The Kinematics of Mixing: Stretching, Chaos, and Transport

The kinematics of mixing: stretching, chaos, and transport

J. M. OTTINO Dcprrrtrnc~nt01' C l ~ r r , ~ i r , uE11qirlrc~ri11g, l Unir.c~r.sitj01' Mu.s.srrc~hu.sc~tt.s

CAMBRIDGE UNIVERSITY PRESS

P U B L I S H E D B Y T H E P R E S S S Y N D I C A T E OF T H E U N I V E R S I T Y O F C A M B R I D G E

The Pitt Building, Trumpington Street, Cambridge CB2 IRP, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 1001 1-421 I , USA I 0 Stamford Road. Oakleigh. Melbourne 3 166, Australia O Cambridge University Press I989

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 Reprinted 1997 Printed in the United Kingdom at the University Press. Cambridge A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data

Ottino, J. M. The kinematics of mixing : stretching, chaos, and transport 1 J. M. Ottino. p. cm. Bibliography : p. Includes index. ISBN 0 521 36335 7. ISBN 0 52 1 36878 2 (paperback) I. Mixing. I. Title. TP156.M5087 1989 660.2 '84292-dc 19 88-30253 ISBN 0 521 36335 7 hardback ISBN 0 52 I 36878 2 paperback

Contents

I I I 1.2 1.3 1.4

Introduction Physical picture Scope and early works Applications and geometrical structure Approach Notes

2 Flow, trajectories, and deformation 7.1 Flow 7.7 Velocity. ;icceler;rtion. Lagrangian and Eulerian viewpoints 2.3 Extension t o multicomponent media 7.4 ('l~issical means for visuali/ation of flows 2.4.1 Ptrr/ic,lr prrfll. orhii, or frcijec'for.~ 2.4.2 S / r ~ ~ t r ~ l l l i ~ ~ ~ ~ . s -7.4.3 Srrc~trlilirlc~.~ 2.5 Steady and periodic flows 7 . 0 1)eformation gradient and velocity gradient 2.7 Kinematics of deformation-strain 7.X Motion around a point 2.9 Kinematics o f deformation: rate of strain 7.10 Rates o f change of material integrals 2.1 I I'hysical meaning of Vv. (Vv)'.and D Hibliography Notes

3 '.I

('onserration equations, change of frame, and vorticity Principle of conservation of mass . 2 F'l.inciple o f conservation of linear momentum 3.-; 7'r:iction t(n. x. 1 ) 3.4 ('auchy's eclu:rtion of motion

Contents Principle of conservalion of angular momentum Mechanical energy equation and the energy equation Change of frame 3.7.1 Objecticity 3.7.2 Velocity 3.7.3 Accelerarion Vorticity distribution Vorticity dynamics Macroscopic balance of vorticity Vortex line stretching in inviscid fluid Streamfunction and potential function Bibliography Notes Computation of stretching and efficiency Efficiency of mixing 4.1.1 Properties of e, and e, 4.1.2 Typical behavior of the eflciencj 4.1.3 Flow classification Examples of stretching and efficiency Flows with a special form of V v 4.3.1 Flows with D(Vv)/Dt = 0 4.3.2 Flows with D(Vv)/Dt small Flows with a special form of F; motions with constant stretch history Efficiency in linear three-dimensional flow The importance of reorientation; efficiencies in sequences of flows Possible ways to improve mixing Bibliography Notes Chaos in dynamical systems Introduction Dynamicdl systems Fixed points and periodic points Local stability and linearized maps 5.4.1 Dejinilions 5.4.2 Stability of area preserriny two-dimensional maps 5.4.3 Fumi1ie.s c.f periodic points Poincare sections Invariant subspaces: stable and unstable manifolds Structural stability Signatures of chaos: homoclinic and heteroclinic points. 1-iapunov exponents and horseshoe maps

Contents 5.8.1

vii

Homoi,linic lend heteroclinic connections und points t o ir~itiulconditions (end Liupuno~:exponents 5.8.3 florseshoe mups 5.9 Summary of definitions of chaos 5.10 Possibilities in higher dimensions Bibliography Notes

111 116 117 124 124 125 128

Chaos in llamiltonian systems Introduction Hamilton's equations Integrability of Ilamiltonian systems General structure of integrable systems phase space of Harniltonian systems Phase space in periodic Hamiltonian flows: Poincare sections and tori 1,iapunov exponents llomoclinic and heteroclinic points in Hamiltonian systems Perturbations of Harniltonian systems: Melnikov's method Behavior near elliptic points 6.10.1 PoincurP--H i r k h o theorem 6.10.2 The Kolmogoroa Arnold-Moser theorem (the K A M theorem) 6.10.3 7'he twist theorr>m 6.1 1 General qualitative picture of ncar integrable chaotic Hamiltonian systcms Bibliography Notes

130 130 131 132 133 135 136 138 140 141 143 144 146 148

5.8.2 Sensiticity

6

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

7

Mixing and chaos in two-dimensional time-periodic flows Introduction 7.2 The tendril-whorl flow 7.2.1 1,occrl c~nu1y.si.s:loctcction und slccbilitj o f perioil-1 and period-2 periodic points 7.2.2 (;loha1 unul~.si.srind interccctions between munifi)kd.s 7.2.3 Formutior1 of horseshoe mccps in the T W mcip 7.3 The blinking vortex flow (BV) 7.3.1 Poinc,ur; seer ions 7.3.2 Sluhilitj of period-I periodic poitlts ctnd conjugate) lines 7.3.3 Hor.se.shoe mrcps iri the BV flow 7.3.4 Liupunot. c~.uponenls,urerage cflic.iencbp, [end 'irrecersihility' 7.3.5 Mclcro.sc~opici1isper.sion o f tracer purtic~les 7.4 Mixing in a journal bearing flow 7.5 Mixing in cavity flows Bibliography Notes

7.1

148 152 152

.. .

Contents

vlll

Mixing and chaos in three-dimensional and open flows Introduction Mixing in the partitioned-pipe mixer 8.2.1 Appro.uirntrtcl 1~~l0cit.v .lirld 8.2.2 Poirlcrrrc; .sec,tiort.s trrlil three-dimertsiond struc'turr 8.2.3 E\-it tirile distributions 8.2.4 L~ocrrlstrercllir~g01' rnirr~rirrllir1e.s Mixing in the eccentric helical annular mixer Mixing and dispersion in a furrowed channel Mixing in the Kelvin cat eyes flow Flows near walls Streamlines in an inviscid flow Bibliography Notes Epilogue: diffusion and reaction in lamellar structures and microstructures in chaotic flows Transport at striation thickness scales 9.1 .I Pirrirrllrter.~rrrttl rtrrirrb1e.s chtrrtrc~rrrizirlytrtrrlsporr ut .srnrrll 9.1.2 Re~jirllc~.~ Complications and illustrations 9.2.1 Distortiorls of Iirrnrllirr structures trrttl rli.stributiort c1/1;.r.t.s 9.2.2 Illustrtrriorts Passive a n d active microstructures 9.3.1 E.~perirnc~rttrrl st~rr1ir.s 9.3.2 Throrrric~rlsr~rt1ir.s Active microstructures as prototypes Bibliography Notes Appendix: Cartesian oectors and tensors List of',fiequently used symbols

Author index

Color section bctn~c~en pages 153 and 154

Preface

1.11~objective of this book is to present a unified treatment of the mixing of fluids from a kinematical viewpoint. The aim is to provide a conceptually clear basis from which to launch analysis and to facilitate the understanding c)fthcnumerous mixing problemsencountered in nature and technology. ['rcsently, the study of fluid mixing has very little scientific basis; processes and phenomena are analyzed on a case-by-case basis without any attempt to discover generality. For example, the analysis of mixing and 'stirring' of contaminants and tracers in two-dimensional geophysical flows such as in oceans; the mixing in shear flows and wakes relevant to aeronautics and combustion; the mixing of fluids under the Stokes's regime generally encountered in the 'blending' of viscous liquids such as polymers; and the mixing of diffusing and reacting fluids encountered in various lqpes of chemical reactors share little in common with each other, except possibly the nearly universal recognition among researchers that they are \cry difficult problems.' Phcre are, however, real similarities among the various problems and the possible benefits from an overall attack on the problem of mixing uhing a general viewpoint are substantial. 7'110poir~t01' c.ie\\. rrdoptetl Ilc>r.ris rl~trr,fi.oru o kir~c~rntrtical rir\z.point ,fluid / ~ r i . \ i r lis ~ ~tho c:[jic,icv~r .srrc~rc~hing rrrltl ,fbltlirlg of' r?~utrriul1irlr.s urld .surCfucc~.s. S~lchu problem corresponds to the solution of the dynamical system tlx,'tlr

=V(X,

t),

l~11crethe right hand side is the Eulerinn velocity field (a solution of the Nal ier Stokes equations. for example) and the initial condition corresponds the initial configuration of the line o r surface placed o r fed into the tlolv ( S represents the location of the initial condition x = X ) . Seen in this light. the problem can be formulated by merging the kinematical foundations of fluid mechanics (Chapters 3 and 3 ) and the theory of dynamical s!\lems (Chapters 5 ; ~ n d6). The approach adopted here is to analyze siml'lc protypical flows to enhance intuition and to extract conclusions

of general validity. Mi.uincl i.s strc~tchinyund ,jOMiny and srrc~tchinycrnd ,ji)ldir~gis r l ~ r,fingc~rprint 01' chlros. Relatively simple flows can act as prototypes of real problems and provide a yardstick of reasonable expectations for the completeness of analyses of more complex flows. Undoubtedly, I expect that such a program would facilitate the analysis of mixing problems in chemical, mechanical, and aeronautical engineering, physics, geophysics, oceanography, etc. The plan of the book is the following: Chapter 1 is a visual summary to motivate the rest of the presentation. In Chapter 2 I have highlighted, whenever possible, the relationship between dynamical systems and kinematics as well as the usefulness of studying fluids dynamics starting with the concepts of motion and ,flow.2 Mixing should be embedded in a kinematical foundation. However, I have avoided references to curvilinear co-ordinates and differential geometry in Chapters 2 and 4, even though it could have made the presentation of some topics more satisfying but the entire presentation slightly uneven and considerably more lengthy. The chapter on fluid dynamics (Chapter 3) is brief and conventional and stresses conceptual points needed in the rest of the work. The dynamical systems presentation (Chapters 5 and 6 ) includes a list of topics which I have found useful in mixing studies and should not be regarded as a balanced introduction to the subject. In this regard, the reader should note that most of the references to dissipative systems were avoided in spite of the rather transparent connection with fluid flows. A few words of caution are necessary. Mixing is intimately related to flow visualization and the material presented here indicates the price one has to pay to understand the inner workings of deterministic unsteady (albeit generally periodic in this work) two-dimensional flows and threedimensional flows in general. However, we should note that the geometrical theory used in the analysis will not carry over when v itself is chaotic. Though mixing is still dependent upon the kinematics, the basic theory for analysis would be considerably different. Also, even though many of the examples presented here pertain to what is sometimes called 'Lagrangian turbulence', the reader might find a disconcerting absence of references to conventional (or Eulerian) turbulence. In this regard I have decided to let the reader establish possible connections rather than present some feeble ones. I give full citation to articles, books, and in a few cases, if an idea is unpublished, conferences. When only a last name and a date is given, particularly in the case of problems or examples, and the name does not appear in the bibliography, it serves to indicate the source of the problem

or idea. I t is important to note also that some sections of the book can be rcgnrded as work in progress and that complete accounts most likely will (‘allow, expanding over the short descriptions given here; a few of the problems, those at the level of small research papers are indicated with an asterisk (*). In several passages I have pointed out problems that need work. Ideally, new questions will occur to the reader.

Preface to the Second Printing In preparing the second printing of this book, several typographical and formatting errors have been corrected. The objectives of the book expressed in the original preface remain unchanged. Owing to space constraint limitations the amount of material covered remains approximately the same. The reader interested in the connection of these ideas with turbulence will find some leads in the article 'Mixing, chaotic advection, and turbulence', Annual Reviews of Fluid Mechanics, 22, 207-53 (1990); a succinct summary of extensions of many of the ideas outlined in this book is presented in 'Chaos, Symmetry, and Self-Similarity: Exploiting Order and Disorder in Mixing Processes', Science, 257, 754-60 (1992). I should appreciate comments from readers pertaining to related articles in the area of fluid mixing as well as possible extensions or shortcomings of the ideas presented in this work.

Notes I Even the terminology is complicated. For example. in chemical engineering the terms mixing, agitation. and blending are common (Hyman, 1963; McCabe and Smith, 1956, Chap. 2. Section 9; Ulbrecht and Patterson, 1985). The terms mixing. advection, and stirring appear in geophysics; e.g.. Eckart. 1948; Holloway and Kirstmannson. 1984. Inevitably. different disciplines have created their own terminology (e.g., classical reaction engineering, combustion, polymer processing, etc.). 2 Kinematics appears as an integral part of books in continuum mechanics but much less so in modern fluid mechanics. There are exceptions ofcourse: Chapters V and VI of the work of Tietjens based on the lecture notes by Prandtl contain and unusually long description 01"deformation and motion around a point (Prandtl and Tietjens, 1934).

/

Acknowledgments /-

This hook grew out from a probably unintelligible course given in Santa EL:. Argentina, in July 1985, followed by a short course given in Amherst, Massachusetts, also in 1985. Most of the material was condensed in eight lectl~rcsgiven at the California Institute of Technology in June 1986, \vhcrc the bulk of the material presented here was written. .rhc connection between stretching and folding, and mixing and chaos, bcc;~rnc transparent after a conversation with H. Aref, then at Brown IJni\crsity, during a visit to Providence, in September 1982. 1 would particularly like to thank him for communicating his results regarding the 'blinking vortex' prior to publication (see Secton 7.3), and also for many research discussions and his friendship during these years. I would like to thank also the many comments of P. Holmes of Cornell University, on a rather imperfect draft of the manuscript, the comments of J . M. Grcenc of G. A. Technologies, who provided valuable ideas regarding syrnnictries as well as to the many comments and discussions with S. Wiggins and A. Leonard, both at the California Institute of Technology, during my stay at Pasadena. I am also grateful to H. Brenner of the Massachusetts Institute of Technology, S. Whitaker of the University of California at Davis, W. R. Schowalter of Princeton University, C . A. Trucsdell of Johns Hopkins University, W. E. Stewart of the University of Wisconsin, R . E. Rosensweig and Exxon Research and Engineering, and J . E:.. Marsden, of the University of California at Berkeley, for various comments and support. I am also particularly thankful to those who Supplicd photographs or who permitted reproductions from previous Pllhlications (G. M . Corcos of the University of California at Berkeley, R . ('hevray of Columbia University. P. E. Dimotakis and L. G. Leal. the California Institute of Technology, R. W. Metcalfe of the University of Houston, D . P. McKenzie, of the University of Texas at Allstin. A. E. Perry of the University of Melbourne, 1. Sobey of Oxford IJni\crsity, and P. Wellnnder of the University of Washington). 1 a m particularly indebted to all my former and present students, but

xiv

A~~know1cdgmc~nt.s

particularly to R. Chella and D. V . Khakhar, for work prior to 1986, and to J. G . Franjione, P. D. Swanson, C . W. Leong, T. J . Danielson, and F. J. Muzzio, who supplied many of the figures and material used in Chapters 7-9. 1 am also indebted to H. A. Kusch for help with the proofs and to H. Rising for many discussions during the early stages of this work. Finally, I would like to express my gratitude to D. Tillwick who helped me with the endless task of typing and proofing, to D. Tranah, from Cambridge University Press, for making this project an enjoyable one, and to my wife, Alicia, for help and support in many other ways. Amherst, Massachusetts

Z

1

/-

Introduction

1.1. Physical picture In .;pite of its universality, mixing does not enjoy the reputation of being scientific subject and, generally speaking, mixing problems in nature ;\nd technology are attacked on a case-by-case basis. From a theoretical viewpoint the entire problem appears to be complex and unwieldy and thcre is no idealized starting picture for analysis; from an applied viewpoint it is easy to get lost in the complexities of particular cases without ever seeing the structure of the entire subject. Figure 1 . I . I , which we will use repeatedly throughout this work, describes the most important physics occurring during mixing. In the simplest case, during mechanical mixing, an initially designated material region of fluid stretches and folds throughout the space. This is indeed thc goal of visualization experiments where a region of fluid marked by a suitable tracer moves with the mean velocity of the fluid. This case is also closcly approximated by the mixing of two fluids with similar properties and no interfacial tension (in this case the interfaces are termed passive, see Aref and Tryggvason (1984)). Obviously, an exact description of the mixing is given by the location of thc interfaces as a function of space and time. However, this level of description is rare because the velocity fields usually found in mixing Processes are complex. Moreover, relatively simple velocity fields can Produce very efficient area generation in such a way that the combined action of stretching and folding produces exponential area growth. Whereas this is a desirable goal in achieving efficient mixing it also implies that initial errors in the location of the interface are amplified exponentially fast and numerical tracing becomes hopeless. More significantly. this is "so a signature ofchaotic flows and it is important to study the conditions under which they are produced (more rigorous definitions of chcios are glvcn in Chapters 5 and 6). However. without the action of molecular diffusion, an instantaneous cut of the fluids reveals a ltrti~c~llcir srrlrctlrrc. (Figure 1.1.1). A measure of the state of mechanical mixing is given by :Ivery

Figurc I . I . I . Basic processes occurring during mixing of fluids: ( 0 )corresponds to the case of two similar fluids with negligible interfiicial tension and negligible interdiffusion: a n initially designated material region stretches a n d folds by the action of a flow: ( h ) corresponds t o a blob diffusing in the fluid: in this case the boundaries become diffuse and the extent of the mixing is given by level curves of concentration ( a profile normal t o the striations is shown at the right): in ( c , ) the blob breaks d u e t o interfacial tension forces, producing smaller fragments which might in turn stretch and break producing smaller fragments. Case ( h ) is a n excellent approximation to ( t i ) if diffusion is small during the time of the stretching and folding. In ( t i ) the blob is pu.s.sii.e, in ( c ) the blob is trc,tii.c.. striation thickness

stretching

@ -'

-

concentration profile

initial condition diffusion

breakup

distance

Scope trtltl eurljs works

3

~ l ~ i ~ . k n e sof s e sthe layers, say s, and s,, and :(s, + s,,) is called the slr.;t,~;o,~ ~liick~ios.s (see Ottino, Ranz, and Macosko, 1979). The amount of intcrfi~cialarea per unit volume, interpreted as a structured continuum prc)pHty. is called the i r l t c ' r r i ~ ~ ~ tLIrett ~ r i ~dcnsit~,, ~l (I,..Thus, if S designates t1,c area within a volume V enclosing the point x at time t , u,.(x, t ) = lim v-0

S

v

.

~ ~ loflthe ~ above c concepts require modification if the fluids are miscible or immiscible. I f the fluids are immiscible, at some point in the mixing process the striations o r blobs d o not remain connected and break into snlnllcr fragments (Figure 1.1.1 (c)). At these length scales the interfaces arc not prrs,sir.c~and instead of being convected (passively) by the flow, they nlodify the surrounding flow, making the analysis considerably more complicated (in this case the interfaces are termed uc.tioe, see Aref and Tryggvuson, 1984). If the fluids are miscible we can still track material volumes in terms of a (hypothetical) non-diffusive tracer which moves with the mean mass velocity of the fluid o r any other suitable reference velocity. Designated surfaces of the tracer remain connected and diffusing species traverse them in both directions.' However, during the mixing process, connected iso-concentration surfaces might break and cuts might rcvcal islands rather than striations (Figure I . l . l ( h ) ) . In this case the specification of the concentration fields of 11 - 1 species constitutes a complete description of mixing but it is also clear that this is a n elusive goal. Thus, it is apparent that the goal of mixing is reduction of length scales (thinning of material volumes and dispersion throughout space, possibly involving breakup), and in the case of miscible fluids, uniformity of concentration.' With this as a basis, we discuss a few of the ideas used to describe mixing and then move to examples before returning to the problem formulation in Section 1.4.

1.2. Scope and early works A cursory examination of a n eclectic and fairly arbitrary listing of some of the earliest references in the literature gives an idea of the scope of mixing processes and the ways in which mixing problems have been attacked in the past." I . Tn>l()r (1034) The f o r m a t ~ o nof cmuls~onsin definable fields o f flow. Pro<,.HI,\,. Sot... A146. 501-23. A. Brc)tllm:lll.
W. R . Hawthorne, D. S. Wendell, and H . C. Hottcl (1048) Mixing and combustion in turbulent gas jets. p. 266-88 in ' I ' h ~ r t Symp. l on C'ot~~hu.\riot~ ut~clFltrtnc, uttd E.uplosion Pltc~t~otnc~t~u. Baltimore: Williams & Wikcns. C . Eckart (1948) An analysis of the stirring a n d m ~ x i n gprocesses in incompressible fluids, .I. M u r i t l r Kc,.\.,VII, 265-75. R . S. Spencer and R . M . Wiley (1951) T h e mixing of very viscous liquids. .I. C o l l . &i.. 6 , 133-45. P. V . Danckwerts (1952) T h e definition and measurement of some characteristics of mixtures, Appl. S ( , i . Ros., A.3, 279 96. P. V . Danckwerts (1953) Continuous flow systems-distribution of residence times, Ch[,m. Eng. S(,i.. 2 , 1 13. P. Welandcr (1955) Studies on the general development of motion in a two-dimensional, ideal fluid, Tollu.s, 7 . 141-56. S. Corrsin (1957) Simple theory of a n idealized turbulent mixer. A . 1 . C h . E . J., 3. 329-30. W. D. M o h r , R . L. Saxton, a n d C. H . Jcpson (1957) Mixing in laminar flow systems, I n d . E n g . C h r m . , 49. 1855-57. T h . N. Zweitering (1959) T h e degree of mixing in continuous flow systems C h r m . E n g . Sci.. I I , 1-15.

It seems at first strange to start a discussion on mixing with Taylor's 1934 paper. However, there are several reasons. The first one is that the problem is of practical importance and was attacked with the best tools of the time, both theoretically and experimentally. The second one is that the natural extension of his ideas to mixing remain largely unfulfilled. Taylor's concerns are obvious from the title of the paper. He distilled the essence of the problem, in general a complex one, and reduced the question to a local analysis: the deformation, stretching, and breakup of a droplet in two prototypical flows - planar hyperbolic flow and simple shear flow.4 Presumably, the long range goal was to mimic a complex velocity field in is similar, in spirit, to the terms of populations of these two flows."his approach adopted in this work, in two respects: ( i ) analysis of simple building blocks which give useful powerful insights into the behavior of complex problems. and (ii) decomposition of a problem in local and global components (this idea is reconsidered in Chapter 9). Brothman, Wollan, and Feldman (1945) had more practical and pressing needs in mind and tried to attack the problem of mixing in a general and abstract way. They spoke of fluid deformation and [fluid] rearrangement, and regarded mixing as a three-dimensional shuminB process. They worked out probabilistic arguments directed predorninantl~ to mixing in closed systems, such as stirred tanks, and obtained kinetic expressions for the creation of interfacial area. Eckart (1948) had in mind substantially larger length scales and started his analysis with the continuum field equations and calculated the 'mixing times' of thermal and

Scope untl earl), works s;,linc

>POIS

5

in oceans without resorting t o a n y mechanistic description

of process. A conceptually similar problem in terms of scales, mixing i n atmospheric flows, was addressed by Welander (1955). Remarkably, he did so by considering the possibility o f applying Hamiltonian mechanics to ideal fluids a n d stressed the need for studying the stretching a n d folding of elements in the flow a n d devised formulas t o follow the process. The grontI1 of a material line by fractal construction is also explained in his work a s well as a treatment of motion of point vortices from a ~ a m i l l o n i a nviewpoint. H e also performed experiments a n d o n e of his visualizations is reproduced in Figure 1.2.1 .h 7'11~ir~tc.racrionst ~ r t w r r nturbulrnce a n d chemical reactions is ofutmost importal~ccin combustion. Although the approach to these problems i n thc .iO's ; I I I C ~ 60's was largely statistical this was not the case and I t I S c o r n t o r t ~ l ~to q k n o b thnt In bell thouqht out expcrlmental papers such as thr orlr h\ Hn\$thornr. il'endcll, a n d Hottcl (1948) one finds d c s c r ~ p tlons c ~ r n p h a s i / ~ l ~thc q qcomc.trlca1 aspects of the problem 7'0 quote I,'~gure1 .:.I. Reproduction of one of the earl? mixing experiments of Welander (10551: evolution of an in~tialcondition in a rotating flow. He used butanol floated o n water and the ~ n ~ t i condition al (square) was made of methql-red; unfortunatelq few addit~onaldetails regarding the experiment were giten in the o r ~ g ~ n paper. al

6

Introduction

from their paper, According t o the physical picture of the turbulent flame. eddies . . . are being drawn in from the surrounding atmosphere and being broken up into particles of various sizes . . . The total area of flame envelope is many times the area available in a diffusion flame but nevertheless the final intimate mixing of the gas and oxygen must occur between eddies as a result of molecular diffusion.'

Subseqently, statistical theory took over and the geometrical aspects of the problem were somewhat lost. Of the many possible offsprings of the statistical theory of turbulence to mixing (e.g., Batchelor, 1953) we might mention the short but influential paper by Corrsin (1959) which proposed a simple model to calculate the rate of decrease of concentration fluctuations in an ideal, yet subsequently widely used mixer.' Another important concept based on statistical reasoning is that of the mixing length theory (Prandtl, 1925; Schlichting, 1955, Chap. XIX), which has found application in an enormous range of problems ranging from chemical engineering (Bird, Stewart, and Lightfoot, 1960) to astrophysics (Chan and Sofia, 1987, Wallerstein, 1988). Even though the statistical treatment does not lend itself easily to visualization, notable exceptions exist and many of the early works focused on the stretching of material lines and surfaces, for example, Batchelor (1952; also Corrsin, 1972) analyzed the problem theoretically, whereas Corrsin and Karweit presented experimental results (Corrsin and Karweit, 1969). Other work focused on local deformation to capture the details of the turbulent motion at small scales. For example, Townsend (1951 ) performed an analysis of deformation and diffusion of small heat spots in order to interpret experimental Eulerian data in homogeneous decaying turbulence. Concurrently, at the other end of the spectrum, the stretching of material lines and volumes was also a concern in the mixing of liquids in low Reynolds number flows. Spencer and Wiley (1951) focused on the mixing of very viscous liquids and stressed the idea of being able to describe the growth of interfacial area between two fluids and the need to relate the results to the fluid mechanics. However, even though the mathematical apparatus, largely developed in continuum mechanics, was already in place for such a program. it was not until much later that such developments took place and most of what followed from Spencer and wiley's work was confined to deformation in shear flows. Two other points worth mentioning from their work. which have a clear relationship with dynamicol systems. are the identification of stretching and cutting or folding as the primary mixing mechanism, the so-called 'baker's transformation' characteristic of chaotic systems (see Chapter 5). and the idea

Scope trnd eurly works

7

ofrcpresenting mixing processes in terms of matrix transformations. which related t o mappings (see C h a p t e r 5 ) a n d transition matrices is (see p, 1 6 4 in Reichl, 1980). A closely related study by M o h r . Saxton, a n d Jepson (1957) also focused o n viscous liquids in the context of polymer p,.oces~illg They considered the stretching of a filament of a fluid in the bulk of another o n e in a shear flow using simple arguments t o account for the Llscosityof the fluids but without taking into account the interfacial tension. T h e problem is similar in spirit t o the o n e treated by Taylor (1934) and much work could follow along these lines. Nevertheless this simplified treatment forms the basis for most of the subsequent developments in the mixing of viscous fluids." ~t 1s probably fair t o say that most of the previous works have the geomctncnl i n t e r p r e t a t ~ o ngiven In Figure 1.1.1. However, a point of departure from this picture of sign~ficantconsequence in chemical engineerIng took place w ~ t hthe papers by Zweitering (1959) a n d Danckwerts (1958). In this case the a p p r o a c h became m o r e 'lumped' o r macroscopic a n d the emphasis shifted t o continuous flow systems a n d the character~zationof m ~ x i n gby the temporal distribution of exit times.'' Whereas the objective of most of the a b o v e works was t o relate the fluid mechanics to the m i x ~ n go r some knowledge of the process t o the o u t p u t . Danckwerts (1953) focused primarily o n the characterization of the mixed state. 1.e.. he devised numbers o r indices t o indicate t o the user how well mlxed a system is (e.g., how well mixed is the system of Figure 1.3.1? Figure 1.3.3? Figure 1.3.4?). Even though we a r e going t o say little a b o u t 'the measurement of mixing'. this is probably the place t o stress o u r opinion o n a few points: ( i ) the measure should be selected according t o the specific a p p l i c a t ~ o na n d i t is futile t o devise a single measure t o cover contingencies. a n d ( i i ) the measurement has t o be relatable t o the fluid mechan~cs. Examplc'.s ( 1 ) The striation t h ~ c k n e s s ,s. Figure 1.1.1. is important in processes in\,olving diffusing and reacting fluids a n d represents the distance that the molecules must diffuse in order t o react w ~ t heach other. In simple Gases. .s can be calculated exactly with a knowledge of the velocity field (see Chella a n d O t t i n o (1985a) a n d examples In C h a p t e r 4 ) . t i i ) Molten polymers are often mixed ( a n oper;ation often referred to a s blending) to produce materials with ~ ~ n ~ properties. que F o r example. In the mnnuf;acture o f barrier polymer films ~t might be desirable to Produce structllrcs with effective permeability. This requires that

Introduction

the clusters of the more permeable materials are disconnected and do not form a percolating structure. However, the details of effective diffusion near the percolation point depend on the ramification of the clusters (Sevick, Monson, and Ottino, 1988). Even though such measurements can be extracted from electron micrographs (eeg., Figure 1.3.4, see color plates) via digital image analysis (Sax and Ottino, 1985), to date there are no models allowing the computation of such details from the fluid mechanics of the process. (iii) A model for the structure of the Earth's upper mantle (Allegre and Turcotte, 1986) postulates that the oceanic crust becomes entrained in the convective mantle where it is subsequently stretched into filaments by buoyancy induced motions. It follows that, on the average, the 'oldest' layers are the thinnest and that the diffusion processes concurrent with the stretching become important over geological time scales when the striation thickness is of the order of 0.1-1 m (the typical diffusion coefficients are of the order 10-1410-l6 cm2/s, which implies diffusion on time scales of the order of 10'6-1020 S. By comparison the time scale based on the age of the Earth is 1.4 x 10'' s). In this case a model describing the entrainment of material in the convective mantle coupled to a model describing the evolution of the striation thickness as function of time is capable of describing the gross characteristics of the process. (iv) Consider Eulerian concentration measurements in a turbulent mixing layer. In principle, the fluctuations can be taken as an indication of the mixing between the streams and an index such as Danckwerts's intensity of segregation2 can be computed. In the ideal case of a non-invasive probe with an infinitely fast response and vanishingly small resolution volume we obtain an indication of the thickness of the striations passing by the point as a function of time. However, even if this were possible, the statistics of the fluctuations would be very complicated and hard to connect to the fluid mechanics of the process itself. What is worse, however, is the inability of the measurements to give a correct global picture of large scale structures, the so-called coherent structures (Roshko, 1976). In this case a flow visualization study based, for example, on shadowgraphs is infintel~ more revealing with regard to the structure of the flow (Brown and Roshko, 1974) (see also Figure 1.3.5; see color plates).

