of Robins, carrying them much further from the state in which Robins left them, but without compromising his ideas. The other, where he introduces his own ideas to the drift of the text, will constitute Euler’s genuine contributions. The majority of these commentaries belong to the first category, but it is in the second where he introduces the streamline model as opposed to the impact one. We shall start with the first category. 58

We have to point out that the list quoted by Euler does not correspond exactly with Robins one, because Euler included two values in group 1, while Robins only calculates one. 59 This happens when v/vm = 1 – CD0/CDm that with CD0/CDm = 3 results v = 1.5 vm. 60 There is an error in the Euler calculations of the first of the three values, that is 23,080 as is pointed out in the reprint in the Vol. 14 (2), p. 311 of the Opera omnia. From now the quotes in brackets will refer to this edition. 61 This will be the height of an air column whose pressure at sea level is the existing one. It is obtained from the equation p = ρgh, and it is 8,480 m. 62 The Robins text we are studying takes eleven pages and in Euler’s forty pages, divided into four Commentaries.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

One of the characteristic features of Robins was the introduction of the rear vacuum which gave rise to a mixed theory. However, as we have seen, for the calculation he went no further than proposing a simple geometric rule (Fig. 3-13). Euler follows in Robins’ footsteps, accepting that when the body moves very quickly in air, the air has no time to fill the vacuum left behind itself, thus the resistance increases because of this depression: ‘then in this case it depends on the velocity at which the air can follow a body, or at what degree of velocity the air can penetrate into an empty space’ [4th Com. p. 271]. In order to determine this velocity, which he supposes to depend on what he calls the elasticity of the air, as we have seen, he defined it as the height of column of air with uniform density which produces at ground level a pressure equal to the real atmospheric pressure. This height now turns out to be ha = 29,100 Fuß (9,160 m).63 He then considers that the air would enter into an empty space at the same velocity that a body falling from that height would acquire. He obtains va = 1,348 Fuß/s (425 m/s).64 Euler does not justify this hypothesis, although in his time people talked of the atmosphere as ‘an ocean of air’. Daniel Bernoulli had used a similar idea when he dealt with the discharges of air from a deposit.65 Bernoulli had noted this when studying the entry and exit of air between a deposit with air and an empty one. If air was a fluid with a constant density, and with a determined depth, its behaviour would follow Torricelli’s Law as any liquid would, and the exit velocity would coincide with that given.66 Thus coming back to the moving body: If therefore a cylinder moves in the direction of its length with the speed of 1348 [Fuß] per second, it can be followed precisely by as much air necessary in order nor to leave and empty behind. But in this case the air does not exercise any force on the cylinder from behind. [p. 271]

That is to say, for velocities greater than the one found, there would be a permanent rear vacuum, as the air would not have time to fill the space; whereas, if the velocity is less there would only be a partial vacuum. On the assumption that the cylinder moved at a greater velocity, Euler states that then ‘not only would no pressure exist behind, but besides this, it would always leave an empty space behind it’ [p. 271]. In this case, the total resistance would be the sum of the front one, which would be the weight of a column of fluid whose height was equal to 63

We note the difference with the value of 8,792 m deduced previously from the 28,845 English feet. 64 As v2 = 2gh, for the previous note nº 61, v2 = 2p/ρ. With current data it results v = 407 m/s. 65 Cf. Hydrodynamica, Sec. 10, §.34–38. 66 This way of thinking matches with the idea of the atmosphere as an ocean of air.

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the kinetics corresponding to the velocity of the body (CD = 1), plus the additional rear resistance. This would be the weight of the other column whose height would be that of the uniform atmosphere. Thus if the height of this equivalent atmosphere is designated as ha, the resistance would have the value: D =

1 2

ρ S C D v2 + ρ g S ha =

1 2

ρ S (v 2 + v a2 )

[3.16]

where v2 = ha/2g is the velocity whose kinetic height is that of the equivalent atmosphere. Obviously the validity of the expression [3.16] depends on the condition that v ≥ va. In the case that the velocity is less than va, a vacuum will not form behind, but neither will the circulating air completely fill it, and as a result a partial vacuum will be created. In order to find the resistance under these circumstances he assumes that the former value [3.16] is diminished by a certain quantity due to a partial inlet of air, which he conjectures to be the equivalent of the weight of an air cylinder whose kinetic height corresponds to the velocity va–v. That is to say ½ρS(va–v)2. Thus the total resistance would be: D=

1 2

ρ S ⎡⎣ va2 + v 2 − (va − v) 2 ⎤⎦ = ρ Svva

[3.17]

This conclusion is somewhat surprising, and makes Euler say: Then if this conclusion was correct, the resistance would not be proportional to the square of the velocity of the body, as we have thought previously, but only to the same velocity. [p. 272]

It is surprising that starting from the hypothesis of ratio of the resistance to the square of the velocity, the process leads to a completely contradictory conclusion. However, he has to recognise that the argument is not completely without basis, and says that: At least, given that the nature of the fluid material is still not totally known, in such a way as to be able to determine all the circumstances only from the theory without experimental proof, it would be useful to continue investigating this concept of the effect of the air and other liquid material on solid bodies, ignoring the fact that this concept cannot be supported or sustained by experiment. [p. 273]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

These arguments, which as we have seen refer exclusively to the air, are extend by Euler to water, disregarding the fact that Robins considered that water never left a vacuum behind it. For water, the equivalent height of the atmosphere would be the depth at which the motion takes place, thus: If the same cylinder moves with the same velocity at different depths below the water, its resistance would be greater in proportion to the depth at which the cylinder is submerged below the water, the resistance would increase with the square roots of the depths below the water. [p. 273]

Nevertheless, here it needs experimental verification, which he proceeds to ask for. He continues to apply this theory to closed bodies, for which he finds a differential formula that he tailors for the case of a sphere. We shall not enter into this argument. He concludes by pointing out that everything found is a consequence of the hypotheses adopted by Robins, although distancing himself from the results he arrived at, saying: Many other intriguing consequences may be deduced from this concept of the resistance of fluid. However we shall pay no attention to them as it is still very uncertain if this concept coincides closely enough with experiment or not. [p. 280].

Having analysed the contributions of Euler based on the earlier ideas of Robins, we can turn to his own contributions. We begin by recalling that Euler describes the resistance generation model [Prop. I, Com. 2nd] that he presented in the Scientia navalis, which he called the quantity of motion, and that he rejected it in favour of live force. Applying this, he arrives at the usual coefficients of CD = 2 and CD = 4 for the cylinder, according to whether or not the rebounds are taken into account. After exploiting and manipulating this model and its results mathematically, he states: However, a given fluid like the one we have considered, not only is not found in the world, but it is even impossible for it to exist. And it is from this point that the resistance that a body encounters in fluids, (as these exist in the real world), must differ from the observations made previously. [Prop. I, 3rd Com. p. 259]

One of the reasons which he brings to bear in order to deny the existence of these fluids is the existence of pressure in all points surrounding the body, thus following the idea of Robins that there is movement of air around the body. A little further on he continues:

THE EVOLUTION OF THE PROBLEM OF RESISTANCE

z0

v0

a

A

d m

m

M d

163

M

D

ds N

R n

z X

B

Y

Fig. 3-15. Body in streamline flow From experience it has also been found that a body in water supports the same resistance as the weight expressed by a column of water whose height = v [v02/2g]; and as the particles of water avoid the moving body in the same way as particles of air, the conclusions reached were that the resistance is made of similar type of particles in both fluids. [p. 262]

With these reflections Euler proposes an explicative model of the motion in which the fluid surrounds the body, without impacts or rebounds, valid for both air and water, that is, ‘streamlines’ as opposed to ‘impacts’. In Fig. 3-15 he imagines a layer of liquid of width Aa, which will be forced to deflect itself following the contour AM just as if it were circulating through a curved channel AaMm. As it flows through the channel, the fluid direction not only changes continuously but also the velocity increases or diminishes according to the width of the channel. It is important to note the radical difference from the impact model. Now there are no impacts, only fluid deviations caused by the surface of the body. In order to analyse the fluid movement through this flowline (Fig. 3-15b), Euler isolates a differential element MmNn whose mass will be ρzds and velocity v, and which will be subjected to a tangential force and another centripetal force whose values are: 2 dF c = v ρ z ds/R

[3.18]

dF t = v ρ z ds = ρ z v dv

[3.19]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The first is caused by the surface curvature, whose radius is R, and the second by the variation in the width of the filament. 67 Projecting over the axis OY and OX, and bearing in mind that the radius of the curvature is R = ds/dθ, and that for the sake of continuity we must verify that z0v0 = zv, the two previous equations are converted into: dF x = ρ z 0 v 0 (v cos θ dθ + sin θ dv ) = ρ z 0 v 0 d (v sin θ )

[3.20]

dF y = ρ z 0 v 0 ( −v sin θ dθ + cos θ dv) = ρ z 0 v 0 d (v cos θ )

[3.21]

Where z0 and v0 are the width and the velocity in the channel that surrounds the body. The total force over one of the halves of the body, between the vertex and a point of the body, will result in the integration of the two previous formulae with the initial conditions θ = 0 and v = v0, leaving us the simple expressions: F x = ρ z 0 v 0 sin θ

⎛

2 F y = ρ z 0 v 0 ⎜1 −

⎝

v ⎞ cos θ⎟ v0 ⎠

[3.22] [3.23]

The first formula expresses the lateral force upon half of the body, which is balanced by that exercised upon the other half. The second represents the axial force, or resistance, also upon half of the body, and it is this that interests us.68 From this expression we can verify that the total force depends only on the geometry of the final point and the local velocity, which is linked to the width of the channel. If, as regards the geometry, the angle θ is a pertinent data, the same is not true for the velocity, which makes it difficult to obtain the resistance value. Euler analyses some specific cases, of which the simplest would be when the geometry forces the fluid to turn 90° from its original trajectory, and where the force would then be F = ρz0v02.

67

The justification of these equations is easy. In the first one v2/R is the centripetal acceleration and in the second dv/dt is the tangential one. 68 Even though the argument is exactly the same as Euler’s, the algebra we have used is completely different and much simpler. The calculus procedures available in those years were much too rudimentary, which does not hamper the Euler’s ingenuity in solving the equation by a simple method.

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165

However, in the case where the body was closed, and the fluid surrounded it completely, the channel would have the same width at the outlet as at the inlet, thus it would have the same velocity and the angle of the flow must be θ = 0. Under these conditions, if the equation [3.23] is applied, a zero resistance will result.69 This surprising result, totally contrary to the reality, does not convince Euler, who tries to find a cause or explanation for it. In order to do this, he analyses the force in the channel surrounding the body, and using the same formula he says that ‘in fact this [force] will increase, and the angle θ will be greater as the distance from the initial [point] Aa, i.e., when channel nearby the body turns downwards, the force generated will increase, and this force pushes the body towards the direction AB’ [p. 268]. That is to say, this force is a resistance, but there is an inflexion70 point in the body contour from which the angle starts to grow smaller, the channel turns its concavity to the side of the body, and consequently the force starts decreasing in such a way that, if at point D the channel becomes parallel to the axis and its width is z0, the force will be zero. Thus, there are two parts separated by the inflexion point: the forward, AM, in which a force will be produced that pulls the body in direction AB, and the rearward, MD, in which an opposite force will push it in the direction BA. However he states that ‘as no body can be moved unless by means of actual pressure, thus also this latter force [pressure] can only exercise an effect upon the body when the pressure of the fluid behind it is strong enough to drive the body forward’ [p. 268]. Here it seems that Euler wants to say that behind the inflexion point the air is in some way so rarefied as to produce a true pressure. That is, he considers that the theory is satisfied in the front part, but not in the rear. He holds this hypothesis at least for known fluids, because he continues: ‘In air and water the pressure at the front not only is not equal to that at the rear, but is usually even greater, therefore it can be seen that the force from the part of the canal MD can have absolutely no effect upon the body, or at the most only a minimum effect’ [p. 268]. But, probably trying to save the basis of the theory, he mentions an imaginary fluid that would meet the theoretical conditions, and thus he rounds off his argument by saying that ‘this case could take place if the fluid matter were infinitely fluid, and at the same time compressed by an infinite force’ [ibid.]. These are very strange properties, but ‘perhaps, of such characteristic is the heavenly aether [Himmels-Materie], in which planets and comets move, and

69

This is the first appearance of the now well known ‘d’Alembert paradox’, so called because it was this author who proposed it as such. Truesdell claims its authorship for Euler [cf. ‘Rat. Fluid Mech.-12(2), p. XL]. For some reason, it deserves some additional comment, that we will make later. 70 For this he understands that dθ/dy > 0, while if dθ/dy < 0 it is concave.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

therefore this is the reason why no change in the motion of these bodies is perceived’ [ibid.]. However, coming down from the heavens to the earth, in the next line he states that ‘this characteristic does not take place in air, water and other known fluids’ [p. 269].71 That is, the theory would only be true for fluids which fulfilled certain ideal conditions which do not occur in reality.

Fig. 3-16. Streamlines over a cylinder

With the intention of providing a qualitative idea of the value of this resistance, Euler presents a figure included here as Fig. 3-16 representing a cylinder surrounded by a fluid flow confined in a tube. He traces the streamlines and adds that resistance undergone by the cylinder will be the function of the curvature of these lines. Although the argument he brings to bear is qualitative, it serves to confirm that the resistance is very dependent upon the form of the head or bow of the body. This is fact which makes manifest his disagreement with Robins,72 declaring to this effect that the resistance of the cylinder has to be less than CD = 2. By way of conclusion, Euler declares that he is very far from Robins in the hypothesis that in a compressed fluid the resistance is independent of the form of the bow.

71

This argument could show that Euler had not fully abandoned the Cartesian hypothesis of planets in motion through the action of vortices in the ether. 72 Truesdell, in the Essay of History of Mechanics, (Chap. IV, p. 225), considers that it is the first manifestation of a motion with ‘fillets’. This is not exactly true, as Daniel Bernoulli had noticed them when he studied the discharge of a fluid from a vessel through a hole (cf. Hydrodynamica, Sec. IV, §.3). However, the conditions are different.

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167

D’Alembert’s paradox D’Alembert is one of the most fascinating figures of the eighteenth century, as much for his contributions to mathematics, mechanics and dynamics as to philosophy and epistemology.73 His work on fluids comprises principally, but not exclusively, two works: Traité de l’équilibre et du mouvement des fluides (Treatise on the equilibrium and motion of fluids) (1744), the second and more decisive Essai d’une nouvelle théorie de la résistance des fluides (Essay on a New Theory of Resistance in Fluids) (1752). There is a radical difference between both works: the Théorie followed impact theory, whereas the Essai changed to a streamlined model from which he obtained the first equations of fluid dynamics. It should be noted from the outset that d’Alembert’s ideas on dynamics were unusual, and in contradiction to the other geometricians of the time. He builds his mechanics on three principles that he explains in his Traité de dynamique: that of the inertial force, that of compound motion, and that of equilibrium. As regards the first, he states that a body cannot put itself into motion, and that the intervention of an ‘external cause’ (cause étrangére) is necessary; and, once established, ‘the motion is uniform by nature’ [Pref. p. ix].74 He says that ‘Like Newton I give the name force of inertia to the property that bodies have of remaining in the state in which they are. Then a body is necessarily in the state of rest or in a state of motion’ [I.I.§.2]. As a corollary he uses the terms power (puissance) or moving cause (cause motrice) for anything that forces a body to move. These external causes are classified by him in two categories: ‘those that are manifested to us at the same time as the effect they are cause of, … the others are known only by their effect, and we completely ignore its nature’ [Pref. p. x.]. Among the first are the mutual actions between bodies, which he says are reduced to impulses, and among the second, the action of gravity. The second principle, that of the composition of motion, refers to the circumstance in which a body is under the action of various different powers, each one of which, in isolation, would provide it with a certain velocity. The principle establishes that the resulting motion will be composed from the individuals following the rule of the parallelogram. A consequence of this principle it is possible to decompose one movement into others. This is a strictly geometric law. The third, that of equilibrium, tells us that ‘if two bodies whose velocities are in inverse proportion to their masses and move in opposite directions, as in such a way that one cannot move without displacing the other, there will be an 73

We recall that he was the author of the ‘Preliminary Discourse’ of the Encyclopedie, an emblematic work of the time. 74 The quotes will correspond with the Traité de dynamique.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

equilibrium between these two bodies’ [I.III.§.39]. On this definition, an impact between two masses of equal momentum, but moving in opposite directions, will result is a state of equilibrium. However, the mechanical impact underlying this definition is understood by d’Alembert to apply also to the case of virtual velocities, i.e., ‘the velocities at which the [two] bodies would tend to move’ [Pref. p. xix]. Of the three principles, this is the one that has most mechanical significance, as he makes equality of the momentum (magnitude in which mass and velocity intervene) the basis of the dynamic equilibrium. A number of points are worth emphasizing in regard to these three principles. The first is the absence of force as an efficient cause of movement, as he explains clearly: ‘I have completely banned forces inherent in a moving body, obscure and metaphysical beings which are only capable of spreading shadows into a science that is clear in itself’ [Pref. p. xvi]. He insists that although sometimes he uses the ‘obscure’ term ‘force’, he only does so in order to avoid certain circumlocutions, but the only valid ideas are those derived from his principles. D’Alembert discards these entities that he considers incomprehensible, and which he substitutes for immediate dynamic effects such as impacts or obstacles impeding movement. This is relatively clear when the moving body changes its uniform motion suddenly due to specific actions. However, when the moment is accelerated or decelerated due to continuous actions, he considers that the increase or decrease of the velocity is proportional to the time interval, that is, du = φdt. He then affirms that ‘therefore it is obvious that when the cause [of the motion] is unknown this always gives rise to the equation φdt = ± du’ [I.I.§.19]; the parameters φ being the ‘accelerating cause’ which he says other authors consider to be the accelerating force. D’Alembert comments this principle, and recalls that for some authors it is a general axiom in one way or another. Concerning this he declares: Daniel Bernoulli (Commentarii petropolitanæ, Vol. I) considers that this principle is only a contingent truth, considering that we ignore the nature of the cause and how it acts, we cannot know if its effect is really proportional, or whether it is like some power or function of the same cause. Euler on the other hand, makes great efforts to prove throughout his mechanics that this principle is a necessary truth. For us, without entering into the discussion of whether this principle is a necessary or contingent truth, we are content to take it as a definition, and we understand accelerating force to be exclusively the quantity to which the increase of the velocity is proportional. [I.I.§.19]

The text is sufficiently clear. D’Alembert observes the motion and establishes a hypothesis about it, but he avoids theorising upon the causes. Let us take note of two things: the parameter φ = F/m is the basis of all Newtonian dynamics,

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169

because certainly the force is something whose ontological content was difficult to explain, and to which d’Alembert is sensitive as a philosopher as well as a mathematician. We should also draw attention to his ‘general principle for finding the motion of various bodies which act among themselves in any fashion’, now known as ‘d’Alembert’s principle’. This principle can be applied to any individual bodies with ties between themselves and with the surroundings. He expresses the problem as follows: Given a system of bodies disposed in any fashion as regards one to the other; and supposing that each one of these bodies is given a specific motion, that the body cannot follow because of the action of the other bodies, [we must] find the motion that each body ought to take. [II-I.§.50]

To which he replies: Having broken down the motions a, b, c etc, imprinted upon each body, each motions into two others a, α; b, β; c, χ; etc, in such a way that if the bodies had only been imprinted with the motions a, b, c, etc. they could have retained their motions without harming each other; and if the bodies had only been imprinted with the motions α, β, χ, etc. the system would remain in repose. [II.I.§.50]

This is to say, if one of the bodies is perturbed by a determined motion, it will respond in a specific way, depending on the nature of the links and the boundaries. It is clear that in order to apply this principle other data and information are required, such as the quoted links in order to determine the breakdown of the initial motion. As J. Morton Briggs75 recalls, Ernest Mach considered this principle as a rule for solving problems, not as a principle in itself. From our present-day perspective, analysing this principle or rule using Newtonian mechanics as a reference is enlightening. Let there be a system of i bodies with a set of links among themselves and limited by boundaries conditions. If this system is perturbed by a set of external forces Fext, each mass will respond with a velocity dvi after a time dt. For the second law of dynamics it has to satisfy: G dvi G = m F ext ∑ i dt

75

In the entry ‘D’Alembert’ in the Dictionary of Scientific Biographies.

[3.24]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Now, if instead of the forces Fext we apply upon each mass a virtual force Fai = –dvi/dm, the system will remain at rest. Hence, introducing this virtual force in the former equation we have: G G dvi G = −∑ Fai [3.25] F ext = ∑ mi dt These virtual forces are called d’Alembert’s forces in some treatises. The substitution of the dynamic effects for these forces gives us Fext + ΣFai = 0, which converts the dynamic problem into a static one. This is how we understand this principle at the present time. Turning now to the application of d’Alembert’s ideas to the impact theory, his approach is that the resistance is produced by the transfer of the corpuscles’ momentum to the body. According to the Newtonian approach, if in a time dt the body is struck by a set of particles, these will transfer a momentum d(mv) to the body, and the resulting force will be F = d(mv)/dt. For d’Alembert, if the body struck has a mass M, after receiving the impact it will acquire a velocity du, so that, in order to retain the movement, Mdu = d(mv) must be satisfied, although he does not infer the existence of any force from this phenomenon. Y

C ds dy dx O

X

Fig. 3-17. Body and small spheres

Returning to the Théorie de l’équilibre, here he analyses the three most important themes regarding fluids: equilibrium, discharge, and resistance. For the moment we will limit ourselves to the third, subtitled ‘On the resistance of fluids to the movement of bodies’. D’Alembert, as we have seen, follows impact theory conceptually, although he expresses it as a variation of the momentum. This is somewhat similar to what Euler did in the Scientia navalis. D’Alembert supposes that the fluid is integrated by spherical corpuscles with δ diameter which,

THE EVOLUTION OF THE PROBLEM OF RESISTANCE

171

when struck by the body, acquire part of their momentum, resulting in the body losing velocity. In order to obtain this decrease qualitatively, he follows two steps. In the first he imagines that after the impact of the first layer of small spheres the body undergoes a finite incremental increase in velocity. In the second, it goes to the limit, converting the incremental increase into an infinitesimal. In order to resolve the first of these steps [§.323–233],76 he uses the mass of the layer of corpuscles surrounding the body as an ancillary parameter. This mass is designated M. Let this be a two-dimensional figure like that shown in Fig. 3-17 whose perimeter is L. If the mass of each particle is represented by µ, the mass of the layer covering this perimeter will be M = Lµ/δ. As regards the momentum that will be transferred for each particle, this will be a function of the local incidence angle. If this is named θ it would be µvcosθ, as he supposes that the corpuscle will rebound in a direction normal to the body. Likewise the axial component will be µvcos2θ which is the interesting part as regards the resistance, since the lateral component is annulled by the symmetry of the body. On the other hand, if the mass of the body is m, and the advance velocity before and after impact with the first layer are u and v, respectively, the total variation of the momentum will be m(u – v). Making this equal to the quantity acquired by the small spheres, we arrive at the following equation: C

m (u − v) =

2

2 M v dy L ∫O ds

[3.26]

If, instead of a two-dimensional body,77 we have an axially symmetric threedimensional body, the equivalent equation would be: m (u − v) =

2π M v Σ

C

∫ O

ydy 2 ds

[3.27]

in which the perimeter has been replaced by the surface Σ, and where M continues to represent the mass of the first corpuscle layer. 76

The reference in brackets corresponds to the Traité de l’équilibre. The steps given to reach this equation are the following: he finds the cosine of the angle as cosθ = dy/ds. Thus it will be: 77

A

∫

m(u − v) = 2v µ 0

dy 2 ds

But as M = Lµ/δ, if ds = δ is identified, the equation is found. Note the value 2 for symmetry reasons.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

This is the first step, and the basis of the method. The following step is to make the radius of the little spheres approach zero so that u – v = –du. Skipping the calculations78 for the two previous cases, we arrive at the following expressions [§.239]: C

− m du = 2 ρ u dx ∫ O C

− m du = 2π ρ u dx ∫ O

dy 3 ds 2 ydy 3 ds 2

[3.28]

[3.29]

Although there is a conceptual difference, if we specify these equations for the case of a flat plate, and assimilate them to a resistance coefficient, we arrive at an equivalence CD = 2, corresponding to the case in which there are impacts without a rebound.79 D’Alembert himself points out that if impact was supposed to be perfectly elastic, the transfer values of the quantity of motion would be double [§.240]. Given these explanations, we can see that up to now the only novelty he had introduced has been to adapt the impacts to his concept of dynamics, although he follows what we have been calling the theory of pure impact. However, a little further on he thinks about it in the following terms. [T]he formulae which we have given … have been deduced from the supposition that each layer annihilates itself as it makes the solid body lose a part of its motion; or what comes to the same thing, that the motion imprinted upon this layer does not communicate itself to its neighbours. [§.243]

He completes this a few lines further on, saying: [T]he fluid particles pushed and put into motion by the body coming into contact with them, do not continue on their way in a straight line, but fall back upon themselves to occupy from behind the space that the body leaves empty, thus forming a sort of vortex around themselves. [§.243]

78

They are similar to the ones already calculated for Newton’s sphere. In the axially symmetric case, a ring 2πyds is assumed, it will intercept unit a fluid mass 2πρdyudx every time, which will be transferred with the factor dy2/ds2. The result is the one shown. 79 The account is relatively easy. Let be a cylinder of radius R an mass m. The drag will be D = ½ρu2πR2CD = mdu/dt. Applying the second formula with dy = ds, it remains mdu = πR2ρudx, that equated drives to CD = 2.

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173

This last point serves him as an introduction to the elastic fluids, a theme which had not been dealt with by any author up to this time and which is the genuine contribution of d’Alembert in this work. The difference with non-elastic fluids is that in these the effect of the impact is confined to the neighbouring layer, while in elastic fluids this effect is propagated to more remote layers up to a certain distance. According to d’Alembert, this causes the fluid to condense in the front part and to rarefy in the rear, which should be understood as compression and expansion. The mathematical treatment that follows is rather confused and poorly justified. In the first place, he supposes that the transmission of the effects follows an arithmetical progression ‘and thus the momentum of fluid contained in the [front] space will be less than half, if the fluid was not elastic’ [§.254]. He offers no basis for the claim of this ‘less than half’. In the second place he introduces what he calls the ‘elastic force’ (‘force élastique’), which is pressure in the present sense of the term, and whose sole antecedent is in the first part of the work [§.69]. To sum up, he considers that the action of the fluid will be the difference between the effects on the front part and the rear part. For the former he points out that the quantity of motion lost by the body is:

ϕ SF

dx u

+ 2u ρ ∫

dxdy dy 2 2

ds 2

[3.30]

The second term of this equation is equal to that found in equation [3.28], with the inclusion of a 2 as divisor, and it appears that he wants to introduce this half value to differentiate the elastic fluid from the non elastic one. As regards the first addend, the coefficient he designated as φ is what he calls the elastic force, and which in present-day language is ‘pressure forces’ affecting the whole of the frontal surface SF, which is not difficult to deduce.80 For the rear part, which he supposes to be flat, he presents the following equation: dx S ϕ SF − u ρ F dx [3.31] u 2 The first member of this equation is the identical to equation [3.30], and the second is equivalent to the integral of equation [3.28] extended to a flat plate with half the surface. This assumption could to be due to an argument equivalent

80

The equality will the following: SFφ = mudu/dx, from it is derived the one shown.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

to one that he proposed when he referred to the fact that only half the quantity of motion was transmitted because of compressibility.81 Upon subtracting in order to find the total effect, the parameter φ disappears. This backs up the idea that when he refers to the atmospheric pressure, as when this acts around the entire body, a null result is produced. This leaves us with the final equation:

⎛

dy 3

∫ ⎝ ds

− m du = u ρ dx ⎜

2

+

1 2

⎞

[3.32]

SF ⎟

⎠

which is the equivalent to equation [3.28] for an non-elastic fluid. In conclusion, he comments that ‘the fluid has no action upon a posterior surface of the moving body, unless the fluid were to have a great elastic force in order to be able to fill up immediately the vacuum left behind by the body’ [§.255]. As can be seen, he repeats his preoccupation with what occurs in the wake forming behind the body. r vr

vx

x

Fig. 3-18. Body in a flow

The argument that d’Alembert provides in his Essai d’une nouvelle théorie de la résistance des fluides follows different paths. The difference is that he brings streamlined models into play and uses general equations of motion instead of impact theory. When we come to discharge, we will investigate this material in detail,82 although to give a sense of the evolution of the problem, we note here that in the new approach, the particles do not impact upon the body, but follow the streamlines surrounding it. The change in direction of these implies a variation in the 81

The d’Alembert derivations are very difficult to follow. He uses the symbol BC for the surface we have called SF, while DC, that is a half, for the one indicated as ½SF. 82 The matter is treated extensively later in Chapter 8, ‘Bodies in flowing currents’.

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175

momentum contributed by the body, and which is responsible for the resistance, as is shown in Fig. 3-18. Now he defines axial and lateral components of the velocity by means of two functions p(r,y) and q(r,y) so that vx = v0p and vy = v0q and their determination responds to a set of differential equations, together with the external shape of the body. Once these are known, the force received by the body turns out to be: − ρ v ∞2 ∫ 2 π y ( p + q ) dy 2

2

[3.33]

The integral is extended from the vertex the final point of the body. He now argues that in the case that the body is symmetrical with respect to its medium plane, functions p and q would also be symmetrical, and thus this integral will be zero, as dy would pass from being a positive in the front part to symmetrical but negative in the rear part, ‘in such a way that the body would suffer no pressure from the fluid, which is contrary to experience’.83 That is to say, there would be no resistance. This is a surprising declaration, as it is contrary to experience, and it questions the validity of the theory. Nowadays this result is known as the ‘d’Alembert’s paradox’, since he proposed it in this form to the geometricians of the time in the Opuscules.84 In seeking a justification of it he assumed that the front part did not completely meet the theory. We have seen that Euler85 arrived at a similar conclusion in his comments to Robins, although his starting point was not the general equations of motion, but a particular case of the streamlined bodies. The solution to the paradox was found only many years later, and although it must have been very disconcerting at the time, it perhaps had the effect of increasing interest in experimental methods. Other works of Euler In 1755, Euler published three papers back to back in the Mémoires de l’Académie of Paris, in which he established the basic equations of fluid mechanics which have passed into history as the century’s most brilliant contribution to the subject. We shall look at these below. At first glance, these equations should have been sufficient to resolve the resistance problem, but it was not so, as d’Alembert’s paradox indicated that there was no resistance, which went against the evidence. As we have indicated in the introduction to this section, although the mathematical 83

Cf. Essai, §.70. In the Vol. V, Mém. 34, I. 85 Cf. previous Note nº 69. 84

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THE GENESIS OF FLUID MECHANICS, 1640–1780

solution was correct, because of this anomaly the same cannot be said for the physical solution, as the obtained equations ignored the limits of viscosity. This latter was not a well-understood phenomenon at this time. Thus impact theory, in one or other of its variants, still has to be used. It is worth underlining that Euler only mentions the resistance problem once in his papers, and that in those same years and in following years he continued using impact theory as an explanatory model even while expressing his doubts. On this score we will analyze two of his works: one on windmills, a theme about which he proposed a mixed theory and which provoked much discussion, to which we will return in the chapter on applications. The other is a specific work on resistance in which he clearly proposes a theory belonging to those we have called hybrids, as he combines the streamlined model with the local predictions of impact theory. Euler dedicated several studies to the theme of windmills, among which are ‘Recherches plus exactes sur l’effect des moulins de vent’ (‘Very precise investigations on the effect of windmills’), published in 1756.86It is an application study, and its fundamental aim is to estimate the force that the wind exerts on the blades of a windmill. Once again Euler uses impact at the basic theoretical level, although he specifies that this is ‘not because I do not believe that this hypothesis conforms completely to the truth, but because the true law of these forces is yet unknown’ [§.I].87 He continues to argue that if, in the case of motion in water there is a certain agreement between experience and predictions of the theory, the same does not occur with effect of wind, where there is a lot of difference. Not much further on he insists on this, considering the theory as a ‘chimera’, so that [W]hen a body moves in a fluid, in the first place a certain motion is put in action by which the body pushes the fluid in front of it, this then circulates immediate around the body to fill the empty space that it has left behind it … and the resistance is none other than the excess of pressure of the fluid exerted by the front part upon that sustained by the rear part. [§.III]

Clearly, the impacts have to give way to the pressures. Euler, while bearing in mind that the real phenomenon is unknown, presents us with an alternative theory belonging to the group we have called mixed. In line with this he continues to maintain impact for the front surface, although with the formation of a wake whose effect in the rear part would be equivalent to having a pressure pt less than the atmospheric pressure (Fig. 3-19). Concerning this he says that ‘it is clear that the air does not know how to fill the spaces behind the plane’ perfectly [§.XII], ‘but in order to determine the exact value of 86 87

In the Mém. Acad. Berlin, Vol. XII. The quote between brackets refers to the ‘Recherches …’.

THE EVOLUTION OF THE PROBLEM OF RESISTANCE

p

177

t

Fig. 3-19. Mixed theory for a flat plate in air

pt, which is where the theory abandons us, we are forced to assume some estimations’ [§.XIII]. The condition that has to comply with these estimations is that where the incident velocity is zero or infinite, the rear pressure being, respectively, that of the atmosphere or zero. On this understanding, Euler presents three formulas that satisfy these conditions from among the infinite number of formulae. These are: pt =

pa 1+ b v2 + c v4

pt =

b pa b + pa 2

p t = p a e - v /b

.

[3.34]

[3.35] [3.36]

If the drag coefficient for the flat plane is CD = 1, the drag, using any one of these formulas, is: 1 D = S ( ρ v2 + p a − pt ) [3.37] 2 Euler, with his usual dexterity in calculation, plays with these formulas and says that the sum of the first two terms ½ρv2 + pa is the pressure in the front part of the plate, which can be rewritten as pa(1 + ρv2/2pa). If furthermore in this last equation the second term is small, then : 88 88

This is for the definition of the number e.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

1+

ρ v2 2 pa

ρv 2 pa 2

≈ e

[3.38]

That in combination with the exponential definition brings him to define the resistance as: ρv ⎛ ρv − p p D = S pa ⎜ e − e ⎜ ⎝ 2

a

t

2

⎞ ⎟ ⎟ ⎠

[3.39]

He declares that this more complex expression does not diverge much from the truth, although he forgets about it quite soon. For the cases in which the wind impinges with an attack angle α, he admits the proportionality of the resistance with the square of the sine of the angle of attack, but he goes back to play with the previous formulas in order to arrive at:

⎛ v2 ⎞ D = S ⎜ ρ + p a − p t ⎟ sin 2α ⎝ 2 ⎠

[3.40]

Which he says is the simplest formula, and which can be transformed into:

⎛ ⎞ ⎟ pa 1 2 ⎜⎜ ⎟ sin 2 α D = ρ v S 1+ 1 2 ⎜ ρ g b + ρ v 2 ⎟⎟ ⎜ 2 ⎝ ⎠

[3.41]

Comparing with the formulas, we can see that the different lies in the correction term which is added to the unit in brackets. Euler will use this new expression further on, but he ends up rejecting this added correction. The second work appeared in 1763 under the title ‘Dilutidationes de resistentia fluidorum’ (‘Disquisitions on the resistance of fluids’).89 Euler starts by asking himself how much resistance a body will support when moving in a fluid, which, as he indicates, ‘is a matter of maximum importance in physics as a whole, as in this world no movement exists which is immune to perturbation of 89

In the Novi comm. petrop. Vol. VIII, 1760/1 (1763).

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179

this type’ [Summ.].90 He goes on to note that this problem has been dealt with in two ways. On the first, which he calls common or the Newtonian manner, i.e., impact, he says ‘this rule [impact] diverges so very little from the truth that we need not fear a great error’ [ibid.]. The second way is the ‘true doctrine of the resistance’, that ‘the Geometricians tried to establish, through their profound investigation of hydrodynamics, [namely] the doctrine of resistance for the nature of the fluids and the pressure they exercise on the bodies’ [§.I]. Here, we come back to the Euler’s basic problem: having developed to the basic equations of fluid mechanics, he still needs to use a false theory, because of its instrumental value. Against impacts he argues that the supposition that the corpuscles bump into each other is false. He says: But what is really true is that the fluid does not impact in this way with the body, but before arriving at the body the it [fluid] is deflected both direction and velocity so that when the corpuscles reach the body they surround it sliding over its surface, and the only other force exerted upon the body is the pressure coming from each point of contact. [§.III]

Euler argues that resistance is a result of the forces of pressure existing around the body, thus the important thing is to deduce these pressure forces as a consequence of motion. He proposes the streamlined model, as is shown in Fig. 3-20 taken from his ‘Dilutidationes’, where the streamlines are shown flowing around the body. He says that the streamlines should respond to a mathematical equation, but it will be very difficult to find the analytical solution: ‘the problem is so great, and the question is so difficult that it would appear that human forces could not overcome the problem’ [§.VI]. Along these lines he quotes the Essai of d’Alembert as well as his own earlier Memoir of 1755. He concludes that in spite of the effort which he had invested in these works, at that time there was no possibility of reduction to analytic equations.

Fig. 3-20. Flow around a body 90

The quote between brackets will refer until new advertisement to the ‘Dilucidationes …’.

180

THE GENESIS OF FLUID MECHANICS, 1640–1780

Using this streamlined model, Euler proposes the following expression as the relation between the pressure and the velocity at a point: p

ρg

=k −

2

v 2g

[3.42]

In which k is a constant for the streamline. This equation is a direct application of what we would nowadays call the Bernoulli theorem. It also follows from it that ‘at a greater velocity the resistance [at a point] is less, but at a lesser velocity the resistance generated is greater, which appears to be directly contrary to the common rule’ [§.IX]. Thus according to Euler, the basic problem resides in calculating the velocity of the fluids at each point. In order to carry out this calculation he makes use of the common rule. In this respect, if the upstream speed is v0 the resistance generated in impact with a perpendicular plate would be p = ½ρv02. And following this line of thinking, for a point of the body whose tangential angle was φ, this last equation is expressed as: p=

1 2

ρ v ∞2 sin 2 ϕ

[3.43]

This indicates that the local pressure is proportional to the square of the normal velocity, i.e., he has introduced impact theory, contrary to his initial postulates, and this is perhaps justified by what he said concerning the fact that the reality of the predictions diverges little from this theory. According to our criteria, the result is a hybrid theory combining the streamline hypothesis with the local effects of impacts. Introducing this pressure into equation [3.39], we arrive at: 2 2 2 v = 2 g k v ∞ sin ϕ

[3.44]

In order to eliminate the parameter k, he supposes that at the vertex, where 2 φ = 90°, the velocity is zero, thus permitting him to determine v ∞ = 2gk, which, introduced in the previous equation leads to the following law of velocity:

v = v∞ cos ϕ

[3.45]

Velocity which only depends on the local slope of the body, is zero at vertex A and equal to the upstream one on boundary E. Thus we see him arriving at his goal, although he notes ‘that it must be remembered that these determinations are not truly rigorous, but only adequate approximations to the truth’ [§.XIV].

THE EVOLUTION OF THE PROBLEM OF RESISTANCE

181

Fig. 3-21. Closed body

Following these developments, he presents the case in which the current flows around a closed body as shown in Fig. 3-21.91 In this connection he claims ‘that certainly this rule fails in the rear part of the body, [for] if we place it as if it were in a front part, the stern of the ship would be propelled with such force as to repel the bow’ [§.XV]. Once again the paradox appears, but in an indirect and qualitative manner. In the light of the deductive process by which the rule has been arrived at [3.45], the total resistance upon the body would be expressed by the integration of the projection of the local pressure on the axis OX, this being: D=

1 2

L

ρ v ∞2 ∫ cos ϕ sin ϕ dx

[3.46]

0

This is an equation which is only zero in certain conditions, like the symmetry equation similar to the one offered by d’Alembert. However, Euler appears to commit himself only to the law of velocity given in the equation [3.45], and to the rule of the relation between this and the pressure, albeit in a qualitative form as shown in [3.42]. Indeed, he asserts that the law of the velocities must diverge from actuality in the rear part. And he continues to say that if the bow did not reduce resistance, this would indicate that the velocity would be constant in the whole of this part of the vessel; and if the velocity were reduced, the resistance would also be reduced, and even if it reached v > v0 there would be a negative pressure which could cause a vacuum in the rear part. He continues with a few notes on the advantages for naval design of the search for a stern that would alleviate the resistance on the prow. Nevertheless, he states that if experience is consulted, such a shape would be hard to find. And calling upon practice, he warns against the problems that this ideal shape could cause in the rudder. 91 This figure consists in the one presented by Euler but complemented with its symmetrical part, so as to show more clearly the set formed by body and flow.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The theory of Jorge Juan To round off this series of theories, we shall comment upon the theory presented by Jorge Juan y Santacilia in his Examen marítimo (1771), which contains novel material of some interest. This work is one of the great naval treatises of the eighteenth century, in which Juan applied Newtonian mechanics to naval engineering. Throughout the work he makes frequent references to practice, as he considers that no theory is valid without this and in this respect he accuses his predecessors of having sublime theories, but lacking a practical dimension. The basis of his theory is the action of gravity, and he uses it to justify both the action of fluid on the rear part and the existence of a unified theory applicable to both fluid dynamics and fluid statics. Referring to Daniel Bernoulli, he says: All this must seem less strange if one considers that the theories expressed do not suppose the fluid except when this is deprived of all gravity, and consequently of all pressure of one particle upon another. This has no place either in our air or in our waters; these fluids, when the velocity of the bodies is not very great, drive the bodies from behind with their remaining gravitational force. This was recognised by Newton himself. [p. XXI]

He not only quotes Newton, but a few lines further on he quotes Robins, in a footnote. m m

z p Vs+V Vs-V

a)

b)

Fig. 3-22. Jorge Juan’s model

Juan starts with the relation between pressure and depth, the well-known formula p = pgz, that clearly demonstrates the relation between pressure and gravity.92 He also uses Torricelli’s Law, v2 = 2gz, which shows that velocity is also a function of gravity. By eliminating the gravity parameter between both 92 A deeper study of this theory is found ‘La Mecánica de los Fluidos en Jorge Juan’ (‘The Fluid Mechanics in Jorge Juan’) by Julián Simón Calero.

THE EVOLUTION OF THE PROBLEM OF RESISTANCE

183

formulas he arrives at p = ½ρvz2. Juan thinks that this equation relates the pressure with the velocity vz at which the liquid would exit if, at a depth z, an imaginary orifice were made to the exterior, a phenomenon which would be due to the existing pressure at this depth. Next, he imagines a flat plate moving at a certain depth with a velocity v (Fig. 3-22a). If a similar orifice were made in this plate, the exit velocity in the front part would be the sum of v + vz, while in the rear it would be the difference v – vz. Juan states that ‘the principle which led us to this was the deduction that the outlet velocity of the fluid for the same differencedifferential would be v + vz , if it had free passage’ [Lib. 2, Prop. 36, Esc., p. 269]. Thus he conjectures that if the imaginary outlet velocity were v ± vz, the pressure would be: p=

1 2

ρ (vz ± v ) 2

[3.47]

Juan does not work with pressures, but with forces, and in this respect his fundamental proposition is that ‘the perpendicular force which a differencedifferential of surface dσ undergoes, when it moves within a fluid in a direction perpendicular to it, will be dF = ½ρ(vz ± v)2dσ, v being the velocity perpendicular to the surface’ [Lib. 2. Prop. 11]. In this expression the positive sign corresponds to the front part of the plate and the negative sign to the rear part. If we substitute vz with its value we obtain: p=

1 2

ρ ( 2 gz ± v) 2

[3.48]

which is the form that he uses in his later calculations. The basic idea behind introducing this fictitious velocity in the formula allows him to cover both the static and the dynamic cases, something which no other theory does, and it is also valid for both bow and sterns. He conserves the formula for incidences different from the normal ones, but substitutes the velocity v for vcosθ. Juan makes extensive use of this equation in his application to naval mechanics. Here we shall only apply it to the resistance of flat plate with low height in comparison to the depth in which it moves. If the surface of the plate is S, the total force will be: F=

1 2

S ρ (vz + v ) 2 −

1 2

S ρ (vz − v ) 2 = 2v ρ S 2 gz

[3.49]

184

THE GENESIS OF FLUID MECHANICS, 1640–1780

This is to say, the force is only proportional to the velocity and to the square root of the depth, which is contrary to all the theories of the time. In spite of this, Juan is not daunted, but attacks everyone, accusing them of neglecting experience. Another point to note about this theory is that it takes into account wave formation, which Juan calls ‘unevenness’. Following the model in Fig. 3-22b, his idea is based on finding the points having zero pressure on each face of the plate. Thus using the equation [3.48], we get: p=

1 2

ρ

(

)

2 gz ± v

2

=0

[3.50]

Whose solution is:

δm =

v2

2g

[3.51]

That would be the height of the wave in the frontal part, and the trough in the rear part. This equation [Prop. 20], together with the height of the plate jutting out of the water, leads him into many cases which are not worth repeating here. The problem of the associated waves was known, but no one had tried to solve it; only Frederik Henrik af Chapman, who was also a naval constructor, had something to say about this. To conclude, we note that Juan also applied his theory to sails. For this he assumes that the sails were submerged in an ‘ocean of air’, whose depth is the height of the layer of air equivalent to a uniform density atmosphere [Vol. 2, §.256]. The conclusions arrived at by Euler in his commentaries to Robins are repeated. In this connection, it is worth pointing out that this particular result on the flat plate coincides with the notes Euler made on Robins, and that there is a certain parallel between both formulations.93 But Juan always acknowledges his obligations to outside contributions, yet makes no mention of this whatsoever,94 in which case it may just have been a coincidence.

93

Cf. previous Section ‘Robins’ New Principles of Gunnery’. Although the Examen Marítimo was published in 1771, we think it was written about 1750, and he probably started it when he was in London in 1749, which is justified by the analysis of the text itself. The original work of Robins was edited in 1742 and Juan was familiar with it. Euler’s commentaries were published in 1745, but in German, a language not spoken by Juan. Nevertheless, it is very likely that Robins had a copy of the German version, and Juan could be aware of Euler’s notes. Jorge Juan was elected member of the Royal Society on 9 November 1749 and he had been proposed on the 6th of April of same year. Among the proponents was Benjamin Robins. However, all this must understood only as a guess. 94

Chapter 4 Experiments on Resistance

In the previous chapters we have witnessed the existence of an experimental world in intimate relation with the theoretical world. Experimentation, which is a questioning of nature, is not a neutral or aseptic activity, but one that in itself implies a theory of the experiment and a theory of its interpretation. The first includes apparatus, circumstances and observers; the second deals with the reading and evaluation of the results, which are not normally as clear or immediate as to allow radical conclusions, requiring instead analyses which are generally complex. These extremes, briefly noted, are especially notable when it comes to the experiments of resistance carried out in the eighteenth century. The discrepancies between theories and measurements were considerable, and it is interesting to see how the experimenters tailored the measurements, interpretations or apparatus to the results they were seeking. However, in the long run, experimental measurements took precedence, and as a result impact theory was discarded in an almost traumatic manner. Nevertheless, it took many years to arrive at an understanding of the multiple aspects of the phenomena of resistance. If from our historical vantage point we observe the progress of fluid mechanics in the eighteenth century, we see that there is a discontinuity in the experimental activity towards the end of the decade of 1750s. Before this, physics experimenters abounded, such as Newton, Bernoulli and others. Afterwards, more ‘purer’ experimenters proliferated, like Jean-Charles de Borda and Charles Bossut. This change is connected with the coming of what we have called the grand theorization, which came to a head around the middle of this decade. As Roger Hahn notes, there was a crisis precipitated by the inability of the hydrodynamic theoreticians to provide a reply to the practical problems.1 This leads him to declare that the second half of this century was marked by an effort of integration with sociological roots. The feeling of the time was not far removed from these preoccupations, and as an example we transcribe the reflections of Bossut: Euler had dealt (after d’Alembert) with the same subject with a depth and abundance that, in the present day state of analysis, does not allow us to go further 1

This is one of the basic thesis defended by this author in L’hydrodynamique au XVIII e siècle.

185

186

THE GENESIS OF FLUID MECHANICS, 1640–1780 without considering the problem under a somewhat different point of view. (Berlin Academy. 1755; St. Petersburg Academy, 1768, 1769, 1770, 1771). Unfortunately, in a manner of speaking, all these formulas do not offer more than speculative truths; and when physical applications are required, it is therefore necessary to simplify them, or to de-naturalise them by employing different suppositions that it would have been equally worthwhile establishing the calculations from the beginning using less strict principles.2

The position of Bossut is illuminating: hydrodynamics is not separated from the applications, and these demand solutions. There were also other causes of the experimentation boom. The theory is not wholly removed from experimentation, since the equations respond to models that participate in both the conceptual abstractions and the observed facts, as abstraction and reality are profoundly interwoven. Therefore, observation, comparison and accurate and rigorous measurements of the phenomena are necessary steps in the development of hydrodynamics. However, the task was slow, arduous and costly and it was necessary to wait until the following century to reap the fruits of the new science. Hahn recalls the words of Alexander Koyré concerning the historical moment in which ‘experience’ was transformed into ‘experimentation’ and hydrodynamics became an ‘active science’. The experimenters In the previous chapters we have seen how many of those whom today we would call physicists did not confine themselves to constructing theories, but were also the originators of experiments associated with their theories. Such were the procedures of Newton, Mariotte and Robins. As regards the rest of the ‘geometricians’, at one time or another they have used the results of the experiments of others, as is the case of Euler or of d’Alembert. In this chapter we shall include the experimental work of three very significant men: Borda, Chapman and Bossut. This will as a complement to the experimental works already described in the previous chapters, which were carried out in direct conjunction with theory. Of these three, the first focused his experiments on the phenomenon of resistance per se, using simple geometrical models in his experiments, Bossut’s models were complex and focused on applications, and Chapman falls between the two. As regards the scope, the last two directed their efforts to maritime and navigational aspects employing authentic towing apparatus. By contrast, Borda operates at laboratory level. 2 Traité théorique et expérimental d’hydrodynamique, cf. Second Part, Chapter V, §.242. The first edition was in 1775 and the second in 1786.

EXPERIMENTS ON RESISTANCE

187

Centring the argument on only three figures does not mean that we have forgotten the labors of many other men; we have chosen three in order to illustrate more clearly the techniques and methods of the time. There are differences among them, but also similarities: the three men carried out their work in the second half of the century, using well-designed and constructed apparatus, and employing well-elaborated methods of reducing data. As regards later authors who come close to them, we must recall Nicolas Charles Romme (1787) and Mark Beaufoy (1794).3 The experiments of Borda Jean-Charles de Borda, better known as the Chevalier Borda, was the author of several theoretical and practical contributions to fluid mechanics in the 1760s, and his experimental work figures among the most elaborate of the entire century. The results appeared in two reports with the same title ‘Expériences sur la Résistance des Fluides’ (‘Experiments on the Resistance of Fluids’) in the Mémoires de l’Académie of Paris. The first appeared in 1763, the second in 1767, although in reality they can be considered as a single work. Their object is to measure the resistance supported by moving bodies with simple geometric forms in air and water. The first report is almost completely devoted to air, ending with a sketch of an apparatus for water which he uses for this medium in the second report. The first begins with a few words recalling that the theory adopted by the majority of the geometricians in order to explain the phenomena had been that of Newton,4 about which he says: That theory even served as the basis for several knowledgable investigations on shipbuilding and Navigation, but famous geometritians considered it to be very uncertain. They had seen how difficult it was to submit to calculation the laws of resistance of an infinite number of fluid particles, which all strike each other in a different way on an opposing surface, and these laws seem to them to be too complicated to be in accordance with the simple rules of the ordinary theory. [p. 358]5

3

Romme used Bossut’s results complemented with additional experiments. It is remarkable that one of the models was a exact copy of the ship L’Illustre. Beaufoy, a British army officer, carried out a very extensive task under the auspices of the Society for the Improvement of Naval Architecture. His apparatus recorded the velocity automatically by means of a mechanism linked to a clock. This apparatus was in use until 1798 and with it about 10,000 trials were made (Cf. Stoot in ‘Some aspect of Naval Architecture in the eighteenth century’, 1959. pp. 31–46). 4 Borda quoted the Principia’s Prop. 34, which the one concerning resistance in a rare medium. 5 Until new indication the quotes between brackets will refer to the 1763 Mém. Acad.

188

THE GENESIS OF FLUID MECHANICS, 1640–1780

It is a recurring question: what they imagined real behavior to be, and the impossibility of dealing with it mathematically. Faced with this, Borda declares that ‘It would be useful to know if the ordinary theory strays far from the truth, and if it can be used without considerable errors in the practice of the arts depending on it.’ [p. 358] As regards the work he is going to present, he says that the experiments that he will contribute are not many in number, although they are very accurate and cry out to be extended. In this respect he states: Perhaps a well-chosen set of experiences, made with precision, could supplement in some way the true theory: this method would even have the advantage of carrying with it a more general conviction that illustrates it better, because the facts inspire almost all men more confidence than the calculations. [p. 359]

Here he demonstrates an excessive faith in that facts supplement a theory, and gives us to understand, quite clearly, the uncertainty in which the men of his time were affected as regards the problem of resistance. We present his work in two parts: the first concerns air, and corresponds to the report of 1763, and the second part deals with water, and except for some rough drafts, corresponds to the report of 1767. In order to perform the measurements of the aerodynamic resistance, he built the apparatus shown in Fig. 4-1. It consists of a rod CB that turns round an axis AE which is also joined to a drum acting as a winch, around which a cord with a weight is wound. Two equal specimens whose resistance will be measured are located at the ends of the rod. Its functioning is quite simple. When the weight descends the axis turns and the whole accelerates until the moment produced by the weight is equal to the one due to the resistance. From this instant the system turns at a constant speed6 that is easily measured. The force in each one of the two specimens is determined by the geometry of the apparatus and the rotary weight. There is a remarkable similarity between the principle of this machine and that of Robins described in the previous chapter. Borda was very conscientious. First, he studied the dynamic behavior of the apparatus, and he found that the period of acceleration ended before the fourth turn, where the stationary condition was already produced. On the other hand, he estimated the friction forces experimentally, finding them to be negligible, and the effect of the aerodynamic resistance of the rod, which also turns out to be proportional to the square of the velocity. 6

This is a problem equivalent to the limit velocity one. Strictly speaking, this equality is never reached, but there is an asymptotical approximation. However, in practical terms, that condition is reached in a few turns of the blades.

EXPERIMENTS ON RESISTANCE

189

Fig. 4-1. Borda’s apparatus

In order to measure the velocity he marked two points on the cord: the first became visible when the cylinder had made the first four turns, and the second 22 turns later. He determined the time by counting the oscillations of a pendulum that beat in half seconds, in its swing between the two marks. He repeated the measurements several times, finding the differences to be less than half a vibration, i.e., 0.25 s. He even registered the temperature and the atmospheric pressure of the test days: 4° Réamur and 28 pouces (5°C and 1,011 mb), thus demonstrating a concern for accuracy and the repeatability of the tests. As he had doubts as to whether the circular movement could induce another in the surrounding air that could falsify the measurements, he made a third mark on the cord the same distance from the second one as this was from the first, and in several of the tests he measured in addition the time taken to pass between the last two. He affirms that the time was the same as that taken to pass the first two, which removed his doubts, and led him suppose that the movement of the specimens located at the ends was as if they were moving in a rectilinear fashion. Borda’s aim was to analyze simple geometric forms with the aim of establishing comparisons. To this end he studied the resistance of plates, prisms, pyramids, cones, semi-cylinders, spheres and semi-spheres. We shall revise the most important, giving at first some details of the measurements in order to make his qualifications as an experimenter quite clear.

190

THE GENESIS OF FLUID MECHANICS, 1640–1780

Square plates He made the first sets of tests with three square plates whose sides were 9, 6 and 4 pouces (244, 162 and 108 mm) dragged by weights of 8, 4, 2, 1 and 0.5 livres (3.916, 1.958, 0.979, 0.489 and 0.245 kg). He corrected the results, counted in half seconds, in order to compensate the effect of lengthening the cord, which, according to his calculations affects less than 0.7 oscillations in the most unfavourable case. Figure 4-2 reproduces his table with these results already corrected: the first column represents the weight and the other three are the vibrations measured, which are the semi-seconds inverted in 22 turns.

Fig. 4-2. Table of results

From the measurements shown in the former table he first deduces the proportionality of the force with the square of the velocity, which as he tells us is plain to see, as when the drag weight is reduced by four, the velocity multiplies approximately by two. The next step is to deduce the absolute resistance generated by the plates. To do so, the parasitical resistance of the rod must be subtracted. Therefore, he dismounted the plates and made the rod turn freely dragged by weights of 2, 1 and 0.5 livres, measuring 26, 27 and 52 oscillations, respectively, that as he makes clear, also follow the law of the square of the velocity. With the help of these data, the dimensions of the apparatus and of the table data, he reduced the absolute forces7 which are shown in another table in his Report [p. 364]. Here we have chosen to show them in the form of the resistance 7 The reckoning is as follows. If W is the applied weight and r the drum radius, the total moment of that weight will be C = Wr, which is two-fold in the corresponding to the resistance of the rod and the resistance to measure. This one will be 2DR = Wr – Cf, being D the absolute resistance. The corresponding to the rod will follow the law Cf = kf ω2; for determining this constant he takes the case when it turns with the rod only (from the three cases with 2, 1 and 0.5 livres which give 26, 37 and 52 vibrations, he chooses the third one). Designating it with the sub-index 0, kf = C0/ωo2 will be obtained, which resolves the problem.

EXPERIMENTS ON RESISTANCE

191

coefficient (Fig. 4-3) as they are thus easier to interpret.8 The CD is practically constant for each of the three plates, which indicates the proportionality with the square of the velocity. But, also, as Borda emphasizes, the CD increases with the size of the plate, which means that the forces grow in greater degree than the proportionality to the surface. This does not altogether satisfy him, as he explains that cannot be accounted for by measurement errors, and concludes ‘that the resistance of the flat surfaces that move in the air with equal velocities, grows in more proportion than the extension of these surfaces’ [p. 365].9 2

Drag coefficient (CD)

Plate 9 pul

1,8 Plate 6 pul

1,6 Plate 4 pul

1,4 1,2 1 0

2

4 6 Velocity (m/s)

8

10

Fig. 4-3. Resistance coefficients of plates 8

The data has been taken from the Table of p. 364. We note that this table has an erratum, because for the 9 pouces surface and 7.57 pieds/s velocity the resistance should be 0.0773 livres, not the 0.733 indicated. Obviously it is a typographical error. 9 After reflecting on this matter and not finding any explanation, I consulted with my brother Javier Simón (doctor in aeronautical engineering, expert in flight physics and with more than thirty years experience in aerodynamic design of modern aircraft), who after a in-depth analysis of the Borda’s text did not reach anything conclusive or clear. In the first place, the experimental value of the resistance coefficient for a squared flat plate diminishes down to 1.14, which is reached when the Reynolds number greater than 1·103. All the Borda’s cases are far greater (1.5·104 up to 6.0·104), as therefore they have to approach that value. Another possible explanation for his measurements could be the non-uniform distribution of velocities over the plate. However, the effect is not significant because the variation of the mean squared velocity compared with the impulsion center is very small for the given geometry. Repeatedly examining the question, the only plausible explanation would be that Borda took the velocity of the end of the rod instead the plate center velocity. In this assumption, which is improbable because he declares explicitly that he takes the velocity of the “impulsion center”, the difference encountered of 17% between the big and small plate would be reduce to 5%. At the end of all these unfruitful explanation attempts, Javier Simón concluded asserting, not without a certain irony, that after so many experiments the figure found by Borda was further from the reality (CD = 1.14) than the one found by his peers.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Prisms With the object of checking the validity of the law of the proportionality of the resistance with the square of the sine of the angle of incidence, he substituted for the previous plates two sets of prisms (Fig. 4-4) with a square base of 4 pouces (108 mm) and semi-angles in the vertex of 30° and 45°, that move with a weight of 2 livres (0.98 kg). The results he obtained were 57 and 65 vibrations which he compared with the 75 found in the case of the flat plate, before discounting the effect of the rod, which for this weight inverted 22 vibrations.10

Fig. 4-4. Prisms

From the reduction of these measurements he extracts the result that the relation of the resistances between each prism and the flat plate is of 0.728 for the 45° prism, and of 0.520 pouces the case of the 30° one. Now, if the hypothesis of the proportionality with the square of the sine was true, the resistances ought to be 0.5 and 0.25 instead of those found. This makes him say that the ‘resistances are not proportional to the squares of the sine of the angles of incidence, rather they are almost proportional to the sine of these angles’ [p. 367]. Pyramids This is an extension of the previous case. He placed two sets of pyramids instead of prisms, but with the same angles in the vertex. The result is the same: the resistances appear to follow the law of the sines instead of the squares.

10

Now the procedure he uses for discounting the parasitic resistance of the rodr is different, because what he makes is to find what would be the turn velocity if that resistance had not existed. For one case let ωm be the velocity measured, then C = kd ωm 2 + k f ωm2. But for this very torque C, the rod rotating freely the velocity ωf was reached, so that C = kf ωf2, where kf is obtained. Now, if this last resistance did not exist, the rotating velocity would be ω0, so that C = k0 ω0 2. Combining both formulae he obtains the following one: 1/ω02 = 1/ωm2 – 1/ωf2, and as he measures times which are inverses of the velocities, he finishes in the simple equation t02 = tm2 – tf2, that indicates the number of vibrations that he would have measured if there were no resistance in the rod.

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Spheres and hemispheres Following the same method, he placed two spheres of 4.5 pouces (122 mm) in diameter that he made turn with a two livres load. The resistance coefficient obtained from the results that he shows is CD = 0.561. The second step was to substitute hemispheres for the spheres, and he finds that when the current impacts on the rounded part, the result obtained is exactly the same as for the spheres, thus leading him to conclude that the bottom parts have no influence. On changing the direction, and impacting the current on the flat part, the results of the plates are repeated with a CD = 1.368.11 He compares his results with those found by Newton in the falling spheres, attributing the differences between both sets of results, which he says are a fifth part, to possible temperature differences that could alter the properties of the air.12 The tests with his rotating machine finish at this point. In addition to these, he describes how he had carried out many more experiments in water, but the majority gave poor results, and he only quotes the ones that appear to him to be accurate. These were the experiments concerning the towing of floating cubes and the rotating machine. Towing of floating cubes The experiment is simple. He constructs a cube whose side is one pied (324 mm), which he loads suitably, so that it submerges one pied in the water, and it is towed it by a weight connected to a cable running through a pulley. After making a series of measurements he observes that the law of the square of the speed continues to be hold. He analyses two towing conditions: one when the advance direction is parallel to the axis of the cube, and another to the diagonal. He finds that the resistance increases in the latter case, instead of becoming less as the impact theory predicts, since, according to this theory, thee should be a reduction by the factor of 1 / 2 , whereas in the experiments it increased by 1.32 times.13 At this point, Borda reflects that if the proportionality in this theory was with the sine of the angle, as would appear in air, the resistance of the cube 11 According to current knowledge, and as the trial velocity was v = 5.97 m/s, the Reynolds number was Re = 4.9·105, indicating that the sphere is in the transition zone between the first and second regime. The result of CD = 0.47 corresponds to the former. That justifies the equivalence with the semi-sphere case. See Chapter 2, note 51 of this book. 12 To justify a variation of 1/5 a large difference of temperature is required. Nevertheless, the mention of this possibility shows his methodological preoccupation. 13 The result, although doubtful, is explainable. If it is true that the resistance coefficient diminishes in the case of diagonal advance, the frontal section presented increments, thus increasing its absolute magnitude.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

would always have to be the same,14 which does not happen either. He attributes the possible causes to the fact that in one case the body is submerged in the air and in the other it is floating in water. Likewise he sets out to discover whether it follows the law of densities between air and water, and finds that it does not. Rotating arm in water At the end of the 1763 Memoire, Borda presents a sketch of a new rotating machine for water, somewhat similar to the one he used in the tests with air, and which he will use for the experiments he presents in the 1767 Memoire, shown in Fig. 4-5. It consisted of a horizontal rotating arm BC that carried the specimen to be tested at one end G, which moved in a tub full of water. The wheel was driven by a weight P, using a cable wound inside a drum, whose diameter he found to be 11 pouces and 11 ½ lignes (161.2 mm). The diameter of the tub AD was 12 pieds (3.9 m) and the point at which the specimen BC was fixed was 8 pieds (2.6 m). He did the majority of the work with a sphere of 59 lignes (133 mm), either whole or divided into two halves, which he tested in two positions.

Fig. 4-5. Borda’s rotating arm

The driving forces varied from 4 onces (122 g) to 8 livres (3.92 kg). The time was measured by a pendulum that beat semi-seconds, and registered the vibrations during two revolutions of the arm. His methodology was as careful as that used in the case of air, identifying the magnitude of the friction in order to discount its effect on the measurements. 14

If the current makes an incident α with one of the faces, it will make ½π-α with the other one. The first one will present a frontal surface as sinα and for the former proportionality the total force will be as sin2α. This occurs in a similar way with the other face, that will be sin2(½π – α) = cos2α. As result, the total will be sin2α + cos2α = 1. If the proportionality with the squares of the sines were followed, the last equation would have resulted in sin3α + cos3α, a function which has a maximum for α = π/4.

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Fig. 4-6. Table of measurements in water

The table of measurements prior to the corrections [p. 497]15 is reproduced is Fig. 4-6. The driving force is in the first column, and the other three are the number of vibrations during two turns for the hemisphere with the flat front, the convex one and the complete sphere. The coincidence of the measurements of the last two columns leads him repeat that the resistance is due only to the shape of the front part of the body. In order to find the absolute value, he took the case of the 2 livres weight and the columns of the hemispheres. The corrections convert the 132 1/7 and 83 vibrations into 132½ and 83¾, respectively. With the geometry of the apparatus he found them to be equivalent to 0.124 livres (0.595 N). He compares this value with what given by Newton, which was CD = 0.5, that according to his calculations corresponds to 0.1127 livres. He therefore concludes that ‘thus the result of my experience is as to that of Newton, as 1240 is to 1127, or as 10 to 9 at the most’ [p. 499]. As regards the relation of the plate with the sphere he gives a relation of 5 to 2. We have reduced all the data presented by Borda, showing them in Fig. 4-7,16 in which the results corresponding to the hemisphere were omitted, for their values were almost identical to those of the complete sphere.17 On the other hand, we draw attention to how he continues to make use of the experimental results of Newton. To complement to the previous measurements he also investigated the effect of the depth in which the sphere moved upon the resistance using the same apparatus. As in all the former experiments, the upper part of the sphere was 6 pouces (166 mm) below the surface of the water; he complemented them with a new series in which he went to 3 lignes (7 mm). The results he obtained [p. 500]

15

The brackets in this section indicate the second Memoire, en the Mém. Acad. 1767. The values have been taken from table p. 498, which are those of p. 497 corrected for friction. 17 We remind here again what has been said for the two regimes of the sphere. See Chapter 2, note 51. 16

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THE GENESIS OF FLUID MECHANICS, 1640–1780 1,6

Drag coefficient (CD)

1,4

Flat hemisphere

1,2 1 0,8

Superficial

0,6

Sphere

0,4 0,2 0 0

0,2

0,4

0,6

0,8

Velocity (m/s)

Fig. 4-7. Resistance of spheres in water

are also shown in Fig. 4-718 where it is evident that while it is at the surface, the resistance is greater, and it increases quickly with the velocity. He concludes: ‘Although only a little and not very extensive knowledge exists as to the theory of resistance of fluids, nevertheless it is easy for me to agree with these two facts that I have just emphasized’ [p. 500]. For the first of the facts, the increase in the resistance, he alludes to his Memoire on the movement of fluids,19 in which he said that a loss of live force must be admitted in the calculation of the resistance, and he now sums this up by stating that: ‘this resistance is always equal to the sum of the live forces lost in each instant’ [p. 501]. When the body is submerged at great depth, the quantity of water displaced by the sphere when it moves is less, as the water surrounds it completely, whereas when it advances superficially, the water displaced cannot go over the upper part. For the second fact, which was the increase of the resistance with the velocity, he says: ‘when a body moves on the surface of the water at great speed, a hollow is formed behind it into which the fluid is rushes’ [p. 501]. As the velocity increases this effect is more noticeable, whereas when the depth increases this does not occur.

18

In this figure only the results corresponding to the sphere towed at the surface have been presented, because those of the submerged are almost coincident with those obtained in the previous trials. 19 Possible he refers to his ‘Mémoire sur l’ecoulement des fluides par orifices des vases’, Mém. Acad. Paris, 1766.

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The quality of Borda’s work and methods, and the care he takes in reducing the data are striking. At this point in the century it was clear that only control and rigor in the conditions and performance of the experiments could lead to results usable in theoretical comparisons. We extract the following three conclusions from his experiments: • The verification of the hypothesis of the proportionality of the resistance with the square of the velocity. • The increase of the resistance of flat surfaces in air is in greater proportion than that of their own surface. • The falseness of the law of proportionality of the resistance with the square of the sine of the angle of incidence. We particularly wish to quote the closing paragraph of his second Memoir [p. 503 ]: That the ordinary theory of the impact of fluids yields only results that are absolutely false, that these results are far from the truth, and that in consequence it would be useless, apart from dangerous to wish to apply this theory to the art of shipbuilding.

This paragraph expresses his habitual precaution, but does not offer any alternative. Chapman’s experiments In 1775, the Tractat on skepps-byggeriet (Treatise on Shipbuilding) was published in Stockholm, written by Frederik Treatise Chapman,20 author also of the Architectura navalis mercatoria (1768). Although the work of Chapman is concerned with ship design and construction, it contains a chapter21 in which he presents a set of experiments dealing with the resistance of bodies in water, which he includes in order to confirm his method of calculating the ships’ resistance to motion. The part of the section that interests us is just a few pages, plus a table and a figure,22 which is little compared to the total extent of the work. But despite Chapman’s words, what he 20

The Tractat was translated to French in 1779 and 1781 by M. Vial du Clairbois. We follow this translation. Respect to the Architectura navalis mercatorta, we note that is worth consulting this book , not only for the technical aspect but also its truly magnificent illustrations. 21 Tractat, Cap. IV, §.15. 22 In the French translation it is enlarged with some comments.

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has done is more an insertion of something which he should have done previously, as it has no relation either to what comes before or to what comes after, and it does not confirm his calculation method as was intended. Proof of this is that the models that he tests are not ships hulls, as one might infer23 from the context, but bodies of revolution that move almost completely submerged. However, in spite of the brevity of the section, and its being almost a parenthetical comment in the work, the significance of what Chapman does is due to his methodology and to the fact that his works are some of this first of this kind.

Fig. 4-8. Chapman apparatus

He performed the tests in a basin using an apparatus as depicted in Fig. 4-8, where the model moved attached by two cables hanging from two opposite towers bordering the pool and separated by a distance of 100 Swedish fots (29.7 m) between them. Each one had a system of pulleys for driving the cable and a weight to produce the force: one tower generated traction tension in the cable, and the other braking tension, so that the action of both upon the model followed a rectilinear trajectory and produced any side-to-side motion. He made two marks on the cables, each separated by 74 fots (21.98 m) that enabled him to determine the times, which he measured using a clock. He tested seven models, all of wood, whose plan-view shape can be seen in the heading of the table of results that we reproduce in Fig. 8-9. The first three consisted of two parts consisting in two paraboloids of revolution at a tangent to their widest point, which was in the middle in No. 1, at 2/7 in No. 2, and at 1/7 23

In the text, after describing the dimension of the bodies he defines them as “all round” (aldeles runda in Swedish), which leads to confusion, as occurred to the French translator, who did not capture the nature of the models.

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in No. 3. In numbers 4 and 5 he substituted the parabolic shape of the stern for a cone; in number 6 both bow and stern were conical; and in number 7 the model took the shape of a inverted cone. He performed the tests moving the model in two directions, so that there was a total of 13 possibilities. Except for No. 7, the length of the models was 28 tum (693 mm) and the width 8 tum (98 mm). Number 7 had the same width, but was only 20 tum long (508 mm). Regarding the weight he says ‘as these bodies are lighter than water, they were loaded with lead until the weight was almost equal to that of the sea water, so that they could hardly float and the axis was parallel to the surface of the water’.24 He does not specify that they were solids of revolution, but the analysis of the weights confirms this fact.25 Moreover, if we attend to the reproduction of the apparatus we can see a dotted line that appears to indicate the water level.

Fig. 4-9. Table of results

He performed four tests with each body, the first three with traction masses that were a fraction of the mass of the model,26 specifically 3/4, 1 and 3/2, keeping 1/2 for the counterweight, whereas in the fourth the traction mass was 37 skålpund (15.65 kg) and 12 (5.08 kg) for the counterweight. He showed the results in transit times between the marks in the table depicted (Fig. 4-9). 24

Ibid. In this point the French translator, thinking he was talking about ship hull models, says in a footnote: ‘However one must believe that they floated sufficiently for the elevation of water in the front part to take place, in the case of high speed, without submerging it.’ 26 The model masses are written in the upper part of the table in Swedish skålpund. These masses are 27 skålpund (11.42 kg), 22 (9.31), 19.75 (8.35), 1.75 (7.09) and 12 (5.08). 25

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THE GENESIS OF FLUID MECHANICS, 1640–1780

As far as accuracy in concerned, he declares that he repeated each case six times and took the average, but that the differences never exceeded a half seconds. Chapman draws three exclusively qualitative conclusions from the results: 1. If the motion is slow, the body goes at higher speed when the most pointed part is at the bow. 2. If the velocity is high, the body moves more quickly when the blunt end is foremost. 3. If the velocity increases by a certain value, the position of the body makes no difference. Really this is not much, and it is difficult to imagine that he did not make any additional calculations.