A p p l i c t ~ t i o n suncl yeometric.nl structure

9

1.3. Applications and geometrical structure I t Is clear that even restricting o u r attention t o fluid fluid systems." o r ~mmiscible,diffusive o r non-diffusive. reacting o r not, the scope of mixing problems IS enormous and it is not possible t o develop a complete and useful theory encompassing all the above situations. It is nevertheless evident that. in spite o f the enormous range o f length a n d time scales. the underlying geometrical structure associated with the process of reduction of length scales is that of Figure 1.1 . I . In this section we highlight this aspect by means of a few examples. Mixing is relevant in processes ranging from geological length scales ( ] O h m ) and exceedingly low Reynolds numbers (10-20). such as in the mixing processes occurring in the Earth's mantle, t o Reynolds n u m b e r o f order 10" corresponding t o mixing in oceans a n d the atmosphere.'* An example of a simulation of mixing in the Earth's mantle is shown in Figure 1.3.1. in which the flow is modelled a s a two-dimensional layer heated from below.13 In these cases, actual mixing experiments are of course impossible. However, laboratory models of large scale circulation in oceans can be carried out with liquids in containers placed o n a rotating turntable sometimes involving combinations of sources and sinks. Figure 1.2.1, from the early paper by Welander (1955). shows a n example (it is worth noting that the output of similar experiments, were described as 'chaot~c',e.g., Veronis, 1973). Undoubtedly. c.htlotic is a n a p t description of stretching in truly turbulent flows. Figure 1.3.2 shows the stretching of material lines in a turbulent, nearly isotropic, flow (Corrsin and Karweit. 1969), where the e x ~ e c t a t i o n 'is~ that of exponential growth (Batchelor, 1952). Note, howevcr. the inherent limitation of experimental techniques in resolving the smallest scales. The deformation of material lines has been studied also in the case o f chaotic Stokes's flows. Experiments focusing o n deformation of material lines were carried out by Chaiken et trl. (1986) and Chien, Rising, and Ottino (1986). In the case o f Chalken pt (11. the flow consists in a n eccentric Journal bearing time-periodic two-dimensional flow which is described in detail in Chapter 7. Figure 1.3.3 shows the shape adopted by a material line of a tracer by the periodic discontinuous operation of the inner and outer cylinders in ;I counter-rotating sense. Note the absence of 'corners' "d 'branches' in the folded structure even though the flow is essentially d i s c o n t i n l l ~ u(compare ~ with Figure 1.2.1 ). Figures 1 .?.I a n d 1.3.3 show complex stretched a n d folded structures. characteristic c,f mixing in two-dimensional flows. In both cases the

miscible

10

Introduction

structure formed is lamellar and an indication of the state of mixing is provided by the striation thickness. However, in other cases the structures obtained are considerable more complex. If the fluids are immiscible and sufficiently different, interfacial tension plays a dominant role at small Flgure 1.3.1. Deformation of a tracer In a numerical experiment of motion In the Earth mantle The sides of the rectangle are Insulating but the bottom is subjected to a constant heat flux while the temperature of the top surface is kept constant. The motion is produced by buoyancy and internal heating effects (the fluid is heated half from below and half from within). The Rayleigh number is 1.4 x loh, the time scale of the numerical simulation corresponds to 155 Myear, and the thickness of the layer is 700 km. An instantaneous plcture of the streaml~nesreveals five cells. (Reproduced with perm~sslonfrom Hoffman and McKenzle (1985).)

Applic3utions ~rndgeometricul structure k.jfurc I .3.2. Cirowth of a 'material line' composed of small hydrogen bubbles a platinum wirc stretched across a n decaying turbulent flow behind a g r ~ dplaced at the extreme left. The Reynolds number based on the diamcter is 1.360. (Reproduced with permission from Corrsin a n d Karweit

produced by (I96llJ.)

Figure 1.3.3. Mixing in a creeping flow. The ligure shows the deformation of a material region in a journal bearing flow when it operates in a time-periodic hishion lexpcriment from Chaiken rt rrl. (1986)), for a complete description see Scction 7.4. (Reproduced with permission.)

11

12

Introduction

scales. Figure 1.3.4 (see color plates) shows an image processed twodimensional structure produced by mixing and preserved by quenching o f two immiscible molten polymers. In this case there has been a complex process of breakup and coalescence. As indicated earlier, the character-

Figure 1.3.6. Concentration of a turbulent round jet fluid injected into water at Reynolds number 2,300, measured by laser induced fluorescence; the cut is along a plane including the axis of a symmetry of the jet. (Reproduced with permission from Dimotakis. Miake-Lye, and Papantoniou (1983).)

Approuch

13

iwtion of this structure depends on the intended application of the blend and a large number of mixing measures are possible." interplay betwecn chemical reactions and mixing is nowhere more evident than in the case of fast reactions (Chapter 9). In many cases of interest the flows are turbulent and careful studies have been carried out to probe the interplay between the fluid mechanics and the transport processes and by using prototypical flows such as turbulent shear flows and wakes, perturbed or not. The reaction itself can be used to map out the interface of reaction. For example, Koochesfahani and Dimotakis (1986; see also 1985) have used laser induced fluorescence and high speed real-time digital image acquisition techniques to visualize the interface of reaction between two reacting liquids undergoing a diffusion controlled reaction (see also Chapter 9). A similar technique can be used in the case of diffusing scalar. For example, Figure 1.3.5 (see color plates) was obtained by measuring the concentration of a fluorescent dye, initially located in one of the streams of the mixing layer, whereas Figure 1.3.6 shows the mixing of turbulent jet containing a fluorescent dye with a clear surrounding fluid. Lamellar structures (Chapter 9) are clearly seen, even at Kolmogorov length scales (Dimotakis, Miake-Lye, and Papantoniou, 1983). Another instance of interplay between mixing with diffusion and reaction. but at smaller length scales and lower Reynolds numbers (approximately 200 5 0 0 ) , well known in polymer engineering, is provided by the impingement mixing of polymers (Lee et al., 1980) where the objective is to mix two viscous liquids (reactive monomers) in short time scales (order 1 0 ~ 2 1 0 -s1) with reaction time scales of the order of 10'~-1o2s. The geometrical picture is similar to the previous cases; in this case the mixing requirement is to produce striations of the order of 20-501tm so that the reaction can take place under kinetically conditions.'h Mixing at even smaller scales might take place due to spontaneous emulsification (Fields, Thomas, and Ottino, 1987; Wickert, Macosko, and Ranz, 1987).

1.4. Approach In spite of its overwhelming diversity, fluid mixing is basically a process ln~olvinga reduction of length scales accomplished by stretching and folding of material lines or surfaces. In some cases the material surface Or line in question is placed in the flow and then subseqently stretched le.%Figures 12.1, 1.3. I 3). in others, the surface is continuously fed into 'he flow (cg.. Figures 1.3.5 6). Thus. at the most elementary level (i.e..

without averaging but at the continuum level) rnising c.onsists of' stretching lrr~d,fbldii~~g of ,fluid filtrmc~nts,lrrld distribution tllrou~ghoutspuce, accom. panied by breakup if the fluids are sufficiently different, and simultaneous diffusion of species and energy (Figure 1.1.1 ). In the most general case, various chemical species might be reacting. We seek an understanding of this process in terms of simple problems which can serve as a 'window' for more complicated situations. O u r approach is to combine the kinematical foundations of fluid mechanics with dynamical systems concepts, especially chaotic dynamics. The objective throughout is to gain insight into the working of mixing flows. The goal is not to construct detailed models of specific problems but rather to provide prototypes for a broad class of problems. Nevertheless. we expect that the insight gained by the analysis will be important in practical applications such as the design of mixing devices and understanding of mixing experiments. Mixing is also inherently related to flow visualization. However, contrary to popular perception the 'unprocessed' Eulerian velocity field gives very little information about mixing and the typical ways of visualizing a flow (streamlines. pathlines, and to a lesser degree. streaklines) are insufficient to completely understand the process. As we shall see, our problem begirls rather rharl ends with the specificurion of v(x, r). The solution of dxidr = V ( X , I ) with x = X at time t = 0. x = @,(X),which is called the flohv or motion," provides the starting point for our analysis. In even the simplest cases this 'solution' might be extremely hard to obtain. Actually. the impossibility of integrating the velocity field in the conventional sense is the subject of much of Chapters 5 and 6, where the modern notion of irlteyrability is introduced. The kinematical foundations lie in an understanding of the point transformation x = @,(X). We consider the following sub-problems: ( 1 ) Within the framework of x = @,(X):mixing of a single fluid or similar fluids. The basic objective here is to compute the length (or area) corresponding to a set of initial conditions. As we shall see only in a few cases can this be done exactly and in most of these the length stretch is mild. The best achievable mixing corresponds to exponential stretching nearly everywhere and occurs in some regions of chaotic flows. However. under these conditions the (exact) calculation of the length and location of lines and areas is hopelessly complicated. As we shall see in Chapter 5 even extremely

simpliljcd ~ O R might S be inherently chaotic and a corripleie char;~cterization is not p o ~ i b l c .F o r example. from a dynamical systems viewpoint Re shall ,,,that ifthe system possesses horseshoes we have infinitely many periodic and with it the implication that we cannot possibly calculate ;dl of t h e m Fortunately. a s far ;IS mixing is concerned we are intercstcd in low period events, since we want to achieve mixing quickly. ~ ~ ~ ~ ~there t h is calways l e ~the~intrinsic limitation of being unable t o ca]culatc p ~ c i s information e (most practical problems involve stretchings of 10' o r higher) such a s length stretch a n d location of material surf;lccs, Note that this is true even though none o f the flows discussed in Chapters 7 and 8 is turbulent in a n Eulerian sense. Rather, the previous findings should be used t o establish the limits o f what might constitute reasonable answers in more complicated flows (real turbulent flows come immediately t o mind). The prohlcm /irr.cl is /io\c, to hcsr ~ ~ / 1 1 1 1 . 1 1 c ~ r c ~ 1 ~t/]o ; : 1 ~ t i ~ i - ~ i kt~o\vit~q i~q, hgf:)rP/lut1dt/iiit u c o t ~ i p l ~~ ~i ~/ ~ ~ ~ r ~ ~ ~ ~ t i,s ; l ~ l p o , ~ , ~ ; l ~ l ~ ~ . ( 2 ) Within the framework of a family of flows x, = @,,(X,), s = 1, . . .. ,1'; each of the motions is assumed t o be topological (see Section 2.3): mixing of similar diffusing and reacting fluids. This case corresponds t o the case of mixing of two streams. composed ofpossibly several species that are rheologically identical, i.e., they have the same density. viscosity. etc., a n d have n o interfacial tension. Concurrently with the mechanical mixing there is mass diffusion, and possibly. chemical reaction. However. for simplicity, w e will assume that neither the diffusion nor t h e reaction affects t h e fluid m o t i o n . ' T h i s case is discussed in C h a p t e r 9 a n d corresponds t o the case of lumellur structures. ( 3 ) Mixing of different fluids; case in which the motions are nontopological. i,e,. there is breakup a n d o r fusion of material elements. In this case. the mixing of two o r more fluids leads t o breakup a n d coalescence of material regions. This problem is complicated a n d only a few special crlses belonging t o this category are discussed in Chapter 9. by decon~posingthe problem into I o ~ o land qlobiil components. An outline of the organization o f the rest o f t h e chapters is the following: 2 describes the kinematical foundations a n d Chapter 3 presents a b r i e f n i c r v i e , o f fluid mechanics. Where;~sthe material of Chapter 2 is ' n d i s ~ e n ~ i i b l eI;~rge . p;lrts o f Chilpter 1 were ; ~ d d e dt o provide b;ll;~ncc. 4 focuses on ;l f e ex;lmp]es ~ which can be solved in detail and ends in ;I riit11cr dcfc;ltist note t o provide a bridge for the study o f chaos. 5 iwcrcnts ;I pener.11 discussion o f dyni~mic;~l systems ; ~ n dC h i ~ p t ~ r

16

Introduction

6 focuses on Hamiltonian systems. Similar comments apply in this case. We use more heavily the 1,laterial of Chapter 6 but omission of Chapter 5 would result in serious imbalance and a misleading representation of facts. Chapters 7 and 8, by far the longest in this work, give examples of chaotic mixing systems in an increasing order of complexity. Chapter 7 discusses two-dimensional flows, Chapter 8 focuses on three-dimensional flows. Chapter 9 discusses briefly the case of diffusing and reacting fluids and active microstructures.

Notes 1 Obviously, one o f the diffusing species can be temperature, as is the case of mixing of fluids with different initial temperatures or processes involving exothermic chemical reactions. 2 A gross, but popular, measure of the concentration variation is given by the intensity of srqreqution, I . If ~ ( xdenotes ) the concentration at point x and ( . ) denotes a volume average, I is defined as [(((.- ( c ) ) ~ ) ] ' / ' (Danckwerts, 1952). 3 Many ofthese references inspired additional work. Some, however, were largely ignored. 4 Subsequently, this problem took a life of its own and much research followed. See for example Rallison (1984). 5 This idea was not widely followed and most of the mixing work in the area of drop breakup and coalescence in complex flow fields resorts to population balances where breakup and coalescence are taken into account in a probabilistic sense. 6 This paper contains many good ideas, however, it has remained largely ignored by the mixing community. 7 In the past few years there has been a revival of this idea (e.g., Spalding, 1976, 1978b; Ottino, 1982). 8 The theory also found use in two-phase mixing. An early reference is Shinnar (1961). 9 The analysis of mixing of viscous fluids has been largely confined to polymers (Middleman, 1977; Tadmor and Gogos, 1979).A follow up paper, written in the context of the mixing of glasses. is Cooper (1966). Even though Cooper's treatment of the kinematics is at the same level as Mohr, Saxton, and Jepson (1957), there is substantially more, since it deals explicitly with mass diffusion. This paper, however, has remained largely ignored. 10 This approach works well for pre-mixed reactors with slow reactions, but it is not suited for diffusion controlled reac!ions, such as in combustion. Curiously enough, the participation of chemical engineers in subjects dealing with non-pre-mixed reactors has been relatively minor and in spite of complex chemistry, the area has become largely the domain of mechanical and aerospace engineering researchers. See the discussion (pp. 100102) following the paper by Danckwerts (1958). Much work followed along these lines. For a summary, see Nauman and Buffham (1983). I I With the possible exception of the very last example of the last chapter. we do not consider fluid-solid systems. 12 See for example Veronis, 1973: Rhines. 1979; 1983. and the articles by Holland ('Ocean circulation models'. pp. 3 4 5 ) . Veronis ('The use of tracer in circulation studies'. pp. 169188), and Rhines ('The dynamics of unsteady currents'. pp. 18%318), in Goldberg er a l . , 1977.

nllld dynaniical aspcct"arc discussed by McKenzic, Roberts, and Weis., see MeKenfie 11983). cxpcct;ltion was not confirmed in their study. It is apparent that there is a 14 for more cxpcr~mcnthadopting primarily a 'Lagrangian viewpoint'. F ~ ex:lmple. , ~ ;lver;igc cluster s i x and cluster size distribution, intcrfiIcial per llnit ;Ij-c;~,ctc. (see Sax :und Ottino. 1985). 16 ,mpingcmcnt mixing I similar t o 'the stopped flow' mcthod devised to study the kinctl~sof very fklst r c i l c ~ i ~An review ~. of the technique by the inventors o f t h e method by Roughton and <'hanee (1963). C h a p . XIV. is 17 flow.ia the tcrm preferred in dynamical systems. Inorro,! the tcrm ofchoice in continuum mcchan~cs.The re;ider should bc warned about possible confusion o f terms; the word /lo\, i \ used often in the convcntioni~lfluid mechanical scnsc, the term f,r;.r;frc, has :I prcclsc definition in dyn:lmical systems and ergodic theory (see Waters. 1 9 ~ 2 , p. 40). To avoid confusion wc will often use the term /luic] ~ > t ; . \ - , , ! ~ . 18 That is. the fluid mechanics governs the transport processes but not the (Ither way a r o u l ~ dT. h c r c i 110 clean way of incorporirting thcaccouplings at the present timc. 13

( 1974,,

Flow, trajectories, and deformation

In the first part of this chapter we record the basic kinematical foundations of fluid mechanics, starting with the primitive concept of particle and motion, and the classical ways of visualizing a flow. In the second part we give the basic equations for the deformation of infinitesimal material lines, planes, and volumes, both with respect to spatial, x, and material, X, variables, and present equations for deformation of lines and surfaces of finite extent.

2.1. Flow The physical idea of flow is represented by the map or point transformation (Arnold, 1985, Chap. 1 ) x = @,(X) with X = @,=,(X), (2.1.1) i.e., the initial condition of particle X (a means of identifying a point in a continuum, in this case labelled by its initial position vector) occupies the position x at time t (see Figure 2.1 .I ). We say that X is mapped to x after a time t.' In continuum mechanics (2.1 .I ) is called the motion and is usually assumed to be invertible and differentiable. In the language of dynamical systems a mapping @,(XI x (2.1.2) is called a Ck diffeomorphism if it is 1-1 and onto, and both a,(.)and its inverse are k-times differentiable. If k = 0 the transformation is called a homeomorphism. In fluid mechanics, k is usually taken equal to three (see Truesdell, 1954; Serrin, 1959). Also, the transformation (2.1.1 ) is required to satisfy +

or alternatively, J

= det(D@,(X))

whert! D denotes the operation i, (),/ax,, i . e . , derivativeswith respect to the configuration, in this case X. I f the Jacobian J is equal to one flow is called isochoric. ~ h requirement c (2.1.3) precludes two particles, X, and X,, from occup),ing the same position x at a given time, or one particle splitting into ~LI'O: i.e., notz-topologicul motions such as breakup or coalescence are not allowed (Truesdell and Toupin, 1960, p. 510). In the language of dynamical systems, the set of diffeomorphisms (2.1.1 ) for all particles X belonging to the body Vo is called the flow (i.e., a one-parameter set of diffeomorphisms) and is represented by r

1 - 1

- ,@,(XI; = @ , ( X i where X) is the set of particles belonging to V, (Vo = {X} and V, = ( x ) ) , Thus. we say that V,, is mapped into Vr at time t , f V 1 - @ rv 1 ,XI

i

ri

-

rt

01

or that the material line L o is mapped into L, at time t , (L,) =

@,I

Lo:.

Figure 2.1.1. Deformation of lines and volumes by a flow x = Q , ( X ) .

motion

-

20

Flow, trujrc.tories ur~tltlt$)rmtrtior~

Note that flows can be composed according to @, +,(XI = @,(@AX)), i.e., X is taken to position @,(X) and then to @,+,(X). The flow can also be reversed, @,-,(XI =@,(@-,(X))= @ l ( @ l - l ( x )= ) x i.e., X is taken to x and then back to X.'

2.2. Velocity, acceleration, Lagrangian and Eulerian viewpoints The velocity is defined as

-

and it is the velocity of the particle X. The acceleration, a , is defined as a (c'.'@,(~)/i?t')I~ = a(X, t). Any function G (scalar, vector, tensor) can therefore be viewed in two different ways: G(X, t ) = Lagrangian or material i.e., follows the motion of a particular fluid particle, or G(x, t ) = ~ u l e r i a n % r spatial i.e., the property of the particle X that happens to be at the spatial location x at time t. Thus, v(X, t ) is the Lagrangian velocity and v(x, t14 is the Eulerian velocity. In most classical problems in fluid mechanics it is enough to obtain the spatial description. The material (or Lagrangian) derivative is defined as DGIDt (('G/?t)1, representing the change of property G with time while following the motion of particle X, whereas the standard time derivative is

-

?G/?t = (?G/?t)lx, and represents the change at a fixed position x. The relationship between the two is easily obtained from the chain rule DG/Dt

= ?G/it

+ V-VG,

where V is defined as V = (?/?.u,)e, (see Appendix). The expression allows the computation of the acceleration at (x, t ) without computing the motion first. The expression for the material time derivative of the Jacobian of the flow is known as Euler's formula (Serrin, 1959, p. 131 ; Chadwick, 1976, p.65). DJIDt = J(V.v) = J tr(Vv) and is the basis of a kinematical result known as the trtrnsport theorem.

Veloc,ity,acceleration, Layrangian and Eulerian consider the integral

1,

Gix, t ) d i

where V, represents a material volume, that is. a volume composed always of the same particles X belonging to the body Vf. Making reference to Figure 2.1 I V,] = " 1 : V,}, i.e.. the flow 0, transforms {V,} into { V f } at time t. The integral can be written, using the definition of the Jacobian, a s

:

r

P

G(x, r ) dc = J

G(X, t ) J d V

where d V represents a volume in the reference configuration V,,. The time derivative.

(which is a material derivative since all the particles in V, remain there) can be written as

and since the domain V, is not a function of time, expanding the material derivative we obtain

r:

I",

G(x. r ) d r =

jv![& +

G(V. v)] dc.

By means of the divergence theorem.

where n is the outward normal to the boundary of V,, denoted ?v,. In general this result holds for any arbitrary control volumes V; moving with velocity v,,

Problem 2.2 .J

Show that if A(r) is invertible, d(det A)/dr = idet A) t r [ ( d A / d r ) ' A ' l . Problem 2.2.2

that if I I ; / = e l [v~,]then ;iv,\= @ , ; i V , , ] , i.e., the boundary is mapped into the hound;iry. Usua]]y, this is taken to be that the surfiice Of a material body consists of the siime p;irticles (von Mises and

22

Flow, trajectories and defbrmution

Friedrichs, 1971). This result is called 'Lagrange's theorem' by Prandtl and Tietjens ( 1 934, p. 97).

2.3. Extension to multicomponent media In the case of multicomponent media we envision material surfaces moving with the mean mass velocity (see Chapter 9). If the system has several components, s = 1 , . . . , N , we assume the existence of a set of motions a?), and Equation (2.1.I ) is generalized as X, = @?)(X,) (no sum) where X, represents a particle of species-s and x, its position at time t.5 Each species is assigned a density p, = p,(X,, t ) such that p = p,, where the sum runs from 1 to N. Individual velocities are defined as

-

x

v, (a@ls)(x,)~at)l,= v,(x,, t ) and the average mass velocity is defined as

-x

v (P,/P)~,. The time derivative of any function G following the motion of the species denoted s, is given by DG'"/Dt = aG/at + v;VG, and the relative velocities are defined by Us = v , - V .

The simplest constitutive equation for us is

us = - W, DVO, (i.e., dilute solution or equimolecular counter-diffusion, Bird, Stewart, and Lightfoot, 1960, p. 502) where w, is the mass fraction (=p/p,) and D is the diffusion coefficient. Other quantities, such as individual deformation tensors for species-S, etc., can be defined analogously (Bowen, 1976) but they are not used in this work.

2.4. Classical means for visualization of flows There are several ways of visualizing a flow. In this section we record the three classical ones. 2.4.1. Particle path, orbit, or trajectory

Given the Eulerian velocity field v = v(x, t), the particle path of X is given by the solution of dxldt = v(x, t ) with x = X at t = 0. Physically it

Cla.ssicu1 meuns ,for visualization of Jows

23

orrcsponds to a long time exposure photograph of an illuminated fluid parti~l~. A S seen in texts of differential equations, the solution to the above x = a , @ ) , is unique and continuous with respect to the initial data if v(x) has a Lipschitz constant, K > 0.' Under these conditions, if we denote x l = @ , ( X I ) and x2 = @,(X2),we have the trajectories evolve according to K > 0. - x21 G IX1 - X21 exp(Kt),

Ix,

we shall see there are many systems (Chapter 5) that diverge from the initial conditions at an exponential rate, i.e., the non-strict inequality becomes an equality.' AS

2.4.2. Stveamlines

The streamlines correspond to the solution of the system of equations dxlds = v(x, t ) where the time r is treated as constant and s is a parameter (that is, we take a 'picture' of the vector field v at time t). Physically, we can mimic the streamlines by labelling a collection of fluid particles and taking two successive photographs at times t and t + At. Joining the displacements gives v in the neighborhood of the point x. The streamlines are tangential to the instantaneous velocity at every point, except at points where v = 0. 2.4.3. Stveaklines

The picture at time t of the streakline passing through the point x' is the curved formed by all the particles X which happened to pass by x' during the time 0 < t' < t. Physically, it corresponds to the curve traced out by a non-diffusive tracer (i.e., the particles X of the tracer move according to x = @,(X))injected at the position x'. Example 2.4.1 Compute the pathlines, streamlines, and streaklines corresponding to the unsteady Eulerian velocity field

+

"1 = x l l ( l t), To compute the pathlines we solve

d.u,/dt = x1/(l + t), with the condition x, = X,, x, = X,, at t .Y, = X l ( l and, eliminating t, we obtain .Y1 -

+ t),

v 2 = 1.

8

dx2/dt = 1 = 0.

s, = X,

x 1 x 2= X1(l - X2).

+t

24

Flow, tr~ljectories~ l n ddeformrrtion

i.e., the particles move in straight lines. The streamlines are given by the solution of dx,~d= s x,/(l + t), dx,/ds = 1 with the condition x, = x y , x, = x;, at s = 0, while holding r constant. Thus, the streamline passing by x, = xy, x , = x;, is given by x, = x,0 + s, x , = xy exp[s/(l + t ) ] , and eliminating the parameter s, ( 1 + t ) ln(x,jxy) = x , - x,,0 which shows that the streamlines are time dependent. T o get the streakline passing through x',, x; we first invert the particle paths at time t'

+

X , = ,u; - t ' , X I = xi/( l t'), which indicates that the particlt X , , X , will be found at the position x i , x; at time t'. The place occupied by this particle at any time t is found again from the particle path as

+

+

+

x, = .u; - t' t , x , = x',(l t ) J ( l t ' ) , and is interpreted as: the particle which occupied position x;, x; at time t' will be found in position x,, x , at time t . Eliminating ['we get the locus of the streakline passing by x',, x;:

x , ( l + x ; + t ) + x',(I + t ) = 0 which shows that the streaklines ure ulso ,functions of' time (plots corresponding to this example are given by Truesdell and Toupin, 1960, p. 333). x,x,

-

2.5. Steady and periodic flows A flow is steady if it lacks explicit time dependence, i.e., v = v(x). Note that the concept of steadiness depends on the frame of reference. An unsteady flow in one frame can be steady in another frame (moving frames are studied in Chapter 3 ) . When the flow is steady in a given frame F, the streamlines and pathlines coincide when viewed in the frame F . Furthermore, the streaklines coincide with both streamlines and pathlines provided that the position of the dye-injection apparatus is fixed with respect to the frame F. It is obvious that these statements are a property of dynamical systems in general and they are not confined to fluid mechanical systems. A point x such that v(x) = 0 for all t is called a ,fired or ,siri
Steady and periodic flows point in fluid mechanics). A point P is periodic, of period T, if P = @,.(P)

,

for = T but not for any r < T. That is, the material particle which happened to be at the position P a t a time r = 0, without loss of generality, will be located in exactly the same spatial position after a time T," i.e., if the flow at different times we have the sequence Po, P,,P,, . . ., p,, r p . Note that the concept of periodicity depends also on the frame of reference. "' ~.rtcrnp/d.5.1

consider the streamlines, pathlines, and streaklines in the shear flow V, = 0 V, = 1 + t a n h .u,, subjected to a time dependent perturbation I.', = 2ri sech 27-r.~~ tanh 27-r.~~ sin[2rc(.u1 - r)], I,>

= 2ri

sech 2 n s 2 cos[2n(.uI - t)].

where tr is the amplitude of the fluctuations. This problem was analyzed by Hnma (1962) to serve as a warning in the interpretation of flow visuoli~ationstudies since it is a case where the streaklines are considerably more complicated than the streamlines and pathlines. The motion of the fluid particles is governed by the system of equations rls,ldr = I . , ( x , ,.Y,, t), d.u2/dr = r',(.u,, x, t ) where I . , = V, + I.', and I . , = V, + r;. Examples of computed streaklines and pathlines are shown in Figure E2.5.1 (computation by Franjione, 1987). Exurnple 2.5.2

Calculate the possible streamlines corresponding to the class of flows V = (Ov)' ex with V . v = 0 (this material is used repeatedly throughout the book ). For simplicity consider first two-dimensional flows. By continuity ( 9 . v = 0. the sum of the eigenvalues being equal to ~ e r o )we , can infer of the character of the three-dimensional flow. We consider a cut of\. = ( v \ . )xI by settings, 0. In general, the two-dimensional version of

-

( V v ) ' . (Vv);, = L. has all non-zero components, but by a suitable transformation R. R-L-R- '. L can be written in one of the three possible ways (we follow Hirsch and Smale. 1974. Chap. 5. Section 4):

[I,

:I,

[): -:I,

[I I].