T mt g

m

D

mc g

V

Fig. 4-10. Physical model of Chapman’s apparatus

In this respect, we have tried to exploit his tests further by reducing the results to resistance coefficients, with the cross section as reference surface. We must confess that several difficulties were found in the initial attempt. The first arises from the mechanical configuration of the towing system employed. This can be likened to a diagram of masses, blocks, and tackles such as the one represented in Fig. 4-10. Due to the configuration of the blocks and tackles, a horizontal displacement of the model would have repercussions on the movement of the weights at a height equal to this displacement, divided by the number of cables attaching it to the pulley. In order to find out how many cables there were, the only thing we have available is the figure that he included in his work, reproduced in Fig. 4-8, and about which he says that ‘everything is as shown in the figure’, although a clear-cut reply cannot be obtained from his examination. It seems that there are four cables, but we are not completely sure of this. The equation governing this mechanical system is the following:

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1 g 1 ⎡ ⎤ 2 ⎢⎣( mt + mc ) n 2 + m ⎥⎦ v = ( mt − mc ) n − 2 ρ SCD v

201

[4.1]

In which mt and mc are the traction and the countertraction masses, respectively, and n the number of cables. The solution for the speed is: v = vl tanh

ρ SCD vl t 2me

[4.2]

Where me = m + (mt+ mc)/n² is an equivalent mass. The previous formula indicates that the velocity starts from zero, and increases asymptotically towards a limit velocity vl defined as: vl =

2( mt − mc ) g n ρ SCD

[4.3]

The practical time in which this velocity is reached is relatively short, thus Chapman is justified in taking the velocity as a constant. Important factors that are not considered in the model, but which influence the results, are the friction, the inertia of the cables, their flexibility, etc. In this respect Chapman points out ‘the number of pulleys over which the cable must pass renders the experiments less accurate because of the friction. But as this friction is the same for all the circumstances, the variations in the speed must be of the same type’.27 This is very optimistic, as we shall see. The motion of the water that goes over the flotation line is also an influence, as Chapman himself notes. For the evaluation we have considered the first three models only (A, B, C), due to their geometrical similitude, which, together with the four traction cases, give rise to five sets of tests, as models numbers 2 and 3 are also tested inverted. The results found, expressed in the form of CD by the formula [4.3] is shown in Fig. 4-11, in which each symbol corresponds to a different model. It can be seen that the data are not distributed following a pattern according to the shape of the body, but rather they are distributed according to the traction masses. This means that there were dominating perturbation elements in the experiments, which would (in modern procedures) disqualify the measurements. On the other

27

Ibid.

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Drag coefficient (CD)

0,8 Model D Model B Model A Model C Model E

0,6

0,4

0,2 0,5

1

1,5

2

2,5

Velocity (m/s)

Fig. 4-11. Evaluation of Chapman’s results

hand, the values obtained tend to be high, as the order of magnitude, by the light of present-day knowledge, would be approximately CD = 0.15, which corroborates what we have already said about the errors. To conclude, we say that the concept was advanced, but the methodology, the theory of the instrument, and perhaps the construction of the apparatus, were not very well worked out. The experiments of Bossut In 1775, Turgot, who was at the time the General Manager of Finance of France, directed d’Alembert, Condorcet, and Bossut to study improvements in French inland shipping.28 As the Prologue of the work Nouvelles expériences sur la 28

This commission was related with the problems that arose in the Picardy canal. A detailed analysis of the controversy with its social, technical and political implications is given by Pietro Redondi in ‘Along the water: the genius theory. D’Alembert, Condorcet and Bossut and the Picardy canal controversy’, 1997. Let us say a few words about the matter following Redondi’s work. In the eighteenth century navigation by rivers and canals was very common. By the middle of the century France had a extensive canal system connecting the rivers. However, in the north, in order to link the Paris Basin with Flanders it was necessary to connect the Crozat canal, completed in 1738, with another canal from Saint-Quentin to Cambrai. To do that it was necessary to cross the heights of the Vermandois, a distance on the map of 13.68 km. This link was not only of a very high commercial value, but also of military interest, and was called the Picardy canal. The first project was due to De Vicq, who in 1727, proposed to dig two underground canals of 5.5 and 1.1 km. The second one was due to other brilliant engineer, Joseph Laurent, who proposed a single tunnel of about 14 km. His project had several innovations going far beyond the standard practical hydraulics. The technique was bored shafts situated at 200 m from each other, which maintained ventilation and water supplies. The tunnel had 20 pieds (6.48 m) in width, including two lateral footpaths of 2 pieds (0.65 m) each for towing the ships. The excavation of this impressive work began in 1768, and in 1773 two thirds of the length had been dug. But the work did not lack criticism

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résistance des fluides (New experiences on the resistance of fluids) declares, they were concerned to analyze the problem of resistance, for which Turgot provided them with the economic means, and allowed them to use the basin of the Military School in Paris. The aforementioned prologue, written by Charles Bossut, also known as Abbot Bossut, says: Before applying geometry and calculus, we thought we should consult with experience, be it in order to verify the known elements of theories concerning this subject, or be it to procure data that could serve as a basis for a new solution. [Prol.]29

This recommendation is said to come from d’Alembert, whom Bossut believes to have already solved the problem by a new and direct analytical method. The book gathers together the results and comments of all this work. It seems that d’Alembert only acted in an advisory capacity, and it was the other two who carried out the real work.30 It was written by Bossut, and a ‘test on the method in order to find the laws of phenomena from observations’ whose author is said to be Condorcet, is included at the end. As regards the problem of resistance, in the Nouvelles expériences it is states that: The resistance suffered by a solid body that divides a fluid may be the most important problem of hydrodynamics, be it due to the difficulty, be it due to the applications to naval architecture, the construction of docks, to hydraulic machines etc. [Prol] based on the weakness of the vaults and tow paths, the strong air current, and the financial problems. Laurent died in October 1773 and was succeeded by his nephew and assistant in January 1774. But in August the same year, Turgot reached the position of Minister of Finances, and he entrusted Condorcet with the examination of the France’s waterways, specially the Picardy Canal. As result of his reports the canal works were interrupted in October 1774, and the controversy involved all of the French institutions. In this situation Turgot established a commission composed of d’Alembert, Condorcet and Bossut, all members of the Académie Royale des Sciences, in order to seek advice on these matters. D’Alembert was chosen more for his notoriety in order to give the commission some political respectability, and the resulting works bear little of his imprint. One of those works was the Memoires we are presenting here; the other is the report Observations sur le canal de Picardie in July 1776, which was never made public (it is included in the Redondi work). They do not recommend the Laurent tunnel for several reasons, one of those being the increase of the resistance by a ratio of 5:3. They preferred an open canal through the river Sambre even though it was much longer than the proposed canal. We do not enter into more detail, except to say that the commission members were even accused of being traitors to France. The question of the underground canal was opened several more times until July 1802, when under Napoleon Bonaparte it was decided to carry out the project, opting for De Vicq’s project, i.e., two tunnels. Finally, Napoleon inaugurated the canal on 28 April 1810, and still it is in service today. 29 The calls between brackets are refer to Nouvelles expériences. 30 J. Morton Briggs says that he only put his name on it. Cf. his article about d’Alembert in the Dictionary of Scientific Biographies, Ed. Charles C. Gillispie. However, according with it is said in note 28, we guess that the political role of d’Alembert should be very relevant.

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This is followed by an acknowledgement that none of the proposed solutions to this problem had solved it: some formulae were simple but imperfect; others based more on the properties of fluids, led to complicated calculations useless in practice. They divided the research into two parts: one, directed towards resistance in what they called undefined media, and another for movement in narrow channels. The first class corresponds to ships sailing on the sea or on large and deep rivers; the second refers to shallow rivers or narrow channels. The interest shown in the latter arises from the fact that apparently more force is required to navigate in them than in unlimited fluids, and in this respect they were concerned to find out whether ‘the rowers were right when they say that its requires much more effort to move their ships when the waters of the river are lower’ [Prol]. This fact is further explained in a second report, with the same title, that appeared in 1778, and which strictly speaking, is the second part of that published in 1777. The installation was built in a basin of some 100 × 53 pieds (32.5 × 17.2 m) in the middle, and of a variable depth up to a maximum of 6.5 pieds (2.1 m). It is reproduced in Fig. 4-12. On one side a vertical mast of 76 pieds (24.7 m) was located, provided with a double set of pulleys as depicted in the figure. These served as the guide for a cable that towed the model on one side, and on the other carried a weight generating the motion. Due to the dimensions of the basin, the maximum run of the pulley was 66 pieds (21.4 m). The basis of the system was simple and similar to that of Chapman. The tension in the cable towed the model with an accelerated motion until it reaches the limit speed when the resistant force equals the weight.

Fig. 4-12. Test basin

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The preparation was meticulous. In order to counteract the effect of the weight of the cord, for example, they dropped and extra cord from the other side of the upper pulley to the ground. In order to measure the velocity, they put in place a set of observation stations separated by 5 pieds (1.625 m) with an observer in each one of them. They measured the time by a half-second pendulum, each observer taking note of the passing instant. These series of measurements allowed them to assess the effect of the acceleration. Although the installation had stations every 5 pieds, in the table of results they only offered one every 10 pieds. For the experiments that simulated movement in both open and closed channels, they adapted a canal in the basin for the purpose, using lateral wooden panels between which the model had to navigate.

Fig. 4-13. Bossut’s models

They used 20 different models, classifiable in various groups, some of which are shown in Fig. 4-13. The simplest was a parallelepiped box, and the most complex was similar to the hull of a ship ‘built according to a model from the fine collection of all kinds and sizes of vessels that Mr. Duhamel had gathered together in a room adjoining that of the Academy of Sciences’ [I.§.27]. At the end of the work they added two more models which they had rejected at first because they were quite different from the rest.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

As an example of the size of these models, the one quoted as being the simplest was 6 1/12 pieds long and 19 8/12 pieds wide and high (1976 and 528 mm, respectively). Evidently they were relatively large. The models were designed according to a definite pattern, with a view to analyzing the relative influence of the different constitutive elements, such as the bows, sterns etc. Of the entire group, there were only two that did not follow a pattern, one imitating the hull of a ship (no. 20), and another whose cross section was also ship-like, but cylindrical (no. 6). The remainders were all prismatic.31 On the plan-view, No. 1 was formed by a rectangle closed by a triangle, acting as the stern. Model No. 7 was the same, but in the opposite direction, i.e., the triangle acting as bow. Number 2 was similar, but with the triangle shortened into a trapezium, acting as stern and with different dimensions. Five more were derived from these (Nos. 8–12), which had triangles as bows, with semi-angles that went from 63.4° to 21.8°, with which they later tried to carry out a sweep in the tests. The shortened ones, Nos. 4 and 5, were derived from No. 3 with its rectangular base. Numbers 13 and 14 with triangular bows were also derived from No. 3, as also was No. 19 with its semi-circumference. In turn, from No. 4 we get Nos. 15 and 16, whose bow did not resemble a prism but was rather a sloping plane, and Nos. 17 and 18 which had the same bows, only inverted. They observed in the experiment that when being towed the models yawed and swayed from side to side. In order to avoid these undesirable effects, they provided the models with a small rudder whose function was to stabilise the advance of the model in the motion.

Fig. 4-14. Results card

The book compiles 293 experiments, each one registered on a card. One is presented by way of example in Fig. 4-14, in which the data of the model, the driving weight and the instants of transit through the stations 0-10-20-30-40-50 31 We understand as prismatic when the plan-view is the same in any cross section parallel to the base.

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207

are noted. As well as this, it shows the elevation of the associated wave (remou) at the front and sides. Concerning the height of the waterline, they kept it constant in each set of tests, although some models carried more load in order to analyze the effect of the depth on the waterline. The time is expressed in semiseconds and the weight in marks.32 They present the results in four groups: (a) (b) (c) (d)

Movement in undefined surfaces (201 cases) Movement in narrow open channels (66 cases) Movement in closed channels (9 cases) Models numbers 21 and 22 were added (17 cases)

A second classification superimposed on the previous one is the separation of what they call ‘direct’ resistance and ‘oblique’ resistance, which they define as: Direct resistance, that is, the resistance of the plane surfaces that strike the fluid perpendicularly … [and] … oblique resistance or that of the surface placed obliquely with respect to the direction of the movement. [II.§.30]

We shall use these when we come to analyze the effect of the sharp-pointed bows as opposed to the flat ones. An inspection of all the data presented shows that the model moved at speeds between 0.44 and 1.42 m/s. In order to analyze the results, they first compared them with the predictions of the theory in use at the time, and specifically with the three basic suppositions of the theory: the proportionalities with the square of the velocity [V.§.4–9], with the surface [V.§.10–21] and with the square of the sine of the angle of the bow [V.§.22–26]. In the second place, they did the same with the assumptions of the absolute value of the force [V.§.27–35]. We shall analyze these four cases separately. Proportionality with the square of the velocity After a number of considerations with respect to the acceleration phase of the model, they noted that, according to the measurements, the motions may be considered as uniform in the last 20 pieds (6.4 m). Therefore, they took the time taken to traverse this distance as the basis for calculating the velocity. They constructed 29 different tables using the cards of the 201 experiments. Each table was defined by the model and its load, and we depict one by way of example in Fig. 4-15. In each table they took a test as a reference point, and for the reminder 32

A mark is half livre of Paris (0.245 kg).

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THE GENESIS OF FLUID MECHANICS, 1640–1780

they determined the resistance that each one ought to have if it fulfilled the square law, comparing this value with the real weights used in the test. They thus obtained a relative value for each case.

Fig. 4-15. Table of results

Reflecting on it, they say that the resistances ‘follow, more or less, the ratio of the squares of the speeds’ [V.§.9]. Nevertheless, they observed that strictly speaking the resistance increases more than the square of the velocity, which they explained by the fact that as the body moves it causes the fluid to divide and cede its place to the body. Proportionality with the surface They found that establishing this presented more difficulties, as they had to look for comparable cases, and since the velocities of each test were not the same, to unify them they had to introduce corrections according to the supposition of proportionality with the square of the velocity. Taking into account these considerations, they analyzed two groups: one, when the submerged depth of the model is noticeably equal, and the other when this does not occur. For the first they had three cases, Nos. 1, 2 and 3 (second draught). They made a table with paired comparisons, of which there are 13 cases, and ended up by concluding that the resistance is noticeably proportional to the surfaces. They

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also made an attempt to refine this by trying to take into account the fact that during the movement the bow rises and the stern sinks. For the case where the draughts were different, they selected model No. 3 which they had tested with three draughts. They obtained 16 comparison pairs that complied with the rule very roughly. Proportionality with the square of the sine For this verification they used sets of models with the same body, but with different bow angles, as Fig. 4-16 shows. They had two sets available. The first comprised six units (No. 2 plus No. 8 to 12) whose base was 2 × 1 pieds (650 × 325 mm), and tapered to a triangular bow whose height varied from 6 pouces (162 mm) to 30 (812 mm) in steps of 6 mm, besides the flat case. The second set comprised two single units whose bases were 1 × 1 pieds (325 × 325 mm), one had a flat base and the other a triangular base of 2 pieds (650 mm).

2'

x

4'

2'

Fig. 4-16. Test model template

As a reference test, they took the case of the flat front at a determined velocity. Then, for each bow they selected the test whose velocity was closer to the previous one, and using the law of proportionality to the square of the velocity, they calculated the value of the traction force that was required in order to obtain the reference velocity.33 Using this procedure they found the relative resistances in function of the semi-angle in the bow, and these are displayed in Fig. 4-17. Likewise, supposing that the law of the square of the sine is correct, they obtained the theoretical relative resistances, also represented in the same figure. As can be seen, the more pointed the bow, the greater the difference. In the same figure the sin2θ is also depicted, showing that even his calculation for the theoretical ones are not very accurate. Finally, combining the two previous 33 They took the experiments 10, 86, 91, 97, 105 and 113 for the models 2, 8, 9, 10, 11, y 12, respectively.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

series, and supposing compliance with a law of the type sinnθ, they calculated the value of the exponent n taking the case of the flat bow (θ = 90°) as n = 2. Figure 4-18 includes the values of n we obtained by Bossut, which separate clearly from the value n = 2 that predicts the impact theory.34 1 sin2θ

Relative coefficients

0,8 0,6

Exper.

0,4

Theory (sen2θ)

0,2 0 0

10

20

30 40 50 60 Semi Bow angle (º)

70

80

90

Fig. 4-17. Angular bow effect

2 Exp. (n)

0,8

1,6

0,6

1,2

CD

0,4

0,8

0,2

0,4

0 0

10

20

30

40

50

60

70

80

Exponent (n)

Drag coefficient (CD)

1

0 90

Semi bow angle (º)

Fig. 4-18. Effect of angular bows

34 The values found by them are: 2, 1.79, 1.59, 1.29, 1.08 and 0.92, but an examination of Bossut’s data leads to 2, 1.55, 1.78, 1.35, 1.31 and 1.07.

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211

At this point we shall complete their data reduction in greater depth, although the final results will be similar. If, instead of taking only one case for each type of model, that is to say one single velocity, all the cases corresponding to all velocities from each table are taken, [Table I, (Chap. II)], and they are reduced to a curve following the criteria of the least squares,35 the CD of the set would be obtained. This is represented in Fig. 4-18 in function of the angle. For comparison purposes, two coefficients have been drawn that would be obtained by the law of sin 2 θ. It is evident that there is no proportionality between them. They end by saying that this resistance “is a very difficult object of research and well worth the attention of the geometricians [V.§.26]. Up to this point all the analyses have been relative, because they were referred to the flat bows. Now, in order to find the absolute value of the resistance they begin noting that: As regards the absolute measurement of the resistance, the authors give it differently. Some claim that the resistance perpendicular to a plane is equal to the weight of a fluid column that has this plane as its base, and by height, the height due to the velocity with which it impacts. Others make the resistance double that of the same column. [V.§.2]

In order to make an absolute determination of the resistance, they needed to eliminate the effects and parasitic resistances of the auxiliary components, such as the pulleys, cords, etc., and even the estimated the resistance of the air. To do this they made complementary determinations in which they loaded weights without a model, and thus derived a table of corrections, which was finally reduced to estimation that the effects to be discarded were 3.125% of the total force applied. [Sec. IV]. Next, they selected three models with flat bows, Nos. 1, 2 and 3 (three draughts) and No. 21, and performed one test for each one. They carried out the calculations and corrections needed to find the measured and theoretical water heights, which turned out to be very close. They concluded from this that the resistance is equal to the height of a column whose base was the frontal surface, the height being equal to the kinetic height. If we repeat the calculations using the values of CD that we obtained previously for the two plane cases, and apply the correction for the parasitic resistances, we arrive at values of 0.862 and 0.934, both close to one. Finally they sought to determine the resistance arising from 35 The starting point is that we want to arrive to a straight line that links the forces and the squares of the velocities, that is, of the type D = kv2. To do so the pairs of experimental data {Wi,vi2} will be used. The regression line for these points will have a value k that verifies that ∆ = Σ(Wi – kvi2)2 is minimum.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

the ‘tenacity’ of the fluid and friction of the mechanism. Once they had analyzed several experiments, they ended by saying that it was very small. As far as the tests in closed canals were concerned, the aim of these tests was to calculate the increase in resistance when the ships navigated in canals, a theme of great interest at that time for inland navigation, not only in open air canals but also in underground canals with the idea of digging through mountains36 [Chap. VI, §.14]. In order to carry out these tests they placed lateral planks of wood in the centre of the basin to simulate these canals. In total they carried out 66 experiments in what they called open canals, and six in closed ones. They varied the configuration of the canal, i.e., its depth and breadth, giving rise to various subdivisions. The general conclusion, the only one that interests us as the methodology of the tests is similar to that of the undefined case, is ‘that the resistance of the fluids contained in narrow or shallow canals is greater than that of the undefined ones in all respects’ [VI.§.12]. Lastly, we draw the reader’s attention to the fact that the CD they found for the ship-like models in two draught conditions are 0.244 and 0.295. These values are very close to present measurements. After the tests described herein, in 1778 the second report appeared in the Mémoires de la Académie, repeating the title of the first one, ‘Nouvelles expériences sur la résistance des fluids’, of which Bossut was the sole author. In the introduction he declared that the new experiments are a continuation of those collected in the previous Memoir, which had been performed by Condorcet and himself. Although the site of the experiments was not the same, but in an old reservoir commonly known as the Grand-égoût,37 and the instruments were improved, the test methodology was similar to that already described therefore it is not necessary to dwell upon it. The questions he tried to answer with the new experiments were the following four [II. Schol.]38: (a) Whether the resistances of polygonal or curvilinear bows follow the same rules as those of angular bows. (b) Whether a longer or shorter stern influences the resistance of the ship. (c) Whether the length of the ship makes a difference or not to the advance of the ship, that is to say to the resistance. (d) What changes does the resistance undergo if a triangular point is placed in the middle of a flat bow? 36

See the previous note 28. Located across the Seine from the present-day Eiffel Tower. 38 We refer to the ‘Nouvelles expériences’. 37

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213

In order to answer these questions he selects a coherent set of models, of which we present the most significant in Fig. 4-19. Of those ending in a point he prepared 15 models with variations in the bow angle. In total he performed 86 experiments, registering for each one the time it took to traverse the base of 96 pieds (31.2 m) according to the tow force.

Fig. 4-19. New models of Bossut To reduce the new experiments, the first thing he prepared is the table of resistances for the 15 types of bows in relative magnitudes, whose results we have represented. He approached the relative resistance for a function of the type f = sin 2 θ + mθ n. The first of the two addends will be the corresponding to the impact theory and the second a correction.39 To determine m and n he used a series of approximations that lead him to n = 3.25 and m = 1.051. Once this was done he went on to answer the four questions posed. For the first he found that ‘whatever the law of resistance of the simple angled bows is, it was not applicable to the polygonal or simple bows’ [III. §.X]. He justifies this in terms of the results for the case of the semi-circular bow. For the second, which deals with the effect of the sterns, one of the controversial points of the time, what he does is to reverse the direction of motion of the models, so that the bows function as sterns. He does not make any detailed analysis, but after a general comparison of several cases, he says ‘that a lengthened stern noticeably increases the speed of the model’ [III. §.XI]. However, he compares the case of a rectangular cube with and without a triangular stern of 48°, and finds that this stern decreased the resistance by a factor of 0.89. As regards the third, the influence of the length of the ship on the resistance, his comments are also qualitative [III. §.XII]. That the resistances are almost proportional to the surfaces, as he recalls having said already in the tests of 1775, is true, in so far the relation length to width is respected. Comparing the 39 Bossut uses the cosine instead the sine. We have changed to sine for congruency with the oft repeated sentence: ‘square of the sine of the angle of incidence’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

experiments when there is a considerable difference between them, he finds that the resistance is proportionally greater when the hull is shorter. An optimum length will exist, that, in accordance with the tests for ships with flat bows and velocities of 2 or 3 pieds/s (0.65 or 0.97 m/s), the said length must be at least triple the width and this relation will increase with the velocity. Finally, the last question that dealt with the reduction of the resistance due to the effect of a point, he found to be case [III. §.XIII].

Chapter 5 Fluid-driven Machines and Naval Theories

In concluding Part I, we examine those machines whose functional basis was the theory of impact, or more generally, the forces generated in a body in motion relative to a fluid. Apart from these machines, there is an additional category of applications, perhaps the most significant, corresponding to naval theory, i.e., to ships. The analyses that we shall undertake will pay attention to the discussion in the Prolegomenon concerning applied science, which in brief is the extension of the theories to models and the theoretical functioning of machines, but without including the technology required for their manufacture. This chapter is divided into two main parts: the first, devoted to hydraulic machines, starts with general considerations, before turning to hydraulic wheels, then going on to some technological experiments that were carried out with the intention of establishing theories or disquisitions on machines, and we end with windmills. The second part deals with naval mechanics, and includes the generation of resistance by the hull, and the production of force in the sails. Fluid machines The hydraulic machines used in the eighteenth century may be classified from two points of view: one according their functionality, the other according to their operative principles. The first reflects the user’s viewpoint, the second that of the expert or engineer. According to the functionality principle, the group includes water-elevating machines, which meet an important social demand, and which constitute a part of the system of water distribution or field irrigation. Among these are hydraulic pumps, the Archimedes screw and the scoop–bucket waterwheels. However, of these three types only the pump is a true hydraulic machine based purely on principles of fluid mechanics. The scoop–bucket waterwheel is not, as the liquid to be raised is equivalent to a simple mass load. We have not found any studies in the known literature examining the Archimedes screw, probably because it is a very complex machine. Hydraulic wheels comprise another group, employed in streams of water and waterfalls in order to produce mechanical energy from the water flow. In this group we must also make some fine-tuned

215

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THE GENESIS OF FLUID MECHANICS, 1640–1780

classification distinctions. Machines belonging to this category would be the ones where a driving element is constituted by a body undergoing a force by the action of the current of the fluid. By contrast, those employing water as a mass that moves the wheel due to its weight, and that only profit from their fluid condition by making loading and unloading easier, would not fall into this category. As a consequence, driving hydraulic wheels and windmills would be fluid machines, but excluded are gravity wheels that function like a bucket wheel but in reverse. In the eighteenth century it was common to find these machines working together, in particular using the driving capability of the hydraulic wheels to move pumps or water-bucket wheels. An example was the wheels of the horizontal axis moved by the current of the rivers. These wheels were provided with scoops fixed to the outer rim of the wheel in order to lift water from the river.1 More highly developed machines were those in which rotary action was used to drive pumps (Fig. 5-1).2 A last group to take into consideration would be the hydraulic presses, whose function is to amplify forces, and which we would classify as static machines, as in their case motion of the liquid is only an intermediate means in order to achieve the main goal.

1

Julio Caro Baroja carried out a vast study of these wheels in the chapter ‘Norias, azudas, aceñas’, which is how these machines are known throughout the Iberian geography, in his book Tecnología popular española (Spanish Folk Technology), Editora Nacional, Madrid, 1983. There is an abridged edition in Ed. Mondadori, Madrid, 1988. These wheels, whose main characteristic is laterally fitted buckets—also known as ‘arcaduces’—are known from ancient times. In this respect Caro Baroja quoted Vitruvius, who studied them in the De Architectura, X.V.I, and he also explains how they were introduced to the Iberian Peninsula by the Arabs, from whom the terminology comes. Caro’s book does not limit itself to Spain, as it title would indicate, but extends to other geographical areas as Syria, Egypt and China. In the Muslim times these machines were common, and many have survived until the present century. Some of the wheels reach up to 9 m diameter, as depicted in illustrations in Caro’s book. In the present day, he quotes a numbered few in irrigation ditches of the Segura river in Murcia. In the Portuguese city of Tomar there is a wheel still working in a park as a reminder of the old ones. The tradition of wheels in this area has been preserved until not many years ago. The reader can consult the delightful book O rio, os açudes e as rodas (The river, the channels and the wheels) by Fernando Ferreira, edited by the Junta Distrital of Santarém. It shows photographs of wheels and several existing wheels in the Nabão River are enumerated. Some of these ‘rodas’ reached diameters similar to Spanish ones. However, they have disappeared progressively exchanged for electric motors. 2 The machines sited in London Bridge and in Marly (France) were of the driving pumps type. A comparison between both is found in ‘A Description of the Water Works at London Bridge’, Phil. Trans. (1731) by H. Beighton. The one constructed by Juanelo in Toledo would be of the same type. The illustration shown has been taken from the History of Technology by T. K. Derry and Trevor I. Williams.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

217

Fig. 5-1. Machine in the London Bridge (1749)

The functioning principles of the machines The object of a machine is to transform one kind of energy into another, which in turn will be used for other purposes. In the case in point, part of the energy existing in a fluid current will pass to the mechanical movement of an axis. The main problem is therefore to relate the geometrical characteristics of the machine and the velocity of the current with the energy produced. These studies, mathematized by the machine analysts of the time, required the introduction of new specific concepts such as the ‘produced effect’ corresponding to the present-day concept of power.3

3

In applied mechanics the concept of work is previous to the power one. The former is defined as:

T =

GG

v∫ Fds

Where F is the applied force and the integral is extended over the travelled trajectory. Its physical meaning corresponds to the energy needed to move a body inside a force field. The work is a scalar magnitude, and it is the result of the scalar product of two vectors: the force and the space. The power represents the work in a time unit, that is W = dT/dt. If the body moves with the velocity v over the trajectory, then ds = vdt, therefore:

G G ds G G dT W = = F· = F ·v dt dt

Said with single words: power equals force times velocity.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Vm V0

Fig. 5-2. Basic motor element

The basic driving element of a hydraulic wheel is a scoop or blade upon which the forces are induced by the fluid current (Fig. 5-2). The phenomenon is the same as that of resistance, with a few slight differences. Thus, when dealing with motion in a fluid, the resistance, as its very name indicates, is an undesired force and the less it is, the better the moving body will behave. In a machine, on the other hand, it is just the opposite. The resistance becomes the driving force, and the greater it is, the better is the use made of it. Another difference is that in the cases of the resistance, either the body or the current are usually immobile, which is not the case in the driving production, as the blade or scoop also move. Using the terminology we have established, if the velocity of the current is v0 and that of the driving element vm, the force exercised by the current would be: D=

1 2

ρ SCD (v0 − vm ) 2

[5.1]

Therefore, the total power delivered by this element would be given by: W = D vm =

1

ρ S C D ( v0 − v m ) v m 2

[5.2]

2 Upon examining this expression it can be seen that the power is zero when vm = v0 and when vm = 0. The first case corresponds physically to a zero force, while the second corresponds to zero velocity. The extremes indicate that there will be an intermediate value of the velocity in which the power will be maximum, and the treatise writers of the eighteenth century considered this point to be reached4 at vm = v0/3. The machines should be made to work at this velocity in order to achieve the maximum power, which would be:

4 To reach to this result the velocity terms in equation [5.2], which are v ²v – 2v v ² + v 3, are 0 m 0 m m derived, obtaining v0² – 4v0vm + 3vm² = 0, whose solution is vm = v0/3.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

Wm =

4 1 27 2

ρ SC D v0

3

219

[5.3]

Some treatise writers of the eighteenth century took the factor 4/27 as an indicator of the maximum energy that can be recovered from the current, which is not strictly true, as the part affected by this factor in the previous formula is not the available energy, but a term with energy dimensions.

Vm V0 Fig. 5-3. Elemental hydraulic wheel

The normal configuration of a hydraulic machine consisted in a wheel with blades inserted in its circumference, generally in a radial direction, as is shown in Fig. 5-3, although sometimes they were sloped with respect to the radius. Of the machines based on the hydraulic wheel, the water mill had been known for some 3,000 years. Its primitive use was to grind grain, and later on it was extended to forges, sawmills, milling machines etc. According to Derry and Williams5 the first mills we know of were those Greek or Scandinavian, whose axis was vertical, suitable for rapid currents, and which we reproduce in Fig. 5-4. Vitrubius in his Architectura,6 written in the first century, proposed a solution almost identical to the earlier one, and was perhaps inspired by the Persian wheels or others that came from Mesopotamian cultures.

5 6

Cf. History of the Technology, Vol. 1, p. 361-ff, where the figure has been taken. Cf. Book X, Chap. IV y V, where he deals on machines to raise water and mills.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 5-4. Scandinavian mill

At first these machines were moved by the current of a river, while later on the water was dammed up in basins or ponds whence it fell on to the scoops or buckets of the wheel. The difference between both types of supply is that in the latter, the motion is generated by the weight of the water, while in the first it is generated by the pull of the current. A third variation arises when the water falls from a certain height in the form of a jet that strikes the blades. Until well into the nineteenth century, hydraulic wheels were the main source of energy throughout the world. With this background it is not surprising that the hydraulic wheel occupied an important place in the studies of the application of hydraulics in the eighteenth century. As Parcieux notes,7 ‘Once the sciences took on a new appearance, and new calculations were applied to all the physical and mathematical sciences, first-rank scientists paid attention to machines moved by water in order to determine the quantity of force maintaining them in movement’. The first references to the estimation of forces produced in a wheel are due to Mariotte, who dealt with the mill wheel of the Seine,8 which we have already quoted, and which he very probably used as the basis for his theories. However,

7

Cf. Mém. Acad. Paris in 1759, with a lengthy title: ‘Mémoire dans laquel on prouve que les aubes de Roues mûes …’. 8 Cf. Traité du mouvement des eaux, II Part, 3er Discurse, V Regle, p. 201.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

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the first works with a more mathematical application did not appear until 1704 with Parent, followed by another by Pitot in 1725. One of the problems most often posed in the eighteenth century was the optimization of the dimensioning of this machine, i.e., the number of blades, their size as a function of the radius, and their angle. A second question was the comparison of the wheels moved by currents and those moved by the fall of jets striking the blades. Those who wrote about these machines used La Hire’s rule to calculate the force of the water from its velocity. This rule is: Take the square of the number of feet traveled by the water in one second, and divide this square by 56, which is a fixed number, and which serves in all the proposed cases, and the quotient of the division will be the height of the feet of water required to be put on top of the given surface t, in order to support the effort or load caused by the water.9

Taking into account that this 56 is twice the acceleration of the gravity expressed in pieds/s², the height is precisely h = v²/2g, or what we have called kinetic height. As we have already stated, this rule was one of the first estimations of the absolute value of the resistance, i.e., another sample of the link between applications and the theories. Parent’s studies In 1704, Parent published ‘Sur le plus grande perfection possible des Machines (‘On the greatest perfection possible in machines’) in the Mémoires de l’Académie of Paris, where he studied an archetype hydraulic machine (Fig. 5-5). It consisted of a hydraulic wheel that transmitted motion by means of a gear to a second axis, which in turn had a drum from which a weight hung. This machine integrates the most significant elements of mechanics and hydraulics: the hydraulic wheel, the gears, and the hoist. The first acted as a force generator, the second as a passive converter of force and motion, and the third as a link with the lineal movement of the weights.

9

‘Examen de la force necessaire pour mouvoir les bateaux tant dans l’eau dormante que courrante, soit avec une corde qui y est attachée & que l’on tire, soit avec des rames, ou par le moyen de quelque machine.’ Mém. Acad. París (p. 259), 1702.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 5-5. Parent’s machine

At the beginning of the work, Parent clearly states its purpose: Given a machine whose driving power is any fluid body whatsoever, such as for example water, wind, flames etc. and which should serve to lift solid or liquid weights, such as stones, minerals, water etc, it is proposed to find the load required for this Machine, and the proportion that its different parts must have in order to produce the greatest possible effect, that is to say to raise a greater quantity of weight in the same time as with all other loads and all other possible proportions.10

In the text he defines a set of basic concepts such as the ‘possible effect’, which is equivalent to the present-day concept of power. For a given machine, with absence of friction and for a specific velocity of the current, Parent indicates that a weight P0 will exist, which he defines as the ‘balance weight’, that applied to the drum will keep the entire apparatus immobile. If the external load P were less than P0, the wheel would turn and raise the load, i.e., it would function like a machine, while for loads bigger than P0, the wheel would turn against the current. About the product P0v0, which is what he calls the natural effect, he says that ‘it will serve to easily determine the degree of perfection of this class of Machines when carried out at random’.11 It would be a property of the machine, a function of its geometrical dimensions, and of the velocity of the current.

10 11

Cf. ‘Sur le plus grande perfection …’, p. 323. Op. cit. p. 332.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

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On the other hand, the force P0 is proportional to the square of the velocity of the current P0 = kv0², and P = k(v0 – v)2 will also be where v is the lineal velocity of the center of the blade as the wheel turns. Joining both in order to eliminate the parameter k, one arrives at: P=

P0 v02

(v0 − v ) 2

[5.4]

The weight P will move at a velocity proportional to v, depending on the geometry of the machine, and Parent takes the product Pv as a measurement of the efficiency, which nowadays we would call power,12 and would be expressed as:

W = Pv =

P0 2 0

v

v (v0 − v) 2

[5.5]

This function is similar in structure to the one analyzed in the previous paragraph. The maximum value of the v will be v = v0 /3, which will give a maximum power of W = (4/27)v0 P0 . As a final annotation, Parent ends his report quoting the experiences of Mariotte and Sebastien on resistance. A current of 26 pieds/s (8.445 m/s) produces a force of 910 livres (4,365 N) over a surface of one square pied (0.1055 m²). Its conversion to a resistance coefficient is CD = 1.16, which was the value given by Mariotte. Pitot’s works

Henri Pitot was a student of almost all hydraulic machines. He devoted his attention as much to hydraulic wheels as to pumps and ships.13 Here we shall consider the three reports that he dedicated to wheels, published in the Mémoires de l’Académie of Paris. In the first of these, ‘Nouvelle methode pour connoître & déterminer l’effort de toutes sortes de machines mûes par un courrant, ou une 12 Strictly speaking, for a machine with gears without losses, this value is proportional to the power. 13 It is often stated that Henri Pitot was an experimenter, which is not true. His only contribution to this field is what nowadays is called ‘Pitot tube’, or simply ‘Pitot’, and which has been the most used instrument for measuring the fluid speed until the new technologies were attained, not many years ago (see Chapter 10, ‘Pitot’s tube’). Pitot was the author of numerous mathematical works, and fluid application studies to machines as well. We have found up to nine of those works in the Mém. Acad. of Paris between 1725 and 1740, apart from his naval treatise in which he completed the Johann Bernoulli works.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

chûte d’eau’ (1725) (‘New method of finding and determining the effort of all kinds of machines driven by a current or waterfall’), Pitot adopted a theoretical tone regarding the machines, similar to one that Parent had displayed, and while he quotes La Hire’s rule, nevertheless he makes no mention of his work. Pitot’s analysis is based on the reduction of complex machines to simpler ones, and in this respect he states: ‘The force of all kinds of machines, whatever their composition or wheels be, can be reduced to that of the simple lever.’14 Likewise, he invokes what he calls the fundamental rule of mechanics, which he will also use in his studies on pumps: In all the machines, the product of the driving force, or (which comes to the same thing in machines driven by water) the driving force of the water against the scoops or blades, multiplied by the velocity of the blades themselves, is always equal to the product of the weight moved by the machine and the velocity.15

The rule is an extension of the one postulating the equality of the product of the force multiplied by the displacement, and which nowadays we call conservation of work.16 The handling of these concepts leads him to the well-known result that the maximum power arises when the velocity of the blade is one third of that carried by the current. Four years later, in 1729, other two articles appear. In the first, ‘Remarques sur les Aubes ou Pallettes des Moulins & autres Machines mûës par le courant des Rivieres’ (‘Notes on the scoops or blades of mills, and other machines driven by the current of the rivers’), he compares a wheel with radial blades with another where the blades are canted with respect to the radius. We shall not make any comments on his comparison as his arguments are somewhat qualitative. In the second, ‘Comparaison entre quelques Machines mûës par les courant des Fluides’ (‘Comparison of machines driven by the current of fluids’), deals with various themes. He studies the force that the water exercises on a plate submerged at an angle, and to do this he uses impact theory, and La Hire’s rule, i.e., CD = 1, which accompanies the tables. He also compares two hydraulic machines, those of Boulogne and Caron, with that of Duguet (Fig. 5-6). Although it is not very clear from his explanation, it appears that the latter, as well as being more up to date, was smaller in size than the other, as he says: ‘In order to make

14

Cf. ‘Nouvelle méthode …’ , p. 78. Op. cit. p. 79. 16 In the ancient books it was said as the ‘golden law of the mechanics’, and it was stated ‘what is gained in force it is lost in path’. 15

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Fig. 5-6. Boulogne and Caron-Duget machines

an exact comparison between Duguet’s machine with those of Boulogne and Caron, let us suppose that that Duguet’s was made on a large scale, presenting as much blade surface or wings to the current as that presented in fact by Boulogne, whose surface is around 130 square pieds [13.7 m²].’17 The difference between both machines is that Duguet’s is of the helical type, with its blade angles at 54° 44′ with respect to the current, which is parallel to the turning axis of the machine.18 The other one is of the type of wheel with radial blades. After analyzing both artifacts, he concludes by preferring neither, as he finds no reason to do so. However, he does point out an important difference: at optimum working condition, the helical one must go more quickly than the other. The reason is simple: due to their configuration, the Boulogne and Caron machines have only one single blade submerged at any given instant, while the other has several. Therefore the driving centre is further from the turning axis in the former than in the latter, which, being more compact in size, will be nearer. Thus he concludes that it will go more quickly and will lose more force.

17

Cf. ‘Comparation entre quelques …’, §.II.

The angle 54° 44′ corresponds to arctan√2, and it was obtained as the optimum inclination according to the impact theory. The reason is that the force following this theory is proportional to sin²α, being the component that produces the turn proportional to cosα, therefore the total will be to sin²α·cosα. The maximum of this expression will be at tan²α = 2, as stated. We must note that Renau d’Elizagaray originally found this value as the optimum angle for the ship’s rudders, because they are governed by the same equations. See Chap. VII of De la Théorie de la Manœuvre des Vaisseaux, 1689. 18

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The studies of Bossut and others

Bossut, in his work ‘Détermination générale de l’effet des roues mûes par le choc de l’eau’ (‘General determination of the effect of wheels driven by the impact of water’) which appeared in the Mémoires de l’Académie of Paris (1769), doubted that the velocity of greatest efficiency arose when the wheel turned with a tangential velocity equal to one third of the current. His argument was based on the fact that the previous studies supposed that each blade of the wheel was independent, and that the current impacts perpendicularly on it, when in reality neither of these two premises was true. The correct way, according to Bossut’s thinking, would be to consider all the blades as a set, each one with its impact angle, and to sum the individual effects, taking into consideration the mutual interferences.

Fig. 5-7. Bossut’s Wheel

Figure 5-7 shows the principles of his method. One can see how some blades cover others, which have repercussions reducing their real surface. Besides, the angles relative to the current in each blade reduce the ideal force, and therefore the momentum with respect to the turning axis. Although the idea is simple, its mathematical development is long and complicated, taking up several pages. The final expression he arrives at for the power of the system is the product of the load times the velocity, and results in a third-degree equation of the tangential velocity of the wheel of the type Qv = Avt + Bvt² + Cvt3, in which the three coefficients are a function of the geometry and makeup of the machine. He finds the optimum value for derivation of the former equation to be a second degree equation.

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Fig. 5-8. Horizontal wheel

As well as the type of wheel shown, Bossut also studies the model of a wheel driven by the force of a jet, as Fig. 5-8 illustrates. He himself recognizes that he deals with this problem ‘in a less general fashion than the previous one, in order to arrive at simpler and more satisfactory results in practice’.19 The solution he proposes takes as its basis that the force produced by a jet of water against a plate at rest, is the equivalent of a column whose height is double the kinetic height, that is CD = 2. The mathematical treatment is similar to that given by Parent, arriving at the optimum velocity as a third of that of the fluid, and at a power of W = (8/27)MH,20 where H is generating height, and M the flow impacting against the blade. After this equation he declares that ‘it can be seen that the greatest effect of the machine is reduced, when the quantity of water that makes it move is raised to a height 8/27 of the height due to the current velocity’.21 Parcieux’s works are less important than those of Bossut, although it is worth mentioning those in which he shares with Bossut and others the preoccupation with the comparative yield and efficiency of the wheels, depending on how they are moved by the current, by scoops or by jets. Deparcieux sent two works to the Academy of Paris, which appeared in 1754 and in 1759. Their lengthy titles were ‘Mémoire dans lequel on démontre que l’eau d’une chûte destinée à faire mouvoir quelque machine, moulin ou autre, peut toûjours produire beacoup plus d’effet en agissant par son poids qu’en agissant par son choc, & 19

Cf. ‘Détermination générale …’ , §.XVIII. Comparing with the formers we see that is a double. The reason is to have supposed the jet force to be double of the generatrix height. Although he does not explain it, this would be justified for being a shock rebounding of all the water. 21 Op. cit. §.XXII. 20

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THE GENESIS OF FLUID MECHANICS, 1640–1780

que les roues à pots qui tournent lentement, produisent plus d’effet que celles qui tournent vîte, relativement aux chûtes & aux dépenses’ (‘Report in which it is demonstrated that the water from a fall which moves any machine, mill or whatever, can always produce more effect acting by its weight rather than by its impact, and that wheels with scoops that move slowly in relation to the falls and deposits produce more effect that those that move quickly’) and ‘Mémoire dans lequel on prouve que les aubes des Roues mûes par les courants des grandes rivières, seroient beaucoup plus d’effet si elles étoient inclinées aux rayons, qu’elles ne sont étant appliquées contre les rayons mêmes, comme elles le sont aux moulins pendants & aux moulins sur bateaux qui sont sur les rivières de Seine, Marne, Loire, &c’. (‘Report in which it is proved that the blades of the wheels moved by the currents of large rivers, will be much more effective if they are angled with respect to the radius, as they are in overhanging mills and in the mills upon ships located on the banks of the Seine, Marne and the Loire, etc.’). In the first of these works, almost qualitative in nature, he says that when a fall is over four pieds (1.30 m) the scoop or bucket wheel is more convenient. The reason he gives as argument is the effectiveness, as, taking the calculations of Parent and of Pitot, only 4/27 of flow can be raised, while this limitation does not exist with a scoop wheel.22 The second work is experimental, and is devoted to analyzing and comparing the force and velocity of a wheel whose blades are angled. He gives a set of measurement tables which conclude that when the blade is radial, the efficiency is less than when it is angled. In spite of the fact that the apparatus was well built, and that the measurements appear to be correct, his work suffers from lack of reference to a theory. The measurements appear to be simple measurements only. Smeaton’s experiments

In May 1759, John Smeaton read before the Royal Society the work entitled ‘An experimental Enquiry concerning the natural Powers of Water and Wind to turn Mills, and other Machines, depending on a circular Motion’,23 which was published in the Philosophical Transactions of the same year. Smeaton’s approach is neither an application study, in the style of those of the authors mentioned in the previous paragraphs, nor an experimental one, such as those discussed in the foregoing chapter, but an experiment on specific machines: the hydraulic wheel and windmills.24 22

Cf. ‘Mémoire dans lequel on démontre …’, pp. 605–606. The following quotes between brackets will refer to this work. 24 Smeaton is one of the first engineers in the modern sense of the word. In contrast to the craftsmen, the distinctive characteristic of the new engineer was the possession of theoretical knowledge 23

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229

Smeaton works with models, which, he tells us, ‘I look upon as the best means of obtaining the most significant results in mechanical enquires’ [p. 100], although he does not overlook the fact that working with models must de done with care, as ‘it is very necessary to distinguish the circumstances in which a model differs from a real large machine; otherwise a model is more apt to lead us from the truth than towards it’.

Fig. 5-9. Smeaton’s machine

He presents the work in three parts, and each one was read in a different session of the Royal Society. The first two refer to hydraulic wheels: one moves with and under the current, and the other with an upper current which is driven they applied to practical problems. Their solutions not only came from ingenuity and inventiveness, but from a prior theoretical reflection. We can compare the approaches to a practical problem of two prototypical persons, like Euler and Belidor. The first presents some impressive mathematical constructions, the second creates imaginary machines. Smeaton falls between the two.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

by impulsion and gravity. The third refers to windmills. Although, according to what we understand by a hydraulic machine, we have not gravity wheels in this category, we will comment upon the results obtained by Smeaton as he makes some interesting comparisons between the two. He calls the impulse hydraulic wheels undershot waterwheels. Figure 5-9 shows a general view of the machine model that he uses in his experiments, and Fig. 5-10 shows a section. We shall spend a while explaining the most relevant aspects of the model. The water is contained in an upper deposit and its level is indicated by a stick F attached to a float G. The outlet flow is regulated by the hatch b, moved by the handle H. The current advances boxed in by a channel, where the blades of the wheel are, to end up in the bottom deposit, from whence it is raised to the upper one by a piston Y, moved by the lever M. The driving energy is used in raising a weight by means of a set of pulleys.

Fig. 5-10. Section of Smeaton’s machine

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The machine operation is simple: once the movement is set in motion, the level in the upper deposit is maintained by the manual action of the pump, and at the same time the rotating velocity of the machine is registered. The determination of the auxiliary parameters, that enable the data to be reduced later on, is an operation preliminary to the tests. These parameters are the velocity of the water on the blades, the loss due to friction and the effect of the air, and the volume consumption. For the first two parameters he measured the velocity of the wheel when it turns freely, driven by the water current and without a load. Then, without the flowing water, he placed a counterweight that made the wheel turn in the same direction, and at the same velocity. He repeated the first step with the counterweight, and adjusted this until the velocities were equal. In these conditions he says the velocity of the water will be equal to that of the blades, and that the friction will be likewise equal to the counterweight. The basis of this argument is, that under these conditions the effect of the water in the wheel is null, as with or without it, the turning velocity is the same. He deduces the third parameter, the volumetric flow, from the number of piston-strokes per minute required for the level of the deposit to remain constant. In the report, Smeaton began by explaining the first tests in great detail, whereas for the remainder he only presents the table of results. We shall do the same. In the first set of tests he maintains the water level at 30 in. (76.2 cm), and in order to maintain this with the position of the outlet gate selected he requires 39.5 piston-strokes per min. He deduces the volume of each piston-stroke as 6.703 lb (3,034 cm3 are obtained by independent means), therefore the mass flow will be Q = 1.997 kg/s. In order to find the velocity of the water in the blades as he has explained it, he allows the wheel to turn, driven by the water without a load and he measures 80 rpm. He then removes the water and the counterweight with 1.5 lb (0.68 kg), the result being 85 rpm. With this counterload and the water flow, the velocity is 86 rpm, which he takes as a final value. As the circumference described by the center of the blades is said to be 75 in. (190.5 cm, corresponding to a radius of 30.3 cm), the linear velocity, which will be that of the water, will be 107.5 in/s (2.73 m/s), i.e., the equivalent of a kinetic height25 of h = 15 in. (0.380 m). With the latter, and the mass flow he finds what he calls effect, that is, W = Qhg = 7.44 W, and which will indicate the maximum power available.

25

That Smeaton says virtual or effective height.

THE GENESIS OF FLUID MECHANICS, 1640–1780 50

280

40

240

lb×rpm

200

rpm

30 rpm 20

160

10

120

0 2

4

6

8

10

lb×rpm

232

80 12

Load (lb)

Fig. 5-11. Hydraulic wheel performances

Next he places consecutive weights on the small plate and registers the velocity for each one. The results are presented in Fig. 5-11, and the loads applied are marked on the x-axis. One of the curves indicates the turning velocity, descending as the weight to be raised is increased, and the other shows the product of this weight times the velocity, which is the measurement of the power, and here the point of maximum efficiency can be seen (for 8 lb). For this velocity he measures the counterweight equivalent to the friction, which is 2 ounces (56 g). As the circumference of the cylinder where the cord going through the pulleys is wound is 9 in. (22.86 cm), and taking into account the arrangement of the set of pulleys, he finds that the mass to be raised will travel H = 135 in/min (5.715 cm/s). The real mass will be that already mentioned, plus that of the small plate and the pulley consisting of 10 ounces (283 g). Thus the total mass is m = 9.375 lb (4.253 kg). From all this data he deduces that the real power or ‘effect’, is Wr = mgH = 2.38 W, which is 32% of the virtual ‘effect’. 26 Following up these accounts, there is a table in which he presents the results of 27 sets of experiments in which the heights and flows were varied. He sums up the results he deduces from these in four maxims of a general nature. These are: That the virtual or effective head being the same, the effect will be nearly as the quantity of water expended. [Max. I, p. 116] That the expanse of water being the same, the effect will be nearly as the height of the virtual effective head. [Max. II, p. 117] 26

Smeaton calls the virtual power, power and the real one effect.

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That the quantity of water expended being the same, the effect is nearly as the square of its velocity. [Max. III, p. 118] The aperture being the same, the effect will be nearly as the cube of the velocity of the water. [Max. IV, p. 120]

These maxims illustrate Smeaton’s interest in obtaining parameters and rules for designing machines. We only wish to underline a particularly arguable point: the encasement of the blades. He himself warns how the water rises inside the blades [p. 114], which he uses as a justification for the discrepancies he finds between the results and the theoretical predictions. For the gravity wheels (overshot wheels) he uses a modification of the previous machine. He changes the wheel for another with the same diameter, but with 36 scoops of 2 in. (50.8 mm) in depth, he closes the lower water outlet, raises the turning axis, and he incorporates an upper outlet that is represented in Fig. 5-10 by the dotted lines. With this configuration, and the water at the edge of the deposit, the water level will be 6 in. (152.4 mm) above the outlet, the total height of the fall of the water thus being 30 in. (762 mm). In these conditions he carries out a test series similar to those of the previous case. The main difference consists in that the efficiency obtained, which is the ratio of the real power to the virtual power, is almost double that of before, a fact that is repeated in the remaining test series that he performed. In his own words, ‘the effect therefore of overshot wheels, under the same circumstances of quantity and fall, is at a medium double to that of the undershot’ [p. 130]. In order to explain these circumstances he argues that ‘non-elastic bodies, when acting by impulse or collision, communicate only a part of their original power’ [ibid.]. The explanation of these differences resides in the different conception of both machines, which, we repeat, only have in common a certain similarity and the use of water. After a series of reflections, he ends by proposing the case of mixed wheels, that is, part gravity and part propulsion. Following Smeaton point by point, let us look at the question of windmills. Their use, like that of hydraulic wheels, dates from antiquity. However, their use was in practice restricted to raising water or to grinding grains. References to these mills exist in the first century in the Pneumatica of Hero of Alexandria, but more as a toy or recreational machine than as an industrial machine. Mills whose aim was to generate energy appeared towards the end of the first millennium, like the Persian vertical windmill.27

27

Cf. the Historical Development of the Windmill, NASA CR 4337, 1990 by Dennis G. Shepherd.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 5-12. Experimental windmill

Smeaton posed a problem similar to that of the hydraulic wheel: to define a model mill, susceptible to some variations, with the capacity to raise a variable load, to submit it to a current of air and to measure the behavior of the system. But it was not as easy to have a regular current of air available as it is to have one of water, which is not only found in nature in the form of rivers and steams, but there is also no difficulty in preparing an artificial current. As for the wind, given the difficulties involved in using natural wind, the dilemma is whether to create an air current against a fixed mill or to move the mill. Smeaton adopted the latter solution, as he himself states, it is more practical to produce a circular movement as shown in Fig. 5-12. The apparatus consists in a mill proper, G, with four vanes situated on a turning mast. The vanes are set in motion by a weight P that drags a cable which, by means of pulleys M, N and O, winds itself around the shaft I of the mill. The current is created by the movement of the mast around its shaft DE, which carries a barrel H where another cord is wound that is pulled by an operator. In order to keep the velocity constant, there is a pendulum VX which the operator endeavors to synchronize by pulling the cord so that there one turn for two pendulum vibrations. This operation needed a skilful operator, and Smeaton says that ‘a little practice renders it easy to give motion thereto with all the regularity that is necessary’ [p. 141].

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The test procedure consists in placing a weight in the small plate, moving the machine during a fixed time and counting the number of turns given by the sail wheel during this time. He registers 19 main rounds in which he varies the angles, twist and shapes of the vanes. In the first set of tests which he comments on in detail, the radius of the mill was 21 in. (533 mm), the span of the sail 18 in. (457 mm), with a cord of 5.6 in. (142 mm) and the duration of the experiment 52 s. The angle of incidence with respect to the plane of movement was variable, with 10° at the tip and 25° in the base.28 The velocity of the central point G with respect to the air, that is, the wind velocity, was 6 ft/s (1.83 m/s). Under these conditions, he loaded the small plate with weights (6, 6.5, 7, 7.5, and 8 lb; 2.72, 2.95, 3.16, 3.40, 3.63, 4.08 kg) and counted the turns of the sails in the 52 s of the test duration. He also did another test without a weight in the scale and increased the weight up to the sails’ detention. With the data obtained he produced a table, his main interest being to have the load for maximum turns multiplied by weight and the detention load. In this case they were 7.5 and 9 lb, respectively. He also corrected these values by introducing friction, scale and pulley masses, obtaining 8.69 and 10.37 lb, respectively, and took the ratio between them, 8.69/10.37 = 0.84, as an efficiency indication.29 Figure 5-13 is the table in a graphic form, where the reduced data are also depicted. This graph shows similar characteristics to the one corresponding to the hydraulic wheel (Fig. 5-11). 150

600

rpm lb×rpm

400

rpm

lb×rpm

100

50

200

0 0

2

4

6

8

0 10

Load (lb)

Fig. 5-13. Windmill performances 28

Today we say that the sail had twist. Smeaton takes the inverse that says 10:8.4. It has been changed in order to approach our view point, in which we normally take the unity as the maximum relative value. 29

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THE GENESIS OF FLUID MECHANICS, 1640–1780

This first round is followed by others until the 19 tests are completed which are presented in a Table.30 Smeaton takes three parameters for characterizing each case; one is the aforementioned ratio of load for maximum product to load for detention, other is the ratio of the velocity in unload to velocity in maximum product, and the third one corresponds to the quotient of the sail surface to the maximum product. Although we will not go into his extensive discussions and comments, which cover many pages, it is worthwhile emphasizing, as one of the most interesting, the effect of the different twist of the sails, which he compares using the parameter resulting from finding the product of the maximum efficiency load multiplied by the turns. The cases analyzed were: • Flat sail and with Parent’s optimum angle, which was 55° with respect to the wind, or 35° with respect to the plane. This value is obtained by applying the theory of impact.31 • Flat sails, but with smaller angles, according to what he calls ‘common practice’. He analyses 12°, 15° and 18°. Of the three, the best results correspond to 15°. • The third series followed McLaurin’s criterion, which offers a twist formula deduced for a constant attack angle. He tested three cases which varied from 9° to 26.5°, 12° to 29.5° and 15° to 32.5° between the tip and the root. • A set of six cases follows which he says are in accord with the ‘Dutch manner’, i.e., a twist law given by practice. In accordance with his statement, he finds a distribution which he tests later in the aforementioned six cases. The limits waver between 0–15° and 12–27°. The most efficient of the group, and also of all the sets, is that of 7.5–22.5°. • A variant of the latter consists in increasing the external surface of the sails by adding a triangular surface. As the total surface increases, so does the power, but when it is reduced to the surface of the other cases he finds that the efficiency improves.

These experiments are followed by an extensive set of considerations and maxims, that are more complex than those concerning the impulsion wheel in water. Here we conclude our treatment of this interesting author, who, while aware that a model is never the same as a real apparatus, tries, with a laboratory apparatus, to obtain rules that will help in the rational construction of machines.

30 31

The specimen described occupies place nº 12 in that Table I. See previous note 18.

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Windmills in Euler

In 1752, there appeared in the Mémoires de l’Académie of Berlin a report of Euler entitled ‘Maximes pour arranger le plus avantageusement les machines destinées à elever de l’eau par le moyen des pompes’ (‘Maxims for the arrangement to maximum advantage machines dedicated to raising water by means of pumps’), whose objective was to carry out a brief summary of the possible energy sources available for working pumps for raising water. He looked at waterwheels, windmills, and animal and human traction.32 We shall confine our attention to windmills, for which he offers a rule for adjusting the angle of the vanes, namely, ‘the velocity of the wings at their end must be (8 – √10)v·tanφ/9 = 0.537525v·tanφ’,33 which he says he obtained from earlier studies, where v is the wind velocity and φ its angle against the surface of the sail. Using this value, he determines the maximum available power in which the geometrical dimensions of the windmill’s components intervene.34 Four years later in 1756, and also in the Mémoires de l’Académie of Berlin, Euler returns to windmills with his ‘Recherches plus exactes sur l’effect des moulins à vent’ (‘More exact research on the effect of windmills’). In this work he alludes to a letter of Lulofs, professor at the University of Leyden, concerning measurements which this gentleman had made in some of the windmills used in Holland to dry out marshes. He says that when the wind was 30 pieds/s (9.75 m/s) 35 the machine raised 1,500 pieds3/min (51.5 m3/min) of water to a height of 4 pieds (1.3 m). The windmill in question had four sails whose span was 43 pieds (13.97 m) and 5.5 pieds (1.79 m) of rope with an angle of incidence of 73° (17° with respect to the plane). The application of the rules given by Euler in the report of 1752 indicated that the water raised ought to be 757 pieds3/min, almost half of the 1,500 registered. On the other hand, Lulofs also noted that the power (effet) of the machine was not proportional to the cube of the velocity, but that it hardly exceeded the square of the velocity. Accounting for these discrepancies, Euler insisted on the inexactness of impact theory and proposed a mixed theory.36 To sum up, let us recall, he came round to saying that the force produced on a plate of surface S, upon which a current of velocity v impacts with an incidence α, could be expressed by the formula:

32

The estimations he makes upon the power that men and animals can develop are quite interesting. He calculates that a man produces a power of 80 W and a horse 1,090 W (Cf. Maximes...’, §.XI–XII). 33 Cf. ‘Maximes pour …’ , §.VIII. 34 Op. cit. §.XVI 35 He does not say where the measurement units come from, we assume they are from Paris. 36 About this theory see Chapter 3, ‘Other works of Euler’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

D=

pa ⎛ ρ v 2 S ⎜1 + 2 2 ⎝ ρ gb + 1 2 ρ v

1

⎞ 2 ⎟ sin α ⎠

[5.6]

We observe that a corrective term has been added to the expression as usually given by impact theory, which is represented by the unit inside the brackets. This term results from the effect of the back face which is the second addendum in the brackets, where pa is the atmospheric pressure and b a parameter whose dimension is a length. Euler began his study of windmills with this expression, although he ended with simplifications. In order to find the force in the sails, Euler, as is usual for him, became absorbed in an extensive mathematical study of the problem, with great profusion of details, ending up covering 70 pages.37 He defines the windmill by a set of N vanes that he takes to be four, with a span b. Both the rope and the angle of each airfoil will be variables and functions of the radius, that is c(r) and ω(r). The wind velocity, w, will be parallel to the turning shaft that will turn at an angular speed Ω. Euler approaches the study in a general fashion.

w vr w v

vr

w

Fig. 5-14. Euler’s windmill

The first point that he emphasizes, [§.XXIV]38 is that a profile, or chord, is subject to the combination of the wind velocity and the velocity induced by the 37

The developments that he carried out are based on the kinetic heights, which make it bothersome to follow the mathematical steps. Besides, to express the velocity of rotations he used the lineal velocity of the tip sail also as a height. For all these reasons we have changed to present day notation. 38 The quotes between brackets refer to the ‘Recherches plus exactes …’

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

239

rotation, as is shown in Fig. 5-14 both generally and in detail. Therefore the real incident velocity vt will be: vt =

w2 + Ω 2 r 2

[5.7 ]

Likewise, the induced airfoil angle of attack will be: sin α i = sin(α 0 − ω ) =

w vt

cos ω −

Ωr vt

sin ω

[5.8]

An important consequence can be deduced from examining this expression for a given configuration. As the rotational velocity increases, the incidence decreases, so much so that it can reach zero and even be negative. In order to prevent this, it is necessary to check that wcosω > Ωr·sinω in the entire wing, that is: w [5.9] tan ω < Ωr This equation limits the maximum angle of any cord for a given wind. In order to find the total turning torque produced by all the forces, he takes a differential section of the vane, whose area will be cdr, which, introduced in equation [5.6] will produce a force dD whose projection on the rotating plane will cause a torque dM = r·sinω·dD. Once all the values are substituted we arrive at: dM =

pa ⎡ ⎤ 2 ρ r sin ω ⎢1 + ( w cos ω − r Ω sin ω ) dr 2 2 2 ⎥ 1 2 ⎣ ρ gb + 2 ρ ( w + Ω r ) ⎦

N

[5.10]

To simplify this formula he assumes the term in first bracket as constant [§.XXVI], which he justifies by saying that ρgb is a lot greater than ½ρ(w² + Ω²r²). Therefore, the mentioned bracket will be replaced by a factor f defined as: p f = 1+ a [5.11] ρ gb With this precision the final formula for the torque, which expresses the relation between the vane geometry, the wind speed, and rotation, will be [§.XXVII]:

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THE GENESIS OF FLUID MECHANICS, 1640–1780

M =

2

rΩ ⎛ ⎞ ρ w2 f ∫ cr sin ω ⎜ cos ω − sin ω ⎟ dr 2 w ⎝ ⎠

N

[5.12]

This formula is valid only if the condition expressed in equation [5.9] is satisfied in the entire vane. If this is not so, then a function sign of α i will have to be included in the sub-integral function, whose value is +1 if α i > 0 and –1 if αi < 0.39 Euler goes on to consider a series of geometrically simple cases. The first one, of considerable practical interest, is for rectangular sails with a constant angle of incidence. This simplifies the resolution of the previous integral, as only r remains as an independent variable. The result is: R

⎡1 ⎤ 2r 3Ω r 4Ω2 M = ρ w fc sin ω ⎢ r 2 cos 2 ω − sin ω cos ω + sin 2 ω ⎥ 2 2 3w 4w ⎣2 ⎦r N

2

[5.13]

0

In this expression we gave as integration limits the span, designated as R, and the distance at which the sail starts from the shaft r0. Euler took this last value to be zero, which is not possible geometrically, although it could be considered as an approximation. We shall suppose it so from now on. Thus the formula becomes: M =

⎡ 1 2 RΩ ⎤ R 2Ω2 ρ w2 fcR 2 sin ω cos 2 ω ⎢ − tan ω + tan 2 ω ⎥ 2 2 4w ⎣ 2 3w ⎦

N

[5.14]

He analyses a new case in particular [§.XXXII], in which the angle of the sail is such that the incidence is zero at the tip, that is: tanω = w/ΩR. It is obtained under these circumstances: M =

N

ρ w2 fcR 2 sin ω cos 2 ω

[5.15]

2 He notes that ‘if one increases the velocity even more, the drive will certainly be reduced’ [§.XXXII], as the angle of attack starts to become negative from the tip inwards. This is due the fact that the force produced by the flow is proportional to the squared of sinαI, but also its direction changes with this angle, a fact concealed by the formula because the square is always positive. 39

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Up to now his dissertation concerns itself with the wheel on its own. The next step is to convert it into a machine, i.e., to provide a driving torque that can be used, and which will be called Mm. Euler treats this topic very prolifically, starting the argument from the condition that the machine be capable of starting up from the initial rest situation, defined by Ω = 0 [§.XXXIII], and following with the influence of the wind velocity, until he arrives at the calculation of the angle producing greater power, which he calls ‘of the greatest effect’. The power is the product of the driving torque times the angular velocity, W = MmΩ, and once the velocity is introduced in equation [5.14] we arrive at the power. The maximum is found by derivation of the function in ω and making the result equal to zero. The operation is tedious, and so we will give the result immediately [§. XLVIII]40: ΩR w

tan ω =

8 − 10 9

= 0.537514

[5.16]

Substituting this value in the expression of moment we will obtain the maximum power available: W max = µ

N 2

ρ w3 fcR cos 3 ω

[5.17]

The factor µ is worth approximately 0.114967. These numerical values justify the value he gives in his report ‘Maximes pour arranger le plus …’ of 1752. At this point Euler applies these results to the windmill mentioned by Lulofs, and in order to get consistent data he finds that the coefficient f must be 1.70. However, he does not commit himself, declaring that he does ‘not wish to say anything yet about the true value of f, as in the case of the experiment it does not agree sufficiently with what I applied to the calculation’ [§.LIV]. As he believes that friction plays an important role, he supposes a constant torque for friction that will be subtracted from the torque produced by the machine. He repeats all the operations, and we shall examine this point. When the twist angle is not fixed, the most interesting study is the one where he tries to find the optimum distribution of the incidences in order to obtain the optimum power. Part of the equation expressing this is: 40

The solution presented follows Euler’s footsteps, but it is not correct. As what he does is to take ΩRtanω/v0 as the parameter upon the derivation is based, but he left out a factor in cos2ω that he considers as constant.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

W=

2

rΩ ⎛ ⎞ ρ w2 f Ω ∫ c r sin ω ⎜ cos ω − sin ω ⎟ dr 2 w ⎝ ⎠

N

[5.18]

The search for the distribution ω(r) that will maximize this integral is what is called a variational problem. In this case the problem is relatively simple, and boils down to solving the following equation: ∂

⎡⎣ rc sin ω ( w cos ω − r Ω sin ω ) 2 ⎤⎦ = 0 ∂ω

[5.19]

The result of this derivation, and making it equal to zero, leads to the following distribution: tan ω =

3 Ωr 4 w

+

9 r 2Ω2 16 w

2

+

1 2

[5.20]

Where it can be seen that the parameter Ωr/w continues to appear, and that it is independent of the chord. MacLaurin, had already arrived at a similar solution, as Smeaton stated. 41 As Euler notes, in the axis, r = 0, the angle will always be arctan(1/√2) = 35.26°. On the other hand, he also indicates that there will not be a negative incidence in any point of the vane, as in the entire one, where tanω < w/Ωr. He now supposes a vane with a constant chord, and with a distribution of this type, and he calculates the impulse it would give and also the maximum power. The calculation is laborious, and we shall jump to the final result. The expression is equivalent to equation [5.17], but the factor µ = 0.114967 is greater than the previous one. Euler does not end here, but continues to extend his calculations by introducing friction, thus leading to the repetition of some of the cases already seen, but with a specific casuistic. However, we judge that we have said enough to make Euler’s contribution to this problem clear. Naval applications

Naval theory is one of the fields where the applications of the resistance problem theory were most widely known. This should not surprise us, as a sailing ship is the fluid machine par excellence. Its essential components are the hull and the 41

Cf. ‘An experimental enquiry …’, Part III, p. 146.

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243

sails. The sails collect the driving force of the wind, the hull maintains and stabilizes the vessel on the water; the balance between them is the interaction of two basic fluids. However, these two elements, without doubt primordial, are not enough to explain the theory of ships. This, in today’s words but with ancient echoes, is a system in which these elements join others, be they theoretical, material or even human, in order to integrate a whole with one goal; to navigate in a safe and stable manner at the command of the Captain’s wishes. We would be bold enough to say that the ship of the line, understood it in its precise sense as a naval fighting ship of the eighteenth century, was the emblematic symbol of its time. The interest in ships demonstrated by scientists, experimenters and construction engineers of the century was very great, and can be explained by two points: one, the importance of the navy in both its commercial and military aspects; the other, the attractiveness that a ship held per se for the eminent minds of the time, as a machine that can be rendered in mathematical terms and analyzed. Both aspects merit a few words. A ship is an entity that has formed part of human culture, being as it is an exemplar of the degree of technical development of those civilizations that had contact with the sea. It suffices to cast a glance over the primitive launches and canoes to the ocean navy of the European powers of the eighteenth century, via the Mediterranean navy of the Phoenicians, Greeks and Romans, the Renaissance galleys and vessels, the caravels and galleons of the beginnings of the modern age, in order to appreciate how, in this the long run, naval construction followed a slow but steady evolution. The ship of the line of 120 cannons attained up to 4,900 tons of displacement, with a crew of 1,000 men,42 with its three decks rigging, sails, cannons, etc., is a sample of the very elaborate technique achieved by naval construction, whose basis consisted almost exclusively of artisan knowledge and experience. Maritime activity was a sign of economic development, and as such was in constant increase in parallel with the latter. This increase was more noticeable in the eighteenth century due to both the trade aspects, particularly with America, and to the European conflicts, which, although they continued to be fought in the old continent, extended to American waters. Great Britain and France, the two hegemonic powers, with Spain as a third and frequent ally of the second, maintained a continual dispute, carried on in Europe both by land and sea. In the New Continent the fight was almost exclusively naval, and the combatants were Spain

42 The most descriptive work of a ship of the line is the Le vaisseau de 74 canons by Jean Boudriot. Editions des Quatre Seigneurs. Grenoble, 1973.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

and Great Britain, as French presence in those lands had almost disappeared by the beginning of the eighteenth century.43 In the extensive area of European influence, a ship was the only suitable method of transport for bringing the riches and products from America, Africa and the Far East, and was almost without substitute in European trade. To trade interests is added the fact that the navy is an extension of the force and capability of domination. It retains this characteristic even to this day. Important installations such as shipyards were required for naval construction, backed up by numerous and complex ancillary industries for manufacturing, sailcloth, cables and ropes, anchors, carpentry, bitumen, etc., i.e., many and varied raw materials and crafts are needed. For one reason or another, the navy represented a high percentage of the economy of European countries, and it was a basic instrument of international relations. Thus it is not strange that the various nations stimulated and promoted any study whatsoever that would serve to improve the capability of ships, as is reflected in the studies, books and prizes published and awarded by the different Academies of Science. Another motive justifying the interest shown in ships is that in themselves they hold a great attraction for any mathematician of the time. The sailing ship is a machine where ordinary mechanics and fluid mechanics meet and cross. Essentially, and it is worth repeating, a ship consists of a hull floating on the water, propelled by the force of the wind in the sails. The two fluid principles, air and water enter into action, although each one acts on very different bodies and in almost opposing conditions. As regards the wind, the maximum force is sought, while for water the aim is to reduce the resistance. As far as the dynamics is concerned, a ship is conceptually much more difficult than any land vehicle. Rolling, pitching and vertical oscillations appear naturally, and the sails intervene in the control of direction just as much as the rudder. The theoretical analysis of all these aspects provides numerous general and specific studies where the calculation is made to serve naval theory. Such were the studies on the shapes of the sails, on the optimum forms for reducing resistance, the distribution of the rigging in order to obtain the maximum advantage from the force of the wind, among others.

43 At the end of the century France had a population of 28 million, Great Britain 16 million and Spain 11 million. In 1772, the fleets were 63 ships of the line for France, 128 for Britain and 55 for Spain (Cf. Antonio de Ulloa, La Marina. Fuerzas navales de la Europa y Costas de Berbería, book written in 1773 but not published until 1995 by the University of Cádiz, Spain).

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The hull, on the other hand, is the structure that supports the system. The forces produced by the water impact directly upon the hull, and indirectly the effect of the wind transmitted by the masts, together with the inertial loads of the structure and cargo. Materials with a structural function intervene in its construction, whether to support the force in the sails or to avoid the ship breaking up from the effect of the waves. To sum up, the ship is not an abstract entity, but an engineering machine whose problems are not limited to fluid dynamics. Therefore, the authors, whether mathematicians or engineers, felt tempted to ask and theorize about forces and maneuvering, and also about the mechanical resistance of the rope lines and stays, timbers or other material integrating the vessel.44 Apart from the historical interest of naval mechanics, there is an aspect we would like to mention. Although it is not the main object of our investigations, it could constitute an example of evolution in the complex world of sciencetechnology relations, and will merit more attention later on. The question refers to the development process in a technical field ranging from handcrafted items to technological ones. In our opinion, this process goes through three steps. First, there is the application of rational theories of a modelling process to existing, or potentially possible handcrafted items. Second, there is the modification and construction of new models applying the results of these theories. This implies a jump, as making new models requires the presence of other natural elements not

44

The list of works about naval matters in the eighteenth century is quite long. We give a chronological relation of the most significant: -

De la Théorie de la Manœuvre des Vaisseaux, 1689, by Bernard Renau d’Elizagaray. Théorie de la construction des Vaisseaux, 1697, by Paul Hoste. Essay d’une nouvelle théorie de la manoeuvre des vaisseaux, 1714, by Johann Bernoulli. De la mature des Vaisseaux, 1727, by Pierre Bouguer. La théorie de la manoeuvre des Vaisseaux reduite en practique, 1731, by Henri Pitot. Traité du navire, de sa construction et de ses mouvemens, 1746, by Pierre Bouguer. Scientia navalis seu tractatus de construendis ac dirigendis navibus, 1749, by Leonard Euler. ‘Solutions des principaux problémes de la manoeuvre des Vaisseaux’, by Pierre Bouguer in the Mémoirs de l’Academie de Paris, en 1754 y 1755. De la manœuvre des vaisseaux, 1757, by Pierre Bouguer. Architectura navalis mercatoria, 1768, by Fredrik Henrik af Chapman. Examen Maritimo Theórico Práctico ó tratado de mechanica aplicado á la construccion, conocimiento y manejo de los navíos y demas embarcaciones, 1771, by Jorge Juan y Santacilia. Théorie complète de la construction et de la manœuvre des vaisseaux, 1773, by Leonard Euler. Tractat Om Skepps-Byggeriet, 1775, by Fredrik Henrik af Chapman.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

contemplated in the theories. Third, there is the interrelation between these results and the theory that gives rise to a technological discipline, with the application of the scientific method according to the scheme hypothesis-deductionexperimentation. The theorising movement that took place in the eighteenth century in the naval environment is a sample of the first phase in the process of ‘technologising’ an artisan activity. Until that century there was no naval theory. All the development was handcrafted and the scholars testify to this, as they complain that the design and construction was in the hands of carpenters or workmen with limited knowledge. The same scholars justify their own work, saying that these throw light on the matter and thanks to them the performance and construction of ships would improve. This is the first phase: the application of available theories in order to understand or explain the functioning and behavior of ships. We cannot be sure that the second phase arose in the eighteenth century: perhaps it started at the end of this period, applying these theories to models in order to obtain rules or laws acting directly on ship building. The third came later in the next century, consisting of the interrelation between science and technology, giving rise to modern technological and scientific techniques. Thus the study of naval activities of the eighteenth century could throw light on the analysis of technological development, although, we repeat, it would form the object of a specific study.45 The forces in the hull The applications of the resistance theory act on two complementary aspects. The first is to calculate the resistance, whether as an absolute or as a coefficient, both in magnitude and in direction, for a given hull. The second searches for the optimum form of the hull in order to reduce the resistance value. In accordance with the stipulations of impact theory, the resistance will be found as an integral of the surface, in which the normal velocity and the local resistance coefficient intervene, as expressed by the following formula:

F=

∫

1 2

ρ vn2C D dS

[5.21]

Σ

45

About the evolution of the naval architecture in the seventeenth and eighteenth centuries, cf. Ships and Science. The Birth of Naval Architecture in the Scientific Revolution, 1600–1800, by Larrie D. Ferreiro (Cambridge: MIT Press, 2007).

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The integral should apply to the entire hull except the shadow zone, i.e., where vn > 0. There were two criteria for the resistance coefficient: one adopting an experimental value, the other adopting a theoretical one. As we have shown, Bouguer was in the first group,46 and in his Traité du navire took an experimental value which translated to a coefficient was CD = 1.21.47 For the case of air, he states that the density is 576 times less than that of water, a figure that he attributes directly to Mariotte. However, in the De la manœuvre des vaisseaux he takes 850 for this value, and says that sea water at 10 ft/s exercises a force of 120 lb, which translated gives CD = 1.01.48 The fact that he uses an experimental value instead of a theoretical one indicates the practical focus of his treatise quite clearly. By contrast, Euler, as it has been also explained,49 in the Scientia navalis surprisingly suggests two theories for the resistance that follow the values of CD = 2 and CD = 1. Between the two, he chooses the last without much justification. In the Théorie complète de la construction et de la manœuvre des vaisseaux (Theory complete of ship construction and manoeuvering), which was an abridged and updated version of Scientia navalis, he maintains the value CD = 1 but with a new line of thought.50 He also makes a criticism when he says that ‘the resistance theory we have expressed here is still very defective, and we only know enough to make a rough evaluation of the results deduced from it’.51 He quotes the effect of the stern as the main defect, as its shape influences how the water comes back over the ship, and it will increase the value of the resistance in such a fashion that ‘if one thinks about it a little, it is easy to understand that this increase depends almost entirely on the shape of the stern, which has been completely ignored up to now in research on the resistance’.52 As a result of the action of the fluid, upon integrating the equation [5.21] extended to the submerged surface of the hull, a force in the direction of the keel will be obtained and another in the lateral direction, both as a function of the drift angle, as is shown in Fig. 5-15. The resulting equations are of the following type: f a = A cos 2 λ + B sin 2 λ ;

46

f l = 2 B cos λ sin λ

[5.22]

Cf. Chapter 3 ‘The naval work of Pierre Bouguer’. Cf. Traité du navire, Lib. III.I.II.§.I. 48 Cf. De la manœeuvre des vaisseaux, I.III.I. 49 Cf. Chapter 3, ‘Euler’s Scientia navalis’. 50 He says that if the surface had a velocity v and it was perforated with a small hole, the water would egress with such velocity, that this will be the one reached falling from the kinetic height. 51 Théorie complète de la construction … II.III.§.23. 52 Op. cit. II.III.§.24. 47

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THE GENESIS OF FLUID MECHANICS, 1640–1780

H

FS RT R

Fig. 5-15. Forces due to the leeway

A and B being parameters depending on the form of the ships bottom, and λ on the leeway or drift angle. These equations were deduced for the first time by Jakob Bernoulli for a two-dimensional figure, although he did not take into account the limits of the shadow zone. This depends on the angle λ, thus the constants will not be the same ones, but a function of this angle. Later, with Bouguer a ship is no longer considered as a plane element, but acquires a threedimensional nature in his work De la mature des vaisseaux. From the point of view of calculus, this shape requires the mathematical process of integration to be extended to the entire surface of the hull instead of a line, arriving at equations similar to those given. However, Bouguer considers that there is no shadow zone as such, but there is water flowing,53 a fact that induces him to correct the formula of a lateral force with an additional term after declaring that ‘it is not enough to consider only the bow with respect to the lateral drive perpendicular to the keel. The entire flank of the ship that does not contribute anything to the direct drive is pushed by its side’.54 The formula is left as: f l = 2 B cos λ sin λ + C sin 2 λ

[5.23]

This complicates his calculations a little more. The numerical solution of the integral follows the idiosyncrasies of each author. Thus Euler presents long discussions whose starting point is the definition of the ship’s bottom as a mathematical surface to which he applies the concepts of analytical geometry in order to solve the problem. Bouguer had divided the surface into small triangles, 53 54

Cf. Chapter 3 ‘The naval work of Pierre Bouguer’. Cf. De la manoeuvre, III.II.§.VIII.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

249

adding the forces acting on each one. Jorge Juan divides this surface in panels by means of planes, which allows him to carry out a numerical integration based on the formula for a spatially defined and orientated panel. In spite of the elegant and ingenious procedures they use, we shall not consider them here. As regards Juan, his theory, as we have seen,55 is based on considering that the differential force is given by the following expression: dF = 12 ρ (vz ± vn ) 2 dS

[5.24]

Here the positive sign corresponds to the bow and the negative to the stern. What he does next is to divide the hull by a series of horizontal and vertical planes in the keel direction at intervals ∆z and ∆y, respectively, as shown in Fig. 5-16. Each pair of adjoining planes defines two rectangles, one in the bow and the other in the stern. If the normal velocities on one and the other are vn1 and vn2, the pressures will be p1 = ½ρ(vz + vn1)² on the bow and p2 = ½ρ(vz – vn2)² on the stern. The projections of these forces on the axis parallel to the keel will be p1∆z∆y and p2∆z∆y, respectively, and the resulting axial force that will be the difference between them, ∆F = (p1 – p2)∆z∆y, will be: ∆F = ρ (2vz vn1 + 2vz vn 2 + vn21 − vn22 )∆x∆y

[5.25]

Jorge Juan considers the difference vn1² – vn2² to be negligible; thus substituting the value of vz = ∫2gz, vn1 = vsinθ1 and vn2 = vsinθ2, we will end up with: ∆F = ρ 2 gz v (sin θ1 + sin θ 2 ) ∆x∆y

[5.26]

This formula is suitable for tabulation in the form of hull data. He does not forget to add other effects, ‘comprising the planking, the keel, astern post, stern, cutwater and rudder’56 about which he makes acute side comments. The total resistance to advance is expressed by Juan as ρRu, where R is a factor equivalent to what we have termed the resistance coefficient, and u is the ship velocity or advance according to the keel, as Juan calls it. If the ship were to advance with a leeway whose lateral velocity perpendicular to the hull was v, the lateral force

55 Cf. Chapter 3, ‘The theory of Jorge Juan’. A more detailed study of this theory is given in Julián Simón, ‘La Mecánica de los Fluidos en Jorge Juan’ (‘Jorge Juan’s Fluid Mechanics Theory’). Asclepio, Vol. 53-2, CSIC, Madrid, 2001, pp. 213–280. 56 Cf. Examen marítimo II.§.181.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

induced would be expressed by the ρrv equivalent to the previous one.57 Note, finally, Juan’s introduction of the resistances of the unevenness in the calculus, which is what he calls the associated waves. He carries out all the formulae, although he ends up by considering their effect to be negligible.

Vn1

Vn2

Fig. 5-16. Hull section in Jorge Juan

Studies on the shapes of minimum resistance enjoyed great popularity at the time, due to the mathematical nature of the problem. The search for solids offering the minimum resistance to the motion of a fluid was not a new problem,58 as previously Newton, Fatio, Johann Bernoulli, l’Hôpital and some others had worked on this topic. Nevertheless, both Euler and Bouguer established a new field in applying it to the hull of ships. From the point of view of analysis, these studies were among the first applications of variational calculus. This was one of the favorite themes of Bouguer, who realized that the minimum resistance solid, or the bow, in the case of a ship, did not necessarily imply that it was the best from a naval viewpoint, but that additional factors had to be introduced. These had to be defined as a consequence of other requirements or characteristics of the vessel. This led him to define several classes or families of bows. As he himself says in the Traité du navire: We have forgotten the distinction that there is among these figures and those which are really the most advantageous for making the day’s run rapid. In our investigation of the first, we have not taken into consideration the weight of the vessel or the form it should have in order to sustain the sails. It is not only a question of determining the shape that offers the least resistance, … but … we must add another consideration, to pay attention to the weight of the vessel, or rather to its moment with respect to the metacenter. [III.V.V.§.I]

57 58

Op. cit. II.§.337. In this respect cf. Chapter 2, ‘The solid of minimum resistance’.

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The first work that Bouguer devoted to these matters was De la mature des vaisseaux. Later on he produced the monographic ‘Une base qui est exposée au choc d’un Fluide étant donnée, trouver l’espece de Conoïde dont il faut la couvrir, pour que l’impulsion soit la moindre qu’il est posible’ (‘Given a base that is exposed to the impact of a fluid, to find the type of conoid required to cover it so that the drive be the least possible’),59 he continued this theme in the Traité du navire, and returned to the topic in the ‘De la impulsion des Fluides sur les Proues faites en pyramidoïes, dont la base est un trapèze’ (‘Of the driving of fluid on bows constructed in pyramids whose base is a trapezium’).60 In all of these he presents the process of minimizing surfaces using what we call nowadays variation calculus, and which began to be developed at this period. Bouguer, in De la mature des vaisseaux, made use of the minimum resistance solid with a circular base. This was one of the examples most studied by mathematicians, but as he himself reflects: A discovery that is apt for application, should, it would appear, be rapidly put into practice. This one has not yet, and the reason might be that it has been acknowledged that it is only apt for one extremely special case.61

With this premise he enlarges his studies to include solids with non-circular bases in his work of 1733. He assumes a body, like that represented in Fig. 5-17, formed by a base ABC of a given shape, and a length DE also defined in such a way that the figures abc are geometrically similar and parallel to the base, leaning on a curve CE that has to be determined. He initiated the calculation process by finding the force that appears in a surface element, and which he later integrates

D

A a

B b

d E

C

c

Fig. 5-17. Minimum resistance bow 59

In the Mém. Acad. of París, 1733. Mém. Acad. de París, 1746. 61 Cf. ‘Une base qui est …’ , p. 85. 60

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THE GENESIS OF FLUID MECHANICS, 1640–1780

for the entire surface in order to obtain the total resistance, which is a function of the curve CE. He ends with an integral equation from which he deduces the differential equation using the variational calculus method. We shall not go into the mathematical details, but confine ourselves to noting that he arrives at the general application formulas, which he specifies in one case by way of example. In the remarkably extensive last article of 1746, he centres his attention on the case with a trapezoidal base with a pyramidal structure that constitutes another case specific to the first group. He writes: Besides the fact that they are most apt for procuring the stability of the floating bodies, they are preferable to the circular ones and other figures with the same extension, as what is required is cut-off pointed shapes cleaving through the medium with the least possible resistance.62

From the study of these pyramidal bows, he infers interesting geometrical relations with respect to the effects of the lateral currents and other properties. In the Traité du navire, Bouguer devotes a lot of work to this matter, almost the whole Section V of the Lib. III. He knows that a ship is a complex system and the optimum bow has to be related to the rest of the system. In this idea he defines two classes, one of least resistance (de la moindre résistance) and one with the greatest velocity (de la plus grande vitesse) [Lib. III.V]. The first corresponds, within given dimensions, to the bow that allows the greatest speed in a day’s run, i.e., whose resistance is minimum. Nevertheless he says: From the diverse facts offered, it appears that it is very doubtful that the form that cleaves the water more easily is absolutely the most advantageous in order to obtain a rapid day’s run, as it is possible that a bow giving a little more resistance will enable the Vessel capable of supporting a proportionally greater quantity of sails. [Lib. III.V.I.]

He bases this line of thought on the fact that in order to maintain the velocity of the advance, the force of the wind is required in the sails. Therefore, the best hull shape will be that which combines little resistance with a greater sail bearing capacity, and this is what he calls having greater velocity. In order to determine this, he has to introduce other vessel data such as the mass distribution, and the stability or metacentric height, which, at the end of the story is what limits the amount and size of the sails to be used. He concludes these studies with two types of the aforesaid bows, a flat one and a pointed one.

62

Cf. ‘De impulsion des fluides …’, p. 237.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES A

253

C A Y B

H X

W

I

Fig. 5-18. Complex bow

In the Traité du navire he again touches upon the case of the circular base [Lib. III,V.II] as well as those having triangular shape [Lib. III.V.I], which he now generates for horizontal planes instead of vertical ones. However, the more interesting case is when he tries to find other figures, closer to the real one. We only present one case as an example [Lib. III.V.III.§.I]. Geometrically the shape he seeks is the intersection of a cylinder of a given base ABA and axis CH (Fig. 5-18), with a prism of base AHA and height HI. This bow is pointed and its vertical sides diminish progressively towards point A. He sought the equation of the line AHA, y = f(x), to obtain the minimum resistance. He also assumes the contour ABA of the cylinder as a function w = (bm – cym)1/n. He obtains the mathematical solution of the problem and carries on some discussion for son particular values of the parameters (n = 1 and m = 2 for an elliptical shape, c = 0 for a rectangular one). He follows with more types (Lib.III.V.IV-VI), including the stern shapes (Lib.III.V.VII-VIII) and the chapter IX for ship of war and frigates, which shows his interest in real ships. However, Bouguer does not stop here, as once the theories for these two types of bow are developed, he faces another naval problem: that of transporting large cargoes. The need for this causes him to arrive at another form: that of the greatest motion (du plus grand mouvement) [Lib. III.V.X.§.I], which he also details. We shall not dwell any longer on these aspects, but from the little already said, the complexity of an apparently simple problem and the need for mathematical analysis for finding the solution is quite clear.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Force in sails

In order to obtain the force of the wind on the sails, impact theory was also used. However, a sail is flexible element whose curvature is in function of the wind striking it, and of its lashing and tension. The problem was very difficult to solve, thus nearly all the authors likened the sails to flat plates, although almost all tried to determine the real shape the sail would adopt considered as a flexible element. This gave rise to a type of curve that they called ‘sailing curve’ (velaria). Renau carried out the first studies on the shape of the sails, and he made some comments on the curvature of the sails at the end of his Théorie de la manœuvre, pointing out that the total force of the wind upon them was the sum of the elemental forces upon each point of the sail. Later on, the brothers Johann and Jakob Bernoulli maintained an active discussion about the shape of the sail that included cutting comments and biting allusions, and they did not tread lightly when it came to showing up the errors of others. Johann, in his Essay d’une nouvelle théorie de la manœuvre des vaisseaux, quotes the successive formulae contributed by Jacob between 1692 and 1695.63 Of these, Johann admits the last,64 repeating the development of his brother in the Essay. Figure 5-19 represents a sail fragment ABH, about which he says that it is like a: Perfectly flexible cord pushed or pulled in all its points so that it forms a curved line due to an infinity of equal or unequal forces, each one following a direction perpendicular to the curve. [XV.§.V]65

If the tangents are drawn in two points of this curve, such as A and B, the resulting force on the arc AB will follow the direction of the bisector in C of an angle formed by the said tangents. D

B

F A

C

Fig. 5-19. Sail curve or ‘velaria’ 63

Jakob had studied the sag problem, and he made use of it for the ‘velaria’ studies. Cf. Jakob’s Opera, p. 656. 65 The quotes are referred to the Essay d'une nouvelle théorie …. 64

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255

As a complement to this Johann states ‘a new method of for determining the nature of the curves of the sails, sailcloth, lines and in general all the flexible material dilated in a curved line by any action of a fluid’ [XVI.§.I]. His theoretical starting point was the isolation of an element of the curve, and the application of equilibrium conditions between the normal force and the internal tension of the thread at the ends of the curve. As a result he arrives at the conclusion that the curvature at a point must be inversely proportional to the pressure at the same point. [XVI.§.III]. He exemplifies this condition in three assumptions. The first is where the pressure is uniform [XVI.§.V], and therefore the curvature constant, thus giving rise to a circumference. He gives the example of a soap bubble that acquires a spherical surface. Although his demonstration refers to plane figure, which is not the case with the soap bubble, the example is valid. The second is the case of the sail, where he considers that the winds slides off it, with the result that the local pressure is proportional to the square of the sine at the impinging point, that is to say the impact law. He arrives at the equation dy3 = adsd²x [XVI.§.VI], which, being subjected to some transformations, yields dy = adx 2ax + x 2 , which he says coincides with what he had already obtained for the sag. As a third example [XVI.§.VIII] he analyses the curve that will be formed by a bag made by a flexible cloth filled with a heavy liquid. The pressure in this case will be proportional to the vertical height of the liquid at each point of the bag, and the resulting equation is ydyds = ½a²d²x, which, reduced to a first order equation, will become dx = y 2 dy a 4 − y 4 . All this makes quite clear the talent and acuteness of Johann in applying differential analysis, which is the kernel of his contributions to the analysis of motion of fluids.66 Bouguer does not make important contributions to this theme, which is somewhat surprising given his ability, and the effort he puts into his studies on optimizing the ships’ hulls. His only analyses are to be found in the De la manœuvre, as in the Traité he hardly dedicates any effort to this problem. In the first of these works he acknowledges that the curvature affects the total resulting force, ‘because the sails are struck on each point of their surface at a different angle due to their curvature’.67 However, he esteemed that a curved sail can be substituted for an equivalent flat surface.

66

One of the most important contributions to the analysis of the fluid motion was the introduction of internal pressure and the partitioning of the fluid in differential elements. One and other have their root here: in the internal tension of the thread, and in the separation of a differential fragment of it. 67 Cf. De la manœuvre des vaisseaux, I.III.II.§.II.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

a

A

A

C

e

E

C

D

b a)

B B

H b)

I

D

Fig. 5-20. Equivalent sails

In this respect, given a sail (Fig. 5-20a) of length L attached at A and B, its equivalent length Le is related to the former as Le = Lsinα, where the angle 2α is the angle formed by the tangents to the sail at its lashing points. In the case where the attachment is not symmetrical (Fig. 5-20b), he supposes that the part AB of the sail is an arc of a circle with its center in C, the intersection of the perpendicular at B and the perpendicular at DI, tangent at D. The equivalent sail will be AH, BH, being an arc of a circle with its center in C. As regards the rest of BD he says that it is subject to another law. As one can see, the construction is quite artificial. Euler in the Scientia navalis is much more explicit in his theory on the shape of the sails, but what he says is surprising, for after supposing that the sail is equivalent to a flexible thread [II.§.728-ff],68 and having devoted various lengthy analyses to this supposition, he concludes that the initial hypothesis is not probable. Following this, he embarks on a new path that resembles to the former, but leads nowhere. We shall examine both approaches. As is frequent is his proceedings, Euler begins by flat rigid sails, followed by rigid curved ones, and ends with flexible ones. As we have already noted, he likens these [II.§.754-ff] to a single thread with a point fixed at B and retained at the opposite end A by a force whose components will be E and F (Fig. 5-21). In the intermediate points of the thread the force of the wind will give rise to a normal component that he designates as p. As the thread must be in static equilibrium, Euler stipulates the condition that at any intermediate point M, the moment of the real wind forces acting between A and M must be in balance with the force of tension at A. The latter is Fx – Ey, which, made equal to one another, leads to the equation:

68

Quotes refer to Scientia navalis.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

257

x B

v

M F

y

A E

Fig. 5-21. Flexible sail in Euler

Fx − Ey = ∫ p ( y − ς ) d ς + ∫ p ( x − ξ ) d ξ

[5.27]

This, by a transformation that we omit, can be expressed as: Fx − Ey = ∫ dy ∫ pdy + ∫ dx ∫ pdx

[5.28]

The value of the force of the wind according to impact theory is expressed as p = ksen²α, having included all the constants in k. Euler’s aim is to arrive at a more general equation with a simpler expression, therefore he looks for transformations ‘that prepare the general equation in its simplest form and lead to its liberation from the integral sign’ [II.§.758]. These consist in taking the third derivative of equation [5.28] in order to eliminate F and E, which ends in69: 69

For the first derivative it is:

F−E

dy dy = dx dx

∫ pdy + ∫ pdx

For the second:

−E=

∫

⎛ dy 2 ⎞ dx 2 pdx + p ⎜⎜1 + 2 ⎟⎟ 2 = dx ⎠ d y ⎝

∫

pdx + p

For the third:

0= p

dy dx

+

dpds 2 d2y

+

2 pdsd 2 s d2y

− p

ds 2 d 3 y

(d y ) 2

2

As ds2 = dx2 + dy2, it easy to see that: d 2sds = d 2ydy, then:

ds 2 d2y

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THE GENESIS OF FLUID MECHANICS, 1640–1780

p=C

d 2 ydx

[5.29]

ds 3

It must be admitted that this expression is manifestly very simple, but it is not very useful. He submits the expression to multiple mathematical transformations, obtaining that C = E 2 + F 2 and that the force must be tangential to the sail in point A. z

B

D

t

H g

a A

K

f

Fig. 5-22. Symmetrical sail

In the application to the symmetric case, similar to that given by Bouguer, he defines the sail by its length 2g and lashing extension 2h (Fig. 5-22). After a series of calculations, which we omit, he arrives at the following equation for the sail: z = a ln

3dy dpds 2 pds 2 d 3 y − 2 − =0 dx d y (d 2 y )2 That is:

dp 3d 2 s d 3 y = − 2 p ds d y Whose solution is equation [5.29].

a + t + at + t 2 a

[5.30]

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

259

where a is the radius of the osculator circle at the vertex, or point H.70 The value of this is related to the sail data for the following equations: g=

2af + f 2

h = a ln

g + a2 + g 2 a

[5.31]

The zone equivalent to the flat sail will 2acosα [II.§.766], which he approached by c(3h – 2g)k after a series expansion. After all these ponderings, Euler reaches a point where he questions the validity of the starting hypothesis, which was the supposition that the sail was like a flexible thread. Concerning this he says: Although it appears that the sail is made of infinite threads parallel to each other, in fact each single thread is affected in the same way as if they were alone, therefore what we have obtained concerning the curvature of threads driven by the wind cannot be transferred in all strictness to the sail. [II.§.787]

He develops various arguments in this respect. The first is that in the case of the thread, the wind flows freely around it, which does not happen in the case of the sail, as this retains a great quantity of air in its bosom. In con sequence the sail does not advance ‘by driving [of the wind], but is required by the single pressure of more condensed air present in many parts of it, in such a way that the force does not follow the same laws we attributed previously to the wind’ [II.§.787]. Here he voices a strong criticism of the impact model, doubting

F P w Fig. 5-23. Flat and rigid sail

70 The radius of the osculator circle to a curve is the curvature radius of the curve in the point. It is a third-order contact, i.e., the first and second derivatives are equal.

260

THE GENESIS OF FLUID MECHANICS, 1640–1780

its validity, but he does enter into the matter in detail. A last argument is the double curvature of the sail, which causes the solution to lie outside the capacities of the analysis. Faced with this undoubtedly peculiar situation (since until he reached it, Euler had filled almost twenty pages with mathematical treatments), he adopted a curious partial-experimental method, in which he went back to following the original plan by studying flat plates in order to go on to the flexible ones. The starting point is the effect of the wind on a flat rigid plate with mass which balances on a pivot (Fig. 5-23). The force of the wind will incline it to a certain angle, a function of the velocity and weight. With the help of the impact theory it is very easy to obtain this angle. After this, he supposes that the table is a perfectly flexible lamina, which, upon receiving the wind will assume a form similar to the case of the thread (Fig. 5-21). On applying the balance conditions he arrives at the same equations [5.27], with the addition of the weight as a moment producing component. He obtains: Fx − Ey = dy ∫ pdy + dx ∫ pdy + ρ sdy

[5.32]

where ρ is the lineal density of the material. He devotes a considerable number of pages to studying this equation, but at the end does not reach any conclusion. In his second work on naval theory, the Théorie complète de la construction, he adopts the flat form after affirming that In this case the sails are more or less curved, having a shape which geometricians have successfully determined. … However, as it is always possible to conceive a flat sail that produces the same force as a curved one, we shall not busy ourselves further with the curvature of the sail, and in the following investigations we shall consider them all as perfectly flat, and suppose them to be proportionately smaller.71

This quote is sufficient. As we have already argued,72 for Jorge Juan when a plate moves in the midst of a fluid, the local force on a point is proportional to the velocity and depth of the point in question. This depth has a well defined magnitude in water, but for a movement in air it is not so clear, as there is no level from which to measure the depth. Regarding this Juan says in the Examen maritimo: ‘For this reason we will suppose that instead of air there is a non elastic fluid with the 71 72

Cf. Théorie complète … 3ª Part, §.4. Cf. Chapter 3, ‘The theory of Jorge Juan’.

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

261

same density as the air that acts on the sail’ [II.§.256].73 What he proposes is to substitute the atmosphere, which he knows is made up of an elastic fluid whose density decreases with the altitude, by an incompressible uniform layer of a definite height, somewhat like a sea of air, whose bottom will be the surface of the Earth.74 The depth of this sea of air is easily obtained from the equation of p = ρgh. For the density of the air Juan takes 1/1,000 of rain water, and for the pressure at sea level 2.5 ft of mercury (762 mm), thus arriving at a 35,000 ft (10,668 m). But this value does not please him, although he does not justify this clearly,75 and in its place he holds to the experiments had made in the equator.76 These are based on the fact that for each 86 ft (26.21 m) of ascent, the barometer falls 1 line (0.1764 mm), from which he concludes that 2.5 ft of mercury will require 30,960 30,960 ft (9,437 m) of altitude, a value that he adopts and designates as D [II.259]. 77 Applying his equations, the resistance undergone by a plate moving at a depth D, with a velocity v and with an incidence α, is: p = 2 ρ Dv sin α

[5.33]

That is the basic equation for his calculation. Juan begins by supposing that the sail is curved from the beginning, and his first preoccupation is to find the shape that it will adopt. He says: Let us suppose, in order to facilitate the calculation, that this is a rectangular canvas with two vertical sides, with the sail firmly resting on them horizontally, and with the force of the wind and due to its total flexibility, the curvature that comes naturally to it, and let us speculate. [§.261]

It is remarkable that he considers the vertical sides to be fixed, when in reality the sails are attached to the yards, which are horizontal. However, it is valid as a model. Figure 5-24 represents a cross section of the sail upon which a wind acts parallel to the x-axis. He takes a differential element EA that is approachable by 73

From now on the quotes will refer to second Volume of the Examen marítimo. This model had been used in some studies of atmospheric refraction. 75 In this point, and without going in detail, we must remember that one of the problems found in measurement of the arc of latitude at the meridian in Quito (as part of the Geodesic Mission in which Jorge Juan participated) was the reduction of data to the sea level by the use of a barometer. 76 Cf. the Observaciones Astronómicas y Physicas, book 5, Chapter 4. 77 If we make this calculation with present values we obtain a height of 8,100 m. 74

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THE GENESIS OF FLUID MECHANICS, 1640–1780

G x

B d T A

E y

+d

ds

T

Fig. 5-24. Sail curvature

an arc of a circle whose curvature center will be G. The internal tension, represented by T, is taken to be constant for the entire sail, which is justified by being the forces produced by normal winds on the surface. The normal force in this element, according to [5.33] will be: dFN = k J v sin α = k J vdy

[5.34]

Where in kJ all the constants are together. This normal force ought to be balanced by the tension component T, caused by the curvature of the element, which he says is Tsin(θ + dθ) – Tsinθ = Tdθ. If this is introduced in the previous equation the following equation results: ds

ρc

=

k J vdy

[5.35]

T

Unifying the characteristics of the sail and the wind in a constant Q = T/kJ v, and su bstituting the value of the radius of the curvature by his differential expression ρc = ds3/dyd²x, the equation [5.35] is transformed into: Qd 2 x = ds 2 = dx 2 + dy 2

[5.36]

Whose solution is:

(

y = Q arccos e x / Q

)

[5.37]

FLUID-DRIVEN MACHINES AND NAVAL THEORIES

263

Jorge Juan follows a more complex procedure,78 as he introduces an ancillary parameter φ, so that x and y will be independent variables linked to the previous equation dx = sinφds and dy = cosφds. The parametric solution he obtains is: x = Q ln cos ϕ ;

y = Qϕ

[5.38]

Which, he affirms, represents a universal function true for any sail, and which is equivalent to [5.37].79

K h

K

F A

„

O

A

Fig. 5-25. Angles and forces in the sail

The curvature found is a universal standard in which any sail can be inscribed, and he makes use of it in order to find the direction of the actual resulting force and the dimensions of the equivalent sail. Thus, we are given a sail (Fig. 5-25) whose yard is the segment AK, whose length is h, acted on by a wind with an angle α. At the two ends the tangents to the sail form the two angles χK y χA, respectively. The resulting force F will have its application point at the intersection of the said tangents, and its direction will be the bisector of the angle

To solve the previous one is not complicated. Equation [5.36] can be written as: Qx″ = 1 + x′2. The substitution x′ = t results in t = tan(y/Q), that restituting t plus a second integration leads to [5.37]. 79 Its mathematical development requires a few comments. According with the starting hypotheses, its plan and final solution are correct. However, the curvature radius he adopts is dsdy/dx2, which is differs from the real one ds3/dyd²x, by the factor ds²/dy². On other hand, in the equation of the normal component of the tension T, this same factor is missing, because the triangle EAG is isosceles and not rectangle. At the end, both errors compensate each other, giving the correct solution, even then it seems strange to us as he assumes that x and y are independent. 78

264

THE GENESIS OF FLUID MECHANICS, 1640–1780

formed by them. This direction will not be perpendicular to the yard, but will form an angle with it represented by ε. After a series of algebraic transformations, Juan [II.§.288-ff.] arrives at the following expression for the force: FN =

k J va ρ h sin α sin 12 ( χ K − χ A ) 1 2

(χK − χ A )

[5.39]

Which differs from a flat sail with the same width by the factor: G=

sin 12 ( χ K − χ A ) 1 2

(χK − χ A )

[5.40]

That is to say, the equivalent sail depends on its tension, and tends to unity when this is increased. Juan concludes that the force in the sail depends on the ‘curvature’ on each side. On the other hand, the direction ε with the perpendicular to the yard is given by:

ε = 12 ( χ K − χ A )

[5.41]

With these elements he states that ‘once the theoretical principles of the sail are established, we must investigate the angles observed in practice, with the aim of applying them to the sail in their proper place’ [II.§.274]. And he continues with long disquisitions about what sailors do, in which he says that these are the usual angles, demonstrating the extensive practical knowledge he had of sailing.

PART II The Problem of Discharge

The Problem of Discharge

We have called the second major line of research into theoretical fluid mechanics in the seventeenth and eighteenth centuries the ‘discharge problem’ because its origin lay in the study of the discharge from vessels through orifices. To be more exact, two types of studies fall under this rubric: the first is the discharge we have just mentioned, the second is the motion of fluids through pipes. The second derives directly from the first, and it ranges from the rationalisation of fluid mechanics to deriving its fundamental equations. Unlike the resistance problem, rather different from the discharge problem in many ways, its social relevance is not so obvious or urgent—even in spite of the problems of water supply and distribution to towns—as the problem is more or less within the compass of experimentation. However, its theoretical interest was in fact richer and further reaching than the case of resistance, the most significant milestone being Euler’s equations, which have shone out from the middle of the eighteenth century right up to the present day. Compared to the conceptual simplicity of the impact model, the problem of discharge led first to the discovery of the relation between pressure and velocity, thus linking two hitherto separate phenomena, and later on they led to the introduction of the concept of internal pressure, a major milestone that allowed differential calculus to be applied to a fluid. Thus the dynamics of the three major categories of bodies, rigid, flexible and fluid, were unified under the same umbrella. It is true that the intervention of differential calculus was decisive, but in its turn the entire set of queries and questions constituted a stimulus for the advance of calculus. Authors like Bernoulli, a name that includes an entire family (though sometimes this is forgotten), Euler, and d’Alembert are associated as much with the mechanics or dynamics of fluids as with mathematics. Moreover, each of these complained of the difficulty in advancing further, due to the limitations of the mathematical instruments at their disposal. We must not forget that it was at that time that the frontier of theoretical mechanics passed through fluid dynamics, and note that some of the important problems in areas such as meteorology and aerodynamics, that nowadays require the integration of the equations presented by Euler and later completed by Navier and Stokes in the nineteenth century, are examples of this.

267

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The beginnings of the discharge problem go back to the decade of the 1640s, when Evangelista Torricelli and his master Benedetto Castelli studied the outlet velocity of a jet through an orifice in a tank. In 1644, Torricelli published the first known law concerning the subject, which carries his name, and indicated that this velocity was equal to that acquired by a heavy body falling from a height equal to the depth of the outlet orifice. The experimental verification of this law appeared to be very simple: to open a spigot of a known exit area and to measure the volume of liquid that flows out in a certain time, and from this data to infer the outlet flow velocity of the liquid. From recorded comments, it seems that a fair number of researchers must have devoted themselves to this task, the most outstanding being the Academy of Sciences of Paris, and the so-called Italian School of Hydraulics. The former sponsored the study of this problem in the first years following its foundation in 1666. Figures like Huygens, Mariotte and other lesser-known ones collaborated in this task. As regards the Italian School,1 a lot less is known, and this is perhaps an opportune moment to say something about it. Apart from belonging, not to one but to the multitude of states that constituted the ‘Italy’ of the era, a characteristic feature of the ‘Italian school’ was its preoccupation with currents of water and experimental hydraulics, fields with great practical interest due to the profusion of floods and overflows that took place in the Italian lakes and rivers. Galileo Galilei can be taken as the first point of reference, followed by Benedetto Castelli, Evangelista Torricelli, Domenico Gugielmini, Giovanni Poleni, Bernandino Zendrini and many more. The greater part of the work of these authors was collected in the Raccolta d’autori che trattano del moto dell’acque (Collection of authors dealing with motion in water)2 which collected together a wide range of activities, the majority of which were experimental, carried out in Italy during the second half of the seventeenth century and the first third of the eighteenth. The Raccolta constitutes an opus magnus, not only for its intrinsic interest, but also because it shows the level that hydraulics studies in Italy had attained. The decline of this School at the beginning of the eighteenth century coincides with the ascent of

1 We took this denomination from Rouse, who names it thus, and justly so, in his History of Hydraulics, p. 113. A work on the same theme exists written by Carlo Maccagni, ‘Galileo, Castelli, Torricelli and others. The Italian School of hydraulics in the 16th and 17th centuries’. 2 Edited for the first time in Florence in 1723, and the second between 1765 and 1771. This second edition, which consisted of nine volumes began with the treaty on Des corps flottants of Archimedes and the discourse Intorno alle cose che stano in sull’acqua, o che in quella si mouvono of Galileo, followed by authors of the time, some of whom were not Italians, such as Mariotte and Couplet. However, some important works of the time are missing in the Raccolta, such as De Castellis of Poleni.

THE PROBLEM OF DISCHARGE

269

the French School, which as we have said began its experiments precisely with discharge in vessels. In a certain manner, by the end of the seventeenth century the French School took over the relay baton. Let us begin with the experiments verifying Torricelli’s Law. The identification of the outflow velocity with the fall of a heavy body is mathematically expressed by the law v2 = 2gh which implies two hypotheses: that the velocity is proportional to the square root of the height, and that the proportionality constant is the root of 2g. The first experiments verified the first hypothesis, but not the second, which was interpreted as follows: the height equivalent to the fall was not the depth, but rather only a fraction of this, and furthermore, the results pointed towards this fraction being only half. Therefore, and considering the belief—we repeat, the belief—that nature seeks whole numbers, the conclusion was that the height of the equivalent fall was half the depth. Even Newton fell into this error, as the first edition of the Principia shows, and he rectified it in the second edition by introducing the phenomena of the stream contraction, about which we have already spoken. The reality of the problem lies precisely there: in the formation of the outlet jet, as the field of velocity in the neighbourhood of the orifice is quite complex, and in which the geometry of the outlet spout, should it exist, plays a determinant role, along with other properties of fluid not well known at the time. In this contradictory situation, Daniel Bernoulli tried to study motion through pipes. He considered the fluid as a group of corpuscles that advanced, subject to the principle of conservation of lives forces, in a form that we might identify today as the conservation of mechanical energy of the fluid. The novelty of this approach was that it invoked a regulating principle of a general nature involving the entire fluid. Bernoulli’s ideas followed the main lines of dynamics: the reduction of problems to laws of a general nature. Newton introduced what we know today as the laws of dynamics, years before Huygens had introduced the conservation of lives forces, and later d’Alembert was to do this with his principle of dynamics. Both Bernoulli and Newton imagined the fluid to be a grouping of corpuscles, but contrary to Newton, who needed to know the individual interactions in order to resolve the problem, Daniel applied a law covering the entire fluid in an overall fashion, with the single hypotheses that the internal interactions were elastic. Bernoulli’s ideas appeared for the first time in 1727 in the Commentarii Petropolitanae, but where they attained their maximum development was in another of the memorable works of the eighteenth century: the Hydrodynamica (1738), in which he not only obtained the basic formula of motion through ducts, with specific particularisation in the case of outlet through an orifice, but he also dealt with the relation between velocity and pressure, which is nowadays known as Bernoulli’s Theorem.

270

THE GENESIS OF FLUID MECHANICS, 1640–1780

The Hydrodynamica was answered by another no less important work: the Hydraulica written by his father Johann Bernoulli, which appeared in 1742. The history of these two works is worth telling. The basic difference between them is that Johann took Newtonian principles, already transformed into his differential formulation, as the regulating ones, and he introduced the concept of internal pressure. Using both of these, he divides the fluid into differential elements capable of being analysed individually, which he employs to reformulate the analysis carried out by his son, in particular what we call today ‘Bernoulli’s Theorem’, which in the process comes very close to its current version. Actually, we are of the opinion that the theorem ought to be called ‘the Bernoullis’ theorem’, as it is due as much to the father as to the son. Johann’s works are in fact the prelude to the general equations of fluids which came to light in the following decade. D’Alembert intervenes also in motion in ducts in his Traité de l’equilibre et du mouvement des fluids, but his solution is similar to that of Daniel Bernoulli for the conservation of live forces; the only difference is that d’Alembert says that this principle is deduced as a consequence of his general principle of dynamics. These studies facilitated a period of activity that at a conceptual level was spectacularly fertilite over a short period of time. These were the years between 1743 and 1755, in which three eminent mathematicians: Clairaut, d’Alembert, and Euler, established the first mathematical formulation of this science. Its merit was to set out a body of theory based on rational mechanics, which has sustained fluid mechanics since then, providing it with meaning and purpose, and serving as a means of explaining its phenomena. However, the magnificent equations reached by them, known today as the Euler equations, and which have undergone practically no modification since then, could not be exploited until the nineteenth century for a lack of suitable tools. They constituted the highest peak of the mechanics of their time, but they remain there really as a unique monument, apt only for contemplation. As Roger Hahn justifiably puts it,3 the realisation of the uselessness of these equations forced the increase in experimentation. Part II will begin with the first experimenters who tried to verify Torricelli’s Law. It will be followed by a chapter dedicated to two great works, Hydrodynamica and Hydraulica, to the extent to which these deal with motion in ducts and vessels, complemented with a short review of the works of d’Alembert and Borda. What we call the ‘grand theorisation’ has been divided into two chapters, and the last chapter will be devoted to applications, where we identify two important groups of machines: hydraulic pumps and impulsion machines. 3

L’hydrodynamique au XVIIIe siècle, aspects scientifiques et sociologiques.

Chapter 6 Discharge from Vessels and Tanks

Torricelli’s Law Torricelli’s Law occupied central place in the first works on fluid mechanics, resolving the question how to determine the velocity with which a jet of water comes out of an orifice located in the lower part of a tank or vessel. The law, which relates velocity to the height, was established by Torricelli in his De motu gravium (On the movement of heavy bodies) (1644),1 a work that dealt with motion of projectiles. However, the problem was not new, as according to what he states and what other authors confirm, Benedetto Castelli had already investigated this matter. In the work quoted there is a set of propositions entitled ‘De motu aquarum’ (‘On movement in water’), in which he analyses the motion of a jet when it is vertical or angled. Torricelli begins with the following statement: [T]he waters that come out violently have the same impetus at the outlet point as any heavy bodies, or a drop of the same water, would have if they fell naturally from the highest surface of the water to the outlet orifice.2

Torricelli does not offer any experimental justification for this statement, although there is an experiment, which we call that of the jets, that does support it. As is shown in Fig. 6-1a, taken from De motu, it consists of a tank whose lower

C

D

A

Fig. 6-1. Torricelli’s jet experiment 1 2

The De motu gravium forms part of the Opera geometrica. Cf. De motu gravium, p. 191.

271

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THE GENESIS OF FLUID MECHANICS, 1640–1780

part has an outlying part with an orifice from which the water spouts upwards. The jet, as Torricelli observes, reaches almost the same height as the water level, indicating that the output velocity of the water jet would be the same as that which a heavy body falling from this height would have. The fact that the upper part of the jet does not actually reach this level is attributed by him to two causes: the resistance of the air and the breaking up of the jet into small drops. The translation of this law into a present-day formula is:

vs = 2 gh

[6.1]

In the case where the discharge orifice is located in the lateral wall, as Fig. 6-1b also shows, the jet would follow a parabola similar to that of a projectile, which Torricelli explains according to the Galilean principles he had already used when studying their movement. This is shown in an additional corollary which is also interesting in that he makes a direct reference to Castelli in it. Referring to the lateral outlet in the tank and following the figure here reproduced, he says: From here it is deduced, according to the doctrine of the Abbot Castelli, that the relation of the quantity of water coming out of orifice C is to that coming out from orifice D as CE is to DF (in the case that the two holes were equal). That is to say, water coming out from equal orifices has the ratio of the square root of the sublimitas or their heights. The first person of all those to investigate the truth of this Corollary experimentally was that extremely erudite gentleman, Rafel Magiotto, who dominated all the sciences equally. He confirmed our truth with the joy of success.3

In order to make it clear, the curve AEF is a parabola and by the ‘ratio subdupla of the sublimitas’4 we understand the square root of the height. According to the text, Torricelli clearly states that Castelli knew that the fall velocity was proportional to the square root of the fall height. However, later authors such as Poleni,5 Guglielmini,6 Daniel Bernoulli7 and du Hamel8 deny this, stating that he maintained the velocity was proportional to the height. Another notable point of 3

Op. cit. p. 196. The magnitudes CE and DF are the chords of a parabola whose vertex A is on the surface of the water and its vertical axis passes through C and D. These have been left out in the figures. 4 The ‘sublimita’ is what we have called kinetic height. 5 Cf. ‘Del Motu misto ….’ §.XXIV-ff. 6 Cf. Introduction to Aquarum fluentium mensura. 7 Cf. Hydrodynamica, Sect. I, §.2. 8 Cf. Regiæ Scientarum, Book I, Chap. V.

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the text quoted is the reference to Magiotto, an experimenter who was the first in a long line who tried to verify this law. It should be noted that the Law is not so obvious, as the phenomenon of a drop coming out from the bottom of a tank is not directly assimilated to the fall of a heavy body from the surface to the bottom of the tank. The law was important for three reasons: it was the first to refer to water in motion; it established a method not only for finding the velocity, but also the water flows or discharge rates; and it held for both solids and liquids. This justifies the important role played by Torricelli9 in the development of fluid mechanics. The work of the Paris Academy

Among the first projects carried out in the Paris Academy were the attempts to confirm Torricelli’s Law. In this connection, Jean-Baptiste du Hamel, in his Regiæ Scientarum Academiæ Historia Parisis (History of the Royal Academy of Sciences of Paris) (1698)10 when dealing with activities related to hydrostatics11 between 1668 and 1669, cited several experiments undertaken by Picard and Huygens to prove that the outflow velocity was proportional to the square root of the height, as Torricelli said, and not to the actual height, as was attributed to Castelli. According to du Hamel, several comparisons between water coming out of two vessels of equal height but different widths were made, and it was found that the flow was the same, i.e., that it was a function of the height of the water. In order to measure it he recorded the time the vessels took to empty completely, which, in theory, ought to be double the time taken to discharge the same volume, if one kept the water level constant.12 Specifically he quoted the demonstration of an experiment of this type carried out on the 28 August 1668.

9

Ernest Mach considered Torricelli as the founder of hydrodynamics. This work is a summary of the most outstanding activities carried out in the Academy classified by subjects. We have followed it although an examination of the Academy’s acts and registers would give a more precise chronology. 11 Cf. Regiæ Scientarum, Book I, Chap. IV. 12 This is easy to demonstrate. The flow poured in a time dt, assuming a uniform velocity in the outlet orifice, will be dq = vS e dt = 2 gz S e dt , Se being the orifice area. If the vessel is not refilled, this flow will be equal to the emptied volume, that is Sdz, where S is the vessel area. Equating the following differential equation 2gzS e dt = Sdz is found. The solution for an initial height of z0 is: T = 2 S z 0 S e 2 g . In the assumption of maintaining the volume at the same height z0, which means that the flow rate will constant, the time needed to egress an equivalent to the total volume, that is V = Sz0, will be: T = S z 0 S e 2 g . Which is half the former. 10

(

)

(

)

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THE GENESIS OF FLUID MECHANICS, 1640–1780

As far Huygens’s activities in this field, apart from what du Hamel recounts, we have available the information collected in his Œuvres Complètes. According to these, around October 1668, Huygens accepted Torricelli’s Law, as his manuscripts indicate: After having considered the experiments we made concerning the flow and springing of water, I think we must conclude that the theory given by Torricelli based on similar experiments is true, and that although sometimes one finds that practice does not correspond exactly to speculation, this depends only on some special circumstances, which upon examination allow the cause of this difference to be seen.13

Now, although this text clearly shows that Huygens was in favour of Torricelli, it seems that by the following year things had changed, as indicated in his Œuvres. In these he relates an experiment 14 consisting of the discharge of a cylinder of 35 pouces (947 mm) high and 5 pouces 9 lignes (156 mm) in diameter with an orifice at the bottom of 4 lignes (9 mm) in diameter. The time taken by the cylinder to empty completely was 2 min 57 s, and if he kept it full it took 1 min 35 sec to empty a content equal to its capacity. Applying Torricelli’s Law he found that this last time ought to be 1 min 5.4 s. As he says, ‘these times … are found as 2 to 3’.15 To which he adds: This demonstrates that the water flowing out through the orifice of the reservoir does not have as much velocity as a body falling from the surface of the water would have, but only a part of that springing at the height of the surface, which is not easy to predict.16

Thus began the very active attempts to revise Torricelli’s law. There were many more experiments of this type, but what we have said already is enough to highlight the significance of this Law, and the tension that existed between the predictions and the experimental measurements.

13

Cf. Œuvres, Vol. XIX, Chap. XI, §.1, p. 166. According to the same source, the experiments he relates appear to have been carried out in August of this year (1668), because a register in the Academie dated the 8th of this month mentions them [ibid. §.2]. 14 The date is 16 February 1669. 15 Ibid. The calculation is not very precise. He says that the egress velocity should be 15 pieds 1 pouce/s (4.9 m/s). But with the present day values it would result in 4.31 m/s, which would give 14.4 s. 16 Ibid.

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The modifications analysed maintained the square root of the height, but varied the proportionality constant, as Huygens implies in the previous quotation. That is, the outflow velocity vs would be represented as: vs = k h = 2 ghe

[6.2]

The first equality of this equation shows that the proportionality with the square root of the depth is maintained, but with the new constant, which can be interpreted as the velocity corresponding to the fall from an equivalent height he < h, which is expressed in the second of the equalities. From our perspective it is easy to explain the deviations they found, as on the one hand Torricelli’s law is only valid when the surface of the outflow orifice is very small compared to that of the tank. On the other, and this is more significant, the outlet stream is the result of the motion of fluid in its neighbourhood. This results in a stream contraction at the outlet which depends very much on the shape of the orifice, which in turn gives rise to large variations in the discharge flow rate. Dominico Guglielmini

Domenico Guglielmin’s work is collected in the two volumes of his Opera Omnia,17 and we are particularly interested in the Aquarum fluentium mensura nova methodo inquistia (New method of measuring currents of waters) published in Bologna in 1690, where he relates his experiments and results,18 comprising six books and an appendix. In the first of these he establishes the basic doctrine, and he dedicates the rest to the movement of water in channels, be they horizontal or sloped, single or networks. In the appendix he provides tables in order to find the egress velocity of the water. To give an idea of the way in which Guglielmini approaches his work, we reproduce the opening words: Once I directed my attention to knowledge of the measurement of running water, then, as anyone would, I set myself to examine with all eager haste as far as I could, everything that had been written by other authors about this science, with the aim of penetrating their methods and demonstrations. However, it cost me little effort to leaf through their treaties, as being few and brief they could not occupy me for long. [Vol. I, p. 315]19 17

The Opera omnia mathematica, hydrodinamica, medica et physica was published in Geneva in 1719. 18 There is a version of this work in Italian in Vol. I of the Raccolta (pp. 313–415) under the title of ‘Misura dell’acque corrienti, ricercata con nuovo metodo’. 19 The quote between brackets refers to the Aquarum fluentium.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

We shall examine fully his second book, subtitled ‘Where the measurement of running water in single sloping channels is proposed’. Note that he is dealing with single channels as opposed to networks and combinations of the same, which he analyzes in later. As a preliminary step, and this is what interests us most, he details an experiment on velocities of water coming out of the tanks [Prop. I], which serves as the starting point for the following studies of channels. This experiment, quoted later by Poleni and Zendrini,20 refers to the comparative study of the lateral egress of water jets at different depths. It consisted of a cylindrical tank of 4 piedi (1.55 m)21 in height and 2 piedi (0.77 m) in diameter, as shown in Fig. 6-2. He made 16 holes in a generatrix of the cylinder, of approximately one pollici (32.3 mm) in diameter and with a separation between their centres of 3 pollice (96 mm). He placed a metal plate with another 16 holes drilled in it in this generatrix, opposite the holes made in the wood, but with a diameter of 0.25 pollici (0.87 mm), whose purpose, we surmise, was to pinpoint the outlet orifices.

Fig. 6-2. Discharge tank

The experimental procedure consisted in sealing all the orifices except one, and measuring the quantity of water that flowed out during a specific time, which was 15 strokes of a pendulum of 28 5/7 pollici (926.6 mm), while the tank was maintained constantly full. As he encountered some difficulties in the process, which he does not specify, he divided the experiment into two sets: one for the eight upper holes and another for the lower ones. 20

Which Guglielmini says took place on 4 October 1683. Measure of Bologna. In a footnote in the Book II, Prop. I, (p. 336) of the translation of the Aquarum … in the Vol. 1 of Raccolta (Cf. ibid. note 18), it is said that 3 piedi and 10 7/8 pollici of Bologna are equivalent to 4 pieds 7 36/41 pouces of Paris. We used this value to convert the Bologna piedi to metres. However, in the same note, it is given a discrepant value when converting the speeds. 21

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150

1,2

Quantity (lb)

125

1

Second round

100

0,8 First round

75

0,6 Square root

50

0,4

25

0,2

0 0

10

20 30 Height (in)

40

0 50

Fig. 6-3. Results of the discharges

The results he found are represented in Fig. 6-3, where the two sets are shown separately and the quantity which would have gone out, according to the rule of the square root taking the first value of each series as base, is also shown.22 That is to say, it is a partial application of Torricelli’s Law. Observing the figure, the good agreement with the square root distribution is perfectly plain, although this only refers to relative magnitudes, and not to the real outlet velocity of the water. He says nothing about this specific point until the Appendix, where he presents some tables showing the velocity as function of the depth of the outlet orifice. The tables are extensive, as they go up to 30 piedi (11.6 m) at intervals of a pollice (32.2 mm). Now, these velocities were calculated starting from a single experimental determination, whose result he extended to all the heights for the square roots law. In order to do so, he took the former vessel and placed a hole with a square cross section of 0.25 pollici (8.08 mm) on the side of the lower part of the tank, at 3 piedi 10 7/8 pollici (1.513 m) in depth. He let the water flow out for 65 strokes of the pendulum, and collected 32 libbre and 10 once (14.3 kg) of water. The change to volume was carried out weighing one cubic oncia of water (33.62 cm3) which turned out to be 786 grani (33.6 g). Finally, in the transformation of water collected at an egress velocity, he obtained 427 piedi 9 11/393 pollici per min (2.76 m/s), which is equivalent to a kinetic height hk = 0.389 m, which is 0.257 times the water depth of the orifice.23 If we

22 23

For the first set it would be 123√(x/48) and for the second one 93√(x/24). Daniel Bernoulli comments on this in the Hydrodynamica, Sec. I, §.2.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

calculate the fall velocity of a heavy body from 3 to 10 7/8 (1.513 m), we find v = 5.445 m/s, almost double the value obtained by him.

Fig. 6-4. Single, sloping channel

Guglielmini does not limit this law to the outflow of liquids, but he supposes it to be also applicable when dealing with currents of water in channels, be they single or in networks. In the simplest case of the former, following Fig. 6-4, he declares that: The velocity of the water flowing through any section of a sloping channel is the same as if it were flowing from a vessel whose opening is similar and with the same section, and is separated from the surface of the water inasmuch as the section is separated from the horizontal at the beginning of the channel. [Book II. Prop. II]

The argument reminds us of Galileo’s principles of movement through sloping planes, which follows the procedure, already begun by Torricelli, of extending mechanical principles to fluids. However, continuing with this method he arrives at the conclusion that ‘at any section of a sloping channel the velocity is greater at the bottom than at the surface of the water’ [Prop. III], which is a consequence of the former since the fluid of the upper layers is at a greater height, and therefore it will fall more slowly. Nowadays we know that this is not true, but if fluids are considered as an aggregate of particles without any force existing among them, and without any rubbing together, his surmise would be true. The remainder of the work then moves on to the extension of these hypotheses to channels of various types. Mariotte’s experiments

In turning to Mariotte we must return to his Traité du mouvement des eaux, which we looked at when we examined resistance. Remember that this work was an authentic manual of hydraulics intending to cover all the practical problems of the time. In keeping with this, the third part of the work is subtitled ‘About

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the measurement of flowing waters according to the different heights of the tanks’, in which he devotes the second discourse to the egress of water from tanks. He uses a phased approach: beginning with the verification of the flow rate law, followed by their measurement, and ending with the effect of the area of the outlet. The process has a practical focus, a fact common to the entire work. The first question that Mariotte considers is to check that the outlet flow follows the law of the square root of the heights. At this point he does not quote Torricelli, although he will do further on. The method is the usual one of carrying out discharges from two heights, measuring the quantities of water flowing out, and comparing both with the theoretical rule. He describes two experiments. In the first [p. 263],24 he perforated an opening of 6 lignes (13.5 mm) at 30 lignes (888 mm) under the surface of the water, measuring 8 6/7 pintes (8.43 dm3). He repeated with another orifice of 7 lignes (15.8 mm) in depth and found 15 ‘demiseptiers’, (5.57 dm3).25 The comparison of these two cases using the rule of the square root was satisfactory.26 In the second [p. 264] the height was 16 pouces (433 mm) and the opening 3 lignes (6.8 mm) and in 30 s 2.5 pintes and around 2 ‘cuillerées’ (spoonfuls) came out.27 The same tank, with an opening of 64 pouces (1.732 m), ought to give double. He found that the quantity was 5 pintes and 4 or 5 ‘cuillerées’. He ends by saying that he had undertaken other experiments up to 5 or 6 pieds (1.62 or 1.95 m), and always with favourable results. These results allowed him to assert the proportionality of velocity with the square root of the height, but not to assert that this would be equal to the fall of a heavy body from this height; i.e., he needed the flow rate. For this: [I]n order to find all the quantities of water that the tanks give easily by calculus no matter what their heights, he chose an average height to which he could refer all the others. [p. 265]

This height was 13 pieds (4.22 m), i.e., he took this height as reference, proceeding in the same way as Guglielmini. The apparatus, Fig. 6-5a [p. 266], consisted of a 20 pintes (19.9 dm3) tank, with a pipe coming down from its lower part that 24

The quotation in brackets belongs to the Traité du mouvement des eaux. From the context one can deduce that a ‘demiseptier’ is a quarter of a pint. Regarding these, he says that they are ‘those that do not weigh more than 2 livres minus 7 gros’ this would give 0.952 kg. Later on he speaks about pintes of 2 livres (Paris) or of 35 pouces per cubic pied (Paris), that is, 0.979 dm3, which apparently was the normal measure. 26 Resulting in a ratio of 3.752 pintes, according to the calculation, as opposed to 3.75 in the measurements. 27 The author does not specify the capacity of a ‘cuillerée’. 25

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THE GENESIS OF FLUID MECHANICS, 1640–1780

curved and ended in an orifice G of three lignes (6.77 mm) which was at the aforementioned 13 pieds from the level of the water in the tank. The internal diameter of the pipe was greater than that of its outlet, although he does not say by how much. With these dimensions the discharge measured in 1 min, which he says he found ‘through various very exact experiences’ [p. 265] was 14 pintes (13.70 dm3).28 He repeated the experiment lengthening the pipe up to 35 pieds (11.37 m) and comparing the discharges. According to what he says, he obtained a loss of 1/17 to 1/18 with respect to the value resulting from the application of the rule of the heights in both cases. In the opposite situation, reducing the length to 6 or 7 pieds (1.95 or 2.27 m), the contrary happened, which he explained as coming ‘from the greater or lesser friction against the edges of the orifice of 3 lignes, and the greater or lesser resistance of the air’ [p. 269]. However, he considers these differences to be small, and ignores them in the table he encloses for use at heights between 6 and 52 pieds (1.94–16.9 m), limiting the outlet to a diameter of three lignes, and which he calculated basing himself on the 14 pintes collected in the experimental determinations using a pipe of 13 pieds.29

E 13' G

a)

F

b)

Fig. 6-5. Egress through pipes

As a complement to this, Mariotte carried out more discharge experiments with the upper reservoir as shown in Fig. 6-5b. The reservoir was 1 pied (0.325 m) in width and height, and although he did not say so, it also ought to have been 1 pied in depth. The first of the tests that he relates had a discharge pipe of 3 pieds in length (0.975 m) and 3 lignes (6.8 mm) at the intersection E, and 3.5 (7.9 mm) at the outlet F. He says that if the pipe did not exist and the discharge 28 Here is where he says that they are 2 livres or that there are 35 pouces the cubic pied. Cf. note 25. 29 The rule he uses is expressed by the following equation Q = Q0√(h/h0), with Q0 = 14 pintes and h0 = 13 pieds.

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was carried out through a bare orifice with the fall from 1 pied, according to his calculations ‘a little less than four pintes’ would be collected. Exactly 3.89 pintes (3.8 dm3) we add, while with the pipe, and according to the same calculation with 4 pieds, he would collect 8 1/3 pintes (8.16 dm3). However, ‘it did not yield more than around the mean proportional between 4 pints and 8 1/3 pintes’ [p. 270], although he did not provide the exact data of the quantity. He proposed that the reason why more water came out with the pipe than without it is as ‘a result of the acceleration caused by the water circulating through the tube’ [p. 270]. In order to clinch the acceleration hypothesis, he placed a new 6 pieds (1.95 m) pipe with one pouce (27.1 mm) in diameter to empty the tank, which took 37 s. He then cut the pipe in half, therefore the time measured 45 s, and cutting the pipe at the base the time went up to 95 s, so that ‘from this it can be seen that the longer the tube the greater the acceleration’ [p. 271].30 However, this did not make him forget the effect of friction, because with a pipe with a half width of 5/4 lignes (5.64 mm in diameter) and 2 pieds (0.65 m) length he only obtained 1/8 of advantage over the case where it was one pouce long, ‘which is due to the friction throughout the narrow pipe preventing the water from increasing its velocity during the fall’ [p. 272]. As a third point he analysed the effect of the outlet area on the total discharge. According to his results ‘it was found that different openings always give a noticeable amount of water, and they give a little more in proportion to their surfaces’ [p. 275]. Consequently, he presents the table for diameters between 1 ligne (2.26 mm) and 12 lignes (27.1 mm) and a height of 13 pieds (4.22 m). With this table and the ones presented previously, given the height of the fall and the outlet area, the problem of calculating the discharge of water would be solved. Nevertheless, in spite of the apparent consistency between discharges and surfaces, he carried out other very interesting experiments which we shall only note here. These consisted in a vessel of 6 pieds (1.95 m) high and 6 pouces (162 mm) in diameter in which he makes two holes in the base, at 1 pouce (27.1 mm) each one from the edge, one of 4 lignes (9 mm) and another of 12 (27.1 mm). He prepared some tanks so that both could be uncovered at the same time, and collected the water in both reservoirs. As one is three times larger than the other, the quantity of water collected ought to be nine times more, but he found it was only eight times more. At first he tried to explain the effect by arguing that the orifice was very close to the wall, and for this reason he repeated the experiment with another 30 It is comprehensible that it was difficult for him to understand the action of the pipe. If he had applied his same rule to this last case, i.e., 95, 45 and 37 s for 1, 4 and 7 pieds in height, assuming the first as 95, he would have found 47.5 and 35.9 s for the other two, which was very close to his measurements. This indicates that his experimental method was excellent.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

bigger tank. Finally he arrived at the conclusion that the large holes allow greater discharge compared to the small ones. He explains this by three reasons: one, because there is more friction in the small holes, another because at the outlet a small jet finds more air resistance than a large one, and a third is that when trying to maintain the level of the tank constant by means of adding water, as the expenditure is greater so is the quantity, and this causes a ‘greater impact’. Later on he cites Torricelli and experiments on the emptying times as a function of the height, but he does not mention that the outlet velocity had to correspond to the kinetic height of the depth. If we carry out this calculation for what he took to be the reference case, we recall it had a height of 13 pieds (4.22 m), 3 lignes of diameter (6.77 or 36 mm2) and an outflow of 14 pintes per min (13.7 dm3). The result is an outlet velocity of 6.34 m/s, which corresponds to a kinetic height of 2.05 m, almost half (0.485 times) the 4.22 initial height. This fact would justify the rule that the velocity was that corresponding to half the height. Discharge in Newton

We analysed Newton’s position on discharge when we dealt with resistance, because, as we have already said, his study of motion of a solid immersed in a liquid was carried out by submerging it in the flow of a discharge. In the first edition of the Principia he supposed, and presumably demonstrated, that the outlet velocity of the fluid was equal to the fall of a heavy body from half the depth. He appears to change his mind in the second edition, but after Duillier’s criticisms, and faced with those of Cotes, he continued to accept the rule of half, although with a trick which was the equivalent of accepting Torricelli’s Law. This consisted in the introduction of the stream contraction as a corrective mechanism between theory and observation.

21/40 5/8 1/2 Fig. 6-6. Stream contraction

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We have already dealt with and we shall not repeat it.31 However, it is worth highlighting how he introduces the contraction effect. In the Principia, when he analyses the cataract, when dealing with the outflow of water through the inferior orifice and explaining the movement of the water, he notes: For now the particles of water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with oblique motions; and in tending downwards they meet in a stream whose diameter is little smaller below the hole than in the hole itself. [Prop. XXXVI, Case 1]

He says that the first contraction value is around 5–6, or as 51/2–61/2 approximately. He goes on to prepare an experiment that is a discharge but from the side of the vessel instead of from the bottom (Fig. 6-6), ‘so that the stream of running water might not be accelerated in its fall and by that acceleration become narrower’ [ibid.]. The diameter of this orifice was 5/8 in. (15.9 mm), and at a distance of 1/2 in. (12.7 mm). The diameter of the stream measured ‘with great accuracy’ was 21/40 in. (13.3 mm). From these measurements he deduces the ratio 25 to 21 in diameter, which in surface terms will be 1.4172, ‘that is, in about the ratio of √2 to 1’, and which he takes as being 17 to 12. In conclusion, he ends his arguments by inferring that inside the vessel the water acquires, at the hole, ‘a velocity downwards nearly equal to that a heavy body would acquire in falling through the half the height of the stagnant water in the vessel’ [ibid.]. However, once the water leaves the vessel: [I]t is accelerated by converging, till it arrives at a distance from the hole that is nearly equal to its diameter, and acquires a velocity greater than the other in about the ratio of √2 to 1; this velocity a heavy body would nearly acquire by falling freely through the whole height of the stagnant water in the vessel. [Ibid.]

He introduced these arguments into the second edition of the Principia, which together with the cataract merit detailed discussion. Giovanni Poleni

Giovanni Poleni, also known as the Marquis of Poleni, is probably the best known author of the Italian School. Of his various works, we would like to highlight two: De motu aquae mixto (On the mixed movement of the waters) and De castellis per quae derivantur fluviorum aquae habentibus latera convergentia 31

Cf. Chapter 2, ‘Acuiform fluids (II)’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

(On the constructions by which the water of the rivers is derived with lateral convergence), that appeared in 1717 and 1718, respectively. The first has a vulgate text in Italian in the Volume III of the Raccolta, entitled ‘Del Moto misto dell’acqua, e di molte cose appartenenti all lagune, ai porti ed ai fuimi’32 (‘About the mixed motion of water, and many things relates to lakes, ports and rivers’) in which he expounds a curious theory about how water in motion tows the becalmed waters. The second deals with discharge of tanks, a theme he tackles experimentally following the guidelines of the time. His explanations prove him to be a connoisseur of Newton, whom he follows in his hypothesis on the constitution of liquids, even using some figures very similar to Newton’s. We shall go on to analyse both works in succession, although from the point of view of experimental contributions to the verification of Torricelli’s Law, we are only interested in De castellis. However, in order to help us understand Poleni’s frame of mind we shall devote a few comments to De motu aquae mixto. According to Poleni, the mixed motion of water is produced when a current of water impinges on waters at rest. In this process he calls the former as ‘live waters’, while the others are termed ‘dead waters’. In a certain way, both are reminders of live and dead forces, which at the time were a matter of dispute among a good number of mathematicians. Concerning the latter he says: I will therefore call dead water that where all the particles of the upper surface (if it is not too small) are at the same distance from that average point to which all heavy bodies tend, and thus, the motion of the parts being equal, all the water is at rest. [§.III]

By contrast: Live water is said of that which receives the motion of the pressure of water it has above it, and to the motion of which, while this part moves against any other, no other water opposes it with a contrary motion. Thus, by way of example, live water is that which falls freely from the open orifice in any vessel. [§.IV]

Said simply: dead waters are at rest while live ones are running waters. If these impinge on the former and induce a motion, the result is a ‘mixed motion’. A case which makes these concepts clear is the movement of water in a canal going through a lock, that Poleni includes a little later on, and which we reproduce as Fig. 6-7. Two canals are shown with a lock in the channel. In the one on the left water flows at a lower level and one on the right with water a higher level. When the lock is opened the water in the upper compartment erupts 32

In the quotes we follow this version, has we were unable to obtain the original in Latin.

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Fig. 6-7. Live and dead waters

into the lower, mixing with it and setting into motion the water that already there. The first is ‘live water’, the second is ‘dead water’, and the final motion is denominated ‘mixed’. He writes: [F]rom this it follows that the water GPZR [located behind] is moved by the section CEFD [that is, the passage] because it is live, and if it moves it pushes the dead water PEFZ [located in front], because this will be pressured by the live water lying over it: but all the water will move GEFR, and such a motion will be called (for greater brevity) the motion of mixed water, which comprises the movement of living water and the movement of dead water. [§.X]

With this explanation the analogy of live and dead forces can be clearly seen. His intention is to analyse these motions, but prior to this he devotes some attention to simple motions, such as the outflow of liquid. At this point [§.XXIV-ff] he goes over the historical contributions to the law of the outlet velocity. He says that the first to announce the law was Castelli, although he supposed a proportionality with the height, and it was his disciple Torricelli who ‘3 years later’, in 1644, corrected it as a square root.33 Likewise he briefly recalls the experiment of the vessel of Guglielmini [§.XXXVI-ff].

33

However, Torricelli, as we have already mentioned, said that Castelli had already pronounced it thus. We do not know if Torricelli was right, or if he wished to save the reputation of the man he considered his maestro.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Q P D

A

I

L

C

Z G

B E

Fig. 6-8. Mixed motion experiment

In order to study mixed motion, he first carried out a curious experiment and then produced a formula which adjusted itself to the results of the measurements he had found. Figure 6-12 reproduces the illustration presented by Poleni, while Fig. 6-8 shows a diagram of this. It consisted in the cylindrical vessel P, closed at the bottom and opened the entire length of a generatrix, named ILEG in the figure, which is partially submerged in a very large tank. The water of the tank that penetrates to the interior of the cylinder is dead water, whose ‘dead height’ will be that of the level of the tank ZG. If a constant flow of water is poured into the vessel the vessel P, this will be live water, and it will mix with the water below, and come out of the tank through the vertical slot, forming a mixed motion. The effect is perceptible due to the elevation of the water level inside the tank, forming what we call ‘live height’. Poleni’s aim was to relate the live and dead heights, the width of the opening, and the flow of the water coming in. The experimental apparatus had two tanks S and T, whose function was to regulate the input flow, represented in Fig. 6-8 as a pipe Q. The first tank, named T, was the main supplier and pours a sufficient flow of water to the second tank, named S, which acts as a regulator in order to produce a constant flow to P. The arrangement is depicted in Fig. 6-12 (right) taken from the original work, but not in Fig. 6-8 for simplicity. The vessel P was 30 pouces (812 mm)34 in diameter, and was submerged in the tank whose depth was some 3 pieds (0.97 m) and 11 pieds in length (3.57 m).

34

Measure of Paris.

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This tank was feed by an intermediate one S, which was 42 pouces (1.14 m) in diameter, and boasted 16 holes K of 8 lignes (2.25 mm) in diameter in is base, through which the water is poured into P. The intermediate tank also had a rectangular window M at a height of 21 pouces (568 mm) which acted as an overflow with the object of maintaining the height of the inside water constant when filling S from tank T. The regulation of the flow was achieved by sealing some of the 16 holes with wax, the discharge therefore being proportional to the number of holes open. In the first of the experiment he describes, the width of the vertical opening was 15.5 lignes (35 mm), the tank was submerged 55 lignes (124 mm), which would be the dead height, and three orifices were uncovered. His description of the phenomenon is: Therefore, the water that fell from three orifices K into the vessel P [equivalent to pipe Q in Fig. 6-8], started to flow upwards through the section IGEL of the same vessel P. but as all the water entering the vessel P did not come out immediately through the aforesaid section, then the water in the vessel P kept growing higher than the surface of the water of the tank ABCD (going over Z). When the water that came out through the section IGEL was equal in quantity to the water transmitted through the three orifices [pipe Q], then this water remained high, until a surface which reached point X, and the live height was XZ = 35:4 lignes [19.7 mm], or what is the same 35:48 of a pouce. [§.XLII]

According to his interpretation, the experiment shows how the simple motion of water falling from the intermediate tank is transformed into mixed motion after mixing with the stagnant or dead water. He continues opening twelve and fifteen of the holes K, pertaining to 25, 42, 58 and 73.5 lignes (56.4, 94.7, 130.8 and 165.8 mm), respectively. He presents a total of eight sets of experiments, in the last three of which he raised the vessel P until its base was level with the tank, ‘which will give the result that the height of the water (inside vessel P) running through the straight section IGEL, was all living height’ [§.LII]. The object of all these measurements was to obtain a formula with which he could analyse the assumptions of the mixed motion. He states that he ‘had looked for several rules, and not without a great number of calculations he tried to adjust the said rule to each one of the experiments’ [§.LXVII]. He did contribute a formula, but to reproduce it here is not of great interest as it was purely empirical.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The second work, the De castellis per quae derivantur fluviorum aquae habentibus latera convergentia, which appeared 1 year later in 1718, is in some way a continuation of it, and its aim as the title indicates, was to analyze the motion through a channel that emerges from one side of a river and becomes progressively narrower. Figure 6-12 shows a composition based on the original drawings of Poleni, and in the bottom left part this channel is seen. It is of clear practical interest. He says there are three things to observe in this phenomenon: the area of the opening, the velocity of the water, and the flow rate of the water in a given time [§.23]. Although the first two are known data, recourse to experiment is needed to determine the third. He goes on immediately to the study of discharge of reservoirs, considering various height and several spouts.

M

S

P

Fig. 6-9. Experiment of De castellis

The apparatus shown as a diagram in Fig. 6-9, consisted of a tank S that he kept full up to a set level with the help of overflows M that discharged over another vessel P with a known capacity. Poleni’s idea was to analyse the influence of the outlet spout, therefore he used various types, as well as its relation to the depth of the water. His operational method was to measure the time taken to fill the vessel P. This latter has an upper diameter of 3 pieds 5 pouces 8 lignes (1.127 m), and a lower one of 4 pieds 4 pouces 4 lignes (1.417 m) and is 3 pieds 5 pouces 11 lignes (1.135 m) in height. He demonstrates that it results in a volume of 73,035 cubic pouces (1.499 m3) [§.24].