26

Flow, trujectories und deformution

The character of the flow is given by the eigenvalues of L, 1.' - tr(L)i. + det(L) = 0. The discriminant is:

A and the eigenvalues are

= [tr(L)I2 - 4

det(L)

*

i(tr(L) All2). Thus, A > 0 corresponds to real eigenvalues, and tr(L) < 0 correspond to eigenvalues with negative real part. If none of the eigenvalues lies on the imaginary axis, the flow is called hyperbolic. The possibilities are the following (see Figure E2.5.2(a)):

I : All veal eigenvalues of different signs

[:

,"I

with i. < 0 < p corresponds to a saddle.

Figure E2.5.1. Streaklines and pathlines in Hama flow corresponding to an amplitude u = 0.05. The vertical scale corresponds to u, = -0.2 to x, = 0.2. ( a ) Streaklines injected at 0 . 1 5 , -0.10, -0.05, 0, 0.05, 0.10, 0.15, the time goes for 5 units; ( h ) pathlines of the particles injected at time t = 0 (tip of the streaklines), injected at 0 . 1 5 , -0.10, -0.05, 0, 0.05, 0.10, 0.15, the total time goes for 5 units.

2

Steudy and periodic flows l..lgurc E2.5.2. (11) Summary of the portraits of the linear two-dimensional ,ul(,city field v = l..x. T h c parabola corresponds t o d e t ( L ) = (I/4)Ltr(L)]'. T h e cilscriminnnt is A = Ltr(l.)I2 - 4 det L. ( h ) Examples of three-dimensional ,cl(,uity fields with t r ( L )= 0 in the neighborhood of v ( P ) = 0 (see also Figure i.6.IJ.

focus o r

focus o r

center

improper node

improper node spiral sink inside parabola A

spiral sourcc

<0

tr L

simple shear

saddle

outside parabola A > 0

28

Flow, trujectories und deformution

11: All eigenvalues have negative real parts II(u). L diuyonl~l

[:;I

[I,

;I

with i< 0 corresponds to a focus

with i. < p < 0 corresponds to a stable node.

with

j.

< 0 corresponds to an improper node.

l l ( d ) , eigencu1ue.s c?f' L ure c,omplrs conjugate

I l l : All eigenvalues have positive real parts Same as I1 but with all arrows in Figure E2.5.2(u) reversed. 1 E All eigenvalues are pure imaginary

corresponds to a simple shear. Two three-dimensional cases are shown in Figure E2.5.2(b). Note that if v~ R' and tr((Vv)i,)= 0, only centers, saddles, and simple shear are

allowed (see Figure P2.5.3). Problem 2.5.1 In this book the acceleration is given by a a = i v , i t +Vv-v')

=

c'v:it

+ v . Vv.

Why not

Problem 2.5.2 Consider a motion x = cb,(X) given by: .x, = X,(I + r2). >.' = X'(l + t)', . X I = XI ( I + t), Compute the Lagrangian velocity, v(X, t ) , and acceleration, a(X, t ) . Compute the Eulerian velocity. v(x, t), and acceleration a(x, t ) . Verify that a = i v i t + v.Vv. Similarly, consider a motion given by .xl = X, exp(-r). s 2 = X2 exp( - t ) , s3 = X-j and verify that DJ,'Dr = J ( V . v ) . Is the flow steady?

Steudy und periodic flows

29

problem 2.5.3 consider the flow c, = Gx,, c2 = KGx,, where - 1 < K < 1. Show that the are given by x: - Kx: =constant, which corresponds to ellipses with axes ratio ( l . ' ( ~ ( ) "if~K, < 0, and to hyperbolas forming an angle / j = arctan(l.'K)"' between the axis of extension and x,, if K > 0 (see Figure P2.5.3). Prove that this flow is the most general representation of a linear isochoric two-dimensional flow.'

'

Problem 2.5.4 show that the pathlines corresponding to the flow v, = ux,, c, = -u(x, - bt) represent a circular motion about a center moving with a velocity = (hr, h,!u). Find the streamlines and particle paths corresponding to = ( . ~ , t ,- . Y , ) Show that the streamlines are given by x,x; = const. (for this and other examples, see Patterson, 1983). Pvohlrm 2.5.5 Consider the one-dimensional time-periodic Eulerian velocity field, v, = C; cos[k(.u - ct)]. Find the velocity and time averaged velocity experienced by a material particle. Pvohlrm 2.5.6 Given the Eulerian velocity field v = x-L,

L = const., obtain the Langrangian velocity field. Generalize for L

= L(t).

t'lgurc P2.5.3. Portraits of two-dimensional isochoric linear veloc~tyfields: = G s 2 .r 2 = K G Y , .as a function of K . ( u ) K = - I , pure rotation; ( h ) K = 0. unld~rectionalshear: and (c,) K = I , orthogonal stagnation flow. 1 ,

30

Flow, trujectories and dejbrmation

Similarly, consider the Lagrangian velocity field v = X-K, K = const. Obtain the Eulerian velocity field. Is the velocity field steady? In both cases verify that a = c?v/iit+ v Vv.

2.6. Deformation gradient and velocity gradient The basic measure of deformation with respect to X (reference configuration) is the deformation gradient, F : F

= (V,@,(X))T

i.e., FiJ= (?.ui/?Xj), or F

= D@,(X)

where Vx and D denote differentiation with respect to X. According to (2.1.3) F is non-singular. The basic measure of deformation with respect to x (present configuration) is the velocity gradient Vv (V denotes differentiation with respect to x).

2.7. Kinematics of deformation-strain By differentiation of x with respect to X we obtain d.u, = (?.ui/?XJ)dX, or dx = F - d X (2.7.1 ) which gives the deformation of an infinitesimal filament of length ldxl and orientation M ( = d ~ / l d ~from I ) its reference state to the present state, dx. with length ldxl and orientation m ( = ddldxl). That is, F - d X 4 dx (see Figure 2.7.1).12 This relation forms the basis of deformation of a material filament. The corresponding relation for the areal vector of an infinitesimal material plane is given by (2.7.2) da = (det F ) ( F - ' ) ' . d A and can be obtained similarly. In this case the area in the present configuration is da = Ida1 and the orientation n (=da/ldal), the area in the reference

31

Kinemutics of deformution-struin

c,,nfigurati~nis dA = ldAl and the initial orientation N ( = d ~ / l d A l )(see Figure 2.7.1). The volumetric change from dV to dtl is given by dt'=(detF)dV. l 3 (2.7.3) problem 2.7.1

Obtain (2.7.2) by defining da = d x , x dx,, dA dx = F - d X .

=dX,

x dX, and using

problem 2.7.2 Obtain (2.7.3). The measures of strain here are the length stretch, i., and the area 11. They are defined as

Figure 2.7.1. Deformation of infinitesimal elements, lines surfaces. and volumes.

32

Flow, trajectories and deformation

and can be obtained from 2 = (C:MM)'I2 q = (det F)(C- ' : NN)'I2 where C ( - F T - F ) is called the right Cauchy-Green strain tensor.14 The vectors M and N are defined by M = dX/ldXI N -= d ~ / l d ~ l . For comparison, the volumetric change J

-

(dv/dV) is given by

J = det F.

(2.7.6)

The orientations of the vectors dx and da are given by

The relationship of m and M, and n and N, to the motion is given by: m = F-MIL

n

= (det

F)(F-')T-N/y.

Since F is non-singular, the polar decomposition theorem states that it can be written as F = R.U = V-R where U and V are positive definite and R is proper orthogonal. A tensor T is positive definite if T: uu > 0 for all u # 0. R is proper orthogonal if R - R T= 1 and det R = + 1. Thus, locally the motion is a composition of rotation and stretching. Example 2.7.1 Consider a flow @,(X)that transforms the body V,, into V,; i.e., @,( V") = { V,)

and a set of motions transforming j V,) into j V , ) , j V , ) into j V,), . . .

@:"'(v,- ,)

=

(V") = { V,)

Motion uround u point or cqu~valently,In a different notation, @;"(x,,; = j x l ; @;"(xl; = (x,; @;"(x2; = (x3i @;"'(xn- = (x,); = (x,j w h ~ c h\bows that j x , , j is the reference configuration for the flow The motions are composed of @;n).@;n-I). . . . .@;2).@;1)( ) =@,( ) and the deformation gradients are composed as ~ ( " 1F(". 1 ) . . ~ ( 2 1 F( . I )( ) = F( ). I t is easy to see that the lineal stretch is given by j.2 = ( F ( I ) ) T . ( F ( Z ) ) T . .( F ( nl l ) T . ( ~ ( n ) ) T .F(ll).F(n1 ) . . . . .~

@I"'.

. ..

, , ,

~ ( 1 MM ).

( 2 ) .

P r o b l ~ ~ 2.7.3 m Obtain ( 2 . 7 . 4 )through (2.7.8). Problem 2.7.4 Prnvc that DF/Dt

=(vv)~-F.

Pvoblcm 2.7.5

Calculate the optimum orientation M for maximum stretching in a given time I , in a simple shear flow I., = 7w2, L., = 0, r , = 0.

2.8. Motion around a point The Toylor series expansion of the relative velocity field around a point P (which represents either a fixed position x, or a particle X,) is v = v, + dx (Vv), + l~igllrrorder rrrr?l.s where dx is a vector centered on P. The velocity gradient VV can be decomposed uniquely into its symmetric and antisymmetric parts. Vv = D + R,

--

D ;(Vv + ( V V ) ~the ) . stretching tensor (symmetric) R ;(Vv - ( V V ) ~the ) . vorticity or spin tensor (antisymmetric) Let us now consider the physical meaning of D and R . The relative velocity, v - v,. at dx is v,,, = d x . ( D component in the direction dx is v,,, (dx/(dx()= ( D

+n ) .

+ R ) :dxdxildx(

34

Flow, trqjectories und defirmation

and v,,, - n = ( D : nn)(dxl, where n = dx/ldx(. Note that since R : nn = 0, R does not contribute to the velocity in the direction normal to the sphere ldxl = constant and the contribution of R is wholly tangential to the sphere (see Figure 2.8.1 ).

Problem 2.8.1 Where is v,,,.n maximum'? Pvoblem 2.8.2 Prove that R : nn = 0. Problem 2.8.3 Show that the vector R - n can be written as :(o x n) where w = 2(Q,,, Q,, , Q , ,) and w = V x v (ois called the vorticity vector, see Section 3.8). Present an argument to show that o represents angular rotation with speed 10/21 in a plane perpendicular to o . Pvoblem 2.8.4 Show that v,,, can be written as v,,, = fV(dx D . dx) + i ~ xpdx.

.

Pvoblem 2.8.5 Show that near a surface with normal n such that v can be written as D = (ox n)n + n(w x n)

if V.v

=0

=0

(Caswell, 1967; Huilgol, 1975). Figure 2.8.1. Velocity field around a point

(and o - n = 0 ) D

Kinemutics of deformation: rute of' struin

35

prohlrm 2.8.6

s t L l d \ the possibility of expanding the relative velocity field near a surface il S

v = $h(x)I-. x (/)(x)is a scalar function and I- is a matrix (Perry and Fairlie, 1974).

p l . ~ h l c n2.8.7 ~ Obtain the general form of the velocity near a solid surface.

2.9. Kinematics of deformation: rate of strain The companion equations to (2.7.4)-(2.7.8) are D(dx)/Dt = dx.Vv D(da)/Dr = da D(det F)/Dt - d a ( V V ) ~ D(dt.),iDt = (V.v)dv.

(2.9.1) (2.9.2) (2.9.3)

The specific rate of stretching of iand v are given by: D(ln i.),iDt = D : mm

(2.9.4)

and t hc t olumetric expansion by D(ln J)/Dt =V.v The L~~griingian histories D(ln ;.)/Dl and D(ln v)/Dt will appear repeatedly throughout this work and are called srretchiny ,functions. They are typically denoted by x . The companion equations to (2.7.7) and (2.7.8) are: (2.9.7) Dm/Dt = m.Vv - ( D : mm)m for the rate of change of the orientation of a material filament, and (2.9.8) Dn/Dt = ( D : nn)n - n e ( V ~ ) ~ for the rate of change of the orientation of the areal plane. Note Equations (2.9.4)--(2.9.6)can be written as

D(ln ;.) Dt = Vv: mm D(ln ' I )Di = Vv: ( I - nn) D(ln J ) i . ~ = r vv:1

projection of Vv onto line with orientution m projection (fVv onto plune nlirli orientution n projection qf Vv onto colume, Oflerin!?21 somewhat different interpretation of the equations. Problem 2.9.1 P

th;it Equiition (2.9.5)can be written as D ln(t,/p)/Dt = - D : nn where is the density,

Flow, trujrctorirs untl defirmution Problem 2.9.2 Starting with

D(ln q)/D/ = V-v - D: nn, show that D(ln q)/D/ = (V,-v,)

- KnU,,,

where the velocity is written as v = v, + Nr.,,, such that I:,, is the speed normal t o the surface, Vs is the surface gradient operator, and K, is the mean curvature, K A = - VS . n.

2.10. Rates of change of material integrals In Section 2.2 we computed the rate of change of the integral over a material volume. It is relatively easy t o generate similar versions for material lines and surfaces (see Figure ?.10.1). For example, for a material line joining x , ( =@,(X,)) and x, ( = @,(X,)) with configuration L,, the Figure 2.10.1. Deformation of linitc material lines and surfaces.

Rutes ?j" chunge of muteriul integruls

37

,,itablc formula is

lvherc G denotes a scalar, vector o r tensor property. Similarly, for a material surface. with configuration A,,

speci;~lcases correspond to the evolution of length of material lines and area of material surfaces. Thus, the evolution of length can be computed as

and since ( m e ( V v ) ) . mIdxl.

Noting that we obtain.

where the h is the normal of the Frenet triad (m, tangent; h, normal; b, binormal. See Aris, 1962, p. 40 o n ) and the curvature ti, is defined as dml(dx(= ~ , h .

Special case If x , = x z or if x2

v.m/

=O

x1

(the first case corresponds to a loop, the second to a line attached to nOn-moving walls for example),

S,.

d [Length(L,)] = v.htiL Idxi. dt Thec~mp;inionequation for the rate of area increase is computed to be ri n) . v Idxi(ven)rAIda (2.10.4) -

rlt

Where h

, IS the mean curvature,

SA,

38

Flow, trujectories und dejormution

with V, being the surface gradient operator (V,( ) = V( ) - ?( )//r?n,where n is a coordinate normal to the surface). The vector a is a tangent vector to the boundary of the surface, ?A,, such that it is right-hand oriented with respect to n. Thus, a x n is tangent to A,. Special case

Closed surface, i.e., ?A, = 4

Problem 2.10.1 Denote by G(x, t ) a scalar function. Show that

Problem 2.10.2

Obtain the evolution of a finite area (Equation (2.10.4))starting with D(ln q)/Dt = V. v - D:nn. Hint: A rough derivation, without using the theory of calculus on surfaces, can be obtained by using the identity v x [f x v] = (Vf).v - (Vv).f + f(V.v) - v(V.f). Another possibility is to integrate D(ln q)lDt = (Vs.vs) - KAU,. Problem 2.10.3

Using Equation (2.9.4), prove (2.10.3).

2.1 1. Physical meaning of Vv, ( V V ) ~and , D Consider a case such that Dm lim = 0, m,, = lim m r-T Dt 1-x where the subscript ss denotes steady-state orientation. Then, by (2.9.71 we have m,,.(Vv)= (D:m,,m,,)m,, or m,;(Vv)

= (Vv: m,,m,,)m,,

Physical meaning of Vv, ( V V ) ~and , D

39

which shows that (Vv: mssmss)= y,,, is an eigenvalue of Vv (2.1 1.1) mns.ie(VV)= Yss.imss.i . . and mss.ithe eigenvectors (i = I to 3), where mss represents the steady,talc orientation of the material filament.Is similarly, for areal vectors we have Dn = 0, n,, = lim n lim 1-cc Dt 1- x and the steady-state orientation corresponds to the eigenvalues problem &,.is ( V V )=~Yss.inss.i. On the other hand, the solution of the eigenvalue problem D .di = yidi, Id,! = 1 ,<..I'

gives cigenvectors di which are the maximum directions of stretching. The physical meaning of di can be appreciated in the following way. If m coincides with di we have, Ddi/Dr = di (D + 0)- (D:didi)di

.

and using the result of Problem 2.8.2 we obtain Ddi/Dt = di.O i.e., the rate of change of the d,s is due only to 0 , giving also an alternative interpretation to the spin tensor. Examplc 2.1 1.1 Apply the above ideas to the linear flow 0 , = Gx,, v, = K G x , , where - 1 < K < I , considered in Problem 2.5.3. Note that the maximum directions of stretching are independent of K . Problem 2.1 1 .I Compute the rate of rotation of the maximum and minimum directions of Stretching in the shear flow L!, = $.u,, u, = 0, 0, = 0. DO they actually rotate? Problem 2 .I 1.2 Find the crror in the following reasoning (given in Ottino, Ranz, and Macosko. 1981): Since D(ln ;.)/Dt = D:mm and D(ln rl)/Dt = V.V - D: nn. n = I. we have D(ln ;.)/Dl + D(ln q)/Dr = V .v. Hence. if V-v = 0. we Obtain ;.I/ = 1. Problem 2.1 1.3 ''plain why i t is possible to 'integrate' D(dx)/Dl =dx.vv but not D(di)!~I= di.O,

40

Flow, trujectories und dejbrmation

Bibliography Even though the kinematical foundation of fluid mechanics dates from the seventeenth century, little is given nowadays in standard works in fluid mechanics. For modern accounts the reader should consult works in continuum mechanics. The presentation given here is based on a much larger body of work. The most comprehensive account is given by Truesdell and Toupin (1960, pp. 226-793). A brief and lucid account is given by Chadwick (1976) in Continuum mechunics. A more advanced treatment is given in Chapter I1 of Truesdell (1977). Other accessible accounts, slanted towards solid mechanics and fluid mechanics are given by Malvern (1969) and Aris (1962), respectively, whereas for historical references the reader should consult Truesdell (1954). A classical work in fluid mechanics with an unusually long discussion on the kinematics of flow around a point is Fundamentals of hydro- and aerodynamics by Tietjens, based on the lectures by Prandtl (see Prandtl and Tietjens, 1934). One of the best visual demonstrations of Lagrangian and Eulerian descriptions is given in the movie Eulerian and Layranyian descriptions in ,fluid mechanics. One clear visual treatment of deformation around a point is given in the movie Dqformation in continuou.~media. Both films are by J. L. Lumley. The scripts are given in Illustrated experiments in ,fluid mechanics, produced by the National Committee for Fluid Mechanics Films, MIT Press, Cambridge, 1972. The 'kinematical reversibility' of creeping flows is illustrated in the film Low Reynolds number creeping flows, by Taylor, also in Illustrated experiments in fluid mechanics, and in the article 'An unmixing demonstration', by Heller (1960). Both works focused on Couette flows, which are the exception rather than the rule among all twodimensional flows. In most cases reversibility is impossible in practice due to unbounded growth of initial errors.

Notes I Throughout this work X E R3.However, most of the results of this chapter are valid even if x E iWn The extension. however, does not carry over to vorticity since the relation between an antisymmetric tensor and an axial vector is valid only in iW" (Truesdell. 1954, pp. 58. 59). 2 See Arnold. 1985, p. 4. Distinguish carefully between the label of the particle. X. and its position, x . 3 It is well documented that the association of names is incorrect but it is apparently too late to set the record straight. Both descriptions are actually due to Euler. For the historical account of credits see Truesdell and Toupin. 1960. p. 327 and Truesdell, 1954. p. 30.

4 In the framework of Equation (2.1.1) v(x. 0 should be interpreted as the velocity of the particle X which happens to be at x at time I . The solution of most fluid mechanicill problems yields the Eulerian velocity since, usually, the equations are solved after bcing formulated in this viewpoint. Historically, the very earliest formulations of fluid mechanics were formulated using this viewpoint, taking v(x, I ) as a primitive quantity (for a n authoritative account. see Truesdell. 1954, p. 37). For a n alternative dcfinition ofthe Eulerian velocity, considering it as the primitive quantity, see Problem 3 3 1. i I'articles of different species can coexist at the same position x (see Bowen. 1976. for ~ a r i o u sextensions of this concept). (, Rcciill that the non-autonomous case can always be transformed into a n ;~iltonomousone by defining I = .u,. 7 rhere have been claims that the veloc~tyfield of a n ~ n v i s ~ c ifluid d in turbulent motion t n ~ g h tnot be Lipschitz. An early Indication is glven by Onsager, 1949). i; I o r reasons that w ~ l be l painfully apparent in Chapters 5 and 6, theexamples cannot be. In scncral, much more complicated than this one. 0 As in Equation (2.1.1 ), we are labell~ngparticles by their initla1 positions. however. 111 this case one has to distinguish carefully between the luhrl of the particle and it\ pl~c~rnrn atl an arbitrary time, since it can be a source of confusion. 10 It iscommon practice in continuum mechanics to usecapital letters to denote reference \[ate and lower case letters t o denote present state (see Figure 2.1 . I ) . However, as I \ also universal practice in dynamical systems in the case of periodic flows. we denote \ a \ x,,. and subsequent states as x , , x,, etc. I I 1'111sflow can be realized by means of a four-roller apparatus: the concept is due ro (iicsckus (1962). See also Section 9.3.1 and Bentley and Leal (19X6a). 12 I'quation (2.7.1) can be interpreted as a matrix multiplication where F operates o n ,I ~ . o l u m nvector d X . We shall not make any distinction between row a n d column icctors by means of a transpose since the meanlng is always cle;ir from the context 13 [ ' h ~ srelation gives the physical meaning t o the J a c o b ~ a nof the flow ( = d e t F ) . I4 ['he ho-called left Cauchy-Green strain tensor is F e F ' . The tensor C ' is called tile Piola tensor. IS Note that a steady state orientation need not exist. For example the rate of rotation can be constant o r periodic in time (see Chapter 4). Also, note that the steady state orientation might be unstable.

Consevvation equations, change of' frame, and vovticity

In this chapter we record for further use and completeness the equations of conservation of mass and linear momentum and laws of tri~nsformation for velocity. acceleration, velocity gradient, etc., for frame transformations involving translation and rotation. We conclude the chapter by studying thc equations of motion in terms of vorticity and the streamfunction.

3.1. Principle of conservation of mass l n t e g r u l cersion Thc mass contained in a material volume V, is given by I( v.1 =

St,,

pix. tl dl..

where p(u. t ) is the density (see Figure 2. I. I ). The principle of conservation of mass states that: t1( ,LI( V,) ). rlt = 0 or M ( V,) = M ( V,,), where C',, is the reference configuration. Microscopic cersions Denoting the reference mass density, p ( x , t

= 0 ) as

p,,(X), we have

Jt,,,

p(l,(X).t)JdV=

p,,(X)dV,

and since V,, is arbitrary, p ( x , t = po(X)iJ (Ldtrqrtrnqilrni.c)r.siorl). Taking the material derivative of p,,(X) we obtain

(3.1.1)

D(p,,(X))lDt= 0 = (DpjDt)J + pDJIDt. Using Euler's formula and the condition J # 0, we obtain, DpjDt = - p ( V - v ) . (3.1.2) Expanding Dp,'Dt we obtain ( / I / ( t = - V e ( I)V 1, (I.:~rlerirrrl i~c~rsiotl). (3.1.3) Both Equations (3.1.2) and (3.1.3) are known as the continuity equation or mass balance. ?

,?

/'r.ohlcm 3.1 .1 lsing the continuity equation prove that tll

jL,

I ~ ~ 1() dr i .=

jl

DG

, 11 Dr d r>

~ i h c r eG is a n y scalar, vector. o r tensor function

3.2. Principle of conservation of linear momentum /rttcgrul cersion (or Euler's uxiom) This principle states that for a material volume V, linear m o m e n t u m is conserved. i.e.. rate of changc of forces ; ~ c t i n gon rnomcnturn body

jb, jL,

[ ~ v d i .= pfdi. dt ;~ccclerat~on bod! Corccs

+

4

*b,,

td.s

(3.2.1 )

contact Corcca

\illere t(n. x. t ) is a s yet a n undefined vector called the trrrclion which depends o n the placement x o n the boundary, CV,, the instantaneous o r ~ c n t a t i o n .n, and time, t (see Figure 3.2.1). T h e vector f is the body force. which in this work is assumed to be independent of the configuration o f t l i e body, V,. Using the transport theorcm (Section 2 . 2 ) we obtain

k > r a n yregion V,, a t any time (where V, can be a n arbitrary control volume). Pr.ohlcm 3.2.1 f'l.o\e that the Principle of Conservation of Linear M o m e n t u m implies the continuum version of 'Newton's third law': t(x. n, 1 ) = t ( x . -n. t ) .

3.3 Traction t(n, x , r ) It can be proved that t(n.x. I ) = T' where T is a tensor (note convention). T h ~ implies s that thc information a b o u t tractions a t the point x a n d a n y '.ll~fi~cc with normal n is contained in the tensor T. T h e proof consists of tllrcc steps which \ye will briefly repeat here. e n ,

(i) Cuuchy's theorcm ('onsider a (small) region V, surrounding a particle X . T h e n , if L reprehents s o m e length scale o f V, we have:

C'onserlution rquutions, chunge of irutne, untl lwrtic.it.r

14

Volume ( V,) = const., L" Surface (c? V , ) = const., L2 T h e mean value theorem states that for a n y continuous function G(X, I ) defined over C', a n d i?V, we c a n find X' a n d X" (at a n y time t ) such that I-

J ,.

G do = const., L-' G ( X f ,r),

(;

d s = const., L2 G(X", t ) ,

F~gure3.2.1

(ti)

Material region

In

where X' belongs t o V, where X" belongs t o ?V,

present configuration. V,, ind~catingnormal momentum.

n and tritctlon I; ( h ) construction for balancc of angular

Truction t(n, x , t) Applying the theorem to Equation (3.2.2) we obtain D ( v ( X 1 ,r ) const., L")= pf const., "L

~r

Thcn, dividing by L2 and letting L go to zero (preserving geometrical similarity),

That is, the tractions are locally in equilibrium (see Serrin, 1959, p. 134).

(ii) Traction on an arhitvary plane (Cauchy's tetvahedvon constvuction) By means of this construction, which we will not repeat here (see Serrin, 1959, p. 134, for details), Cauchy was able to prove that the components of the traction t(n), [t([n])] = ( r n l , rnZ,rn3),are related to the normal n, In] = ( n l ,n,, n,), by the matrix multiplication:' I"' = 7'..)1.

11 I o r equivalently [t([n])] = [n][T], [t([n])] = [T1][n] whcrc the brackets [ ] represent the display of the components of the vector n and the matrix representation of the tensor T .

(iii) The third step is to prove that [TI is indeed the matrix rc~presentutionof a tensor T (see Appendix) In order to prove that the T,,s are the components of a tensor T , we need to prove that the components Ti, transform as a tensor. Since n is just a frcc vector, which is objective,

n' = Q - n , and since the traction t transforms as t'

=Q.t,

then. T transforms as' T'= Q.T.QT.

Problem 3.3.1 (Jsc Cauchy's construction in conjunction with the 'mass balance' to show that the mass flux in the direction normal to a plane n is given by j - n . I+licrc j is the mass flux vector. This allows the definition of the Eulcrian \clocity as v = j/p which can now be regarded as the primitive quantity. I-athcr than the velocity obtained by means of time differentiation of thc 'notion (Section 2 . 9 ) .

3.4. Cauchy's equation of motion Using Gauss's theorem, Equation (3.2.2) can be written as

Since V, is arbitrary, invoking the usual conditions.

we obtain Cauchy's Equation of Motion or Cauchy's First Law of Motion. This equation is in terms of spatial co-ordinates ( x , t ) . For a formulation in material co-ordinates (X, t ) see Truesdell and Toupi?. 1960, p. 553. Pvohlem 3.4.1 Show that (3.4.2)can be written as

and interpret the result physically

3.5. Principle of conservation of angular momentum In the simplest case (without body couples, or equivalently, for non-polar materials, see Truesdell and Toupin, 1960, p. 538; Serrin, 1959, p. 136) this principle states that (see Figure 3.2.1 ) (3.5.1) The interrelation between linear momentum, angular momentum (without body couples, as above), and the symmetry of the stress tensor is the following: ( 1 ) Principle of conservation of linear momentum

+ Principle of conservation

( 2 ) Principle of conservation + T of linear momentum ( 3 ) Principle of conservation of angular momentum

+

T

= TT

of angular momentum = T T -+

+ T = TT

+

Principle of conservation of angular momentum Principle of conservation of linear momentum

For example. ( 2 ) is proved in Serrin (1959. p. 136). We will assume that

M e r h u n i r u l energy equution und the energ!, equution

47

'I' = T~ but we note that non-symmetric tensors are indeed possible and have highly non-trivial consequences as well as practical importance (Rrcnner, 1984; Rosensweig, 1985). The tensor T is normally written as T = -pl+t where p = - tr(T)/3, so that t r ( t ) 0. For a fluid at rest t 0, and T = - P I . These conditions ident~fyalso an inviscid fluid. The simplest constitutive equation involving viscosity is the incompressible Newtonian fluid which is defined as T = 2 p D h n d where p is the shear viscosity. Replacing T = - p l + 2 p D into (3.4.2) and assuming V p = V v = 0, we obtain the Navier-Stokes equation

-

-

.

If

/ i = 0,

the equation is traditionally called Euler's equation.

Prohlem 3.5.1 Thc ~zormulstress on a plane n is T : nn (i.e., components Tjj). The sheur slrsss on a plane n is T : n t where t is orthogonal to n (i.e., components T,,. i f j ) . Prove that if T = - p l there are no shear stresses on any plane and that the normal stresses are non-zero and independent of n.

3.6. Mechanical energy equation and the energy equation Thc ~ncchuniculenergy equation is not an independent principle but a consequence of the Principle of Linear Momentum. Taking the scalar product of Cauchy's Equation of Motion with v and integrating over Vt h c obtain Dv (3.6.1) j v , PV. Dt dl- = pv.f do + (V.T).v do.