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400 350

Time (s)

300 Hole 250 200

Tube

150 100 100

200

300 400 Height (lin-P)

500

600

Fig. 6-10. Discharges results

He sets out ten sets of experiments [§.29–43], three with a tubular spout and three different heights (128, 256 and 542 lignes: 289, 577 and 1223 mm), three with the simple orifice and the same heights, and the other four frustumconical spouts, and all four with a height of 256 lignes, measuring the time taken to fill the reservoir. The results obtained for each of the first two set satisfied the square root law, but not were comparable between sets. The quantities of the waters that flow due to force of the water on top of them, always correspond to the height of the water that is on top; however, the different quantities of waters flowing out from several and different openings … can not be compared with each other. [§.83]35

In Fig. 6-10 we present a graph of the results obtained, together with two dash lines showing the inverse square root functions. In both cases the adjustment and the different between them can be seen. Poleni assumes that the spouts play an important role in this behaviour, which is why he also investigates their effect. In order to do this he prepares another set of experiments in which he sets the outlet cross section and the height of the discharge. The last one is 1 pieds, 2 pouces, 10 lignes (402 mm), and he takes a circular orifice of 9 lignes (20.3 mm) for the outlet, or instead a square outlet whose side measure 7 2/3 lignes (17.3 mm). To collect the water he uses a new vessel S of 2.560 cubic pouces (50.77 dm3). In total he lists the 35

The quotes refer to De castellis.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

results of 13 experiments [§.46–61] with spouts ranging from a fine lamina, another of 4/5 lignes thickness with a sharp edge, a similar one with a conical edge, and several tubes from 18 up to 108 lignes (46.6–243.6 mm). Some of these tubes had a conical interior, others were squared, and other channelled. The findings were very uneven, and we have grouped them in the graph shown in Fig. 6-11, which illustrates the time he obtained as a function of the lengths of the tubes or channels. 110 Cylindrical tubes Squares tubes Conical

Time (s)

100

90

80

70

60 0

25

50 75 Tube length (lignes)

100

125

Fig. 6-11. Discharge with tubes

In his comments he underlines that the conical spouts are better and he asks himself if this is due to the fact the cylinder of water which is above pushes more as it has a greater base [§.67]. As regards the tubes he notes little difference among them [§.69]. He also analyses the contraction of the stream, a datum that he notes down in all his experiments, and comments on the results give by Newton in the Principia, and pointing out that there still remained much to be studied. Returning to motion in the channel, he also asks about the force opposing the motion of water, which allows him to enter the dispute over the live forces [§.89-ff], but we shall not pursue this question. He ends his work by reflecting on Torricelli’s Law, although in this context he quotes is Johann Bernoulli. For this he recalls that a heavy body in free fall travels a space of 15 pieds, 1 pouce, i.e., 181 pouces (4.899 m) in 1 s,36 and therefore the water flowing from a reservoir of this depth will have the same velocity as this heavy body. However he states: 36

This value is precisely half the acceleration of gravity. We note that an error exits, as he says 181 lignes when they are pouces.

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‘the experiments have made me see that the velocity of this water is much less, bearing in mind the contraction of the stream’ [§.130]. Throughout the De castellis, Poleni always measures the time take to fill the vessel, but he never offers any calculation of the egress velocity of this water, although undoubtedly he must have made them. As a backup to his observation we have used the data from his first experiment with the discharge cases at 256 lignes (0.577 m). From this height the fall velocity will be v = 3.363 m/s. Reducing the reservoir filling times to the outlet velocities through the spouts, v = 2.868 m/s is obtained for the tube, v = 1.943 m/s for the simple orifice, and v = 3.030 m/s for the larger frustumconical spout. Comparing these values with the aforementioned theoretical freefall velocity, they represent 85.5%, 57.8% and 90.1%, respectively. A difference certainly exits. He concludes his reflections and his book thus: [T]herefore some things ought to be investigated further, such as the fall velocities of heavy bodies, velocities of out-flowing water, the fluidity of water (to which I have given little consideration) and similar things presented hereby in this way, not only for their own sake, but also for the illumination of the entire Science of Mechanics, which can throw no darkness. [§.130]

Fig. 6-12. Poleni’s apparatus

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THE GENESIS OF FLUID MECHANICS, 1640–1780

We have seen that Poleni sought mathematical laws to justify his observations, and faced with a discrepancy he acknowledges that there still remains a lot to study. As a closing piece, in Fig. 6-12 we reproduce an illustration which includes the two apparatuses and the channel as Poleni presented them in the ‘Del Moto misto dell’acqua’. The one on the right is used for mixed movements, and the one on the left is for the quantity of out-flowing liquid.

Chapter 7 The Hydrodynamica and the Hydraulica

Daniel and Johann Bernoulli

This chapter is dedicated to two men, two works and a plagiarism. On the face of it, the plagiarism should come as no surprise, as the history of science is replete with similar conflicts from time to time. The unusual thing here is the special relation that existed between these two men: the father and son, and the father plagiarised the son. They are Daniel and Johann Bernoulli, members of an outstanding family of mathematicians.1 Both are actors known to us, as they have previously appeared in these pages, and they will continue to show up after this chapter that discusses their principal works. These works are the Hydrodynamica, sive de viribus et motibus fluidorum comentarii (Hydrodynamics or Commentaries on Forces and Motions of Fluids) and the Hydraulica, equally valuable and famous milestones along the road of the science of fluids. But the conflict does not alter the outcome of the science, for each of them represents a particular way of approaching science: that of the mathematician and that of the physicist, or in eighteenth-century terminology, the way of the geometer and the way of the natural philosopher. Johann comes in the first category, as do d’Alembert and Euler. Daniel comes under the other, along with Galileo and Newton. The tension between these points of view is a constant in science, and perhaps they are two intrinsic poles of science, a fact illustrated by Daniel’s own thoughts expressed at the beginning of Hydrodynamica: How little hope there is that at sometime the Laws of motions for fluids will be reduced to the rules of pure Geometry without any physical hypothesis. … The principles of the Theory are physical, and are to be accepted, not without generosity, as approximately true. But, after the principles have been accepted, all will be Geometry: they will be interconnected by the necessary links, without being subjected to any restrictions whatsoever.2

These words contain the eternal hope, almost unattainable, of reducing physical reality to a symbolic set by means of a hypothesis. Once the reduction has been accepted with ‘generosity’, the theoretical manipulation allows us to project the 1 2

See L’école mathématique baloise des Bernoulli a l’aube du XVIII siècle, by J.O. Fleckenstein. Cf. Hydrodynamica, Chap. I, §.3.

293

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THE GENESIS OF FLUID MECHANICS, 1640–1780

image of reality towards the future in a precise and necessary manner. The only thing he did not do was to record the experiment as a questioning of nature, a link between the real and symbolic universe. Let us begin with the dispute over who was the originator of the ideas. The Hydrodynamica was published in Strasbourg in 1738.3 Daniel began work on it around 1729 in St. Petersburg, where he stayed for several years. When he left for Basle in 1733, he left an incomplete handwritten copy with instructions for it to be destroyed when the book was published, although it appears that his wishes were not carried out.4 At the beginning of the preface he says: ‘Finally our Hydrodynamica is published, after all the obstacles, which delayed its printing for almost 8 years, have been overcome’, i.e., the work must have been finished in 1730. However, in the work itself, reporting certain measurements made with a gas thermometer, he mentions the town of St. Petersburg and various dates, of which the latest is the 21 December 1733. This discrepancy is actually not really significant as it may be due to the last-minute introduction of these dates in the manuscript that he sent to Strasbourg, which bears out the hypothesis that it was this year in which the work was considered as finished. Some of the themes treated in the Hydrodynamica have already been dealt with in the ‘Theoria nova de motu aquarum per canales quosqunque fluentium’ (‘New theory on the motion of water flowing through any channel’) which appeared in the Commentarii petropolitanæ, 1727 (1729), and there are also some letters from the 1720s in which Daniel makes comments on the most important points. Therefore it is plausible to conjecture that he had already formed his basic ideas in this decade, and that part of them passed to the Hydrodynamica, perhaps in a extended form, while others did so after undergoing greater revision and elaboration. The Hydrodynamica is not a homogeneous treaty, and its thirteen chapters can be grouped into several areas. So much so, that there are even changes in the conceptual treatment of the phenomena, and in the symbols used to define the physical magnitudes, especially those related to velocity. Broadly speaking, the first eight chapters deal with the motion of fluids in pipes in general, and on discharges as a particular case. It continues with an application to the case of machines; and then with the behaviour of fluids in moving vessels, and even 3

This book was written in Latin. There is an English edition published in 1968 with the title Hydrodynamics by Daniel Bernoulli & Hydraulics by Johann Bernoulli, translated by Thomas Carmody and Helmut Kobus, respectively, and prefaced by Hunter Rouse. There is also a German version by Karl Flierl as Des Daneil Bernoulli Hydrodynamik oder Kommentaire über die Kräfte und Bewegungen der Flüssigkeiten published in Veröffentlichchungen des Forschungsinsttuts des Deutschen Museums für die Geschichte des Naturwissenschaften ud der Technik, Munich 1964. Here we have taken as reference the English version. 4 According to Rouse, this copy is in the archives of the Soviet Academy of Science; see the introduction of the English translation of the Hydrodynamica.

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includes chapter devoted to gases, in spite of the work’s title. A key chapter, which is often the only thing for which the work is remembered, is the study on the mutual relation between velocities and pressures, which motivated the coining of the term ‘hydrodynamics’ and was the starting point of the later developments. It is here that what came to be called ‘Bernoulli’s theorem’ was formulated the first time. The work follows a theoretical–practical approach. Each section begins by announcing the theme or problem to be dealt with, and goes on to demonstrate or develop it for the chosen case. It then proceeds to detail the results of more specific and practical cases, and usually ends with a collection of experiments relevant to the topic in question. The publication of Hydrodynamica was a great success in the scientific circles of the time. But not everything was rosy: [The] publication of the Hydrodynamics, and the success it achieved, brought upon its author the envious criticism of two of the principal geometers of the century, one near the end of his life and the other at the beginning of his career: the devious d’Alembert, and his own father, the mighty, jealous and irascible John Bernoulli.

With these words Truesdell5 links Daniel’s Hydrodynamica with the Hydraulica of Johann, his father. D’Alembert’s criticism appeared in the Traité de l’equilibre et du mouvement des fluids, and he bases it on the fact that Daniel did not adequately justify the principle of conservation of live forces, apart from some corrections of the cases presented. Concerning Johann’s intervention, the matter is somewhat shadier, as the father tried to appropriate the son’s discovery; which is why he wrote the Hydraulica. The Hydraulica deals with the motion of water through ducts, and the forces or pressures associated with this motion. It is a much shorter work, divided into parts the second of which is a generalisation of the first. The style is direct and does without experiments. There is a considerable difference from the Hydrodynamica. The Hydraulica first appeared in 1742 as part of his Opera Omnia, when Johann was 75 years old. It was also published in the Commentarii petropolitanæ, divided into its two parts: the first in Volume IX, which corresponds to the year 1737, but was published in 1744, and the second in Volume X of 1738, published in 1747. The first thing that surprises one on contemplating the cover of the work, is that after the title comes the following comment: 5

Cf. ‘Rat. Fluid. Mech.-12’, p. XXXI.

296

THE GENESIS OF FLUID MECHANICS, 1640–1780 Discovered and demonstrated for the first time from purely mechanical foundations. Year 1732.

We recall that it was in 1733 when his son Daniel delivered the manuscript of the Hydrodynamica, and this explains why Johann expressly indicates the year 1732, trying to make us believe that his work was prior to that of his son. This becomes even clearer in the text, when he explains that the method of live forces does not satisfy him, and declares that he was trying to look for something that was based only on the principles of dynamics. He says: Finally, after a rather long meditation, I achieved my aim in the year 1729, when I saw that the crux of the whole matter lay in contemplating the whirlpool, hitherto unnoticed by anyone. And so now I propose to make public my discoveries, 6 already explained privately to certain friends.

It has since been demonstrated that the dates given by Johann are false, and that, the son was plagiarised by the father, who tried to take the credit for the discovery of the relation between velocity and pressure. The controversy was bitter, as the letters, written by one and the other to Euler testify, who was always held in great esteem by both, and who surprisingly answered both on the same day. The letters of Daniel are very bitter. Truesdell7 and Rouse8 offer a more detailed analysis of this controversy. Nevertheless, while we condemn the fact that he plagiarised his son, we acknowledge in Johann’s defence the contributions he made: the direct application of the equations of Newtonian dynamics instead of the method of the live forces, and the introduction of the concept of internal pressure. This new approach is what Johann wished to underline when he made the comment that he had ‘now demonstrated directly from purely mechanical foundations’. The use of live forces appeared to him to be an indirect road, ‘which is most certainly true and was proven by me as well, but is still not accepted by all philosophers’.9 Of even greater importance is the concept of ‘internal pressure’ in a fluid, which in fact opened up the development of hydrodynamics. The concept of internal pressure and the division of the fluid into mentally delimited differential elements allowed an element to be isolated, its evolution to be followed throughout a duct, and the equivalent local forces to be determined.

6

Cf. Hydraulica, Part I, §.VIII. Cf. ‘Rat. Fluid-Mech-12’ pp. XXXII–XXXV 8 Cf. The preface to the translations of both works to English. 9 Cl. Hydraulica, Preface, p. 392. 7

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But apart from these conceptual innovations Johann also improved the terminology and symbolism, therefore paving the way to a better understanding of the phenomenon. Specifically, Bernoulli’s theorem became formulated in a way very similar to the current one. Truesdell concludes that ‘For the first time, fluid mechanics appeared in its proper and yet current station in the great system of classical mechanics. In John Bernoulli’s hydraulics, not in its progenitor, the hydraulico-statics of Daniel Bernoulli, lie the roots of Euler’s hydrodynamics’.10 This is true, but we must consider that Euler was also in debt to d’Alembert. To sum up, and without detracting from the value of Daniel’s concepts, we consider that Johann could well have acknowledged the work of his son, which would have done him no discredit. On the other hand, we believe what is currently called ‘Bernoulli’s theorem’ should rather be called ‘Bernoullis’ theorem’, as it is due to both. As regards Johann, history has not done him justice regarding his contributions in the field of hydraulics,11 possibly because of the natural revulsion of historians towards his plagiarism; and also, perhaps, because ‘he never explained himself with any clarity, … lived in a world of challenges, enmities, secret methods and anagrams, … [and] abstained from expressing it from fear lest the English accuse him of borrowing the “cataract” of Newton’.12 There remains only one final point to make in these commentaries, which have been somewhat long but necessary. The basic merits of Daniel’s and Johann’s work is that they analysed the fluid as a moving mass shaped by the effect of the walls, and thus they obtained the laws of motion as a result of applying a property in an integral manner to the entire mass. Looking back, we see that Torricelli’s Law referred only to an isolated mass point, although the result was extended to an entire volume, and that Newton also referred to point masses when he deals with the resistance using impact theory. In order to develop these ideas we begin by setting out the basic principles of motion, as expressed by Daniel in the Hydrodynamica. We continue with ‘hydraulico-statics’, detailing how the relation between velocities and pressures was reached. We then go on to the Hydraulica, and with Johann’s contributions. Finally, we shall deal with the ideas of d’Alembert and Borda. Although the latter presented his works quite a few years later, they are a logical continuation of the earlier ones.

10

Op. cit. p. XXXVII. As confirmation of this omission, Dugas, in the Historie de la méchanique, only mentions it almost in passing, and Rouse and Ince, in the History of Hydraulics ignore it. 12 Truesdell, ‘Experience, Theory and Experiment’, Proceedings of the Sixth Hydraulics Conference, 1956, p. 10. 11

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The basic principles of the Hydrodynamica

The contents of Hydrodynamica can be grouped into several broad areas13 all dealing with the motion of fluids, except for the Introduction and a chapter devoted to hydrostatics. Moreover, even though the title in a certain way implies it is devoted to liquids, there is a chapter dealing with the air. The types of movement he studied can be grouped into several areas: discharges or outflow from vessels, oscillations in pipes, liquids passing between reservoirs through orifices, and what he calls hydraulico-statics, which turns out to be discharge. To this class of motion, caused solely by the action of gravity, he adds a chapter in which he considers the action of other external forces, from which he derives applications to hydraulic machines. A penultimate variant, which he also deals with, is the case where motion takes place in moving vessel, and he ends with studies of the reaction to a jet when it egresses, also applicable as the driving source of motion, as well as the effect of these jets against plates, a theme already dealt with. In spite of this wide range, the basic element upon which he builds his theories is reduced to the motion of the fluid mass through a duct with a variable cross section. He conceptualises this, once given the contour through which the fluid has to circulate, in the demarcation of a volume whose evolution he analyses as a function of the live force it contains, and whose conservation is the first of his basic principles. The second is the hypothesis of motion by plane sections that, due to the law of continuity, relate the velocities and the areas of passage. Following on from what he had already stated, ‘by these two principles we resolve all the problems’.14 However, in accordance with his feeling that ‘the principles of physics have to be accepted as approximately true’, he does not refrain from questioning the field of application, be it qualitatively or, in certain applications, quantitatively. We go on to analyse these principles in the way he presents them in the Hydrodynamica, as they are already fully developed in this work, although he had demonstrated them previously in the ‘Theoria nova’. There, the live force imprinted on a body, or set of bodies in motion, was understood to be the sum of the product of the mass of each multiplied by the square of its velocity, that is Σmivi2. A fundamental principle was that the live force had to be maintained constant if the forces causing the movement were 13

We do not know of any comprehensive study of the entire Hydrodynamica, although it is surely worthy of one. The recent ‘Introduction to Daniel Bernoulli’s Hydrodynamica’ by G. K. Mikhailov can be consulted in Vol. 5, Hydrodynamik II of Die Werke von Daniel Bernoulli, Birkhauser Verlag AG, 2004. 14 Cf. ‘Theoria nova de motu aquarum per canales quoscumque fluentium’ Comm. acad. petrop. Vol. II (1727) p. 114.

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gravitational. In the ‘Theoria nova’15 Daniel says that Huygens had demonstrated that it also had to remain constant in the case where there were elastic impacts between several moving bodies. To this Daniel adds: Thus I state, that the diminutive corpuscles composing the fluid are perfectly elastic; as unless they were very hard, and therefore provided with high elasticity, they could be subdivided into others.16

From this he concludes that this principle must be applied to fluids. Likewise he recalls that in the case of a single body, Galileo had been the first to show that a heavy body that falls from a height, be it vertically, or following any type of curve, always acquires the same velocity. But it was Huygens who extended this proposition to more general cases, as when the various bodies exist with different velocities, and even with elastic impacts between them. He reproduces the axiom given by Huygens: If any masses begin to move in any way through the force of their own gravity, the velocities of each one will always be such that the sum of the products of the squares of these [velocities] multiplied by their appropriate masses will be proportional to the vertical height through which the centre of gravity of the set of the bodies descends multiplied by the mass of all of bodies. [I.§.19]17

With present-day notation this will be expressed as:

∑m v

2 i i

= khcg ∑ mi

[7.1]

k being the proportionality constant to which he alludes, and with the units currently in use its value is k = 2g. This principle will be applicable to water or any other liquid, provided that its elemental components satisfy the condition of the axiom, i.e., they can collide and impact elastically. Although in the Hydrodynamica there is no explicit hypothesis on this, in the ‘Theoria nova’ Daniel considered that liquids were constituted by moving particles, in the form of an aggregate of elastic units similar to those supposed by Newton in the Principia. Following the thread of the definitions, it is clear that instead of using the expression ‘conservation of live forces’, he prefers to use what he calls ‘the equality between the real descent and the potential ascent’ [I.§.18]. In justification he 15

Op. cit. p. 112. Ibid. 17 The quotations inside brackets will refer to the Hydrodynamica from now on. 16

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THE GENESIS OF FLUID MECHANICS, 1640–1780

states that he will ‘make use of this last expression because, although it means the same as the other one, with time perhaps it will be found to be more suitable (benignam), although now it is considered strange (sortem) by some philosophers who precisely let themselves be carried away by the very name of live force’ [I.§.18].18 That is, Daniel warns those who are familiar with the first of expressions, that he wishes to express the concepts more clearly and concisely, avoiding vague or common notions. Referring to the potential ascent (ascensus potencialis), and to the actual descent (descendus actualis), he recalls that: The potential ascent of a system, the individual portions moving at any velocity whatever, indicates the vertical height to which the centre of gravity of that System reaches. This supposes that the individual particles ascend all they can, once the direction of their velocity has turned upwards; and the actual descent denotes the vertical height through which the centre of gravity descends after the individual particles have come to rest. [III.§.1]

In order to understand this proposition better, we imagine a set of bodies (Fig. 7-1) each one with a mass mi and provided with velocities vi, that differ in both magnitude and in direction. In Fig. 7-1a, at any given instant the centre of gravity of the set will be at a specific level, which is taken as a reference. Here, the potential ascent of this set will be the height reached by the centre of gravity of the system as the result of a movement in which the velocities of each individual body turn and point upwards (Fig. 7-1b and c). Obviously, the potential ascent represents an instantaneous property of the system.19 On the other hand, the set will evolve in time in agreement with the velocity of each mass in particular (Fig. 7-2a), in such a way that the configuration, centre of gravity and potential ascent will vary in accord with one another (Fig. 7-2b). The descent of the centre of gravity of the set is what is termed the actual descent, and according to theorem, it must be equal to the increase of the potential ascent. That is:

Ap (t ) = Ap 0 + Da (t )

18

[7.2]

The translation of this passage is difficult, as Daniel uses the terms ‘benignam’ and ‘sortem’, that are difficult to interpret in the context. 19 Potential ascent of an isolated body is ξ = v2/2g which is what has been called the ‘kinetic height’ of its velocity. We maintain this denomination, reserving potential height for a set of mobiles or a fluid volume. We shall use the symbol ξ for this magnitude with the aim of avoiding confusion with the h, which usually designates the real height of particle.

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Ap

a)

c)

b) Fig. 7-1. Potential ascent

Ap

Ap

0

Da

a)

b) Fig. 7-2. Actual descent

Seen from today’s perspective, is it is easy to check that we are dealing with the principle of conservation of mechanical energy. According to the definition of the potential ascent, its numerical value is:

Ap =

∑m v 2g ∑ m

2 i i

[7.3]

i

As regards the centre of gravity, its position is:

hcg =

∑m h ∑m

i i i

[7.4]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

And if we let the gain of one equal the loss of the other, we obtain the wellknown equation:

∑

1 2

mi vi2 + ∑ gmi hi = 0

[7.5]

This indicates that the sum of the kinetic energy and the potential are maintained constant throughout the evolution of the system. This whole approach rests on the assumption that the impacts between the particles of the fluid are elastic, because if this is not so there will be a loss of live force. In effect, he says that there are movements ‘where sometimes it is clear that a portion of the potential ascent is lost continuously’ [I.§.20], but he does not give a rule for discerning these cases a priori.

Fig. 7-3. Plane motion

In order to calculate the live force, or potential ascent of a set of moving bodies, we need to know the velocity of each individual mass, which is generally very difficult. When a fluid moves through a duct that is symmetrical with respect to its axis (Fig. 7-3), Daniel supposes that it advances by flat sections of which he says: [A]fter we grasp mentally that the fluid is divided into layers perpendicular to the direction of the motion, we declare that the particles of fluid belonging to the same layer move at the same velocity, in such a way that the velocity is inversely proportional to the cross section of the vessel. [I.§.22]

He is aware that this ‘plane sections’ hypothesis is only an approximation of reality, and this is made obvious when he declares that ‘it is well known that the fluid moves slightly more slowly at the sides of a vessel but faster in the middle’ [I.§.22].

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It is clear that if the duct narrows, the velocity must have a component in the direction of the axis of the tube, even though it be small in magnitude. With this hypothesis and the continuity principle it must be shown that:

Q = S 0 v0 = Sv

[7.6]

However, in the neighbourhood of the outlet orifices, when there are abrupt jumps in the section, this hypothesis is not very likely. Concerning this, he makes a few very interesting and acute comments about the structure of motion. He imagines [IV.§.2] a cylinder placed vertically, and perforated by a small outlet orifice in its bottom part. During the discharge process, and in accord with what has been said previously, he declares: ‘let us conceive the internal water to be divided into horizontal strata, and in this hypothesis we judged that the motion of each was the same, and such that the horizontal position is kept’ [IV.§.2], with the exception of the region around the outlet orifice, which, although it will not affect the results much, does limit the range of the hypothesis. The most interesting part comes in the following paragraph, where he adds: Moreover, it seems to me that the motion of the internal water can be considered as if the water were carried through infinitely small tubes placed next to each other. Of these, the central ones descend almost straight down from the surface to the orifice, and the remaining ones curve gradually near the orifice. [IV.§.3]

Fig. 7-4. Streamlines

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THE GENESIS OF FLUID MECHANICS, 1640–1780

That is to say, the trajectory of a particle, which will move as though led through a pipe, first goes straight, and when as it approaches the orifice it curves. This is the first known description of the streamlines, explained in the figure reproduced here in Fig. 7-4. He notes that the phenomenon can be visualised with water and particles of wax.20 Motion through tubes in the Hydrodynamica

Daniel Bernoulli applies these principles to an elemental construction on which his further developments are based. This consists in calculating the potential ascent of a fixed mass of fluid circulating through a pipe, together with a variant of this ascent as it moves through the interior of the pipe. Let there be a pipe (Fig. 7-5) with a straight section S(x) through which liquid circulates with a velocity v(x). By way of reference, he takes any section of the pipe designated by the sub-index O, whose section will be S0 , and the velocity of passage through the pipe v0. Inside the pipe the volume of fluid is limited by means of two imaginary planes S1 and S2, and this volume will move with time to reach a new position, that was represented by the dotted line in the previously mentioned Fig. 7-5. Daniel first seeks to determine the potential ascent of this volume when it finds itself in the position bounded by S1 and S2. The elements of the section S(x) will have potential ascent ξ(x) = v2(x)/2g, that, when reduced to the reference section, will be transformed into21:

⎛ S ⎞ v 2 ( x) v02 S 02 ξ ( x) = = = ξ 0 ⎜⎜ 0 ⎟⎟ 2 2g 2 g S ( x) ⎝ S ( x) ⎠

2

[7.7]

Where ξ0 is the potential ascent corresponding to the reference section.

20

We note that, in the Essays in the History of Mechanics (Fig. 81, Chap. IV, §.11), Truesdell says that Euler was the first to describe streamlines in his commentaries to Robins’ Gunnery, when he translated it into German in the year 1745. However, as we can see, Daniel Bernoulli had already alluded to them in 1738. 21 We note that Bernoulli did not work with real velocities in this part, but with the corresponding kinetic heights which he frequently designated with the letter v, which might lead to some confusion. Besides, the formula that that he uses in the equivalence between both is v = √ξ, i.e., the system of units that he implicitly uses is such that g = 1/2 [IV.§.9]. Following our criteria of rendering the reasoning of Daniel Bernoulli easily intelligible, we shall employ present-day notation and units, and will make appropriate comments when necessary.

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S0

305

S 2 S'2

S 1 S'1

Fig. 7-5. Motion through ducts

The potential ascent of the entire mass of liquid enclosed between the control planes, and designated as Ξ, is defined by the following equation:

∫

∫

x2

x2

Ξ = Sdx = Sξdx x1

[7.8]

x1

With the appropriate substitutions, this reduces to:

Ξ=

ξ0 S0

∫

x2

x1

dx S

[7.9]

∫ Sdx x2

x1

Bernoulli made this calculation using geometrical constructions in use at this time. He defines the two following functions:

N=

∫

x2

x1

S 02 dx ; S

M =

∫ Sdx x2

[7.10]

x1

which, introduced in equation [7.9], enable him to find the potential ascent ‘of all the water affected’ [III.§.2], that is:

Ξ=

ξ0 N M

[7.11]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

We note that these two integrals M represented the volume of the fluid enclosed within the fixed domain.22 We recall that the potential ascent is an instantaneous property of fluid enclosed between the fixed limits, i.e., as if it were frozen at the very instant of the calculation. A further step is to determine the variation of the potential ascent of the fixed fluid while moving through the pipe, which implies a variation in the domain limits. In Fig. 7-5 the new limits in t + dt were represented by a dotted line. He makes the calculation differentiating the last equation, resulting in:

dΞ=

ξ 0 dN + Ndξ M

[7.12]

He does not consider any variation of M, as it is invariable, representing as it does the volume of liquid in evolution, which is constant. This equation points towards non-stationary phenomena, i.e., those in which the velocity at any point varies with time. This is implicit in the presence of dξ0, as in the stationary supposition it would be ξ0 = const and dξ0 = 0.

dx

S

x

S0 Ss

(S/Ss)dx

Fig. 7-6. Egress of a droplet

Daniel approached several problems with the help of these formulas, and we will examine some of the most significant ones. The most interesting problem is the draining of a vessel through an orifice in its base [III.§.6]. The calculation method consists in obtaining the variation of the potential ascent of a certain quantity of liquid coming out of the reservoir, and which he likens to a ‘drop’. 22 In the developments that he makes, he specifies these equations for when the sections of the duct under study are circular or two-dimensional. This requires a certain degree of caution when following Daniel Bernoulli’s text, as he frequently changes from one to the other without previous warning.

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Later he equates this variation with the actual descent of the entire volume of liquid. Thus, (Fig. 7-6) let there be a reservoir whose cross section S is filled up to level x in the process of discharge. Between two consecutive instants, t and t + dt, a drop will flow out, which in turn will make the level of the reservoir fall dx, thus its volume will be Sdx. He defines the droplet as a small cylinder with base Ss, and consequently its height will be Sdx/Ss. With S0 as the reference surface, and using the continuity equation we will obtain:

Sv = S 0 v0 = S s vs

[7.13]

And for the potential ascent, we will obtain:

S 2 h = S 02 h0 = S s2 hs

[7.14]

The calculation of the variation of the previous equation, designated as dH, is made with the help of the equation [7.12]. Previously, it is necessary to obtain dN, which in this case is the variation of the magnitude N before and after the egress of the droplet, which is found bearing in mind the increase of the potential ascent of the out-flowing fluid, and the decrease due to the lowering of the surface level. After the corresponding calculations, which we will not detail here, the result is:

dN = − S 02

ds dx + S 02 S 2 S Ss

[7.15]

and after applying equation [7.12], we obtain:

⎛ S 1⎞ d Ξ = Ndξ 0 + ⎜⎜ 2 − ⎟⎟ S 02 ξ 0 dx ⎝ Ss S ⎠

[7.16]

Regarding the potential descent, its evaluation is a lot simpler, as its magnitude is that of the descent of the centre of gravity of the set when the droplet falls, which will be23:

dD = − S

23

dx M

In the calculation he leaves aside, with justification, terms of the order of dx 2.

[7.17]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Making the last two expressions equal, reducing the values to the surface level and the egress level, and after some algebraic operations, we end up with the following expression:

⎛ ⎞ S s2ξ s NS dξ s = ⎜⎜ ξ s − 2 − x ⎟⎟ SS 02 dx S ⎝ ⎠ 2 s

[7.18]

This is a somewhat abstruse formula which links the variation of the potential ascent of the droplet (dξs )—which in a certain way is the equivalent to the velocity—with the descent of the free level of the liquid (dx). The difference between this treatment and that given by previous authors is clear: this is the first time that dynamic concepts have been introduced, although this has been done indirectly through live forces.24 The specification and application of this expression to geometrically defined cases occupies him in the following Sections. The first group deals with the drainage process of cylindrical vessels through simple orifices, or ones provided with nozzles. The next group deals with the time taken to empty a reservoir which is a variant of the previous problem. At this point [Chap. IV], he faces the phenomena of the stream contraction. Bernoulli, goes beyond taking a fixed value of contraction coefficient, which is what Newton does, and he introduces a factor, which he denominates α, which appears in all the formulas from then on. It is following this train of thought that he makes his penetrating observations on the streamlines and the descent of fluid in horizontal strata (Fig. 7-4). If we pass now to the discharge of the cylindrical glass placed vertically, the equations are simplified, given that N = Sx = Sx0, and with the introduction of r = S0/Ss, the expression [7.18] converts itself into the following differential equation:

⎛ S2 ⎞ S2 xdξ + ⎜⎜1 − 02 ⎟⎟ξ s = − x 02 dx Ss ⎝ Ss ⎠

[7.19]

That combined with the initial condition that, at the moment in which the discharge starts, the level of liquid is a, leads to the following expression: 24

For curiosity’s sake we reproduce below the last formula found, just as Daniel Bernoulli obtained it [III.§.9]: dx nnNdz = −mmzydx + mmnnz = mmyxdx y This is to note the error in the original, consisting in the lack of the subtraction sign in the final quantity. It should say –mmyxdx.

THE HYDRODYNAMICA AND THE HYDRAULICA

ar 2 ξs = 2 − r2

⎡⎛ a ⎞1−r x ⎤ ⎢⎜ ⎟ − ⎥ a⎥ ⎢⎣⎝ x ⎠ ⎦

309

2

[7.20]

Which represents the variation of the potential ascent of the fluid at the outlet as a function of the level. If, instead of this, we had used that of the surface of the fluid as Daniel did [III.§.13], we would have obtained:

⎡⎛ a ⎞1−r x ⎤ ⎢⎜ ⎟ − ⎥ a⎥ ⎢⎣⎝ x ⎠ ⎦ 2

a ξ= 2 − r2

[7.21]

The discharge velocity is obtained from the previous equation starting from v s = 2 ghs , which, in combination with equation [7.20], yields:

r2 v s = 2 gx 2 − r2

⎡⎛ x ⎞ r ⎢⎜ ⎟ ⎢⎣⎝ a ⎠

2

−2

⎤ − 1⎥ ⎥⎦

[7.22]

The first term 2 gx of this equation is Torricelli’s formula, which we see is affected by another factor whose value is zero in the initial instant, i.e., when x = a, and this will grow to a maximum, and then will descend again. A subsequent application is the determination of the discharge time, which is related to the potential height by:

dt = −

dx dx =− v 2 gξ

[7.23]

This expression, once we introduce the ξ of equation [7.21], yields to another differential equation, which with a simple integration resolves the problem. If the reservoir was maintained constantly full, the discharge would be stationary, and he devotes an entire chapter to this. We will not go into the mathematical details here; we will only say that, according to Daniel, the mathematical solution depends upon how the vessel was kept full. He distinguishes two cases: either by a vertical supply of liquid, so that it reaches the surface of the upper level at just the precise velocity, or it is kept full by a lateral entrance with zero velocity. In one case there will be a contribution of potential ascent, while in the

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THE GENESIS OF FLUID MECHANICS, 1640–1780

other case not, which will determine the result.25 Finally, simplifying, he concludes that when the outlet orifice is small compared to the surface of the vessel, that is, r → ∞, the potential ascent to the outlet is the same in both cases, and its value is26:

r 2a ξs = 2 →a r −1

[7.24]

That is to say, it tends towards Torricelli’s Law. Another example of the advantage that Daniel obtained from his method is shown in the study he made of the oscillation of fluids inside pipes. This problem had already been analysed by Newton,27 and by Daniel’s father Johann28 as Daniel himself recalls. The studies of Johann, also with the help of live forces, referred to pipe of uniform cross section whose lateral arms were inclined, as is shown in Fig. 7-7a. The apparatus was filled with water to a specific level, and then made to oscillate. Johann demonstrated that the ascents and descents of water in the branches of the pipes were isochrones, with a frequency equal to that of the pendulum of length: L/(sinθ1 + sin θ2), where L represents the total length of the duct with water. Newton’s studies were even simpler, as he had supposed that the two lateral branches of the pipes were both vertical, which reduced the previous formula to ½ L. 1

1

2

a)

2

s

A(s)

b)

Fig. 7-7. Oscillating fluids in pipes

Daniel approaches the problem in a completely general way: Because truly, since I have resolved to expound a more complete theory concerning the motions of water, it will be to the point to follow this type of argument to its full extent. Therefore, I shall inquire at length into what to what extent unequal 25

D’Alembert criticised the case of the lateral supply of liquid, arguing, not without reason, that this liquid that enters with zero velocity, is conferred with a velocity instantaneously. 26 The calculations he makes [Chap. V] are somewhat confused, due to the models that he uses. 27 Cf. Principia Book 2, Prop. 44. 28 Cf. Comm. acad. petrop., Vol. I ‘Teoremata selecta pro …’

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oscillations of a fluid become isochronous, and likewise to what extent they do not. I shall go on to give the length of the simple tautochronous pendulum, and for the other I will indicate the time of duration. Moreover, I will consider pipes bent in any way and with a variable cross section. [VI, §.5]

And this he does. The formulas he arrived at are quite complex and difficult to apply. One of the cases in which he specifies the results corresponds to the pipe represented in Fig. 7-7b. The formula he arrived at for the length of the equivalent pendulum is the following:

∫

s2

ds s1 A( s ) L= sin θ1 sin θ2 + A1 A2

[7.25]

The cases of Johann Bernoulli and Newton are derived from this formula in an immediate manner. But apart from these oscillating motions, in which the fluid stays in the pipe, he also pays attention to other types of oscillations more directly related to the process of discharge through orifices. We refer to the situation represented in Fig. 7-8, which is a vessel with a hole in its bottom, supposedly plugged, and which is partially submerged in a very large tank full of a liquid. X

Y

X

Y

J

K

Fig. 7-8. Charge or discharge of a submerged vessel

In the initial instant, which is when the orifices is unplugged, the liquid erupts in the interior of the vessel, ascending progressively until it reaches a maximum height above the exterior level designated as XY. From this situation it starts to discharge, with continuous descent until a minimum point, lower than the level of the exterior reservoir, represented by JK. The process will continue in alternates phases of entrance and egress, but in each cycle the difference between the maximum and minimum heights will become less, until it arrives at a situation of a final balance, where naturally it will correspond to the same level of the external reservoir.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

These oscillations, as he himself says, ‘differ completely from the oscillations considered in the preceding chapter [liquids in pipes]’ [VII.§.1]. He calculates that the damping of the amplitudes is not only due to the friction or to other obstacles, but it is an undeniable fact that in each period part of the live force is lost, passing to the internal motion of the liquid (motus intestines). In order to try to determine the laws of motion in this case, he adduces that if he could determine how much live force is dissipated in each cycle, the problem would be solvable. Following this line of thinking he recalls a somewhat similar procedure used in mechanics when deformable bodies are studied. For just as the rules of motions are determined correctly in deformable bodies, if after collision that part of the live force expended in the compression of the bodies is considered as lost (for even this is not restored to the progressive motion as occurs in elastic bodies), in the same way the ascent of the fluid will be defined no less correctly if one examines accurately what quantity of live force transferred instantly to the internal motion of the aqueous particles, never to return to the progressive motion, which is precisely what we are dealing with. [VII.§.1]

The hypothesis introduced by Daniel is that in the egress phase of the liquid from the vessel to the reservoir the live forces are preserved, as the existing force is transferred to the remainder of the water, while in the entrance phase part of the live force is lost. In order to quantify this loss he considers that: As the water enters through the orifice at greater velocity than that present in the internal rising water, this excess produces a certain internal motion in the same internal water, contributing little or nothing to the ascent. [VII.§.2]

That is to say, during the water entry phase the ascent velocity of the liquid inside the reservoir will be v, which implies a potential ascent of ξ = v2/2g. Now the velocity will be ve = vS/Se just as it passes through the orifice, due to the widening effect and the ascent ξe = ξ(S/Se)2, where ξe = ξ(S/Se) < ξ. That is, potential ascent is lost. Seen from his point of view only a part of the through velocity ve will contribute to the potential ascent, and this part will be precisely v, as it is equal to the velocity of the remainder of the water in the reservoir. Only the potential ascent ξ, corresponding to this velocity, is saved and the remainder, that is ξe – ξ, is lost. He thinks that this part should be evaluated as being transmitted to the movement of the internal particles. These criteria allow him to resolve the problem by means of some rather long and troublesome calculations.

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Fig. 7-9. Multiple discharge

It is important to note how he raises the hypothesis of conserving the live forces when he needs it, and how he tries to define the problem for physical phenomena. In this respect he says: This hypothesis, although it is physical and only approximately true, nevertheless is most useful for determining the motions of fluids without noticeable error whenever the uniform continuity of a vessel is broken, which we have supposed up to now, in the same way as occurs with the water passing through many orifices. [VII.§.2]

Regarding the experiments belonging to this section, he makes clear the difficulties in measuring the successive amplitudes of the oscillations, and the even greater difficulties for measuring the periods. He likewise uses the hypothesis of the partial loss of live force in the cases of dumping and discharge through the various cavities or diaphragms, be it with the same or different liquids. The various cases that he studies are complex, (Fig. 7-9 here shown by way of example), and though the conservation principle is always clear, being as it is applicable to all the cases, the assumption of losses is rather more contrived, taking into account that he has to conjecture how much is retained and how much is lost in each case, trying to carry out the evacuation that seems to him more probable. In accordance with the case of the oscillating fluid, he usually supposes that part of the potential ascent is lost when passing through a diaphragm, as it is absorbed by the mass of the liquid. On the other hand he insists that the hypothesis of the plane sections continues to be useful.

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Hydraulico-statics

When Daniel Bernoulli brought up the subject of hydrostatics (to which he made no important new contributions), he made the following declaration concerning its relationship to hydrodynamics: The pressure of water at rest must be clearly distinguished from the pressure of flowing waters, although no one, as far I know, has been aware of this. Hence it is that the rules presented by other authors are only valid for water at rest, although they employ terms that might lead us to believe that such rules refer to flowing water. [II.§.17]

Certainly, no previous author had noticed this fact, one of such overriding importance that it constitutes the base of all hydrodynamics. What is more, hydrostatics and hydraulics had hitherto been two separate disciplines, the only nexus of union being the fluid. From now on the separation will disappear. We have seen that in the chapters devoted to movement in pipes and to discharges, his sole aim was to obtain velocities, but he said nothing about pressures, which constitute an element of hydrostatics. From the moment at which they proceed side-by-side, a new discipline appears which he baptises as ‘hydraulico-statics’, as both participate in it. He says that ‘in this hydraulico-statics it is surprising that the pressures of waters cannot be defined without previously having grasped their motions’ [XII.§.2], and he devotes himself to this, arriving at the first definition of what is now known as ‘Bernoulli’s theorem’, which is the first theorem that any student of fluid mechanics encounters.29

Fig. 7-10. Water manometer 29 From my own experience, I can state that it is difficult to understand something of the abstruse science of fluids if this theorem has not been fully understood.

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But Daniel’s contributions are not limited to the theoretical field. We have already said that he was a great experimenter, in this respect we must acknowledge that he used the column water manometer in order to measure pressures30 (Fig. 7-10). This consists of a narrow pipe placed vertically in a small lateral orifice of a pipe through which water circulates, and through which the water will ascend until it reaches the height proportional to the pressure.31 It is not surprising that Daniel was the first to use this instrument, as there is actually a close relation between the phenomenon of ‘hydraulico-statics’ and this measuring instrument. So much so, that we believe these pipes and measurements existed before the theoretical formulation of the theorem, as we shall go on to show.

a

c Vs

Fig. 7-11. Discharge of a reservoir through a tube

Employing a methodological procedure standard in his study of the relation between pressure and velocity, he proposes a study of motion based on an apparatus that he uses as a model. It is a reservoir that discharges through a horizontal tube closed in turn by a perforated seal, as shown in Fig. 7-11. In principle the discharge is like those already studied in the previous chapters. The difference in level between the free surface of the water and the outlet orifice is designated by him as a, which will remain constant as he considers the surface of the 30

We must remember that Daniel was trained as a doctor, and first he developed this manometer to measure the pressure of arterial flow. Perhaps following Varignon’s invention of the manometer in 1705, Bernoulli also experimented by puncturing the wall of a pipe with a small open-ended straw and noted that the height to which the fluid rose in the straw was related to fluid’s pressure in the pipe. Based on this observation, doctors began to measure blood pressure by sticking sharpened glass tubes directly into their patients’ arteries. The less-painful sphygmomanometer (bloodpressure cuff) was not invented until the close of the nineteenth century (This comment is due to Larrie Ferreiro). 31 The pressure at the base of column of water is expressed by the well known formula p = ρgh, i.e., if h is known, p will also be know. These apparatus have continued to be used up to the present-day, be it with water or another liquid, and so much so that in experimental aerodynamics the pressures are frequently quoted as heights of water or mercury.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

reservoir to be very large. According to Torricelli’s Law, the speed of the liquid flowing out through the orifice will be vs = 2 ga ,32 and consequently the velocity in the tube, obviously less, must be vs/r, where r is the relation between the sections of the tube and the orifice. From what has already been said, it follows that the velocity inside the reservoir is zero, given its large size.

c

c v=0

a)

Vs dx

b)

dx

Fig. 7-12. Separation of the spout

Given this, let us imagine that the right part of the horizontal tube (Fig. 7-12a) was to disappear suddenly. It is clear that the liquid in the tube would accelerate from its previous velocity vs/r to vs, which would be the velocity accorded it by Torricelli’s Law. Therefore, according to Daniel, the effect of the perforated plug can be interpreted as if its presence were compressing and retaining the water, pressing it against the walls of the reservoir and preventing it from expanding. This retention pressure will be greater as the velocity of the water circulating through the tube is slower, because the water will have greater acceleration capability upon the disappearance of the plug, which is the obstacle preventing free movement. The result of this compression and retention (nisus et renisus) is that the water is compressed along the axial hub of the tube, and this pressure is transmitted to the lateral walls. We see that there is a likeness between this containing pressure and accelerating force that appears when we remove the plug. He writes:

32

From this point on of the Hydrodynamica, Daniel changes the conceptual sense of the velocity, which goes from being represented by the kinetic height to its intrinsic sense of a space travelled in a unit of time.

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It seems that the pressure of the lateral walls is proportional to the acceleration or the increment of velocity which the water would receive if the entire obstacle to motion were to vanish in an instant, so that [the water] might pour out directly into the air. [XII.§.5]

In order to reduce this reasoning to equations he makes use of the principle of live forces, not as a potential ascent however but in the form Σmv2 = 2gΣma.33 After the disappearance of the plug in an intermediate moment of acceleration process, the velocity of the liquid in the tube will be v, and this will be increased in dv in a time dt. In this interval the mass of water ms = ρSdx will egress the tube through the cut section (Fig. 7-12b), and will be substituted for an equal mass coming from the reservoir. As this entering mass has no velocity, it will pass from repose to the velocity v + dv. The increase of live force of the entire set will be the sum of that acquired by the mass of water entering, which is ρSv2dx, plus the increase corresponding to the mass that was inside the tube that passes through v to v + dv, and which will be 2ρScvdv. On the other hand, the real descent will be that corresponding to the fall of the mass ms from the height a, which is 2gρSadx. Equalling will result in:

v 2 dx + 2cvdv = 2 gadx

[7.26]

2 ga − v 2 vdv = 2a dx

[7.27]

which yields:

He says that in all motion the increase of velocity is proportional to the pressure multiplied by the increase in time, which would be dv = kpdt, or rather dv = kpdx/v. At this point we note that he uses pressure instead of force,34 which would be justified if he were talking about the internal pressure on the bases of the cylinder of fluid that he has isolated, as this would act in the same direction as the variation of the velocity. However, he indicates that the pressure on the walls is what he is looking for, and in some intuitive way he likens these.35 On 33

If one follows the original text, one observes that the factor 2g does not appear, and that the formulas are not non-dimensional. It is also appreciated that the relation between the velocity and the height is simply v = √a, as the same factor is also missing. This is explained because in his system of units it is verified that g = 1/2 as he warned in Note no. 20. 34 This will be later criticised by d’Alembert, in the Traité de l’équilibre et mouvement des fluides. 35 The concept of internal pressure would be introduced by his father later on, but Daniel here has an inkling of this.

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the other hand, the equation he gives Newton’s second law,36 which he introduces as a differential equality between the impulse and the variation of the quantity of motion. With these clarifications the previous equation changes to:

2 ga − v 2 = kp 2a

[7.28]

This equation refers to an intermediate moment between the separation of the tube and the final condition. In the initial instant the velocity was v s / r = 2 ga / r , and the pressure was that which existed before the separation of the tube, and which he likens to a height of z as ρgz. Introduced into the last formula, the result is:

2 ga ⎛ 1⎞ ⎜1 − 2 ⎟ = kp = ρ gz 2c ⎝ r ⎠

[7.29]

In order to eliminate all the unknown parameters that still remain in this formula, he imagines the case in which the outlet orifice is infinitesimal, i.e., r → ∞. In this condition the outlet flow will be practically null, and therefore the pressure would be ρga, then c = 1/ρ, which introduced into the last equation leads to:

z=

r 2 −1 a r2

[7.30]

This is the final formula presented by Daniel, in which a relation is established between the velocities, determined by the magnitude r and the height of the pressure z. If we make a small transformation with the aim of updating of formula37 we arrive at:

p = p0 −

1 2 ρv 2

[7.31]

an equation which is much more familiar to the present-day student. 36

We recall that Newton formulated his law in an integral manner, not differential. Therefore, the approximation made by Daniel Bernoulli in this point should no surprise us. See above in previous Chapter 2, ‘Resistance in aeriform fluids’. 37 On one hand p = ρgz, and p0 = ρga, which is the pressure that would exist without egress or repose pressure. On other hand r = ST / Ss = vs /vT = vs /√(2ga). Substituting, p = p0 – p0/r2 = p0 – ½ ρvT 2 .

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Fig. 7-13. Manometer and jet

Daniel suggests several experiments and corroborations for this formula, and here we reproduce one of them (Fig. 7-13). We can see the manometer tube and a small jet, and in both the water reaches a height z, which is a function of the velocity v. This apparatus clearly testifies to the intimate relation existing between the height of the water in the manometers and the formula, which was the reason why we venture to suggest that perhaps the instrument existed prior to the law. It is common to try to pinpoint an exact date for an important factor. However, this desire is frequently impossible to satisfy, not through ignorance or lack of documentary proof, but because the thought follows a line of evolutionary maturity, and there is no definite crystallisation point. Something like this happens with Bernoulli’s formula. What is more, there are a couple of indications that the idea was mulling around in Daniel’s head quite a long time. We know of a letter that he sent to Golbach, dated 17 July 1730, in which he writes that: In these past days I have made a new discovery which can be of great use for the design of ducts for water, but which above all will bring in a new day in physiology: it is to have found the statics of running water, which no one before me has considered, so far as I know ….38

He encloses a drawing of an apparatus similar to that of the Hydrodynamica, and the final formula without any type of demonstration.

38 Comment by Truesdell. Cf. ‘Rat. Fluid Mech-12’, p. XXX, that specifies that the mention of physiology is due to the fact that Daniel was a medical practitioner. See also previous note 31.

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Another note on this is found in the article ‘Experimenta coram societate instituta in confirmationem theoriae pressionum quas lateera canalis ab aqua transfluente sustinent’ (‘Experiments made before the Academy in confirmation of the theory of pressures exercised by the waters circulating in the laterals of chanels’) published in the Commentarii petropolitanæ (1729), in which there is a drawing of an experimental apparatus, that we reproduce in Fig. 7-14, and also the formula, as well as some paragraphs that are later repeated almost word for word in the Hydrodynamica. The volume is from the year 1729, but it was published in 1735. The drawing shows the tube he used as a manometer, and the removable plug with the orifice used in the experiments. It is clearly the precursor of the model he will use to demonstrate his theory.

Fig. 7-14. The Commentarii apparatus

It is curious that in both cases he gives the formula of the height of the pressure as (1 – 1/r2)a without proof. This seems to indicate that he obtained this formula from experimental data, which is not unusual, as it is very simple and easy to conjecture, especially for an excellent experimenter like Daniel was. If this is so, then his work consisted in looking for a basis for this equation, which would constitute another of the many cases in which theory has to justify experiments. This could also explain his father’s anger, as he very probably had the empirical formula, perhaps even before the publication of the Commentarii in 1735 and he could have been very close to its theoretical reduction. Nevertheless, this is a simple conjecture, interesting and even likely, but not verified. The discharge of elastic fluids

In the Hydrodynamica there is a chapter subtitled ‘Concerning Properties and Motions of Elastic Fluids, but especially of Air’, the only chapter that does not deal with liquids. This section begins with a kinetic theory of gases, one of the

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first in history,39 and goes on to the problems of the discharge of gases. We shall pay some attention to his kinetic theory because of its intrinsic interest and because it serves as an introduction to the phenomena of the discharges of gases. Daniel underlines three main properties of the elastic fluids by contrast with the liquids: they are heavy, they extend all directions, and they can be compressed more and more depending on the force applied to them. In order to expound his hypothesis on the internal constitution of a fluid of this type, he supposes a receptacle like that represented (Fig. 7-15), closed with a mobile lid by way of a plunger, upon which he places a weight. ‘Let the cavity ECDF contain extremely small bodies agitated hither and thither by very rapid motion; thus the small bodies, while they strike the lid EF, also support it by their continually repeated impacts’ [X.§.2], i.e., the force acting on the wall is a the result of the change in the momentum of the particles. This model is also designed to explain the tendency to expand, because if the weight on the lid is reduced, the agitation of the corpuscles would raise it to a new position of equilibrium. P0 E

C

P0 + p F

D

e

f

C

D

Fig. 7-15. Compressed air

39

Although the atomic theory of matter goes back to Democritus and Leucippus, the kinetic theories of gases began in the eighteenth century. The first was presented by Hermann in 1716 [Phoronomia, Chap. XXIV] in which he supposed that the gas was formed by different particles and these were in movement, establishing the pressure as a result of the impacts. This would be proportional to the density and the square of the velocity, and this in turn is in function of the temperature. Euler provided a second theory in 1729 (‘Tentamen explicationis phaenomenorum aeris’, Comm. acad. petrop., Vol. II, 1727), taking the revision of the vortex theory of Johann Bernoulli as its basis, supposing that the air was constituted by equal sized and equally spaced molecules. Each one had a spherical nucleus of ether covered by a first coating of the true substance of air, then a second coating of water. These molecules rotated at a velocity which was a function of the temperature. As regards pressure, this was proportional to the lineal velocity, and was constant for all the molecules. Euler calculated the external velocity of rotation, finding 447 m/s. Both this model and that of Hermann account for Boyle’s law. For more details see the study presented by Truesdell on the kinetic series of gases in his Essay in the History of Mechanics. [Chap. VI].

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In order to obtain the relations regulating the phenomena, Daniel starts out from an equilibrium state where there is a weight P0 over the plunger. If this increases up to a value P, the plunger will descend from the previous position, determined by EC, to another eC, so that σ = eC/EC < 1. He explains the effect of compression in the force on the plunger as being double. On the one hand, the number of particles per unit volume increases, and on the other hand so does the number of impacts against the plunger. In order to evaluate the first effect he says that ‘we shall consider the particles as being at rest’ [X.§.4], and distributed in layers. If in the initial position there were n of them in contact with the lid, now, after compression there will be σn-2/3, a higher number. As for the second effect, now with moving particles, he states that ‘of course, the number of collisions will be inversely proportional to the mean distance between the surfaces of the particles’ [X.§.4], although he offers no justification whatsoever for this statement. On this reasoning, if in the initial situation the mean distance between two particles was D, and the diameter of each one d, the aforementioned distance would be D – d. After compression the distance between the centres becomes Dσ 1/3 , and between the surfaces Dσ 1/3– d. According to the statement of the ratio of the forces with the distances, his basic equation will be:

P D−d = σ −2 / 3 1/ 3 P0 Dσ − d

[7.32]

Given the difficulty in handling the parameters D and d, Daniel proposes substituting other more manageable ones for them. For this he imagines that the compression increases progressively until the particles are touching. This would happen for a compression ratio σm. In this condition Dσm1/3 = d will hold. Introducing this equality into equation [7.32] transforms it into:

1 − σ m1 / 3 P = P0 σ − σ 2 / 3σ m

[7.33]

Now in practice σm → 0, which leads to a simple proportionality between forces and displacements. To this, which is exclusively mechanical, he adds: Meanwhile, the elasticity of air is not only increased by compression but also by an increase in heat, and since it is established that heat is spread out everywhere by increasing the internal motion of the particles, it follows that an increased elasticity of air in an unchanging space is proof of a more intensive motion in the particles of air. [X.§.6]

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In current terms we should understand ‘elasticity’ here as pressure measured by the barometer, as he says later on, and ‘heat’ as temperature. The velocity of the air particles depends on the temperature, moving more quickly as this augments. Due to mechanical factors, the force transmitted by the particles to the plunger during the impacts must be proportional to the square of their velocity, and in turn is equal to the weight P placed on top. On the other hand, there must be proportionality between this magnitude and the degree of heat.40 Once the problems of dimensional analysis are resolved, the fundamental thing is that the behaviour of the fluid will be expressed as P = kP0v2/V, where v indicates the agitation velocity. When densities and pressures are introduced, the former are inversely proportional to V, and the latter are directly proportional to P, therefore the previous formula can be expressed as:

p = kRv 2 ρ

[7.34]

where all the constants have been included in kR. The fundamental thing about this equation is that the relation between pressure and density is solely a function of v2 and therefore of the temperature, a fact that he indicates by saying that the height of the equivalent cylinder of homogeneous air ‘is always the same, because the elasticity and density of the air diminish in the same proportion, if we assume that the temperature (calor) remains unchanged” [X. §.34].41 As a result of this he comments on the gas thermometer presented by Guillaume Amontons42 40

The current kinetic theory of gases coincides in many points with that of Bernoulli. The main difference is that the velocity of particles is not the same for all of them, but there is a statistic distribution for which various models have been suggested. However, considering the average velocity, it is estimated that ½mv2 =(3/2)kT, where k is the Stefan-Boltzman constant and T the absolute temperature. 41 As a continuation of the previous note, the formulation of the perfect gases equation for a specific gas is p/ρ = RT, where R is the constant of this gas, which is the universal of the perfect gases divided by the molecular mass of the gas in question. Just as Daniel says, the ‘pressure’ and density vary in the same proportion, and comparing with the formula of the previous note, the velocity of the particles of gas only depends on the temperature. For Daniel the velocity of all the corpuscles was the same. Today we assume a statistic distribution, but his hypotheses will be valid for average values. 42 Amontons presented experiments relating heat and pressure. Today these are known as the GayLussac law. Besides, we must consider that the concept of temperature, as we know it today, was still not well outlined. It was normal to speak of the quantity or grade of heat. Cf. Manuel Sellés García ‘La ley de Amontons y las indagaciones sobre el aire en la Academia de París (1699–1710)’ (‘The Amontons’ law and the inquiries about air in the Academy of Sciences of Paris (1699– 1710)’).

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in the Paris Academy in 1702, discussing its usefulness for measuring the ‘degree of heat’ of air. Having presented his kinetic theory, he interprets the discharge of elastic fluids. Daniel studied three cases: the discharge of a reservoir through an orifice to an empty space; the same case but to a less dense space; and the entrance of air from the outside to a reservoir. The three cases are fairly similar, and the theoretical basis is common to all three, therefore he proposes comparing this phenomenon with the discharge of an uncompressible fluid. In order to do this he defines a non-elastic imaginary fluid equivalent to the real elastic one, from which the problem can be reduced to cases already known. For this reduction he considers: If we consider a vertical column of air of uniform density and adjusted to the mercury of the barometer, then the height of that column will be what I call the height of homogeneous air for the given density. [X.§.32]

The pressure at the base of column of homogeneous fluid will be determined by the equation p = pgh, and therefore the equivalent height will be:

he =

p ρg

[7.35]

Now if the relation p/ρ depends only on the temperature, as we have seen, this would mean that the equivalent height will depend solely upon this, and will not vary once the initial conditions were given, if the pressure or density were modified.43 We now take the first of the three cases: the discharge of the vessel into an empty space, and which may be considered the most explanatory of the three. Once the height of the equivalent non-elastic air is calculated, the problem is reduced to the outflow of the liquid from a reservoir having this height. However, unlike the liquids, the height of the discharge will not decrease while the temperature is kept constant: what will decrease is the density and internal pressure.44 43 These interpretations signify likening the atmosphere to an ‘ocean of air’, a widely spread idea at the time. 44 Is interesting to note that Daniel always makes the observation that ‘the same grade of heat is maintained’ [§.34–38]. Nowadays we know that in order to maintain the temperature constant in a process of this type it is necessary to transfer heat to the recipient. If the discharge was adiabatic, that is without any heat transfer, the relation between pressure and density would be p/ργ = cte, γ being the relation between the specific heats of constant air pressure and volume, whose value at

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In order to obtain the variation of the internal pressure, Daniel calculated the mass of air expelled in a time dt, which will be vs ρSs dt that is also equal to Vd dρ, Vd being the total volume of the reservoir. The solution of the differential equation that results from equalling the previous magnitudes is:

ρ = ρ0e

− 2 gh e

Sst Vd

[7.36]

where in the egress velocity has been replaced by its expression as a function of the equivalent height he. The other two discharge cases received similar treatment, with the difference that two heights should be borne in mind: one of the outgoing fluid and the other to where it flows. Therefore, in the expression of the outflow velocity, the difference between both should be introduced. There is no conceptual difference, only in the calculations.

m

m

M m a

V

x a

Fig. 7-16. Intrinsic live force

We would like to comment on just one more aspect of Daniel’s treatment. As he says, one of the differences between elastic fluids and non-elastic fluids is that the former possess a live force at rest which he calls ‘the live force contained in a compressed elastic body’ [X.§.38]. This live force is what produces the potential ascent that can be communicated to another body when the elastic force, i.e., pressure, is lost. In order to estimate the value of this intrinsic live force he presents an apparatus (Fig. 7-16) consisting of a cylinder with the plunger upon which lies the weight p = mg, which equilibrates the interior pressure to a height a. If an additional weight, p = mg, is placed on it, the plunger will start to descend towards a new position of equilibrium, and at an intermediate point of its run, room temperature is γ = 1.4 Therefore, in order to verify that the phenomena is isothermal, that is, p/ρ = cte, heat must be introduced.

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when it has descended a – x, it will carry a velocity v. At the same time, supposing an isothermal compression, the internal pressure will have augmented, producing an internal force of value mga/(a – x). Equating, the expression governing the system is found to be:

(m + M )

vdv a m) g = (m + M − dx a−x

[7.37]

Which once integrated, with the initial condition of null velocity, leads to:

1 (m + M )v 2 = (m + M ) gx − amg ln a 2 a−x

[7.38]

The interpretation he offers of this formula is clear. If no fluid existed in the cylinder, the masses m + M, on descending a space x, would generate a real live force of ½(m + M)v2, deriving from the transformation of the potential live force (m + M)gx. However, in the case of the plunger, a part of it was used in compressing the air, and this part is the subtracted from the second element in the formula, which is, precisely, what would be generated by the descent of the weight p through the distance a·ln(a/(a – x)). Therefore, this distance is a measurement of this intrinsic force. As regards the interest that this force holds, he says: [T]his is an argument worthy of attention, since to these are reduced the measurements for machines driven by air or fire or other similar driving forces of this kind, many of which, perhaps new, could not be developed without considerable improvement and perfection of practical mechanics. [X.§.39]

Regarding these possible interventions, a little further on [§.43] he quotes Hales’ works in the Vegetable staticks, in which he finds the ‘equivalent air’ produced by a quantity of coal. The chapter has an appendix describing experiments with gunpowder igniting in firing cannonballs.45 Johann Bernoulli’s Hydraulica

The Hydraulica is a relatively short treatise, somewhat less than a hundred pages, divided into a short preface and two parts. Having established the general principles in the Preface, he then he deals with the movement of water through pipes and vessels of simple geometrical shapes in the first part, whilst in the second he generalises this to any type of duct, allowing a continuous variation in 45

The experiments were published in the Comm. acad. petrop. Vol. II.

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their sections. By contrast with Daniel’s Hydrodynamica, the general approach and focus of the work is directed towards a general theory of the motion of fluids undertaken in a comprehensive manner. It is not accurate to say that the Hydraulica is only an extension of the hydraulico-statics of the Hydrodynamica46 since, as well including additional problems such as that he calls the ‘hydraulic problem’ [I.§.XXIV],47 and the calculation of the resistance of a cylinder moving axially in a liquid [I.§.XXVI], the new approach gives it a new significance, as we shall see from what follows. At the beginning of the work, in the preface, Johann puts aside his interest in hydrostatics, as he judges that its laws have already been demonstrated and rationally deduced, and he devotes his efforts to hydraulics, concerning which he states: This Science, commonly called Hydraulica, is extremely difficult and up to this time has not been subjected to the laws and rules of Mechanics. Everything that authors have written on this subject is based on experience alone, on theories that are wholly uncertain, and with insufficient foundation. [Preface]

He goes on to mention the work of his son: In the book Hydrodynamica which my son published not long ago, he undertook this subject under luckier auspices, but he relied upon an indirect foundation, namely the conservation of live forces, which although certainly true, and proven by me, is still not accepted by all philosophers. … Thus far no one has given a direct method by which, a priori and only using the principles of dynamics, the nature of the motion of water issuing forth from vessels through orifices through ducts of no uniform size can be investigated. [Preface]

For him, the direct method is based on the principles of mechanics, i.e., on Newtonian principles. After these initial thoughts, and in order to have a clear understanding of things, he includes in the Preface eleven basic definitions, ‘the validity of which is manifest for dynamics as well as hydrostatics’. In the Definition IX he sets out how to divide the liquid held in a vessel (Fig. 7-17) into very thin horizontal layers. Daniel had already outlined this idea, but he only used it to explain the movement of water through the outlet of a vessel,48 whereas in Johann these very thin layers became differential elements where ‘every one of these strata is pressed down just as if an aqueous cylinder of water, whose height equals that 46

This is what Daniel said in one of his letters to Euler. To which he later dedicated a monograph. Cf. Opera Omnia, CLXXXVII. The quotations in brackets refer to the Hydraulica. 48 Cf. Hydrodynamica. Chap. IV, §.2 and cf. supra §.7.2.2. 47

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corresponding to the depth of stratum itself in the vessel, were lying over it’ [Def. IX], i.e., they could support forces and other properties even when the division into layers is effected only by the mind and does not respond to a physical reality. Concerning the weight of each one of these layers, which will be proportional to its surface (amplitudine), he says that: Their own gravitational forces can be mentally separated from the strata, so that only their mass remains without its weight. But if instead of removing the gravitational forces, these are substituted by others that press on the upper surface of the water, but keeping for each one [force] the analogy of being just in the same relation that exists between the surface of each stratum and the surface of the upper surface, so each layer gravitational force will be equal to the gravitational force by which it is substituted. Therefore, the same pressure will arise in the individual stratum, just as if all the strata had remained in their natural state. [Pref. Def. X]

What he proposes with this operation, which he calls ‘translation’, is the substitution of the weight of the layer by an equivalent pressure applied at the free upper surface of the liquid (Fig. 7-17). While he now translates the weight, later on [I.§.IX] he will extend it to any other force, with the result that all the internal forces will move or ‘translate’ to the outside boundaries.49 F0

F

Fig. 7-17. Strata movement or ‘translation’

According to his ‘translation’ rule, if a force F acts on an intermediate layer whose surface is S, the force translated will be F0 = FS0/S. The justification, from the point of view of our present-day knowledge, is simple. According to Pascal’s Principle, the pressure at a point is transmitted to the entire fluid. The force F is 49

The only difficulty which we must resolve in order to understand correctly what he is saying, is that he gives the name pressure to the total force acting on the surface, and not to the force by surface unit.

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equivalent to a pressure p = F/S, which would be p = F0/S0 in the upper surface. Johann does not identify the pressure as a force by unit surface, but as a total force: therein lies the reason and necessity of this rule. What must be underlined as being particularly important in this reasoning is the notion of internal force, which would be better described as an internal stress. Therefore, when taking a layer, he is tracing an imaginary plane that divides the fluid internally, so that one of its parts can be substituted by the force that it causes on the other part. The whole will remain balanced, because both forces are equal and opposite, but the elimination of one of the parts is possible if interaction forces are placed upon the other. Up to this point, pressure is understood to be an action against the walls, but not inside the fluid. The new concept has a mechanical background, and therefore it is not surprising that it is Johann Bernoulli who introduces it, as he had also divided solid bodies into differential elements. Let us return to the strata. In the definition quoted he considered them as being horizontal, as he is only concerned with the weight as mass force, but at the end of the Preface he indicates that this does not have to be so, as, in the case of movement in the pipes, the strata must be perpendicular to the direction of the movement. Later on, as we shall see, he fails to follow his own recommendation. At the end of the Preface, he also continues to point out that the motion under consideration is an abstraction of the real motion, which has impediments that modify it, such as ‘imperfect fluidity’ and adherence to the walls. Of the two parts into which the Hydraulica is divided, the first dealing movement in cylindrical ducts, is headed by the following subtitle: Dealing with the movement of water through vessels and cylindrical ducts, which are composed of several cylindrical pipes joined together.

In order to study this motion he supposes a construction comprising two cylindrical pipes joined together with different diameters, which he calls channels (Fig. 7-18),50 through which the liquid circulates. The liquid will move at both ends far from the narrowing with a uniform velocity according to the plane sections. In order to maintain this motion, he imagines that whirlpools (gurges) are formed in the transition between the pipes causing a funnel effect, and forcing the fluid to accelerate progressively in an inverse ratio to the sections, as ‘no change is sudden, but successive and gradual’ [I.§.III], but maintaining the plane motion. Evidently the whirlpool hypothesis is somewhat artificial, although it is backed up in a certain way by the observational evidence that the natural flow of water through open channels produces phenomena of this type in the corners. On 50

The original figure is presented as Fig. I-14 in the ‘Introduction’ of this book.

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the other hand, the whirlpool reminds us of Newton’s ‘cataract’, as Johann himself notes when he comments on this later on [II.§.XX]. At this time few manageable models of this phenomenon could be conjectured, therefore including these ‘whirlpools’ is certainly a wise choice, not least by reason of its instrumental nature.

E

A

V0

S0

dx

F

V1

G

S1

C

B

Fig. 7-18. Whirlpool generation

With this model, the question proposed is how to find the law of acceleration that the liquid has when flowing through a channel limited by the whirlpool. On this topic he says: Let us suppose that the entire channel BE is full of a liquid with no weight of its own, but which is pushed through the orifice AE driven by a given force p, that uniformly presses upon over the entire surface of the liquid AE. With this we shall try to find the law of acceleration with which the liquids flows through the channel. [I.§.I]

In this exposition, Johann delimits an imaginary fluid dominion, and supposes the existence of a uniform force, which when applied over one of the surfaces of the dominion is capable of generating a movement. In present-day terminology this force would be p1S1 – p0S0. The existence of the funnel produced by the whirlpool partially obstructs the passage of the fluid, making it accelerate, a fact which requires the existence of a force pushing the liquid, a force which will be the pressure. In this respect he states: From hydrostatics I assumed that the immaterial driving force p pressing upon the surface of the liquid AE is instantly propagated to the surface GE of the liquid in the pipe BF, and that this occurs whether the liquid is standing still or flowing in the entire channel, as long as it remains full. [I.§.II]

where we have understand force p as being what we have already expounded.

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In order to resolve the problem mathematically, he supposes the fluid to be divided in layers perpendicular to the motion. He takes one of these layer located in the centre of the whirlpool, in which the acceleration of the liquid is a, and which will be submitted to a force whose value must be the product of this acceleration times the mass of the liquid of the layer, which is ρSdx. Therefore the elemental force (vis motrix) applied to the same will be Df = ρSadx. But for kinetic reasons we must demonstrate that adx = vdv, and therefore dF = ρSvdv, a formula that relates the force only to the velocity and acceleration of the layer.51 According to the Definitions of the Preface that we have already commented on, this force (vis motrix particularis) is applied to the frontal surface S0 resulting in ρS0vdv. The total force will be the integral throughout the length of the whirlpool, at whose ends the velocities are v0 and v1. The final result will be:

F0 =

∫

v1

ρS 0 vdv =

v0

⎛ S2 ⎞ 1 1 ρS 0 (v12 − v22 ) = ρS 0 ⎜⎜1 − 12 2⎟⎟v1 2 2 ⎝ S0 ⎠

[7.39]

Johann names this total force, which we have represented as F0 , pressure p. It turns out to be independent of the shape of the funnel formed by the whirlpool. He affirms that its role is to maintenance of the whirlpool: If the velocity in the pipe BF remains constant with the liquid flowing continuously in the pipe, it is obvious that the other velocity in the pipe HE will also remain constant; and accordingly the driving force, or pressure p, makes no contribution whatsoever to accelerate the motion in either pipe. Therefore it is clear that the force p is only applicable to forming the whirlpool, and to maintaining it in this state. Thus p=(hh-mm)vv/2h. [I.§.IX, Cor. II]

The formula included in the quotation is the one given literally by Johann, and corresponds to the one we have obtained as equation [7.39]. After the general deduction of the equation, he goes on to specify it, using a model in which his son Daniel obtained the formula relating pressure and velocity [I.§.IX, Cor. III]. As the motion is vertical, the force or pressure will be the weight of the fluid of the reservoir, which is F0 = ρghS0, where h is the height from the level of the liquid until the outlet. As regards the outlet velocity, which will be the unknown to be determined, he takes it as that corresponding to a

51

We have included the density in the developments so that that the formulae were coherent, although he does not do this as he very probably supposed that the density was the unit

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kinetic height z, i.e., v12 = 2gz. Introducing both equations in the general one, the height equivalent to the velocity is obtained:

z=h

S 02 S 02 − S12

[7.40]

Johann considers this formula to be a theorem [I.§.X]. According to it, when the outlet section S1 is null, the outflow velocity of the fluid will be that determined by Torricelli’s Law, concerning which Johann says: ‘This is a very well known Theorem, but up to now it has not been demonstrated from dynamic principles’ [I.§.X, Cor. I]. Likewise, he concludes that when the outlet section is increased, the outlet velocity of the water also increases, with the result that if S1 = S0, it will be infinite, and even negative if the outlet section is greater than that of the reservoir. Faced with these surprising conclusions, he indicates that throughout the demonstration he has assumed the water level to be constant, and therefore without the existence of driving forces (vis motrices). In order to eliminate these restrictions, he embarks on an extensive study with a discharge starting from a state of rest, and here the anomalous cases noted disappear. If we compare the previous formula [7.40] with the one found by Daniel in equation [7.30], certain likeness can be seen. However, each represents a different phenomenon. Daniel’s one refers to the variation of the pressure in movement through tubes, whilst Johann’s represents the outlet velocities of a vessel as a function of its relative section and the outlet orifice. In view of what has already been said, we are in a position to be able to compare the methods and the hypotheses of father and son. First, Johann takes as the starting point of his deductions Newton’s second law of mechanics, although with some additional hypotheses, while Daniel starts from the principle of the conservation of live forces. Second, Daniel’s calculations are based on a hypothetical transient phenomenon that would happen if part of the outlet pipe were to disappear suddenly whereas Johann supposes a real and stationary phenomenon. Third, Daniel identifies the forces or pressures on the lateral walls of the ducts with those that ought to be required to accelerate the fluid after this hypothetical disappearance. By contrast, for his father the forces are those applied to a fluid domain bounded by abstraction. As a fourth difference, Johann’s solution is more general and includes the other as a particularisation. In conclusion: there is a notable conceptual distance between both. As well as these considerations, the new focus, together with the depth of the new approach, comes into its own in the second part of the Hydraulica, where the case of ducts with continuous variation is applied.

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333

Apart from the studies on the outlet of vessels, whether they are completely full or not, in the first part he also analyses the case when there are more than two tubes forming the duct. The solution, although more complex, does not bring about any theoretical novelty and therefore we shall go on to the second part of the work. He begins the second part, whose object is motion through continuous ducts, with the following statement: Containing the direct and universal method for solving all hydraulics problems, whatever they may be, that can be formed and proposed concerning water flowing through ducts of any shape.

As the heading indicates, in this second part he lifts the restrictions imposed on the first part. The duct can be of any form, straight or curved, continuous or composed of several cylindrical tubes, it can be in a vertical or horizontal position, or each part can have a different slope. The whole may be totally or partially filled with any liquid, which may be allowed to flow until it empties, or may be kept constantly full by providing the water required. Some cases may be more complicated to evaluate than others, but the theory will be valid in whatever circumstance. The generality to which he aspires includes both stationary and transitory cases. Once the duct and the liquid have been defined, the question to be elucidated is at what velocity the liquid will flow when going through any given section and how much pressure will be brought to bear on the lateral walls. He supposes that the fluid descends through a channel whose section and slope are variable (Fig. 7-19a),52 and submitted to the action of gravity. He divides the fluid into horizontal layers, and with this configuration carries out the calculation process, which due to the fact that the layers are horizontal turns out to be long and laborious. With a view to rendering it more accessible and easier, we have taken layers perpendicular to the centre line (Fig. 7-19b) which is what he himself had recommended in the Admonition with which he ended the Preface: ‘I would also wish it to be noted that it is not absolutely necessary that the strata are always considered in a horizontal position. It is more accurate to assume them to be perpendicular to the direction of the movement of the water.’ [Pref. Admon.] While following his suggestions, we have also modernised the presentation, although of course without altering the sense.

52

The original figure is presented as Fig. I-15 in the ‘Introduction’ of this book.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

V0

V0

A0

A0 s

y

ds

dy

h

V

V A1

A1

V1

a)

b)

V1

Fig. 7-19. Motion through continuous tubes

The geometry of the duct is defined by the centre line s(y) and the normal area A(s). The motion follows the law of plane sections, i.e., the motion is onedimensional. The fact that the variation in the area of passage is slight avoids the need to postulate the whirlpool, which he found necessary in the first part. He solves the problem in two stages: the first for stationary motions, and the second for the non-stationary ones. In the first of these cases, an intermediate layer whose mass will be ρAds, will find itself submitted only to weight and an inertial force due to the variation of velocity arising from the change in the transit area throughout the duct. The value of the first force is ρgAds in a vertical direction, it is an absolute force (vir absoluta) and its component along the centre line becomes ρgAcosθds. In order to calculate the second force, he takes the tangential acceleration a = vdv/ds, which give the force ρAads = ρAvdvds/ds = ρAads, which in turn is translated to the upper surface as ρA0vdv. The final result is obtained integrating both into the upper surface and the inferior surface and equalling them53:

∫

s1

ρgA0 cos θ ds = s0

∫

s1

ρA0 vdv

[7.41]

s0

Whose integration is elemental, yielding the expression:

p = ρgA0 h =

53

1 ρA0 (v12 − v02 ) 2

Angles are used instead of differential quotients and sub-indexes.

[7.42]

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335

The total force existing between the two control surfaces is represented as p, which he calls ‘hydrostatic force’ (vis hydrostatica), and which corresponds to the stationary phenomenon. When the movement is not stationary, and the velocity in a section is variable with the time, a new force appears that he designated as ‘hydraulic’ (vis hydraulica) in order to differentiate one from the other. These two forces, the hydrostatic and the hydraulic, compose the total force which undoubtedly generated by the action of the primitive p, which was found in §.III to be equal to gha [p = ρgha]. [II.§.VI]

In order to calculate the hydraulic force, he takes as reference the acceleration of the fluid in the outlet section, and he refers the acceleration of any other fluid to this one. Starting from the continuity equation:

Av = A1v1

[7.43]

He introduces the acceleration as a derivative with respect to the time

a=

dv A1a1 = dt A

[7.44]

The elemental force arising from this acceleration upon the layer in question is ρAads = ρA1a1ds/a, which, translated to the upper surface, becomes ρA0A1a1ds/A, which once integrated and added to the stationary case, finishes in the following final formula:

1 ρ A0 (v12 − v02 ) + ρ A0 A1 a1 2

∫

s1

ds = ρgh s0 A

[7.45]

Which is the Bernoulli’s equation for a non-stationary motion. Although we have updated the symbology of the original expression, there is not much difference from how it is written in the Hydraulica. We see that there are three terms. The first and third represent the variation of the kinetic and potential energies. The second is the non-stationary component, which enters through the acceleration in the reference section (a1), and affects the entire mass of fluid enclosed within the control sections. This equation, obtained after some brilliant hypotheses and deductions, clearly marks the differences between Johann and Daniel.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

One of the most outstanding contributions of the Hydraulica, if not the most outstanding, is the introduction of the concept of internal pressure. The importance of this concept lies in that it makes it possible to isolate and analyse the evolution of a differential element of fluid as a function of time, or more generally, of the specific domain delimited by some procedure, be it physical or imaginary. In order to apply Newtonian mechanics correctly to fluids, it is necessary to know the mass conditions of the volume to be studied as well as the forces applied upon it. These forces will be mass or surfaces generated. The first ones are a lineal function of the mass, such as the weight or the inertial forces, and the second ones are perpendicular to the surface, as with the pressure. Therefore, until the introduction of the internal pressure, the most that could be studied was the evolution of a fluid volume limited by a physical contour, as the only thing that seemed feasible was the introduction of the forces exercised by the walls. The result was the Bernoulli’s theorem, just as Daniel obtained it, while as regards the internal pressure we arrive at Johann’s solution. Although Johann Bernoulli was the person who introduced the internal pressure, previous indications and traces of this existed. We can perceive its existence already in Stevin’s theorem, as on substituting a portion of the fluid by a solid body with the same contour, it can be inferred that the forces exercised by the substituting body should be the same as those of the fluid substituted.54 Another indication is found when Daniel identifies the forces on the wall with the axial thrust, as in this operation he isolates a drop of liquid and supposes the existence of inertial forces on it that are transmitted to the surface. In any case, the existence of these internal pressures was the additional hypothesis needed for applying Newtonian laws to fluids.55 E F

e f

F

f c

c C

C

Fig. 7-20. Internal force

54

We recall that Stevin stated his ‘principle of solidification’ as when in a fluid at rest, part of it was replaced by a rigid solid, the forces exercised by the rest of the fluid are not altered (cf. Truesdell ‘Rat. Fluid Mech.-12’, p. XI). 55 Cf. Truesdell, Essays of History of Mechanics, Chap. II, §.11.

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337

Johann Bernoulli clearly states the priority of the internal pressure over the lateral pressure, noting: In order that we might comprehend clearly and correctly in what that force consists which is exerted on the sides of a duct while liquid flows through it, we must know that that force is nothing more than the force originating from the compression force by which, certainly, neighbouring portions of the fluid, … are driven one against the other. [II. §.IX]

In order to determine this force (Fig. 7-20) he says: Imagine for the time being that a part of the duct EFfe (during flow) disappears suddenly, the remaining CFfc staying in its place with all its circumstances, and [consider] that at the same moment at the area Ff a new driven force equal to π itself is applied. [II.§.XI]

This force is calculated by applying equation [7.45] between the planes Ee and Ff. After this he obtains the formula that links π with the rest of the variables, and which we leave out as being obvious. He does not fail to consider that the pressure could be negative, in conditions that are easily determined mathematically. Having admitted this possibility, he states: However, in whatever manner it may occur, it is plain that in cases of this sort the pressure is changed into relaxation, which causes the walls of the duct near Ff not only not to be pressed outward, but to be contracted inward everywhere (if the rigidity of the walls does not prevent it). [II.§.XIV]

After all these deductions, and perhaps to demonstrate his mastery of the subject or to silence possible criticisms, he repeats all the developments starting out from the principle of live forces: For a more productive confirmation of the validity of our direct and universal method, it now pleases me to propose an indirect solution, deduced from the theory of the conservation of live forces, of the principal Proposition concerning the velocity of water flowing out of a vessel or duct which is always full, just as we proved it by the equation shown in § VII. [II.§.XXIX]

And so he does. In order to finish our review of the Hydraulica, we mentioned the criticism of Newton’s cataract made by Johann based on the new theories. The argument is the impossibility of forming an imaginary wall so that on one side of it the fluid is at rest, and on the other in motion, which is what Newton

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THE GENESIS OF FLUID MECHANICS, 1640–1780

imagined. The argument is that the pressure of water descending cannot be equal to that of the water at rest, therefore there will be a discontinuity in the pressures between both sides of this surface, and this difference between the internal forces will give rise to a motion which will make the cataract disappear and will mix the water: Moreover, as the sides of the cataract are not rigid, and the waters at rest are not subject to any opposing pressures due to the flowing waters, it is reasonable and necessary that the waters supposedly at rest and continuously exerting pressure make their effect felt: i.e. they enter the cataract and mix with the flowing water itself. Therefore, the shape of the cataract will be destroyed and thrown into disorder, and the water descends differently from our explanation. Therefore the Newtonian explanation, since it is contrary to the laws of hydrostatics, cannot stand. [II.§.LX]

The criticism cannot be more crushing and justified. Later authors also mention it. D’Alembert’s account of motion in tubes

D’Alembert’s Traité de l’équilibre et du mouvement des fluides was published in 1744. This work deals with three fundamental themes: equilibrium in fluids, motion in vessels and ducts, and resistance. These are the topics of the times. We have commented on the third of these already, when we dealt with the problem of resistance, and now we will deal with the first two, which are quite closely linked together. D’Alembert, like other authors of the period, sought to base fluid mechanics on mechanics in general, and in this respect he tried to adapt what he considered to be its general principles, already set out in the Traité de Dynamique (1743).56 However, this was not an easy task, so much so that he expressed his caution in the preface to the Traité de l’équilibre in the following words: As the mechanics of solid bodies are based only on metaphysical principles which are independent of experience, one can determine exactly which of these principles should serve as a foundation for the others. The theory of fluids, on the contrary, must of necessity be based on experience, from which we derive only limited enlightenment. [Pref. p. vi]57

56 57

In Chapter 3, ‘D’Alembert paradox’, we have resumed the dynamic concepts of d’Alembert. The quotes in brackets correspond to the Traité de l’équilibre.

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D’Alembert compares the certainty upon which mechanics is based with the need for experiment in the case of the fluids. Such a statement does not mean to say that it is impossible to reduce fluids to the same principles of mechanics, but rather that, as they are composed of a large number of individual particles, it is almost impossible to deal with them. He reinforces this a little further on when he says: As regards, fluid particles, in as much as they are bodies, there is no doubt that the principle of the inertial force and that of compound motion are suited to each of these parts. The same applies to the principle of equilibrium, if we could compare the fluid parts individually; but we can only compare the masses as a whole, and their mutual action depends on the combined action of the different parts, which are unknown to us. Experience can thus only instruct us about the fundamental laws of hydrostatics. [Pref. p. viii]

Clearly, the particles forming the fluids follow the laws of mechanics just like any other body. However, the behaviour of the whole can not be determined, because of the impossibility of knowing all the forces acting among them. Therefore he has to look for a way to sidestep this difficulty and provide an alternative law. We remember that Newton found himself with the same problem when analysing resistance in liquids. As an alternative, d’Alembert introduces what he calls ‘principles of experience’ which take the place of the metaphysical truths that rule mechanics. For the equilibrium of fluids, the principle of experience tells us that the parts of the heavy fluids are under pressure, and create equal pressure in all directions [§.1]; while for Hydrodynamics, which he interprets as a discharge phenomenon, the principle is that in this process the free upper surface stays noticeably horizontal [§.10]. The constituting particles are thus forgotten, or rather subsumed in other more global principles. With these two principles, together with his general principle of dynamics, his aim is to reduce fluid dynamics problems to hydrostatics; this being the corollary of the goal expressed in the Traité de dynamique in order to reduce general dynamics to statics. The application of everything that he has said to a discharge process leads him to demonstrate that the law of conservation of live forces is satisfied: But one of the greatest advantages that follow from our theory is that of being able to demonstrate that the famous Law of Mechanics, called the conservation of live forces, takes place in the motion of fluids as well as in that of solid bodies. [Pref. p. xvi]

At this point he criticises Daniel Bernoulli in his ‘Theoria Nova de motu aquarum’, for using this principle in his deductions without offering any proof of its

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THE GENESIS OF FLUID MECHANICS, 1640–1780

certainty, just assuming that the fluid was composed of an aggregate of small elastic corpuscles: ‘But it seems to me that such a demonstration cannot be regarded as carrying great force: also the Author seems to have given it only as an induction’ [Pref. p. xvi]. To be fair, the criticism was not justified, because, as we have already shown, d’Alembert arrived at this demonstration only after experience leading him to propose that the fluids travel in parallel planes, while Daniel Bernoulli made this assumption with the object of being able to apply a more general law of mechanics, namely the conservation of live forces expounded by Huygens. If, as d’Alembert states, ‘this principle (live forces) is recognised today as true by all mechanics’ [Pref. p. xvi], he did not need to ask anyone to prove it. He also criticises Johann Bernoulli, calling his theory ‘uncertain and arbitrary. Besides, his general [Newtonian] Principle is so easily demonstrable from that of the live forces, that it appears to be none other than this latter Principle in another form’ [Pref. p. xx]. Let us now analyse in greater depth the aspects related to the discharges contemplated in the Traité. Having established the validity of the principle of live forces, he obtains results similar to those of Daniel Bernoulli, with some variations. Where there is a conceptual advance, it is in the extension of the theory to the discharge of elastic fluids, and motion through flexible ducts. From the first book, where d’Alembert studies the equilibrium of fluids at rest, we note only some points that enable us to understand the cases of movement better, as these are dealt with in the second book. An initial theorem [§.14] is related to the pressure acting on the bottom of a closed vessel of any shape, submitted to variable vertical ‘accelerating forces’. Let this be the vessel represented in Fig. 7-21a, with the distribution of the accelerating forces, φ(x), shown on the left. He establishes that the ‘pressure’ on the base DE is equal to the product of its surface times the integral of φ(x).58 In order to demonstrate this he divides the vessel into horizontal layers (tranches). Each one will be submitted to its particular weight (pesanteur particulière), which it will transmit to the following layer, so that they all gravitate upon the last one. In this respect, he affirms that the pressure upon this will be the same pressure as when he replaces the vessel with another cylindrical one whose base is DE. We understand from the text that what he calls the particular weight is the product of the mass times the accelerating force.

58 One of the major difficulties found in d’Alembert is how to define what he means by words like ‘pression’, ‘puissance’, ‘resistance’, ‘force’ and others.

THE HYDRODYNAMICA AND THE HYDRAULICA O

P A

A

B

341

D

L

dx

D

E

Z

B

x

a)

T

C

Q

x

b)

Fig. 7-21. Vertical vessels

A variation of this theorem is that where the vessel is open at both ends [§.22], as shown in Fig. 7-21b, where part of the fluid is delimited by two parallel surfaces, AD and ZC. Given the distribution of the accelerating forces, φ(x) on the right, in order for the fluid to be in equilibrium in this case it is necessary that ∫φdx. However, apart from this, he adds two additional conditions: the first that ‘the force acting on the surface AD stretches from L to B, and the one acting on ZC, stretches from B to L … [and] there is no part negatively pressured’ must be demonstrated. [§.24]; the second that as the results of the integration of φ(x) from L downwards ‘there is no part pressed negatively’ [ibid.]. He also observes that the second condition enclosed the first. After this, d’Alembert makes an important addition: ‘if dv is the small velocity which, in a constant time, would be proportional to the accelerating force φ, we would have ∫dvdx/dt=0, or simply ∫dvdx=0.’ [§.25]. The increase dv is obviously a virtual velocity, and it is in the handling of virtual velocities that d’Alembert bases his principles. The formula:

∫ δvdx = 0

[7.46]

is basic in his entire treatise. He replaces dv by δv, with the aim of expressing the virtuality of the velocity. He had already obtained this equation, as he loses no opportunity to remind us, in the Traité de dynamique.59 Concerning the movement of liquids in vessels, d’Alembert begins by assuming a vessel of any shape, through which a liquid circulates without weight, and is not submitted to any other external force [§.84] (Fig. 7-22). When divided into 59

Op. cit. §.175.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

layers, as was done in the case of equilibrium, at a given instant any one of these layers will be moved by a velocity v. This layer, understood as being a constant mass of fluid, will increase this velocity up to v + δv in a subsequent moment.60 His exact words are ‘that v±δv expresses the velocity of each layer, as it takes the place of the one immediately below it’ [§.84]. Next, following his general principle of dynamics, he goes on to say that ‘if one supposed that each layer tends to move with a single infinitely small velocity ±δv, the fluid would remain in equilibrium” [§.84]. Let us analyse what d’Alembert says. We have seen that he spurns forces because he does not know how to explain them ontologically, placing the foundations of his mechanics on the variations in the velocity. Therefore, what has to be introduced into the motions of the liquid is a fictional force, or an inertial force, whose value will be the product of the mass multiplied by the acceleration, in such a way that the equality ∫δvdx = 0 continues to be satisfied. If we compare this procedure with that of Johann Bernoulli, we shall see that he interprets the phenomenon as the motion of a differential fluid element submitted to two pressure forces, one in the lower part and another in the upper, whose result is an acceleration such that Adp = ρAdX(dv/dx). The upshot is a dynamic equilibrium between both forces is determined by the boundary conditions that comprise the shape of the duct and the forces at the ends. Returning to the motion of the liquid, he says that ‘the fluid CDLP is assumed to be divided into infinitely small portions CDdc, KZzk, &c. that contain an equal quantity of fluid, and the height is called dx [δx], and [A] is the indeterminate width of each of these portions, ydx [Aδx] will therefore be constant’ [§.90].

V dx V+ V X

Fig. 7-22. Fluid motion in a duct 60

D’Alembert says, and quite rightly, that the velocity may increase or decrease according to the variation of the section, which obliges him to carry the signs more or less continuously. For the sake of simplicity we shall only consider one of them.

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343

Taking the mass of a layer as a reference, which we denominate dm, it is clear that dm = Adx, and introducing continuity with a reference section designated R, the result will be:

δx=

vδ m vR AR

[7.47]

That allows the equilibrium condition to be expressed as:

1

∫ dmδv = 2 ∫ dm(v

2 R

− v2 ) = 0

[7.48]

Eliminating the constants the equation becomes:

∫ δvdx = ∫ v A vδv = 0 dm R

[7.49]

R

So that finally:

∫ v dm = Cte 2

[7.50]

He judges this formula to be important, as it allows him to assert that ‘one then sees that the principle of conservation of live forces holds for fluids’ [§.92], and he criticises Daniel Bernoulli for having failed to demonstrate this. D’Alembert manipulates the last formula in a few respects when he refers the velocities to the reference surface, thus transforming it into:

v

2 R

∫

x1 x0

AR2 dx = vR2 N = Cte A

[7.51]

The constancy of this formula implies that d(vr2N) = 0, which after a few operations leads to the following condition [§.90]:

2 A12 A0 Nv R dv R + v R2 dx0 AR2 ( A02 − A12 ) = 0

[7.52]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

This is a formula just as abstruse as that given by Daniel Bernoulli, relating the variation of velocity in the reference section with the two limiting surfaces, and with the descent of the liquid in the upper one. He obtains the last formula by using two additional procedures that we will not dwell on here. All these calculations cover the case where weight is absent. Should it exist then, since it is an accelerating force, the equilibrium condition would be ∫(gdt – δv)dx = 0. Repeating all the steps of the calculation would lead to the following expression:

Nv R2 = 2 gMh + K

[7.53]

Which is the equivalent of equation [7.52], and where M is the mass of the fluid under control, h the center of gravity and K a constant depending on the initial conditions. He studies several cases with the help of this formula, and compares their solutions with those of Daniel Bernoulli. Almost all the cases are in agreement. It is worth distinguishing two groups among the few discrepancies between them. The first refers to the discharge of a vessel that is kept constantly full.61 Bernoulli had analyzed this case with two assumptions for maintaining the vessel full: one where the liquid supply is vertical, and another with a horizontal entrance. The final results were different in the two cases, as in the first the water had a velocity, while in the second it did not. D’Alembert criticised the last solution, because in it an instantaneous increase of live forces is applied to the mass of replacing water. The second group corresponds to the outflow of fluid in a submerged vessel, and to the movement through various diaphragms, cases in which Daniel supposed that there was a loss of live force. D’Alembert states that due to the irregular shapes, the phenomena could not be calculated [§.145]. He does not pay much attention to the pressure of the fluid moving in the vessel. He applies the condition gdt – δv, transformed as gx/vt – δv, of which he says that: [I]t represents the small velocity at which each layer tends to move in order to remain in equilibrium. It follows that gdx/vdt-dv/dt represents the indeterminate accelerating force, in virtue of which each layer would remain in repose. [§.146]

As a result the pressure will be:

61

Cf. previously in this chapter ‘Motion through tubes in the Hydrodynamica’.

THE HYDRODYNAMICA AND THE HYDRAULICA

∫

gr = gdx −

∫

2 NvR dv R + v R2 dN dxδv = gh − dt 2vR AR dt

345

[7.54]

Where r is the height equivalent to the pressure at the point under consideration. Although he does not apply this equation to any particular case, here we have applied it to a receptacle similar to that used by Daniel, with a steady flow and finite dimensions. Once all the operations are made, we arrive at the following formula:

1 1 gr = gh + v 2 − v02 2 2

[7.55]

Which agrees with that of Johann Bernoulli. As a corollary to the foregoing [§.149], he comments on the possibility that the pressure is negative, the situation in which Daniel said that there had to be suction instead of pressure, which d’Alembert rejects. He notes accurately the effect of the environmental pressure which would be added to the internal pressure. That is to say, he distinguishes the absolute pressure from the relative pressure.62 The first is the force times the surface unit, whilst the second is the former minus the atmospheric pressure. The entire body of formulae and calculations refers to the relative pressure.

A V

Fig. 7-23. Elastic fluid

The treatment of elastic fluids constitutes a novelty with respect to his predecessors, as only Daniel had done the calculations for the gases escaping to the exterior. The problem is more complex and more casuistic. 62

The matter is treated in greater depth in the Opuscules mathématiques, Vol. 1, p. 160-ff. It was also the motive of a letter to Euler, who replied saying that d’Alembert was right, who quotes it in the second edition of the Traité.

THE GENESIS OF FLUID MECHANICS, 1640–1780

346

He imagines a receptacle closed above, and analysed its discharge (Fig. 7-23). The most significant feature of the argument is his conclusion that under these conditions the theorem of the conservation of live forces should not be applied, because of the variation of the density. The equation ∫ϕ dtδ v = 0 becomes the following:

∫

Adx dx =

∫

AdxVvδv

∫

[7.56]

Adx

Where V is the volume of the recipient [§.201] Finally, we note that in the Traité, he also deals with the analysis of the speed of sound, an important theme in this period, but one that lies out with the objectives of the present work. Borda’s works

In 1788, Jean Charles Borda published his ‘Mémoire sur l’écoulement des fluides par des orifices des vases’ in the Mémoires de la Académie de Paris. This work linked up directly with those of Daniel Bernoulli and d’Alembert, even though more than 20 years had gone by since the appearance of their work, and the Memoirs of Euler had already appeared, although he omits to mention, as he does Johann Bernoulli. He begins his Report by quoting the Hydrodynamica and the Traité des Fluides [sic], commenting on the discrepancies between the two: I am known to have praised highly the two works I have just quoted, but I must confess that the solutions found therein are not always in agreement, there is still great uncertainty in this part of the theory of fluids. Uncertainty which it is astonishing that nobody has attempted to remove by examining more closely the hypotheses and use of the principles upon which the solutions are based. This task seems to me to be worthy of the attention of the Geometricians, and I am determined to undertake it. [p. 579]

The Report studied several problems, all related to the outflow of liquids: motion through orifices, the contraction of the stream, the entrance of fluids into submerged tanks, and the effect of spouts. We shall only analyse the first two cases, which seem to us to be the most original ones. The third is an excellent exercise in which he tries to make d’Alembert and Bernoulli agree with respect to the loss of live forces. Likewise

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he also tries to calculate these forces when the discharge takes place through spouts, in which he also follows the steps of the Hydrodynamica. For the outflow through an orifice, he begins by recalling the problem proposed by Daniel Bernoulli, and his hypothesis on plane movement (Fig. 7-24), although he admits that the hypothesis cannot be satisfied near the outlet. Borda supposes that only the upper surface (AB in the figure) and the lower surface (EF) descend vertically maintaining the horizontality. In this way, the fluid will move following flow tubes (branches) in such a way that at any horizontal surface, except the ones already mentioned, the velocity will not be uniform. We recall that the idea of the tubes, or streamlines, was Daniel’s own idea, although he did not use it.

Fig. 7-24. Borda outlet

Borda takes a stream tube limited by the upper and bottom surfaces. After an infinitesimal moment, the upper surface will have descended dx0, and consequently the bottom one will also have descended a magnitude dxe, both interrelated as a function of the cross section of the stream tube in both surfaces, which he designates as a0 and ae, respectively. We have noted that he does not use the time as an independent variable, but instead used the space travelled. The calculation procedure that he followed is to equate the variation of the live force of the fluid in the tube with that lost at the outlet, which is the same method that Daniel had used. An intermediate element PSTR of the stream tube which has a cross section a, and is has a velocity v, will have a kinetic height63 ξ = v2/2g. Therefore the live force of the element will be dε = ξdm, taking dm to be the mass.64 Reducing the variables to the output surface, the total live force of the stream tube will be: 63 64

Borda is the first person that we know of, to use 2g in the denominator of this expression. We note that Borda does not include the density in his calculations.

THE GENESIS OF FLUID MECHANICS, 1640–1780

348

ε=

ρae2 ve2 2g

∫

H 0

ds a

[7.57]

The variation of the former, during the time in which the upper surface descends dx0, must be equal to the loss of potential energy of the out-flowing droplet during this same time. This latter will be ρa0H0 dx, where H is the height of the water level in the vessel. That is:

⎡ ae2 ve2 a0 Xdx0 = d ⎢ ⎣ 2g

∫

X

0

ds ⎤ ⎥ a⎦

[7.58]

Although Borda, as we have already warned, does not use time, strictly speaking the previous equality should be interpreted as d/dt. Considering the relation ae/a0 = Ae/A0, we obtain:

2 ve

dve Ae2 dx0 A02

∫

X 0

⎛ ds A2 ⎞ + ve2 ⎜⎜1 − e2 ⎟⎟ − 2 gX = 0 a A0 ⎠ ⎝

[7.59]

This is the equation that Borda arrives at [§.3].65 As he himself declares, this equation does not differ from those given by Bernoulli or d’Alembert, except in the integral term66 that these authors also include, but as an integration according to horizontal surfaces, and not following the stream tube. Therefore in order to find the outlet velocity this term is required, ‘that is to say, [to know] the movement of the molecules in the vessel, but this is what the geometricians have been unable to achieve, thus the solution we have given is still very incomplete’ [§.5]. Therefore, the solution would only be valid when this term is negligible or rather when it is small there will be no noticeable differences in the ejection velocity when other solutions are employed. This is especially true if the outlet is small, as in this circumstance the phenomenon of stream contraction is important. 65

The left-hand side of equation [7.58] is expanded as:

⎡ d ⎢v e2 ⎣

∫

X

0

ds ⎤ = 2v e dv e a ⎥⎦

∫

X

0

ds + v e2 d a

∫

X

0

ds a

Also, we have to take in account:

d

∫

X

0

ds dx e dx0 = − a ae a0

That with the previously quoted a/a0 = Ae/A0 leads to the search. Cf. equations [7.45] and [7.54] by Johann Bernoulli and d’Alembert, respectively.

66

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Fig. 7-25. Measuring the flow

Concerning contraction of the stream, he recognises that Newton was the first to observe this phenomenon, and he agrees with him that the cause of it lies with the lateral movements of the molecules that the outlet, which tend to reduce the thickness of the stream. He recognises that ‘to determine the quantity of contraction for any vessel and any orifice would be an extremely difficult problem’ [§.6]. However, he states that there is a case that can be resolved fairly easily, i.e., a vessel with a pipe of an infinitely small diameter that penetrates its interior (Fig. 7-25 left) in such a way that the fluid contracts upon entry into the pipe, and remains in this state of contraction, sliding along a horizontal perfectly polished plane. Newton had calculated that the section of the contracted stream was half that of the orifice. Borda justifies this by a curious mathematical argument [§.7]. In the first place, he supposes that the velocity of the liquid in the narrowest point of the contraction will be due to the total height, so that if this is h, the former will have the value vs2 = 2gh. Therefore, if the pipe section were to be As, and µ were the contraction coefficient, the reaction of the jet would be R = 2ghρµAs , a formula that has been obtained on several occasions. This force would have to push the reservoir towards the left, and if the supporting plane of the reservoir was perfectly polished, this would move driven by this force. Next, in order to determine the coefficient value µ, he calculates the reaction force by another method. He considers that when the motion starts, all the molecules of the fluid that are against the walls of the vessel and at the foot of the pipe would not move except at an infinitely small velocity, and as a result of the absence of movement, the fluid as a whole could be considered as if it was perfectly at rest, ‘from which it follows that the difference in pressures that the fluids exerts on the sides AB and CD of the vessels come only from the single part O opposite the orifice’ [§.7]. The pressure at this point O is ρgh, and therefore the force will be R = ρghAs. Equating it to the other expression we obtain

350

THE GENESIS OF FLUID MECHANICS, 1640–1780

the result µ = 1/2, which is what he wanted to demonstrate.67 Obviously the reasoning is not valid, as the latter is a static force without change in momentum, and different in nature to the other. He now moves from theory into the realm of experiment. We recall that Borda was a great experimenter. He prepares a vessel 3 pieds in diameter (974 mm) with a lateral hole of 15 lignes (33.8 mm) and measures the horizontal and vertical diameters of the stream, saying that the vertical is always smaller. He finds an average value of 0.647, as opposed to the 0.707 found by Newton.68 The difference between both, which he judged to be quite large, is attributed to the fact that perhaps Newton used a very small orifice. As these measurement of the dimensions of the stream is not very accurate, he changes his method, preferring that proposed by Daniel Bernoulli, which consists in measuring the outflow time of a certain quantity of water, and comparing the measurement with the time it would have taken if contraction had not existed [§.9]. He proposes an apparatus (Fig. 7-25b) consisting of a vessel with an interior spout that we sketch in Fig. 7-26. The diameter was 3 pieds (974 mm) and the outflow was through a tube T of 6 pouces (162 mm) length and 14 1/10 lignes (31.8 mm) of inner diameter. The total water depth was about 367 mm and to measure the discharge he placed two needles NN separated by 4 pouces (108 mm), the first of these was at 2 pouces (54.1 mm) from the initial level of the water. In order to compare two types of contractions, he prepared an additional piece whose aim was to place a disc PP (dotted line) of 1 pied (325 mm) with the same inner diameter as the tube. What he intended was to modify the entry conditions. In the discharge the stream does not have to touch the inner tube walls. The time taken in emptying the water between the needles was measured with the pendulum that beat in half seconds that he compared with the time calculated theoretically for the case of no contraction. For the experiment without the plate PP he found a contraction of 0.515, and with that plate it was 0.625. He said that the first came very close to the theory of 0.5, the second diverged from it, giving rise to a rather vague explanation. [§.10] Borda continued with the cases referring to filling submerged vessels, as Daniel Bernoulli had already done69 in both entry and exit of water. He analysed the phenomena by the conservation of live forces, supposing that these underwent a loss, and by the principle of d’Alembert. After obtaining the equations

67

Here it is worth recalling the repeated intent to assimilate dynamic forces static forces in order to find always, this factor of 1/2. 68 He presents the values inversely, that is, 154.66/100 and 141.43/100, respectively. 69 Cf. previously in this chapter ‘Motion through tubes in the Hydrodynamica’.

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Fig. 7-26. Borda experiment

that give the height the water reaches in its ascent and its descent, he notes that his solution is very different from those reached by Bernoulli and d’Alembert, solutions in turn very different from one another. In view of the discrepancies he states: ‘I believe that I should not content myself with opposing demonstration against demonstration, and it would be good to decide the question by some experiments made to this end’ [§.18]. For this, he took a tube of 18 lignes (40.6 mm) in diameter and one pied (325 mm) in length which he submerged in a large tank. After several attempts, he found that when the tube sank 8 pouces and ½ ligne (218 mm), the water ascended up to the edge, i.e., 4 pouces minus ½ a ligne (107 mm). According to Bernoulli’s theory, it should have ascended up to 8 pouce (217 mm). Applying his own theory, and with a contraction coefficient of µ = 0.515, which he took as the experimental one, it ought to have been 49.5 pouces (117 mm), only somewhat greater than that measured. He attributes the difference to friction against the walls. Moreover, he says that even with the value of µ = 0.5, his solution fits very well to the tests. Another problem he deals with is the discharge of a reservoir through a cylindrical spout (Fig. 7-27a). On the account he gives, an internal contraction in the spout ought to exist, as the figure shows. If the outlet velocity through F is v, the velocity at the point of maximum contraction will be v/µ, with a loss of total velocity of v/µ – v, that will imply a loss of live forces of (v/µ – v)2. Therefore, the total evaluation of live forces between the outlet and the height h of the tank will be: 2

⎛ ⎛ 1⎞ 2 1 ⎞ 2 gh = v + ⎜1 − ⎟ v 2 = ⎜ 2 − + 2 ⎟v 2 µ⎠ µ µ ⎠ ⎝ ⎝ 2

[7.60]

352

THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 7-27. Discharge through spouts

From this equation the height of the equivalent fall can be deduced. Specifically, with the experimental value of µ = 0.625 that he found for these spouts, the height turns out to be 0.735 times the real depth. In the case where he uses an entrance spout, (Fig. 7-26b), for which he calculated that µ = 0.5 it would be just half, a value, we recall, that was one of the most used and referred to in order to explain this phenomenon.70 Borda is a late follower of the live forces principle, exploiting this method and trying to explain the limitation of this principle regarding the loss of live force. He is worth quoting, not only for his work, but for the great effort he makes comparing the solutions given by Daniel Bernoulli and d’Alembert. By way of summary

The works of the Bernoulli family, as well as those of d’Alembert, constitute a preliminary step in dealing with fluid mechanics through the use of analytical methods, culminating in the grand theorization. If we think the attention devoted to the phenomena of discharge as being divided into three periods, this would be the second, preceded by the derivations of Torricelli’s Law, and succeeded by the differential equations of hydrodynamics. In examining where the credit must be given, we have seen how an experimenter, Daniel Bernoulli, moved by purely observational considerations, led the attack by integrating pressure and velocity in the same phenomena. He was followed by a mathematician, Johann Bernoulli, who exploited this result, and connected it with mechanics by introducing new concepts opening up new paths. Another mathematician, d’Alembert, rounds this off, because although he did 70

Let us just say that this type of spout is known as Borda’s.

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not contribute any new solutions, he did reflect on the nature of the fluid condition, in spite of the fact that in order to enter the theoretical universe, he has to make use of experimental facts. Finally, Borda made an attempt to harmonise Daniel and d’Alembert. This happened more than 20 years later, which shows that in spite of the brilliant decade of the 1750s, the basic problems continued to be in force. This entire body of work—new concepts, new thoughts, new methods— open up the way to the world of theory which will lead to the apex of the eighteenth century, as we shall now see.

Chapter 8 Theoretical Constructions (I): Clairaut and d’Alembert

For a brief period in the middle of the eighteenth century, there was a moment of extraordinary brilliance in the development of fluid mechanics. Three men, Clairaut, d’Alembert, and Euler, raised the theory of fluid mechanics to the height where it remained almost unchanged for more than 50 years. The process began with Clairaut’s Théorie de la figure de la Terre, tirée des Principes de l’Hydrostatique (Theory of the shape of the Earth obtained from the Principles of Hydrostatics), which appeared in 1743, and continued with the Essai d’une nouvelle théorie de la résistance des fluides (Essay on a new theory of the resistance of fluids) of d’Alembert, which, though it published in 1752, had been presented in 1749. The process ended in 1755 with three monographs of Euler: ‘Principes généraux de l’état d’équilibre des fluides’ (‘General principles of the state of equilibrium of fluids’), ‘Principes généraux du mouvement des fluides’ (‘General principles of motion of fluids’) and ‘Continuation des recherches sur la théorie du mouvement des fluides’ sur la théorie du mouvement des fluids’ (‘Sequel to the researches on the theory of motion of fluids’). Less than 13 years had elapsed from the appearance of Clairaut’s work. That this brief period should produce such contributions is an indication that the time was ripe, and the theories were ready to bear fruit. This process must be explained as the result of a network of influences, as it is obvious that the key works are not isolated events, but form part of a linked chain. Very probably the distribution and dissemination of the work was accompanied by debates in the Academies and other forums, followed by letters and sundry documents circulating along all the highways and byways of European science. At the same time, this activity must be viewed as being framed by the development of mechanics, which in turn was promoting and making use of mathematical analysis. It is by chance that the three figures involved were great mathematicians. The grand theorisation As an aid in disentangling the network of influences, we present the diagram below based on the most relevant works. Note the uniformity of the dates of the 355

356

THE GENESIS OF FLUID MECHANICS, 1640–1780

basic works: all date from the year 1743. Of these three works, the Hydraulica of Johann Bernoulli had a fundamental influence on Euler due to the concepts of pressure or internal forces, while d’Alembert scarcely quotes it. As regards Clairaut, the Théorie de la figure de la Terre answered a very specific question about the shape of the planet Earth: was it flattened or lengthened at the poles? This problem had placed Cartesians and Newtonians is in different camps, and in order to elucidate it the Academy of Sciences of Paris sponsored two expeditions—one to Lapland, another to the Viceroyalty of Peru—in order to carry out geodesic measurements in both latitudes, and afterwards to compare the results. Clairaut participated not only in the debate, but also in the first expedition, and his work constitutes an analysis within the mathematical branch the problem, obtaining for the first time the conditions of equilibrium of a fluid in a field of forces. D. Bernoulli Hydrodynamica

1738

D'Alembert

J. Bernoulli

Clairaut

Traité de l'équilibre et du mouvement des fluides 1744

Théorie de la figure de la Terre 1743

Hydraulica

1743

Euler Essai d'une nouvelle théorie de la résistance de fluides 1749 (1752)

Principia motus fluidorum 1752

Memoires 1755

THE GRAND THEORISATION

Two works of d’Alembert have achieved distinction: the Traité de dynamique and the Essai d’une nouvelle théorie de la résistance des fluides. An intermediate work exists between the two, of which we have spoken extensively, the Traité de l’équilibre des fluides published in 1744. It reflects an important influence of the Hydrodynamica of Daniel Bernoulli, which, however, had little

THE THEORETICAL CONSTRUCTIONS (I)

357

effect on the Essai. The latter constitutes the main contribution of d’Alembert to the rationalisation of fluid mechanics, and has its own little story. The work was presented to the competition organised by the Royal Academy of Sciences and Letters of Berlin in 1750, whose theme was the theory of the resistance of fluids. D’Alembert sent it in December 1749. The jury, of whom Euler was a member, left the award on the grounds that none of the three works presented contributed experimental evidence for the theoretical developments. This decision annoyed d’Alembert profoundly, and he made this very clear in the introduction to the Essai with harsh and bitter words, where among other things, and with a certain presumption he says: ‘I do not boast having pushed to its perfection a theory that so many great men have scarcely begun’.1 As a result of all this, his relations with Euler, which had never been very good, reached their lowest point. We do not know the reasons which led to the jury to take this decision, although according to Truesdell ‘certainly Euler must have had a hand in the refusal’2 and ‘rather, it is likely that the reason given out [compare results with experimental measurements] was only a pretext, offered in place of the truth’.3 It was finally edited by the Academy of Paris in 1752. Clairaut’s influence can be clearly detected in the Essai, although d’Alembert does not give him credit for this. The method of calculating the channels is, as we shall see in due course, identical in both, although Clairaut limits himself to the non-compressible case. In fact d’Alembert recognises that this particular case had already been treated by ‘someone’, and in the appendix to his work he attributes those aspects related to the surfaces of levels to Clairaut. That he tries to avoid mentioning this, attributing the principle of the reentry channels to MacLaurin, is very probably due to personal matters more than anything else. As regards Euler, although the concepts of mechanics that he takes as his starting point differ from those of d’Alembert, the latter has a perceptible influence on him in the Principia motus fluidorum (Principles of motion of the fluids). This work deals solely with the movement of the non-compressible fluids, and the criterion for obtaining the equation of continuity is the invariability of a differential fluid volume, which had already been used by d’Alembert. But there is more: d’Alembert, in his eagerness to avoid the forces, eliminated them mathematically by means of two successive derivations of the differential equations of motion, which gave rise to solutions of a more general nature, but less practical.4 This 1

Cf. Essai p. xxxviii. Cf. ‘Rat. Fluid-Mech.-12’, p. LVII. 3 Cf. ‘Rat. Fluid-Mech.-12’, p. LVIII. 4 D’Alembert try to reduce the problem to some equations in velocities, which once solved would lead to the pressures. 2

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THE GENESIS OF FLUID MECHANICS, 1640–1780

technique of manipulating equations was also used by Euler in the Principia, but he did not use it in his later reports, where he restored pressure as a variable to be considered right from the start. In these reports, Euler removed the restrictions on irrotationality, and clearly presents the equations of continuity and impulse not only for the non-compressible fluids, but also for compressible ones. The other noticeable influence on Euler is that of Johann Bernoulli, who, through his Hydraulica transmitted, as we have already said, the concept of pressure as an internal force and its application to differential elements. Comparing the basis of one and the other, the superior conception of Euler is noticeable but the idea was already present in the work of Johann. It is also interesting to note that both based themselves on Newtonian dynamics, in contrast to d’Alembert. Finally Euler, with his ability to synthesize, clarified and expanded on these achievements in the three reports that appeared in 1755 in the Mémoires de la Académie of Berlin. These were the final product of a long run, and represented the culminating moment of theorisation of fluid mechanics. They were not surpassed until the nineteenth century. These monographs rendered all previous studies obsolete, among them those of d’Alembert. We cannot but recognise, as serious historians have pointed out, his bad luck he had, as he only enjoyed his achievements for a few short years. The analysis of these papers, due to their extensive nature, will be divided between this chapter, which deals with Clairaut and d’Alembert, and the next chapter on Euler, with a brief mention of the ‘Mémoire sur la Théorie du mouvement des fluids’ (‘Memoir on the Theory of motion in fluids’) of Lagrange of 1781. The shape of the earth The theorisation of fluid mechanics reaches a high point with Clairaut, who presents for the first time the equilibrium condition of a fluid by means of differential equations. In contrast to other authors, the motives which induced him to write his work Théorie de la figure de la Terre, tirée des Principes de l’Hydrostatique do not proceed from the movement of fluids through pipes, but, as the title of the work indicates, from geodynamics. What was the shape of the Earth, as a rotating body, was a polemic engaged in at the end of the seventeenth century and during the first half of the eighteenth century. The question5 began

5

The origins and developments were clearly described in Los caballeros del punto fijo (The knights of the exact point) by Antonio Lafuente and Antonio Mazuecos, (Ed. Serbal/CSIC. Barcelona, 1987) work which was subtitled ‘Science politics and adventure in the French-Spanish geodesic expedition to the Viceroyalty of Peru in the 18th century’. In particular the chapters

THE THEORETICAL CONSTRUCTIONS (I)

359

with the hypothesis of the flattening of the poles expounded by Newton in the Principia, based on his theory of the attraction of gravity, and which he verified by the decrease in the length of a pendulum that beat at 1 s intervals, as that pendulum was brought closer to the equator. Unlike Newton, Huygens adopted the Cartesian theory of gravitation, according to which external vortices were responsible for the attraction of the bodies. These vortices dragged the bodies towards the centre of the earth, an action which he denominated ‘conatus’. According to this theory, a flattening of the poles ought to exist, but its magnitude would be less than the Newtonian one, specifically a difference in equatorial versus polar diameters of 576/577 instead of 230/231. In counterpoint to these values stood the geodesic measurements of the meridian arcs made in France at the end of the eighteenth century by such distinguished geometricians as Picard and Cassini. From the measurements obtained by them, it was concluded that the Earth had an oblong shape, stretched towards the poles, with a difference in diameters of 1/262. This opposition between the theoretical derivations and experimental measurements, together with the difficulty of accepting the theses of Newtonian mechanics, radicalised the scientists’ positions, divided the Academy of Sciences of Paris, and even came to have national and theological implications. In the words of Lafuente, ‘Theory vs. experiment, Newtonianism vs. Cartesianism, laicism vs. scholasticism, savant vs. Academic, England vs. France, all were, in the end, powerful alternatives to stir up controversy and kindle all the passions’.6 The controversies caused rivers of ink to run, but their positive side was that the Academy of Paris sponsored two important scientific expeditions: one to Lapland and the other to the Viceroyalty of Peru, in present-day Ecuador. The aim of both was to measure the meridian arcs in two very separate latitudes.7 With these measurements, together with those made in France, the intention was to obtain the longitude of a meridian degree at various points of the Earth, and thus to clarify dedicated to this topic are ‘Astronomers, geometricians and geodesists’ and ‘London and Paris, two sciences about the Earth’ to which we refer. 6 Cf. Los caballeros del punto fijo, p. 48. 7 The expedition to Lapland lasted a little over a year, beginning in the middle of 1736. Maupertuis, Celsius, Lemonnier and Clairaut himself took part. Godin, La Condamine, Bouguer, Jorge Juan and Ulloa participated in the second between 1736 and 1744. The last two, both Spaniards, were sent by the Spanish Crown as the territory chosen for the measurements belonged to the Spanish Viceroyalty of Peru. There are several works on this theme, among which are: La geometrización de la tierra by A. Lafuente and A. J. Delgado; the collection of monographic papers edited for the 250th anniversary of the expedition under the title La forma de la Tierra. Medición del meridiano Ed. The Naval Museum, Madrid 1987; and the paper ‘L’aventure et la science dans l’expédition au Peru (1735–1743)’, presented by Antonio Lafuente an the Colloquium the ‘Academia de la Ciencia y la forma de la Tierra’, in Paris 1986, apart from the previously mentioned Los caballeros del punto fijo.

360

THE GENESIS OF FLUID MECHANICS, 1640–1780

the value of the Earth’s flattening. It was a crucial experiment about whose was the true theory: whether it was that of Newton or that of Descartes.8 Within the framework of this discussion, and with the expedition to Peru still underway, Clairaut tried to deduce the shape that the planet Earth— supposedly fluid—would acquire in its rotation subject to the laws of gravity, and taking hydrostatics as a starting point. Clairaut writes: But could not the laws of hydrostatics permit this mass of water to be irregular in shape, and flattened at one pole and lengthened at the other, and that the meridians be similar?… Let us see then what the laws of hydrostatics require. [Intro. P. vii]9

The theoretical aspects, which are what most interest us, represent only a small portion of the work, which consists of two parts. In the first he expounded the general principles of equilibrium, which he specifies for several possible types of gravity. He then goes on to deduce the general conditions of equilibrium, which is the best-known part of the work, in order to continue to introduce the concept of level curves, and then to extend these conditions to special cases. He generalises the former points, which refer to bodies of revolution, to the threedimensional case. He also applies his findings to an imaginary planet composed of different fluids distributed in layers without mixing, and ends by explaining how to use the measurements of the meridian degrees and longitudes of the pendulums beating the seconds,10 in order to deduce the law describing the action of gravity. The second part is wholly dedicated to determining the shape of the earth, supposing that its parts are attracted to each other inversely to the squares of the distances. Clairaut’s equilibrium conditions As we have already stated, Newton and Huygens, quite separately, tried to find the external shape of the earth, considering the planet to be mass of a rotating liquid. The former took as his criterion the immobility of the liquid contained in two straight channels meeting at right angles at the centre of the earth, one 8 The history of this problem had already been dealt with by Montucla in his Histoire des Mathématiques. However the classical work on the theme is A history of the mathematical theories of attraction and the figure of the Earth. From the time of Newton to that of Laplace (2 vol.) London, 1873 of Isaac Todhunter. 9 The quotes between brackets will refer to La Théorie de la figure de la Terre. 10 The period of a pendulum, for small oscillations is determined by the equation T = 2π√(l/g), where l is its length and g the acceleration of gravity (see Chap. 2, §.3). With the help of this formula it is possible to calculate the latter with a pendulum of no length and whose period is measured experimentally.

THE THEORETICAL CONSTRUCTIONS (I)

361

originating at the pole and the other at the equator, as he had explained in the Principia.11 Given that the weight of a channel at the equator is alleviated by centrifugal force, it had to be longer than the other channel, as pressure at the intersection of both had to be the same. For his part, Huygens, whose hypothesis on gravity was Cartesian vortices, assumed that any point on the surface of the earth had to be at rest, therefore the level of the water had to be perpendicular to the local vertical. Given this condition, at both the poles and the equator, the vertical of the plumb line coincides with the radius vector, but in other latitudes this is not so because of the action of the centrifugal force, giving rise also to flattening of the poles. Both conditions were necessary, as non-compliance with them implied that the liquid would move,12 but they were not equivalent as Bouguer showed in the article ‘Comparaison des deux lois que la Terre et les autres Planétes doivent observer dans la figure que la Pesantuer leur fait prendre’ (‘Comparison between the two laws that the Earth and other planets have to meet in the shape due to gravity’).13 N M'' s u

z G'' G G'

W

H

M

F

t

p

y

M'

D O

E

S

Fig. 8-1. Bouguer’ rotating body

In this work Bouguer made a very acute analysis assuming a distribution of gravity that does not converge in the centre of the Earth. Figure 8.1 presents the meridian section of an axisymmetric body NESW, which can rotate on its axis 11

Book 3, Prop. 18-ff. There is a previous study of rotating fluid masses. In 1732 a work of Maupertuis appeared in the Phil. Trans, ‘Of the figure of the fluids, turning round on an axis’ in which is supposed a law of gravity of the type rn, arriving at the equation of the contour of the fluid. However he did not establish any internal condition, only that the force on the turning axis was zero. 13 Mém. Acad. Paris, 1734. 12

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THE GENESIS OF FLUID MECHANICS, 1640–1780

NS. In the non-rotating condition, which Bouguer calls ‘primitive’, the gravities p (pesanteur) will follow straight lines, as MG, M′G′’, M″G″, that reach the axis NS at the points G, G′, G″, but not in the center O. Inside the body he defines a spheroid NDSH as the property of being crossed perpendicularly by the gravity lines. The strength of gravity depends on the distance to the axis as p(y), and besides also on the latitude by a factor ζ(s), that is ζ = 1 at the pole N. Therefore, according to the first law, the weight of the channels MG, must be equal to NG at the point G, which in primitive conditions will be ζ∫pdy = ∫pdu.14 When the body rotates the centrifugal force acting over the channel MG has to be taken into account at the point G, resulting in the following equation for the shape of the surface [§.I]15:

ζ

∫

pdy − k

ω2 s 2 = 2

∫ pdu

[8.1]

For the second law the condition for no movement is that the force at the surface must be perpendicular to it, which results in [§.II]:

ω 2 sds = ζpdy − ζpdt

[8.2]

Both equations are completely different, giving as a result different general shapes for the planet surface. As they also seem difficult to resolve, Bouguer takes the simple case when p = const and ζ = 1, with the following results:

( (

1 pt ± p 2 t 2 − kpω 2 z 2 u 2t 1 y2 = pt ± p 2 t 2 − kpω 2 z 2 t 2t y1 =

) )

[8.3]

At first glance they seem equal, but we have to note that they differ in the second term under the root, because one has an u and the other a t as one of the factors. All this makes Bouguer say: ‘they depend so little on one another, that they are

14

Bouguer uses a very complicated system of coordinates: u,y,t,z,s, that are not independent, e.g., z/t = s/y is immediate. 15 For the term of the centrifugal force Bouguer writes: fs2/2a, where f is the centrifugal force at a distant a. We prefer to introduce the rotation ω and also a factor k for making the formula physically consistent. Cf. §.I.

THE THEORETICAL CONSTRUCTIONS (I)

363

almost always in contradiction, they mutually exclude each other, and more often it is sufficient that one is observed for the other not to be’ [§.III]. Next Bouguer sets out to find in what conditions both give the same solution, and he finds the equation [§.III]:

dζ

∫ pdy + ζpdt = pdu

[8.4]

This equation continues to be complicated, so he takes the same hypothesis as before (p = const and ζ = 1) the solution is t = u, according to equation [8.3]. However, to satisfy that equality it is necessary that all gravity lines converge to the centre O, and the spheroid NDSH will be a sphere, i.e., coming close to the real Earth. We stop here, though Bouguer still continues with more specific cases. Faced with all these facts, Clairaut asked himself: But if we see with Mr. Bouguer that these two equally necessary conditions for the equilibrium of fluids do not follow on from each other, could it not be that still other completely different conditions from the first two exist, and which are besides entirely necessary? [p. xxxii]

It is in trying to overcome this doubt that he arrives at the first differential law of hydrostatics. The principle on which Clairaut bases himself is that of ‘equilibrium in channels’, which he sets out in successive steps as we show below. The initial principle is the following: A mass of fluids cannot be in equilibrium, except when the efforts of all the parts that are contained in a channel of any shape that one imagines crossing the entire mass, cancel each other out. [§.I]

In this text, the words ‘to cross the entire mass’ must be understood as starting and ending on the surface. This indicates that if the entire mass, except a channel such as ORS (Fig. 8-2 left), were to solidify, this would only maintain in equilibrium if the efforts of OR towards S were equal to those of SR to O, R being any intermediate point.16

16 His hypothesis vaguely recalls Stevin’s theorem, as if the mass of the fluid is in equilibrium, the solidification of one of its parts does not change it.

364

THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 8-2. Channels in a fluid

Clairaut notes [§.II] that this principle includes as much of Newton as of Huygens. The former would correspond to the channel MCN whose vertex C would be placed in the centre, and in which the weights of the columns MC and NC must be equal. As regards the latter, if FGD located along the surface is chosen, for this to be in equilibrium it is necessary that the weight at each point be perpendicular to the surface, as Huygens said, or that a part of FG counterweighs the other GD. But this cannot take place unless the entire channel FGD is in equilibrium, as its length can be any whatsoever, and what we have said would be applicable to any part of that channel. Therefore both hypotheses must be admitted. Another principle is derived from this hypothesis, referring to the equilibrium of a channel closed upon itself that he calls a re-entry (rentrant) channel. The demonstration is immediate (Fig. 8-2 right). Let there be a channel ITLK, which divides into two, ITL and IKL, which in turn are part of another two ending in the surface, such as FIKLG and FITLG. If these are in equilibrium, so is the part in common, which is the closed part. Thus he shows that this case is a corollary of the first.

Fig. 8-3. Rotating channels

THE THEORETICAL CONSTRUCTIONS (I)

365

In order to proceed to the case of the rotating fluid masses, he assumes first [§.V] two open channels with different routes, but whose entrances and exits are equidistant from the rotating axis Pp (Fig. 8-3 left). He demonstrates that the centrifugal force (force centrifuge) is the same at each end. After this he closes the channels forming a ring crossed by the rotating axis (Fig. 8-2 right), that, as a consequence of the foregoing must also be in equilibrium, ‘as ab & cb will exert equal pressure on each other in b, likewise ad & cd in d: thus the rotation will not disturb the equilibrium of any channel closing upon itself’ [§.VI]. From this we observe that if a fluid mass reaches a figure of equilibrium without rotating, of necessity it will also arrive at equilibrium in rotation, although the shape will be different. This means that we can do without the centrifugal force in the process of determining the conditions of equilibrium. Concerning this he states: When one wishes to examine if a law of gravity is such that a mass of fluid rotating around an axis can keep a constant shape, it is useless to pay attention to the centrifugal force, i.e. if the mass of fluid can have a constant shape without rotating, it can also have one when rotating. [§.VI]

He establishes the last of his principles following the same line of argument, which refers to the case in which the channel is in a meridian plane. For this he assumes that the condition required for this fluid ‘spheroid’, rotating on its axis and submitted to a given gravity law, can conserve a constant shape ‘it is enough that any channel closed upon itself, and located in the meridian plane of the spheroid, will always be in equilibrium, taking into consideration only the sole force of gravity, without the centrifugal force’ [§.VIII]. With this, Clairaut wants to say that if a fluid subject to a given law of gravity, and without rotating, attains an equilibrium shape it will also arrive at this condition if the body rotates, although it will not adopt the same shape. In other words: the rotation does not destroy the equilibrium, it simply modifies it therefore it is possible to establish the condition of equilibrium based only on the gravitational forces. We will return to this point once we have established these conditions mathematically, as then it will be easy to corroborate these statements analytically. Having developed several particular cases with different types of gravity forces, Clairaut devotes himself exclusively to finding the condition of equilibrium of a rotating mass. He defines the problem as: ‘Given the law showing how gravity acts on all parts of a fluid mass rotating around its axis, find out if this mass can have a shape that keeps it constant’ [§.XVI]. In order to resolve this question, he uses the channel theorems previously deduced.

366

THE GENESIS OF FLUID MECHANICS, 1640–1780

Y ds N L X Fig. 8-4. Forces in an element of the channel

Let there be a channel determined by two points L and N (Fig. 8-4) located in a meridian plane of a body rotating around the axis OX. This channel can be interpreted as a fragment of a closed channel, and therefore in equilibrium. Thus, leaving the part that closes the channel unchanged, any other channel passing through these two points will have to withstand the same gravitational forces. Suppose that this field is defined by its components on the axis OX and OY, designated as Q(x,y) and P(x,y), the tangential force on an element ds of the channel will be expressed as ρdσ(Qdx + Pdy), where ρ is the density and dσ is section.17 The magnitude of the whole force over the limited section between L and N will be found by the integral of this last expression along the said curve. It follows from what we have already said that, as its value has to be the same whatever the curve of the channel between the aforementioned points be, ‘Pdy + Qdx must be a exact differential, so that there can be equilibrium in the Fluid’ [§.XVI].18 With respect to the mathematical expressions of P and Q, it might be difficult to check if Qdx + Pdy is an exact differential. In order to solve this difficulty, he recalls that years before he himself had obtained a theorem determining whether such a condition was satisfied solely by examining the functions Q and P.19 It is what is nowadays known as the equality of the crossed derivatives, and it is expressed as:

∂P ∂Q = ∂x ∂y

[8.5]

17 Clairaut leaves out these factors and simply writes: Qdx + Pdy justified for being the constant density. 18 We can also say that the force field is potential. 19 Cf. Mém. Acad. Paris, 1740.

THE THEORETICAL CONSTRUCTIONS (I)

367

This differential equation represents the condition, necessary but not sufficient, of equilibrium. Its merit lies in the fact that it is independent of the shape, type or any other condition of the fluid. It depends only on the nature of the field of forces. In order to find equation of a meridian line [§.XVIII] in the case where the mass rotates around the axis OX, Clairaut considers that in all points of the mass the total weight (poids) of the fluid in a channel coming out from the centre, and discounting the centrifugal force,20 must be constant. Therefore, for a generic point M, this weight will be ∫ Pdy + Qdx and the sum of the centrifugal forces (efforts) is ∫ ω 2 ydy , resulting in:

∫ (Pdy + Qdx) −

1 2

ω2 y = A

[8.6]

ω being the rotation velocity and A a constant. We now go back in order to examine analytically the equivalence of the condition of equilibrium between rotating masses and non-rotating ones. If in the case of immobility, the law of gravity is determined by a vector whose components are Q and P, when there is rotation the centripetal force –ω2y appears in the direction of the OY axis. Thus the total component according to this axis would become R = P–ω2y. However, when the condition of equilibrium is established the following occurs:

∂R ∂ ∂P = ( P − ω2 y ) = ∂x ∂x ∂x

[8.7]

Next follows a demonstration of the general condition [8.5], which shows that the case of rotation falls within that of rest, which is what Clairaut argued in his reasoning that took closed channels as its starting point. A consequence of these formulations is the existence of level surfaces, [§.XIX] which he defines as a surface having the property of being perpendicular at all points to the direction of the weight (Fig. 8-5), which explains the

20

We use the expression ‘centrifugal force’ because it is used by Clairaut and other of the time, but we can not forget that it is a centripetal acceleration or force.

368

THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 8-5. Surface curves

assumption that the volume is divided into concentric layers, each one enclosing the interior ones, and all enjoying this condition. The constitutive equation of these layers will be determined by the condition that the weight component is null21 over any line lying on the surface. This is expressed [§.XXIV] as Rdy + Qdy = 0, where R = P – yω2. Obviously in the case of rest the equation would be Pdy + Qdx = 0. Generalisations

He generalises the fundamental results and hypotheses in several directions, one for bodies with fields of forces in three dimensions and another for the nonhomogeneous ones. The final point, which takes up the entire second part of the work, is to apply these results to the earth, and to contrast them with the available measurements. In order to extend the case of revolution symmetry to a fluid mass subject to a field of three-dimensional forces, Clairaut asks the same question as before: what relation between the three components of this field of forces would make the equilibrium possible? [§.XLVIII]. Likewise, the process is identical. He takes a closed three-dimensional channel, selecting a section of this channel upon which he calculates the existing

21 Clairaut, following Maupertuis, differentiates the weight (pesanteur) from gravity (gravité). The first is the natural force with which all bodies fall, while the second would be the force with which they would fall if the Earth did not rotate. Cf. footnote in p. xi.

THE THEORETICAL CONSTRUCTIONS (I)

Z

ds

369

M

Y X Fig. 8-6. Three-dimensional channel

total force by means of the sum of the elemental forces (Fig. 8-6). He arrives at expressions such as Rdx + Qdy + Pdz, which must be the exact differential.22 When it is difficult to elucidate this condition, we can make use of an extension of the same method employed previously, which is equivalent to fulfilling the following three equalities:

∂P ∂Q ; = ∂y ∂z

∂P ∂R ; = ∂x ∂z

∂Q ∂R = ∂x ∂y

[8.8]

Which are obviously an extension of the two-dimensional case, where we can clearly appreciate the denomination of this theorem as that of the quality of the ‘crossed derivatives’. The developments and theorems shown correspond to homogeneous liquid masses. However, the values calculated for the flattening at the poles with the hypothesis that the Earth behaves like a homogeneous body do not correspond to the determinations existing up to that time. This is true as much for the measurements of the reviews of Paris and Lapland, as for the values of gravity of obtained from experiments with pendulums. This discrepancy led Clairaut to conjecture that the earth was probably not homogenous, but that its density varied with the depth.

22

The expression Rdx + Qdy + Pdz can be expressed as Vds, V being the effect vector field whose components are R, Q and P, and ds is also a vector. The condition of exact differential is equivalent to the fact that the integral ∫CVds is independent of the trajectory C. This is verified if the field is irrotational, i.e., curlV = 0. Developing the expression, we arrive at the three equations of the crossed derivatives.

370

THE GENESIS OF FLUID MECHANICS, 1640–1780

Fig. 8-7. Non-homogeneous body

In order to introduce this variation of density, he supposes that the fluid mass is formed by immiscible layers: each one with a density (Fig. 8-7) that where it were submitted to a central force, would be distributed in concentric layers. He demonstrates that his channel principles continue to hold in this case, and that the layers will follow the level surfaces. The second part of the Théorie de la figure de la Terre is devoted to tailoring all the previous results to planets, given a Newtonian law of gravity: in particular the earth, but with some mention of other planets such as Jupiter. He begins by determining the terrestrial ellipsoid shape, starting out from the theoretical hypotheses, first as a homogeneous mass and then composed of layers of different densities. Here we have an excellent piece of calculation that ends by comparing the results with those predicted. He observes: As the length of the pendulum and the relation of the axis given by Mr. Newton do not agree with the observations we made in Lapland, I have abandoned the supposition of the earth’s homogeneity, and I have searched for its shape, supposing it to be composed of an infinite number of layers, from the center to the surface, whose densities vary following any law. [2nd Part, Intro]

The argument is that on the hypothesis that the earth is made of a homogeneous fluid, the values of the theoretical flattening do not correspond to the ones measured. Finally, we note that Clairaut’s contribution centres on two points: one, the postulate of a field of forces, which he identifies in first instance with the law of gravity, and then generalises for any other type of forces. The second is the

THE THEORETICAL CONSTRUCTIONS (I)

371

determination of a possible equilibrium condition starting only from certain analytical relations between the components of this field. Clairaut does not go into the nature of the fluid, nor its composition or interactions. He only introduces the fact that all the particles are submitted to some specific forces. His results initiate the mathematization of fluid mechanics, although he limits himself exclusively to hydrostatics, and to the case of constant density, even though he does try to transcend this restriction with the assumption of the existence of concentric layers. As well as these two points there is another that requires underlining: his attempt to find the shape of the earth from equations, obtained from other more general principles, and experimental measurements. His insistence, on perceiving the discrepancies between theory and practice led him to postulate a new constitution of the earth as non-uniform body, to which he devotes the entire second part of the work. D’Alembert

The main contribution of d’Alembert to the rationalisation of fluid mechanics is in his book Essai d’une nouvelle théorie de la résistance des fluides, published in 1752, although as we have said, its origins were prior to 1749. The book is regarded as difficult to read, and the qualifications of ‘long and tortuous’ are shared by Dugas23 and Truesdell,24 who comment on it somewhat unfavourably. We begin with a brief review. It comprises nine chapters, devoted to several themes which review the most relevant developments in connection with fluids up to that time. Of the first five chapters, which are of the greatest interest for our purposes, the first studies the general principles of dynamics, the second is devoted to hydrostatics and the third is general in nature, dealing with all fluids. It is in the fourth and fifth where the main subject matter that interests us lies. The first of these deals with the ‘pressure’ that a moving fluid exercises on a still body, while the other, deals with the ‘resistance’ produced by a fluid on a body moving in that fluid. The division of the chapters is illustrative of d’Alembert’s method: it begins with prior foundations, then obtains a general principle of fluids that has its origins in hydrostatics and is extended to fluid dynamics, and he ends by applying this general principle to the fluid-body combination in two ways, immobilising either the body or the fluid. In the remaining chapters, he deals with the oscillations of a floating body (Chap. VI), the motion of a fluid in a vessel (Chap. VIII), and the effect of a river on its banks (Chap. IX). In all these it appears that d’Alembert 23 24

Cf. Histoire de la Méchanique, p. 295. Cf. ‘Rat. Fluid. Mech.-12’, p. LI.

372

THE GENESIS OF FLUID MECHANICS, 1640–1780

also wished to expound his ideas on themes that had been, or were being, treated by his contemporaries. Although the general approach is clear, it becomes more confused when he goes into detail, as he repeats theorems, the mathematical developments are heavy and tedious, he changes his notation frequently, and at some points the reasoning is obscure. Likewise, it is difficult to follow the treatment of the terms referring to pressure and force, which is largely due to his conception of mechanics.25 We will underline this in each case. His tone is frequently pretentious, whether assuring us of the truth or necessity of his arguments, or whether the primacy or simplicity of them, extreme but not always true. In the Introduction, apart from a list of the previous works of other authors,26 some declarations are worth underlining. Thus, he affirms: It turns out that… the theory of the resistance of fluids, … is still very imperfect in its own actual elements. These factors have obliged me to deal with this subject by an entirely new method, and without borrowing anything from those preceding me in this task. The theory I set out, … has, it seems to me, the advantage that it does not rest on any arbitrary supposition: I only suppose, which nobody can deny me, that a fluid is a body composed of very tiny detached particles capable of moving freely. The resistance that the body experiences when it collides with another is, strictly speaking, no more than the momentum that it loses. As the motion of the body is altered, it can be regarded as being composed of the motion the body will have the following instant, and of another motion which is destroyed. It is not difficult to conclude from this that all the laws of communication of motion among bodies can be reduced to those of equilibrium. [Intro. p. xxv]27 25

Between ‘Forces’, ‘forces accélératrices’, ‘causes étrangeres’, ‘pression’, ‘puissances’, and sometimes ‘poids’, the panorama of terms becomes somewhat hieroglyphic. Let us recall what he said about some of these terms in the Traité de dynamique: − − − −

‘By the word force, we should understand the effect produced by overcoming an obstacle or in resisting it’. [Pref. p. 20] ‘In general we call Puissance [power] or cause motrice [driving force] to everything forcing a body to move. [§.5] ‘The uniform motion of a body can only be altered by a cause étrangere [outside cause]’. [§.22, p. 22] ‘We understand as a force accélératrice [accelerating force] as only the quantity to which the increment of velocity is proportional.’ [§.22, p. 25]

26 It is noteworthy that Newton and Daniel Bernoulli constitute the bulk of his references. He quotes Johann Bernouilli only once, and does not quote Clairaut or Euler at all. That he is an enemy of Clairaut is well known and his relations with Euler, which were not always good, turned sour precisely due to his work being rejected by the Berlin Academy, because as we have already explained Euler formed part of the jury. 27 The quotas refer to ‘Essai d’une nouvelle théorie de la résistance’.

THE THEORETICAL CONSTRUCTIONS (I)

373

The existence of the corpuscles led him to admit the difficulty in handling them: Let us suppose in effect, that we were to have the advantage, of which we are deprived, of knowing the shape and mutual arrangements of the particles composing the fluids: the laws of their resistance and their action will be reduced without doubt to the known laws of motion. [Intro. p. xxvii].

We can compare these words with those used to treat the same topic in the Traité de l‘équilibre. There, in order to solve the discharge problem of a glass, he supposed that the fluid moved along plane sections. Now he looks for a more general principle that he encounters in the hydrostatics: As the philosophers cannot immediately and directly deduce the laws of their equilibrium from the nature of the fluids, at least they have reduced them to a single principle of experience, the equality of pressure in all directions; a principle they have taken (for lack of a better one) as the fundamental property of fluids. [Intro. p. xxviii]

This is Pascal’s principle, and although it should be understood in the context of hydrostatics, d’Alembert’s aim is to reduce hydrodynamics to the former, just as he reduced dynamics to statics, as he had proposed in the Traité de dynamique. All the statements concerning these necessary truths remain in the air, as in the end everything remains as a principle of experience. Regarding the comparison of his results with experience, a reason given for rejecting the work in the Berlin prize, he says: I would like to be able to compare my theory of the resistance of fluids with experiments performed by several famous physicists. However, having examined their experiments, I have found them to be in so little agreement with each other, that it seems to me there is nothing perfectly sure on this point as yet. [Intro. p. xxxv]

Nevertheless, in the work we find frequent allusions to experiments. We shall analyse the work of d’Alembert in three parts. The first will be devoted to the general principles of dynamics and hydrostatics that provide him with support later on. The other two will be specific to bodies and fluids: to what distinguishes both of the cases, where the body is immobile in a fluid current, and the opposite, where the fluid is at rest and the body moves. After carrying out the two studies he ends by identifying them. We shall see that d’Alembert uses a different terminology in each case; in the first, the fluid current produces a ‘pressure’ on the body, whereas in the second the body suffers a ‘resistance’.

374

THE GENESIS OF FLUID MECHANICS, 1640–1780

Principles of dynamics and hydrostatics

Starting from his fundamental principle, and taking the specific case of the absence of external forces, he deduces two important consequences: first, the non-dimensionality of the motion with respect to time [§.5] and second, the constancy of the fluid field with respect to the velocity [§.8]. He explains the first by saying that if in a system of bodies at rest, any velocity is applied to each body, the trajectories of each body will follow shapes similar to those that all the bodies would have if all the velocities maintained their direction and were multiplied by a specific factor. If this factor is k, this would be equivalent to varying the time scale as 1/k. The second part is a consequence of the former, as a fluid field is the set of the trajectories of all the particles of the fluid. The advantage of this approach is that it makes the phenomenon independent of the specific value of the velocities. An easily interpreted simile is that when this factor is varied, geometrically similar fields are generated, which are only affected by the scale.28 A corollary to the above is that the resistance to motion (résistance du fluide) in a fluid is proportional to the square of the velocity [§.9]. The demonstration is simple: since the resistance R must meet Rdx = –vdv, then multiplying the velocity by a factor k, means that the resistance will be multiplied by k2, ‘having abstracted the weight, the friction and the elasticity’ [§.10]. In order to obtain the basic hydrostatic equation, he imagines a volume of a fluid submitted to any force whatsoever. An inner particle, located in the vertex of two rectilinear channels going up to the surface, will be submitted to the same pressure by both channels. ‘In effect, nobody is ignorant of the fact that when a fluid is in equilibrium, each particle P undergoes equal pressure in all directions’ [§.13]. Although there is a noticeable similarity with Clairaut’s method, he does not quote him, but refers back to MacLaurin.29 After a consecutive series of particularisations, he ends by applying the principle to an infinitely small rectangular channel [§.19].30 As Fig. 8-8 shows, the element MNQP, whose sides are formed by four channels, is subjected to

28

His general principle of dynamics was formulated for the first time in the Traité de Dynamique, and was used in the Traité de l’équilibre et du mouvement des fluids. He himself says of it that it is very general for questions of Dynamics and is also useful for fluids. To this respect see the summary in Chapter 3, ‘D’Alembert paradox’. 29 Cf. A treatise on fluxions Chap. XIV, §.638–640. ‘On the figure of the Earth and the Variation of Gravity towards it’. MacLaurin does not talk about channels but instead takes a point ‘P’ inside the spheroid and establishes that ‘therefore the particle P is pressed equally in all directions in the meridian plane APDB that passes through P’ [§.639].

THE THEORETICAL CONSTRUCTIONS (I)

Q

O

375

X

M

N

P

Q

R

Y Fig. 8-8. Closed channel

accelerating forces acting on the entire fluid. Starting from point M(x,y) the pressure to which the point Q(x + dx,y + dy), will be submitted will be equal, whether it follows the path MNQ or the path MPQ. Let Q(x,y) and R(x,y) be the components of the ‘force’ depending on the axis OY and OX. Depending on the channels NQ and PQ, the forces Q and R will be:

Q+

∂Q dx ; ∂x

R+

∂R dy ∂y

[8.9]

Bearing in mind that the density in each column will vary following an equivalent expression, the equality of the pressures in Q will be expressed by the following equation:

⎛ ∂ρ ⎞ ∂R ⎞⎛ ∂ρ ⎞ ∂Q ⎞⎛ ⎛ ρQdy + ⎜ R + dy ⎟⎜ ρ + dy ⎟ = ρRdx ⎜ Q + dx ⎟⎜ ρ + dx ⎟ [8.10] ∂x ⎠ ∂y ⎠⎝ ∂y ⎠ ∂x ⎠⎝ ⎝ ⎝ After working on and eliminating the second order terms, the expression is reduced to:

∂ ( ρQ ) ∂ ( ρR ) = ∂x ∂y

30

[8.11]

In op. cit, [§.19], in less than ten lines he uses the terms ‘puisssances’, ‘forces’, and ‘forces accélératrices’. The ‘puissance’ acting on a point like M breaks down into two ‘forces’, and a little further, he imagines that the ‘forces accélératrices’ that act on these selfsame points. …

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THE GENESIS OF FLUID MECHANICS, 1640–1780

In the case of constant density, ρ can be eliminated, yielding the same equation as Clairaut, ‘a proposition’, D’Alembert tells us, ‘which was already known, but which nobody it seems to me had yet demonstrated by such a simple method as that we have just used’[§.20].31 Just a couple of comments are in order before continuing. Both Clairaut and d’Alembert take as their starting point that any point of a fluid in equilibrium is subjected to the same pressure in any direction. This principle is still considered to be the fundamental property of fluids. A now classic text such as the Hydrodynamics of Horace Lamb says ‘The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface’.32 On the other hand, given that the last equation is obtained from this property, it indicates a necessary, but not sufficient, condition for the equilibrium. D’Alembert continues with the deduction of a series of propositions33 for the fluids at rest as well as those in motion. We are interested in the motion in a tube with a very small cross section.