Sv, SV,

For a symmetric T,

+

V . ( T - V )= ( V - T ) . v T:Vv, and since R is antisymmetric, T:Vv=T:(D+R)=T:D. RupIacing into (3.6.1) and using Gauss's theorem we obtain

(yt I,,,irv2

(fl

1,.

Iv,

pf-v d r

+ $.v,

rate of work of body forces

Iv. 1,.

( T - v l - nds -

ipv2 dl. = I v , pf-v dl1 + $-v, t - v d s

of change of L~neticenergy

r;itc'

dl.=

rate of work of contact forccs

-

(T:D)

(T: D ) dl..

expansion of work and viscous dissipation

(3.6.2)

Conservation equations, change of frame, and vorticity

48

An alternative way of expressing the mechanical energy equation is

where

is the kinetic energy in V,, and

is the rate of work on dV, and within Vt. The energy equation is the first law of thermodynamics for a continuum. If we define:

E = Jvt pr dv. where e is internal energy per unit mass and q the heat flux vector, then the Principle of Conservation of Energy states: d (K + E) = Jv, p f v do + $,v, (T-v).n ds dt Combining (3.6.2) and (3.6.3) we obtain, -

i.e., the rate of generation of internal energy within the volume is due to viscous dissipation-expansion within the volume and energy input-output through the boundaries. Problem 3.6.1 With the aid of the mechanical energy equation, show that any fluid of a Newtonian incompressible fluid completely enclosed within a rigid non-moving boundary and acted upon by a conservative body force must approach zero velocity everywhere for long times. Pvoblem 3.6.2 Show that T:D gives rise to p(V-v) (expansion work) and t : D (viscous dissipation). Using the results of Section 3.7, prove that the viscous dissipation is frame indifferent. Problem 3.6.3 Verify that the viscous dissipation produced by a Newtonian fluid is positive.

problem 3.6.4

manipulation of T : D show that for the case of an inviscid fluid ('1' = - p l ) obeying the ideal gas law, the time integral of the rate of work, (' ( ( / W tir)dt, is given by the popular equation j p dl.. Indicate any other ncccssary assumptions. fj,

3.7. Change of frame ('onsider a point P as seen in two frames, F and F', related by x' = x,(t) + Q ( t ) . x (3.7.1) \+herex indicates the position in F and x' in F', x,(t) is an arbitrary vector, , ~ n dQ ( t ) represents a (time dependent) proper orthogonal transformation, 1.e.. with det[Q(t)] = 1 . Q(r)'.Q(r) = 1 If 0 represents the center of co-ordinates of the frame F, then x,(t) represents its position as seen from F' (see Figure 3.7.1). 3.7.1. Ohjectiritj

Denote generic scalars, vectors, and tensors, as 1; w, and S, respectively. f . H . and S are ohjrc,titt. if they transform according to: ( i ) , l '=,f' (ii) H'=Q.w ( i i i ) S' = Q.S.QT. Figure 3.7.1. Change of framc.

f

50

Conservution equations, change of' jrumr, untl uorticity

For example ( i i ) indicates that w transforms as a 'true' vector, and (iii) can be regarded as the definition of a second order tensor (see Appendix). However, as we shall see not every [ I x 33 matrix transforms as a 'true' vector, is., according to ( i i ) above, and not every [3 x 33 matrix qualifies as a tensor, i.e., they d o not transform according to the transformation rule (iii). However, due to usage, the words 'vector' and 'tensor' are used for quantities that d o not transform as above under the change of frame (3.7.1). If the quantities ,f; w, and S, transform according to (i)-(iii) under the frame transformation (3.7.1 ), then they are called ,frume indiffi.rpnt 3.7.2. Velocity

The velocity of a particle X is the time derivative of the position vector x. Thus, the velocity measured in F' is given by dx'ldt: dxi/dr = dx,(r)/dt Since d(QT.Q)/dt = dlldr = 0,

+ d(Q(r).x)/dt.

- + -

[dQ/dr] QT Q [dQT/dt] and since dQT/dr = (dQ/dr)T, then

+

[dQ/dr]. QT Q .(dQ/dt)T = [dQ/dt] and we conclude that

= 0,

- + ([dQIdt]. QT

CdQldrl- QT = - ( [dQ/dtl. QT i.e., [dQ/dr].QT is antisymmetric. Since dxldt = v, replacing x as x = Q T - X' QT-x(, we obtain v'

=Q-v

QT)T

IT,

+ ~ ( t+) A(t).xi

where c(r) = dx,(t)/dr

-

A(r).x,(t)

and A([) = [dQ/dr]. QT. Thus, the important result is that the velocity vector is, in general, not frame indifferent. In principle, this result can be used to establish the conditions under which a velocity field is steady in a moving frame (c:vi/c?r = 0). Special case Ifx, = constant and Q = constant, then v' = Q - v . This implies that the two frames are fixed without relative motion.

problem 3.7.1 s h o w that if A is an antisymmetric tensor A - b can always be written as x b where a is a vector. Interpret a in physical terms.

,

3.7.3. Accelevution

(J,ing (3.7.2) and after similar manipulations as in the previous section, M'L' get a ' = Q - a + 2A-v' + [(dA/dr) - A . A ) ] . x f + (dcldr - A - c ) (3.7.3) ;,c.. the acceleration is not frame indifferent. Spcciul case a ' = Q - a if and only if A = 0 and dcldt = 0 which implies Q = constant (no rotation) and dx,(r)/dt = constant (linear speed of separation between t.' and F'). Such a transformation is called a Gulileun truns/brmution. Problem 3.7.2 Provc that the Navier-Stokes equation is Galilean invariant. Problem 3.7.3 Show that the deformation tensor transforms as F' = Q - F . Problem 3.7.4 Show the transformation rules for the gradient operator V Problem 3.7.5 Verify that the length stretch is frame indifferent using the equations of Section 2.7. P ~ ~ o h l e3.7.6 m Check that the velocity gradient transforms as V'v' = Q - V v . Q T - A where A is antisymmetric. Show also that the stretching tensor and the spin tcnsor transform components as

D' = Q . D . Q ~ .

indifferent

i2' = Q . i 2 - Q T - A,

non-indifferent.

Problem 3.7.7 Compute the acceleration in frame F' using (7v1/('t + v'-V'v'. Compare the rcsult with that obtained by computing Dvl/Dr. Problem 3.7.8 Show that the isovorticity map of a flow is not changed by Galilean transformations.

Conservution equations, change qf frume, und vorticity

52

Problem 3.7.9 Assuming that if in a frame F we have pa = p f + V - T and in F' we have p'a' = p'f' + V' T ' , show that ( 2 ~ ' - A+T x i . [ d A / d t - A . A I T + dcldt - C - A T ) = ( f ' - f . ~ ~ )( ~+ I ~ ) ) v ' . ( T ' - Q . T . Q ~ Note that the first term is independent of the material while the second is dependent on the nature of the material. Present an argument to show that T ' = Q - T . Q T Note that this reasoning assumes only that the linear momentum equation has the same form in any frame and does not invoke frame indifference for T (suggested by Serrin, 1977). Pvoblem 3.7.10 Show that the equation in the moving frame F' is identical to that in frame F if the 'new' body force, f', is:

f'=

Q-f 'old force'

+ p [ 2 A - V '+ (dAldt

-

A . A ) . x 1 + (dcldt - A - c ) ] .

'extra body forces that arisedue to motion of frame'

Interpret the additional terms from a physical viewpoint. It is customary to designate the additional terms as (Batchelor, 1967, p. 140) 2A.v' = 2w x v', Coriolis 'force' - A - A . x l = - w x (w x x ' ) , centrifugal 'force' Euler's acceleration. ( d A / d t ) . x l = ( d w l d t ) x x', Interpret w . Problem 3.7.1 1 Consider the linear velocity field r , = G.x2,v2 = KG-u,. Work out, explicitly, v', for a change of frame such that

[

1

cos tot - sin o t sin w t cos o t Obtain V'v' and verify that V'v' = Q - V v . Q T - A . Obtain D': D' and verify that it is independent of (o. Compute Q':Qf and obtain its dependence with tu. Investigate the possibility of selecting o ( t ) in such a way that R':Q1= 0 . [QI

=

Problem 3.7.12 Consider O(x. t ) = 0, cos[k(s, - c t ) ] .Sample tI in a trajectory x, = R cos o f , s, = R sin mt. Compute the time history of 0 observed following this trajectory.

Vorticity distribution

53

3.8. Vorticity distribution ~t is well known that new insight is obtained when the behavior of fluid motion is examined in terms of vorticity, rather than velocity. Several hard to visualize or understand in terms of velocity and momentum, become clear in this type of formulation.

Origin of vorticity We have seen that the velocity gradient can be written as the sum of a symmetric part (D) and an antisymmetric part (Q). Since in threedimensions an antisymmetric tensor can be expressed in terms of an axial vector we obtain

VV= D

-

1 x 012,

where o is the vorticity, o = V x v (see Truesdell, 1954, p. 3 and p. 58). The vorticity is readily interpreted as twice the local angular velocity in the fluid (see Section 2.8). Note also that v.o=v.v xv=o. Vortex line A vortex line is a line everywhere tangential to the local vorticity (i.e., same relationship as streamlines and velocity field). Obviously, vortex lines cannot cross for that would imply that a given fluid element has two different rates of rotation. Similarly, physical arguments indicate that a vortcx line cannot end somewhere in the fluid. However, they can end at boundaries, form closed loops, and can also, as will be shown in Chapter 7, continuously wander throughout space without ever intersecting themselves (SCC Section 8.7).

Making reference to Figure 3.8.l(a), the circulation on a closed curve C = C ( t )is defined as P

By using Stokes's theorem we have P

r

which gives further insight into the meaning of o, i.e.,

54

Corlsercation equations, chunye of' jrume, urld corticitj

An important kinematical result is

(i.e., replace G by v in Equation (2.10.1 ) and consider L, as a loop with perimeter C ) . Two special cases of this result are of interest. For incompressible Newtonian fluids the acceleration is given by the Navier-Stokes equation and therefore,

Thus, the circulation is constant if r, = 0 or if the vorticity is uniform. For barotropic fluids, p = , f ( p ) (Serrin, 1959, p. 150), a is given in terms of a potential, and we obtain also

A cortex tube is the surface formed by all the vortex lines passing through a given closed curve C (which is assumed to be reducible). Consider the construction of Figure 3.8.l(h) (at an arbitrary instant of time). Applying Figure 3.8.1. ( a )Vortex line. vortex tube, and circulat~onaround a closed curve C; ( h ) construction to prove constancy of circulation in a vortex tube.

1 1 1 ~divergence

S

theorem we obtain:

o - n ' da 'top'. A '

S

o e n " du

+

'bottom'. A"

hcnce

S

o . n du =

'lateral area'

S

V . o dl,= 0

r

J

w . n d u = constant.

0Lcr a vortex tube, which implies that the circulation is constant.

3.9. Vorticity dynamics Consider the Navies-Stokes equation, written a s

with f such that f = -Vcp (i.e., f is conservative given by a potential cp). [,'sing the identity v x (V x v) = IV(v.v) - v . v v and defining the speed y = (v.v)' we obtain

'

T h ~ scquatlon forms the b a s ~ sof m a n y Important results. F o r example, Mc hive ( i ) If c = 0 a n d t h e flow IS steady a n d ~ r r o t a t ~ o n(OJ a l = 0) the Equations (3.9 1) reduce to: V(cp p I' i y 2 ) = 0, cp + p p + iy' = constant (everywhere In the flow) Thls 1s the s ~ m p l e s tverslon of the 'Bernoull~e q u a t ~ o n ' ( 1 1 ) I f the flow 1s steddy a n d r) = 0 In t h ~ scase, uslng the d e f i n ~ t ~ oof n the materlal derlvdtlve we obtaln, D(cp p l p l y 2 ) Dt = 0 , so cp p p iq' = constant over the pathline of a fluid particle" ( a n d ~ i l s o ,since the flow is steady, over streamlines). ( i i i l I f the flow is steady, multiplying Equation (3.9.1) by v

+

+

+

+

+

+

a n d using the definition of the material derivative,

which implies that cp + p p + ~ y ~ e c r e a s following es a fluid particle if (V2v).v < 0, i t . . if the viscous forces decelerate the fluid particle.

56

Conseruution equution.~,chunge of' frame, und uorticity

(iv) If the flow is steady and two-dimensional and I ) = 0 (Crocco's theorem). Taking the dot product with v we obtain (vxo).v=O= -V(~p+p/~+:q~).v which implies that pathlines). Further Since V x V( ) = O obtain an equation

+ p/p + :q2

is constant over streamlines (and manipulation produces a very important result. by taking the curl of Equation (3.9.1) we can in terms of vorticty and velocity gradients.

cp

Using the identity and recognizing that the underlined terms are zero, we obtain

or alternatively,

rate of change of vorticity following 21 particle

interaction between velocity gradients and vorticity

diffusion of vorticity

Remarks Note that the vorticity equation does not involve pressure. Note also that o - V v = 0 in two cases: two-dimensional motion, where o is perpendicular to v, and unidirectional motion, i.e., v = (u,, 0,O). Under these conditions the vorticity diffuses as a passive scalar. Note that the transport coefficient is the kinematical viscosity. Problem 3.9.1 Verify that R = - l x 4 2 . Problem 3.9.2 Show that if V-v = 0 and V x v = 0, then the viscous forces in a Newtonian fluid, V2v, are zero. Problem 3.9.3 There are cases where the intensification of vorticity due to velocity gradients and the vorticity diffusion balance in such a way that there is :I steady state distribution of vorticity. Consider 1 % - = x. r r = - ;%I-.

Vortex line stretching in inviscid jluid 1.ind c , consistent with the Navier-Stokes's Equation for the cases: ( I ) (11= tu(r) (Burgers, 1948; Batchelor, 1979, p. 27 1 ). ( i i ) ( 1 ) ; = o),(r, 0, t ) (Lundgren, 1982).

3.10. Macroscopic balance of vorticity I n order to obtain a macroscopic balance of vorticity, form the scalar I)~.oductof Equation (3.9.2) with o and integrate over material volume I ; . After some manipulations,

' I Sb(a ds = J v , (Vv:ow) d r + s

Jv,

(Vo: (Vw)') du - ;I)

$-",

V(o.o).n d

11,

rate of ;i~~.umulation

generation by stretching

dissipation (always positive)

generation at the boundaries

(3.10.1) Problem 3.10.1 Show that in the case of a fluid enclosed entirely by stationary rigid the energy dissipation decays as j 02d r . boundaries (or v 0 as 1x1 x), 1!i11/ : Prove the kinematical identity -+

-+

V - a = D(V.v)/Dt + D: D - :02. Problem 3.10.2 Ekpnnd the vorticity near a solid wall with normal n. Assume that very near the wall o is parallel to the wall. Show that to the first order approximation v = (ox n): \~llcrez measures distances normal to the wall. Assuming V-v = 0 show that the normal component of the velocity field is proportional to I' (Lighthill, 1963, p. 64). Obtain the same result by direct expansion of the velocity field in a Taylor series expansion, forcing the series to satisfy Impenetrability and no-slip at the wall.

3.1 1. Vortex line stretching in inviscid fluid tor

11

=O

Equation (3.9.2) gives

DolDt = o.Vv. (3.11.1) -1'hc solution of (3.l I. l ) can be computed explicitly (Serrin. 1959. p. 152). 1)cfine a vectorc by w = c . F'. where F is thedefcx-mation tensor. Then. Dcu,lDt = cu-Vv = (Dc/Dt).F1+ c - ( D F 1 / D t ) .

58

Consercution equations, change of frume, and tlorticity

Since DFT/Dt = FT.Vv, and o = c.FT, we obtain W . V V = (DC,D ~ ) . + F O.VV, ~ which implies that (Dc/Dt).FT= 0 and c = c(X). Since at t = 0,o = o o , we obtain o = o O . F T . Note also that (3.1 1 . I ) is identical in form to the equation giving the material rate of change of dx (Section 2.9). Thus, D ( o - ti dx)lDt = (o- ti dx).Vv where K is a constant. Ifat t

= 0,(o- ti

d x ) is equal to (o, - ti dX) then

Hence if (o, - ti dX) = 0 initially, then (o- ti d x ) = 0, i.e., vortex lines move as material lines (Figure 3.1 1 . I ). For a vortex line initially coincident with material filament, we obtain /o///oo/ = E.li.,. Thus, the intensification ofvorticity is proportional to the length ~ t r e n g t h . ~ Many of these results can be easily generalized to the case of barotropic fluids (see Batchelor, 1967). For example, (3.l I . 1 ) becomes D(o/p)Dt = (olp).Vv.

Figure 3 . 1 1 . 1 . Stretch~ngof material lines in invisc~dflow: the material lines and the vortex l ~ n e scoincide.

material line

Streurnjunction und potentiul junction

59

o t h e r results follow within this framework. For example, it can be shown that the helicity (Moffat, 1969) I

=

J"

d'.,

measure of the degree of knottedness of the vortex lines, satisfies DI/Dt = 0 provided that n . o = 0 at the boundaries of V (or o = O ( / X ~ - as ~ ) 1x1 + z ) . Furthermore I is non-zero if the vortex lines are knotted.

,I

Problem 3.1 1.1 llsing Equation (3.1 1.1) show that a body of fluid initially in irrotational rnotion continues to move irrotationally.

3.12. Streamfunction and potential function Thc streamfunction allows the formulation of problems in such a way that the continuity equation is satisfied identically. Such a formulation is possible in two-dimensional flows and axisymmetric flows and for very special classes of three-dimensional flows (Truesdell and Toupin, 1960, p. 479). For example, for two-dimensional flow in rectangular coordinates M.C can write v = V x ($e,), or6 v, = - (!*/?x, c, = i7*/('y, in such a way that V-v = 0. The curve $(x, y, t = given) = constant, gives thc instantaneous picture of the streamlines. Since v.V$ = 0, then if it//:'?t=O, IC/ is constant following a fluid particle.' The streamfunction and the vorticity are related by

v2* = - w . In irrotational motions there are several simplifications. T o start with, $ satislies Laplace's equation

v2* = 0. Also, since the flow is irrotational the velocity can be obtained from a potential, 4 ( x , t), as

v =v4, ( V x V4 = w = 0). Also since V-v = 0, the potential function satisfies v24= 0
?*/?Y ?4/?y

= ?l+b/?y, =

-

(?l+b/?.~),

60

Conserrution eyuutions, chunge c?j' jrume, unil rorticity

which are precisely the Cauchy Riemann conditions for a complex function, w ( z ) = @(x,y) + i$ ( x , y), to be analytic. Thus, any complex analytic function is the solution of some fluid mechanical problem. However, finding the complex potential for a given fluid mechanical situation is a substantially more difficult problem. Example 3.12.1 Consider the complex potential = UI + Uii'iz which corresponds to potential flow past a cylinder of radius ri (Figure E 3 . 1 2 . l ( r 1 ) ) .I~f the cylinder moves a t , speed U. we substract the contribution of the mean flow ( U z ) and the potential function is therefore Uri2,'z. Thus. to an observer mounted on the cylinder the portrait of the streamlines is given by \it(:)

- ( u ri'- sln .' O/r.) = $. which is shown in Figure 3.12.1 ( h ) .The instantaneous speed of the particle at (r, 0 ) is given by r, = ?4/?r

and

l',,

= (l/r)d@/d~

where 4 = UuZ cos I)/r and therefore v " UU'aJ'r4which indicates that all the particles located at r. = const. have the same speed. In particular for r = u the fluid particles slip with speed U. Consider the cylinder at .Y = - Y_ and label a fluid particle in the as yet undisturbed flow by its initial coordinates X I , Y , . As the cylinder passes by. the particle moves as indicated in Figure E3.12.1(c,), undergoing a tr~qectorywhich is the solution of a differential equation commonly known as the c~lti.stic,~i. The particles suffer a pcJrmiinclnr displacement in the s direction (d,. given by a rather complicated expression) and a tctnpor-tir! displacement in the j1 direction, given by d, = ;[(4a2 + Y f ) ' - I Y which is maximum when the particle is abreast the cylinder. Note that to an observer moving with the cylinder the flow is steady."

'

fl].

Example 3.12.2 Assume that we have an ~rtlstcrirl~. description of a flow field

r I = r , ( . v , ,.Y,. t )

and

r, = r2(.vI..v2, 1). How d o we calculate the instantaneous picture of the streamlines? In this case d$

=

(i$,i.\-,) d r , + (i$i s 2 d) s 2 + ( ? $ l i t ) dr

and d$

= I.,

+

d . ~-- I.? ~ d . ~ , ( i $i t d t .

Strrumfunction und potentiul junction Figure E3.IZ.I. Flow around a cylinder; ( u ) streamlines with respect to n fixed frame. [he cylinder 1s at resl and the fluid moves at speed U ; ( h ) instantaneous streamlines as they appear to an observer mounted on the cylinder: and ((,) typical trajectories of fluld parricles as [he cylinder moves from left lo r ~ g h ~ .

position of point with w / i ; n d e r a t + m

,'

position of point with cylinder

is

approximately a circle

62

Conseruution equutions, change of frume, und vorticity

Thus, the portrait at a given time t' passing by the point x ; , x i is $(XI,

~

2 t ' ,) - IC/(x;,x i ,

t ' )=

S

xI.xz.1'

(0,

d x 2 - v2 d x l ) .

x i ,x'2.1'

Such a method was used to analyze experimental data by Cantwell, Coles, and Dimotakis ( 1 978). Example 3.12.3 As will become more apparent in Chapter 7, the visualization of unsteady two-dimensional flows is extremely complicated even in cases where the velocity field is known explicitly (see, for example, Hama, 1962, and Example 2.5.1). An interesting approach to visualize flows was proposed by Cantwell (1978). The basic idea is to reduce, not exactly, but possibly ,for limited scules of' space and time, the system to an autonomous system by studying the flow in a moving frame and with new variables T , ( , , ( , , such that the velocity field is transformed into d 5 l l d ~ Ul(<1, d 5 2 l d ~= U2(<1,5 2 ) . A list of problems (such as jets, boundary layer problems, etc.) for which this approach works is given by Cantwell ( 1 98 1 ) . I 0 5 2 1 3

Problem 3.1 2.1 Compute the location at time r of a material region of radius k R , k > 1, initially surrounding a sphere of radius R outside which there is a fluid moving in potential flow. Compare the result with that obtained for a sphere of radius R in a low Reynolds number flow. Problem 3.12.2 Consider a flow in the frame F with the stream function $. Compute the stream function $' for a Galilean transformation between F and F'.

Bibliography This chapter is a very succinct account of fluid mechanics suited only to our rather specific needs. The interested reader should consult the original works on which this exposition is based. The sections on conservation equation are inspired in the classical article, 'The mathematical principles of classical fluid mechanics', by Serrin (1959). The treatment of vorticity follows Batchelor (1979). For a treatment of change of frame and objectivity the reader should consult the book by Chadwick, Continuum r,ic~c~hclriir..s (1976). A few sections, such as Example 3.12.1. are based on the article by Truesdell and Toupin (1960).

Notes I We havcdroppcd the posltlon x a n d t ~ m rc. It should be understood that all qunntltles ,ire evaluated at x, r .

2 I-clu~c;ilcntly.the transformat~onso f t as above can be consldcred a n altcrnaticc \tatcment of the P r ~ n c ~ p lofe Material Frame lnd~ffcrcncc(see S c c t ~ o n3.7). i Note that In accordance w ~ t hthe results of Chapter 6 , 'z = Q . s . Q ' . All constitutive cqu;itlons must gcncriitc a frame ~nd~fference stress tensor (for details see Schowaltcr. 1079. Chaptcr 6: Chadwlck, 1976). 4 In the language o f c h a p t e r 5 t h ~ quantlty s m ~ g h be t regarded as a 'constant'of the motlon. .; ,Accord~ngto the cl~iss~cnl plcture of u turbulent flow. lnltlally designated vortex I I I I C \ stretch and fold a s matcrlal lmcs u n t ~ lthe folding reaches a scale where the illffus~onof cortlc~tybecomes domlnant. At this point a stat~sticalsteady-state 1s reached. (1 In fluld mechanics b o o k s o n e c a n find sultablccxpresslons in otherco-ordinatesystcms. 7 hhown in Problem 4.7.1, if ;@,';I = 0 the mlxlng is poor slnce fluid particles a r c ~l-appedbetween lcvcl curccs of @. S r l i ~ sexample IS based o n M~lne-Thomson.1955, Section 9.21; and Truesdell ~ n t Toupln, l 1960, p. 332. where additional references can be found. Strcamllnes referred t o a rotatlng cylinder arc analyzed by M~lne-Thompson(1955, S c c t ~ o n9.71) and L a m b (1932 edition. Sectlon 72). A good discussion of streamlines , ~ n dstrenkl~nes.especially from a n cxpcrlmcntal viewpoint, is gicen by Tritton, 1977. Chapter 6. I0 O h c ~ o ~ ~ sifl yt h, ~ s1s poss~blethe flow cannot be chaotic (scc Chapters 5 and 6).

Computution

of' stretching und efficiency

As we have seen in ('hapter 7. the deformation tensor F a n d its associated ctc.. form the fundamental quantities for the analysis of tensors C, C deformation of infinitesinial clenicnts. In most cases the flow x = O , ( X ) is unknown a n d has t o be obtained by integration from the Eulerian velocity field. If this can be d o n e analytically. F c a n then be obtained by differentiation of the flow with respect t o the material coordinates X. T h e chapter starts with examples belonging t o this class. However, this procedure is not always convenient. o r even possible, for reasons that will be apparent in C h a p t e r s 5 through 8, a n d we explore briefly other :~pproachcs which a r c valid for relatively simple but. nevertheless, 'integrnble' flows of widespread applicability. T h e flows of interest belong ( i ) flows with a special form of Vv. a n d ( i i ) flows with a t o two ~I;ISSCS: special form of F. In class ( i ) we havc the subcases D ( V v ) Dt = O a n d D(Vv):'Dt small. In the second class we havc Constant Stretch History Motions ( C S H M ) a n d a n important subset of viscomctric flows, Steady Curvilineal Flows ( S C F ) .

'.

4.1. Efficiency of mixing According to the physical picture described in C h a p t e r I . mixing involves . h u s , if we place a material stretching a n d folding o f material c l c n ~ c n t s T filament d X a n d a n element of area d A . in nn arbitrary initial location P, the specific rates of generation of length. 11 In i 11t. a n d arca, D In rl,'Dt, a r c given by the equations of C h a p t e r 7. W e say that the flow s = O , ( X ) , o r a region o f the flow. mixes well if the time uvcragcd values of D In i D i and D In 11 I)/, d o not dcc:~y t o ~ r ono , matter what the initial placement P a n d the initial orientations 31 and N . However. it is not reasonnblc t o c o m p a r e flows o n the basis of the \ i ~ l u c sof the stl.etchings D In i D/ and I1 In 11 Dt since they a r e ohvic>usl> dependent upon the units of time. It is therefore o f i r n p o r t ; ~ ~t o~ eseek solnc ~.ation:~l\\.ay of cll~i~ntifyinp the efficiency o f

65

I:[/ic.ic~tlc.\. of' nli.uir~y

lllc stretching process in order to h a \ c some basis o f comparison for the 1111\inpability of different flows. Wc note that the specific rates of stretching a r e bounded. T h u s , for 1, In L 111, u\ing the ('auchy S c h \ \ a r ~inequality m.c ha\,c: 1) In i111 = I): m m


m m

=

( D :D ) '

',

,111i.cm m = I . F o r 11 In ti 111 we obtain 1) In ti 111 = V.v - D : n n = [ I ( V . v ) - ~ v ] : n n< J l ( V . v )- V v J n n J = [(Vav)' ,,I..

+ D : D l 1 ',

if 1) In ti 111 is written in a slightly different \\ay,

I1 In rl DI = D : ( I - nn) < D I - n n = 2 ' ' ( D : D ) ' '. I Iicsc upper bounds provide a natural \\ay of quantifying the efficiency ~ i ' t l i stretching. c Thus, m,e define the stretching efficiency, cJj, = cJ;,(X,M,r ) oi' tlic material clement d X ( M = d X d ~ )p l, ~ ~ c catd X. a s el, ( D In iD r ) . ( D :D l ' ' < I . (4.1.1)

-

I lie stretching efficiencies. el,, = c,,(X. N. r ) of the area element d A placed ,I[ N ( U = d.4 d . 4 ) . are defined in a similar \\ay. I f the motion is isochoric \\c. define cl,,= ( D In rl D r ) ( D : D ) ' ' < I , (4.1.2)

,irid if i t is not e,, E ( D In

L D r ) 12' ' ( D : D l '1 < I .

111 this

\\ark \vc will use the first definition since \\e will assume throughout [ l i ~ cthe flo\vs a r c isochoric. W e define the a.vj~r~lptotic efficiencies a s

lim c,,(X. N , t ) ', [ Ilc /irllc. rrr.clr.irqc~rlefficiencies a r e defined a s el,,, =

I

(

J

;

)

=

r

cl,(X. XI. 1 ' ) d l '

corresponding asymptotic \ ; ~ l u e s are denoted as ( ( I , ) , and r c s p c c t i ~ c l y . Efficiencies c a n also be defined \\ith respect t o ,. i \ \ : ( V F ) ' ) '': thus we h a \ c

c ~ ~ l ctheir l (',, \ /

OT (I) In D I ) ( V F :( V v l ' ) ' ' = D : m m (Vv: ( V v ) ' ) ' ' 05 f ( 1 ) 11111 111) ( V v : ( V v ) I ) '' = (Vav D : n n ) , ( V v :( V v ) ' ) ' '. f

/,

-

Similarly. ~ l l ccorrc\ponding time ~ ~ v c r i ~ p;~symptotic cd cfficiencics a r c

66

Computution

oJ' stretching und efficiency

denoted (e;), and (e,*), . Other efficiencies, for finite material lines and surfaces will be used when convenient.

4.1 . l . Properties of e, and e,

Frame indifference The efficiencies e, and e, are frame indifferent, i.e., they are objective, e, = e>,e, = eb, under the frame transformation x' = x,(t) + Q ( t ) . x . Note that D, m, and n, are frame indifferent. Since D, m, and n transform as Q . D.QT, Q - m , and Q - n , respectively, then D: mm, D : nn, and D: D are also frame indifferent. Additionally, since V.v is also frame indifferent, it follows that e, and e, are frame indifferent. Note, however, that e: and e,* are not frame indifferent since Vv is not objective. Physical meaning of e, and e, For purely viscous fluids, ( D : D)'I2 is related to the viscous dissipation. For example, for incompressible Newtonian fluids T : D = 2p(D: D). In this case the efficiency can be thought of as the fraction of the energy dissipated locally that is used to stretch fluid elements. Relationships among e,, e, and eX and e,* Since Vv: ( V V )=~D: D - O : O it follows that e* = e,/(l

+ W:)'I2,

i = j., q

where W i = - (Q:O)/(D:D ) is the kinematical vorticity number.' Although neither the C$( i = i.,q ) nor W: are objective, the product eT(1 + W;)'I2 isobjective. 4.1.2. Typical behavior of the efficiency

According to the time history of the stretching efficiency we divide bounded flows into: flows without reorientation, where the efficiency c.(X, t ) decays as t ' ; flows with partial reorientation, where there is some periodic restoration, but where on the average the efficiency also decays as t '; and flows with strong reorientation, where the time average of the efficiency tends to a constant value (see Figure 4.1 . I ). We shall encounter examples of all these flows; thc first two in this chapter. the third onc in Chapters 7 and 8.

.Ef/ic.irnc~yoJ' mixing

67

4.1.3. Flow classification

~'11cability of flows to stretch material lines and areas provides a useful means of flow classification. The objective of flow classification is to define ;I .measure' which reflects the predominant characteristics of an arbitrary flow. and provides a basis for comparison of flows. Translating this rather loose concept into concrete parameters has led to the use of different crircria which have been used in different contexts. The idea is popular in shoology where it is interpreted in a 'local'sense and has applications such as drop breakup and drag reduction. T'anner and Huilgol(1975) and Tanner ( 1976) proposed a scheme based (In the asymptotic rate of growth of 'test' fluid microstructures. Their frnmcwork was extended by Olbricht, Rallison, and Leal (1982) to account for rhe microstructure shape, interactions between it and the surrounding fluid. and elastic forces resisting deformation, by means of a linearized dynamic analysis (see Chapter 9). However, these criteria are applicable onl!. to flows with D(Vv)/Dr =O. Astarita (1976) proposed an objective measure, applicable to arbitrary flows, related to the ratio of the magnitudes of the vorticity rate to the strain tensor. However, this measure is in general difficult to calculate. Another measure is the vorticity number, M'L. defined many years ago by Truesdell (1954). W i = 0 corresponds to purely irrotational motion, and the value increases with increasing vorticity becoming W i = -x for a purely rotational flow. This quantity hoivcver, suffers from the disadvantage of being, in general, non-frame indifferent (see Problems 3.7.1 1 and 4.1.2). Prohlcm 4.1.1

P r o ~ that c ifV.v = O , V E R " , ( ><~[(n - I ) / I I ] ' ~ ~ , ~ j.,q = (Khakhar, 1986). Prohlcm 4.1.2

Compute the vorticity number W i for the linear flow o, How does W z change under a change of frame?

= G.u2, P ,= K G x I .

I'ipure 4.1.1. Typical behavior of mlxing efficiency: ( 1 1 ) flow with d e c a y ~ n g cficiency: (hjflow w ~ t hpartl;ll rcstor;ltion:and ((,)flowwith strongreorientution.

Computrriion

68

stretching und c.fficirncy

of

Pvohlem 4.1.3 Show that for a two-dimensional isochoric flow such that DmlDi can be written as (1/2119[1- W:]'12 (Fra~ljione,1986).

= 0, eAm

Problem 4.1.4 Consider a two-dimensional isochoric flow. Let O denote the angle between a material filament and the maximum direction of stretching. Show that the instantaneous value of the stretching efficiency is given by (2'12/2)cos 20 (Franjione, 1987).

4.2. Examples of stretching and efficiency Example 4.2.1 Efficiency qf axisymmetric extensional ,flows In this case

,

(1.u /di = i;.u , tl.uz/tlt = ( - i:/2)s2, d .Jdt ~ I f at i = 0, x = X we have dx = dX, M = d ~ / l d ~ , .u, = XI exp(i:t ),

then F and C

=

.x2 = Xz exp[(-i;/2)t], F ' . F are computed to be exp(i:i) 0

[F]

0

=

I

[C]

=

[

exp(-i;t/2)

0 cxp(2i:t )

0

=(

i/2).x3.

.u3 = X3 exp[( - C/2)t]

0 0 exp( - i:t/2)

0

-

0

1

exp( - t t )

0 0 exp(-it) The lineal stretch is given by ., x - = M f exp(2i:t)+ M i exp( - i t ) + M : exp( -i:i) and the specific rate of stretching by, D In i - i;[2Mf exp(3ii)- M: - M i ] 2[Mf exp(3ii)+ M i + M:] Di '

The magnitude of D, ( D : D ) ' ', is equal to ?(3/2)' '. The long time value of the efficiency, (D In i.,'Dt).'(D:D ) ' is equal to (213)' 2 0.816 (which is the i~pperbound for thrcc-dinicnsional flows). unless M is of the form ( 0 . M,. M,). Note that i. = i ( X . M. i ) in this case reduces to i. = i ( M , t). This is [rile for all linear flows.

'

1:'suniple 4.2.2 Efficicvrcy of siriiple slrc~ar.i l o ~ c ' In this case (I.\--3 (I.Y, 111 = f Y : . t l . ~ ,t l r = 0.

'

111 =

0.

w'ith the samc initial condition as before,

.u,

= $s,t

+ XI, [C]

=

['

$1

: ;]

.'2

= X,,

1

+ (7r)Z

similarly, C is computed to be

.u,

= X,.

and 11 In i/Dt is,

D In i-

jM2(Ml

+ M2.jt)

[ I + M2jt(2M,+ M2jr)] In th15 case ( D : D)'I2is equal to $/(2'12).The efficiency reaches a maximum ialue of (2'12)/2s 0.707 (which is the upper bound for two-dimensional flo\z\). For long times, the efficiency decays as t - I . Dt

Erumple 4.2.3 Efficiency in linear two-dimensional ,flow Consider again the two-dimensional flow of Problem 2.5.3. Thc motion is obtained by solving the second order system

dx,/dt = Gx,,

d.u,/dr = K G x , ,

with the initial condition x, = X I and x, = X, (the algebraic manipulations can be reduced considerably using the formalism of Section 4.3). The spccific rate of stretching of a material plane N = ( N , , N,) is

for K < 0 , and D In rl

-

7Ch: exp(3r)- h: exp( - $1)

+

h: exp(?r) h: exp(-$1)

+ 2h,h2

for K > 0, where $ =2G(l~1)"~

I f K < 0, 11 In rl;lIt is periodic: if k' > 0, D In rl!Dt i~ttainsa non-zero limit. 1 lo\vevcr. of more interest than the instantaneous efficiencies are the time a\cragcd v;~l~rcs(the asymptotic values are related to the Liapunov

70

(cJ,)

Computution of stretching untl e1jicienc.y

exponents, see Chapter 5 ) . In this case,

The time evolution of (e,) for different values of K > 0 is given in Figure E4.2.1 for a material plane with initial orientation N = ( I , 0). For K > 0 the , = (2K)'l2/(1+ K ) average efficiency rises rapidly to a constant value (for $1 = 0(1),as expected) and it is relatively insensitive to the value of K ((cJ,,), is equal to 0.666 for K = 0.5 and 0.406 for K = 0.1). This observation is significant and indicates that the cffic.ienc.y tun he significuntly inr-reused hy introducing u smull urnount of irrotutionulity into the ,flow. For K = 0, corresponding to simple shear flow, (e,) decays as t ' . For K < 0 the efficiency oscillates with a period proportional to I/GIKI"~. Thus, flows with elongational character are efficient but the efficiency of shear-like flows is poor unless special precautions are taken. One way to increase the efficiency of rotational flows is by periodic reorientation of filaments or areas to offset the tendency of the flow efficiency to decay as Figure E4.2.1. Evolution of the time-averaged efficiency, ( r , ) , for various values of K in a linear flow (see Figure P2.5.3).

Exumples of' stretching and efficiency

71

I ' . Such a concept is explored in detail in Section 4.6 and forms the basis for the high efficiencies encountered in Chapters 7 and 8.

E.vurnple 4.2.4 Efficiency in the helical annular mixer Hcre we consider the stretching in the mixer of the Figure E4.2.2. The flow is a combination of Couette and Poiseuille flow. For a Newtonian fluid the azimuthal (0) and axial ( z )components of the velocity field are ~lncoupled.The velocity field is of the form ur = 0, o0 = .f'(r), u, = Y ( r ) , :ind the motion is given by the solution of drldt = 0, r dO/dt = f'(r), dzldt = y(r), that is, r= R, z = Z g(r)t, 0 =O [f(r)/r]t, uhcre ( R , O , 2 )represents the initial position of the particle ( r , H, z ) . The deformation tensor is computed using c?r 1 c'r (?r

+

+

2

1 c'z

c'R

RiO

?z (72

Figure E4.2.2. Helical annular mixer. The flow is a combination of a Couette flow (transversal) and Poiseuille flow (axial). T h e length of the mixer is L.

72

Computtrtion of' stretching and c
and

where the primes represent differentiation with respect to K. I t is relatively straightforward to compute the area stretch, ti, and to show that D In tl/ll)t decays as t and forms the basis for the understanding of more complex ~ases.~."or example, the 'mixed cup average', i.e., the average weighted with respect to the axial flow, ((.)) = { . i l , dtr/(i u, da), of the stretching function sc = D In q/Dt, ( ( r ) ) , decays as 2 '. For small gaps (K + I , K = Ri/R,,), the distribution of striation thicknesses, s, is nearly symmetric between the cylinders and the value of s,,,,,/((s j) is 1.5. Also, as K + 1 , i t can be shown that

'

(s/.s0),,,,, % 3(1

-

K)/(2zAE)

i.e., the maximum striation thickness decreases linearly with axial distance. This result corresponds to the case of the inner cylinder rotating and the outer cylinder stationary; the striations are fed radially. The parameters A and E are defined as A = L/KO,E = V,,/V,.,,,, where V, = R,R,/(K - - 1 ) and V_.,,, is the average axial velocity. For small gaps the mixing is dominated by the azimuthal flow rather than by the axial flow. Exumple 4.2.5 Efficiency in the stunduvd two-dimmsionul cuvity flow Here we consider stretching in the configuration shown in Figure E4.2.3, which is a rectangular cavity with width W and height t f , and infinitely long in the .x, direction (perpendicular to the plane s,-.x,). A twodimensional flow is induccd by the motion o f the upper wall at x, = H at a constant velocity U in the direction of negative x , . We consider the case where the flow is described by the creeping flow equations ({I,,

1 3 , )

/1v2v = v p ,

v.v = 0

in a shallow cavity, i.e., tf/W<< I. Defining dimensionless co-ordinates s,+ r , / W , s, + s2/t1and dimensionless velocities I . , and r., with respect to U , the boundary conditions are:

I-,= 0 r.,

=0

(for .x, = 0, I ; .xzE 10. I ] ) (for .x, = O; .u,E [O. I ] )

-I (for .x2 = I ; .x, r 10, I]). In this cnsc an exact solution is not :ivailable and we use the approximate solution of Chellu and Ottino (1985u) obtained using the KantorovichG:ilerkin method (the solution compirrcs very well with numcr-ical si~nulaI-, =

Extimplcs

01' strc~tchinyund e#icirncs

73

tiolls for largc aspect ratios, WIH > 0 (10)).In order to compute the length ,tl-~tchwe use ~ ( t =)

S

(c:M M ) I / ~ ~ ~ X I ,

I.( 0 )

wt~crcthe integration (numerical, but based on an analytical expression rot. thc velocity field) is carried out with respect to the initi~~l configuration of ~ h cline, L(0). For purposes of illustration, we consider two different orlcntations: M = (0, I ) , a horizontal line spanning the cavity at .u2 = H/2, Llnd M = ( 1, O), a vertical line at .u, = Wl2, also spanning the entire cavity (XC Figure 7.5.2 for experimental results). A typical result is shown in Figure E4.2.3 where [d In L(t)/dt]/(U/H), ~ h l c l ican be interpreted as the efficiency of the stretching, is shown as

I.'~gureE4.2.3. Specllic rate of length generation for two different ln~tial orlcntations. In a cav~tyof itapect rotlo W,'lI = 15. Note that the effects of ~ n ~ t orlentatlon ~al d~sappearqulckly.

74

Computution of stretching und eficiency

a function of time for the two initial placements of the line (note that ( D : D)'I2 = O(2UIH)). As expected, for short times the stretching of the vertical interface is more efficient since the line is placed perpendicular to the streamlines. However, rather quickly, the efficiency becomes independent of the initial orientation of the line and, on the average, the efficiency decays as t l . This is a case of mixing with weak reorientation. More complete computational results indicate the aspect ratio WIH, has a very small effect on the (dimensionless) stretch, which is mostly confined to changing the period of oscillation (of order WIU). Example 4.2.6 Extensions of the cavity flow and the helical mixer flow Several flows can be created by combining variations of cavity flows square, trapezoidal, with one or two walls moving in steudy motion with Poiseuille or axial drag flows. One such case is shown in Figure E4.2.4 where an upper plate slides diagonally at an angle 0 over a rectangular ~ a v i t y Provided .~ that the cross-sectional flow and axial flows can be decoupled (as in the case of a Newtonian fluid) the previous results of the cavity flow can be extended to this situation. The mixing characteristics of such a system are similar to those of the cavity flow; fluid particles are -

Figure E4.2.4. Flow in a rectangular channel produced by a combination of pressure induced axial flow and a cross flow produced by sliding diagonally an upper plate. The cross section flow corresponds to the system of Figure 7.5.2(d).

~.onfincdI o cylindrical stream-surfaces characterized by a value of ~ bAs . 11, the helical annular mixer, the mixing is largely dominated by the ;(/itnutha1 flow, since the axial flow is similar to the simple shear flow in 12stns of rnixing and involves n o reorientation. p~.ohlem4.2.1 (';ilculate the time necessary to achieve the maxirnurn efficiency in a simple ,hcar flow as a function of M , , M,, and M,. problem 4.2.2 ['rove that the relative flow around any material particle in a linear flow 15 den tical to the flow itself. Problem 4.2.3 ('alculate the length stretch i in a Couette flow by means of the formulas ~ 1 1 cinn Chapter 2, by using both Cartesian and cylindrical co-ordinates. P~.ohlem4.2.4 Study the stretching efficiency in vortex decay (Batchelor, 1967, p. 2 0 4 ) . I.,, =

(C/2nr)[I

Show that the efficiency decays as

-

exp( - r 2 / 4 0 t ) ] .

t-

'.

Problem 4.2.5 ('onsider the velocity field for r < r,

c,) = ( C r ) / ( 2 n r ? ) r, = Cl(2rrr)

for r > r ,

Show that the striation thickness varies as s .= rrr/[l + ( C ' t 2 / 4 n r 4 ) ] ' " . Thus. s A r 3 for c 2 t 2 / 4 n r 4>> I and d In .s/rlt G t I . Sirni!arly, if r, represents Ihc radial position of the layer with striation thickness s, r, moves outward :is r, .- t 1 / 3(see Figure P4.2.1) (Ottino, 1 9 8 2 ) ) . The striation thickness cllstribution is important in problems involving mixing with diffusion and r~:iction(see Chapter 9).

Problem 4.2.6 Study the efficiency in the linear three-dirnensional flow

-

tlx/tlr = x Vv \\ith Vv constant. We can formally solve for the motion as x =X

. exp(rVv).

I'hc deformation gradient is F = [exp(rVv)]'

= exp(r[Vvlr)

76

Computution qj' stretching und efliciency

and the length stretch, i.= ' FT. F: MM = exp(tVv) exp(t[VvlT):MM. Examine the stretching behavior of this flow by studying the eigenvalues of Vv (note that exp(Q-Vv.QT)= Q-exp(Vv)-Q'.). This flow is studied in more detail in Section 4.5. Compare also with the CSHM of Section 4.4. Problem 4.2.7 Consider Poiseiulle flow between parallel lines of length L. Compute the stretching iand efficiency e, as a function of the placement M and location r. What is the value of the efficiency at the center of the tube'? Obtain the map of the efficiencies of the elements reaching z = L. Obtain the location r,,, corresponding to maximum efficiency as function of L (Franjione and Ottino, 1987). Problem 4.2.8 Compute the stretching for the velocity field

c, = 0 / r ) (-4 ) c, = 0, where K is positive for diverging flow and negative for converging flow, and 4=O/r (Hamel flow, see Figure P4.2.2). Obtain the motion by integrating the velocity field with the initial condition (r. O , Z ) , = ~= (R, O, Z). Prove that F can be written as

r

=(

Figure P4.2.1. Stretching of a material line in a vortex after two and four turns of the vortex core.

Flows with u speciul form of' Vv

77

wllcre t* = 2 j ~ l t / ( ~ s cObtain ) ~ . r12 = C: N N . This problem is analyzed in 5omc dctiiil, by a different method, in Section 4.3.2.

4.3. Flows with a special form of Vv Wc consider the case of stretching of lines; the derivation for areas is siniilar. We know that the rate of change of dx is given by D(dx)/Dt = dx-Vv (4.3.1 ) with dx = dX at t = 0. Formally, this equation can be solved as

-

dx d X - F T , but, if D(Vv)lDt f 0 this route does not offer any advantage over the co~lventional approach, since in general, we cannot compute Vv(X, t ) without knowing the flow. However, for two special subcases this approach is useful, and even though their practical utility is limited, it is convenient to record them here. (i) D(Vv)lDt = 0. In this case the problem can be solved in terms of the cigenvectors of Vv. ( i i ) For slowly varying flows F(t) can be determined approximately by perturbation e x p a n ~ i o n . ~ 4.3.1. Flows with D(Vv)/Dt

=0

In this case consider the eigenvalue problems:

vv = bigi

g, .

and

Figure P4.2.2. Hamel flow in a wedge

78

Compurution of stretching untl efficiency

which generate the dual basis (gii and (gii, which we assume to be linearly independent. (Note that the eigenvalues for both problems are the same.) Taking the dot product of both sides of (4.3.1) with g', (D(dx)/Dt).gi= dx. (Vv).gi and since the gi are independent of time, D(dx.gi)/Dt = g i -( V ~ ) ~ . d x D(dx.gi)/Dt = /Iigi.dx we can solve for dx.gi, dx g' = (dX .gi) exp(/Iit). Since a vector can be written in terms of the dual bases (giJ and jg,) as dx = C (dx-gi)gi we have dx = C (dX-g')exp(/l',t)gi, and in terms of the initial orientation, since M = dX/ldXI dx = C ui e ~ p ( P ~ t ) ~ , l d X I , where ai = ( d ~ / l d ~ I ) Then, . ~ ' . the length stretch is computed as jb2= C 1 aidj exp[(pi pj)t]gij where yij = gi.gj (summations on i and j carried from I to 3). The results for area stretch are obtained in a similar manner except that the governing equation is

.

+

D(da)lDt = da D(det F)lDt - d a . ( V ~ ) ~ .

Example 4.3.1 Linear two-dimensional flow Consider again the flow of Problem 2.5.3. We can compute the stretching without solving any differential equations. In this case

with eigenvalues pi = k G(K)Ii2,0, and eigenvectors gi = [(K/(I + K))li2,f 1/(1 + K)'I2,01, [O,O, I]. Assuming K > 0, the area stretch is computed to be (7, h l , h2, defined earlier, Example 4.2.3) q2 = [(I + K)/2][h: exp(yr) + h: exp(-yt) 2hlh2(K- I)/(K I)].

+

+

Remark If K = 0, {g,) and {g') are not independent and the method breaks down (e.g., the method does not work for shear flows).

Flows with u special form of Vv

79

sp~vciulcase

When Vv is symmetric (as for example in all irrotational flows), {g;) = {g') (i.c.. orthogonal), and the equations simplify considerably. problem 4.3.1 Show that if D(Vv)/Dt = 0, then the length stretch is given by ).' = bibj exp[(Pi /3,)t]hij

xx

+

where hij = gi.g', bi = (dX/ld~I)-g,,and where the gi are obtained from gi. ( V V )=~pigi, Assume D(det F)/Dt = 0. 4.3.2 show that Problem

D In 2. - = /Ima, Dt whcre P,, is the largest positive real part of the eigenvalues of Vv. Note that if the eigenvalues are of the form fit, D In E./Dt oscillates. Since the eigenvalues of Vv are the same as those of ( V V ) ~note , also that the limit of D(ln q)/Dt is also given by p,,,. lim

1-,

4.3.3 Study the general conditions for which D(Vv)/Dt = 0 for steady flows. Problem

4.3.2 Flows with D(Vv)/Dt small

In many cases of interest it is not possible to obtain x = @,(X), and hence we do not have Vv(X, t). However, if the flow is slow,"t might be possible to express Vv in the form vv = vv, + cvv, + E ~ V V+, . . . with the base flow, v,, such that D(Vv,)/Dt = 0. In such cases it is possible to obtain the stretching by means of a perturbation expansion. For example, to obtain the lineal stretch we write dx = dx, + E d x l + s2 d x 2 + . . . Using Equation (2.9.1) and grouping powers oft: we obtain

t:" :

D(dx,,)= dx,, .VV,, Dl

+

,I

dx,, I =1

-

I .

Vv,

80

Computution of' stretching and ~ficiency

As initial conditions for the dx,s we take: dx, dx,

= dX

at 1 = 0 att=Oforn>l.

=0

Since the system of equations is linear they can be solved successively to obtain dx, up to any order in E . Exumple 4.3.2 Consider again the Hamel flow, solved earlier using the conventional approach to compare the exact and perturbation solutions. In this case we have

where ;I=

1K I / ( R ~ ) ~

and R 1s the initial radial co-ordinate of the particle X. Integrating v,, i.e., d(r2)/dt = 2K(I

-

d2)/r

with the initial condit~onr = R, and expanding the series, we obtain (Rlr)' = 1 / ( 1 - ~ t *=) 1

+ i:t* + (ct*)' + . . .

where t* = 27t and the small parameter c is

T ( 1 - d2)r. Thus, in t h ~ scase Vv, is taken to be f 2 ~ 4o E =

[v"ol=[;

-: ;]

and

vv, = (~*)"VV,,. Such an approach was used by Chella and Ottino (1985b). Figure P4.3.1 shows a comparison of the exact solution (obtained as indicated in Problem 4.2.8) and a five-term perturbation solution for two different sets of parameters. Problem 4.3.4 Calculate the stretch of a filament with orientation N = ( N , , 0 ) placed at cf, = 0 (this is the easiest case and requires no knowledge of the material of Chapter 2).

Flows with u speciul form of V v

81

plnhlem 4.3.5 stlow that near the wall, the stretch is similar to that of a shear flow only the case of diverging flow ( K > 1 ) . Show also that away from the walls [he stretch is ulwuys weaker than an elongational flow. problem 4.3.6 ~ ' ~ ~ l c u lthe a t evorticitj number for Hamel flow. Compare l o u r results with thc l~mitingbehavior obtained in the example above. Does the vorticitj number provide a good measure of the balance between shear and elongation in the flow, in terms of stretching abilitb? Studj the l ~ m i cases t of II 0 and W: + 1 . Does W: = 1 mean that the flow is elongational? Note also that W: does not dist~nguishbetween converging ( K < 0) and dl\erglng flows ( K > 0).

:

+

Prohlrm 4.3.7 Calculate the viscous dissipation following a fluid particle in Hamel flow. Figure P4.3.1. Stretching in Hamel flow. Comparison between exact solution (full line) a n d perturbation solution (broken). The values of the parameters ;ire N = ( I , 0 ) . 4 = 0.5. a = 1 '20, K < 0. (Reproduced with permission from Chella and Ottino (1985b).)

Cornput~itionof' stretching untl efficiency

4.4. Flows with a special form of F; motions with constant stretch history For our purposes a constant stretch history motion ( C S H M ) is defined such that the value of the deformation tensor following a material particle is given by7 F ( t ) = Q(t).exp[tH] where H is a constant tensor and Q ( t ) an orthogonal tensor such that Q ( 0 )= 1 (Noll, 1962). H is related to Vv through the relation

where Z is the antisymmetric tensor

Recalling the results of Section 3.7.2, H can be interpreted as the velocity gradient tensor with reference to a frame that moves with a fluid particle and rotates as Q(t). The tensor H is the fundamental quantity in the analysis of deformation in a CSHM.; if H is known, then it is a trivial matter to calculate /. and ) I . However, in general, it is not an easy matter to calculate H , starting with v = v(x) without calculating the flow first. One important case where this can be done corresponds to an important subclass of viscometric flows (see Problem 4.4.2) termed Steudj, Curvilineal Flows (SCF).8 Noll (1962) defined an S C F as a motion whose velocity field has the contravariant components

in an orthogonal system (.uk] for which the magnitudes of the natural basis of (.uki, e, (see Appendix), are constant along the trajectory = <(X,t ) , with 5(X,t = 0 ) = 0,given by 2' = X 1 , t2= X2 + t r ( X 1 ) , = X3 + tw(X1). Thus, in an S C F the surfaces .u' = constant are material surfaces and they slide over one another without net stretching (Huilgol, 1975, Chap. 2). In the language of dynamical systems these surfaces are invariants of the motion. See Figure 4.4.1. For an SCF, H , and therefore F , can be calculated explicitly from the spatial derivatives of v using a theorem due to Noll (1962). In this case and with respect to the fixed orthogonal basis referred to above, the physical components of H are given by:

c3

<

Flows with a special , f i r m of F: constant stretch

83

LI = (02/01), h = (e3/el), and where the eis are the magnitudes of the \cctors of the natural basis and 1 is the generalized shear rate

a ith

x = ( [ ( ~ ' ) ~ +e :(~')~e:]/e:)'I2 (tile primes denote differentiation with respect to x,). For an SCF wc have,

F = exp[tH]

=1

+ Ht,

:111ci C=1+t[H+HT]+t2HT-H,

F-

' = exp[-rH]

C-

=

= 1 - Ht,

1- t [ + ~ HT] + ~ ~ H T - H .

Using the results of Chapter 2 and setting y = ~ t a, generalized shear, :~ftersome algebraic manipulations, we get that

i.c.. i z MI?, for y >> I , and except for the trivial case M, = 0 the stretch gocs linearly with time, and since (D:D)'I2 is constant following a fluid particle, the efficiency decays as t - I . This result is very important since cxnmples of SCF are all rectilinear flow in pipes, Couette flow, helical flow. torsional flow, flow between coaxial cones or concentric spheres, etc. Figure 4.4.1. Representation of a steady curvilineal flow.

84

Computution of stretching and efficiency

Example 4.4.1 Consider the shear flow c,

c2 = 0, c3 = 0. F can be written as F=l+tH where H has the representation = 7x2,

Problem 4.4.1 Show that if D(Vv),/Dt= 0, then F ( t ) is given by F ( t ) = exp[tVv] The flow is called ciscometric if (Vv12 = 0. Show that if (Vv13 = 0, j. z t 2 and therefore D(ln E.)/Dt z t ' for long times. Problem 4.4.2 Consider a velocity field such that C 2 = Ii.Y,, v , = cpx, + K X 2 , r , = 0, where K and cp are constants. Show that F can be written as F ( t ) = exp[tH] with H 3 = 0 (from Noll, 1962).

4.5. Efficiency in linear three-dimensional flow In this section we consider the stretching efficiency of the linear flow v = x.vv which approximates the velocity field around a fluid particle.9 In general, Vv will be a function of time but here we restrict ourselves to the case D(Vv)/Dt = 0 and V.v = 0. The streamlines corresponding to twodimensional flow were examined in Section 2.5.2. From our studies with the linear two-dimensional flow (Example 4.2.3) we anticipate that the three-dimensional flow can give rise to either timeperiodic o r constant efficiencies and our analysis here will be restricted to the set of flows leading to constant efficiencies. We wish to display in a condensed form the numerical value of the asymptotic stretching efficiency in such a way as to put into evidence the roles of both vorticity and elongation. The basic idea is the following: First. we normalize the flow (i.e., V v ) and second. we use the invariants of the normalized Vv to characterire the flow. Since one of the invariants happens to be zero (V.\ = 0). two invariants are enough.

Efficiency in lineur three-dimensionul flow

85

If we orient the axis along the eigenvectors of D , the most general expressions for D and R are:

\+here a, b, y, h, j are all constants. In order to evaluate e, (recall that c J , , = e,, ) we need to determine the eigenvalues of Vv normalized by ( I ) : D ) 1 2 . However, it turns out to be more convenient to calculate the ) ' which gives e*, (which is not invariant) eigenvalues of L Vv/(Vv: ( V V ) ~2, and then use the relation between e*, and e, (Section 4.1) to convert the results to an invariant form. The invariants of L are: I,= tr L = V . V / ( V V : ( V V ) ~ )= ' O 11, = :[(tr L)' - tr L2] = - $ tr L2 = :(I - W:)/(l + w:) 111, = det L = (Vv: ( V V ) ~{ah(u ) ' ~ + b) + [y2(u + h) - j 2 a - h2h]) where

-

W: = (CJ' + h2 + j 2 ) / ( a 2+ ah + h2). [See Olbricht, Rallison, and Leal (1982).] Thus, every linear threedimensional flow can be represented by a point on a plane with coordinates t r ( L 2 )and det(L) and assigned a unique value of e*, o r e.. The axis tr(L2) is related to the amount of vorticity in the flow whereas det(L) can be thought of as related to the three-dimensionality of the flow. The bounds Tor the region tr(L2), det(L) are: (4.5.1)

- 1 ,< tr(L2),< 1 :I 11ci

F ( t r ( L 2 ) )< det(L) < F ( t r ( L 2 ) ) xhere F ( t r ( L 2 ) )is given by -

F ( t r ( L 2 ) )= (1.54' ') sign(h)(l + 3Wi).(1

(4.5.2)

+ W:)3'2.