F

A a

P p D

V0

G

B b

M m

ds

C

Fig. 8-9. Motion in a tube

31 He does not quote Clairaut. However, at the end of the work he includes an Appendix with some thoughts on the law of equilibrium of fluids. When he talks about the surfaces of levels in this appendix he does quote him. 32 The first edition was published in 1879. We quote the reprinting of the sixth edition in the Cambridge University press, Cambridge, 1945. Cf. p. 1. 33 In Chap. III. By the way, Truesdell says that this series of propositions are incomprehensible to him. (‘Rat. Fluid Mech.-12’, p. LII).

THE THEORETICAL CONSTRUCTIONS (I)

377

The approach and solution he arrives at correspond to what is nowadays called Bernoulli’s theorem. Even his development of the theorem is remarkably similar to Johan Bernoulli’s solution, although he does not say so. He supposes a very thin tube (Fig. 8-9) that has a length FA with a constant cross section value, and another AD with a variable one. The fluid, currently without weight, flows down from F to D. He supposes that the speed is constant for the planes, the so-called plane section hypothesis, and d’Alembert attempts to justify ‘since a certain tenacity can be imagined in the particles, in virtue of which the particles next to each other in a same layer PM stick to each other, and have the same velocity’ [§.27]. Using the continuity equation he expresses the velocities in any section as a function of the reference velocity, so v = v0S0/S. Now he argues that on passing from a point P to another point separated from this by a distance ds, the velocity will vary by dv. To apply his fundamental principle of motion ‘it follows that the layer PM being drawn by the indefinitely small velocity dv, or which comes to the same thing, by the single accelerating forces –dv/dt, the fluid contained in the channel ABCD would be in equilibrium’ [§.27]. The minus sign is due to the fact that as the section increases, the velocity decreases, which for us is irrelevant. It is clear that his method of converting dynamic problems into static ones is to include an ‘accelerating force’. Finally, in order to find the pressure (pression) at P, d’Alembert integrates dv/dt, resulting in:

⎞ dv 1 2 1 2 ⎛ S 02 2 P=ρ ds = ρ vdv = ρ(v − v0 ) = ρv0 ⎜⎜ 2 − 1⎟⎟ dt 2 2 ⎝S ⎠

∫

∫

[8.12]

A formula in which we have included the density. He extends the previous reasoning to the non-stationary case, i.e., where v0 is not constant. In this case dv = S0(Sdv0 – v0dS)/S2, and the resulting integral is:

P=

∫

ρ

dv ds = ρS 0 dt

∫

s s0

⎞ ds 1 2 ⎛ S 02 + ρv0 ⎜⎜ 2 − 1⎟⎟ S 2 ⎝S ⎠

[8.13]

A further step is the introduction of weight [§.29], for which the force will be g-dv/dt. The final equation, which we will not examine here, also applies to the case in which the tube is sloping [§.31].34 34 It is interesting to compare these developments with those made in the Traité. Cf. Chapter 7, ‘D’Alembert’s account of motions in tubes’.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

The last case he analyses refers to the assumption that the density is variable, and this is the only case in which we can speak of a significant contribution by Johann Bernoulli. Letting σ = S0/S and δ = ρ/ρ0, to simplify things, the variation of the velocity will be dv = σδdv0 + v0d(σδ). The final equation arrived at is:

P=

dv0 dt

∫

ρσδds + v0 ρ0

∫

d (σδ ) ds dt

[8.14]

Having arrived at these formulas, which he will use to obtain the pressures once the velocities are known, he regards the introduction as finished, and passes directly on to the most interesting part. Bodies in flowing currents

As we have remarked, d’Alembert distinguishes the case in which the body is immobile in a current from the converse case, when the body is in motion and the fluid is at rest. In the first of the two, the body introduces an alteration into the fluid field, which produces pressure upon it, while in the second case the body encounters resistance upon advancing. Let us look at the first case. He imagines the fluid to be without weight and of infinite extension. As regards the body, he considers it to be symmetrical with respect to an axis parallel to the current. The problem to be solved is to determine the fluid field as a consequence of the shape of the body, and then to find the pressure exerted upon the body. He begins with five prior observations which we summarize: (a) The particles of the fluid in motion, well upstream from the body, will describe streamlines parallel to each other. As they come nearer the body, this will induce deviations in the lines which will then travel around it, and become parallel again once downstream. (b) The phenomenon is stationary, i.e., the streamlines are always the same ones. (c) All change is continuous, therefore it is impossible for a streamline to arrive at the vortex of the body (point A in Fig. 8-10a) and turn abruptly. Because of this, the central streamline starts to deviate at a certain distance beforehand (point F), in order to return and adhere to the body further on (point R). Therefore, a stagnation zone will form in the front part (FAM plus the symmetrical part). The same happens in the egress or stern (LCR zone).

THE THEORETICAL CONSTRUCTIONS (I)

379

(d) His general principle of dynamics: if a particle undergoes an increase of velocity dv in a time dt, the pressure of the fluid will be the same as if it were at rest, and the accelerating force dv/dt were to act on the particle. Therefore, the body cannot suffer any pressure other than that proceeding from the particles, which either change their velocity or their direction due to the effect of the body. (e) Elaborating more fully on the previous point, the pressure at a point of the body will be that produced by the fluid circulating through a channel next to it. Therefore, the question boils down to finding the curvature of the channel and the accelerating forces acting upon it. The most relevant comment that we can make concerns the hypothesis of the existence of the two stagnation zones mentioned in point three. Even admitting the impossibility of abrupt jumps in the fluid, they are not strictly necessary. These zones are separated from the rest of the fluid by a tangential discontinuity of the velocities, as the interior of the fluid is at rest, while on the other side the particles advance at a certain speed, i.e., there is a jump. The situation is quite improbable, and reminds one of the cataract of Newton.35 F

F A

m q A

M

b n

M

L C R

a)

b)

Fig. 8-10. Body in a fluid stream

A consequence of the stagnation of the inner fluid is that the velocity on the boundary line FM must be constant and equal to that of the fluid without perturbations. The justification of this is simple: he imagines a number of channels, such as bnqm, represented in Fig. 8-10b, one of whose sides coincides with the designated boundary and the other in the body axis. Given that the inner points are at rest, there can be no difference in pressure between the ends mq and bn, and as this latter is the boundary, this implies that no force can exist on the particle 35 Which, by the way, d’Alembert criticised the cataract with the terms of ‘insufficient’ and ‘erroneous’ in the Introduction p. xvii.

380

THE GENESIS OF FLUID MECHANICS, 1640–1780

describing this current line, and if this is so, the velocity will be constant. He only admits a change of velocity direction, which according to what he says, would only produce a force perpendicular to the channel, which would not affect the pressure according to the direction FM. This is not quite true, as making use of the fundamental principle, which he himself invoked in hydrostatics, the pressure is transmitted in all directions. Therefore, the stagnation zone will be subjected to pressures which are not uniform in their contour and will be at very least unstable.36 As a result of the constancy of the fluid field around the body [§.39], at any given point the particles will always follow the same streamline, independently of the velocity of the fluid upstream and of its density. D’Alembert demonstrates that this is necessary and sufficient. The result is important, as it is a precursor of the unity of the fluid field, and allows him make the problem dimensionless, substituting the velocity at one point for the relation v/v∞ , where the denominator represents the velocity of the undisturbed fluid or infinitely upstream.37 Therefore, given a body, the fluid field produced by it will be defined by two nondimensional functions q(x,r) and p(x,r) which will represent the components of the velocity according to the lateral and longitudinal axes, such as:

v x ( x, r ) = v∞ q ( x, r ) ;

v r ( x, r ) = v∞ p ( x, r )

[8.15]

The functions q and p completely define the problem, and once known, obtaining the value of the pressures is automatic with the help of the previously established theorems. Once the distribution of pressures is known, the resistance, which he calls total pressure, can be found using an integration.

E

S

N

P

dx r

dr X Fig. 8-11. Streamline 36

Here it is appropriate to mention the criticism that Johann Bernoulli makes of Newton’s cataract. See Chapter 7, ‘D’Alembert’s account of motion in tubes’. 37 This procedure continues to be customary in fluid mechanics.

THE THEORETICAL CONSTRUCTIONS (I)

381

He tries two ways to determine the functions q and p, one using lateral forces, and other the invariance of the volume of a fluid element. The two are not wholly equivalent although he takes it that they were. In both cases the result will be two partial differential equations that reduce the physical problem to a mathematical one. It is in this task that the talent of d’Alembert shines to most advantage, and where his major contribution to fluid mechanics lies, without forgetting the establishment of the constancy of the fluid field and the breakdown of the velocity into its two components.38 However, the way in which he approaches the question [§.43–48] is not easy to follow, as although the idea is brilliant, its explanation is somewhat confused and is aggravated by some errors. D’Alembert begins by the plane problem, just as this has been expressed up to now, but before he gets to the end, he has converted it into an axial symmetric one.39 We have tried to present the reasoning of d’Alembert as clearly as possible in what follows. Solution using lateral forces

Following this path, the first question that arises is the kinetic relations on a streamline. As shown in Fig. 8-11, let this be a particle located at point S, which passes through a time interval dt to N. By definition, both S and N belong to the streamline. The variation of the functions q and p between these two points is expressed as:

dq =

∂q ∂q dx + dr ∂x ∂r

[8.16]

dp =

∂p ∂p dx + dr ∂x ∂r

[8.17]

On the other hand, in accordance with the definition of q and p, the increments in velocity when going from S to N will be:

dv x = v∞ dq dvr = v∞ dp

38 39

Rouse says that this is the first time that it happens. Cf. History of Hydraulics, p. 102. Which obviously, is also two dimensional.

[8.18]

382

THE GENESIS OF FLUID MECHANICS, 1640–1780

Moreover, since the process is stationary, the velocity must be tangential to the streamline, which will also be a trajectory. This is expressed as

dr vr p = = dx v x q

[8.19]

Combining the last three groups of equations, and remembering that according to his principle the accelerating forces are opposed to the increments of velocity per unit of time, he determines their value, which turns out to be:

∂q ⎞ ⎛ ∂q a x = −v∞2 ⎜ p + q ⎟ ∂x ⎠ ⎝ ∂r

[8.20]

∂p ⎞ ⎛ ∂p a r = −v∞2 ⎜ p + q ⎟ ∂x ⎠ ⎝ ∂r

[8.21]

We now call attention to the fact that the relations deduced by him are purely kinetic [§.44], although later on these accelerations fulfil the function of accelerating forces.40 v

M

A Q p

M rc(x)

n

dx N m

rc(x) drc

N

n r m r(x) d+d

Fig. 8-12. Flow around the body

40

We use the terms ax and ar that represent the accelerations, in order to make the exposition quite clear, just as Truesdell does, although we point out that d’Alembert does not use the term ‘acceleration’.

THE THEORETICAL CONSTRUCTIONS (I)

383

In order to explain the following steps we shall follow Fig. 8-12, which represents the cutting of the body by a meridian plane.41 Upstream the current advances in streamlines, just as d’Alembert supposes; when the trajectories come near they curve leaving the corresponding stagnation areas in the bow and stern. We have also drawn a streamline very close to the surface, which at the height of the plane passing through P will define a circular crown whose width is δ around the body. Continuing upstream, once outside the body and the stagnant area, this crown will become a cylinder of radius δ∞. Supposing that the width δ is very small compared with the radius of the body, by the condition of continuity the following equation holds:

πδ∞2 v∞ = 2πrδv cos γ = 2πrqδ

[8.22]

From the former it follows that δ = δ2∞/(2rq), an equation which relates the thickness of layer to the geometry of the body, and to one of the functions of the velocity. D’Alembert continues to find the expressions of the slope of the streamline passing through point N by two different paths, whose results he will identify in order to obtain the first of the equations that governs the motion. The first begins with a kinetic basis, considering that if the components of velocity at point M are vx and vr, with a slope ΓM = vr/vx = p/q. If we move to N, separated δ radially from M, the new components will be:

∂q ⎞ ⎛ v x = v∞ ⎜ q + δ ⎟ ; ∂r ⎠ ⎝

∂p ⎞ ⎛ v r = v∞ ⎜ p + δ ⎟ ∂r ⎠ ⎝

[8.23]

And the slope:

∂p δ ∂r ΓN = ∂q q+ δ ∂r p+

[8.24]

By geometry, the slope ΓN = rm/rN is expressed analytically as:

41 Our Figures are taken from the d’Alembert’s ones, but sometimes we have changed his letters and symbols, in order to make our exposition more consistent.

384

THE GENESIS OF FLUID MECHANICS, 1640–1780

ΓN =

dr + dδ p dδ = + dx q dx

[8.25]

Now then, as δ = δ2∞/(2rq), dδ develops in the following manner:

1 dq ⎞ dδ d ⎛ δ∞2 ⎞ δ 2 ⎛ 1 dr ⎟⎟ = − ∞ ⎜ 2 + 2 = ⎜⎜ ⎟ 2 ⎝ r q dx rq dx ⎠ dx dx ⎝ 2rq ⎠

[8.26]

Taking into account that dr/dx = p/q by [8.19], introduced in the former equation, and equalled to the development of equation [8.24], he arrives at:

−

∂p ∂q p = + ∂r ∂x r

[8.27]

This equation is the first of the constitutive formulas of the motion,42 and, as he demonstrates, he unites in a single equation the functions of velocity and position.43 We stress that although the deduction was made for a current line near the body, it is valid for the entire fluid, as the arguments can be generalised. In order to obtain the second equation of motion he calculates the pressures at the points N and M in relation to the undisturbed flow upstream, using the equation [8.12] he had obtained previously. This is:

PM = 42 43

1 ρ( v∞2 − v N2 ) 2

[8.28]

For the case of plane motion this formula will be –∂p/∂r = ∂q/∂x. We show the development in greater detail below. Equation [8.16] is completed as:

dq ∂q ∂q dr ∂q p ∂q = + = + dx ∂x ∂r dx ∂x q ∂r

Introduced in equation [8.25] the next expression is obtained for the slope:

ΓN =

p δ ∞2 ∂q δ ∞2 − − q 2rq 2 ∂x 2

⎛ 1 ∂q 1 ⎞p ⎜⎜ 2 + 2 ⎟⎟ ⎝ rq ∂r qr ⎠ q

On the other hand, the slope obtained in equation [8.24] is approximated as:

∂p δ 2 2 ∂r = p ⎛⎜1 − δ ∞ ∂q + δ ∞ ∂p ⎞⎟ ΓN = ∂q q ⎜⎝ 2rq 2 ∂r 2rqp ∂r ⎟⎠ q+ δ ∂r p+

Which equalled to the former, after eliminating terms, leads to that given by equation [8.27].

THE THEORETICAL CONSTRUCTIONS (I)

PN =

1 ρ( v∞2 − vM2 ) 2

385

[8.29]

Now the velocity at M has vx and vr as components, and the corresponding one at N has those found in equation [8.23]. Therefore, after the corresponding operations, the difference in pressure between the points M and N will be:

∆P =

1 ∂q ⎞ ⎛ ∂p ρ( v M2 − v N2 ) = − ρv∞2 ⎜ p + q ⎟ 2 ∂r ⎠ ⎝ ∂r

[8.30]

On the other hand, this difference of pressure between N and M is the accelerating force between both points, which is found as ar in [8.21], and made equal to the second member of the former equation leads us to:

∂q ∂p = ∂r ∂x

[8.31]

Which is the second of the constitutive formulas, and just like the first one [8.27], it defines the relation between the velocity functions and the position. Strictly speaking, these two equations define the problem. D’Alembert also presents a variant to these, which is the result of introducing the last and equation [8.27] in the formulas for dq and dp given in equations [8.16] and [8.17] resulting in:

dq =

∂q ∂q dx + dr ∂x ∂r

d ( pr ) = r

∂q ∂q dx − r dr ∂r ∂x

[8.32]

[8.33]

He says that these should be exact differentials 44 [§.46], although this would would in fact be an additional hypothesis.

44

In the case of bi-dimensional flow equation [8.33] would be:

dp =

∂q ∂q dx − dr ∂r ∂x

386

THE GENESIS OF FLUID MECHANICS, 1640–1780

Solution by constancy of volume

After obtaining the equations using the method of lateral forces, d’Alembert undertakes a new deduction which employs ‘a somewhat more general method’ [§.48], and which is carried out also in two successive steps. The first consists in isolating a differential element of the fluid which, due to its incompressibility, maintains a constant volume when it evolves from one instant to a later one, an idea which he will use extensively from then on and from which he obtains one of the two conditions. For the second, he uses the principle of the channels, which we have identified as being due to Clairaut, in which he introduces accelerations as a field of forces, considering that his general principle of dynamics allows this. For the first part, he imagines an elemental volume of fluid (Fig. 8-13) whose dimensions in an instant t will be ∆x, ∆r and r∆θ. The volume of this element will be ∆V = ∆x∆r(r∆θ). In an interval of time dt the previous dimensions will evolve in factors of 1 + v∞(∂q/∂x)dt, 1 + v∞(∂p/∂r)dt and 1 + v∞(q/r)dt. As the volume must be invariable when establishing the equality between these, we have:

∂q ⎞⎛ ∂p ⎞⎛ p ⎞ ⎛ ∆x∆r ( r∆θ ) = r ⎜1 + v∞ dt ⎟⎜1 + v∞ dt ⎟⎜1 + v∞ dt ⎟∆θ ∂x ⎠⎝ ∂r ⎠⎝ r ⎠ ⎝

[8.34]

Calculating and eliminating the higher order terms, we arrive at the already known expression [8.27]. Basically, there exists a strong parallel with the other procedure, since in that procedure we also arrive at this equation starting from the continuity condition, only instead of establishing the invariance of a differential element as it is made here, this condition is applied to a layer of fluid from the infinite to the point under study. In both cases it is a kinetic construction.

d x r

r

Fig. 8-13. Fluid element

THE THEORETICAL CONSTRUCTIONS (I)

387

In order to obtain the other constitutive equation of the motion, he proposes a more debatable step. He first extends the condition of equilibrium which he has established for a liquid at rest [8.31] to the dynamic case, substituting the field of forces for the accelerative forces which he says ‘must destroy themselves’ [§.48]. This means he applies the equality of the crossed derivatives to the accelerations [8.20 and 8.21], so that:

∂p ⎞ ∂ ⎛ ∂q ∂q ⎞ ∂ ⎛ ∂p +q ⎟= ⎜p +q ⎟ ⎜p ∂r ⎝ ∂r ∂x ⎠ ∂x ⎝ ∂r ∂x ⎠

[8.35]

Developing this formula we obtain:

∂ 2 q ∂p ∂p ∂ 2 p ∂q ∂p ∂2 p ∂q ∂q ∂ 2 q ∂p ∂q +q +q + +p = +p + ∂x∂r ∂r ∂x ∂x∂r ∂r ∂r ∂x∂r ∂r ∂x ∂x∂r ∂r ∂r [8.36]

Now he says ‘that this equation will be true’ if the two motion equations [8.27] and [8.31] are demonstrated. In order to show this he begins by reminding us that the two expressions [8.16 and 8.17] are exact differentials, which means that the equality of the cross derivatives can be applied to them:

∂ ⎛ ∂q ⎞ ∂ ⎛ ∂q ⎞ ⎜ ⎟ = ⎜ ⎟; ∂x ⎝ ∂r ⎠ ∂r ⎝ ∂x ⎠

∂ ⎛ ∂p ⎞ ∂ ⎛ ∂p ⎞ ⎜ ⎟= ⎜ ⎟ ∂r ⎝ ∂x ⎠ ∂x ⎝ ∂r ⎠

[8.37]

And by the second constitutive equation of the motion [8.31], (∂q/∂r = ∂p/∂x), which he supposes to be true, we have:

∂ ⎛ ∂p ⎞ ∂ ⎛ ∂q ⎞ ⎜ ⎟ = ⎜ ⎟; ∂x ⎝ ∂x ⎠ ∂r ⎝ ∂x ⎠

∂ ⎛ ∂q ⎞ ∂ ⎛ ∂p ⎞ ⎜ ⎟= ⎜ ⎟ ∂r ⎝ ∂r ⎠ ∂x ⎝ ∂r ⎠

[8.38]

∂2q ∂2 p = ∂r 2 ∂x∂r

[8.39]

Which leads to:

∂2 p ∂2q = ; ∂x 2 ∂r∂x

388

THE GENESIS OF FLUID MECHANICS, 1640–1780

Introducing both in equation [8.35], this is simplified as:

∂q ∂q ∂p ∂q ∂p ∂p ∂q ∂p + = + ∂r ∂x ∂r ∂r ∂r ∂x ∂r ∂r

[8.40]

Which, regrouped, yields:

⎛ ∂q ∂p ⎞⎛ ∂q ∂p ⎞ ⎜ − ⎟⎜ + ⎟ = 0 ⎝ ∂r ∂x ⎠⎝ ∂x ∂r ⎠

[8.41]

The second factor is –p/r for the proved equation [8.27]’ therefore, for the results to be nil it is necessary that the first factor be zero, and this factor is the constitutive equation [8.31], which had been assumed to be true. In view of this, we conclude that both the equality hypothesis of the crossed derivatives plus the two constitutive equations are a sufficient condition for the existence of the motion, although not a necessary one. That is to say, there will be possible motions that do not follow these hypotheses.45 In view of the foregoing, the calculation of the forces and pressures for a axial-symmetric body is reduced to determining the two functions p and q, which must verify the two differential equations of motion, and meet the boundary conditions. In spite of the beauty and cleanness of the approach, the practical resolution is more difficult still.46 D’Alembert did not arrive at a solution, although he takes us a few steps further, especially the introduction of complex variable functions for resolving the problem, which is his most noticeable contribution. In the nineteenth century this technique enabled an analytical solution to the problem in certain cases. The question is strictly mathematical, although it will come to form part of the basic study of fluid mechanics.47 In order to do 45 Truesdell had already made note of this, when he criticised the entire process ‘Trat. Fluid Mech.-12’, p. LIII. 46 What is more, a considerable part of all the efforts which have been devoted to this science since then have been dedicated precisely to resolving this problem. 47 Morris Kline, in Mathematical Thought from Ancient to Modern Times (Oxford University Press, 1972) quotes this case as the first historical contribution of the complex variable functions [Chap. 27], although complex numbers had already been used for solving other mathematical problems. Besides we have to say that two-dimensional aerodynamics or more specifically profile theory has been able to be studied thanks to this transition to the complex field. First with the transformations method (of which the best known is that of Kutta-Jukowski), and later by profile assimilation to the continuous distribution of differential sources and vortexes which do indeed have an exact solution in this field.

THE THEORETICAL CONSTRUCTIONS (I)

389

this, d’Alembert abandoned the axial-symmetric motion in which he developed the equations to proceed to the plane motion. Mathematically the plane case is a particularisation of the other when r → ∞, but the solutions that he proposes are only valid for the plane case.48 In order to solve the plane motion, he first proposes the following mathematical problem: ‘Let Mdx+Ndz & Ndx-Mdz be exact differentials. I propose to calculate the quantities M and N’ [§.58]. After a few manipulations which are based on the fact that the exact differential condition was conserved in the addition, and on the introduction of an imaginary unit i49 he arrives at a position in which M and N have to correspond to functions of the type:

M + iN = Φ( F + x − iz )

[8.42]

M − iN = Ψ (G + x + iz )

[8.43]

Where both F and G are constants and Φ and Ψ are any functions. Now x + iz and M + iN are complex quantities, and the previous expressions indicate the transformation of certain differential equations into the field of complex variable functions.50 On the other hand, we repeat the equations for the plane case, which were:

∂p ∂q =− ; ∂x ∂z

∂p ∂q = ∂x ∂z

[8.44]

This implies that qdx + pdz and pdx – qdz must be exact differentials. Therefore, applying the conclusions obtained from the proposed problem we obtain:

48 In reality it is a return to the plane case, as he had started the equation is in this [§.43] in order to abandon it, and go on to the axial-symmetric one [§.45]. One of the difficulties that reading d’Alembert presents, although not one of the worst, resides in the changes he makes jumping from one case to another. 49 In what follows the sign i will be used to express imaginary unit. D’Alembert used the expression − 1 for this. Euler occasionally used the symbol i, although some years later. Its use became general with Gauss in the nineteenth century. 50 The fundament, developed by d’Alembert, is based on the fact that if Mdx + Ndz and Ndx – Mdz are exact differentials, Mdx + iNdz/i and iNdx – iMdz must also be, just like the sum and the difference, that is to say (M + Ni)(dx – idz) and (M – Ni)(dx + idz). If we do du = dx – idz and dt = dx+ idz, it will have that: u = F + x – iz and t = G + x + iz, where F and G are integration constants. He does likewise M + iN = α and M – iN = β, and as αdu and βdt are exact differentials, it means that α = Φ(u) and β = Ψ(t), which constitute the expressions he arrives at.

390

THE GENESIS OF FLUID MECHANICS, 1640–1780

q + ip = Φ( F + x − iz ) q − ip = Ψ (G + x + iz )

[8.45]

With some more considerations concerning the value of the constants F and G, so that the final solution of p and q does not contain imaginary terms, he concludes that these will be functions of the type:

p = −iξ ( x − iz ) + ζ ( x − iz ) + iξ ( x + iz ) + ζ ( x + iz )

[8.46]

q = ξ ( x − iz ) + iζ ( x − iz ) + ξ ( x + iz ) − iζ ( x + iz )

[8.47]

We note that the problem of finding p and q has transformed itself into a search for the functions ξ and ζ which must depend exclusively on the boundary of the body submerged in the fluid. The solution continues to be difficult and faced with this, d’Alembert assumes ‘as an example’ [§.60] that both will be third degree polynomials with unknown coefficients, which he tries to determine in order to adjust them to the boundary. He tries to apply this approximation [§.61] to the axial-symmetric case, and sets about establishing the equations, but their solution was still a long way off. Once the functions p and q are known, the calculation of the velocity and pressure at any point is immediate [§.66], the first as v∞ p 2 + q 2 and the second as ½ρv∞2(1 – p2 – q2), where the density is included for consistency of the equation. As the pressure at each point is proportional to the square of the velocity, he infers that the total pressure must likewise be proportional to the square. But before going into what he understands by total pressure, note the following, apparently contradictory paragraph. He says: Moreover, for this expression to be exact, one needs to suppose that pp+qq is smaller than 1 everywhere, i.e. that the velocity along the flow line MDL [Fig. 8-13] is smaller than v∞ everywhere, or at least it is not larger. Because after having determined p and q by calculation one finds that √(pp+qq) was > 1 in certain points, one would have to first look for the point where the value of √(pp+qq) would be a maximum which would occur on supposing pdp+qdq=0. Then naming K the value of √(pp+qq) at this point, one would obtain v∞2(K²-pp-qq)/2 for the pressure in N. [§.66]

Strictly speaking, what he does is to measure the pressures with respect to the pressure existing upstream, when the current is still undisturbed, which implies that q = 1, p = 0, and that the pressure will be p∞. However, if p2 + q2 were greater than one, the pressure would become negative, and this would influence the supposition of the first part of the quotation. Now, there is no reason obliging

THE THEORETICAL CONSTRUCTIONS (I)

391

it to be so, and therefore he looks for the point with minimum pressure, for which K = p2 + q2, that he takes as reference for measuring the pressures, (i.e., p = 0). With this reference, obviously, no negative values will appear, even when they are possible with respect the upstream conditions.51 With these clarifications, the total force on the body is a relative magnitude in which the surrounding, or external, pressure has no influence on the total result. The weakness of his concept of pressure is almost tangible. The value of the ‘total pressure’ on the body, given that this is axialsymmetric, is found by integrating the following equation throughout its boundary r(x).

2π

∫

1 2 2 ρv∞ ( K − p 2 − q 2 )rdr 2

[8.48]

After several excursuses, due to the difficulty in handling the defined integrals, and taking into account that a constant value is the one corresponding to the addend K2, he reduces the former to the following:

− 2π

∫

xM xL

1 2 2 ρv∞ ( p + q 2 )rdr 2

[8.49]

Where xM corresponds to the point where the stagnation area at the bow ends, and xL is the beginning of the stern one. In short, the total pressure is tailored to an expression of the type φv∞, where φ is always the same for a given boundary. [§.68] Now if the body were to have mirror symmetry with respect to its equatorial plane (DD′ ), then the longitudinal velocity, represented by q, would be equal in symmetrical points while the radial, represented by p, would have the same magnitude, but a different sign. Then p2 + q2 would be symmetrical, and the pressure of a point would be cancelled out by that of the symmetrical one. In conclusion, if the arcs LD and DM (Fig. 8-14a) were equal, no pressure at all would exist. Faced with this surprising conclusion he says: ‘From this it follows that the arcs LD and DM cannot be equal, because if they were, then the quantity –∫2π(p2 + q2)rdr would be equal to zero, in such a way that the body would not suffer any pressure on the part of the fluid, which is contrary to experience’ [§.70]. Besides, as the force has to be directed towards the rear, he concludes that LD > DM. 51 Following this line of argument we recall to mind the criticism that d’Alembert made to Daniel Bernoulli regarding the existence of ‘suctions’ and ‘pressures’.

392

THE GENESIS OF FLUID MECHANICS, 1640–1780 u u A

F A P

p M q

Vol

M

D

D'

D

-p C

L q

a)

I C

L

b)

Fig. 8-14. Symmetrical body and paradox

This statement marks the official appearance of ‘d’Alembert’s paradox’, which we have already had occasion to mention. It is noteworthy that d’Alembert tries to avoid this contradiction by introducing an ad hoc hypothesis on the non-symmetry between the two stagnation zones, a purely arbitrary assumption. Years later, in his Opuscules Mathématiques,52 he returns to this theme with a ‘Paradox proposed to geometricians on the resistance of fluids’ in which he uses arguments tending towards geometry and symmetry, but without making any mention of the stagnation zones. It is not the first time that this possibility was mooted in fluid mechanics. Euler, in his translation of Robins’ Gunnery, had arrived at similar results.53 However, Euler had approached the problem using a method derived from impact theory, while d’Alembert used a more elaborate method, although in both arguments the basis of the paradox is the geometry of the boundary. D’Alembert requires the mirror symmetry of the body, while Euler needs certain geometrical conditions. In spite of their differences, both tried to escape from this embarrassing conclusion, imagining the existence of zones of the fluid in which the theory is not satisfied. The study of the effect of the flow on the body is concluded with the impulsive generation of pressure upon the body [§.51–56]. This corresponds to an initial situation of rest, in which the fluid is put into motion instantly by some procedure or other. He designates this instant velocity as u, and as a result of this an axial and a lateral velocity will appear in each particle. These will be –qu and –pu, q and p corresponding to the previous meanings. Following a line of thinking similar to the foregoing, d’Alembert demonstrates that the resulting field of

52 53

Vol. V, 1768 Memoire 34, I. Cf. Chapter 3, Robins’ New Principles of Gunnery.

THE THEORETICAL CONSTRUCTIONS (I)

393

velocities is only in function of the body shape, and it coincides with the one he found for the case of the moving fluid, even with the same stagnation zones. As a continuation to this instantaneous jump, he goes on to evaluate the pressure of the fluid over the entire body in this first instant. With respect to this he says that ‘this research is absolutely necessary … in order to determine the quantities p and q’ [§.54]. The basis for the calculation is the idea that the system will be in equilibrium if the ‘forces’ u, –uq and –up are destroyed. Regarding the first, he says: Now the pressure resulting from the velocity u common to all the particles and parallel to AC will be µδu [ρVolu], where µ [Vol] is the body mass and δ [ρ] the density of the fluid, and this pressure will be along CA. [§.54]

We observe that ρVol is the mass of fluid displaced, and the resulting ‘pressure’ is the product of this body mass by the velocity. He treats the other two ‘forces’54 in a similar manner, whose result at a point of the body would be u p 2 + q 2 . The total force projected over the axis is expressed by a double integral, one extended along the boundary and the other over the radius, so that the force would be:

ρu ∫ 2πrdr ∫ p 2 + q 2 ds

[8.50]

Taking into account that the velocity over the boundary has to be tangent to the body, ds will have the direction marked out by the components p and q, and then the integral is simplified to adopt the form:

ρu ∫ 2πrdr ∫ pdr + qdx

[8.51]

The calculation of the integral is not very clear, and moreover it is affected by the difficulty with the boundaries. To calculate it, he uses two functions: Ω and Γ. The first corresponds to the integral [8.50] limited from point L up to M (Fig. 8-14b), and the second is defined as Γ = ∫pdr + qdx between the same limits. The result of these forces is the following:

Ω + π Γ( rM2 − rL2 )

54

We insist in the ambiguity of the term ‘force’ in d’Alembert.

[8.52]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

As a final result, because two forces are included—one of total volume and another of surface— for reasons that are not very clear he says that ‘the quantity … πΓAA [πΓrL2] must be subtracted; at the end µδu [ρVolu], must be added’ [§.54]. The final result is:

∆ = ρu (Vol + Ω − π ΓrL2 )

[8.53]

Note that this pressure or force is not congruent with that given in equation [8.49]. It has dimensions of density–velocity to the square-surface, while equation [8.53] is of density–velocity–volume. The only way to make them compatible would be to admit that the instant jump u of velocities was not such, but a jump in finite time, or rather letting u be an infinitesimal. The following paragraph appears to run along these lines. All this arises from the background of the application that d’Alembert makes of his principle of dynamics. He continues: ‘It can be easily demonstrated by experience that µ+Ω-πΓbb [Vol+Ω-πΓrL2] is =0’ [§.55]. The reference to experience is surprising. He continues accordingly, ‘because a weight can be found that is capable, solely by its weight (pesanteur), of maintaining a body ADCE in equilibrium starting from the first instant of the thrust (impulsion) of the fluid’ [§.55], which we interpret as the force of inertia. He continues: ‘the action of a weight that is in equilibrium is equivalent to a finite mass animated by an infinitely small velocity, … or an infinitely small mass animated by a finite velocity’. [§.55] As u is finite, he deduces that the term ‘must of necessity be infinitely small, i.e., zero’ [§.55]. The interest that this justification has for us arises from the fact that when he comes to study the motion of a body in a stationary fluid (which he goes on to do), the annulment of this parameter is what justifies the equivalence of this new case with the immobile body. As the last point, and as one of the reasons why, as he says, these investigations were necessary, he analyses the pattern of fluid streamlines when the velocity varies with the time. Assuming a body which is placed at rest in a stagnant fluid, if the fluid is given a velocity u, parallel to the axis of the body, then all fluid particles will move following the streamlines that are always the same, irrespective of the value of u. If an instant later an additional velocity u ′ is given to the fluid, he says that ‘it is clear that this new velocity does not disturb the streamlines in any way, as if this velocity [u′] was the only one, it would have traced them’ [§.56]. The only thing that happens is the velocity in each point must change the ratio u + u′ to u′. He says he will use this proposition further on.

THE THEORETICAL CONSTRUCTIONS (I)

395

Bodies moving in a stationary fluid

After studying a body at rest in a moving fluid, he turns to the opposite case, with the body now moving. As he conceives it, in the first of the cases pressures are generated on the body and the integration of these gives the total pressure, while in the second case what appears is resistance, that is, the quantity of motion that the body communicates to the fluid. His general approach to this case is more complex due to the variants he introduces, although mathematically it leans heavily upon the other case. He considers that the fluid may be either incompressible (which he calls non-elastic) or elastic, and indefinite (indéfini) or finite (fini) [§.84]. This classification leads him to treat three different cases: (a) non-elastic and indefinite fluids; (b) non-elastic and finite fluids; and (c) elastic fluids. The difference between the first two rests on whether they leave a vacuum behind in the motion or not: the indefinite fluids never leave a vacuum, whereas the finite fluids do. In addition, he introduces lengthy considerations on the resistance produced by friction [§.93], the different sources of resistance, the pendulums oscillating in fluids [§.95], comments on Bernoulli [§.99] and on Robins [§.112], and goes back to present the paradox [§.104–105]: in sum, a review of all the subjects of the period. We will limit ourselves to the most relevant matters in what follows. In the first of the three types, he conceives of the phenomenon in such a way that the advancing body will induce velocities at different points of the fluid which were initially at rest. Concerning this he says: It is clear that the velocity of these particles in each instant can be considered as being composed of another two: an velocity equal and parallel to the one at which the body moves at this instant, and another velocity which will be the relative velocity of these particles with respect to the body. [§.86]

What he does is to convert the phenomenon from a fixed axes to fluid ones, that is to say from stationary axis into moving ones linked to the body, which he tells us does not need to move at a constant velocity. He breaks down the second velocity, the relative one, again into two: one following the axis of the body and the other following the radius. That is, if the body advances with u velocity, and the relative one v is broken down in turn into vx and vr, the absolute velocity will have u – vx and vr as components. The relation between both at the surface of the body will be determined by the condition of tangency, which is just the same as what happened when the current moved [8.19]. In order to continue with the similarities, he introduces the functions q and p, now defined as:

396

THE GENESIS OF FLUID MECHANICS, 1640–1780

v x ( x, r ) = q( x, r )u ;

vr ( x, r ) = p( x, r )u

[8.54]

This differs from equation [8.15] in the substitution of v∞ by u, which is not a significant difference. As before, he finds the first relation between q and p by determining a fixed quantity of fluid enclosed in the differential element. The calculation method is the same, and obviously the solution is equation [8.27]. For the other equation he follows a procedure similar to that described in the calculation of the ‘forces’ with the construction shown in Fig. 8-10. However, here he introduces a small variant, due to the fact that the velocity u does not have to be constant. As a result of the latter, the previous equations [8.20 and 8.21] are transformed into:

a x = −q

du ∂q ⎞ ⎛ ∂q p+ − u2⎜ q⎟ dt ∂x ⎠ ⎝ ∂r

[8.55]

du ∂p ⎞ ⎛ ∂p − u2⎜ q + q⎟ dt ∂x ⎠ ⎝ ∂r

[8.56]

ar = − p

The destruction of both depends on the equality of the cross derivatives, and the following equation results as the substitute of equation [8.35]:

du ∂q ∂ ⎛ ∂q ∂q ⎞ du ∂p ∂ ⎛ ∂p ∂p ⎞ + u2 ⎜ p +q ⎟= + u2 ⎜ p +q ⎟ dt ∂r ∂r ⎝ ∂r ∂x ⎠ dt ∂r ∂x ⎝ ∂r ∂x ⎠

[8.57]

He adds that ‘in this equation p and q do not depend on the indeterminate u’ [§.86], which means that the fluid field is independent of the value of the velocity. This means that:

du ∂q du ∂p = dt ∂r dt ∂x

[8.58]

which is the second equation [8.31] again. In the other case, he repeats the reasoning at [8.31-ff], which we omit here.

THE THEORETICAL CONSTRUCTIONS (I)

397

He ends by saying that ‘the equations [constitutive 8.27 and 8.31] hold just the same, whether in the case where the fluid moves, or that where the fluid is at rest and it is the body that moves’. [§.86] In order to find the resistance manifested by the body, he uses a tortuous argument [§.86] which we can schematise in the following manner: if the body has a velocity u, variable with the time, then an equal but contrary velocity will be brought to the fluid-body system, thus causing the body to remain at rest, and it will be the fluid that moves. He continues, ‘but by the primitive laws of motion, the pressure of the fluid on the body will not change’ [§.88], from which he infers that this force will be proportional to u2. But, in addition, he introduces the accelerating force kdt, with which he wants to include the force due to the changes of velocity. That is to say, he considers two forces: one that includes the stationary effects, and another including the transitory ones. As we explained in the previous paragraph, when dealing with the effects of sudden changes in velocity, the effect of this kdt will be like ∆kdt, ∆ being the magnitude expressed in equation [8.53], which he demonstrated was zero. D’Alembert argues: Writers on hydraulics up to now have all maintained the principle that the resistance of a body moving in a fluid is equal to the pressure that a fluid moving at the same velocity will exert upon a supposed body at rest. But, firstly, they have not paid attention to the fact that when this velocity is variable, the resulting pressure could contain the element du, and consequently would not be proportional to u2. Secondly, taking this velocity to be variable, where they had taken a constant velocity, does not prove, except in a very vague manner, that the pressure is the same as u2 (see art. 10)55 It seems to me that we have fully overcome all the difficulties, demonstrating that the coefficient of du/dt is zero and that the coefficient φ of u2 is always the same whatever the value of u. [§.88]

That is to say, when changing the body’s velocity by du, three types of forces appear: (a) that of the inertia of the body which is ρc Vol du/dt, or the mass times the variation of the velocity; (b) the resistance proportional to the square of velocity which will be ρφu 2, where φ acts as the resistance coefficient just as we have seen it being used56; and (c) the force due to the change of velocity which will be ρ∆du/dt, ∆ being defined in equation [8.53]. In conclusion, according to his premises, [§.89] the equilibrium equation of the system would be:

ρcVol 55 56

du du + ρφu 2 − ρ (Vol + Ω − π ΓrL2 ) = 0 dt dt

In this article he insists in the vagueness of the demonstration. For us this would be φ = ½CDS.

[8.59]

398

THE GENESIS OF FLUID MECHANICS, 1640–1780

But, as he has demonstrated that the last term was zero,57 the final equation would remain as the following:

ρcVol

du + ρφ u 2 = 0 dt

[8.60]

Which is just the expression of the second law of Newtonian dynamics. At this point he affirms that: It is obvious that the resistance of a fluid, … is proportional to ρφu2, that is to say, it is equal to the pressure that this fluid exercises upon a supposed body at rest, if the fluid were to impact with the velocity u. This proposition, as we have said, has been taken as true up to now, but there was no less need to prove it. [§.90]

We shall skip his studies on frictional resistance in order to present the equations he arrives at in the case of elastic fluids. The calculation method continues to be the same, to fix the constancy of the mass of a fluid element and to follow its evolution. He obtains the following equation:

∂ ( ρp ) ∂ ( ρq) ρp + + =0 ∂r r ∂x

[8.61]

Which differs from equation [8.27] in the inclusion of the density as in multiplying factor. For the second he uses the closed channel theorem, in which he cancel the forces by the equality of the cross derivatives. He includes non-stationary terms in the solution, arriving at equation [8.62] [§.116]58

−

dv ∂ ( ρq) ∂ ⎛ ∂q ∂q ⎞ dv ∂ ( ρp ) ∂ ⎛ ∂p ∂p ⎞ − v 2 ⎜ ρp + ρq ⎟ + + v 2 ⎜ ρp + ρq ⎟ = 0 dt ∂r ∂r ⎝ ∂r ∂x ⎠ dt ∂r ∂x ⎝ ∂r ∂x ⎠ [8.62]

which takes the place of equation [8.35] or [8.57]. However, d’Alembert shows that in order to handle it is necessary to assume the existence of a function relating density with velocity and position. 57 58

Cf. §.55 and previous section where we have commented previously. There is an important error in the derivation of this formula [§.116]: the appearance of a ∂ρ/∂t.

THE THEORETICAL CONSTRUCTIONS (I)

399

Conclusion: D’Alembert’s contributions

There is much that could be said about d’Alembert’s work, his defects and contributions. We have already identified some negative aspects: the density and intricacy of his discourse, complicated terminology, lack of precision in his use of dynamic terms, etc. We shall add just two fundamental concepts. We have seen how he deduces the fluid field three times: with a moving fluid, with a sudden impulsion, and with a moving body. This is due to his concept of dynamics, in which he tries to make the forces obvious, an attitude that has an epistemological rationale, but which complicates the whole account. Moreover, it gives the impression that the forces continue to be there as a background. His defects have been more talked about than his virtues,59 so here we will now underline some positive aspects: • The introduction of the fluid field concept, expressed by two functions that express the velocity components. • The demonstration of the invariance of this field with velocity, depending only on the shape of the body. • The reduction of the phenomenon of motion to two equations in partial derivatives that do not solve the problem, but serve only to define or contain it. The simple fact, on the other hand not so simple, of the intellectual assumption of these equations supposes a remarkable advance, as he makes a very profound abstraction of physical reality. • The attempt to transform the problem, in some cases, from the real plane to the complex one. • The attempt to apply these equations to practical problems, particularly resistance.

Finally—and this is Euler’s view—we repeat that in d’Alembert’s equations of motion, the components of velocity are the only ones that figure. He did not adequately and correctly deal with forces or pressures except in deducing dynamic concepts. While it is true that the treatment of forces and pressures is somewhat confused, this should not lead us to overlook the merits of the rest of his work.

59 We have made particular use of Truesdell’s analyses, as other authors do not go as deeply into d’Alembert’s mechanics.

Chapter 9 Theoretical Constructions (II): Euler

Leonhard Euler

Although Euler devoted numerous works to the development of fluid mechanics, his most outstanding contribution to the theorisation of this discipline centres on three monographic papers that appeared in the eleventh volume of the Mémoires de l’Académie de Sciences de Berlin, 1755, published in 1757. Their titles were: ‘Principes généraux de l’état d’équilibre des fluides’, ‘Principes généraux du mouvement des fluides’ and ‘Continuation des recherches sur la théorie de mouvement des fluides’ (‘General principles of the state of equilibrium of fluids’, ‘General principles of motion of fluids’, ‘Sequel to the researches on the motion of fluids’). The phased continuity of these titles, the fact that they follow a careful unity of expression and method, and the clarity of the argument all indicate that they were the result of settled reflection upon which Euler wished to establish the basis of the new theory of fluid mechanics. The ideas he expresses are not completely new with him, as he had already written forerunners to some of the works, particularly the ‘Principia motus fluidorum’ containing the nucleus of the theory, and which had been read in the Berlin Academy in 1752, although it was published in volume VI of the Novi comentarii academiae scientarum petropolitanae, in the year 1756/1757, but which appeared in 1761, i.e., after the three Memoirs.1 The comparison of the contents of this work with the three previous ones enables us to understand the evolution of Euler’s thought.2 The general principles upon which Euler bases himself are the Newtonian laws expressed in differential form, the complete acceptance of the concept of force, the use of pressure as force per surface unit, and the use of clearly defined systems of Cartesian coordinates. All are expressed with an absolute conceptual clarity, and with admirable accuracy in the formulation of the equations, so much so that, although some of the concepts that Euler deploys had already been underlined or used by previous treatise writers, the redefining, concision and accuracy to which he submits them greatly surpasses all his predecessors. Just as 1

Cf. Truesdell, ‘Rat. Fluid Mech.-12’, p. LXII. The date he quotes is the 31 August of this year. The source is Eneström. Euler quotes this work in the monographs. Concerning this see the second one in §.17 and §.29. 2 In translating Euler’s works from Latin or French into English, we have taken as reference the translation made by Truesdell in the ‘Rat. Fluid Mech.-12’.

401

402

THE GENESIS OF FLUID MECHANICS, 1640–1780

in solid mechanics, many of his formulations come down to us almost without any alteration; and what is more, some of his discoveries have been attributed to other authors.3 In our attempts to follow the theoretical revolution of Euler, we begin with the ‘Principia motus fluidorum’, following with the other three works. As with d’Alembert, we shall limit ourselves only to the more relevant matters, as a detailed study would go significantly beyond the goals we have set ourselves. Finally, a note of a general nature: d’Alembert, as we have seen in the previous chapter, arrived at the constitutive equations of motion as a consequence of the study of a particular problem, which was the search for a new theory for resistance. Euler, by contrast, attacked the problems concerning fluids in a general and very pure way, without reference to any specific application. Principia motus fluidorum

It is in this work that the ideas of Euler on how to deal with the movement of fluids appear for the first time with clarity, although its scope is limited to noncompressible fluids. The work is divided into two parts: the first refers to the conditions of existence of motion and the second to the motion resulting when forces are applied. In both parts he begins with the assumption of twodimensional movement, then proceeding to three-dimensional motion. There is no qualitative difference between one and the other, but only one of complexity of the calculations and formulas. The first question Euler asks is how a fluid is to be understood, because the answer to this question depends on how we formulate the conditions of existence of motion, and how we distinguish possible and impossible motions: To this end we must find what characteristic is appropriate to possible motions, separating them from the impossible ones. When this is done, we shall have to determine which one of all possible motions in a certain case ought actually to occur. Plainly we must then turn to the forces which act upon the water, so that the motion appropriate to them may be determined from the principles of mechanics. [§.5]4

3

Specifically, the equations for perfect, non-compressible fluids continue to be used even today. On the other hand, the fluid mechanics equations are formulated nowadays with respect to fixed axis, called Eulerian, or in moving axes fixed to the actual particle, which we call Lagrangian, even when they are also due to Euler. 4 Inasmuch as the contrary is not stated, the quotes between inverted commas refer to the ‘Principia motus fluidorum’.

THE THEORETICAL CONSTRUCTIONS (II)

403

The conditions of fluidity that he uses are contiguity and impenetrability, without any reference to whether the fluid is constituted by corpuscles or another type of particle. He supposes the fluid to be a continuous material, impenetrable and unable to be segregated, which is in accord with the hypothesis of the noncompressibility. Here there is a convergence with Clairaut and d’Alembert, in the sense of escaping from possible physical reality, which all understand as being corpuscular, in order to adopt a continuous mathematical form that persists up to the present day. In the light of the methods of calculation available at the time, the hypothesis of a continuous medium allows differential analysis, which was already well developed, to be used. It is worth mentioning how the three mathematicians distanced themselves from what they believed to be reality, namely the corpuscular nature of the fluid, to go into an imaginary construct, i.e., a continuous fluid. Euler says: I assume the fluid to be such it is impossible for it to be forced into a lesser space, nor can its continuity be interrupted. I establish with certainly that no empty space remains in the middle of the fluid during movement, but that its continuity is conserved uninterruptedly. [§.6]

These conditions have to be established for the entire amount of the fluid and for any point of it whatsoever, and with this aim he calculates the mathematical conditions. In order to study the continuity, Euler begins by looking at twodimensional motion, that is to say motion in a plane. In this plane he takes a differential element of the fluid consisting in a rectangular triangle, and imposes the condition that the enclosed surface be constant during its temporal evolution. This is the equivalent of saying that the quantity of material contained in its interior must remain constant. Remember that d’Alembert had already imposed this condition with his requirement of the constancy of volume during motion. Let the triangle of fluid be designated as NML (Fig. 9-1) at the instant t, and which in t + dt had evolved up to N′ M′ L′ , which would not necessarily be rectangular, but which has the same initial surface. If the velocity of point L is the vector v(x,y) and the components along the axes OX and OY are designated as u(x,y) and v(x,y), the following equations will be verified:

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Y M M L' N' N

L

Fig. 9-1. Triangle evolution

du =

dv =

∂u ∂u dx + dy ∂y ∂x

∂v ∂v dx + dy ∂y ∂x

[9.1]

[9.2]

He supposes them to be exact differentials, therefore the equality of the crossderivatives must be established:

∂ 2v ∂ 2u = ∂y∂x ∂x∂y

[9.3]

∂ 2u ∂ 2v = ∂y∂x ∂x∂y

[9.4]

With the help of the equations [9.1] and [9.2] it is possible to determine the velocities of the two vertices of the triangle, knowing the velocities of the other and its derivatives. Having chosen the vertex of the right angle L as base, he obtains5:

G K G v L = u i + vj 5

[9.5]

Hereinafter vectorial notation and the unitary vectors i and j will be used in order to simplify the presentation, although Euler wrote the components separately.

THE THEORETICAL CONSTRUCTIONS (II)

405

G ∂v ⎞ G ∂v ⎞ G ⎛ ⎛ v N = ⎜ v + dx ⎟i + ⎜ v + dy ⎟ j ∂y ⎠ ∂x ⎠ ⎝ ⎝

[9.6]

G ∂u ⎞ G ∂u ⎞ G ⎛ ⎛ v M = ⎜ u + dx ⎟i + ⎜ u + dy ⎟ j ∂y ⎠ ∂x ⎠ ⎝ ⎝

[9.7]

When a time dt has elapsed, each vertex will travel from its position r to r + vdt, both r and v being vectors. The result for each one of them will be:

G G G G G rL = xi + yj ⇒ ( x + udt )i + ( y + vdt ) j

[9.8]

G G ∂v ⎞ ⎤ G ⎡ ⎛ ∂v ⎞ ⎤ G G ⎡ ⎛ rN = ( x + dx)i + yj ⇒ ⎢ x + dx + ⎜ u + dx ⎟ dt ⎥ i + ⎢ y + ⎜ v + dy ⎟ dt ⎥ j ∂x ⎠ ∂y ⎝

⎣

⎦

⎣

⎝

⎠ ⎦

[9.9]

G G ∂u ⎞ ⎤ G ⎡ ⎛ ∂v ⎞ ⎤ G G ⎡ ⎛ rM = xi + ( y + dy ) j ⇒ ⎢ x + ⎜ u + dx ⎟ dt ⎥ i + ⎢ y + dy + ⎜ v + dy ⎟ dt ⎥ j [9.10] ∂x ⎠ ⎦ ⎣ ⎝ ⎝ ∂y ⎠ ⎦ ⎣

Where the initial coordinates and final coordinates of each vertex are expressed. The area of the initial triangle was ½dxdy, while that of the displaced one, after an involved calculation [§.19], turns out to be:

⎡ ⎛ ∂u ∂v ⎞ ⎛ ∂u ∂v ∂v ∂u ⎞ 2 ⎤ 1 dxdy ⎢1 + ⎜ + ⎟dt + ⎜ − ⎟dt ⎥ 2 ⎝ ∂x ∂y ∂x ∂y ⎠ ⎦ ⎣ ⎝ ∂x ∂y ⎠

[9.11]

Which made equal with the first, leads to the following expression:

⎛ ∂u ∂v ⎞ ⎛ ∂u ∂v ∂v ∂u ⎞ − ⎜ + ⎟+⎜ ⎟dt ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂y ∂x ∂y ⎠

[9.12]

Neglecting the terms of a higher order it simplifies to:

⎛ ∂u ∂v ⎞ ⎜ + ⎟ ⎝ ∂x ∂y ⎠

[9.13]

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THE GENESIS OF FLUID MECHANICS, 1640–1780

To which Euler adds that ‘unless this condition holds, the motion of the fluid cannot take place’ [§.20]. This last equation had already been found by d’Alembert, starting, just as Euler did, from kinematic conditions, although d’Alembert had also extended it to compressible fluids, something Euler would not do until his monographs of 1755. After the development of the plane movement he goes on to the threedimensional case [§.21]. The procedure is the same, now supposing that the initial element of fluid is a rectangular tetrahedron instead of a triangle. The mathematical calculation is considerably more involved and bothersome, as the factor adding a new dimension causes not only one more equation to appear, but the corresponding cross-equations. The final result is the following expression [§.35]:

∂u ∂v ∂w ⎛ ∂ (u, v ) ∂ ( v, w) ∂ (u, w) ⎞ ∂ ( u , v , w) 2 + + +⎜ + + dt = 0 [9.14] ⎟dt + ∂x ∂y ∂z ⎝ ∂ ( x, y ) ∂ ( y , z ) ∂ ( x, z ) ⎠ ∂ ( x, y , z ) We note6 that there now appear terms of the order dt2, which is one order more than in the plane case. Neglecting these just as much as those of an inferior order, and justifying this on the grounds that they are differential quantities, he ends with the equation:

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

[9.15]

With current calculus tools, if the field of velocities v(x,t) is assimilated to a vector field, which is certainly what it is, the last equation, nowadays called a

6

The following algorithm is used in the formula that follows in order to simplify its writing:

∂u ∂ (u, v ) ∂x = ∂ ( x, y ) ∂v ∂x

∂u ∂y ∂v ∂y

∂u ∂x ∂ (u, v, w) ∂v = ∂ ( x, y , z ) ∂x ∂w ∂x

∂u ∂y ∂v ∂y ∂w ∂y

∂u ∂z ∂v ∂z ∂w ∂z

THE THEORETICAL CONSTRUCTIONS (II)

407

‘continuity’ equation, could be written with the help of a divergence7 operator such as:

G G div v = ∇·v = 0

[9.16]

Where div is the aforementioned operator which is the scalar product of the ‘nabla’ operator and the velocity, Up to now, Euler has used kinematic arguments. The next step is to introduce dynamic conditions, that is to say forces, and, as a result, accelerations. In this respect, he makes the following reflection when beginning the second part: Once exposed these things that pertain only to possible motion, we now also investigate the nature of motion that can truly subsist in the fluid. [§.39]

This is the same as saying that the conditions [9.13] and [9.15] are necessary, but not sufficient. The process that follows begins by determining the accelerations, starting from the kinematics of motion, in order to continue to introduce the forces of pressure and gravity. At the end he generalises the results for other classes of mass forces.8 He does this entire first in its two-dimensional aspect, and then afterwards in its three-dimensional aspect. In this approach he makes a clear separation between the kinematic aspects and the dynamic ones, which, according to Truesdell,9 occurs for the first time. However, this separation is possible exclusively because of the incompressibility of the fluid, and in the case of compressible fluids the kinematics must be complemented by the mass properties as can be seen in the monographs of 1755. 7

In a three-dimensional vector field, of which the two-dimensional is a particular case, divergence

G ∂f is defined as: div f = ∂f x + y + ∂f z ∂x

∂y

∂z

The divergence is a scalar magnitude. A physical interpretation of it can be given as the tendency a particle would have of concentrating or diverging when moving throughout the field following its force lines. In the case that the field was of velocities, divergence would measure the tendency of the fluid to vary in volume, in such a way that the volume is invariable when the divergence is G zero;G which is precisely the case we are dealing with. It is customary to write ∇· f instead of div f , where ∇ is the nabla operator which is defined as:

∇=

∂ G ∂ G ∂ G i + j+ k ∂x ∂y ∂z

This operator is applicable to scalar and vectors and we shall make use of it from henceforth. Mass forces are understood to be those whose magnitude is proportional to the mass of the particle upon which it acts. 9 Cf. ‘Rat. Fluid Mech. -12’, p. LXXI. 8

408

THE GENESIS OF FLUID MECHANICS, 1640–1780

In order to find the accelerations of any particle whose velocity components are u and v, he takes the variation of these when they move in a time dt. Deriving with respect to time and space he obtains the result:

du =

∂u ∂u ∂u dx + dy + dt ∂t ∂x ∂y

[9.17]

dv =

∂v ∂v ∂v dx + dy + dt ∂t ∂x ∂y

[9.18]

As the particle in question moves precisely with the velocities u and v, the displacement will be dx = udt and dy = vdt, which introduced into the previous equations lead to:

du ∂u ∂u ∂u =u +v + dt ∂x ∂y ∂t

[9.19]

dv ∂v ∂v ∂v =u +v + dt ∂x ∂y ∂t

[9.20]

Which are the ‘accelerating forces’ along the axes OX and OY, ‘by which the forces that are acting on particle of water must be equal’ [§.41]10 and which he goes on to make equal to the forces acting upon it. Among the possible acting forces he enumerates three: gravity, friction and pressure. Of the first he says its effect,

10

In the text of Euler it says precisely: Vis acceleratirx secundum AL = 2(Lu + lv + L) Vis acceleratirx secundum AB = 2(Mu + mv + M)

that correspond to the formulas above expressed with the exception of factor 2, whose reason to exist he radicates in the peculiar system of units which Euler employs, in which the value of the acceleration of gravity is ½. See his work ‘Découverte d’un nouveau principe de méchanique’ Mém. Acad. Berlin VI (1750), where he expounds for the first time the Newtonian equation of the second principle in a differential form to the time , in the following manner: 2MX = P; 2MY = Q; 2MZ = R; The justification of this value is found in the ‘Théorie plus complete des machines qui sont mises en mouvement par la reaction de l’eau’ that appears in Vol. X of the Mém. Acad. Berlin (1754) He repeats the ‘2’ again now. We shall ignore it. See also Truesdell, ‘Rat. Fluid mech-12’, p. XLIII.

THE THEORETICAL CONSTRUCTIONS (II)

409

[I]f the plane of motion is horizontal, is to be taken as zero. But if instead the plane is inclined, and the axis OY follows that slope, due to gravity a constant accelerating force of magnitude α arises. [§.42]11

He leaves friction to one side for the moment, and concentrates on pressure: Moreover, the pressure must be brought into the calculation. This pressure is the reciprocal action of the water particles upon each other. Each particle is pressed on every side by its neighbours, and as this pressure is not equal everywhere, to this extent motion is communicated to the particles. In all places the water will simply find itself in a certain state of compression similar to that of quiet water at a certain height finds itself. Therefore, this height (at which quiet water is found to be in a state similar to compression) can be conveniently employed to represent the pressure at an arbitrary point l of the fluid. Therefore let that height (or depth) expressing the state of compression at l be p; a certain function of the coordinates x and y, and if the pressure at l varies also with the time, then the time will also enter into the function p. [§.43]

This paragraph is important. On one hand Euler defines the concept of pressure as a force over a unit surface, although he still does not do this with total clarity. It is not that this is new, as it can be detected in d’Alembert, though he does not explain it so clearly, and other authors, such as Johann Bernoulli, liken the pressure to a total force upon a section of a fluid. As regards measuring the magnitude of pressure, Euler identifies it with the height of water column, which is not new either. We recall that Daniel Bernoulli introduced a water height manometer in his experiments, and that Pitot based his experiments on these apparatus. There are inklings of this idea even in Newton, but in Euler the idea of acquires the value of the measurement, and not of an equivalent force, a very important difference. Nevertheless, the concept will become even clearer in the succeeding monographs In order to introduce the pressure equation he defines a differential element of fluid, which will now be a rectangle instead of triangle, designated as NLMO in Fig. 9-2, and which he imagines inside a fluid field whose pressure is the function of time and position. If the pressure on a vertex of this element is p(x,y), then on the others it will be:

11

This value depends on the choice of the system of axis.

410

THE GENESIS OF FLUID MECHANICS, 1640–1780

Y N

O

L

M

Fig. 9-2. Pressure forces

L: p

N:

p+

∂p dy ∂y

O:

M:

p+

∂p dx ∂x

[9.21]

p+

∂p ∂p dx + dy ∂y ∂x

[9.22]

Therefore, the result of the forces produced by the pressures on the sides of the rectangle along both axis, will be: OX Axis:

−

∂p dxdy ∂x

[9.23]

OY Axis:

−

∂p dxdy ∂y

[9.24]

These forces, plus the gravity along the axis OX, will be the ‘accelerating forces’, and by making them equal to the accelerations given in equations [9.19] and [9.20] the following two equations are obtained12:

g−

1 ∂p ∂u ∂u ∂u =u +v + ∂x ∂y ∂t ρ ∂x

[9.25]

12 In the formulas that follows the density, ρ, is introduced as the deviser of the pressure, the aim being to make the equations coherent.

THE THEORETICAL CONSTRUCTIONS (II)

−

1 ∂p ∂v ∂v ∂v =u +v + ∂x ∂y ∂t ρ ∂x

411

[9.26]

Moreover, the variation of pressure with time and space can be written as:

dp =

∂p ∂p ∂p dx + dy + dt ∂t ∂x ∂y

[9.27]

Introducing the values of the two previous equations into this one, we arrive at the following expression for dp: dp

ρ

⎛ ∂u

= gdx − ⎜ u

⎝ ∂x

+v

∂u ∂y

+

∂u ⎞

∂v ∂v ⎞ 1 ∂p ⎛ ∂v dx − ⎜ u + v + ⎟ dy + dt ⎟ ρ ∂t ∂t ⎠ ∂y ∂t ⎠ ⎝ ∂x

[9.28]

Which he says that it must be integrable. He states that ‘the term g is per se integrable and nothing is defined for ∂p/∂t, and by nature the differentials need to be exact’ [§.46]. Therefore, it will be necessary to comply with the equality of the cross-derivatives between the other two terms:

∂ ⎛ ∂u ∂u ∂u ⎞ ∂ ⎛ ∂v ∂v ∂v ⎞ ⎜⎜ u +v + ⎟⎟ = ⎜⎜ u + v + ⎟⎟ ∂y ⎝ ∂x ∂y ∂t ⎠ ∂x ⎝ ∂x ∂y ∂t ⎠

[9.29]

Which, after the corresponding manipulations become the following:

⎛ ∂u ∂v ∂ ∂ ∂ ⎞⎛ ∂u ∂v ⎞ ⎜⎜ + + u + v + ⎟⎟⎜⎜ − ⎟⎟ = 0 ∂x ∂y ∂t ⎠⎝ ∂y ∂x ⎠ ⎝ ∂x ∂y

[9.30]

That says [§.47] that it is completely satisfied by:

∂u ∂v = ∂y ∂x

[9.31]

This is an important question, as it is true that the fluids which fulfil this last condition will also verify the previous one, but the opposite is not true. This means that the condition [9.31] is sufficient, but not necessary. Euler limits the

412

THE GENESIS OF FLUID MECHANICS, 1640–1780

possible motions to a single category, which later would be called ‘irrotational motions’. Later on, in successive works he rectifies having considered only this solution. D’Alembert had also found himself in a similar situation.13 Truesdell insisted that Euler’s mistake was due to d’Alembert’s influence,14 although one can easily interpret it as Euler having chosen the easiest and most obvious solution of the equation. Before continuing, we must introduce a specification which Euler fails to mention. In equation [9.30], the sum ∂u/∂x + ∂v/∂y is zero, as had already been found in equation [9.13] as a result of the continuity, which simplifies the formulation. We take note that having started from the pressure as the only acting force, Euler arrived at some relationships in which this parameter disappeared in favour of the velocities. Now, within the irrotationality hypothesis the pressure returns, for which he introduces the results found in the expression containing the pressure, ‘hence now we shall be able to ascertain the pressure p itself, which is absolutely necessary for the perfect determination of the motion of the fluid’ [§.49]. With the condition [9.31], the pressure equation [9.28] becomes:

dp

ρ

= gdx − udu − vdv −

∂u ∂v dx − dy ∂t ∂t

[9.32]

The condition that udx + vdy is an exact differential allows him to introduce the function S, which is the potential of the velocities.15

dS = udx + vdy

[9.33]

And after some transformations he arrives at:

dp

ρ

= gdx − udu − vdv − d

∂S ∂t

[9.34]

This is already an integrable equation whose result is:

13

Cf. Essai d’une nouvelle théorie de la résistance des fluides, §. 48–49. Although what d’Alembert really did was to demonstrate if the expression [9.31] was substituted by another of the type: ∂v ∂x = ∂u ∂y + λ this will only fulfill the conditions of potentiality if λ = 0. 14 Cf. ‘Rat. Fluid. Mech.-12’, p. LXXIII. K 15 That is to say, it verifies v = ∇S .

THE THEORETICAL CONSTRUCTIONS (II)

1 = gx − (u 2 + v 2 ) − U + Cte 2 ρ p

413

[9.35]

As the total velocity at a point is V = u 2 + v 2 , what he obtains is Bernoulli’s equation for a non-stationary motion. We shall come back to this potential function S once we have analysed the three-dimensional case. If we are dealing with motion in three dimensions, the arguments will follow the same lines although, just as in the case of continuity, with a greater degree of complexity. There will be a third component of the velocity w, corresponding to the projection along the OZ axis, and on establishing the accelerations we shall have three equations that replace the two [9.19 and 9.20].

du ∂u ∂u ∂u ∂u =u +v +w + dt ∂x ∂y ∂z ∂t

[9.36]

dv ∂v ∂v ∂v ∂w =u +v +w + dt ∂x ∂y ∂z ∂t

[9.37]

dw ∂w ∂w ∂w ∂w =u +v +w + dt ∂x ∂y ∂z ∂t

[9.38]

These equations may be written with vectorial notation as:

G G G G ∂v a = v ·∇v + ∂t

[9.39]

In which the nabla operator is used as the generator of the velocity gradient. It would be even simpler to use the concept of the substantial derivative, which would result in16: 16

The gradient function is applied to a scalar field or to each component of a vectorial field. In the first of the cases if the field is represented by φ the gradient would be the vector:

∇ϕ = gradϕ =

∂ϕ G ∂ϕ G ∂ϕ G i + j+ k ∂x ∂y ∂z

The direction of the gradient is the variation of the property φ when it moves through the field in such a way that dϕ = ∇ϕ ·dr indicates the variation of φ when the position changes the distance dr. As regards the substantial derivative of the property φ this is defined as follows:

414

THE GENESIS OF FLUID MECHANICS, 1640–1780

G G Dv a= Dt

[9.40]

The condition that the expression dp/ρ is an exact differential, leads to three equalities among the cross-derivatives, which will be the equivalent of condition [9.29]. When he develops them he obtains the following three equations [§.59] equivalent to equation [9.30] of the two-dimensional case: ∂ ∂ ∂ ∂ ⎞ ⎛ ∂u ∂v ⎞ ∂u ∂w ∂v ∂w ⎛ ∂u ∂v ⎜ ∂x + ∂y + u ∂x + v ∂y + w ∂z + ∂t ⎟ ⎜ ∂y − ∂x ⎟ + ∂z ∂y − ∂z ∂x = 0 ⎝ ⎠⎝ ⎠

[9.41]

∂ ∂ ∂ ∂ ⎞ ⎛ ∂v ∂w ⎞ ∂v ∂u ∂w ∂u ⎛ ∂v ∂w ⎜ ∂y + ∂z + u ∂x + v ∂y + w ∂z + ∂t ⎟ ⎜ ∂z − ∂y ⎟ + ∂x ∂z − ∂x ∂y = 0 ⎝ ⎠⎝ ⎠

[9.42]

∂ ∂ ∂ ∂ ⎞ ⎛ ∂u ∂w ⎞ ∂u ∂w ∂w ∂v ⎛ ∂u ∂w ⎜ ∂x + ∂x + u ∂x + v ∂y + w ∂z + ∂t ⎟ ⎜ ∂z − ∂x ⎟ + ∂y ∂z − ∂y ∂x = 0 ⎠ ⎝ ⎠⎝

[9.43]

A system which is sufficiently established with the following three values:

∂u ∂v − = 0; ∂y ∂x

∂u ∂w ∂v ∂w − = 0; − =0 ∂z ∂x ∂z ∂y

[9.44]

Which correspond to an irrotational motion. The presentation of the equations obtained by Euler is simplified using modern vectorial notation. Firstly, the ‘vorticity’ is defined as the curl17 of the velocity: Dϕ ∂ϕ G = + v ⋅ ∇ϕ Dt ∂t The significance is the variation of the property φ of a particle when this moves following a trajectory. That is to say with axis fixed to the particle. 17 The curl of a vector field is another vector field defined as:

G G curl v = ∇ × v =

G i

G j

G k

∂ ∂x

∂ ∂y

∂ ∂z

u

v

w

THE THEORETICAL CONSTRUCTIONS (II)

G

G

415

G

ω = curl v = ∇ × v

[9.45]

Which is a vector whose three components are precisely the first members of the antepenultimate equation [9.44]. With the help of vorticity, equations [9.41]– [9.43] can be written as the much simpler equation:

G G G Dω = (ω ⋅ ∇)v Dt

[9.46]

Examining the foregoing confirms that it satisfied the cases of ω = 0, that is, an irrotational motion, but it does not do so necessarily, as these cases are only one class of the possible motions that satisfies equation [9.46], but obviously there are more possible motions. For the two-dimensional motion, the previous equation becomes:

G Dω =0 Dt

[9.47]

G

G

That corresponds to equation [9.30]. The disappearance of the term (ω ⋅ ∇)v is easy to explain as the curl vector is perpendicular to the plane of motion, and this plane contains the gradient vector, therefore the scalar product of both will be zero18

In order to capture the meaning of this vector field we suppose that small parallelepipeds move over the vector field. Now then, the curl will indicate the tendency to rotate upon themselves that these elements have. In the event that it as zero, they would shift without turning, which is designated as irrotational. 18 These equations can be deducted starting from the equation of the momentum:

G G Dv 1 = − ∇p + f Dt ρ

[1]

The first member can be written as:

G G G ⎛ v2 ⎞ G G Dv ∂v K G ∂v + ∇⎜⎜ ⎟⎟ − v × (∇ × v ) + ∇ = = ( v )v ∂t Dt ∂t 2 ⎝ ⎠

[2]

Introducing this in equation [1], together with the definition of vorticity, and applying the curl function to both members we end up with:

∇×

G G G G ∂v ∇v 2 ∇p +∇× − ∇ × ( v × ω ) = −∇ × +∇× f ρ ∂t 2

[3]

416

THE GENESIS OF FLUID MECHANICS, 1640–1780

As regards pressures, he repeats the process adding a new variable, thus arriving at the expression:

dp

ρ

= gdx − udu − vdv − wdw −

∂u ∂v ∂w dx − dy − dz ∂t ∂t ∂t

[9.48]

This equation substitutes the two-dimensional one deduced previously [9.32]. An interesting detail, analysed by Euler, occurs where the field of velocities is integrable, S being its integral, i.e., there is a potential function.19 Simplifying his transformations, the velocities will be:

u=

∂S ; ∂x

v=

∂S ; ∂y

w=

∂S ∂z

[9.49]

Therefore, it will also be established that:

d d ⎛ ∂S ∂S ∂S ⎞ dS (udx + vdy + wdz ) = ⎜⎜ dx + dy + dz ⎟ = dt dt ⎝ ∂x ∂y ∂z ⎟⎠ dt

[9.50]

Resulting in the expression for pressure:

p

1 dS = C − gz − V 2 − ρ 2 dt

[9.51]

In the assumption that the forces are derived from potential and that the density is constant, the two add-ins of the left are cancelled out, as is the second on the right. Therefore, recalling the definition of vorticity, we arrive at:

G G G ∂ω = ∇ × (v × ω ) ∂t

[4]

Going on to the substantial derivative we end with:

G G G G G Dω = (ω ⋅ ∇)v − ω∇ ⋅ v Dt

[5]

G

Which like the density is constant ∇v = 0, giving the [9.46] results. We note that Truesdell [Rat. Fluid Mech.-12, p. LXXII] points out this transformation. However, he presents the previous equation [5] as being equivalent to equations [9.41] and [9.42] of Euler, which would only be true if the density were not constant. 19 Euler takes two addends, one of which he calls U, which is variable with the time. We shall skip this step.