The region defined by these bounds is called the crc~,essihlcdoln~li11and is plotted in Figure 4.5.1. The Ls such that tr(L2) = 1 correspond to irrotational flows whereas the value tr(L2) = - 1 corresponds to pure rotation (a = h = 0 in D). T h e upper bound of (4.5.2) corresponds to uniaxial extension ( t r = -2h). with the vorticity vector parallel to the c\lcnsion i~xis.whereas the lower bound corresponds to biaxial extension

86

Computation of stretching and eficiency

(a = b) with the vorticity vector parallel to the compression axis. The

det(L) axis corresponds to generalized shear ,flows (w: = I ) . Inside the region OADC all the eigenvalues of L are real; outside it there is one real eigenvalue and two complex conjugates. The level curves are iso-contours of the efficiency e , ; i.e., e,, =/(det(L), tr(L2))= constant. Note that the relation however, is not unique, and that more than one flow can lead to the same value of e,,. Note also the following points: (i) e,, does not decrease monotonically with increasing W:; instead for any given value of det(L)one can find a value of tr(L2)that maximizes ex. (ii) All flows along AB achieve the maximum value oft., corresponding to three-dimensional flows ( = (2/3)11'). (iii) e , is zero only at the point det(L) = 0, tr(L2)= 0, corresponding to a simple two-dimensional shear flow

Problem 4.5.1 Is the flow a CSHM? Justify Figure 4.5.1. Accessible domain of a linear three-dimensional flow. (Reproduced with permission from Chella and Ottino (1985b).)

det L

lmportunce of reorientution

87

probl em 4.5.2 Obtain the pathlines of periodic points corresponding to the vector field dxldt = x.Vv. L)o all the flows that have closed orbits have poor efficiency'? (Hint: Write Ov in Jordan form and expand the exponential.) Problem 4.5.3 Identify the region in the accessible domain such that v flows are called Beltrami flows, see Section 8.7).

= const. o

(such

P~.ohlem4.5.4 ('onstruct the iso-helicity curves corresponding to Figure 4.5.1

4.6. The importance of reorientation; efficiencies in sequences of flows I t is apparent that for a wide class of flows (e.g., S C F ) the efficiency decays a s 1 . r . Some other flows (e.g., cavity flow) have partial restoration but, on the average, the efficiency decays also as l/t for long times. A possible

u a y to maintain a high average value of the efficiency is to design a sequence of flows (see Example 2.7.1 ) involving reorientation of material elcments; another possibility is to 'reorient' the flow. We discuss several special cases below. T o start with the simplest case consider the flow r , = G.u2, r2= O a n d n line with initial orientation forming an angle O,, with the s , axis I Figure 4.6.1 ( a ) ) .In this case we have D(ln i)/Dt = G(Gt cot Oo)/[l (Gt cot 00)2]

+

+

+

Figure 4.6.1, Initial orientation of filament in a shear flow ( u ) ,and in a vortex flow ( h ) .

Computution of stretching und efficiency

88

which is the behavior expected of SCFs. The average efficiency in the interval 0-y (y = Gr), e,,, is: e;, = (1/211Z) In(yz sinZ0, + y sin 20, + 1)/y and, if all initial orientations are equally likely, averaging over O,, we obtain: ( e ) , = (1/2lI2)In[]

+ y2/4]/y.

(4.6.1) The maximum value of (e), is 0.284 which corresponds to a reorientation strain = 3.98. As might be expected, the same functional relationship holds for other flows. For example, for a point vortex v, = 0 v, = w/r, and an infinitesimal material line oriented forming an angle O,, measured in the counter-clockwise direction with respect to r (see Figure 4.6.1(b)), we obtain D(ln i.)/Dt = y(yr cos2 0, - (sin 20,)/2)/(1 + y 2 t 2 cosZ8, - y t sin 8,) ;i

where y E 2w/r2. If all orientations are equally likely, we obtain, as before, the result (4.6.1). Keeping the assumption of random angle reorientation, consider the effect of a distribution of strains, y, and an infinite sequence of periods.'' Consider first the case in which the distribution of time periods is random within an interval, i.e., for (7, - p) d y d (7, + p) otherwise In this case it is possible to obtain an exact expression for the efficiency , results in an infinite series which is evaluated of the sequence, ( P ) ; . ~which numerically (Khakhar and Ottino, 1986a). The result is shown in Figure 4.6.2(u) as a function of the mean value, y,, and the width of the distribution, p. Note that the curves corresponding to p = 0.05 up to p = 1 almost coincide and that the best efficiency is achieved for p = 0 (uniform period). Another case we might consider is a normal distribution of strains, i.e., t ( 7 ) = (!IN) expC- (7 - 7 m ) 2 / 2 ~ z 1 Figure 4.6.2. Efficiency in shear flow a n d vortex flow a s a function of shear ( u ) random reorientation of the element after a strain 7 . The strains a r e random in a n interval 2p about a mean value 7,. ( h ) Normal distribution of ;. about a mean value ,*; and with standard deviation Zp. The broken line represents the maximum efficiency obtainable from the blinking vortex flow of Section 7.3. (Reproduced with permission from K h a k h a r and O t t i n o (IYX6a).)

:,.;

Importance of reorientation

90

Computution of stretching und efficiency

with

In this case it is not possible to obtain an exact expression for ( e ) y m but , for p,/y, small, (e),,m is given by an expression similar to (4.6.1), i.e., (e>,, = K(Y,, p ) lnC1 + ?i/41im, but where K(y,, p ) is a constant which is a complicated function of the variance (y) and the mean value (y,) of the distribution. A plot of (e),m as a function of y, for various values of the variance p is shown in Figure 4.6.2(h) (note similarity with 4.6.2(a)). In these two examples we imagined that while the flow remains the same, the material elements are somehow reoriented. The idea has practical merit and finds application in the improvement of mixing in devices used to mix viscous fluids, such as extruders. In extruders, which can be modelled by the construction shown in Figure 4.6.3(a), sometimes one finds 'mixing sections' (e.g., protruding pins) that disrupt the flow and reorient material elements at the expense of only a small increase in power consumption. Chella and Ottino (1985) studied the case where material elements are assigned random reorientations when they reach designated planes. As expected, this mechanism greatly increases the mixing ability of the system (the length stretch becomes exponential with the number of mixing sections). A somewhat similar situation is to keep the orientation of the material element, but to somehow change the flow. One possibility is to Ijoin' flows as shown in Figure 4.6.3(h). In this particular case a fluid particle jumps from one streamsurface to another in a periodic manner, between an alternating sequence of flows produced by top and bottom moving walls. This idea is exploited further in Chapter 8, Section 8.2. As we shall see, it leads to very efficient mixing; indeed the mixing in such a device is chaotic. Several other possiblities might occur to the reader after studying the material of Sections 8.2 and 8.3.

Problem 4.6.1 Consider the family of irrotational flows given by the complex potential w = zn. Show that these flows have an asymptotic efficiency equal to 1/2~", except for n = 1 (Franjione, 1986). Problem 4.6.2 Generalize the ideas of these sections to sequences of Steady ~urvilineal Flows. Problem 4.6.3 Generalize these ideas to the linear two-dimensional flow.

Possible ways to improve mixing

91

4.7. Possible ways to improve mixing It is apparent that there are very few velocity fields that can be integrated explicitly to obtain the stretching. Of those that can be integrated, hyperbolic flows produce the most significant stretching (and high values of ( e ) ) but they are unbounded and therefore not very practical to work with since they cannot be realized in the laboratory except in small regions of space. As we have seen in the previous section, time-periodic sequences of weak flows can achieve high efficiency and sequences of flows involving jumping from streamsurface to streamsurface hold similar promise. One might wonder if it is completely hopeless to try to achieve high efficiencies in steady bounded two-dimensional flows, and it is instructive to consider

Figure 4.6.3. (a) Extruder with mixing sections (generalization of Figure E4.2.4); ( h ) idealized flow produced by joining flows such as Figure E4.2.4, with upper and lower moving plates. A fluid particle jumps between streamsurfaces of adjacent sections.

92

Computation of stretching und efficiency

one possible way to increase mixing in such flows. One might start with hyperbolic flow (see Section 2.5) and try to improve the stretching by somehow closing the flow, i.e., by feeding the outflow into the inflow as indicated qualitatively in Figure 4.7.1. By continuity, nearby orbits have to close smoothly and form closed loops. The expectation might be that since the stretching in the neighborhood of the hyperbolic flow is exponential, then every time that a line segment passes by the hyperbolic region the stretches are compounded in such a way that the stretching is exponential. However, this is illusory. As long as the flow is steady, isochoric, and two-dimensional, we can construct a streamfunction $(x, y) such that v . V $ = 0. This implies that a material filament is trapped between curves of constant $. Another way of seeing this is to note that the topology of the streamlines in two-dimensional area preserving flows is composed of basically two building blocks: hyperbolic points (stagnation points) and elliptic points. It is obvious that if the flow is bounded and steady the streamlines join smoothly, and that a n initially designated material very much in filament can spiral between the two curves, $ , and $, the same way as in a Couette flow but that the length increase will grow linearly for long times (see Problem 4.7.1).11 Indeed, as we shall see in Chapter 6, the long time value of the averaged stretching function -

-

2 =

lim r - j

{:

' D In j.(tl) dt,] Dt'

is zero for these flows (termed 'integrable', a term defined in Chapter 6. Loosely speaking it means that it is possible to find the ~ t r e a m f u n c t i o n ' ~ ) . The same argument applies to the combination of several hyperbolic points as shown in Figure 4.7.1. Apparently we face the conclusion that all two-dimensional flows are poor mixing flows. However, this need not be so. and as we shall see in Chapter 6 and demonstrate by means of examples in Chapter 7, if the flows become unsteady (e.g., time-periodic) the mixing can be excellent. In three dimensions the situation is more complicated. If the flow is three-dimensional (Chapter 8). combinations of inflows and outflows belonging to hyperbolic points can produce excellent mixing even if the flow is steady. P~*ohlem4.7.1 Consider a two-dimension:tl steady bounded flow containing a region of closed strc:tmlincs $(.Y.J . ) . Denote by T ( $ ) the time i t takes for a fluid particle belonging to the stre:tmlinc 4'/ to return to its original position.

Possible ways to improw mixing Figure 4.7.1. Attempts at improving efficiencies in two-dimensional flows. By joining smoothly the outflow and the inflow of a hyperbolic point the streamlines form closed curve!, and the ellicicncy of mixing decays as I (sec Problem 4.7.1). In the top figure the broken line is a hotnoc'linic trajectory: in the figure involving two hyperbolic points, A and B, the broken lines form hc,lrroc.linic. trajectories.

'

0 - -

/

outflow

.

,/\ \

\

I

\

I

I

\

/

, point

inflow

\

+elliptic point

--- ------.

outflow of A joins inflow of B

.

\

I--

%?( I\

_--- i r j j

'--_----

-_-A'

94

Computution qf stretching und
Show that a material filament dx at time t , dx(t), with initial orientation m,,, is mapped to dx(t + T ) = dx(t).[l - (dT,!d$)(V$)v] higher order terms in dx, after a time T, and that the orientation m (=dx,!ldx) after a time nT, where t~ is the number of cycles of the flow, is given by

+

m,+ = m,. [ I - (dT/d$)(V$)vIn/~. Show also that, for n -t Y, lim m,,,. x v -, 0 (i.e., the filament becomes aligned with the streamlines) and that the stretching i.is linear in n (Franjione, 1987).

Bibliography The foundations for the material discussed in this chapter can be found in continuum mechanics works. The most comprehensive treatment is given in Truesdell and Toupin (1960). A more accessible presentation can be found in Truesdell (1977). The original material on S C F is from No11 ( 1 962). A particularly clear discussion can be found in Coleman, Markovitz, and Noll (1966). Major portions of this chapter can be found in Chella and Ottino (1985b).The example of the cavity flow can be found in Chella and Ottino (1985a).Section 4.5 is based on Khakhar and Ottino (1986a).

Notes 1 See Sections 56 -61 in Truesdell (1954). 2 This flow allows the computation of many quantities of interest. T h e flow is a 'steady curvilineal flow' (see Section 4 . 4 ) a n d consequently all the mixing functions can be calculated analytically. 3 For example. mixing in many polymer processing applications. See for example, Chella and Ottino (1985a). 4 Such a flow forms the basis of a practical mixing device known a s a single screw extruder. Fairly detailed analyses of the mixing are possible but numerical computations ;ire necessary (see Chella and Ottino. 19X5a). 5 These sections are based o n R. Chella a n d J. M . Ottino (1985b). The notation is slightly different. Also we corrected several unfortunate typographical errors which appear in the paper. 6 This class of flows is relevant t o loil Reynolds number floils, such a s some of those encountered in polymer processes, geophysics. etc. 7 The results of this section necessit;~teuse of the concept of rrltrrir,t, n ~ o r i c . However, ~~. since the concept is not used anywhere else in this work. we d o not discuss i t here even though we lose s o m e rigor in not doing so (see Noll 1962). X A p;~rticularlyclear and succinct discussion is given in Section 13 of C o l c m ; ~ n . Markovitz. and Noll ( 1966). 9 For additional dctails the reader should consult Chella a n d Ottino (19X5b).

Notes

95

10 This involves changing simultaneously the stretching time t and the location r if w is regarded as constant in the vortex flow or the time t and the shear rate G in theshear flow. I I There is substantially more to the analogy with the Couette flow. The Couette flow might be regarded as the prototype of an integrable flow (Chapter 6). 12 This does not imply that the streamfunction cannot be complicated. For example, Berker (1963), mentions three special cases studied by Oseen (1930) (cf. pp. 3 and 4). The streamfunctions studied can be interpreted as the superposition of two mutually orthogonal logarithmic spirals giving rise to Ivl = constant. In this case the portrait of the streamlines is complex enough to deserve the name of 'pseudoturbulent' indicated by Berker.

Chaos in dynamical systems

In this chapter we consider the flow x = @,(X) from a dynamical systems viewpoint. We study continuous and discrete dynamical systems, fixed and periodic points, invariant manifolds associated with hyperbolic points, and various signatures of chaos, such as homoclinic points and horseshoe maps.

5.1. Introduction In Chapter 3 we considered classical ways of visualizing flows using 'classical' fluid mechanical tools such as streamlines, pathlines, and streaklines, which could be regarded as global in the sense that they give some information about the entire flow. The same can be said about instantaneous contour plots of viscous dissipation, vorticity, helicity, and various other quantities. On the other hand, in Chapter 4, we considered descriptors that have to do with stretching at local scales, e.g., specific length stretch, area stretch, and mixing efficiency. However, as we have seen, only in a few cases can the velocity field be integrated exactly to compute the stretching, and in those cases where i t is indeed possible it does appear that the rate of stretching is mild. As we shall see, for most velocity fields with good mixing abilities, i t is often impossible to obtain the flow. Consequently, in many cases of interest we are unable to calculate the local quantities described in Chapter 4. In this chapter we reconsider the central question of our analysis, namely: What are the conditions under which a deterministic flow K = @,(X) is able to stretch as efficiently as possible a material surface :hroughout the space occupied by the fluid?' Even though it is not possible .o give a complete answer to this question, in this chapter we will focus )n some relevant aspects of the general problem, to build intuition when Ne examine the prototypical model flows in Chapter 7 and 8 from a lynamical systems viewpoint. Therefore, in order to provide some back;round, we present here some of the main elements of the theory of lynamical systems.

D.ynumicu1 systems

97

In order t o answer the general queston 'how docs x = @ , ( X ) mix?' we Iiave t o focus o u r attention o n the periodic orbits a n d fixed points of flows (c~otztitzuoussystc~ms)and the discrete systems derived from them (discrete Ij,~t~umic,ul sy.st~m.s).After this is d o n e we will study the invariant sets :~ssociatedwith hyperbolic points. D u e t o its importance in practice, more than passing attention is given to the case of area preserving flows. Other ecneral points should be made clear: In both Chapters 5 and 6 much of the treatment concentrates o n time-periodic flows since this case is the cnsiest t o analyze and gives rise t o central concepts such a s Poincare ,cctions a n d horseshoe maps. F o r convenience, the treatment of ,/lows ; ~ n d~nuppinysis carried o u t in parallel a n d most of the discussion is in tcrms of systems with low number of dimensions. This might require special alertness o n the part of the reader. Note also that from a physical \,icwpoint we deal with ,/lorv.s, but the analysis is sometimes more conveniently carried out in terms of maps. F o r example, the case of time-periodic two-dimensional flow necessitates three-dimensions but in tcrms of mappings it can be reduced t o just two. The discussion is heuristic. F o r the mathematical foundations the reader should consult the books by Hirsch a n d Smale (1974)a n d Guckenheimer and Holmes (1983),a n d especially the article by Smale (1967) which gives a rigorous account of much of the material discussed here.

5.2. Dynamical systems Thc starting point for o u r analysis is the flow o r motion x practice the flow is generated by the Eulerian velocity field

= @,(X). In

dxldt = v ( x , t ) , \vith the initial condition x = X a t t = O . Under fairly non-restrictive cotiditions solutions exist, a t least locally, a n d they are unique with respect to the initial d a t a i f v ( x )is Lipschitz (see Hirsch and Smale, 1974, C h a p . 8).' The motion of fluid particles in a n Eulerian velocity field is a dynamical system. In general we will consider that a continuous dynamicol system is a system of differential equations tix. dt = f ( x , t ) , (5.2.1 \\,hcrc the right hand side is arbitrary a n d X E R". An important specia c ~ i s coccurs when f is periodic in time. i t . , f ( s .t ) = f ( x , t + T ) . .flit spncc x. with t as n parnmctcr o r with x augmented by t (spncc W" x E if tlic titiic t is added a s one of the axes), is called the ~~lltrsc, s1~trc.c~ of thc

98

Chuos in dynumicul sj~stems

flow. Unless explicitly indicated we will deal exclusively with either autonomous systems or time-periodic systems. I f we denote d V = d X , dX, . . . dXN as the volume of the initial conditions, and by dc dx, d.u, . . . dx, as the volume of the initial conditions at time t , we have, as before,

-

where is the deformation gradient in N- dimension^.^ In particular, setting G = const. in the transport theorem (Section 2.2), we obtain Liouville's theorem,

where V(0) denotes the volume of the initial conditions, {X}, and V(t) denotes the volume of the initial conditions at time t , i.e., {x}= @,{X}.4 If V - f < 0 (see Equation (5.2.1)) the system is called dissipative. Dissipative systems contract volume in phase space. In Chapter 6 we will consider Hamiltonian systems, which conserve volume in phase space. These systems have the structure

where H is called the Hamiltonian and is a function of the pks and q,s, and in some cases an explicit function of time. In particular, problems arising from Newton's law of motion can be cast into this structure. Under fairly non-restrictive conditions the solution of the dynamical system (5.2.1) (see Section 2.4) gives rise to a flow x = @,(X), i.e., the initial condition X is found at x at time t . Given any point x, belonging to the phase space iWn x iW at some time arbitrarily designated as zero, the orbit or trajectory based at x, is given by @,(x,) for all times t . Thus, @,,,(x,) denotes the orbit of x, for t > 0 (future times) and @,,,(x,) for all t
Fixetl points und periodic points

99

rccorded at t = T and denoted x , , again at t = 2 T and denoted x2, and so ti. l.e., X I = @,.(x,)

x2 = @.l.(@, (x,)) ...

x,, = @,.[@.[.[. . . (x,)]; or in more compact notation, xn = @y.(x,)

(5.2.2)

where @;(. ) denotes the composition of n mappings a,. Alternative ways of expressing the same concept are

X,,

+

@T(x,,)

(5.2.3)

a~id Xn

+ I =@T(x~).

(5.2.4)

Obviously, such a mapping contains some information about the original f l o w h a d in some instances it is convenient to deal with such a map rather than with the entire flow. In general we refer to the mappings (5.2.2)-(5.2.4) as a discrete dynamical system.

5.3. Fixed points and periodic points Given a flow x

= @,(X), P

is a fixed point qf the flow if P = @,(P) for all time t ( i s , the particle located at the position P stays at P ) , as for example at a stagnation point in a velocity field. On the other hand P is :I periodic point of period T (belonging to a periodic or closed orbit) if P = @,(P) 1.c.. the particle located at the position P with orbit x = @,(P) returns to its initial position after a time T (@,(P) # P for any t < T ) , as for example 111 the orbits of circular Couette flow. For a discrete dynamical system Xn

+ 1 = f(x,,),

the definitions are similar (see Smale, 1967). A point P is a ji.ued point of' tllc mupping f( . ) if P = f"(P) lor ull n. Thus, a ,fi.ued point of' 11 ,flow and its corresponding mapping are in general not the same. Generally, a fixed point of a mapping corresponds to a periodic point of the flow (see Figurc 5.3.1 ).

100

Chaos in dynumicul systems

We say that P is a periodic point of' order n of the map f ( . ) if P =f (P)

i.e., P returns to its initial location after C . Y U ~ III/ Jiterations . (fn(P# ) P for any m < n ) . In this case we say that P is a periodic. poirlt o f ' order n . Note that if we define a mapping f ( .E ) g(.), then P is a fixed point of the map

x,, + 1 = g(x,). The most common way (in this work) of generating mappings from flows is by means of the Poincari. surfilcc. o f section which is introduced in Section 5.5. However, in this work this technique is used primarily in the context of Hamiltonian systems studied in Chapter 6.

5.4. Local stability and linearized maps The behavior near fixed and periodic points is central to the understanding of dynamical systems. In this section we review a few of the most important points. Figure 5.3.1 Fixed point of a map and period~cp a n t of a f l o ~P is a per~odic point of the flow x = @ , ( X ) and a lixed polnt of the mapplng x,,. = @,(x,). At 1 = O the polnt is located at P. @,,(P) denotes the posltlon at 1 , . @,:(P) denotes the posltlon at l 2

,

periodic orbit

Local stability and linearized maps

101

5.4.1. Definitions

1-ixed and periodic points can be stable or ~ n s t a b l e Consider .~ first the dcfinitinns of stability (Guckenheimer and Holmes, 1983, p. 4; Hirsch and Smoie, 1974, p. 185). These definitions arc for local stability since they ~;,cuson the behavior near the points.' We discuss the definitions in terms of fixed points of flows; the definitions for mappings are similar (see Arnold, 1980, p. 1 15). Iiupunov stable The point P is a stable equilibrium of the flow if for all neighborhoods U of P there exists a neighborhood U , of P E U such that x = @,(X)belongs to U for all times if X belongs to U , (see Figure 5.4.l(a)).This behavior is typified by centers; see Section 2.4. ,.I ,\ymptotimlly stable Thc point P is asymptotically stable if and only if there exists a nc~ghborhoodU of P such that for all X belonging to U we have

lim @,(X)= P 1-

I

and for t > s See Figure 5.4.1 ( b ) . This behavior is typified by sinks; see Section 2.4.

5.4.2. Stability of area preserving two-dimensional maps ('onsider the two-dimensional area preserving mapping,

Figure 5.4.1. ( a ) Representation of Liapunov stability. and ( h ) asymptotic stability.

Such mappings arc central to the understanding of flu~d m ~ x i n g in two-dimensional flows. The behavior near P is given by the linearized mapping

t n +I

= ' a t n

where A is 2 x 2 real matrix (Df) and i.,, i, are given by '2

/.

- tr[A]j.

5 = x - P. The eigenvalues of A,

+ I = 0,

since det[A] = 1 . In order to understand the dynamics of the mapping it is convenient to transform A into a Jordan form (i.e., there exists a matrix R such that S = Re A. R - ' is of Jordan form, see Hirsch and Smale, 1974, Chap. 5 . Note that det S = det A and tr S = tr A. See Example 2.5.2). Thus, if A is

we have the following cases:

Hyperbolic. (i.,, jw2real)

,,= r

Elliptic (llij= I for i = 1 , 2, with i ,

+

ic!) = exp( iH)) cos H -sin H sin H cos 8

Prirribolic (i.;= f 1 for i = 1 , 2)

The effects of the different linear mappings S on the square (0, 1 ) x (0, 1) are shown in Figure 5.4.2 (this analysis follows Percival and Richards, 1982, Chap. 2). The parabolic case corresponds to simple shear. Note that if the initial condition x, belongs to one of the eigenspaces, successive mappings are given by x, = Lyx, (i = 1 , 2); therefore all the x,s remain in the eigenspace. It follows that if /tr[A]/ > 2 one of the eigenvalues has modulus greater than one, and x, becomes unbounded. The mapping is said to be unstable. In general we say that the mapping is hyperbolic if none of its eigenvalues belongs to the unit circle; therefore hyperbolic mappings are unstable. O n the other hand if /tr[A]/ < 2 the eigenvalues are complex conjugates and lie on the unit circle (Arnold, 1980, p. 116). In this case the system is called cllipticx and the system is .stirhle. The (degeneratc) case ( i ,= I ) is called ptirribolic..

5.4.3. Families of' periodic points

('onsider a fixed point P of order k. Successive iterations produce the kimily P-P, +P,+. ..+P, (5.4.1 ) until reaching P with the kth iterate. The stability around each point is given by a linearized mapping as indicated earlier. If we want to relate the zeroth and nth iterates in the family (5.4.1) we compose the linearized mappings around each point to obtain X, = J - X , where J is the product of the Jacobian matrices evaluated at each point. The stability of the composition is given by the solution of the eigenvalue problem J - x = i.x

P~noblem5.4.1 Prove that the eigenvalues, and consequently the stability, of all the members of the family (5.4.1) are the same (Lichtenberg and Lieberman, 1983, p. 186).

5.5. Poincare sections The Poincare or surface sections method allows a systematic reduction in complexity of problems by means of a reduction in the number of dimensions (see Hirsch and Smale, 1974, Chap. 13) since it converts the Figure 5.4.2. Effect of linear area preserving mappings on a square; ( a ) hyperbolic. ( h ) elliptic, ( c ) parabolic.

flow into a map. The concept was known to Poincare and Birkhoff. As an example of the idea, consider an autonomous continuous dynamics] system rlxirit = f ( x ) with x E [W.' and a periodic orbit with period T. Cut the orbit transversally with a (small) plane and denote the intersection as P (Figure 5.5.1). Consider the successive intersections of orbits originating from a point x , near to P . The successive intersections define a map x,,, = @(x,,),i.e., the point x , is mapped to x, = @ ( x , ) ,the point x, is mapped to x, = @(x,) = a 2 ( x ,), and so on. Note that the time between successive intersections need not be equal to ~."igure 5.5.1(u)shows this idea graphically in three-dimensions. Thus the periodic trajectory in phase space corresponds to a fixed point of the mapping P = @(P). In the same fashion, a periodic trajectory of period-2 corresponds to a mapping such that

,

P, = @(PI1,

p , =@(P,)

or

P, = a 2 ( P , ) and so on (see Figure 5 . 5 . l ( h ) ) . The ideas can be generalized: for example if dx;dt = f ( x ) with x E R" we consider a surface of dimension N - 1 , x ( x ) = 0, such that the flow is everywhere transverse to i.e.,

x,

V X ( x ) . f ( x )f 0 Figure 5.5.1. Poincare section iterates: ( a )focuses o n orbits near P and shows the intersections ~ i t ah plane X orthogonal to the orbit passing by P. A point close to P. x , , is mapped to x 2 = @ ( x , ) . and then to x, = @ ( @ ( x , l l : ( h ) representation of n period-? point in Poincare section.

Invariant .sub.spuce.s: stable and unstable rnanifi,lds

105

for all X E ~ ( X This ) . defines a m a p R N - ' R N - I . Occa~ionally it is possible to define another surface of section and construct another mapping R N - 2 R N - * , and so on. In this work Poincare sections will be used in two main ways: globally, when the system is periodic in either space o r time, or locally, near periodic or homoclinic orbits. Both ideas will be discussed and used in Chapters 6 through 8. In the case of time-periodic systems the technique amounts to taking stroboscopic pictures of initial conditions placed on C ( x ) at intervals T, 2T, 3T, . . ., etc. Thus Poincare surfaces are a convenient way of displaying in a single plot the churacter qf the solutions belonging to t i l l possible iniriul conditions.1° As we shall see some of the most important properties of dynamical systems are better understood with this viewpoint. In particular, Poincare sections are of utmost importance in the case of periodic Hamiltonian systems. -+

-+

5.6. Invariant subspaces: stable and unstable manifolds Both the fixed (or periodic) hyperbolic points ofdiscrete dynamical systems and the hyperbolic fixed points of continuous dynamical systems have auociated with them invariant subspaces called .stuhlt~ and un.stuhle tr~trn~folds.' ' T o start with the simplest case, note that the linearization of discrete and continuous dynamical systems generates linecrr invariant subspaces. Thus, if dc/dt = D f ( P ) . c is the linearization of dxldt = f ( x ) around a point P such that f ( P ) = 0 with X E R", the stable, E', un.stublr, Eu, and c.c2tltrr,Ec, subspaces are defined in terms of the Jacobian D f ( P ) as:" k''

=

(spcrc.e spt~r~ric~ci by c~iqc~ri~'ec~tors c,orrc).spondiny to eiqrnr.uluccs w~host. <0 )

~'c'trl ptrrt

I t is clear that E", Eu. and E', are i111.rrr.irrntsets; initial conditions remain

[rapped in the set. Examples are given in Figure 5.6.1. Similarly. for the linearization of the mapping x + G ( x ) (or x , , + , = G(x,,)) \r it11 x E R*', 5 -+ DG(P)T, around n fixed point P , the strrhlc. E:', ~ir~.sttrhlc~. I:". and C O I I ~ O ~1;'. , subspaces are defined in terms of the Jncobian matrix I ) G ( P ) as:'-'

106

Chaos in dynamical sy.stems

These ideas can be extended to the non-linear system. For example, for the system dx,rlr = f ( x ) with a flow x = @,(X), the stable, W S ( P ) , and unstable, Wu(P), manifolds associated with the hyperbolic point P are defined by: W'(P) = (all X E R v such that @,(X) -t P as t -t x W u ( P )= :all X E R,' such that @,(X) -t P as t -t - x ) . Similarly, for mappings we have W'(P) = [all X E R' such that G " ( x ) -t P as 11 x 1 CVu(P)= [all X E RV such that G"(x)-+ P as n - x ) . I 4 An interpretation of CV'(P) and CVu(P) in the context of the Poincare maps corresponding to a hyperbolic cycle is given in Figure 5.6.2. Fluid dynamics provides a visual analog for periodic points and stable and unstable manifolds. Consider a two-dimensional time-periodic velocity field and focus on the behavior of a small fluid element near an elliptic cycle and a hyperbolic cycle. If a material point P belonging to the cycle is surrounded by a small blob S the elliptic cycle produces a net rotation

I

-+

Figure 5.6.1. Examples of linear subspaces corresponding to the system dx dl = f ( x ) with f ( P )= 0, X E W'.

Structurul stuhility

107

of the blob when P returns to its original position whereas the hyperbolic cycle contracts the blob in one direction and stretches i t in another. I n this case, the unstable manifold Wu(P)corresponds to the region mapped out by the blob when time runs forward (see Figure 5.6.3). A seemingly minor complication in the blob experiment corresponds to the case when the blob S comes back folded upon its initial position, as shown in Figure 5.6.4. The complication, however, is profound. This construction, which will be studied in Section 5.8.3, is prototypical of chaotic flows. Note that uniqueness of solutions places strong restrictions on the behavior of stable and unstable manifolds. For example, two unstable (or stable) manifolds belonging to two different periodic (or fixed) points cannot intersect; also WU(P)cannot intersect with itself, similarly, Ws(P) cannot intersect with itself. However, intersections of stable and unstable manifolds belonging to the same or different points are permitted and are, in fact, responsible for much of the complex behavior of flows.

5.7. Structural stability Suppose that dxjtit = f(x) originates a flow x = @,(X). An important question is: How 'different' is the flow generated by dx dr = f(x)+ ~ c g ( x ) Figure 5.6.2. Interpretation of stable. W . and ~ ~ n s t a b l eW. u , man~foldsfor a s map. hyperbolic cycle and ~ t corresponding

108

Chuos in dynumicul systems

where,ug(x)is some perturbation. llpg(x)ll i< Ilf(x)ll?'"n order to quantify this statement we give first (rather loosely) the idea of topological equivalence (see Srnale, 1967). Two flows are said to be topologicully equioulent (or topologically conjugate) if there is a continuous invertible transformation and a time parametrizationwhich mapsoneofthe flowsinto theother. Moreprecisely, two C' flows, x = @,(X)and x = Y,(X),are said to be Ck conjugate ( k < r ) if there exists a Ck diffeomorphism h such that a , = h-'.Yr.h. This makes the orbits of the flows coincide. Co-equivalence (i.e., h is a homeomorphism) is called ropoloyicul equioalrncc.. This implies that fixed points of a, correspond to fixed points of Y,, unstable manifolds of x = @,(X)to unstable manifolds of Y,. etc. Figure 5.6.3. Visualization of manifolds W'(P) and W u ( P )by means of a flow visualization (thought) experiment. The point P is a periodic point; a blob of tracer S encircles P. The figure represents the state of the blob after one period @,(S) and two periods @:is). Note that this picture is not possible in a steady flow since there is crossing of particle trajectories. The blob S is mapped to @ , ( S ) after a time T and to cD,,(S)after 2T: @,is)shows the position of the blob for r < T.

A vector field dxldt = f ( x ) ( o r its corresponding flow) has the property of .structurtrl stclhility if all 11-perturbations of it are topologically equivalent. More precisely: A vector field dxldt = f ( x ) (or flow x = @,(X)) is structurally stable if there exists 11 > 0 such that all C' p-perturbations of dxldt = f ( x ) ( o r flow) are topologically equivalent to dxldt = f ( x ) (or flow). Similar definitions apply to maps. Pictorial examples are given in Figure 5.7.1 (based on Abraham and Shaw, 1985, Part 3, p. 37, where additional cxamples can be found). The conditions under which structural stability is guaranteed are only known for the case of two-dimensional flows on orientable manifold^'^ and is the subject of Peixoto's theorem (Peixoto, 1962): A Ck ( k

1 ) vector field dxldt = f ( x ) on an orientable two-dimensional ~nanifoldis structurally stable if and only if: ( i ) tllr tlumhrr of' fixed points und periodic orbits is jinite and euch is hyperbolic, ( i i ) tllere ure no trujgiectories joining suddle points, ( i i i ) the non-wandering set consists of ,fixed and periodic orbits only."

Figure 5.6.4. A complication on the case of Figure 5.6.3: the blob comes back folded (1, < r , < T ) .

110

Chuos in dynumicul systems

Problem 5.7.1 Study the structural stability of the one-dimensional equation dxldt = C x n , where C is a constant, for n = 1 , 2, and 3. Consider a perturbation of the form p g ( x ) . Take y ( x ) = 1 .

Figure 5.7.1. Pictorial examples of topological equivalence, and structural stability and instability.

topologically equivalent

topologically non-equivalent

Base flows structurally stable

structurally unstable

6 +

structurally unstable

perturbations

+

% -

Results

6 % -6

Signutures of chuos

111

5.8. Signatures of chaos: homoclinic and heteroclinic points, Liapunov exponents, and horseshoe maps 5.8.1. Homoclinic and heterodinic connections and points

If the unstable manifold of a hyperbolic cycle joins smoothly with the stable manifold of another cycle, the connection formed is called heteroclinic. If the unstable manifold of a cycle joins smoothly with the stable manifold Figure 5.8.1. Homoclinic and heteroclinic connections. In (a) the stable and unstable manifolds of a hyperbolic cycle join smoothly forming a homoclinic connection; ( h ) shows two hyperbolic cycles before connecting, and in (c) the stable and unstable manifolds of the cycles join smoothly forming heteroclinic connections.

heteroclinic connection

of the same cycle, the connection is called homoclinic (see Figure 5.8.1). Similar definitions apply to maps. A point y is called heteroclinic (Smale, 1967; Guckenheimer and Holmes, 1983, p. 22, etc.) if it belongs simultaneously to both the stable and unstable manifolds of two different fixed (or periodic) points p and q. Thus, y is heteroclinic if Y E W U ( p ) nW"q). The point is called homoclinic if p = q. If the point of intersection of the manifolds at y is transversal, i.e., the manifolds intersect non-tangentially, then y is called a transverse homoclinic point (similarly for heteroclinic). Generally, when no confusion arises, a transverse homoclinic point is called a homoclinic point (similarly for heteroclinic). Figure 5.8.2 shows homoclinic and heteroclinic points for a map; only parts of the stable and unstable manifolds are shown. Such maps can be originated by the transverse intersection of one or two hyperbolic cycles, as shown in Figure 5.8.1. By definition, homoclinic and heteroclinic points belong simultaneously to two invariant sets and therefore cannot escape from them.'' One intersection implies infinitely many. In the case of an orientation preserving flow - as are all the systems originated from differential equations the area A is mapped to A' (if the area A is mapped to B the map does not -

Figure 5.8.2. Transverse homoclinic and heteroclinic connections for maps: ( a ) shows transverse homoclinic points (compare with 5.8.l(a) and 5.8.4(a)); (h) shows transverse heteroclinic points (compare with 5.8.l(h) and 5.8.4(h)). The region A is mapped to A', in turn A' is mapped into A".

\

transverse homoclinic point

preserve orientation). Note that the intersection of invariant manifolds, as shown in Figure 5.8.2, is a projection onto the Poincare section: the point belongs to the intersection at time T, 2T, etc. However, what happens at intermediate times? Again a fluid flow visualization is useful. Figure 5.8.3 (which is a continuation of Figure 5.6.3) shows, schematically, intermediate Figure 5.8.3. Visualization of W u ( P in ) a time-periodic system by means of a fluid mechanical experiment. The manifolds W q ( P )and W u ( P )intersect transversally.

simultaneously to hyperbolic periodic point

blob stretched by WU (lower part not shown) blob moves; position coincides at t and r + T

V)

W 2

.*2

."E" 4-

view in Poincar6 section

114

Chaos in dynumicul systems

times by displaying the region of Wu already travelled by the homoclinic point y and the section of W' to be travelled in future times. A tracer placed near the periodic point maps the unstable manifold (this is equivalent to the computational determination of W". By carefully placing a line of tracer on top of the stable manifold (this is extremely hard to do!) the line will form a blob in the neighborhood of the periodic point and is subsequently stretched by the unstable manifold. A construction using hyperbolic cycles serves to clarify why the area A is mapped to A' in Figure 5.8.3. Figure 5.8.4(u)shows the construction for a homoclinic connection whereas Figure 5.8.4(b)shows the construction for a heteroclinic connection. The dynamics of the homoclinic points is extremely complex and is responsible for much of the behavior of chaotic systems. In order to state a few key results we need to introduce some additional terminology (the Figure 5.8.4. Visualization of transverse ( a ) homoclinic and (b) heteroclinic connections corresponding to the maps of Figure 5.8.2. (a) indicates the mapping of the cross-hatched regions; the reader might compare this figure with 5.8.3.

Signatures of chuos

115

following concepts are due to Birkhoff, see Smale, 1967). A point x is rc~undrringif there exists some neighborhood U of x such that

for some t' > 0. Non-wut~drringpoints are defined to be those points which are not wandering. For example, the points belonging to a closed orbit are non-wandering since portions of the neighborhood intersect the initial location. Non-wandering points do not imply periodic behavior. In fact, it is possible to show that homoclinic points are non-wandering (and they are clearly not periodic). Moreover, in every neighborhood of a homoclinic point there is a periodic point (see Smale, 1967).

5.8.2. Sensitivity t o initiul conditions und Liupunov exponents

One of the manifestations of chaos most readily related to fluid mixing is the sensitivity to initial conditions. The flow @,(X) is said to be sensitive to initial conditions on a domain S if for all X # X,,, with X belonging to an >:-ballaround XI, ( i t . , the set of Xs such that IX - Xl,I < E , with E small), there exists a time, t < such that @,(X)lies outside the >:-ballfor all X,, contained in S (see Figure 5.8.5). A similar definition applies to maps.'" A signature of chaotic systems is that they are characterized by a rapid divergence of initial conditions. Usually the divergence of initial conditions is quantified by means of numbers called Liapunov exponents which are l related to the stretching o f a filament of initial conditions. l f l d ~ represents the length of a vector of initial conditions dX around X with orientation M = dX/dXI, and its length at time t is Idxi, the Liapunov exponent ai corresponding to a given orientation M,, is defined as (Lichtenberg and Lieberman, 1983, p. 2621~':

Another way of writing this, more suitable for computations, is

since /. = I(D@,(X).M,~. An N-dimensional flow has at the most N, in general different, Liapunov exponents ( i t . , there are N vectors Mi, linearly independent, each of which givcs. possibly. a different value). This is so since the eigenvalue problem

F~gure5.8.5. Sensitivity to initial conditions.

, /@,(X) .*'

has at the most N different eigenvectors. In the cases of interest here, V - v = 0, det(D
X

M i = 0.

< - I

For dissipative systems this sum is negative." If one of the Liapunov exponents is non-zero, say a,, then ldxl 5 l d ~ exp(a,t) i and the length of the fi lament grows exponentially with time. The Liapunov exponents are related to the specific stretching rate and lnixing efficiency discussed in Chapter 4. The relation to the stretching rate IS

i.e., the Liapunov exponent is the long time average of the specific rate of stretching, D In ;./Dr. Similarly, the average stretching efficiency

where

e(X, M,, t ) = D:mimi/(D:D)'I2 = (D In i . / ~ t ) / ( DD)'" : can be interpreted as a normalized Liapunov exponent (with respect to ( D :D ) ' '). The relationship between the (maximum) Liapunov exponent and the efficiency is not direct, unless ( D : D ) is constant over the pathlines. In most cases of interest ( D : D)'" is a function of both X and t. 5.8.3. Horseshoe maps

ldorseshoe maps occupy a central position in dynamical systems, and as we shall see they are very relevant to mixing. The existence of such a m a p Indicates that the system is chaotic; in fact they can be regarded as the archetypical chaotic map (see equivalence among definitions of chaos in Section 5.9). Before going into their characteristics let us consider what we mean by 'mixing' from a mathematical standpoint. We will give two definitions. The difference between the two lies in the types of sets that arc allowed to remain unmixed. Consider a flow region K and identify, arbitrarily, two others regions contained in R : a material volume A and another volume, fixed in space, B (see Figure 5.8.6). In a rough sense we hay that the system mixes if there is a time T (or a number of mappings N), such that for any t > T, (or tl > N ) ,
118

Chuos in dynamicul systems

Mathematically, there are two main versions of this idea: ( i ) We may take A and B to be the collection of all measurable sets in the region R. If the above property works for all sets except some of measure zero, we say that the system is strongly measure-theoreric mixing.

(ii) Another possibility is to take A and B to be all the sets with non-empty interior (i.e., we can fit an c-ball inside). If the above property works,

Figure 5.8.6. Sketch for mathematical definitions of mixing. In ( u ) the blob A and the test region d o not intersect, in ( h ) part of A lies within the region B. There is no widely accepted way of defining mixedness from a practical viewpoint. One possibility might be to divide the region R into cubes with sided and to specify that within each cube the volume fraction of the blob A has to be within 4 - r: and 4 + I : (6 is the overall volume fraction of A in R). The dimensions of the cube specify the resolution required and c indicates the desired uniformity. If the materials are miscible, the time scale necessary to achieve uniformity at a length scale d is proportional to dZ. However, large regions of poorly mixed material might persist for long times and can be reduced only by further fluid mechanical action (see Figure E9.2.1).

Signatures of chaos

119

we say that the system is strongly topologically mixing. Measuretheoretic mixing implies topological mixing.22 Measure-theoretic mixing is the best achievable mixing. A physical feeling for measure-theoretic mixing can be obtained by means of a deceptively looking simple transformation which is called the 'baker's transformation', so named by the analogy of rolling and cutting dough. In the simplest case we cut a square and then we stack the pieces as indicated in Figure 5.8.7(a). A theorem asserts that a flow is measure-theoretic mixing if there exists a homeomorphism (i.e., a C0 diffeomorphism) of the region R to a square under which the flow becomes a baker's t r a n ~ f o r m a t i o n Since . ~ ~ measuretheoretic mixing is the best possible mixing then the baker's transformation is the best mixing device. However, generally we are dealing with continuous flows which preserve connectedness and do not allow for cutting and welding.24 The closest that we can come to a 'baker's transformation' is stretching and folding as shown in Figure 5.8.7(b) and hope that not much material is 'lost' in the transformation (i.e., it leaves the quadrilateral). Such a construction is probably the simplest example of the so called Smale Horseshoe Map (Smale, 1967). The forward mapping is denoted ,f(S); the inverse map is denoted f - (S); forward iterates are denoted as f "(S) and backward iterates as f -"(S). Both maps are defined geometrically in Figure 5.8.7 (the reader should check that the mapping of all points A , B, to A', B', etc. in the forward and backward transformations is indeed correct). Note that the horseshoe map can approximate a baker's transformation by restricting the amount of material that goes out of the q ~ a d r i l a t e r a l . ~ ~ In the case of the figure area elements in both horizontal striations (Ho

'

Figure 5.8.7. (a) Baker's transformation; ( h ) horseshoe map.

- D

stretch

A

cut and fuse

Figure 5.8 7 continurd

D'

A'

C'

B'

fold

n

stretch

0

v

t

c

h

)(

backward, f - ' (b)

and H , ) are stretched by the same amount in the vertical direction by a factor i., and contracted in the horizontal direction by a factor i.,. Thus, the Jacobian everywhere in H,, is

whereas the Jacobian in HI is

since it involves a 180" rotation. The values of i., and i., are selected in such a way that 0 < i., < and i., > 2 so that the vertical striations are disjoint. Thus, both f' restricted to f ( S ) n S and f ' - I restricted to f ' - ' ( S ) n S are linear; the non-linearity comes in due to the folding. I t is clear that n-applications of the forward mapping create 2" vertical striations occupying an area 2";"; (Figure 5.8.8 shows f 2 ( S ) ) . Since 0 < i., < the total area goes to zero as n tends to infinity. Something similar occurs for the backward striations (area 2"E.;). Note that if the map is area preserving i.,i., = 1 . Figure 5.8.8 shows the 'loss' of material that does not get mixed by the transformation (i.e., it goes somewhere else in the flow region where it may or may not get mixed by another horseshoe map) and, in fact we might think of the horseshoe map as a baker's transformation with loss (Rising and Ottino, 1985). The set remaining in the original square, denoted I,

Figure 5.8.8. ( u ) , f 2 ( S )and ( h )l n ( S ) n , / ' - " ( S ) n Sfor n

= 4.

is in fact completely cquivalcnt, in terms of its dynamics, to a baker's transformation (Figure 5.8.8 shows , f U ( S ) n , / ' - " ( S ) nfor S 11 = 4). In fact, the set I has several other properties: I is an invariant (Cantor) set consisting of an uncountable numbcr of points which contains an infinite but countable number of periodic hyperbolic points of arbitrarily long periods and an uncountable set of bounded non-periodic motions. Furthermore, if , f ( . )is restricted to I it is equivalent to a 'Bernoulli shift' (see Schuster, 1984, Chap. 2) and it is structurally stable. This last point is very significant and indicates that these properties are not exclusive to the special horseshoe defined geometrically in Figure 5.8.7(h).There are many other types of horseshoes, some of which with their rcspective inverses are shown in Figure 5.8.9 (see Smnle, 1967). In particular the striations need not be perfectly horizontal or vertical, the Jacobian need not be uniform, the striation thicknesses can be different, etc. Homoclinic/ heteroclinic points and horseshoes are intimately related. In fact the existence of one implies the other. There are several qualitative ways of seeing this. Figure 5.8.10 shows one possible way. Figure 5.8.10(tr) shows the forward and backward transformations due to the presence of a homoclinic point. Figure 5.8.10(6) is just a deformed version of 5.8.10(6)

Figure 5.8.9. Various types of horseshoes with corresponding inverses

Signuturrs of chnos

123

which clearly displays the intersections of forward and backward transformations characteristic of horseshoes (see Abraham and Shaw, 1985. Part 3. Chap. 5). Thus, we might visualize the horseshoe as a sort of 'black box' mixing device: material enters and comes out partially mixed. I t is then natural to think of improving the mixing characteristics of the system by feeding back its output; another possibility is to 'connect' several horseshoes (Rising, 1989). Figure 5.8.10. ( 0 )Forward and backward transformations due to homoclinic point associated with a fixed point P, ( h )a deformed version of ( u ) . forward

backward

(a)

124

Chuos in dynumical systems

Problem 5.8.1 Verify that the Jacobian corresponding to H, has the form indicated in the text. Problem 5.8.2 Verify the inverses of Figure 5.8.7(b), and Figure 5.8.9

5.9. Summary of definitions of chaos Throughout this work we will say that a system displays chaos when it satisfies one of the following conditions: (i) There is an invariunt set S (i.e., @,(S) = S ) and the ,Pow is sensitive to initiul conditions on S.26 (ii) The flow has homor-linic and/or heteroclinir- points. (iii) The ,flow produces horseshoe maps. We note that there are other possible definitions2' and that, in a strict sense, definitions ( I ) , (2), and (3), are not equivalent. (In 'general' (1) implies (2) and (3); (3) implies (1); and (2) and (3) are equivalent.) In principle, these three definitions are amenable to mathematical proof. However, in practice, given a flow, ( I ) is extremely hard to prove analytically (although Liapunov exponents are routinely reported via numerical computations); (2) can be proved by means of the Melnikov method,28 and (3) is accessible from both an analytical as well as experimental viewpoint (see Chapter 7).

5.10. Possibilities in higher dimensions Most of the concepts discussed in the previous section were explained in terms of time-periodic two-dimensional systems. The central ideas horseshoes, homoclinic/heteroclinic intersections, etc. - can be generalized to higher dimensions. For example, intersections can involve manifolds of different d i m e n ~ i o n s .Since ~ ~ the implications to fluid mixing are SO obvious we present a few visual examples here (the reader interested in a more complete account should consult Abraham and Shaw, 1985, Part 3). It is clear that all these possibilities can occur in a system of the form tlx/tlt = f(x) with V . f ( x )= 0. (5.10.1) Figure 5.10.1 shows I: transverse heteroclinic trajectory produced by the intersection between a two-dimensional unstable manifold and a twodirne~lson~~l stable m:unifold belonging to two different hyperbolic points.

Biblioy ruphy

125

Figure 5.10.2 shows a heteroclinic connection between the unstable manifold of a hyperbolic point of the saddle type and the two-dimensional stable manifold of a hyperbolic (saddle) cycle. Finally, note that Figure 5.8.3 can be interpreted as a cycle to cycle c o n n e c t i ~ n . ~ ~

Bibliography The origin of most of the material presented in this chapter is mathematical but much of the motivation can be found in physics and astronomy. In a very general sense there are two kinds of chaotic systems: dissipative .sy.stt.ms - with one-dimensional linear mappings being the most pervasive example - and Hu~niltoniunsyste~ns,which will be treated in Chapter 6. Historically, the earliest studies focused on Hamiltonian systems and were concerned with celestial mechanics; e.g., stability of the solar system. The earliest work can be traced back to Poincare (1892, 1893, 1899); the Figure 5.10.1. Connection between two hyperbolic points, P, and P,, in three-dimensions.

126

Chaos in dynumicul systems

concept of intersections of manifolds is already present in his work. Another work originating in the celestial mechanics literature and reaching into the mathematics literature is Moser's (1973) Stable and random motion in dynamicul systems; it contains a thorough and fairly readable discussion of horseshoe maps. Major portions of this chapter are contained in Smale's (1967) 'Differentiable dynamical systems'. The reader interested in the more mathematical aspects of this material is encouraged to consult this work. Much of the recent emphasis in volume contracting flows can be traced to a pioneering paper by Lorenz (1963), 'Deterministic nonperiodic flow' (here is where strange attractors find their home). Lorenz's work remained Figure 5.10.2. Connection between a hyperbolic point (P) and a hyperbolic cycle in three-dimensions.

Bihlioyruphy

127

buried in the meterological literature until it was rediscovered in the physics literature in the last fifteen years o r so. The applications of these ideas in fluid mechanics are many. For leads the reader should consult Landford ( 1982) and Guckenheimer ( 1 986). Fluid mechanical and other applications are reviewed by Swinney (1985) (the Appendix of this paper was published in 1983 in Phjlsicrr, 7D, 3-15). The literature on one-dimensional non-linear mappings and period doubling is abundant. O n e of the earliest general references is May (1976). One of the most influential works in one-dimensional mappings is Feigenbaum (1980). 'Universal behavior in non-linear systems', where references to many of the original works can be found. A simple book describing some of these matters is Schuster (1984), Dett~rministicchuos: r / r ~introduction (the discussion on Hamiltonian systems is very brief). A more mathematical treatment is given by Devaney (1986). The matters discussed in this chapter are closely connected with stability and bifurcations (see for example, 100s and Joseph, 1980), the mathematical foundations of mechanics (e.g., Arnold, 1980; highly recommended, Thirring, 1978), and the qualitative theory of differential equations (e.g., Arnold, 1985; Hirsch and Smale, 1974; Arnold, 1983). The reader unfamiliar with classical mechanics should consult the textbook lntrotluc.tiorz t o rlynr/mic.s by Percival and Richards (1982). This work focuses on the systems with one- and two-degrees of freedom and i t covers Hamiltonian systems and transformation theory. There are a large number of works dealing with the matters discussed in this chapter and i t is not possible to d o justice to the subject in such :I short space. Fortunately most of the above matters have been condensed in the book Nonlinrur o,scillutions, dynamicul systems, and h;furcution o f ~.ec,tor,firlds,by Guckenheimer and Holmes (1983). This book focuses on systems with a few degrees of freedom and has extensive discussions on the Smale horseshoe maps, homoclinic and heteroclinic bifurcations, and a large number of examples. The visually oriented reader should consult the series by Abraham and Shaw (1985). Another book, probably more accessible for the non-mathematical reader, is the work by Lichtenberg and Lieberman (1983). This book is particularly useful for Hamiltonian systems. The reader is warned that this chapter is a rather myopic view of dynamical systems; many important topics, concepts, and methods, are not covered. Missing are symbolic dynamics. normal fc)rnis. symplectic manifolds. and presentations of well studied flows such as the logistic equation, Henon's map. the standard map. and Lorenz's equations, etc.

Chuos in dynumicul systems Notes 1 Thus, the goal is not to he able to integrate the flow exactly but rather to find the general conditions for efficient stretching in tltly flow. 2 Recall that ifv E IW" the non-autonomous systems can beconverted into a n autonomous system by defining .x,, = r. See also Section 2.4. 3 Recall that, with the exception of vorticity, most of our results carry from IWZ to IW". 4 The relationship between the Reynolds theorcm and the Liouville theorcm is mentioned by Synge (1960. p. 174). The d a t e of Liouville theorem is 1838; the date of Reynolds ( o r transport) theorem is 1903. 5 For example if the flow is smooth, say !i-times continuously differentiable with bounded Jacohian, then the mapping is smooth and has a smooth inverse. 6 We restrict the discussion to autonomous systems. 7 Stability can be defined also in a global sense. but it is usually much harder to prove (see Hirsch a n d Smale, 1974. Section 9.3.). X Sometimes called cross-section. 9 In some cases, such a s time-periodic systems, we can select surfaces in such a way that the timc between successive intersections is the same 10 It sould be mentioned that there is a converse construction whlch is also of importance. allows the reconstruction of the flow starting with the m a p (Smale. 1967). A s~r\pc,t~.\iot~ 1 I A set S is called a n i ~ ~ r w i r rset t ~ rof the flow x = @,(XIo n a mnn~foldM ( S c M ) i f @ , ( X ) € S for all X ~ S , f o all r time t (for our purposes we can regard a manifold as a smooth surface). Thus, if a point belongs to a n invariant set, then when acted o n by the flow, it remains in the set. The definition for mappings is analogous. 12 F o r additional details see Section 2.4. 13 See Section 5.4.2 for the two-dimensional case. 14 According to the stable manifold theorem near the fixed point the stable and unstable manifolds a r e tangents to the linear eigenspaces E ' a n d Eu(for references see Smale. 1967, also Guckenheimer and Holmes. 1983, p. 18).This theorem was known to Poincare and Birkhoff. 15 F o r conditions o n g(xlsee Andronov, Vitt. and Khaiken (1966).T h e concept ofstructural stability is largely due t o Andronov. A similar question is addressed in Section 6.10. 16 That is, the surface has two 'sides'. 17 Note that in many cases this theorem is of rather limited use t o us since we are generally interested in area ( o r volume) preserving perturbations a n d not just any perturbation. 18 The implications of this invariance are extremely important when the flow is area preserving as in the case of Hamiltonian systems (see Chapter 6). 19 Note that sensitivity to initial conditions is compatible with continuity with respect to initial conditions 20 A similar definition. with timc t replaced by 11, applies to mappings (Lichtenberg and Lieherman, 1983, p. 267). 21 In general the numerical calculation oft111 Liapunov exponents is complicated. Straightforward application of the definition produces the value of the maximum Lii~punov exponent for almost all initial Mis (Lichtenberg and Lieberman. 1983, p. 280). See also Greene a n d Kim. 1987. 22 The term mixing has a precise meaning in ergodic theory. T h e term has been used in a fluid mechanical sense throughout this work (stretching plus difTusion). This is the only section in which it is used, unavoidably, in a mathematical sense. T h e terms 'recurrence'. 'wandering', 'mixing' (measure-theoretic). and 'ergodic', are obviously related but their relationship will not he discussed in detail here. Forexample,ergodicity implies recurrence, but the converse is not true. Both recurrence and ergodicity d o not imply mixing. +

,

Notrs

129

23 Tlic relationship bctwccn Iluid mixing ant1 a hakcr's transformaton was pointctl out in the 1050s hy Spencer and Wilcy (I951 ) hut its full mathematical implications were apparently not rcali/cd. 23 Static mixers provide. po\sihly. the closest experiniental and practical approximation tt) a hahcr's tranaformati~,nin the context of mixing (see Section 8.1 ). T h c c devices arc used in the polymer processing i n d u t r y tt1 mix viscous liquids ( w e Middleman. 1977). The most popular is the Kcnics" mixer which consists of a tuhc with internal helical s~~bdivisions. a twisted plane. of alternating right hand and left hand pitches. is callcd a n clement. Ideally. after each element the s t r e a m fed Fach s~~bdivisit)n into the mixer arc suhd~videdinto two a n d after 11 elements the striation thickness d c c r c a s c a s 3 ". Expcrime~itscarried out hy the Kcnics corp~)rationshow indeed a layered structure. I'hotographs a r e reproduced in Middlcnian's hook (op. (rr). Some of these matters arc discussed in the context of the 'partitioned-pipe m ~ x c r ' described in Chapter 8. 25 Many other types of horseshoes a r c possible; the o n e depicted in Figure 5.8.7(h) i.; the simplcst and most popular one. 1h In dissipative systems a system is callcd chat)tic if it possesses a .slrr1116/crrlrrrlc./or ( a n attractor is called strange if it is a n attractor. and it is sensitive to initial conditions. i.e.. ~t pt)sscsscs at Icast one positive Liapunov exponent). Attractors arc impossible in Hamiltonian systems. ]..or dcfin~tionsscc Ci~lckcnhcirncrand Holmes. 1983. For applications of strange attractors concepts in fluid mechanics. scc Landford ( 1083) and Gl~ckcnheimcr( 1986). 17 Otlicr possihlc definitions of (temporal ) c h a o s arc to ohscrvea signal .x(r)asa function of time a n d t o compute the power spectrum of.\-(r1. An indication ol'chaos IS a continuous .;pcctrum. Another possibility is t o compute the correlation function. r.(r). of thc signal \ - ( I ). i.c..