THE THEORETICAL CONSTRUCTIONS (II)

417

which is the generalisation of the equation of Bernoulli for non-stationary motions. On the other hand, he recalls the existence condition of the motion, expressed in equation [9.15], in which he introduces the velocities derived from the potential S, as presented in equation [9.49]. The resulting equation is:

∂2S ∂2S ∂2S + + =0 ∂x 2 ∂y 2 ∂z 2

[9.52]

This is an equation which is called the potential equation or ‘Laplacian’ and which is usually written as ∇ 2 S = 0 or ∆S = 0 . As a complement to this last formulation, Euler tries to find some kind of solution [§.68-ff] supposing that the function S takes the form:

S = ( Ax + By + Cz ) n

[9.53]

He applies the previous condition [9.52] to this one and he arrives at the following relation between the parameters:

n(n − 1)( A 2 + B 2 + C 2 )( Ax + By + Cz ) n−2 = 0

[9.54]

He devotes a lot of attention to these functions, in particular to the solution corresponding to n = 1, where he finds that it is the equivalent to a shift in space at constant velocity, as can be easily deduced by applying equation [9.49] to the function S = Ax + By + Cz; and where the fluid behaves like a rigid solid. Following this thread Euler ponders whether ‘it is legitimate to suspect in other cases that the motion of the fluid can also be assimilated to the motion of a solid body, whether rotational or with any other anomaly’ [§.75]. With this aim, he launches himself into the search of relations having the velocities that makes possible the motion of the fluid as a solid rotation, A situation complementary to that of the shift. After a series of calculations, he finds that for this type of motion to be possible, the matrix ∂vi / ∂x j must be anti-symmetrical, that is:

∂u ∂v ∂z = = =0 ∂x ∂y ∂w

[9.55]

418

THE GENESIS OF FLUID MECHANICS, 1640–1780

∂v ∂u =− ; ∂y ∂x

∂u ∂w =− ; ∂z ∂x

∂v ∂w =− ; ∂z ∂y

[9.56]

Now the last three equalities contradict the condition found for the existence of fluid motion, which was [9.44], which indicates that this motion will not be compatible with these conditions unless the velocities are constant. In response, he ends by saying: ‘thus, it is obvious that it is only in this case [vi = cte] where the motion of a fluid can be assimilated to that of a solid body’ [§.77] The interest of this statement lies in the fact that the solid rotation is an example of motions that are not covered by the conditions of existence.20 The next step he takes is to extend the forces to ones other than weight and pressure, these being the only ones he has handled up to now. In order to do this, he extends the theory to assumptions where other external forces exist. Instead of using a new approach to the equations, what Euler does do is to introduce an acceleration potential T, so that:

T=

1 2 ∂S (u + v 2 + w 2 ) + ∂t 2

[9.57]

If the new external forces are of the type Qdx + qdy + φdz, the expression for pressure will result in:

∫

p = C + (Qdx + qdy + dz ) − T ρ

[9.58]

In concluding our treatment of the ‘Principia motus fluidorum’, the last point of significance is the specification he makes for fluids moving in ducts, declaring that ‘everything which has hitherto been said concerning the motion of a fluid through tubes is easily derived from these principles’ [§.87]. The final equation he arrives at is:

dV S2 p = C + (Qdx + qdy + dz ) − 02 V02 − 2 0 ρ dt S

∫

20

∫

S0 ds S

[9.59]

Truesdell (‘Rat Fluid Mech-12’, note 2, p. LXXIV) quotes Professor Kuert’s remark that makes plain his puzzlement that neither Euler nor d’Alembert found counter examples to their theories in these motions. That a solid rotation is not irrotational is very easy to see. A irrotational motion requires that a particle does not turn in its movement, which in turn requires that the law of velocities be inverse to the distance for the center of rotation, that is to say of the v = k/r type. Now, in a solid rotation the velocity is proportional to the distance, that is, v = kr.

THE THEORETICAL CONSTRUCTIONS (II)

419

Where S0 and S(s) are the cross sections of the pipe and V0 the velocity in the cross section S0 taken as a reference. This equation is an extension of the one given by d’Alembert, and of course turns out to be the equation of Bernoulli, an equation that begins to occupy second place with respect to the general hypotheses of hydrodynamics. Finally, it is useful to compare Euler’s method with that of d’Alembert. They clearly have differing approaches to the problem. The latter begins by obtaining an equation that links the pressures21 to the velocities for a pipe with a very narrow current tube; next, using kinematic and dynamic considerations, he obtains the field of velocities defined by differential equations in which only the velocities intervene, and whose solution depends on the shape of the body. Once these are solved, the pressures at each point in the fluid can be deduced. By contrast, Euler first introduces the pressures as forces, and with these, together with the equations of general dynamics, he establishes some relations among the velocities alone, just like d’Alembert. He goes back to introduce the pressures at the end of the Bernoulli’s equations, which will give the pressures at specific points once the velocities are obtained. The elements brought into play are the same, although in a different order, and with a different methodology, in which Euler deals with the dynamic concepts with greater clarity. Apart from this, Euler tackles the three-dimensional problem, while d’Alembert limits himself to this last case, be it in the plane or axisymmetric case. General principles of the state of equilibrium of the fluids

This first of the three monographs of the series is dedicated to hydrostatics. Euler begins by a declaration of his aims: Here I propose to develop the principles upon which all hydrostatics, or the science of the equilibrium in fluids, is founded. … I include in my investigations not only fluids that have the same density in all their parts … but also those fluids composed of particles of different density. … Moreover, I shall not limit my investigations to the single force of gravity, but will extend them to any forces. [§.1]22

As a consequence of the general nature of the research, the earlier explanations ‘are only a very particular case of those which I am going to establish here’ [§.2] he notes however

21

See Chapter 8, ‘Body in flowing currents’ of this book. The quotes between brackets follow the monograph ‘Principes généraux de l’état d’equilibre des fluides’. 22

420

THE GENESIS OF FLUID MECHANICS, 1640–1780 The reproaches often justly directed at those who have undertaken to bring the researches of others to a greater generality. I agree that too great a generality often obscures rather than enlightens. … When the generalizations are subject to this inconvenience, it is very true that it would be infinitely better to abstain from them entirely, and limit one’s investigations to specific cases. [§.3]

After this fascinating comment he continues: [T]he generality which I undertake, rather than dazzling our lights, will reveal the true laws of Nature more to us in all their brilliance, and we shall find therein even stronger reasons to admire her beauty and simplicity. [§.4]

The basic point or foundation stone upon which the remainder of his arguments rests involves the nature of fluidity, as the laws of equilibrium of fluids should not differ from those of solid bodies more than solids and fluids differ among themselves. The science of mechanics is unique, and the laws are the same for both solids and liquids: the differences lie in the nature of the bodies. We see that the doubts and hesitations of d’Alembert have already disappeared. Euler begins by defining fluidity as ‘the first idea, the foundation for all those arguments that we need to make in order to arrive at our goal’ [§.5]. But, contrary to the ideas that he had followed in the ‘Principia motus fluidorum’, where he established as his starting point the impenetrability and inseparability of fluid bodies, he now considers that for an isolated fluid mass, the essential property is that equilibrium cannot be found ‘unless it is subject at all points of its surface to forces equal and perpendicular to the surface’ [§.9]. The nearest precedent is d’Alembert, who had established the ‘equality of pressure in all directions’.23 Regarding the possible internal constitution of the fluids, he only shows that it cannot consist in a set of loose particles, arguing that if this were so, the system would lack stability. Specifically he declares: Hence it is clear that fluidity cannot be explained by a swarm of solid corpuscles, even when one supposes them to be infinitely small, completely separate from each other, and infinitely great in number, and it seems still very doubtful if an internal motion would be capable of making good this defect. [§.8]

That is to say, Euler defends the fluids as a continuum, and from his words one can deduce that he does so not only methodologically, but also as an expression of reality. 23

Essai, p.xxviii.

THE THEORETICAL CONSTRUCTIONS (II)

421

Just as he had done in the ‘Principia motus fluidorum’, he represents the pressure as the height of the cylinder of homogeneous fluid, which he designates by the letter p, normally used henceforth for this magnitude. The resulting force upon any surface will be the product of the pressure times the surface ‘[given] an element [of surface] ds2, which is pressed on by a force pds2 acting perpendicularly to the element’ [§.11]. He considers this to be the best way of representing the pressure, and as he declares, ‘this pressure cannot be better represented than by a certain height, which refers to gravity, with a homogenous material that will be judged the most suitable for application in this measurement’ [§.14]. This procedure indeed gives us a very intuitive way of representing pressure, and that it has been used in science since.24 That the pressure external to a domain be constant implies that it is also constant inside it. In order to demonstrate this, he supposes that the domain is divided by an immaterial diaphragm separating it into two parts (Fig. 9-3). Now that both are in equilibrium, according to his hypothesis, this requires that the pressure on this diaphragm be equal to the external pressure. As this has to be established for any diaphragm, ‘it follows that each element of the fluid mass IKki will undergo pressure in all parts from similar forces’ [§.13]. We see that d’Alembert’s condition follows on from that of Euler.

Fig. 9-3. Element equilibrium

In the absence of other forces acting on the particles of the fluid, an element of the fluid will be subjected to lateral pressure forces that will tend to compress it and reduce its size. Therefore, Euler concludes, the pressure is the last magnitude where both the elasticity or the compressibility, and the degree of heat, 24

Daily case is the measuring of atmospheric pressure in millimeters of mercury. This has to do with the measurement apparatus: the barometer of a column of mercury, still in use up to the present-day.

422

THE GENESIS OF FLUID MECHANICS, 1640–1780

terminates, as one or the other is made manifest in the amount of pressure measured. He deals with these concepts several times in his monograph, and they are very clearly expressed when later on he declares: ‘let r be the degree of heat … and let the elasticity p be a composite function of the density q and the heat r; in this case p = αqr is obtained where it is easy to determine α by the absolute measurements’ [§.40]. Here Euler is expressing the equation of perfect gases, if we identify what he calls the degree of heat with the absolute temperature, and the other two variables with the usual ones of pressure and density.25 What he has dealt with up to now has been solely the equilibrium of an isolated fluid, without any external force. When this is not the case, so that any type of force exists, then the pressure will not be constant, but will vary from point to point according to the nature and magnitude of the forces. He sets out the problem in these terms: The forces, for which all the elements of the fluid are sought, together with the relation subsisting at each point between the density and the elasticity of the fluid: find the forces acting on all points of the fluid mass, for it to be in equilibrium. [§.21]

In order to obtain the basic equations, he takes a differential element subject to the field of mass forces26 whose components will be P, Q and R along the three axes. If the dimensions of the element are dx, dy, dz, their mass will be ρdxdydz, and the resulting ‘accelerating forces’ corresponding to each axis will be: OX Axis:

ρPdxdydz

[9.60]

OY Axis:

ρRdxdydz

[9.61]

OZ Axis:

ρQdxdydz

[9.62]

As regards the pressure, it will be a differentiable function p(x,y,z):

25

Guillaume Amontons worked on this subject at the beginning of the eighteenth century, measuring temperatures by means of pressure sin what he calls a gas thermometer. See ‘La Ley de Amontons y las indagaciones sobre el aire en la Academia de Ciencias de París (1699–1710)’ (‘The Amontons Law and research on the air in the Paris Academy’). On the other part, the relation between the pressure and the density was established experimentally by Boyle. 26 In this case the components of the mass force P, Q and R will have a dimension of force/mass.

THE THEORETICAL CONSTRUCTIONS (II)

dp =

∂p ∂p ∂p dx + dy + dz ∂x ∂y ∂z

423

[9.63]

Now the difference of pressure between two opposite faces of the element will have to be equal to the accelerating force in this direction. After the corresponding operations, he arrives at the following three equations:

∂p = ρ P; ∂x

∂p = ρ Q; ∂y

∂p = ρ R; ∂z

[9.64]

Which are the general hydrostatic equations, and introduced into equation [9.63] the resulting pressure will be:

dp = ρ ( Pdx + Qdy + Rdz )

[9.65]

At this point he assumes that in order to integrate it the equality of the crossderivatives must be established, that is:

∂ ( ρP ) ∂ ( ρ Q ) = ; ∂x ∂y

∂ ( ρP ) ∂ ( ρR ) = ; ∂x ∂z

∂ ( ρQ ) ∂ ( ρR ) = ∂y ∂z

[9.66]

‘Without these conditions it is impossible that the fluid mass could be reduced to equilibrium by the extraneous forces P, Q, R’ [§.28]. These equations are an extension of those of Clairaut and d’Alembert, and more specifically an extension of those presented by the latter in three variables. However, once he has established these conditions, he did not use them to discover what type of solutions could exist, but returns instead to the differential expression of pressure [9.63], where both p and ρ are functions of the position (x, y, z). He states: [W]hen the forces P, Q, R are real, whether they constitute natural gravity or forces directed to fixed centers, and each is an arbitrary function of the distance from its center. In all these cases I notice that the formula Pdx+Qdy+Rdz expresses a real differential which results from the differentiation of a finite quantity, function of the x, y and z. [§.31]

For Euler, there are certain phenomena which considers ‘natural’ or real, and whose field of forces is limited to two classes: gravity and forces directed towards fixed centers. Although he does not mention other types, it could be

424

THE GENESIS OF FLUID MECHANICS, 1640–1780

inferred that those force fields differing from these would enter into theoretical hypotheses but not natural ones. Leaving to one side the epistemological nature of this hypothesis, it allows him to define a potential function s:

ds = Pdx + Qdy + Rdz

[9.67]

This function reduces the condition [9.63] to the simple expression:

dp = ρ ds

[9.68]

He declares that ‘therefore the three variables x, y and z, are united in the single s… . In all the other cases, … the equilibrium is impossible, and the components of such a fluid would of necessity be set into motion’ [§.33]. For these same cases, for which this equation is resolvable, the density must either be a function of the potential function, or of the pressure, or of both. That is, p(s), ρ(p) or ρ(p,s). In what follows we shall only outline his arguments. He considers two assumptions: the incompressible and the compressible, and he divides each into two in turn: homogeneous and non-homogeneous, thus arriving at four basic cases. In the first of these [§.37], he supposes that the fluid is homogeneous and incompressible, so that the density ρ is constant and the integral of [9.68] will be p = ρ(s + a), where a is a parameter. The pressure will be constant upon the level surfaces, that is when s = cte. An obvious case is when only gravity exists, where the level surfaces are horizontal planes. The second case [§.38] is when, although the fluid continues to be incompressible, the density is not uniform. In these circumstances each level surface must have the same density, that is ρ = ρ(s), whose solution would be p = ∫ρds. A particular solution of this equation is the previous one referring to gravity. The ones analyzed by Clairaut in the Théorie de la figure de la Terre come into this category. The third assumption covers cases of compressible fluids where the density is a certain function of pressure [§.39] and whose solution will be:

∫ρ

dp

=s

[9.69]

which will obviously require an integration constant. If the relation between pressure and density were proportional, the solution to this integral would be of an exponential type. A particular case in this category is air, whenever the air is of uniform temperature.

THE THEORETICAL CONSTRUCTIONS (II)

425

The fourth and last case he considers [§.40] is that in which the pressure does not only depend upon the density, but where the ‘heat’ is also variable within the fluid mass. In this case, equality cannot exist unless the temperature is the same in each layer of s = cte, and therefore the pressure is a function of the temperature and the density in the form that he designates as p = αρr. In this formula α is a constant and r is the ‘degree of heat’ (temperature). This equation currently known as the state of perfect gases, and with the constant R = ℜ / M , where ℜ is the universal constant of the perfect gases, and M the molecular mass of the gas, is expressed as:

p

ρ

= RT

[9.70]

which, introduced in equation [9.68], leads us to:

dp =

pds RT

[9.71]

an equation which, once integrated leads to the following expressions for pressure and density, yields: ds

p = p0e

∫T

;

ρ=

ρ0 RT

e

−

ds

∫T

[9.72]

After presenting these formulas he insists that: ‘everything concerning the equilibrium of fluids is derived as easily as it is naturally from our formula dp = qds, which is thus rightfully to be regarded as the unique foundation of all the theory of the equilibrium of fluids’ [§.42]. He devotes the rest of his monograph to two specific cases which he deals with at considerable length [§§.43–94]. The first is the case in which only ‘natural gravity’ exists: i.e., dp = –ρdz, in which the variable s coincides with the height z, and which is applicable to the atmosphere, and from which he will deduce the distribution of pressure with height. The second example refers to equilibrium where gravity is directed towards one or several centres, as is the case of planetary figures. It is well known that both have a direct and immediate relation with what we nowadays call the earth sciences, in consonance with the name of ‘natural’ that he gives to this type of forces. In order to solve the first of these cases, he mentions once more the relation between heat, pressure and density

426

THE GENESIS OF FLUID MECHANICS, 1640–1780

which he now writes as: p:qr = h:gc [§.49]. With the version of this equation given in equation [9.70], the solution for our atmospheric pressure is given by the following integral:

p = p0 e

−

1 dz R T

∫

[9.73]

Which can only be solved if the relation between temperature and height is known. Euler offers various hypotheses, but this relation was not known until the twentieth century. General principles of the motion of fluids

Whereas the previous monograph dealt with fluid statics, the following will deal with fluid dynamics, which he handles as if it were the next step of the same subject. Having established the principles of the equilibrium of fluids much more generally in my preceding report, … I propose to deal with the motion of fluids on the same footing, and to investigate the general principles upon which the entire science of the motion of fluids is based. [§.1]27

His goal, as he explains at several points, is to bring to light those principles by which movements can be determined, when we know what the external forces are. The entire process, as it concerns motion, must start from the initial conditions of the fluid, which he refers to as ‘the primitive state of the fluid’. It will evolve from this state, depending on the forces and the boundary through which it is allowed to flow, until it arrives at the instant under study, i.e., the present state. There are cases in which this initial state is unknown: he gives the motion of a river as an example of this circumstance [§.3], where the permanent state in which a fluid will end up is deduced from the external conditions. Nevertheless, the calculations will be the same in both cases: only the integration constants will change. The goal he proposes is ambitious and daring; and although the key lines of this study had already been given in the ‘Principia motus fluidorum’, the clarity, refinement and length now greatly exceeds that. As fundamental differences between both, he emphasises the three-dimensional treatment from the beginning, including compressible fluids together with the incompressible ones, the

27

The quotes in brackets correspond to this second monograph.

THE THEORETICAL CONSTRUCTIONS (II)

427

consideration of all types of forces, and the search for solutions differing from the irrotational ones. Euler uses a system of co-ordinates in which each point is defined by its three co-ordinates28 x, y and z in a moment t. A set of forces act upon the mass particle found at this point, and these are expressed by their three components according to the axis. The other intervening variables are pressure, density and what he calls ‘heat’, which we can interpret as temperature. In addition, he supposes that these three variables are inter-related by a law [§.2] and that the heat is known at each point and instant.29 Z

Y X Fig. 9-4. Continuity

The present-day reading of this monograph surprises us by its modernity: as Dugas outs it, it is ‘a memoir so perfect that not a line has aged’.30 In order to find the equations of motion he follows a process similar to that already developed in the ‘Principia motus fluidorum’,31 therefore we will not go into much detail. Let there be a fluid, and in this a point defined by its position r and velocity v (both vectors) in an instant t. A point near the aforementioned one will have a velocity defined as:

G G G G G v + d v = v + ∇v ⋅ d r

[9.74]

28 Which nowadays are designated precisely as Euler did so. An alternative system is the so-called Lagrangian system, which is also due to Euler, in which the axis is fixed to a particle in order to follow its evolution over time. 29 Euler lacks thermodynamic concepts, which will still take longer than a century to be introduced. 30 Cf. Historie de la Méchanique, part 3, Chap. VIII, §.6. 31 As the method is similar to the previous work, and with the aim of not repeating the calculations which would take up too much room without contributing anything new in the way of concepts, we will use vectorial calculus, although we shall emphasise specific points where it is merited.

428

THE GENESIS OF FLUID MECHANICS, 1640–1780

But as a time dt elapses, both the position of the moving particle r and dr will vary following the expression:

G G G G dr ' = (1 + ∇v ⋅ dr )dr

[9.75]

Now if we start from elemental parallelepipeds (Fig. 9-4) whose dimensions are dx, dy and dz, when we apply the previous formula these will become:

⎛ ∂v ⎞ ⎜1 + dt ⎟dy ; ⎝ ∂y ⎠

⎛ ∂u ⎞ ⎜1 + dt ⎟dx ; ∂x ⎠ ⎝

⎛ ∂w ⎞ dt ⎟dz ⎜1 + ∂z ⎠ ⎝

[9.76]

And the initial volume dxdydz will be:

⎛ ∂u ∂v ∂w ⎞ dt ⎟dxdydz ⎜1 + dt + dt + ∂x ∂y ∂z ⎠ ⎝

[9.77]

If the fluid were incompressible, as he supposed in the ‘Principia’, the initial volume and the final one would be equal, which would lead him to the equation G already obtained, div v = 0 , even though he has now started from a parallelepiped instead of a tetrahedron. 32 But if this is not so, and the density is also variable, then it is the mass that will have to be kept constant, that is to say ρdV. In this case the variation of the density is expressed by the equation:

G

G

G

ρ ( r + dr , t + dt ) = ρ ( r , t ) +

G G Dρ ∂ρ dt + v ⋅ ∇ρdt = ρ ( r , t ) + dt ∂t Dt

[9.78]

He arrives at the following expression by applying the constancy of the product of the density multiplied by the volume.

∂ρ ∂ ( ρu ) ∂ ( ρv ) ∂ ( ρw) + + + =0 ∂t ∂x ∂y ∂z

[9.79]

With the help of vectorial calculus it can be expressed in a simplified form:

32 It is true that in the new calculation from the outset he despises the higher order terms. He himself warns us referring us to the ‘Principia motus fluidorum’ [§.15].

THE THEORETICAL CONSTRUCTIONS (II)

429

G ∂ρ + ∇ ⋅ ( ρv ) = 0 ∂t

[9.80]

He considers this expression as basic, and says about it that this formula, ‘having been provided by considering the continuity of the fluid, already contains a certain relation that must hold among the quantities u, v, w and p’ [§.18] Nowadays it is known as the continuity equation, and it is an extension of d’Alembert’s equation to three variables and to non-steady motion, although the method followed by Euler to reach it was much more elegant and precise. In order to take into consideration the dynamic conditions, he first finds the accelerations along the three axes in a manner identical to that used for the density, and he obtains the following three equations: [§.19]

∂u ∂u ∂u ∂u +u +v +w =X ∂t ∂x ∂y ∂z ∂v ∂v ∂v ∂v +u +v + w =Y ∂t ∂x ∂y ∂z ∂w ∂w ∂w ∂w +u +v +w =Z ∂t ∂x ∂y ∂z

[9.81] [9.82] [9.83]

Following a vectorial notation these could be written in a single expression as:

G G ∂v G G a= + v ⋅ ∇v ∂t

[9.84]

Once he has found the accelerations, he goes on to look for ‘the accelerating forces’ which will consist of two addends. The first are the pressure forces, and the second forces those of external origin such as gravity or any other that may exist. In order to obtain the first ones, Euler repeated the steps taken in the ‘Principia’, which ended in the expressions [9.23 and 9.24] for the axes OX and OZ. These will be completed with the components corresponding to OZ, and to which he added the forces of external origin expressed as force per mass unit. The sum of both turns out to be:

fx = P −

1 ∂p ; ρ ∂x

fy = Q −

1 ∂p ; ρ ∂y

fz = R −

1 ∂p ρ ∂z

[9.85]

430

THE GENESIS OF FLUID MECHANICS, 1640–1780

And expressed as a vector:

G G 1 f = F − ∇p

ρ

[9.86]

Which, when made equal to the previous expressions, results in the following three equations: [§.21]

P−

1 ∂p ∂u ∂u ∂u ∂u = +u +v +w ρ ∂x ∂t ∂z ∂x ∂y

[9.87]

Q−

1 ∂p ∂v ∂v ∂v ∂v = +u +v +w ρ ∂y ∂t ∂z ∂x ∂y

[9.88]

R−

1 ∂p ∂w ∂w ∂w ∂w = +u +v +w ρ ∂w ∂t ∂z ∂x ∂y

[9.89]

With these three equations,33 together with that of continuity and the one relating density, elasticity and heat, ‘we have five equations enclosing the entire theory of the motion of fluids’ [§.21], a brief statement whose transcendence merits contemplation for the truth it holds. If, when comparing these equations with those found in the ‘Principia’,34 little or no substantial difference is found, what must be emphasised is the subsequent treatment given to them. He now no longer follows the method of imposing the equality of the cross-derivatives, which is what led him to identify the irrotational condition as the only one possible, but instead tries an attack on various fronts, without much success as the simplicity of these equations in no way implies the possibility of their solution. Faced with this, Euler caries out a series of attempts not only to resolve some simple cases, but also to obtain the conditions that must be met by the functions representing the components of the velocities. He initiates the process by supposing that the integrability of the expression Pds + Qdy + Rdz, which is that the forces derive from a potential, is such that: 33 The ‘2’ has already disappeared. It affected the equations of the forces in the ‘Principia’. See previous note Nº 10. 34 Expressed in the equations [9.25] and [9.26] for the two-dimensional case with gravity as the only external force.

THE THEORETICAL CONSTRUCTIONS (II)

dS = Pdx + Qdy + Rdz

431

[9.90]

This, together with equations [9.87]–[9.89] and the identity of equations [9.81]– [9.83] lead him to a single equation35:

dS = Pdx + Qdy + Rdz −

dp

ρ

= Xdx + Ydy + Zdz

[9.91]

Which, in turn can be reduced by using a vector formula to the following:

G G G ∂v G K dS − = dr + ( v ⋅ grad v )dr ρ ∂t

dp

[9.92]

Here we mention one of the frequent allusions that Euler makes about the drawbacks of mathematical analysis: ‘But as we have still very little work on the resolution of such differential equations of three variables, we can only await a more complete solution of our equation, before the boundaries of Analysis have been extended considerably further’ [§.25].36 In consequence, he declares that he has to go on to specific solutions, as from them we can judge the route to be followed in order to reach the complete solution [§.26]. Euler initiates a series of particularisations of this idea, among which predominate the supposition that the density is constant, or rather that the forces are derived from a potential. In the case where the density is constant, and the function udx + vdy + wdz can be integrated, a potential function W(x,y,z,t) can be supposed. So that:

dW = udx + vdy + wdz + Πdt

35

[9.93]

Euler expresses that equation in its components as follows:

dS −

dp

ρ

= dx

∂u ∂v ∂w ⎛ ∂u ∂u ∂u ⎞ + dy + dz + ⎜u + v + w + ⎟dx ∂t ∂t ∂t ⎝ ∂x ∂y ∂z ⎠

⎛ ∂v ⎛ ∂w ∂v ∂v ⎞ ∂w ∂w ⎞ + ⎜ u + v + w + ⎟dy + ⎜ u +v +w + ⎟dz ∂ ∂ ∂ ∂ ∂ x ∂z ⎠ y z x y ⎝ ⎠ ⎝ 36

This wish has still not come true, although the numerical calculations and the computers make part of this task.

432

THE GENESIS OF FLUID MECHANICS, 1640–1780

Now, combining the development of the expression [9.91] with the conditions derived from the equality of the cross-derivatives,37 which will be six equations, we end with the following:

dS −

dp

ρ

= dΠ + udu + vdv + wdw

[9.94]

That in the case where the density is constant can be integrated in the form:

⎛ u 2 v 2 w2 ⎞ ⎜ ⎟⎟ − − p = ρ ⎜ C (t ) + S − Π − 2 2 2 ⎝ ⎠

[9.95]

But he takes one more step, identifying the terms in brackets with a function V, that in both the cases of constant density and where it is related to pressure, end in the equality:

dp = ρdV

[9.96]

Of which he says that ‘it generally includes all the fundamentals of the theory of motion in fluids’ [§.29]. Next he analyses ‘an example of a real motion which is perfectly in accord with all the formulæ which the principles of mechanics have furnished, without, nevertheless, the formula udx + vdy + wdz being integrable’ [§.30]. That is, a non-irrotational case. In this example he supposes that the external forces to be zero, P = Q = R = 0, and that the velocities follow the law:

u = − Zy ;

v = Zx ;

w=0

[9.97]

In which Z is any function of Z ( x 2 + y 2 ) . This causes the expression udx + vdy + wdz to change into –Zydx + Zxdy, which can only be integrated if Z = 1/ (x 2 + y 2 ). On the other hand, if the density is constant, the equation of continuity [9.79] remains as:

37

The cross-derivatives are the following:

∂u ∂v = ; ∂y ∂x

∂u ∂w ; = ∂z ∂x

∂u ∂Π = ; ∂t ∂x

∂v ∂w = ; ∂z ∂y

∂v ∂Π = ; ∂t ∂y

∂w ∂Π = ; ∂t ∂z

THE THEORETICAL CONSTRUCTIONS (II)

− ρZ '

xy x +y 2

2

+ ρZ '

yx x + y2 2

=0

433

[9.98]

That is, any function Z meets the equations, but only one class of these establishes that udx + vdy + wdz is an exact differential. This is what makes him say: ‘Thus the assumption of the possibility of the differential formula udx + vdy + wdz furnishes only a particular solution of the formulæ we have found’ [§.32]. He returns to equation [9.91], now written as:

dp

ρ

= ( P − X )dx + (Q − Y )dy + ( R − Z )dz

[9.99]

He now analyses this for the compressible and incompressible assumptions and the latter for the homogeneous and non-homogeneous cases. The core of this analysis lies in the differential:

( P − X )dx + (Q − Y )dy + ( R − Z )dz

[9.100]

If the motion is incompressible and homogeneous, this expression can be integrated. If it is compressible, there must be a relation between the density and the pressure.38 In either case a function V will exist, so that dV is equal to the expression [9.100], and V = p/ρ or rather V = ∫dp/ρ. However, he warns that the continuity equation has to be satisfied also. In the case of incompressibility and heterogeneity, V must be a function of density. He also contemplates the possibility of introducing an integrating factor. We appreciate the parallel between these manipulations and the ones he made in the first of the memoirs when he dealt with fluid statics. After a considerable effort he insists once again that: Since a general solution must be judged impossible from want of analysis, we must be content with the knowledge of some special cases, and that all the more, since the development of various [special] cases seems to be the only way of bringing us to a more perfect knowledge at last. [§.41]

Among these specific cases, he examines those where the velocities are zero, which correspond to the state of rest studied in his preceding memoir. Another is

38

Euler says ‘that the density is expressed by any function of the elasticity’.

434

THE GENESIS OF FLUID MECHANICS, 1640–1780

the case in which the velocity is constant, with the variation that the two components are zero, and the motion is a shift along an axis. The third, which we will look at in more detail, is that where he changes the coordinates, taking the new ones to be the total velocity of a particle and its three angles along the OXYZ axes, instead of the three components of the velocity [§§.60-ff]. In this respect he defines the velocities u, v and w as a function of the total V, as:

u = αV ;

v = βV ;

w = γV

[9.101]

Making the following identification:

dV =

∂V ∂V ∂V ∂V dt + dx + dy + dz ∂t ∂x ∂y ∂z

[9.102]

We will have for the former X, Y and Z:

∂V ∂V ∂V ∂V +α2 + αβ + αγ ∂z ∂t ∂x ∂y ∂V ∂V ∂V ∂V Y =β + αβ + β2 + βγ ∂z ∂t ∂x ∂y ∂V ∂V ∂V ∂V Z =γ + αγ + βγ +γ 2 ∂z ∂t ∂x ∂y X =α

[9.103] [9.104] [9.105]

If in these three equations one takes:

Φ=

∂V ∂V ∂V ∂V +α +β +γ ∂z ∂t ∂x ∂y

[9.106]

Then we can simplify the three previous ones, resulting in the formula:

X = α Φ;

Y = β Φ;

Z = γΦ

[9.107]

And the two basic equations, impulse and continuity, become the following two:

dp

ρ

= Pdx + Qdy + Rdz − Φ (αdx + βdy + γdz

[9.108]

THE THEORETICAL CONSTRUCTIONS (II)

∂ρV ∂ρV ∂ρ ∂ρV +α +β +γ =0 ∂t ∂x ∂y ∂z

435

[9.109]

Taking these new equations as a starting point, he makes new simplifications that we shall not go into. We only wish to underline the new point of view. Already in the last paragraph of the memoir he stresses how the determination of the trajectories of the particles is of paramount importance, making special mention of the case of ships. However he laments that ‘it is not the principles of mechanics that are lacking, but only those of analysis which is still not sufficiently advanced for these ends. On starting out one sees clearly certain gaps that must still be filled in this science, before we can arrive at a more perfect theory of the motion of fluids [§.68]. We must add that, without what Euler says has some truth, there still remained phenomena that were already known but which needed to be put into mathematical form, such as viscosity; and there were other fields to be discovered such as thermodynamics. Each new discovery implied greater needs for mathematical analysis. Sequel to the researches on the motions of fluids

After the theoretical effort of the two previous memoirs, he starts the third one almost like a steamroller: Since in my two preceding memoirs I reduced the whole of the theory of fluids … to two analytic equations, the consideration of these formulae appears to be of the greatest importance, for they include not only all that has been discovered … but also all that one could further desire in this science. However sublime are the researches on fluids which we owe to the Messrs. Bernoulli, Clairaut and d’Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this agreement of their profound meditations with the simplicity of the principles from which I have drawn my two equations, and to which I was led immediately by the first axioms of mechanics. [§.1]39

The structure of this ‘Sequel’ is that of a final version of the whole theory. As he states in the paragraph quoted, all the previous theories are derived from his two equations, and this is what he aims to demonstrate. In the first paragraphs [§§.2– 18] he repeats and perfects the arguments already expressed, with commentaries upon the deduction of the formulae and the conditions of existence of the solutions. He then goes on to direct his arguments to demonstrating how the two 39

The quotes inside bracket correspond to the third memoir.

436

THE GENESIS OF FLUID MECHANICS, 1640–1780

most relevant lines of theory existing up to then, namely the motions of fluids in narrow ducts and the two-dimensional case of d’Alembert, are to be derived from his formulas. Z

Z A(s)

s r

v

v0

Y

V(s,t)

X

X

a)

b) Fig. 9-5. Moving tube of fluid

Concerning the first of these two themes, that of the narrow ducts, he says: For almost all that has been contributed on this subject can be reduced to the motion of fluids in infinitely narrow tubes, or at least [tubes] which can be regarded as such, so that in these cases one imagines only a single dimension both for the fluid and for it motion. [§.18]

Here he refers especially to the works of Johann and Daniel Bernoulli, and the later works of d’Alembert. He treats the problem extensively [§.19], including not only the tubes at rest, but also moving ones, and he extends the results to motion in the streamlines. He imagines a tube in a space (Fig. 9-5a) that follows a curve of a known equation, with a variable cross section A(s), and through which a fluid circulates at a velocity V(s,t) and density ρ(s,t). In order to apply the continuity condition he marks off a fragment of the tube, in which there will be a mass of fluid at a given instant ∫ρAds When a time interval dt elapses, this mass will augment by dt∫A(∂ρ/∂t)ds, while fluid masses V0S0ρ0dt and V1S1ρ1dt, respectively will have gone in and out of this tube fragment. The continuity equation is established by a simple balance:

ρ1 A1V1 = ρ 0 A0V0 −

∫

s1

A s0

∂ρ ds ∂t

[9.110]

THE THEORETICAL CONSTRUCTIONS (II)

437

As well as this, the velocity V can be broken down into its three components along the Cartesian axes, which are denominated u, v, and w, and are related to the first by:

u =V

dx ; ds

v =V

dy ; ds

w =V

dz ds

[9.111]

Now, in the general momentum equation, the values of X, Y and Z are the accelerating forces which are expressed as:

∂u ∂u +u ∂x ∂t ∂v ∂v Y= +v ∂y ∂t ∂w ∂w Z= +w ∂z ∂t

X =

[9.112] [9.113] [9.114]

And which once introduced into the equation quoted [9.99] will make it become:

dp

ρ

= Pdx + Qdy + Rdz −

∂V ds − VdV ∂t

[9.115]

Here we recall that P, Q and R are the components of the field of forces. This new equation, together with equation [9.110], substitutes the general ones for the case of motion in a duct. He goes on to specify first the incompressible case (ρ = Cte) and later includes the steady condition (∂ρ/∂t = 0), another set of equations that we shall not repeat here. He recalls that the case of motion in tubes turning around an axis had already been dealt with by other authors, and also by himself,40 ‘but, given that the tube can receive an infinity of other movements, I shall make the application to any movement of the tube’ [§.32]. The continuity equation has the same expression as the equation found, as, in its deduction only local conditions intervene in the tube. By contrast, in the momentum equation, the acceleration under consideration must be absolute with respect to some fixed axes. In order to find it, Euler adds the local acceleration to the translation one, which as we shall see is 40 He refers to the quoted ‘Théorie plus complete de machines qui sont mises en mouvment par al reaction de l’eau’. Mém. Acad. Berlin, Vol. X, 1754.

438

THE GENESIS OF FLUID MECHANICS, 1640–1780

incorrect. In his calculations Euler supposes (Fig. 9-5b) that the tube has one translation velocity and another of rotation, respectively v0 and ω, both being vectors and variable with time.41 Therefore, the velocity of any point P of the tube, defined by its vector of position r with respect to the origin, is expressed as:

G G G v = v0 + ω × r

[9.116]

He obtains the acceleration by deriving the previous equation with respect to the time, resulting in:

G G G G dω G G dr a = a0 + ×r +ω × dt dt

[9.117]

As dr/dt is the velocity expressed in equation [9.116], when he introduces it into the previous equation it leads to:

G G G dω G G G G G G a = a0 + × r + ω × v0 + ω × (ω × r ) dt

[9.118]

These results are included in equation [9.115], which, when written as a vector would be:

G ∂V K G = fds − ds − ads − VdV ρ ∂t

dp

[9.119]

But besides this, for kinematic reasons:

G G VdV = a f ds

[9.120]

Where af is the acceleration of a fluid particle with respect to the tube, and which introduced in the previous equations leads to:

G ∂V K G G G = fds − ds − ads − a f ds ρ ∂t

dp

41

[9.121]

We note that the formula used by Euler is v = v0 -ωxv with a minus sign instead of the plus sign we have used. The reason is very simple: his reference trihedron is inverse, ours is direct, and is the one almost exclusively employed nowadays.

THE THEORETICAL CONSTRUCTIONS (II)

439

Which clearly manifests that what Euler does is to introduce a + af as the total acceleration of the point under study. This is a mistake, as he still has to introduce an additional term, nowadays known as the Coriolis acceleration, whose value is 2ωv. His mistake consists of the fact that when calculating the derivative of the vector of the velocity of the fluid inside the tube with respect to the time in the relative motion, which was V, he should have also taken into consideration the temporal variation of the intermediate system—which is the one fixed to the tube—with respect to the fixed system itself. Euler simply assumes that this last variation does not exist. It is not that he makes a mistake in the calculation, as Truesdell would have it,42 but that he does not even consider it.43 What is strange is that in the work ‘Théorie plus complete des machines qui sont mises en mouvement par la reaction de l’eau’ (1754), he had indeed considered this term.

Z z a F b

Y X Fig. 9-6. Streamline

42

In a footnote to §.31 (‘Rat. Fluid. Mech-12’, p. 109), he points out ‘What nowdays is called acceleration of Coriolis is calculated here incorrectly’. In the point he quotes of his introduction he says that the relative acceleration is false, in all Euler’s attempts to calculate it, as he obtains ‘ω × v’ instead of ‘2ω × v’. This interpretation is incorrect, as Euler makes no mistake in his calculations, but conceptually ignores the system’s motion with respect to the fixed one, which causes the aforementioned acceleration of Coriolis. See the next note. 43 The theme merits a more detailed commentary, as it is a composition of movements, designated as relative motion. Let it be a frame of fixed or absolute axis and another translating one, which moves with respect to the former with an angular velocity ωa, and in which in turn a mobile point M exists with the velocity vr and acceleration ar with respect to the translating frame, and therefore designated as relative. Now the absolute velocity of this point is that some of the former relative velocity plus the shift velocity, which is which is the velocity that this point has as a member of the intermediate system with respect to the fixed system. The resulting velocity is v = vr + va. But concerning the acceleration, apart from the sum of the relative plus the drag ones, an additional term appears whose value is 2ωa × vr, or the so-called Coriolis acceleration. Then, a = ar + aa + 2ωa × v results for the total.

440

THE GENESIS OF FLUID MECHANICS, 1640–1780

After these analyses Euler declares ‘that I have already noticed that almost all the cases of motion of fluids dealt with here, can be reduced to an infinitely narrow duct, which I have just developed’ [§.44]. At the limit this tube will become blends with the streamline. Bearing this in mind, and with some simplifications such as those of supposing the stationary motion and an incompressible fluid, he tries to integrate the equations once again. In stationary conditions, as he acknowledges, all the particles passing through a point will always follow the same path, i.e., the same streamline or trajectory (Fig. 9-6). He expresses each streamline44 by a pair of implicit equations:

Φ1 ( x, y, z, b, c) = 0

[9.122]

Φ 2 ( x , y , z , b, c ) = 0

[9.123]

Strictly speaking each one represents a surface, and the intersection of both will be the line in question. In these expressions c and b represent two parameters that will identify each stream-line, and Euler identifies with the coordinates of the intersection point with the plane OXY. The two previous equations can be transformed into another two of the type:

b = B ( x, y , z ) ;

c = C ( x, y , z )

[9.124]

As the velocity is intrinsically tangential to the stream-line, the following equations must be established:

dx dy dz = = u v w

[9.125]

When operating on the functions [9.124] and the previous one, this becomes [§.47]:

dx dy dz = = ∂ ( B, C ) ∂ ( B, C ) ∂ ( B, C ) ∂ ( y, z ) ∂ ( x, z ) ∂ ( x, y ) 44

[9.126]

In an steady motion the stream-lines coincide with the trajectories. We recall that a trajectory is the path traveled for a particle with time, i.e., it has a temporal significance. On the other had, a stream-line is the geometrical place of the tangent to the particle velocities in a given instant.

THE THEORETICAL CONSTRUCTIONS (II)

441

Which can be written as a function of the parameter K as [§.47]45:

u=K

∂ ( B, C ) ; ∂ ( y, z )

v=K

∂ ( B, C ) ; ∂ ( x, z )

w=K

∂ ( B, C ) ∂ ( x, y )

[9.127]

He adds that ‘it is not yet certain whether the common factor K depends on the constants b and c, or in addition on the co-ordinates x, y, z. This will have to be decided by the equation drawn from the continuity of the fluid’ [§.47]. In fact, he does not get around to establishing whether this is true. What he does is to start from the hypothesis K(a,b) and operate, finding that he does not arrive at a contradiction, which itself indicates that there is a class of solutions of this type, but the possibility of another class is not excluded. In order to apply the impulse formula to the length of the streamline, and given that it is defined by equations, he only requires a coordinate which he designates as x. The problem is similar to the case of motion through a slim tube. Manipulating the two previous equations and considering that the forces are derived from potential field, i.e., that they correspond to an exact differential, he arrives at the expression [§.54]:

p

1 =U − V2 + D ρ 2

[9.128]

Which once again turns out to be Bernoulli’s equation, although he does not identify it as such, now applied to the streamline. The potential of the forces is represented by U, and D will be a constant function D(b,c) for each stream-line. He dedicates considerable effort to the solution of this equation, and does not confine himself to the cases of non-irrotationality which, we recall, were those in which udx + vdy + wdz could be integrated, and they were the only ones which he had dealt with in the ‘Principia’. Now he does not confine himself to this condition, but instead declares that ‘one must take good care, lest one give out this solution as general, since there are an infinity of possible motions in which udx + vdy + wdz is not integrable’. [§.65] Nevertheless, he also analyses the specific case of irrotational motion [§.66] on the same assumptions of incompressibility and fixity, and he arrives at the following equations quite easily:

45 Euler details the calculus. We do not think it is worth repeating it, as it can be found in any book on differential geometry.

442

THE GENESIS OF FLUID MECHANICS, 1640–1780

p

1 =U − V2 +C ρ 2

[9.129]

There is only one difference between this formula and in the previous one: in this one the constant C is the same for all the streamlines existing in the fluid field, while D in the previous equation was different for each one, being, as we said, a function of a and b. After analysing motion in ducts, Euler goes on to the two-dimensional case with constant density, of which he said at the beginning of the memoir: ‘I shall make plain how all that has been written on the motion of fluids in two dimensions flows very naturally from these same formulas’ [§.18]. And in effect, this it does, although in order to introduce forms of a complex variable, he remembers the ‘the very ingenious’ method of d’Alembert. The starting point is the particularisation of the equations already obtained, eliminating one variable. These are:

udx + vdy

integrable

∂u ∂v + =0 ∂x ∂y

[9.130]

[9.131]

Regarding the first, he recalls that ‘one must not think that these two conditions cover all the motions possible in the same plane, as, in effect there are motions where the formula udx + vdy cannot be integrated’ [§.69]. That is to say, he goes back to insist upon the existence of rotational movements. He likewise points out that in order to meet the second of the equations, that of continuity, it is sufficient that udy – vdx can be integrated, thus the resolution of the problem boils down to the integrability of this one, and of udx + vdy.46 At this point he recalls that ‘which one achieves by the very ingenious method of Mr. d’Alembert’ [§.70], which, we remember, is based on the use of functions of a complex variable. Following steps similar to these he arrives at the following two functions47:

1 1 1 1 u = ϕ ( x + iy ) + ϕ ( x − iy ) + ψ ( x + iy ) + ψ ( x − iy ) 2 2 2i 2i 46

[9.132]

This arises from the condition that for a function to be an exact differential. The equality of its cross-derivatives when applied to this expression lead to the continuity equation. 47 The annotation is different on both. Apart from this, Euler introduces the factor ½. We shall also use the imaginary symbol i instead of − 1 just like we did with d’Alembert.

THE THEORETICAL CONSTRUCTIONS (II)

v=

1 1 1 1 ϕ ( x + iy ) + ϕ ( x − iy ) + ψ ( x + iy ) + ψ ( x − iy ) 2i 2i 2 2

443

[9.133]

In order to resolve these equations he makes the variables change x = s·cosω and y = s·sinω, jointly with the following formula:

( x ± iy ) n = s n (cos nω ± i sin nω )

[9.134]

The latter allows the separation of real and imaginary components in a power operation.48 And with its help, Euler attempts to obtain the functions φ and ψ as an expansion in power series [§.72], so that:

ϕ ( p) =

∞

∑A p k

k =0

∞

k

;

ψ ( p) =

∑B p

k

k

[9.135]

k =0

After some calculations he arrives at the following expressions for the velocities:

v = B + B1 s cos ω + B2 s 2 cos 2ω + B3 s 3 cos 3ω + ... A1 s sin ω + A2 s 2 sin 2ω + A3 s 3 sin 3ω + ... u = A + A1 s cos ω + A2 s 2 cos 2ω + A3 s 3 cos 3ω + ... B1 s sin ω + B2 s 2 sin 2ω + B3 s 3 sin 3ω + ...

[9.136]

[9.137]

If the coefficients Ai and Bi were known, it would be possible to obtain u and v, given that s2 = x2 + y2, sinω = y/s and cosω = x/s. The inverse problem, i.e., the calculation of the coefficients, will be done in theory, from the geometry of the body shape under analyses. We say in theory, because in practice the method is only feasible for a few simple forms. Finally, when Euler talks about reproducing the shape of a body using these methods, he says that ‘this research will serve to discover the true resistance which a body of arbitrary shape will experience from a current in which it is

nωi n A common form of expressing this equality is: e = (cos ω + i sin ω ) = cos nω + i sin nω which are known as Euler’s formulas.

48

444

THE GENESIS OF FLUID MECHANICS, 1640–1780

placed’ [§.79]. In principle the idea is sound, if the d’Alembert paradox were not involved. This, we recall, says that the resistance of a body in a fluid current is nil. Truesdell says that Euler appears to have forgotten this, and also the fact that he himself had demonstrated the existence of this paradox in his comments on Robins’ Gunnery.49 We add that he also forgot what d’Alembert said in this respect. Euler’s contribution

The first thing one notices, or rather does not notice, on reading these monographs of Euler, is his modernity. Apart from a few formulations and the absence of a few symbols50 currently in use, the approach, the hypotheses, the notation, symbolism and almost the style of writing would pass for present day for the vast majority of readers who ignore the provenance of these works. This indicates an almost qualitative jump with respect to his predecessors: he opens the gateway to the maturity of the discipline. Along these same lines we must emphasize his clarity in expressing concepts, although a slight evolution can be appreciated between the first work, the ‘Principia motus fluidorum’, and the three monographs. All this is in line with the principles upon which he bases his theories, i.e., the Newtonian ones, with forces as basic entities, and therefore the pressures as well. He interprets these as a force acting perpendicularly to a surface, in the exterior of the fluid as much on any internal imaginary surface that might be traced. Although these ideas had been formulated before, they had never been expressed with the characteristic clarity Euler brings to them. Together with the pressures, it is worthwhile underlining how he includes the field of mass forces as generators of accelerating forces, or simply accelerations, given that the relation between them is necessary and not contingent. This is done through Newton’s second law, which Euler himself helped to establish in its differential form, by which it is normally known nowadays. The resolute use of the forces and the law of momentum allows him to avoid complex mathematical transformations in order to eliminate the troublesome parameters. On the other hand, his handling of the analysis is clean, clear and powerful. The mathematical resources appear in a natural and appropriate manner, although he frequently laments that analysis was underdeveloped for his needs.

49 50

See Chapter 3, Robins’ New Principles of Gunnery. Such as the sign indicating the partial derivative.

THE THEORETICAL CONSTRUCTIONS (II)

445

Lastly, and this is one of his major achievements, he manages to reduce hydrodynamics to a set of differential equations completing and perfecting the work initiated by d’Alembert in particular. For this he starts out from a fluid model, and rather than look for real models, he idealises it in a continuum, although he himself believes that fluids are constituted by particles. We have here a separation between reality and modelling backed up by hypotheses and experience. The only thing to be regretted was the difficulty in resolving these equations, which became a reference point outside practical reach until almost 100 years later. Lagrange’s paper on fluids

After the three monographs of Euler, we find the following significant works on the theory of fluids by d’Alembert: ‘Remarques sur les loix du mouvement des fluides’ (‘Notes on the law of motion of fluids’), and the Mémoires XXXXXXIV contained in his Opuscules mathématiques, Vol. 1 (1761) and Vol. 5 (1768), respectively. Both constitute an extension of his previous work, and certainly they perfect it. Later on, four new monographs of Euler appeared in the Novi commmentarii academiae scientarum Petropolitane, 1768 to 1771. These corresponded to another four sections of a Tractatus hydrodynamicus that Euler intended to prepare, and whose titles were: ‘Sectio prima de statu aequilibrii fluidorum’, ‘Sectio secunda de principiis motus fluidorum’, ‘Sectio tertia de motu fluidorum lineari potissimum aquae’ and ‘Sectio quarta de motu aeris in tubis’. As Truesdell says, ‘they are careful and detailed expositions of the simpler results from Euler’s earlier papers’,51 but apart from an improved level of formal presentation, they introduce no new concepts. The next work is the ‘Mémoire sur la théorie de mouvement des fluides’ (‘Memoir on the Theory of Fluid Motion’) of Joseph Lagrange, published in the Nouveaux Mémoires de l’Académie Royale de Sciences er bellers-Lettres of Berlin in 1781.52 We shall make a brief comment upon these. Lagrange divides his work into two parts, the first dedicated to general equations, and the second to the movement of fluids in ducts. Both are preceded by a short introduction in which he

51 52

Cf. ‘Rat. Fluid Mech.-13’, p. X. It is also in the Memoirs of the Academy of Torino.

446

THE GENESIS OF FLUID MECHANICS, 1640–1780

praises d’Alembert, and omits to mention Euler, although certainly his work is much more of continuation of the latter than that of the former. 53 He begins by obtaining equations of continuity, [§.1–5] which he calls the ‘density equation’, and which, once specified for the incompressible case, calls it ‘incompressibility equation’. The procedure he follows is to isolate a fluid parallelepiped that evolves with time, establishing the condition that its mass be constant. He then goes on to introduce the internal pressure and the external pressures on this parallelepiped, thus arriving at the momentum equation [§.6– 11]. This is Euler’s method, so we shall not repeat the equations. Nevertheless he makes an important contribution when he deals with the case in which the fluid is confined by a surface A(x,y,z,t) = 0, which also evolves with time [§.10–11]. He imposes the condition of movement, supposing that this surface will behave like the fluid particles in contact with it. He states that for this he assumes that the particles will never be separated from the wall. This is expressed analytically as:

∂A ∂A ∂A ∂A +u +v +w =0 ∂t ∂x ∂y ∂z

[9.138]

If the fluid is restricted to moving inside a given boundary, then the function A would correspond to the boundary, which would have to support the internal pressure of the fluid. But if the fluid is free, the external surface will not support any pressure, its shape will be variable, and to determine it forms part of the problem. The difficulties encountered by Lagrange continue to be mathematical ones, and this leads him to say, in words that not only recall d’Alembert but also Euler, that: Such are the principles and general formulae of the theory of fluids. The difficulty does not consist in their application; but this difficulty is so great, that up to now even in the solution of the simplest questions, one is content to employ specific methods founded on very limited hypotheses. [§.12]

That is to say, we have to go on to specific cases. Among these, important cases are those where exists a velocity potential. On the assumption that the field of forces is of the type dV = Pdx + Qdy + Rdz, i.e., an exact differential, the impulse equation becomes the expression:

53

The equations will refer to the ‘Mémoire sur la Théorie du mouvement des fluids’.

THE THEORETICAL CONSTRUCTIONS (II)

447

⎛ ∂u ⎛ ∂v ∂u ∂u ∂u ⎞ ∂v ∂v ∂v ⎞ + w ⎟dx + ⎜ + u + v + w ⎟dy ⎜ +u +v ∂x ∂y ∂z ⎠ ∂x ∂y ∂z ⎠ ⎝ ∂t ⎝ ∂t ⎛ ∂w ∂w ∂w ∂w ⎞ dp +⎜ +u +v + w ⎟dz = dV − ρ ∂x ∂y ∂z ⎠ ⎝ ∂t

[9.139]

That had already been found by Euler.54 Here Lagrange introduces the new transformation d(u2 + v2 + w2), and after various operations ends in: ∂u ∂t

dx +

∂v ∂t

dy +

∂w ∂t

⎛ ∂u

dz + ⎜

⎝ ∂y

−

∂v ⎞

⎛ ∂u ∂w ⎞ (vdx − udy ) + ⎜ − ⎟ ( wdx − udz ) ⎟ ∂x ⎠ ⎝ ∂z ∂x ⎠

[9.140]

dp 1 ⎛ ∂v ∂w ⎞ +⎜ − ( wdy − vdz ) = dV − − d (u 2 + v 2 + w 2 ) ⎟ ρ ∂ z ∂ y 2 ⎝ ⎠

Also an equation similar to the one arrived at by Euler.55 In this one Lagrange introduces a potential function of velocities, which Euler had already done (equation [9.49]), supposing that for this the expression udx + vdy + dz is an exact differential. The relation of the velocities with a potential φ will be:

u=

∂ϕ ; ∂x

v=

∂ϕ ; ∂y

w=

∂ϕ ∂z

[9.141]

Introducing the potential function, this equation will remain [§.5] as:

∫

2

∂ϕ 1 ⎛ ∂ϕ ⎞ 1 ⎛ ∂ϕ ⎞ 1 ⎛ ∂ϕ ⎞ =V − − ⎜ ⎟ − ⎜ ⎟ − ⎜ ⎟ ∂t 2 ⎝ ∂x ⎠ 2 ⎝ ∂y ⎠ 2 ⎝ ∂z ⎠ ρ

dp

2

2

[9.142]

This equation, together with that of the continuity, now written as [§.15]:

∂ ⎛ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞ ∂ ⎛ ∂ϕ ⎞ ∂ρ =0 ⎟ + ⎜ρ ⎜ρ ⎟+ ⎜ρ ⎟+ ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ ∂t

54

[9.143]

Cf. ‘Principes généraux du mouvement des fluides’ [§.24]. We have modified the symbols used by Lagrange to make them compatible with those of Euler. See also the upper equation [9.99]. 55 Cf. ‘Continuation’ [§.54].

448

THE GENESIS OF FLUID MECHANICS, 1640–1780

Will allow, at least in theory, the elimination of density, leaving a differential equation in φ, and the movement of the fluid will be found from its solution. Another simplification that he introduces consists in supposing that the density is proportional to the pressure, because ‘in known elastic fluids the density is always proportional to the pressure’ [§.16]. We shall not give the new equations he arrives at, noting only that the incompressible case ends in the following:

∂ 2φ ∂ 2φ ∂ 2φ + + =0 ∂x 2 ∂y 2 ∂z 2

[9.144]

That is very similar to the equation obtained by Euler for the potential in the ‘Principia motus fluidorum’. A significant contribution is the introduction of what we would nowadays call small perturbations method, and which has been so fruitful in fluid mechanics. Their basis is the supposition that the velocities produced u, v, w, ‘are very small, and that the very small quantities of second order and the following orders are ignored’ [§.22]. From this it follows that in the equation of the potential [9.139] all the products of the velocities and their derivatives are eliminated, resulting in a simplified form:

∂u ∂v ∂w dp dx + dy + dz = dV − ∂t ∂t ∂t ρ

[9.145]

Here we must point out that Lagrange not only supposes the velocities to be small, but also their spatial derivatives. In that last equation the first member can be integrated and its value is dφ and also φ was small. Given the small value of the velocities, the coordinates x, y, z that will follow a particle will be almost constant, thus it will be shown that:

∫

x = x0 + udt ;

∫

y = y0 + vdt ;

∫

z = z0 + wdt ;

[9.146]

Where x0, y0, z0 are the initial values of x, y, z. Defining Φ = ∫φdt, the previous equations can be expressed as:

THE THEORETICAL CONSTRUCTIONS (II)

x = x0 +

∂Φ ; ∂x

y = y0 +

∂Φ ; ∂y

z = z0 +

∂Φ ; ∂z

449

[9.147]

He goes on to apply this method to the case where the density is proportional to the pressure, which he says is the case corresponding to the propagation of sound. An additional simplification would be the case in which one of the dimensions was considerably smaller than the other two. In these circumstances, the rest of the variables, u, v, w, ρ could be expanded in power series of the former coordinate, adopting the generic form Σfn(x,y) zn. Lagrange also deals with motion in channels of incompressible fluids, where he supposes that only the force of gravity acts on the fluid, as this has the surface z = a free. If the angles of the each axis with the vertical are designated by ξ, η, ζ the potential of the forces will be expressed as:

V = gx cos ξ + gy cos η + gz cos ζ

[9.148]

The equation to be solved is the potential one, ∇ 2ϕ = 0 , which was found in [9.139]. He proposes to expand the potential φ in a series of powers of z, so that: ∞

φ=

∑

φn z n

[9.149]

n =0

an equation that once derived and introduced in the Laplacian, ( ∇ 2ϕ = 0 ), ends in a recurrent expression with φ0 and φ1 as initial values. So that:

φ2 n =

∂ 2n ⎞ ( −1) n ⎛ ∂ 2 n ⎜⎜ 2 n + 2 n ⎟⎟φ0 ∂y ⎠ ( 2n )! ⎝ ∂x

φ2 n +1 =

∂ 2n ⎞ ( −1) n ⎛ ∂ 2 n ⎜⎜ 2 n + 2 n ⎟⎟φ1 ∂y ⎠ ( 2n + 1)! ⎝ ∂x

[9.150]

[9.151]

The velocities and pressures whose mathematical structure is similar are a result of these equations.

450

THE GENESIS OF FLUID MECHANICS, 1640–1780

He applies these formulas to the case of a narrow vessel and vertical in the x direction, so that its potential of forces will be V = gx. The solutions are [§.37]:

φ0 = θ (t )

∫ λ ( x ) + Θ( t )

[9.152]

φ1 = θ (t )

d ⎛ µ( x ) ⎞ ⎜ ⎟ dt ⎝ λ( x ) ⎠

[9.153]

dz

where λ, µ, θ are functions depending of the vessel. Lagrange applies them to four cases, only outlined here: (a) a given quantity of fluid running through an infinite vessel, (b) the vessel is finite and the fluid exits from its lower part, and (c) similar to the first, but the vessel is always kept full. As a final application he applies the motion of waves to the case of shallow channels. If the depth of the channel with water at rest is h, after applying the formulas found he arrives at the following equation:

⎛ ∂ 2φ 0 ∂ 2φ 0 ⎞ ∂ 2φ 0 ⎟= gh⎜⎜ 2 + ∂y 2 ⎟⎠ ∂t 2 ⎝ ∂x

[9.154]

Which is a form of the equation known as the wave equation, and from this it follows that of the velocity of wave propagation in this type of channel is:

c = gh

[9.155]

a formula which has survived up to the present day. Although Lagrange does not contribute anything new conceptually, he does indeed introduce important advances in the methods of analysis. This has great value, as we have seen how the other great mathematicians of the time floundered when faced with the limited resources of mathematical analysis of the period. In this respect, Lagrange opens up new avenues which will be exploited in the two following centuries. In particular we must note: • The introduction of a potential of velocities, which is, so to speak, equivalent to a reduction in the number of the variables to be managed.

THE THEORETICAL CONSTRUCTIONS (II)

451

• The use of approximation by small perturbations, a very rich method which extends with successive simplifications. • And the application of expansion in power series. This is not a new method but had never been used in fluid mechanics.

For all these reasons, we consider that Lagrange provides a splendid finishing touch to the grand theorisation that took place in the eighteenth century.

Chapter 10 Application of Fluid Mechanics to Pumps and Turbines

In the eighteenth century, fluid mechanics was studied in its application to existing machines and equipment, or those that were theoretically feasible (if not necessarily practical). These can be classified into two categories: pumps for raising water, and machines that generated movement by reaction. There was another group that was important due to its interest for city life, which was the distribution of liquid through pipes. However, it was only dealt with in an incipient form, given that its theoretical bases were linked with viscous phenomena, still not understood in this century.1 Therefore, we shall not include it in our analysis. Pumps for raising water (hydraulic pumps) have been known since antiquity. As Pitot rightly pointed out, ‘pumps occupy the first rank among all the machines used to raise water. Their usefulness and the great and widespread use made of them in all countries has caused many excellent mechanics to work on perfecting them’.2 They are not the only machines used to raise water, however, and in this respect we ought to remember the waterwheel with buckets and its derivations. However, as opposed to these, pumps enjoyed numerous advantages such as energy yield, size, versatility, etc., which explains the interest devoted to them by the sages of the time. Although descriptions of pumps were frequent, the first analyses subjecting them to theoretical consideration are found in Pitot, even though he mentions previous attempts which, for several reasons, were unsuccessful. We find a precedent in the Hydrodynamica of Daniel Bernoulli,3 but he refers to somewhat more elemental machines, almost static, rather than pumps, and only makes use of the mechanical law of conservation of energy. Chronologically, studies on pumps progressed from these analyses of Bernoulli, almost within hydrostatics, up to the inclusion of narrowing and losses in 1

Euler has a study that deals with this ‘Tentamen theoriæ de frictione fluidorm’ Novi comm. acad. petrop., Vol. VI (1761), in which Euler makes a mistake in the Basic Law of friction. 2 Cf. ‘Essais d’une théorie nouvelle de Pompes’, Mém. Acad. Paris, (1735), p. 327. 3 Cf. Chapter IX, ‘Concerning the Motion of Fluids that are Pushed forth not by their own Weight but by an Outside Force, and particularly concerning Hydraulic Machines and their Ultimate Grade of Perfection that can be Attained, and how this could be Perfected further through the Mechanics of Solids as well as of Fluids’.

453

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THE GENESIS OF FLUID MECHANICS, 1640–1780

movement of the liquid in their interior, clearly within the dominion of hydrodynamics. In this respect we emphasize the works of Pitot, Euler and Borda. As regards the second group of machines, which we have designated as reaction machines, these include all those that generate movement by the reaction of a jet of fluid launched to the exterior,4 and whose theoretical justification is the principle of action and reaction explained in the Newton’s third law.5 There are two very significant applications: propelling ships by a water jet, and the reaction turbine. The first is described in Hydrodynamica,6 and the second we have called Segner-Euler turbine, as it was proposed by the first, and perfected by the second. We shall describe both. The hydraulic pump

As we have mentioned, the hydraulic pump is a device known from ancient times. Its main usefulness is in raising water, whether for the use of the water or to drink, or to pump water out of the mines. The elements defining the pumps are: a cylinder, a piston which moves inside the cylinder, a pair of one-way valves, and pipes or ducts that communicate the appliance with its feed tank and with its collecting tank. Technologically the pumps have a critical point: the sealing between the piston and the cylinder. The pumps can work as suction or driving pumps. In the first condition (Fig. 10-1), the apparatus is located at a certain height above the feed tank and at the same level as the collecting or drainage tank. In the first phase of functioning, when the piston rises, the water goes up through suction from the feed tank, and will enter through one of the valves to fill the cylinder. In the second phase, as the piston descends, the water will go out through the other valve towards the other tank. The maximum height at which the pump can be placed is the height at which this pressure balances the weight of column of water, whose height is approximately 10.3 m above sea level.7

4 Historic precedent of reaction machines is Hero’s turbine, which almost did not pass from a toy or recreational appliance in Roman times. However, in both hydraulic turbines and in Hero’s turbine, the force is produced by the reaction of the fluid. The difference is that the latter is a thermodynamic machine, while the others are only mechanical. 5 As we know, the principle of generating motion by reaction is now, in the twenty-first century, one of the most fruitful methods for propelling aircraft as well as high-speed ships. 6 Cf. Chap. XIII, §.20. 7 This figure is obtained from the hydrostatics equation p = ρgh with p = 101.3 kN/m2 and with ρ = 1,000 kg/m3.

APPLICATION OF FLUID MECHANICS

455

Fig. 10-1. Suction pump

In the pressure pumps the cylinder is at the level of the feed tank, and may even be submerged (Fig. 10-2). The water will enter almost freely in the first phase, but will require the application of a force in order to raise it to the collecting deposit in the second phase. There is now no limit as regards to height, except those imposed by the construction of the machine. The intermediate cases between both configurations are evident, and do not require any commentary. Comparing both, in the first the force is applied during suction while in the second it is during pressure.

Fig. 10-2. Pressure pump

In the one just as in the other, if the velocity of the motion of the piston were to be very small, we would find ourselves in the domain of hydrostatics with the rule of the equality of pressures as the only regulator of the process. However, it is not so in reality, as the dynamic effects arising from the level of the liquid are significant. It is in the analysis of this flow through the pipes or

456

THE GENESIS OF FLUID MECHANICS, 1640–1780

through the valves, or of the mass flow driven by the applied powers, where the application studies which we shall now take a look at, come into play.8 Pitot’s theory of pumps

Henri Pitot devoted three essays to the study of pumps. These appeared in the Mémoires de l’Académie of Paris for the years 1735, 1739 and 1740, entitled: ‘Essais d’une théorie nouvelle de Pompes’ (‘Essays on a new theory of pumps’), ‘Suite de l’essai d’une théorie nouvelle de Pompes’) ‘Continuation of a new theory of pumps’) and ‘Suite de l’essai d’une théorie de Pompes’ (‘Continuation of an essay) on the theory of pumps’. Pitot claimed to be the first person to study pumps, but according to him both Mariotte and Parent had proposed writing about the theme, but their deaths had prevented this, and he recounts that de la Hire had searched among the papers left by Mariotte without finding anything. As for Parent, he only left a proposal of eight problems on pumps to the wise men of the time, whose solutions were presented by Pitot. Due to these circumstances, Pitot was really the first person to write scientifically about applying fluid mechanics to water pumps, as he notes in the second of his essays: I have said at the beginning of the essay on the theory of pumps that up to now nobody has contributed specific treatises on these machines, even though they are the most widely used and most useful of all the hydraulic Machines. [Ess. 2, p. 393]9

This seems to be true, and the surprising thing is that in later years we have only found two more works on this subject: one by Euler and another by Borda, dated 1752 and 1768, respectively. Of Pitot’s three essays, the first was a study focused on the pump itself, in which he establishes a series of basic principles, giving some examples and proofs, including the answers to Parent’s eight problems. The second complements the first by adding water inlet and outlet pipes to and from the pump, introducing differences in level between the pump and the tanks. The third, which is briefer, collects the former findings, adding the applied power as a data, and presenting a final formula with its application to several typical problems. 8

Even more, for working at a very low speed a pump needs to have very tight tolerances in construction. This means without leakages in the valves and in the cylinder–piston assembly, something very difficult to achieve in the seventeenth century. On the contrary, at higher velocities the tolerances can be looser, because the fluid leakages take some time, although that will have a slight effect on system efficiency. 9 Given the thematic unity of the three essays, the quotes shall refer in an ordinal mode to each one.

APPLICATION OF FLUID MECHANICS

457

Though the whole theory of the machine is to be found in the first of the three essays, there is a progressive perfection of this subject throughout the three. We will analyse the problem as a whole, without dividing it among the three essays, in order to avoid unnecessary repetitions.

F vp

Fp vp Sp Sv

vv

vv

Fig. 10-3. Basic pump

For the basic pump theory, Pitot considers the body of the pump, consisting in a cylinder with its piston and two valves, the same as those already explained and shown as diagrams in Fig. 10-3. The force applied to the piston is designated by Fp and its velocity by vp. When the piston moves the water will leave through one of the valves with the speed of vv, while the other valve will remain closed. The value of this velocity will be defined as a function of the geometry of the pump as a whole, and of the velocity of the piston. The goal of Pitot’s analysis was to obtain a mathematical expression, relating force and velocity, applied to the performance of the pump as a function of its geometry. For this he established the following three basic principles: • • •

The force applied to the piston, Fp, is inversely proportional to the square of the surface of passage in the valves. [Ess. 1, §.1] The force applied to the piston is directly proportional to the square of the velocity of liquid in its passage through the valves. [Ess. 1, §.V] The ratio between the force applied to the piston and the force of water in the valve is equal to the ratio between the surfaces of the piston and the valves. [Ess. 1, §.VI]

We know that the last principle mentions ‘the force of the water during its passage through the aperture of the valve’, and he tells us that ‘it is more than obvious that the ratio of the force moving the piston to the force of the water in its passage through the aperture is the same ratio as that of the surface of the base of

458

THE GENESIS OF FLUID MECHANICS, 1640–1780

the piston to the aperture [Ess. 1, §.I]. We can interpret this as a consequence of the constancy of the internal pressure, where the result of the pressure existing on the passage surface of the valve is what he calls the force of water through the aperture.10 Therefore, if p was the pressure inside the cylinder, then it would follow that:

p=

Fp Fv = S p Sv

[10. 1]

Where Fv is that force. This argument justifies the third of his principles, although the supposition is not strictly accurate, as on the one hand, the pressure in the lower part of the pump differs from that existing in the upper part in the weight of the column of water, and on the other hand the dynamic effects of the current come into play. But as a first approach his hypotheses are acceptable. In the second of his principles, he establishes that the force of the piston is proportional to the square of the velocity of the fluid passage through the valve. He assumes that the force in the valve is the weight of the column of water, whose height is the kinetic height corresponding to the passage velocity. Designating this one as vv the kinetic height would be ξ = vv2/2g and the force Fv = Svρgξ. From those we would obtain:

Fv =

1 Sv ρvv2 2

[10. 2]

Substituting Fv by Fp with the help of the third of his principles we obtain:

Fp =

1 S p ρvv2 2

[10. 3]

That responds to the statement given by Pitot. It is worthwhile emphasizing that this formula is a forerunner of Bernoulli’s theory. We recall that the work of Pitot is from 1732, while the Hydrodynamica of Bernoulli dates from 1738. Now, if in the last equation we substitute velocity of water in the valve for that of the piston, for which we make use of the condition of continuity Svvv = Spvp, we would end up with the following equation:

10 For the readers of Pitot’s work we note that he supposes the surfaces to be circular, and that he works with the radius of the circles instead of the surfaces themselves.

APPLICATION OF FLUID MECHANICS

Fv =

S3 1 Sv ρv 2p p2 2 Sv

459

[10. 4]

That is the expression of the first of his principles, and Pitot uses it to evaluate the volume of the flow and the power. He continues with several comments and applications to specific cases,11 ending with the comparison between the two pumps made in the Royal Gardens before various Commissioners of the Academy. vp

F

vv

Fig. 10-4. Pump and tank

Up to this point Pitot had considered the pump as an isolated machine. Now, in the second of his Essays, he will provide it with two pipes going to the feed and discharge tanks (Fig. 10-4). In this case, the force acting upon the piston is used not only in the passage of fluid through the valves, but also in raising the liquid. In connection with this, he says: Besides the weight of the column of water that the power driving the piston must overcome, there is yet another resistance that the selfsame power should overcome, that where the pumps are in motion; this resistance or force comes from the velocity of the water that is raised, mainly in its passage through the opening of the valves or flappers. [Ess. 2, §.VI]12 11

In his calculations the value he takes the acceleration of the gravity as 28 pieds/s2 (9.1 m/s2), taken from de la Hire, cf. ‘Examen de la force necessaire pour mouvoir les bateaux tant dans l’eau dormante que courante’ Mém. Acad. Paris, 1702. 12 We must point out that this depends on how the machine is conceived, because he only considers this force in the suction cycle.

460

THE GENESIS OF FLUID MECHANICS, 1640–1780

In this commentary he insists on the existence of a force due to the motion of the water, perhaps to underline the difference with what he called a ‘perfect pump’ (les plus parfaits), i.e., the pump in which this force does not exist. He obtained the solution to the case by virtue of what he calls the fundamental law of mechanics: In all machines, the quantities of momentum are always equal. That is to say, the product of the driving force multiplied by its velocity is always equal to the product of the weight moved by the machine multiplied by its velocity. [Ess. 1, §.XIII].

Including this force, the last formula [10.4] becomes:

Ft = ρghS p +

S3 1 Sv ρv 2p p2 2 Sv

[10. 5]

which takes its place in the calculations of performance. As a final contribution to the theory of pumps he introduces the power, i.e., energy applied per unit of time. The concept of power was still not accurately outlined in the eighteenth century, and Pitot did not apply it as such, but rather applied the product of the forces multiplied by the velocity, which is exactly what we nowadays call power. Once again he recalls the fundamental principle of mechanics: ‘in all machines, the product of the driving force multiplied by its velocity is always equal to the product of the weight moved by the machine multiplied by its velocity’ [Ess. 3, p. 512],13 and using this equality between the power supplied (W) and having obtained (Ftvp), Pitot, after some mathematical operations, ends with the following equation:

v 3p +

ghSv2 Sv2 v − W =0 p S p2 2 ρS p3

[10. 6]

in which we see how the geometry of the system (Sv and Sp), the position of the body of the pump (h), the power applied (W) and how it is applied (vp), are all

13 The difference with the text previously quoted is that the former appears to identify the momentum with the product of the force multiplied by the velocity.

APPLICATION OF FLUID MECHANICS

461

related. Pitot declares that this equation enabled him to resolve all the problems of the pumps, and he offers various examples.14

a+b x

b

Fig. 10-5. Pump with an empty space

Extending the study of the set of the piston/cylinder ensemble, Pitot incorporates the assumption that an air chamber exists between the piston and the water.15 The chamber will vary its volume during the functioning of the pump due to the elasticity of the air. Let us suppose, as is shown in Fig. 10-5, the piston is separated by a distance b from the surface of the water, forming a cavity containing air. When the piston ascends to an additional height a, a certain quantity of water will come in which will be raised up to x, which is less than the run made by the piston. He obtains the ratio between these magnitudes assuming that the air follows the law nowadays known as Boyle-Mariotte, i.e., it maintains the product of the volume multiplied by the pressure, and that the pressure at the base of the pump (before and after) is equal to that of the atmosphere. Designating the latter by pa and the inner as pi, he establishes the following relation:

pa = pi + ρgx

[10. 7]

Now pi = pab/(a + b – x) must be demonstrated by the previous law. In order to unify the magnitudes, Pitot substitutes for the atmospheric pressure pa its equivalent height of water f,16 obtained by the relation pa = ρgf introducing this and p in the previous relation, so that he obtains:

14

In some of these the driving force is human, and in this respect he estimates that the power that may be yielded by a man is around 175 watts [1st Essay, p. 337]. 15 This supposition might well indicate Pitot’s preoccupation with real pumps, as due to leakage in the joints air always entered into the cylinder. 16 Pitot takes f = 32 pieds (10.4 m).

462

THE GENESIS OF FLUID MECHANICS, 1640–1780

x 2 − ( a + b + f ) x + af = 0

[10. 8]

with which all the parameters are determined. A complimentary case occurs when the pump has not only an empty space but also what he calls a ‘suction space’. That means that the level of water in the inlet tanks is placed at a distance c below the base of the pump. The arguments are similar and the final resulting equation is:

x 2 − ( a + b + f + c ) x + af = 0

[10. 9]

An interesting consequence, which can be deduced from the last equation, is the run the piston will have to make in order for the water to ascend just up to the entrance of the valve, i.e., c = x. The result is:

xmax =

af a+b

[10. 10]

whose limit is xmax = f, which leads him to conclude that ‘a pump will be better, … when the suction and the empty space are less with respect to the piston run’ [Ess. 1, ‘Princp. et Régles’]. The focus that Pitot gave to the study of pumps is quite obvious, and reflects very precisely the sense of transfer from theory to practice. We shall now proceed to Euler, who goes into this approach in greater depth. Pumps in Euler’s works

In 1752 Euler published the work ‘Sur le mouvement de l’eau par des tuyeux de conduite’ (‘On the motion of water through ducts’) in the Mémoires de l’Académie of Berlin. Euler’s intention was to study the behaviour of water in the ensemble formed by the pump and the tubes using the principles of hydrodynamics. As he states, his concern centered on two themes: In the first one seeks the quantity of water that the pumps will be able to supply to the tank in a given time. The other question is about the forces, which must be sustained, as much in the bodies of the pumps as in the pipes of the ducts, while the machine is functioning. [§.II]17

17 Quotations between inverted commas are taken from the memoir ‘Sur le mouvement de l’eau par des tuyeaux de conduite’.

APPLICATION OF FLUID MECHANICS

463

Euler insists that his approach is dynamic and not static, alleging that this difference had not been understood by previous authors, and noting that ‘the principles of hydrostatics can be explained by simple geometry with the help of elementary analysis; but it is not the same with the principles of hydraulics, where the real motion of water is dealt with’ [§.VIII]. From the earlier authors he quotes only, and in glowing terms, Daniel Bernoulli and d’Alembert. G y

Fp

z ds

y

M z D Fig. 10-6. Diagram of pump and tube

His diagram of a pump and tubes, shown in Fig. 10-6, consists of the body of a pump where a piston M moves driven by a force Fp, which raises the water through a tube which starts at the base of the pump at D, and goes up to a higher tank where it discharges through the spout G. The tube will have variable cross section which he defines by its section A(s), measured as a function of the distance s counted from D. As a result of the configuration and geometry, the piston will move at a velocity vp and acceleration dvp/dt depending on the run x of the piston from the base. In order to determine the dynamics of the system, he takes an element ZY of the tube located at a height y over the base. The fluid contained in the element will be subject to acceleration a composed of two terms, one due to the variation of the cross section of the tube and the other to the acceleration of the piston. The first is obtained from the equation of continuity Av = Cte, which yields the result Adv + vdA = 0. Including the time in this, he obtains the first of the two terms quoted:

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THE GENESIS OF FLUID MECHANICS, 1640–1780

dv v dA ds =− dt A ds dt

[10. 11]

an equation in which both A and dA/ds are geometrical data of the tubes. The second term is generated by the non-uniform movement of the plunger, defined by its acceleration dvp /dt, which transferred to the element ZY becomes (Sp/A)(dvp/dt). From the sum of this and the previous equation the total acceleration of the liquid turns out to be:

a=

S p dv p A dt

−

v dA ds A ds dt

[10. 12]

The forces acting on the isolated ZY element are the pressures and the weight. The first are p and p + dp acting on the bases, giving a resulting value Adp. The weight, whose action is vertical, will only contribute a component in the direction of the pipe, that is Aρgdscosθ = Aρgdy. The total force will be the sum of the two, and as the mass of the element in question is dm = ρAds, by applying the second law of dynamics he arrives at:

⎛ S dv v dA ds ⎞ dp + ρgdy = −⎜⎜ p p − ⎟ ρds A ds dt ⎟⎠ ⎝ A dt

[10. 13]

That by substituting ds = vdt = vpSpdt/A, and the subsequent integration yields the following expression:

dv p = − ρgy − ρS p p dt

∫

2 ds 1 2 S p + ρv p 2 + C A A 2

[10. 14]

In order to determine the integration constant C, he applies to the former formula to a path that runs inside the cylinder from the bottom part of the piston to point D. The pressure on the piston will be pp = Fp/A, the section Sp = A, and the height y = x. The final result is the following expression [§.XV]:

⎛ p = p p − ρg ( y − x ) − ⎜ S p ⎝

∫

s 0

2 ⎞ ds ⎞ dv p 1 2 ⎛⎜ S p − x⎟ρ + ρ v p 2 − 1⎟ ⎜A ⎟ 2 A ⎠ dt ⎝ ⎠

[10. 15]

APPLICATION OF FLUID MECHANICS

465

Equation that takes the place of the [10.5] given by Pitot. On the other hand, given that the pressure in the discharge spout G is zero,18 if in the previous equation is substituted p = 0, y = h, the velocity vp(t) of the piston can be determined. This, as Euler noted, is not arbitrary. The differential equation he arrives that is the following one:

dt = ρ( x + H )

dv p 2 ⎞ 1 2 ⎛⎜ S p p p − ρ g ( h − x ) + ρ v p 2 − 1⎟ ⎜S ⎟ 2 ⎝ G ⎠

[10. 16]

Where H = Sp∫ds/A is extended throughout the entire length of the tube from D to SG, which is the section at the outlet. The resolution of this equation will complete the solution to problem. Once he arrives at this point, and in view of the complexity of the theme, Euler introduces the simplification that the height of the tank be very great in comparison with the run of the pump [§.XVI], that is to say h >> x and H >> x, which is reasonable, and it simplifies the formulas. With this hypothesis he obtains times, flow volumes and other variables as well as the rule that the efficiency is better as the ascending tube is shorter. And in order to complete the examination [§.XXXI], he carries out a parametric study based on the minimum static force needed for the pump to begin functioning, which is that required to sustain a column of water between the pump and the drain, that is, Fmin = ρSh. With the introduction of a parameter λ so that Fp = λFmin he obtains non-dimensional tables for the section, flow volume of the water and pressure in several points, whose usefulness he justifies with various examples. We can clearly see the difference that exists between the solutions of Euler and Pitot. Not only are the 12 years that lie between them evident, but also Euler was already applying his theory of fluid dynamics, therefore his warning concerning the difference between the static and dynamic solutions was fully justified. In spite of Euler’s superior concept, we should not forget that Pitot was on the very edge of Bernoulli’s equation.

18 In reality it is the atmospheric pressure. However throughout the entire development he works with relative pressures, which is zero at G.

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THE GENESIS OF FLUID MECHANICS, 1640–1780

Borda’s studies

In 1768, Borda’s paper in the Mémoires de l’Académie of Paris entitled ‘Mémoire sur les pompes’ (‘Memoir on Pumps’) appeared, and in spite of the generality of its title he says that ‘the object of this report is to examine the effect of the narrowing or contractions that the columns of water moving in the pumps experience when going through the passages of the valves’.19 In order to bring his ship into harbour, so to speak, he used the theoretical estimation of the loss of live force produced in the narrowing of ducts as the calculation basis. This must be equal to that supplied from the exterior.

S r

Mg

A

B

C

F

H M

I Q

K L N

O

Fig. 10-7. Borda’s pump

In order to develop his theory, Borda used the model of the pump shown in Fig. 10-7, and in his view the results found can be generalised. The ensemble consisted of a cylinder with a plunger which in the ascending phase sucks the water up from a lower tank MQ through the bottom part of the pump NO. In this process the water passes through a valve KL to fill the cylindrical body. 19

‘Mémoire sur les pompes’, p. 418.

APPLICATION OF FLUID MECHANICS

467

Simultaneously, the piston raises the liquid found in the upper part of the cylinder, which enters into the upper tank through the aperture CF. In the descending phase, the previously suctioned water, now retained by the closed valve KL, passes through the upper part of the cylinder through a valve HI, located in the piston. Definitively speaking, water transportation from the lower tank to the upper one takes place in the ascending phase of the cycle, while in the descending phase there is only the passage of liquid through the valves and a small portion of liquid goes to the upper tank. This portion corresponds to the volume occupied by the rod of the piston. The set as a whole is moved by a rod and crank mechanism, whose driving force is generated by a mass M which descends due to the force of gravity. Borda’s objective was to find the relations among all the parameters, geometric and mass, with special emphasis on the loss of live forces originating in the passage through the valves, narrowings, and widenings. In order to calculate these he says: ‘I shall use the principle of conservation of live forces in the same way as I did in my report of the theory of fluids’.20 By contrast with earlier work, in which he studied the instantaneous loss of live force in the outlet of the fluid through an orifice, he now integrates the losses during the complete revolution of the crank. He states that in this period the total increase of live force has to be zero; then the sum of all the losses in the valves and stages has to be equal to the variation of the applied moments. According to his theory, if the velocity of the liquid at the entrance of a narrowing is v, and at the outlet it is κv, κ being a velocity loss factor which is less than the unit, the loss of live force, measured as a height, will be v2(κ – 1)2/2g. In establishing this formula, he recalls the report quoted where he set out this demonstration.21 The basis of this argument is founded on the supposition that the fluid entering collides with the fluid in the inside in an inelastic manner, this being what causes the loss. Borda supposes that the angular velocity of the crank is constant,22 therefore the linear motion velocity of the piston is v = ωrsinθ, which will be that of 20

Cf. ‘Mémoire sur l’écoulement des fluides’ pour les orifices de vases’, Mém. Acad. Paris, 1766 (p. 419). 21 In the §.12–13 of the op. cit. he explains the argument. If a body of mass m1 and speed v1 collides with another of m2 and v2, in the collision they become embedded in each other, the resulting velocity will be v = (m1v1 + m2v2)/(m1 + m2). Consequently, the loss of live force in the process will be expressed as:

∆Fv =

m1v12 m2 v22 (m1 + m2 )v 2 m1m2 (v1 − v2 ) 2 + − = 2g 2g 2g m1 + m2 2g

He estimates that this phenomenon also takes place when a liquid enters a tank through an orifice. This is not justified, given the geometry of the whole.

22

THE GENESIS OF FLUID MECHANICS, 1640–1780

468

the water upon entering the valves. In the ascending phase, the fluid goes through the section NO of the inlet and through the valve KL, whose velocity loss factors are designated as µ and ν. Moreover, in the transit CF to the upper tank the quantity of fluid that ascends is less than that which enters the lower part due to the cross section of the rod. The proportion will be (A – a)/A, A and a being the areas of the cylinder and the rod. Therefore the remaining fraction a/A will enter the descending phase. On the other hand, the surface of the collecting tank is theoretically infinite, therefore κ = 0, as the final velocity is zero. With these details the live force lost in a given instant of the ascending phase will be:

v2 ⎛ A−a⎞ 2 2 ⎜ ( µ − 1) + ( ν − 1) + ⎟ A ⎠ 2g ⎝

[10. 17]

Now the volumetric flow in an elemental time is dq = Avdt, so the total loss in the ascending half cycle will be expressed as:

ω 2 r 2 sin 2 θ ⎛ A−a⎞ 2 2 ⎜ ( µ − 1) + (ν − 1) + ⎟ Ar sin θ dθ = 2g A ⎠ 0 0 ⎝ 4 Aω 2 r 3 ⎛ A−a⎞ 2 2 = ⎜ ( µ − 1) + (ν − 1) + ⎟ A ⎠ 3 2g ⎝

∫

π

v2 dq = 2g

∫

π

[10. 18]

During the descending phase, there is only one transit through the piston valves HI, whose factor is σ, and at the same time—because of what we have said concerning the surface of the rod—part of the water will pass into the tank. For similar reasons, the live force lost in this descending phase will be:

4 Aω 2 r 3 ⎡ ⎛ A ⎞ a⎛ a ⎞ ⎟ ⎟ + ⎜ ⎢⎜ σ − 3 2 g ⎢⎣⎝ A−a⎠ A⎝ A− a ⎠ 2

2

⎤ ⎥ ⎥⎦

[10. 19]

which, added to the former, will give the total loss in the cycle. On the other hand, the work supplied to the system by the weight P is 2πRP, and the quantity of water raised is 2rA, where r is the crank radius. If the height at which the collecting tank lies with respect to the feed tank is H, the energy contributed to the water is 2ρgrAH, thus giving a final result:

APPLICATION OF FLUID MECHANICS

469

2πRmg − 2 ρgrAH 2 2 4 Ar 3ω2 ⎡ A−a ⎛ A ⎞ a⎛ a ⎞ ⎤ 2 2 = + ⎜σ − ⎟ + ⎜ ⎟ ⎥ ⎢( µ − 1) + ( ν − 1) + 3 2 g ⎢⎣ A A−a⎠ A ⎝ A − a ⎠ ⎦⎥ ⎝

[10. 20]

a complex formula, in which the losses are contained in the term on the right.23 Borda completes his work by assuming a similar pump, but moved by a linear force, where the treatment of the losses in the narrowing is the same. He ends his memoir by applying his theory to three types of pumps then existing: one used in a mine24 moved by a steam engine, another used to dry out a lake, and the third of the type used in ships. In the three he concludes that with the use of his theory several of the geometric parameters of the pumps could be varied, thus improving performance. Daniel Bernoulli on jet propulsion

In the eighteenth century there were only two methods of propelling ships: by sail or by oars. There was a search for alternatives, however, as evidenced by the theme of the prize of the Academy of Paris of 1753, won by Daniel Bernoulli, with a device which was a sort of propeller. Years before, in the Hydrodynamica, he had suggested a procedure based on making use of the reaction produced by a jet of water ejected backwards from the stern. Daniel developed the idea at the end of Chapter XIII of this work, which he subtitled ‘Reaction of fluids flowing out of vessels’. He writes: It occasionally has occurred to me that these things I had thought about concerning the repelling force (vi repellente) of fluids while they are ejected, the majority of which I have expressed here, could be usefully applied to establishing a new method of navigation. For I do not think there is anything preventing ships being moved without sails and oars by this method: the water is raised continually to a height and then flows out through orifices in the lowest part of the ship, causing the direction of the out-flowing water flowing to face towards the stern. [XIII, §.20]25

We represent his idea in Fig. 10-8, and it consists of a ship provided with a water tank which discharges through a pipe in the stern. The tank is maintained at a constant level by some procedure or other, which he initially supposes could 23

The point out that in Borda’s original work there are several errors. He says that it was the mine of Montrelais near Ingrande-sur-Loire. 25 The references inside inverted commas correspond to the Hydrodynamica. 24

470

THE GENESIS OF FLUID MECHANICS, 1640–1780

even be by rainfall, but by the end he identifies it with the raising of water by means of operators. The propulsion force will be due to the jet reaction, although part of it will be absorbed in the process of replacing the water. What we need to establish is ‘how much resistance the ship experiences from that perpetual and uniform inflow of water and its inertia’ [XIII, §.22]. The magnitude of the reaction force of the outlet jet had already been determined by Newton26 as Bernoulli acknowledges. However, Bernoulli used a different method from Newton to make the calculation, and we shall give a brief summary of it. The outlet velocity of the water, in the supposition of a tank having a very large surface with respect to the outlet orifice, is vs = 2 gh . In a time t, a quantity of water should have flowed out which would be the equivalent of the cylinder whose length is L = vst. If the area of the outlet orifice is S, the mass of the cylinder will be m = ρLS. Daniel asked himself what force would be required so that the velocity vs would be imprinted upon the cylinder in this time t, and gives the answer immediately, F = mLvs2 = ρSvs2, which he expresses as a function of the kinetic height ξ as F = 2ρgSξ.27

vs v0

Fig. 10-8. Ship propelled by a jet

This calculation would be valid if the ship was still, but if it is in movement the water replacing the outgoing water (would either come from rainfall or would be taken directly from the sea) passes from rest to a velocity equal to that of the ship. That is, it must be provided with a certain momentum that is equivalent to a resistance, which is what he questioned previously. He makes the calculation using the procedure similar to the previous one, although we shall simplify it. The quantity of outgoing water in a time dt is dm = ρSvsdt. If the velocity of 26

Cf. Principia, Prop. XXXVI, Cor. 2. One quick way of making this calculation is simply to say that the force is equal to the change rate of the momentum ejected. The mass flow is ρSvs, and the rate of momentum ρSvs2, which what he found. 27

APPLICATION OF FLUID MECHANICS

471

the ship is v0, the replacement water taken from the water at rest will acquire a momentum of d(mv) = v0dm = ρSvsv0dt, which is equivalent to a resistance force FR = ρSvsv0. The resulting propulsion force will be the difference between the impulsion and the resistance:

F = FI − FR = ρSvs2 − ρSvs v0 = m ( vs − v0 )

[10. 21]

That is what he calls the absolute advance force (potentia absoluta). From a simple inspection of the formula it can be appreciated that the advance force is greater when the vessel is stationary (vs = 0), and that the maximum velocity that the vessel can acquire is equal to the outflow velocity (vs = v0). Having established the theory, Daniel goes on to study how to fill the tank using operators. He supposes that these workers are capable of generating a driving power W, which he interprets as the capacity of raising N cubic feet of water to a height of one foot in one second. Using this power as data he looks for the most suitable relation between the outflow velocity and the ship’s advance velocity, or what comes to the same thing, the most suitable height that the tank must have in order to achieve maximum efficiency. The power W is used to raise the mass flow per unit time q to the height h; i.e., W = qgh, and substituting h by velocity he ends in q = 2W/vs2. Substituting in equation [10.21], he finds the relation between the power and the outflow velocity and advance:

F=

2W 2W ( v s − v0 ) = 2 vs v0

⎛ v0 v02 ⎞ ⎜⎜ − 2 ⎟⎟ ⎝ vs vs ⎠

[10. 22]

Whose maximum value is verified by vs = 2v0, and from which the advance force results, so that:

F=

W W = vs 2v0

[10. 23]

Starting from this result it is easy to obtain the height of the tank. This last equation allows us to offer a pretty clear physical interpretation. Given that the power supplied to a moving body is equal to the product of the force driving it multiplied by the speed, i.e., Fv0, the last equation [10.23] indicates that this is exactly half of the total power applied, W. This means that half of this power is used in the advance, and the other half in contributing the momentum to the water used as a propelling agent.

472

THE GENESIS OF FLUID MECHANICS, 1640–1780

These calculations assume that the ship is not subject to any other resistance upon advancing. However in real life circumstances, he states that ‘the relation between the velocities of the ship and forces propelling the ship is to be assumed as known; here it is commonly stated that velocities are in proportion to the square roots of the propelling forces’ [XIII, §.26], which means that a relation of the type F = kv02 exists28 between F and v0. The introduction of this in equation [10.23] leads to the following expression of the optimum velocity:

v0 = 3

W 2k

[10. 24]

He ends with a long scholium [§.29] in which he discusses the application of his new method to a galley with 260 oarsmen who raise the water instead of rowing. On the assumption that the power contributed by each operator will be 87 Watts,29 his calculations lead him to an advance velocity of 2.75 m/s, which he says is very high for a ship of this type. In conclusion, although this method is suitable in theory for propelling a vehicle, the true usefulness of a jet system appears when the outlet velocity of the jet is high, thus implying the needs of a greater energy contribution to the jet itself. Unfortunately, in the eighteenth century, the only source of such energy imaginable was those that came from a fall from a height.30 Segner’s turbine

A rotational machine with the same propulsion basis as that proposed by Daniel Bernoulli was presented by János-András Segner in 1750. We analyse this machine according to his description of it in his Theoria machinæ cuisdam 28

Daniel introduces this not so obviously, using a kinetic height which he designates as C, but which we shall avoid in order to facilitate the understanding of his arguments. 29 These calculations are difficult to verify. On the other hand in Chap. IX, §.43, he makes an experiment which results in the power that a man may supply is 56 W. 30 It worth pointing out that Benjamin Franklin thought a lot about Bernoulli’s jet propulsion. In a letter he wrote to Mr. Alphonsus Le Roy in August 1785 entitled “Sundry Maritime Observations” [later published in the Transactions of the American Philosophical Society for Promoting Useful Knowledge, Vol. II, pp. 294–329] he went over that system, and proposed improving it by the addition of one pipe and a pump for collecting and raising the water to the reservoir. Franklin asked whether the work for pumping would be less than for rowing, and added that ‘a fire-engine might possibly in some cases be applied in this operation with advantage’. But he also explained the possible use of air instead of water; where the mechanisms would be more complicated due to the necessity of valves and an auxiliary reservoir, but the system at least in theory, could work properly (This comment is due to Larrie Ferreiro).

APPLICATION OF FLUID MECHANICS

473

hyraulicæ (Theory of a certain hydraulic machine), Göttingen, 1750.31 The Fig. 10-9 shows a diagram of the machine drawn according to the figures of his work. Segner describes it as a ‘machine constituted by any rotating vessel, placed so that it can turn freely around its axis in a vertical position. Tubes come out from the bottom of the vessel, two, three, four or more tubes up to any number [§.2]’32. H Q C

A

L

B F

G D

Fig. 10-9. Segner engine

The figure only shows a tube sloping down, at the end of which is an orifice F, from which the water comes out tangentially, generating the motion around the CD axis. Likewise it shows the weight Q that is raised by means of a cable and pulley H. The ejected liquid, apart from generating a driving torque, also generates a damping torque due to the combined effect of the rotation and the motion of water through the pipe, which Segner considers as resistance, as it counteracts the driving effect of the former. We call the former a damping torque because it is proportional to the rotation velocity of the machine—as we shall see—, thus it is zero if this is not rotating and limiting the maximum rotational velocity. 31

According to what A.P. Youschkevitch and A.T. Grigorian say in their article about Segner in their Dictionary of Scientific Biographies, there are two more works on this subject: Programma in quo computatio formae atque virium machine nuper descriptae (Göttingen, 1750) and Speciment theoriae turbinum (Halle, 1735). In addition there are several letters to Euler. 32 The quotes refer to the Theoria machinæ cuisdam hydralicæ.

THE GENESIS OF FLUID MECHANICS, 1640–1780

474

R

vt

vs F P

r P'

vF

dt

A vs dt Fig. 10-10. Rotating arm

Segner first calculates the force due to the outflow of the liquid from the spouts of each one of the pipes. If the outlet velocity is vs, the outlet section A and the density ρ, this force will be F = ρAvs2. However, given that Segner works with kinetic heights, he did not arrive at this expression, but at F = 2ρgAξs, being vs2 = 2gξs, i.e., the reaction force is equal to double the weight of the kinetic column [§.8].33 In the case where there were n pipes and the arms of these were R, the engine torque would be given by the expression M = QRH = 2nρgAξs. This will be met ‘if the machine is maintained in equilibrium’ [§.12]; when this is not so, and the machine turns, the aforementioned damping moment must be introduced. In order to evaluate the latter, he supposes a horizontal pipe (Fig. 10-10) that turns around one of its ends so that the velocity at the other end is vF = ωR and through which a fluid circulates at a velocity vs. After a series of arguments and theorems he arrives at: The absolute resistance of a particle, which we suppose to move from A to F, is equal to the force acting with a constant value, which during the time of the motion can imprint upon the particle the velocity at which F moves in the circle. [§§.23– 24]

Let dm = ρAdx be this particle located at point P, the time of stay in the pipe from A to F is tR = vs/R, therefore the acceleration in order to reach the velocity vF will be a = vF/ts. Therefore, the resistance will be dF = vFvsdm/R, which integrated throughout the length of the pipe will give a total damping torque: 33

Let us recall how this force is calculated. That is F = d(mv)/dt = vdm/dt. In a time dt the flow of water ejected is dm = ρAvdt, which introduced into the former equation leads to F = ρAv2.

APPLICATION OF FLUID MECHANICS

Ma =

1 1 ρARv F vs = ρAR 2ω vs 2 2

475

[10. 25]

resulting for the total torque in:

1 ⎞ ⎛ M = nρARvs ⎜ vs − ωR ⎟ 2 ⎠ ⎝

[10. 26]

With his system of kinetic heights, Segner obtains the expression ρgAR(2ξs2 – ξsξF) for this last equation. However, there is an error in this argument. The real damping is just exactly double that calculated. This is because the velocity of the particle which at the instant t is at point P, has two components: a radial one which is precisely vs, and another tangential one, ωr. After a time interval dt, the point P passes to P′, located at a distance r + vsdt from the origin. The radial velocity will be the same in magnitude, although rotated through the angle ωdt, that is, equal to the first plus a tangential component of value ωvsdt, which is what Segner calculates. But in addition, at the point P′, the tangential velocity will have increased with respect to the velocity it had at P in ωdr = ωvs dt. whose value coincides with the former. In other words, in the step from P to P′ , the particle is submitted to a tangential acceleration whose value is 2ωvs, which nowadays known as the Coriolis acceleration.34 With this detail, the expression of moment given in equation [10.26] becomes:

M = πρ ARvs ( vs − ωR )

[10. 27]

Which leads to a limit velocity vs = ωR, which indicates that in this case the fluid will flow out at a relatively null velocity.35 In spite of his detail, Segner’s work is of extraordinary value, not only intrinsically, but also because of the application that Euler made of it. Euler’s analyses

Segner sent Euler several letters about his machine, which motivated Euler to devote three works to it, published in the Mémoires de l’Académie of Berlin, in 34 Apart from this a radial acceleration exists which is the centrifuge force. For more details please see note number 36. 35 This formula can be deduced from the one found later by Euler, shown in general form as equation [10.32].

476

THE GENESIS OF FLUID MECHANICS, 1640–1780

which he not only applied his wide theoretical knowledge to the apparatus, but also proposed a substantial improvement to it. In the first, ‘Recherches sur l’effet d’une machine hydraulique proposée par Mr. Segner professeur a Göttingue’ (1750), Euler analyzed the movement of fluids through the pipes and proposed modifications to the machine. In the second, ‘Théorie plus complette des machine qui sont mises en mouvement par la réaction de l’eau’ (1754), he continued he extended his examination significantly, and proposed another machine, now greatly evolved. The third, entitled ‘Détermination de l’effet d’une machine hydraulique inventé par M, Segner, professeur à Göttingue’ appeared only in his posthumous works.

Fig. 10-11. Euler version

Euler, as usual, carried out the very general and careful mathematical analysis which surpassed the field of applications in order to enter into the dominions of pure theory. We shall examine the analysis he made of Segner’s machine, and then look at his studies on the movement of fluids in rotating tubes, which ended with a proposal for improving what we have called the Segner-Euler machine. In Fig. 10-11 a diagram of Segner’s machine is shown, made from the figures presented in Euler’s first report. The apparatus consists of a cylindrical tank which can turn around its central axis. In the lower part it has a series of radial tubes fixed to it, each one with the lateral orifice at its end. When the water comes out of these orifices, it introduces a tangential reaction which makes the entire ensemble rotate. In his analysis, Euler warns that the forces acting on the fluid are very different when the ensemble is at rest and when it is in motion. For this reason, he

APPLICATION OF FLUID MECHANICS

N

t

ds

T

477

M n

r

v0

R r0

Fig. 10-12. Horizontal rotating tube

begins by isolating an element of fluid inside the tubes, and obtains the kinematic and dynamic conditions to which they are subject. Once this was done, he goes on to deduce the force and total moment that act in the tube for an integration process along the length of the tube. Figure 10-12 shows a rotating tube with a variable cross section, contained in a horizontal plane, from which the circular tank comes out, and where he isolates the differential element under study. In this model Euler proposes a successive series of problems tending towards the goal stated. The first is to obtain the law of pressures throughout the tube, supposing that the system rotates with an angular velocity ω, and that at its entrance the water has a velocity of v0. The second is to determine the rotation torque produced by the water when it circulates and flows out, causing the machine to turn. The third is the application in the specific case in which the tank has a certain depth. He ends with some particularisations, and additional optimisations that hold no significance for the theory of the system. We shall examine each one of the three steps in some detail.36 Pressure in the tube

The fundamental point is to establish the dynamic equilibrium of the water contained in the differential element of the tube, which is the result of the inertial forces due to the acceleration of the water circulating and of the external forces,

36

In his works he uses the notation of the kinetic heights for the velocities, and also interprets the rotations as lineal velocities at the ends of the tubes, which in turn he reconverts into kinetic heights. All this makes it quite difficult to follow his arguments.

478

THE GENESIS OF FLUID MECHANICS, 1640–1780

which, as the motion is horizontal, will consist solely of pressures. The process is it similar to the case of motion that he has already studied tubes in pumps, although now there is the additional complication of motion. The route followed by Euler in order to find the acceleration is to determine the velocity of a differential element (Fig. 10-12) referred to a fixed frame of axes. To do so he takes the entry velocity of the water in a tube, v0, and the angular velocity ω. Once he finds the velocity, he obtains the acceleration by means of a derivation with respect to the time.37 The process is very laborious, made worse by a complicated system of coordinates used to define the tube. The following equations are the final result, and they represent the radial and tangential accelerations, obtained by a different procedure, but equivalent term by term to those found by Euler [pp. 318–319]:

37 The calculation of the final acceleration can be considerably simplified by using the kinetic method of the composition of relative motions. We recall that in a motion composed of a fixed axes system and other mobile ones, the absolute acceleration of a point is determined by the following expression:

G G G G G a21 = a20 + a01 + 2ω01 × v20

Where the fixed frame is indicated by the sub-index 1, the mobile by 2 and the intermediate one by 0. In the case being studied, the latter is linked to the vessel and turning jointly with it at the velocity ω, and the water moves in the vessel. The term a20 represents the acceleration of the particle of water with respect to the tube, or cylinder, which comprises two components: one in a tangential direction and the other in the perpendicular one. The former is subdivided in other two: one is due to the variation of the water entry velocity v0, which will have the value A0dv0/Adt, and the other to the variation of the lumen, which, given that Va = Cte, will be dv = –vdA/A. Concerning the normal component, caused by the curvature of the tube, that will be v2/ρc, the curvature is expressed as:

dθ 1 sin θ = + cos θ r dr ρc

The second term a01 is the acceleration at the point in question, but considering the latter as belonging to the tube with respect to the fixed frame, and it is composed by a component following the normal to the radius vector OM, that due to the angular acceleration whose value is rdω/dt, and another centripetal along the radius vector, which is –ω2r. When this two components are referred on to the perpendicular and tangential directions they break down according to the sine or cosine of the angle. The third component 2ω01 × v20, is nowadays known as the Coriolis acceleration and is a vector perpendicular to the plane formed by θ and v, that is to say, in the positive perpendicular direction rR and the module 2ωv. The equations given by Euler are reached through using these elements with their signs, reducing the local velocity v to the entry velocity v0 by the simple artifice of the continuity, and by eliminating the time by the equation ds = vdt.

APPLICATION OF FLUID MECHANICS

an = −ω2 r sin θ + 2ωv0

479

A0 v02 A02 + − ω r cos θ A ρc A2

[10.28]

A0 v02 A02 dA − 3 A A ds

[10.29]

at = − ω2 r cos θ − ω r sin θ + v0

The directions of the vectors tangent and the normal are Tt and Nn as depicted in Fig. 10.12, just as Euler showed. In order to find the distribution of pressures, he makes use of the equality dF = Adp= –Aρatds, which, integrated with the condition that the outlet pressure has to be zero,38 and bearing in mind the geometric relation dr = cosθds, yields [p. 322]39:

∫

se 1 p = − ω 2 ( R 2 − r 2 ) − ω r sin θds + v0 A0 2 ρ s

∫

se s

ds 1 ⎛ A02 A02 ⎞ ⎟ [10. 30] + ⎜ − A 2 ⎜⎝ Ae2 A2 ⎟⎠

Where the sub-index e represents the outlet conditions. At this point Euler warns about the possibility of negative pressures appearing, which can be explained by the fact that he is working with relative, and not absolute pressures. An immediate case in point is that of uniform motion. In this case, the former equation is reduced to:

1 1 ⎛ A2 A2 ⎞ p = − ω2 ( R 2 − r 2 ) + ⎜⎜ 02 − 02 ⎟⎟ ρ 2 2 ⎝ Ae A ⎠

[10. 31]

We note that on this assumption, it is simply a question of the geometry in the inlet and outlet of the tubes and the solution is independent of the intermediate configuration. By way of an aside, we comment that on this point [Esc. II, p. 324] Euler introduces a few considerations regarding the losses through friction. He says:

38

Refers to the relative pressure. He recalls that the absolute pressure is the coefficient of the force divided by the surface, while the relative, or differential pressure, is the same but minus the atmospheric pressure. 39 Les references between brackets correspond to the ‘Recherche sur l’effect d’une machine hydraulique proposée par M. Segner professeur a Götingue’.

THE GENESIS OF FLUID MECHANICS, 1640–1780

480

The water passing through the tubes experiences a type of friction [while in the tubes] like all solid bodies, and experience teaches us that when the tubes are very narrow, the water in them undergoes a very considerable reduction. However I do not know of anyone who has as yet discovered the rules governing this friction: therefore I shall make an attempt to arrive at this goal, though I am sure that this question requires more profound investigations. [p. 324]

While awaiting this promised study, he supposes that the losses are proportional to the ratio between the pressure and the diameter. He operates upon this basis and obtains a collection of formulas that we shall pass over, given the falsity of the hypothesis. This is one of the a few attempts made in the eighteenth century to undertake the analysis of the losses through friction in a pipe. The solution to this problem had to wait until much later.40 Turning torque

Picking up the thread of the argument again, we now proceed the second of the steps taken by Euler. This consists in determining the turning torque. In order to find it he follows two routes. The first and longer route directly integrates the forces of pressure on the fluid element. The second and faster route uses the two accelerations at and an projected on the tangent to the radius vector in the form:

dM = ρAr ( at sin θ + an cos θ )ds

[10. 32]

An equation which once integrated yields [p. 226]:

M = ρv 0 A0 ∫

se 0

r sin θds − ρω ∫

R sin ζ +ρv A Ae 2 0

se 0

r 2 Ads − ρ ω v 0 A0 ( R 2 − r 2 ) [10. 33]

2 0

where ζ is the angle of the water outflow, and, when it is 90°, it produces the maximum effect. If the movement is uniform the first two terms disappear.

40

A work that analysed the movement through tubes was the ‘Tentamen theoriæ de frictione fluidorum’ Novi. comm. petrop., 1756/1757. In this one he supposes that the losses by friction follow this law precisely [cf. §.9–11]. Perhaps this is the study he promised in 1750.

APPLICATION OF FLUID MECHANICS

481

Tank height

Upon introducing the height of the tank, he, the machine is completed. Constructively, Euler also supposes that the inner radius r0 = 0, which is easy to estimate by prolonging to the vertical shaft the tubes on the inside. With height he, the pressure at the entrance of the pipes will be p = ρghe – ½ρv02, which identified with equation [10.31], and with inner radius r = 0, yields:

A02 2 ghe = −ω R + v 2 Ae 2

2

2 0

[10. 34]

and as the outlook velocity is ve = v0A0/Ae, he ends with the simple formula [Cor. II, p. 339]:

ve2 = 2 ghe + ω2 R 2

[10. 35]

which when introduced in the torque yields expression such as:

M = ρAe R( 2 ghe + ω2 R 2 ) − ρωAe R 2 2 ghe + ω2 R 2

[10. 36]

a formula which relates the torque produced with the rest of the parameters, and which Euler continues to manipulate until he concludes with parametric tables for use. [p. 341] Segner-Euler turbine

Four years later, Euler returned to the idea of tubes rotating in space with the second of his memoirs, ‘Théorie plus complete de machines qui sont mises en mouvement par la reaction de l’eau’ (‘A more complete theory for machines put in motion by the reaction of water’). He begins this with the statement that, ‘Having already explained in some reports the effect that the machine projected by Mr de Segner is capable of producing, I here propose to develop the same subject in greater detail’ [p. 227].41 And fulfilling his promise he devotes nearly 70 pages to the topic. However, another work dealing with this matter exits, ‘De motu et reactione aquæ per tubos mobiles transfluentis’ (On the motion and reaction of water flowing through moving tubes) which appeared in the Novi 41 The references in bracket correspond to the page numbering of the ‘Théorie plus complete de machines qui sont mises en mouvement par la reaction de l’eau’.

482

THE GENESIS OF FLUID MECHANICS, 1640–1780

commentarii petropolitanæ, Vol. 6, 1761, and which seems to be a preliminary sketch of the former. The basis of his arguments is similar to the previous case: he first attempts to find the equations of motion in a narrow tube, and then to apply them to the machine. The difference is that now the tube is not held on a horizontal plane, but is a warped curve, which complicates the kinetic formulation and causes the weight to intervene as a component of the external forces. Nevertheless, although he continues to employ the same method to find the acceleration, he now uses cylindrical coordinates (Fig. 10-13) instead of the components along the local tangent and normal,42 which makes the formulation somewhat easier. In the cylindrical coordinates the first of the three components follows the radius vector, the second the vertical axis, and the third goes in the direction of the tangent to the radius vector, not the tangent to the tube. He states that these last two form an angle β, which he introduces to make the formulation easier. The figure shows the descending tube MC which turns around the vertical axis Z with the three coordinates r, ω and z, and the angle β. Using a procedure similar to the plane case already seen, and which we shall not repeat here, the three components of the velocity in the point M are obtained [§.XXXV] as:

vr = v

dr ; ds

vθ = v cos β + ω r ;

vz = v

[10. 37]

Where v is the local velocity of the fluid with respect to the tube and s the length of the tube. uz

z

ur u

y x

r

Fig. 10-13. Tube in three-dimensional space 42

This system is denominated intrinsic coordinates.

APPLICATION OF FLUID MECHANICS

483

As far as the accelerations are concerned, Euler obtains them by direct derivations, although they can be obtained more quickly by a composition of movements. We shall transcribe his results [§.XLVIII], once the corresponding transliterations have been made. These are the three following equations:

dr v 2 cos 2 β d dr a r = −ω r + v − + v2 A − 2ω v cos β ds r ds Ads 2

az = v

aθ = ω r + v cos β +

d dA dz + v2 A ds ds ds

dr d cos β v 2 dr cos β + v 2 A + 2ω v ds ds A r ds

[10. 38]

[10. 39]

[10. 40]

Of these, only the last will give rise to a rotating torque, as:

∫

M = ρ vθ rAds

[10. 41]

In order to carry out this integration, Euler takes the outlet Ae as the control section, to which he reduces the velocities so that v = veAe/A. We recall that in the plane case the reference section was the input of the tube. He ends up with [§.IL]:

M = − ρω µ − ρ v e v − ρ v e Ae (v e re cos β e − v 0 r0 cos β 0 ) + ρ ωAe v e (re2 − r02 )

[10. 42]

In which the sub-index 0 represents the conditions at the inlet, and e at the outlet.43 This equation is equivalent to equation [10.33] of the plane case.44 As regards the parameters µ and ν, their values are:

43

The following equality was made use of in the calculations:

⎛ cos β ⎞ r cos β − ⎟= A ⎠ A

∫ rd ⎜⎝

∫

cos β dr A

44 If one tries to reduce the plane case of the previous equation [10.42], differences in the sign will be found that affect the angular velocity, as he has taken the contrary in each case.

THE GENESIS OF FLUID MECHANICS, 1640–1780

484

µ=

∫

se

Ar 2 ds ;

∫

se

ν = Ae r cos β ds

0

[10. 43]

0

The velocity v0 appears in these equations, and he says of it that ‘the velocity of the water does not depend on our wishes; it must be determined by the principles of hydrodynamics and its value substituted immediately in the formula found’ [§.LXIV]. By this he means that the velocity is an input data of the problem, and when he applies these formulas further on to the turbine, subject of this memoir, this is what he will do. In order to obtain the pressure throughout the tube he uses the tangential acceleration. That is, he goes from the cylindrical coordinates to the intrinsic ones. We note the inverse route of the plane case. Without going into algebraic details, the final result turns out to be [§.LXXII]:

at = ω r cos β + v − 2ω r

dr d 1 − v 2 A2 ds ds A2

[10. 44]

to which gravity must be added in order to balance the forces of pressure in a differential element, i.e.:

dp = − ρ ( vt ds + gdz )

[10. 45]

which once substituted and integrated, leads to the following final equation [§.LXXV] : s

s

0

0

p = ρg (a − z ) − ρω ∫ r cos βds − ρ v e Ae ∫

ds + ρ ω 2 (r 2 − r02 ) A [10. 46]

⎛ A2 A2 ⎞ − v ⎜⎜ e2 − e2 ⎟⎟ A0 ⎠ ⎝A 2 e

Euler analyses each one of the terms in detail [§.LXXVI-ff]. In the outlet section the pressure is zero and the height z = 0, and therefore an additional equation will be obtained. In the case of stationary motion, the solution is fairly simple:

2 ga − ω2 ( r02 − re2 ) v = 1 − Ae2 A02 2 e

[10. 47]

APPLICATION OF FLUID MECHANICS

485

Whose similarity with equation [10.35] is evident. Bearing in mind that veAe = v0A0, this equation can be reorganised as:

ve2 − ω2 re2 − ( v02 − ω2 r02 ) = 2 ga

[10. 48]

which he will use in the machine we shall describe next.

Fig. 10-14. The Segner-Euler machine

Once the theoretical analysis, which takes up 50 of the 70 pages of the report, is finished, Euler describes his new machine that we have called SegnerEuler. Figure 10-14 shows a diagram of it, and Fig. 10-15 a reproduction of the proposal made by Euler.45 From this figure, it can be seen that the mobile tank has been reduced to a crown of minimum dimensions, from which radial pipes go out, descending to a lower plane in order to gain potential energy. The fixed feed tank is located above the mobile tank, and supplies the latter with water by means of a set of sloping tubes. The idea is that the water supply should arrive at the tank with a tangential velocity equal to that of the tank, therefore the relative inlet velocity will be zero. In Fig. 10-14, for the sake of simplicity, only five outlet pipes have been shown, while Euler proposed placing as many as would be allowed by the geometrical dimensions of the crown. The set is enclosed in a 45

There is a preliminary sketch in the ‘De motu et reactione …’, as Fig. 4.

486

THE GENESIS OF FLUID MECHANICS, 1640–1780

trunk-conical flange that provides the whole with rigidity, and according to what he says, reduces the air resistance. The basic dimensions are represented in the figure. These are the height a of the rotor, the height k of the replacement water, the total height h, being the sum of the other two, the angle φ the water inlet to the rotor, the radius rc , and re at the crown and the outlet, the outlet area Ae of all the pipes, A c the area of the annular entry crown and Ai the intake ones. Other parameters linked to the foregoing are the angular velocity ω, the water flow rate Q, and the torque M to be driven by the engine.

Fig. 10-15. Euler’s proposal

Euler applied the results of the theory to a machine with these dimensions that functioned in a steady regime, and he arrived at a set of formulas that he called ‘the necessary requisites for constructing such a hydraulic machine’ [§.CV], consisting of the following:

a=h−

Q2 2 gAi2

1 ⎞ rc2 Q2 ⎛ 1 ⎜ 2 − 2 ⎟⎟ = 2 2 2 ⎜ re ω re ⎝ Ai Ac ⎠

[10. 49]

[10. 50]

APPLICATION OF FLUID MECHANICS

⎛ 1 1 ⎞ ve2 = ω2 re2 + 2 gh − 2Q 2 ⎜⎜ 2 − 2 ⎟⎟ Ac ⎠ ⎝ Ai Ae =

Q ve

487

[10. 51]

[10. 52]

To which he adds the slope of the feed tubes:

sin φ =

Ai Ac

[10. 53]

And three additional conditions:

Ae < Ac ;

Ai < Ac ;

rc2 > A0

[10. 54]

Equation [10.52] indicates the continuity condition, expressed as the constancy of the flow rate in the three series of orifices, the rotor outlets, the feeder outlets and the inlets to the tubes in the crown, i.e., Q = veAe = viAi = vcAc. Introducing the velocity vi in equation [10.49] he obtains vi2 = 2g(h-a), which is the expression of Torricelli’s Law. This velocity vi enters in a sloping fashion, forming an angle φ with the horizontal so that its horizontal component is ωrc and the vertical vc, so that the sine of this angle must be sinφ = vc/vi, which through continuity leads to equation [10.53]. Moreover, vi2 = vc2 + ω2rc2 is established, which is precisely equation [10.50]. With respect to the latter, re2 could be eliminated from both sides of the equation, although it is written thus respecting Euler. For the missing equation we take [10.49] that, as we recall, expressed the condition of steady motion. If we substitute a, and ωrc in this equation for the values already found, we arrive at ve2 = 2gh + ω2re2 – 2vi2 + 2vc2, which is equation [10.51]. With this set of equations, Euler declares that the design of this machine can be undertaken. The machine is defined by its seven parameters, which are: a, h, rc/re, Ai, Ac, Ae and φ. To these we have to add another three dynamic ones, Q, ve and ωre, which gives a total of ten,46 and as there are five equations that leaves us with five independent parameters. Euler takes as given the flow rate and the 46

If had separated re and rc it would have appeared one more obviously.

488

THE GENESIS OF FLUID MECHANICS, 1640–1780

height (Q and h), leaving Ac , Ai and ωr, to ‘be taken at will’ [§.CVI]. There is another additional condition, however. When it is functioning, the machine must supply the exterior with part of the energy that it receives from the fall of the water. This is materialised by means of a driven load or external torque, which in turn will condition the rotation velocity. The energy for time unit or power extracted is measured as being the product of the resistance torque multiplied by the angular velocity. The new condition introduced by Euler is that this power will be the maximum. The torque is expressed in equation [10.42] and therefore the power will be:

W = Mω = − ρAe ve2 reω + ρAe ( re2 − r02 )veω2

[10. 55]

Which with the same parameters used in the other formulas becomes:

⎡ ⎛ r 2 ⎞⎤ W = ρ Q ⎢− ω re ve + ω 2 re2 ⎜⎜1 − c2 ⎟⎟⎥ ⎝ re ⎠⎦ ⎣

[10. 56]

He now introduces ve and rc/re into this equation. These values are taken from the initial group [10.50 and 10.51] obtaining a complicated expression in ωre and Q2(1/Ai2 – 1/Ac2). In order to find the optimum value of W he derives equation [10.56] with respect to ω2re2 and makes the result equal to zero. We omit the complicated calculations.47 He ends by showing that this condition is determined by the simple formula:

⎛ 1 1 ⎞ Q 2 ⎜⎜ 2 − 2 ⎟⎟ = gh ⎝ Ai Ac ⎠

[10. 57]

The surprising fact is that this expression only relates the geometrical parameters of the system. As regards the power in this condition, it will be:

W = ρ Qgh

[10. 58]

which signifies that all the energy of the fall is transferred to the exterior, that is to say, the theoretical efficiency will be unity. Euler specifies all the formulas in 47 Euler, in his treatment of this matter [§.CXI-ff], drags an angle which he designates as ζ and which correspond to the one we have named βe.

APPLICATION OF FLUID MECHANICS

489

this condition, but we note only that the outlet velocity turns out to be ve = ωre; in other words, the absolute velocity of the water at the outlet is zero, a fact that is consistent with the efficiency obtained. Two unknowns still remain, which he takes as Ac and ωrc, and for which there are no further equations. Here Euler analyses the concept of the machine as an appliance which must function, and he obtains what is missing. He takes the angle φ, and starting from equation [10.53], and with the contribution of the other conditions he finds:

tan φ =

Q gh

[10. 59]

With this angle already defined, Ac = Ai/sinφ will be found and in addition a = ½h(1 – tan2φ). For the first, the angle ‘should not be too small as this would require too many diaphragms’ [§.CXVIII], where by diaphragms we understand the separation between the entrances of the tubes. But in the second he says that ‘it can be seen that the angle must be absolutely smaller than a half right angle’ [§.CXIX]. As with a given value of φ there remains only one parameter, which he calls λ, so that λ = Ac/Ai, and from which results:

ω re = λ gh tan rc 1 = re λ tan

[10. 60]

[10. 61]

From the latter [§.CXXII] and the definition of λ itself, he estimates as good that λtanφ = 3/2, with λ = 3 and tanφ = 1/2, i.e., φ = 26°34′. Euler continues with a table and some lengthy examples, although the length is justified as the work provides an excellent example of how theoretical knowledge is applied to improve a machine. It is a pity that at the time the technology for putting it into practice was not available.48

48

In the twentieth century, Jakob Ackeret studied Euler’s works in this area in detail, and he set them out in the ‘Editorial Introduction’ to the Volume 15(2) of the Leonhardi Euleri Opera Omnia, 1957. In this Introduction he relates the construction of a model of the turbine [p. XL-VI-ff] in 1944, which was submitted to testing and with which he managed to measure a efficiency of around 71%.

THE GENESIS OF FLUID MECHANICS, 1640–1780

490

Pitot’s tube

In 1732, in the Mémoires de l’Académie of Paris a work appeared entitled ‘Description d’une Machine pour mésurer la vitesse des Eaux courantes, & le sillage des Vaisseaux’ (‘Description of a machine for measuring the velocity of running water and the day’s run of ships’) by Pitot. In the preamble, he talks about the currents of rivers and the friction with the banks and the riverbed, a theme that was then under discussion. In order to elucidate these questions, he indicates that the velocity at several points in the current needs to be known, which was not possible with the customary procedure of the time, which consisted of throwing small pieces of wood or wax balls, and measuring the distance travelled by them in a certain time. He says: All these equally useful and curious questions can be cleared up in the field very easily by means of the instrument that I propose, as the operation is as simple as submerging a stick in water and then taking it out. This machine will measure the exact quantity of the velocity of the water at the depth required, and will do so just as easily as at the surface. 49

What he proposes is not a machine in the strict sense of the word, but a measuring instrument. However, it is true that it is an application, and it is no exaggeration to say that it has been one of the basic instruments in experimentation and practice of fluid mechanics. For many years it has been used regularly, and was named ‘Pitot’s tube’ or simply ‘Pitot’ in aeronautical terminology. For this reason we have not hesitated to include it in this chapter, although we have placed it at the end as a way of separating it from the other machines.

a)

b) Fig. 10-16. Pitot’s tube

49 Cf. ‘Description d’une Machine pour mésurer la vitesse des Eaux courantes & le sillage des Vaisseaux’, p. 366.

APPLICATION OF FLUID MECHANICS

491

The apparatus consists of two pipes, one elbowed, the other straight, which are fixed to a rod with a sliding rule (Fig. 10-16) When the apparatus is submerged in still water (Fig. 10-16a) the water will reach the same level in both tubes, but if it is submerged in a current (Fig. 10-16b) and the opening in the bend is directed towards the current, the water will ascend in the first tube depending on the speed of the water. The basis of this machine, according to Pitot, is the assimilation of the velocity of the fluid to the velocity acquired in a fall. In this respect he declares: Since, following the first principles of this science, the velocity of running water must be considered as the velocity acquired by its fall from a certain height, and if the water moves from the bottom to the top at a wholly acquired velocity, it will climb up to precisely the same height, or to a height equal to that of the fall from where it would have fallen in order to acquire this velocity. [p. 369]

This argument is in agreement with the custom of the time of measuring the speed by heights. Expressed in modern-day terms, the difference is that the straight tube measures the so-called static pressure, while the elbowed tube measures the stagnation pressure, thus named because the fluid has decelerated from its speed until stagnation, which is how it is inside the tube. Bernoulli’s equation gives the relation between the two.

p0 = p +

1 2 ρv 2

[10. 62]

The term ½ρv2 is called dynamic pressure, and it is that measured by Pitot’s apparatus, as the difference between the stagnation pressure and static pressure, represented as p0 and p, respectively. Pitot goes into the use of his apparatus at considerable length, how to use it for measuring the velocity of the current and rivers at different depths and positions, and in addition he suggests a version for application in ships. As an aid he supplies tables in which he links the measured heights to the velocities. These consist in the tabulation of h = v2/2g, and upon establishing his accuracy, we observe that he takes g = 9.09 m/s2, which corresponds to the 28 pieds/s2, a value that coincides with the one he uses in the pump theory. Finally, we shall make just two comments on the apparatus. First, it is striking how close Pitot comes to discovering the relation between velocities and pressures; what he lacked was the concept of pressure. Second, that even though Pitot proposed the device as a ship’s log to measure speed, the system would

492

THE GENESIS OF FLUID MECHANICS, 1640–1780

have not worked in practice because the naval constructors did not want tubes penetrating the hull, and the solution to place it in the stern would have made the readings useless due to the continuous rise and fall of the stern through the water.50 Recapitulation: hydraulic machines

We have set out to show the interest that the applications to fluids invoked in the eighteenth century. Nevertheless, from a sociological point of view, there is a difference between pumps and turbines. The first were appliances in use, primarily of urban and industrial necessity, and it is very probable that the incipient engineers, who were then appearing upon the scene, used the writings of the great sages in order to improve their pumps. The specific results are unknown to us, as the technological interest lay more in the machine itself than in writing about it. Turbines or power-generating machines, by contrast, responded to a growing need for sources of energy traditionally generated by waterwheels, and the steam engine began to appear on the scene as a great revolution. The road to jet turbines was theoretically acceptable, but there was an important conditioning factor, the necessity of high velocities at water outlets, which could only be achieved by discharging from a height or increasing the pressure. The former is what Euler studied, while the latter was somewhat removed from the technological capabilities of the time. Finally, in these studies we ought to perceive the precedent of modern engineering, which as we have already warned, implied theory plus material in order to arrive at technical devices. Its first step is theoretical analysis in the style of the eighteenth-century studies. The difference between the analysis of Euler and what is done nowadays lies in the quantity of empirical data, means of calculation and testing, and the range of working equipment. These are definitely not qualitative differences.

50 Pitot’s proposal to the Minister of the Navy resulted in a flurry of back-and-forth memos with navy officers on its pros and cons (these memos are in ANF Fonds Marine G/100/2, folios 26–55). The solution of attaching a set of Pitot tubes to the stern of the ship was proposed by French constructor Alexandre Savérien in his book L’Art de mesurer sur mer le sillage du vaisseau, Paris, Jombert 1750 pp. 40–44, which contains several figures outlining his proposals. (This comment is due to Larrie Ferreiro.)

Appendix

Units of measure One of the main difficulties we have find in analysing the texts of the epoch, specially those relating to experimental data or applications, has been the existence of different systems of units of measurement, which not only varied from one country to other, but also from city to city. In order to compare results and to make the measurement easier to understand for present-day reader, the equivalent in the International System, which is the prevailing in the present scientific and technical world, has been adjoined to each measure. With the exception of the Paris or British units, the search for the conversion factors has not been easy. We have consulted data coming from current texts, Encyclopaedias and different tables, and also primary references of the eighteenth century, because frequently the author referred his local units to the Paris or British ones. This variety of sources have occasionally given rise to discrepancies, in which case we have taken the value we have consider as more reliable. In this appendix we present the conversion factors used in two tables: one for the lengths and other for the masses. For the forces, expressed in Newtons, the factor 9.8 m/s2 has been used to obtain the weight associated with a mass.

493

494

THE GENESIS OF FLUID MECHANICS, 1640–1780

Length Paris

Toise

6,000 pied

Pied

British

324.83 mm

Pouce

1/12 pied

Ligne

1/12 pounce

2.256 mm

Point

1/12 ligne

0.188 mm

Yard

3 ft

914.4 mm

Foot Rhenish

27.069 mm

304.8 mm

Inch

1/12 ft

25.4 mm

Meile

2,400 Fuß

7,541 m

Fuß Dutch

Voet

Castilians

Vara

Swedish

1,949 m

314.2 mm 283 mm 3 pie

835.9 mm

Pie

278.6 mm

Fot

296.9 mm

Tum

1/12 fot

24.75 mm

Hamburg

Fuß

286 mm

Bologna

Piede

387 mm

Police

1/12 piede

32.25 mm

APPENDIX

495

Mass Paris

British

Livre

0.4895 kg

Once

1/16 livre

30.594 g

Gros

1/8 once

3.824 g

Grain

1/72 once

0.0531 g

Pound

0.4536 kg

Ounce

1/16 pound

28.35 g

Dram

1/16 ounce

1.772 g

Grain

1/7,000 pound

Dutch

Pound

Castilians

Quintal

0.494 kg 100 libras

Skålpund

Bologne

Libbra

46.00 kg 0.4600 kg

Libra Swedish

0.0648 g

0.425 kg 0.3619 kg

Oncia

1/16 libbra

Grano

1/640 oncia

27.31 g 0.04267 g

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498

THE GENESIS OF FLUID MECHANICS, 1640–1780

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THE GENESIS OF FLUID MECHANICS, 1640–1780

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Dramatis Personæ

Simon Stevin Galileo Galilei Benedetto Castelli Evangelista Torricelli Edme Mariotte Blaise Pascal Jean-Baptiste du Hamel Christiaan Huygens Ignace Gaston Pardies Philippe de la Hire Isaac Newton Jakob Bernoulli Paul Hoste Domenico Guglielmini Antoine Parent Guillaume Antoine de l’Hopital Nicolas Fatio de Dullier Johann Bernoulli Jakob Hermann Bernardino Zendrini Jean Théofile Desaguliers Giovanni Poleni Henri Pitot Pierre Bouguer Daniel Bernoulli Pierre Louis Moreau de Maupertuis George Wolffgan Kraft János-András Segner Benjamin Robins Leonhard Euler Jorge Juan y Santacilia Alexis Claude Clairaut Jean le Rond d’Alembert Frederic Henrik Chapman John Smeaton

511

1548–1620 1564–1642 c1577–c1644 1608–1684 1620–1684 1623–1662 1623–1706 1629–1695 1636–1673 1640–1718 1642–1727 1654–1705 1652–1700 1655–1710 1636–1673 1661–1704 1664–1753 1667–1748 1678–1733 1679–1747 1683–1744 1683–1761 1695–1771 1698–1758 1700–1782 1698–1759 1701–1754 1704–1777 1707–1751 1707–1783 1713–1773 1713–1765 1717–1783 1721–1808 1724–1792

512

THE GENESIS OF FLUID MECHANICS, 1640–1780

Charles Bossut Jean Charles Borda Pierre Louis Georges du Bouat Joseph Louis Lagrange

1730–1814 1733–1799 1734–1809 1736–1813

Index

A

342, 345, 346, 348, 352, 356, 358, 372, 377, 378, 380, 409, 435, 436 Discours sur les loix de la communication du mouvement, 21, 129, 131, 137 Essay d’une nouvelle théorie de la manœuvre des vaisseax, 20, 22, 123, 128, 129, 245, 254 Hydraulica, 26, 28, 29, 33, 103, 270, 293, 295, 296, 326, 329, 332, 336, 337, 355, 358, 364 Oscillationibus penduli. In medio quod resistit in ratione simplici velocitate, 131 Problema ballisticum, 22, 131 Bernoulli, Nicholas, 140 Bernoulli’s theorem/equation, 30, 36, 180, 269, 270, 295, 297, 314, 336, 377, 413, 417, 441, 458, 465 Borda, Jean Charles, 3, 40, 185-191, 193-195, 197, 270, 297, 346-353, 453, 454, 466, 467, 469 Expériences sur la Résistance des Fluides, 187, 194, 195, 197 Mémoire sur l’écoulement des fluides par des orifices des vases, 196, 346, 467 Mémoire sur les pompes, 466 Bos, H.J.M., 63 Bossut, Charles, 3, 40, 185-187, 202, 203, 205, 210, 213, 226, 227 Détermination générale de l’effet des roues …, 226, 227 Nouvelles expériences sur la résistance des fluides, 202, 203, 212 Boudriot, Jean, 243 Bouguer, Pierre, 3, 30, 31, 34, 40, 41, 51, 121, 122, 126-128, 143, 144, 245, 247, 248, 251-253, 255, 258, 359, 361-363 Comparaison des deux lois que la Terre…, 361 De la impulsion des Fluides sur les Proues…, 252 De la manœuvre des vaisseaux, 245, 247, 248, 255

Ackeret, Jakob, 489 Amontons, Guillaume, 323, 422 Archimedes, 7, 215 B Beaufoy, Mark, 187 Beighton, H., 216 Belidor, Bernard Forest, 4, 9, 229 Bernoulli, Daniel, 2, 4, 7, 10, 11, 21-26, 28, 30, 32, 40, 50, 52, 58, 72, 107, 114, 126, 128, 132, 133, 135-145, 150, 160, 166, 168, 182, 267, 269, 270, 272, 277, 293-300, 302, 303, 306, 308-310, 312, 314-325, 327, 331, 332, 335, 336, 339, 340, 343-347, 350, 352, 353, 356, 372, 391, 409, 435, 436, 453, 458, 463, 469, 470, 471, 472 De legibus quibus mechanicis..., 138 Disertatio de actione fluidorum in corpora solida..., 21, 132, 133, 137, 138 Experimenta coram societate..., 23, 320 Hydrodynamica, 11, 21-24, 26, 28, 29, 33, 72, 107, 138, 160, 166, 269, 270, 272, 277, 293-299, 304, 316, 319, 320, 327, 346, 347, 356, 453, 454, 458 Theoria nova de motu aquarum per canales quoscunque fluentium, 22, 294, 298, 299, 339 Bernoulli, Jakob, 2, 18, 20, 50, 113, 114, 118 121, 123, 128, 133, 136, 149, 254, 267 De Celeritate & Declinatione [Dérive] Navis, 115, 118 De resistencia figurarum in fluidis motarum, 115, 119 Bernoulli, Johann, 2, 19, 20, 21, 25, 26, 28-30, 37, 50, 103, 113, 114, 122, 123, 128, 129, 130, 132, 223, 245, 248, 251, 254, 255, 267, 269, 290, 293-298, 310, 311, 317, 321, 326, 327, 329-332, 335-337,

513

514 De la mature des vaisseaux, 143, 248, 251 Traité du navire…, 30, 51, 121, 143, 144, 245, 247, 250-253, 255 Une base qui est exposée au choc…, 251 Boyle, Robert, 76, 321, 422, 461 Briggs, Morton, 169, 203 C Cajori, Florian, 73 Carmody, Thomas, 312 Caro Baroja, Julio, 216 Castelli, Benedetto, 7, 8, 10, 58, 268, 271-273, 285 Chapman, Frederic Henrik af, 40, 184, 186, 197, 198, 200, 201, 204, 219, 245 Architectura navalis mercatoria, 197, 245 Tractat on skepps-byggeriet, 197 Celsius, Anders, 359 Clairaut, Alexis Claude, 2, 34-37, 270, 355-360, 363-370, 372-374, 376, 386, 403, 423, 424, 435 Re-entry/closed channels, 35, 357, 364 Théorie de la figure de la Terre…, 34, 355, 356, 358, 360, 370, 424, 442 Cohen, I. B., 73, 97, 99 Condorcet, Jean-Antoine-Nicholas, 202, 203, 212 Continuity equation, 407, 429, 432, 433, 436, 437, 442 Cotes, Robert, 74, 75, 99, 103, 282 D D’Alembert, Jean Le Rond, 2, 4, 33-39, 45, 52, 74, 95, 113, 127, 128, 143, 167-170, 172-174, 179, 181, 185, 186, 202, 203, 267-270, 293, 295, 297, 310, 317, 338, 339, 341-346, 348, 350, 352, 353, 355-358, 371, 373, 374, 376-381, 383, 385, 386, 388-390, 392, 393, 397-399, 402, 403, 406, 409, 412, 419-421, 423, 429, 435, 436, 442, 444-446, 463 Essai d’une nouvelle théorie de la résistance des fluides, 36, 37, 52, 143, 167, 174, 175, 179, 355-357, 371, 412 Opuscules mathématiques, 37, 175, 345, 392, 445 D’Alembert paradox, 37, 52, 165, 167, 175, 181, 338, 374, 392, 395, 444

INDEX Traité de dynamique, 33, 167, 338, 339, 341, 356, 372-374 Traité de l’équilibre et du mouvement des fluides, 33, 74, 167, 170, 171, 317, 338, 340, 346, 356, 373, 374, 377 Democritus and Leucippus, 321 Derry, T. K., 216, 219 Desaguliers, Jean Théofile, 20, 21, 111 Descartes, 360 Díez Martínez, Amparo, 158 Dugas, René, 10, 13, 106, 297, 371, 427 Duhamel, Henri-Louis, 205 E Euler, Leonhard, 2, 5, 27, 30, 32, 34-41, 46, 47, 51, 52, 122, 126-128, 143, 145-150, 153, 155, 156, 158-166, 168, 170, 175-181, 184-186, 229, 237, 238, 240-242, 245, 247, 248, 251, 256, 257, 259, 267, 268, 270, 293, 296, 297, 303, 321, 327, 345, 346, 355, 357, 358, 372, 392, 399, 401-404, 406-409, 411, 412, 414, 416-421, 423, 426, 427, 429-431, 433, 435, 437-448, 453, 454, 456, 462, 463, 465, 473, 475-479, 480, 481, 483-489, 492 Continuation des recherches sur la théorie du mouvement des fluides, 355, 401, 435, 447 Dilutidationes de resistentia fluidorum, 178, 179 Maximes pour arranger le plus avantageusement les machines…, 237, 241 Principes généraux de l’état d’équilibre des fluides, 355, 401, 419 Principes généraux du mouvement des fluides, 355, 357, 401, 426, 447 Principia motus fluidorum, 37-39, 357, 401, 402, 418, 420, 421, 426-430, 441, 444, 448 Recherches plus exactes sur l’effect des moulins de vent, 126, 176, 237, 238 Recherches sur l’effet d’une machine hydraulique… Segner..., 476, 479 Sur le mouvement de l’eau par des tuyaux de conduite, 462 Tentamen theoriæ de frictione fluidorum, 321, 453, 480 Théorie complète de la construction et de la manœuvre des vaisseaux, 245, 247, 260

INDEX Théorie plus complete de machines qui sont mises en mouvement par la reaction de l’eau, 408, 437 Scientia navalis…, 30, 51, 143, 145, 149, 162, 170, 245, 247, 256 F Fatio de Duillier, Nicolas, 19, 50, 74, 97, 99, 113, 122, 250, 280 Ferreira, Fernando, 216 Ferreiro, Larrie D., 129, 246, 315, 472, 492 Fleckenstein, J.O., 293 Flierl, Karl, 294 Franklin, Benjamin, 472 G Galileo, Galilei, 10, 67, 69, 100, 268, 278, 293, 299 Godin, Louis, 359 Golbach, 319 Guglielmini, Domenico, 8, 10, 11, 268, 272, 275, 276, 278, 279, 285 Aquarum fluentium mensura..., 10, 272, 275, 276 H Hahn, Roger, 52, 185, 186, 270 Hales, Stephen, 76, 326 Vegetable staticks, 76, 326 Hamel, Jean-Baptiste du, 8, 57, 58, 139, 272, 273 Regiæ scientarum Academiæ Historia Parisiis, 8, 57, 58, 139, 272, 273 Hankins, T. L., 90 Hermann, Jakob, 321 Phoromonia, 321 Hero of Alexandria, 233, 454 Hire, Philippe de la, 19, 64, 123, 224, 225, 456, 459 Examen de la force necessaire…, 221, 459 Hooke, Robert, 76 Micrographia, 76 Hoste, Paul, 245 Huygens, Christiaan, 2, 4, 7-10, 12, 14, 20, 34, 35, 40, 49, 50, 55, 57, 58, 60, 61-64, 66-68, 70, 82, 122, 128, 135, 137, 140, 268, 269, 273-275, 299, 340, 359, 360 Œuvres Complètes, 57, 58, 60, 70

515 I Impact theory, 13, 16-22, 25, 30, 31, 33, 36, 40, 41, 48, 49, 51-53, 55, 57, 70, 75, 91, 92, 113, 114, 116, 121, 123, 125-130, 132, 137, 139, 143, 145, 167, 170, 174, 176, 180, 185, 193, 210, 213, 225, 237, 246, 254, 259, 297, 392 Irrotational motion, 38, 412, 414, 415, 418, 427, 430, 441, 442 J Johnson, W., 149 Juan y Santacilia, Jorge, 3, 27, 30, 40, 41, 52, 126, 128, 145, 182-184, 245, 249, 250, 260, 261, 263, 264, 359 Examen marítimo, 41, 52, 128, 182, 184, 245, 249, 260, 261 Juanelo Turriano, 216 K Kármán, Theodore von, 112 Kline, Morris, 388 Kobus, Helmut, 294 König, Emmanuel, 140 Koyré, Alexander, 97, 99, 186 Krafft, George Wolffgan, 25, 141-143 De vi venæ aquæ contra planum..., 142 L L’Hôpital, Guillaume Antoine de, 19, 50, 103, 113, 122, 250 La Condamine, Charles Marie de, 359 Lafuente, Antonio, 358, 359 Lagrange, Joseph Louis, 39, 143, 358, 445-451 Mémoir sur la théorie de mouvement des fluides, 358, 445 Sur la percussion des fluides, 143 Lamb, Horace, 376 Lemonnier, Pierre, 359 Live force, 4, 21, 23, 25-29, 34, 40, 45, 57, 146, 147, 196, 270, 290, 295, 296, 298-300, 302, 308, 310, 312, 313, 317, 325, 326, 337, 340, 343, 344, 346, 347, 351, 352, 466, 468 Lulofs, 237, 241

516 M Maccagni, Carlo, 10, 268 Mach, Ernest, 169, 273 Magiotto, Rafel, 272, 273 Mariotte, Edmé, 2, 5, 7, 10-16, 19, 21, 49, 50, 55, 58, 63-66, 68-70, 76, 99, 123, 135, 137, 144, 186, 220, 223, 247, 268, 278-280, 456, 461 Traité de la percussion ou choc des corps, 64, 68, 70 Traité du mouvement des eaux, 11, 13, 49, 64, 68, 76, 144, 220, 278, 279 Maupertuis, Pierre Louis Moreau de, 4, 359, 361, 368 Mayow, John, 76 Mazuecos, Antonio, 258 McLaurin, Colin, 236, 242, 236, 242, 357, 374 Mikhailov, G.K., 298 Momentum equation, 52, 437, 445 Montucla, Jean-Etienne, 360 Motte, Andrew, 73 N Newton, Isaac, 1, 2, 4, 5, 9, 13-20, 22, 31, 34, 35, 37, 49-51, 55, 73-86, 89-96, 99, 100, 102-104, 106, 107, 109-114, 121-123, 125, 128, 129, 132, 135-139, 144, 146, 148, 149-158, 167, 172, 182, 185-187, 193, 195, 250, 269, 282, 284, 290, 293, 297, 299, 308, 310, 311, 318, 330, 332, 337, 349, 358-360, 364, 370, 372, 379, 444, 454, 470 Cataract, 20, 99-101, 103, 104, 297, 330, 337, 379, 380 Opticks, 79, 80 Principia mathematica, 1, 2, 5, 13, 15, 17-19, 49, 73, 74, 76, 78, 82, 86, 94, 95, 97, 99, 106, 111-113, 122, 135-137, 148, 187, 269, 282, 283, 290, 299, 310, 359, 361, 470 Questiones, 112 Rare medium, 16, 18, 74, 75, 78, 91, 105 P Parcieux, M de, (Deparcieux), 220, 227 Mémoire dans lequel on démontre que l’eau..., 227

INDEX Mémoire dans lequel on prouve que les aubes …, 220, 228 Parent, Antoine, 19, 128, 221-224, 228, 236, 474 Sur le plus grande perfection possible des Machines, 221, 222 Pascal, Blaise, 7, 24, 328, 373 Picard, 58, 272, 359 Pitot, Henri, 25, 221, 223, 228, 245, 409, 453, 456-462, 465, 490-492 Comparaison entre quelques Machines mûës par les courant des Fluides, 224, 225 Description d’une Machine pour mesurer…, 490 Essais d’une théorie nouvelle de Pompes, 453, 474 La théorie de la manœuvre des vaisseax reduite en pratique, 25 Nouvelle methode pour connoître…, 223 Remarques sur les Aubes ou Pallettes des Moulins…, 224 Suite de l’essai d’une théorie de Pompes, 454 Suite de l’essai d’une théorie nouvelle de Pompes, 454 Plane sections, hypothesis, 23, 26, 298, 302, 303, 329, 334, 373 Poleni, Giovanni, 7, 8, 21, 268, 272, 276, 283, 284, 286, 288, 289, 291, 292, 340 De castellis per quæ derivantur…, 21, 283, 284, 288, 289, 291 De motu aquæ mixto, 21, 283, 284 Del moto misto dell’acqua..., 272, 284, 292 R Rada García, Eloy, 73 Redondi, Pietro, 202 Renau d’Elizagaray, 20, 128, 225, 245, 254 De la théorie de la manœuvre des vaisseaux, 128, 225 Roberval, 58 Robins, Benjamin, 2, 22, 31, 40, 51, 126, 128, 149-160, 162, 166, 175, 182, 184, 186, 188, 304, 392, 395, 444 New Principles of Gunnery, 31, 32, 51, 149, 150, 156-158, 184, 392, 444, 462 Romme, Nicolas Charles, 187 Rouse of Harborough, 157 Rouse, Hunter, 268, 294, 296, 297, 381

INDEX S Savérien, Alexandre, 492 Segner, János-András, 454, 472-476, 479, 481, 485 Theoria machinæ cuisdam hyraulicæ, 472 Sellés García, Manuel, 323 Simón Calero, Julián, 182, 249, 266 Simón, Javier, 191 Smeaton, John, 157, 228-231, 233-236, 242 An experimental Enquiry…, 157, 228 Steele, Brett D., 149 Stevin, 7, 234-236, 242, 336, 363 Stream lines, 32, 36, 46, 158, 163, 174, 176, 180, 303, 378, 380-383, 394 436, 439, 440, 442 T Todhunter, Isaac, 360 Torricelli, Evangelista, 2, 6, 8-10, 17, 23, 268, 271, 274, 278, 279, 282, 285 De motu gravium, 6, 271 Torricelli’s Law, 9, 11, 17-19, 21, 57, 58, 60, 71, 97, 99, 100, 103, 107, 145, 160, 182, 268, 270, 271, 273-275, 277, 282, 284, 290, 297, 309, 310, 316, 332, 487

517 Truesdell, Clifford A., 73, 74, 127, 150, 165, 166, 295-297, 304, 319, 321, 336, 357, 371, 376, 382, 388, 399, 401, 407, 408, 412, 416, 418, 439, 444, 445 Turgot, Robert-Jacques, 202, 203 U Ulloa, Antonio de, 30, 244 V Varignon, Pierre, 315 Vitrubius, 216, 219 W Westfall, Richard S, 9, 45, 90, 97, 99, 103 Whitman, Anne, 73 Williams, I., 216, 219 Wilson, James, 157 Wren, Christopher, 82 Z Zendrini, Bernardino, 268, 276