-

\rlicrc \ . ' ( I ) is the difference hctucen . \ ( I ) a n d the time average value of .x(r ). If c ( r )+ 0 as 7 / the system IS considered chaotic. S~)mctimcs. the visual appearance of n~~rncrically computed Poincarb scctit)n is taken as cvidcncc of c h a m . If possible. \\c prefer t o designate as chaotic any system uhicli satisfies any of thc conditions ( 1 ) ( 3 ) abobc. sincc they are. in principle mathematically based and anienahlc t o ~I-ooI'.If tlic\e cannot he proved we then might consider any of these alternative dclinitions. In any case we will clcnrly specify which criterion is being used. 1 S S ~ n c c1111stechnique \*ill be uscd mostly in the context of Hamiltonian systems tlic di\cush~oni\ I-escr\cd until Section 0.10. 7') S ~ ~ i i e t i ~ ithe i c s\vord 'ilidc\' i \ ~ l s e din this co~itcxt.The index of It'" is the numbelof tlimcnsions of It"'. .30 The situ;~tionsslioun Ilere ;il-e commtjn sincc hyperbolic I ~ n c a rvolume prcsel-bing IIo\*.; a r c dense ant1 open a m o n g all i o l u m c prcscl-\ing flous (Smale. 1007). H ~ v . c \ c r . it is a n unsolved prohlcm whether o r not this is true when the right hand side of the c \ o l u t ~ o ncclu;~tion(I\ (11 \I\. I ) is c o n t r a i n e d to satisl'x the Navicr S t o h c ecluationa 01. tlic 1:uIcr ec1ui11ionh. -

Chaos in Hamiltonian systems

The equations describing the trajectory of a fluid particle in a twodimensional isochoric flow are a Hamiltonian system, a special case of a volume preserving dynamical system. The objective of this chapter is to discuss the general structure of chaotic Hamiltonian systems, with an emphasis on periodic systems with one-degree of freedom and timeperiodic Hamiltonians. Central to the understanding of these systems is the study of the flow near hyperbolic and elliptic fixed and periodic points. Hamiltonian systems conserve volume in phase space and in PoincarC sections. As we shall see, this restriction has important implications in unravelling the general structure of these systems, especially near elliptic points, and is the most important point of departure from the previous chapter.

6.1. Introduction It seems at first contradictory to focus on Hamiltonian systems in the context of mixing of viscous fluids. However, the connection is purely kinematic and involves no approximations. As seen in Chapter 3, the equations describing the trajectory of a fluid particle in an isochoric two-dimensional velocity field are dr2/dt = - i?$/?.u,, dsl/dr = (7$/i?u2, where $ is the streamfunction. Such a system of equations, regardless of the form of $ is a Hamiltonian system (Aref, 1984) and therefore it is profitable to study fluid mixing from such a viewpoint to exploit the substantial body of theory focusing on these systems. The system is said to have one degree of freedom if the flow is steady, $ = $(.u,, x,), and two if the flow is unsteady, $ = $ ( r ls,, , t). If $ is time-periodic we say that the system has 'one and a half' degrees of freedom. As a confirmation of thc ideas of Section 4.7 we will prove that two-dimensional steady flows arc poor mixing flows and that time-periodic flows are, most likely, effective mixing flows at least in some region of space.

Hamilton's ryuutior~s

131

In the following sections we discuss Hamiltonian systems from a general iicwpoint. Even though most of o u r applications will be t o systems with one and a half degrees of freedom, some of the results will be given for \\stems with il'-degrees of freedom. T h e objective is t o show the underlying katurcs shared by all Hamiltonian systems. We shall see that Hamiltonian \ystems with one-degree of freedom a r e intc~qrahlrand hence they cannot hc c~hirotic~. O n the other hand. Hamiltonian systems with one and a half degrees o r two-degrees of freedom stand a very good chance of being non-integrable and chaotic (non-integrability is a necessary but not \~~fficient condition for chaos). The implications of the results presented in this chapter in the context 01' mixing a r c mainly two: ( i ) all steady bounded two-dimensional flows I>a\,ezero asymptotic mixing efficiencies; ( i i ) many time-periodic twodimensional flows have positive mixing efficiencies. a t least in some part the flow. Some of the theorems and methods sketched in this chapter describe \\hat happens under .sn~irllperturbations of the integrable case. It should he noted that we are primarily interested in the behavior for lurqc) 12crturbations, since it is where the best mixing occurs; however, much less is known for this case from a mathematical viewpoint and standard analytical techniques d o not work (see Section 6.9). Nevertheless, the concepts presented h e r e should constitute a reasonable basis o n which to launch such a n analysis. In Chapters 7 a n d 8 we discuss specific examples \\ hich serve t o clarify some of the points discussed here. It should be clear that a s long a s the system is Hamiltonian, the general features will appear c \ c n in the simplest representatives of the class.

6.2. Hamilton's equations 31s sccn in Chapter 5 a Hamiltonian system has the structure given by IJIC

2.Y first order differential equations

(6.2.1a.b) ilil, ilt = ? H Cp,, dp,jt/t = - i H ill,, \vllcre the 11,s a r c the componcnts of a vector q = (11,. . . . 'I,,.),the 'positiotl', and the p,s, the components of, p = ( p , . . . .. p , ) , the '~,~on~ct~tlrr,~', and H i h thc Hami1toni;in. which is a scalar function of p and q a n d , in some casts. a s wc sh:ill consider here. a n explicit function of time. In mechuniciil systems q can actually represent the position and p the Illorncntum understood in the usual way.' 111 other npplicntions p and q tahc different physical significance.' I f p a n d q have .Y componcnts each. the Hamiltonian system is said to

.

have N-dcgrces of freedom. I f H is a n cj\-plicil function of time the system has a n additional degree of freedom (i.c., N + I ). In the case of timeperiodic Hamiltonians, time is regarded a s a n i ~ d d i t i o n ~ ~degree l of freedom.

:

6.3. Integrability of Hamiltonian systems Central to the discussion of chaos in Hamiltonian systems is the concept of integrability. By 'integrability' we d o not mean the computation of the solution in terms of known functions but rather the ability of finding sufficient n u m b e r of constants of the motion s o as t o bc able t o predict qualitatively the motion in phase space. As we shall see, if a system is I ' I I ~ P < / L I I . L ~ ~ / Pit cannot be chaotic. Thcrc a r c t w o cquivalcnt ways of defining integrability; the first o n e rcqi~ircsthe use of Poisson brackets; the second action-angle variables. T h e Poisson bracket between the two functions 11 = lr(p, q ) a n d 1% = r(p, q ) is defined a s (Arnold, 1 9 8 0 ) :

With this definition Hamilton's equations c a n be written as: dqiidl

=

[q,, I11

(6.3.2a) (6.3.2b)

d p , / ~ l l= [ p i , t 1j. A Hamiltonian t l , with N degrees of freedom, is called it~tc
tlamiltoninn systems with two-degrees of freedom arc rlon-intc$qrrrhlr,i.e., i t is in general not possible to find F, and F2 such that dF,/dt = O and rlk'2/dt = 0 along a trajectory. Another way of defining integrability is by means of rrc.tior1-rrr~glc r.trritrhlcs, which is a special case of a canonical transformation. In general transformation of variables p, q 8, 1 0 = 0(0,, . . . , O N ) , I = I ( / , , . . . , with Ii = I;(p, q, t ), Oi= Oi(p, q, t ), is called canonical if the structure of the system in the I, 8 variables has the I'orm +

dl,/rlt

= ?H/i0,,

dO,/rlt

=

-

c?H/il,,

(6.3.3a,b)

where H = H ( I , 8 ) . That is, a transformation is called canonical if the Ilamiltonian structure is preserved (the appearance of (6.3.3a,b) is identical lo (6.2.1a,b)).4 A very useful subset of canonical transformations is the rrc.tion-rrn~qlc~ [ransformation p, q + 8, 1 the Hamiltonian in the 8, 1 co-ordinates is not a function of 8. (This is indeed one of the definitions of c.yc.lic. \'ariable. Conversely, it can be stated that the action-angle transformation is possible only in cyclic systems.) Under these conditions the system hceomes

dI,/dt = (?H(l)/?Oi)= 0,

rlOi/dt = - (?H(I)~(;,)= (,)(I)

which can be readily integrated to

+

I ( t ) = I(O), O i ( t ) = ( ~ ) ~ ( l )Oi(0). t (6.3.4a,b) Thus, the action-angle transformation allows us to trivially integrate the Hamiltonian system (hence the justification of the name integrable). ('onversely, if the action-angle transformation is possible the system is called integrable.5 The mapping (6.3.4a,b) is called an integrable t\z.i.st rtllrppir~~q, the rrc.tion r; remains constant while the trn~qleOi rotates with \peed ~ ~ ( 1BJ,) tlc~finition, . rrll ir~tcyrrrhleHtrr~liltor1irrr1.swith hoirrltlrtl orhits" c.trr1 he rc.rluc,c~dto irltcvqrtrhlc tn'ist rllrrppir1!q.s.

6.4. General structure of integrable systems ('onsider the simplest case of a Hamiltonian system. namely dy/rlt = i H / i p . tlp/tlt = - iHii'cl. (6.4.1a.b) with H = H ( p , 11). i t . , a system with one-degree offreedom. The linearization of the system around a fixed point, tlq/tlt = tlp/tlt = 0 , shows that the systcrn

134

Chuos in Hamiltoniun svstrm.~

admits only three kinds of fixed points: saddle or hyperbolic points, centers or elliptic points, and the degenerate case of parabolic points (see Example 2.5.2, two-dimensional area preserving case). In the most general case the unstable and stable manifolds of hyperbolic points join smoothly by a curve of constant H. The essence of any integrable system is embodied in Figure 6.4.1. To see why this is so, note that all integrable systems with N-degrees of freedom can be transformed into N-uncoupled one-degree of freedom systems, i.e., the system

yields N uncoupled linear equations for Oi as a function o f t . Since this is generic to all integrable systems, all integrable systems can be non-linearly transformed into each other, and are in this sense equivalent. Since the prototype of a one-degree of freedom is a simple pendulum, the essence of an integrable system with N-degrees of freedom is embodied in a system of N non-interacting pendula. Thus, we expect the phase plane of any integrable system to be some non-linearly deformed version of the phase portrait of Figure 6.4.1 or any of the streamfunction portraits of Figure 4.7.1 .'

Figure 6.4.1. General picture of an integrable Hamiltonian system. The prototype is a pendulum without friction; in this case q represents the angle from the vertical ( - r r , rr) and p is proportional to the angular momentum. The level curves correspond to H(p, q ) = constant.

Ap

hyperbolic point

separatrix

I

elliptic point

Phtrsr sptrce

Hrrmiltor~itrr~ .s~~.sien~.s

of'

135

6.5. Phase space of Hamiltonian systems ..I\ in Chnptcr 5 the state of a Hamiltoninn system can be represented in the 2 N - d ~ m e n s i o n n lphase space by n vector x.

x 11

= (1'1..

. ., p,.i 41. . . . , q y )

ith time i being a parameter. T h e initial condition is represented by X.

X

= ip:),

..

.. p,.0

q 0l . . . ., 4 ; )

such that the solution t o the system ( 6 . 1 . l a . b ) c a n be represented by the 110 I\ with X = @ , = , ( X ) . x = @,(X) Phc velocity of a point x In phase space is given by v = tlx 'tli = (tlp,, rli, . . . , rip, di, rlrl 'tlt. . . . , dy ,, d i ) . If I 1

=

H ( p . q, t ) a n d V H is defined a s V H = (dHiap,, . . ., d H ~ a p ,dHIdq,,. ~, . ., dHidqv),

then by the chain rule. tli

irl, tli rlH tlt

i p , dt

+i H

=V-VH

it

-

hich is interpreted as the rate of change of H (energy in the usual case) I'ollowing the flow. F o r Hamiltonian systems v.VH 0, which implies (hat v belongs t o surfaces in phase space of constant H . Furthermore, if [I is not a n r.\-plic,it function of time H remains constant for a given inilia1 condition X. Another way of seeing this. a n d a t the same time adding some physical ~ n e a n i n gt o the Poisson brackets, is to note that (6.5.1) c a n be written a s ir

I'hus. the result [H. H I = O corresponds to the statement ( V H ) - v = 0. T h e ~ ~ a m i l t o n i aphase n space has also several other properties: If V ( 0 )denotes tlie volume of a set of initial conditions. according t o Liouville's theorem (('1i:ipter 5 ) . C'(i) evolves according t o

1'01. Hamiltsniiili systems V - v = 0. by substitution. kind I/(()= V ( 0 ) .Hencc lllc system conserves volume in phase space.

136

Chaos in Hamiltonian systems

The Liouville theorem has many implications in Hamiltonian mechanics, here we will mention only two: (i) Poincart's recurrence theorem (see Arnold, 1980, p. 71): Consider a region {Dl. in phase space such that @,(Dl.= {D) for all t. Then any trajectory in D returns infinitely close, infinitely often, to its initial location (i.e.,all the points are non-wandering).The reason for this is easy to see. Consider an 8-ball around X, denoted B,(X). Since the initial conditions are trapped in {D),conservation of volume requires that after a finite time t, @,(B,(X))nB,:(X)# 4. (ii) A Hamiltonian system cannot have asymptotic equilibrium positions and asymptotically stable limit cycles in phase space (for asymptotically stable equilibrium and limit cycles see Hirsch and Smale, 1974).

6.6. Phase space in periodic Hamiltonian flows: Poincark sections and tori Consider a system with a time-periodic Hamiltonian with period T. In this case it is traditional to represent the flow of the system on a torus with the time being a cyclic coordinate around the torus (Figure 6.6.1(a)). A surface transversal to the flow gives a good idea about the behavior of the trajectories for long times. For example, consider a surface of section (or Poincare section) C such that

,

Such a section defines a return map in the plane C, x,, = G(x,). Note that all trajectories originating from C return to C after the period T. Note also that area is conserved in the Poincare section. In the case of an integrable system, the cross-section of the torus can be regarded as an integrable twist mapping (Equation (6.3.4a,b)).An initial trajectory starting on C wraps around the torus as indicated in Figure 6.6.1(b). The trajectory may or may not join with itself depending on the value of toi(r). If the trajectory comes back to its initial position p in the Poincare section after going around the torus nz-times, then p is a periodic point of order m. As seen in Chapter 5 the stability of fixed and periodic points corresponds to the solution of the eigenvalue problem

where DG(p) is the Jacobian matrix of the mapping evaluated at the point

Phase space in periodic Hamiltonian ,flows

137

in question. In a Hamiltonian system with N degrees of freedom DG(p) is a 2N x 2N matrix. In general, in Hamiltonian systems the eigenvalues appear as four-tuples jv,2,

l/j., 1 /Z

where the overbars represent the complex conjugate (see Lichtenberg and Lieberman, 1983, p. 181 ). For N = 1 ( x E RZ)the only possibilities for fixed points of the map G(x,) = x,+ are (Moser, 1973, p. 54, see Section 5.4.2):

,

Hyperbolic Elliptic Parabolic

..

~ j ~ , ~ > l > ~ j~. .2, ~/ ., ~ = l = I (i = I , 2 ) but ii # 1 j.i= f I (i = I , 2 ) .

1.1

Figure 6.6.1. (a) Flow on a torus and correspondingmap on Poincare section; (h) rational trajectory.

138

C ~ U O Sln

Humiltonlun systems

Problem 6.6.1 Prove that area is conserved in the Poincart: section of a Hamiltonian system.

6.7. Liapunov exponents As was seen in Chapter 5, chaotic systems are characterized by a rapid divergence of initial conditions and usually, the divergence of initial conditions is quantified by means of Liapunov exponents. We know that a 2N-dimensional flow has at the most 2N, in general different, Liapunov exponents (i.e., there are 2N vectors Mi, linearly independent, each of which gives, possibly, a different value). In Hamiltonian systems, due to volume conservation, 2N

1 o,(X, M i )= 0. 1

As seen in Chapter 5 , for dissipative systems this sum is negative. Integrable systems have all ois equal to zero. T o see why this is so recall that all integrable systems can be reduced to an integrable twist mapping. From our discussion in Chapter 4 it is easy to see that in a twist mapping a filament of initial conditions ldxl grows linearly with time and the Liapunov exponents are zero (a twist mapping is essentially a Couette flow which is also a SCF). Furthermore, since chaotic systems have to have at least one positive Liapunov exponent, integrable systerns cannot he c,haotic.. An important consequence is that all steady bounded twodimensional flows are poor mixing flows. Example 6.7.1 Consider a flow given by the streamfunction

I)(.Y~,.u2, t ) = U.x2 + [AUh/2] In(cosh(u,/h) + A cos[(.ul - Ut)jh]) representing a train of vortices moving in the direction u , at speed U. The non-dimensional parameter, A , represents the concentration of vorticity. The value A = 0 corresponds to parallel flow with the classical hyperbolic tangent velocity profile. The meaning of the other terms is the following: AU is the velocity difference across the layer and h is proportional to the vortex spacing (for details see Stuart, 1967). This expression was used by Roberts (1985) to model the stretching of material lines and the roll-up of streaklincs in shear layers. The evolution of streaklines and material lines corresponds to the solution of

In a moving frame k"(.\-',. .\->) moving at speed U in the direction the streamfunction becomes

.\-, > O

+

I,!/'(.\-',. s > )= (AUlli2) In[cosh(s>//l) A COS(.\-;//I)] and the flow is alrtorlor,~olrsin this frame, and therefore integrable. with a .;[reamline portrait similar to Figure 6.4.1 (this flow is usually referred to ;IS 'Kelvin cat eyes', see Lamb, 1932, p. 225). If

+

I ~ ' ( . Y ' ,, .\->)/(AU/1,'2) < ln(1 A ) rhc orbits are closed and periodic. Since the flow is integrable we can concludc that a material line marked in the flow will stretch linearly. Also, c\ery segment of a streakline fed into the flow will stretch linearly. Howevcr, as we have seen in similar examples (Example 2.5.1) the streaklines can be quite complicated. This flow is reconsidered in Chapter 8. Section 8.3.

E-rumple 6.7.2

In the simplest case, the basic equations for the motion of point vortices in the plane form a Hamiltonian dynamical system (Batchelor, 1967, 1'. 530). Thus if we consider point vortices of strengths r i , . K ? , . . .. K,,. with y1), (.\-?, y 2 ) , . . . (.Y,,y,,) the instantaneous value of the positions (s,, \trcamfunction is

.

Since there is no self induced motion, the movement of the vortex of strength r i i is equal to the velocity of the fluid at the point (u,, due to all the other vortices. The result is jsi)

rl.\-j

c'w

tlt

?r; '

=

I<,

aw

(lyi Ki

tlt

=--

ax,

'fhus, the system is Hamiltonian with a number of degrees of freedom equal to the number of vortices. Note, however, that in this case the value of It' (Hamiltonian)associated with a fluid particle is, in general, a function 01' time since it depends on the instantaneous positions of all the vortices (contrast with previous example) and i t is not obviously clear that invariants should exist. Clearly, nothing interesting happens in the case

1 40

Chaos in Humiltoniun systems

3fjust one vortex (one-degree of freedom) and that the case of two vortices In the plane can be integrated has been known for a long time. In this case, for example, the distance between the two vortices remains constant, and details can be found in Batchelor (1967, p. 530), and various other places. However, it was only recently that the case of three vortices in the plane was shown to be integrable, and that a system of jbur, o r more than four, vortices in an unbounded two-dimensional region is in general chaotic (Aref and Pomphrey, 1982; for a review see Aref, 1983).

6.8. Homoclinic and heteroclinic points in Hamiltonian systems As seen in Chapter 5 a point y is called hrreroclinic if i t belongs simultaneously to both the stable and unstable manifolds of two different fixed or periodic points. In Hamiltonian systems we have volume conservation and this fact has important consequences in the behavior of homoclinic and heteroclinic points (see Figure 6.8.1 ). Since the manifolds are invariant sets (see Section 5.6) mappings of the heteroclinic point y belong to the intersection of manifolds. The only difference now is that since area is conserved, the shadowed regions of Figure 6.8.1 transform as indicated (Area A = Area A ' ) ; a point in WYQ) moves asymptotically slowly as it approaches the stable fixed point Q 8 ; the trajectory must Figure 6.8.1. System of Figure 6.4.1 after a perturbation showing a transverse heteroclinic point; the region A is mapped to A ' , the point y is mapped to y'. The area of A is equal to the area of A ' . Some curves survive the perturbation ( K A M curves, see Section 6.10.2). KAM curvc

Perturh~ltion.~ of Humiltoniun systrm.s: Mrlnikoc's method

141

wander greater and greater distances normal to the stable manifold of Q. Similar behavior arises from the stable manifold of Q. Similar behavior arises from the stable manifold of P in the neighborhood of Q. Multiple ~ntcrsectionsoccur and other hctcroclinic points appear.' As seen in ('hapter 5 the prc~.sc~nc8c~ of' trLlt1.si.c.rse hotnoc+lit~ic. undlor heteroc,linic points is or1cJ of' the) ~rc,c+c~ptc)d dqfit1ition.sof' c+h~~o.s. Furthermore, transverse homoclinic points imply horseshoes (Smale Birkhoff theorem, Guckenheimer and Holmes, 1983, p. 252).

6.9. Perturbations of Hamiltonian systems: Melnikov's method The method of Melnikov (1963) provides one of the few analytical ways 10determine the existence of intersections of stable and unstable manifolds. I n this section we present the method in its simplest possible form (for a more complete treatment the reader should consult Guckenheimer and tlolmes, 1983, Section 4.5; generalization to systems with 11-degree of l'recdom are possible). Consider a system of the form dxlrlt = f ( x ) with u

=(

s ~.yZ), . f = ( f'l.

f2)

+ r:g(x, t),

i:

small

such t h i ~ tf is Hamiltonian

= ?H/?.Y?.

f'2 =

-?H/?.Y,,

ant1 g = ( g I ,g 2 ) time-periodic but not necessarily Hamiltonian. Both f

-

a n d g are smooth. In this work we will identify f with a n Eulerian velocity ficld and H with the streamfunction. Consider that the unperturbed system I;: 0 ) has a hyperbolic saddle point H with a homoclinic orbit, q"'(u1, .uz) = const. o r q"'(t), and with stable, W ' , and unstable, Wu, manifolds surrounding a n elliptic point E as shown in Figure 6.9.1(~1). Assume also that the interior of the homoclinic orbit q"' is filled with elliptic orbits q"', where r is a parameter ranging from 0 to 1. Denote by T(q'"') the period corresponding to the orbit and assume also that T+ x

:IS

ql"' + qlO).

Consider now the behavior of the perturbed system using a Poincari: section with the period of g (see Figure 6.9. I ( h ) ) . It can be shown that the perturbed system has a unique hyperbolic saddle point with stable and unstitblc manifolds close to those of the unperturbed system. I f r,, denotes n p:iriimcter which measures length along the unperturbed orbit,

.2

Chuos in Humiltoniun systems

,e distance in the Poincare section between the stable. W:. and the Istable manifold, W,",01' the pertlrrhed system is given by

here M ( t , ) is the so-called Melnikov's integral and is given by

F ~ g u r e6.9.1. ( t i ) H o m o c l ~ n ~orb11 c before p e r t u r b a t ~ o n(h)after : p e rttu r b a t ~ o n . qcO),unperturbed manifolds

d(t,), distance between

perturbed manifolds

Behu~liornerlr elliptic points

where

'A '

is the wedge product, defined by

f A g=1'1<12-.1'2.1/1. -Phcrefore the manifolds cross when M(t,,)= 0. I f the perturbation g is Hamiltonian,

y, = iC/c'r,, gz = -?G/?X,, (he Mclnikov integral can be written as M(to) = n hcrc [

1 denotes

f-+,'

I H ( ~ ( O-) (,,)I, ( ~ ~ ( q ( ~-to), ) ( t t ) i dt

the Poisson bracket (Section 6.3). The applications of this method are numerous. An example is discussed In Section 7.2; problems are suggested in Sections 7.3, 8.3, and 8.4 (for rcccnt developments see Wiggins, 1985a,b). Thc relation between the information provided by the Melnikov function and Poincare sections is not trivial. A measure of the 'extent of chaos' is provided by the area A (Figure 6.8.1) which is the integral o f the distance between manifolds. The dynamics of A governs the transport in the system. Such an approach was adopted by Leonard, RomKedar, and Wiggins (1987) and extended by Rom-Kedar, Leonard, and Wiggins (1990).

6.10. Behavior near elliptic points ('onsider an integrable twist mapping in the neighborhood of an elliptic point in the Poincare section ofa time-periodic Hamiltonian system, i t . ,

(Defining the \t,irlrliilg r~~rrnhcr. or rotrrtiorl nurnhcr. a as c o i = 2x0, see 6.3.4a.b). According to the value of r., a(r.,,)can be a rational or irrational number. If a(,;,) is irrational the trajectories wrap densely (in the mathematical sense) around n torus, never intersecting themselves (see Figure 6.10.1 ). On the other hand, if ~ ( t ; , )is rational ( =rrl:r7) the tr:~jectory returns exactly to its initial position after III turns around the torus. In the Poincari: section (Figure 6.6.l(h))the rational orbits (periodic) produce a finite number of intersections with u circle (according to the \:tlue of 1 7 ) . The intersections of the irrational orbits, also culled quasiperiodic, f i l l the circle densely.

Chuos in Humiltoniun systrrns

144

The central question is what happens to the twist map under a perturbation of strength ,D such that (6.10.2a) rn+I = rn + /d(rn3(',,) (6.10.2b) O n + , = On 2na(rn)+ ,DH(Y,, 0,). The perturbations ,f' and g are such that they are 271-periodic, area preserving, and vanish faster than r as r tends to the origin. Over the years several theorems have been discovered that produce a fairly complete picture of the events that occur after the perturbation. Three theorems will be described here and in chronological order: the Poincare Birkhoff theorem, the Kolmogorov-Arnold-Moser theorem, and Moser's twist theorem. The discussion is mostly qualitative.

+

6.10.1. Poincave-Bivkhoff theorem

This theorem describes the fate of the rutionul tori upon perturbations of the twist mapping. A necessary condition is that a' # 0 (the prime denotes the derivative with respect to r). Consider the integrable twist mapping of Equations (6.10.2a,b). Select a radius r, such that o(r,) = m/n and an initial angle 0,. After n mappings we get and 0,, = Oo + n2na(ro)= 0, + n2n(m/n) 0, (up to 271) ro = r,,. On the other hand, in the non-integrable case we get, in general, and 0,, = 0, n[2na(ro)+ ph(r,, Oo)l # Oo, rn # ro

-

+

Figure 6.lO.l. Integrable twist mapping.

I

rational (periodic)

Behurior neur elliptic points

145

,+llcrc 1.1 is a perturbation and h(r, 0,) is some complicated but calculable fullction that can be obtained as a function of the perturbations ,/'and q in the non-integrable mapping (Equation (6.10.2a,b)).In general the point ( I , , , , ; , will return neither to its initial radial position nor to its initial illlgular position. However, if both the functions h(r. 0,) and a ( r ) have a Tilylor series expansion in r (condition a ' # 0) we can find a point 0,. I)(, !]car the initial O,, r,, such that 2na(p,) + ,ull(p,, 0,) = 2n1~7~n. Thus. for this value of / I ,we d o get 0 , = Or,, bur p,, # I),,. Since the same i~rgumentholds for any 0,) we obtain a curve p,,(O) of points that are mapped rudiull!'. Since this happens after exactly n mappings and the %inding number is monotonic in the neighborhood of r,,, we can envision rhar the circle corresponding to r, does not rotate and that the neighboring orbits (r > r,, and r < %) rotate in opposite directions, as shown in Figure 6.10.2 (see Helleman, 1980, p. 185). Also. since the map preserves area, not all the points belonging to the carve p,(O) can move inwdrd or ouruard; some have to move inward and some have to move outward. Also since the mapping is continuous some points d o not move at all (i.e.. tlicy cross the circle r, or in other words, they are fixed points) and due to periodicity and conservation of area the number of such points has to bc even. say 2k. Examination of the flow in the neighborhood of the fixed points shows that k points are hyperbolic and k are elliptic (see Figure 0.10.2). Thus the Poincare-Birkhoff theorem (Birkhoff, 1935) tells us under iin 1.1-perturbationthe rational orbits break into a collection of k-hyperbolic Figure 6.10.2. Sketch used in thedescrlption of the Polncart-B~rkhofftheorcm. hyperbolic

irrational

146

Chuos in Humiltonbt~sv.stetn.s

and X-elliptic points. Such a prediction is confirmed by numerous computational studies. Note that this theorem does not say anything about the irrational orbits. 6.10.2. 7he Kolmogorov-Arnold-Moser theorem (the K A M theorem)

The local picture of what happens to the ii.rlrtiorltr1 orbits in the neighborhood of an elliptic point is the context of the celebrated KAM theorem. I t says basically that most of the invariant irrational tori of the integrable system are conserved in the perturbed system. This theorem was outlined by Kolmogorov (1954a,b) and proved by Arnold in a series of papers starting in 1961. A detailed exposition is given in a very long paper written in 1963 (Arnold, 1963). This paper gives probably the best account since it gives background, examples, a non-rigorous heuristic derivation, and concludes with the rigorous proofs. The results were generalized by Moser in 1962 and subsequently by Riissman in 1970. As indicated by Arnold, the proof is rather cumbersome and it is based on a very large number of inequalities. The technical aspects of the proof go back to methods developed by astronomers in the nineteenth century and, as indicated by Kolmogorov, on earlier work by Birkhoff (1927) and a method of successive approximations developed by Kolmogorov himself. Keeping in line with the previous sections we will not attempt to reproduce the theorem in detail here but rather give an outline of some of the central ideas involved in the arguments. Consider for simplicity an integrable Hamiltonian system with twodegrees of freedom i.e. the phase space is four dimensional. The Hamiltoninn is of the form H(p, q ) = HO(p) with p = ( p , , p,) and q = (11,. ~ 1[the ~ results ) can be generalized to a system with N-degrees of freedom]. In this case the equations of motion take the form (with k = 1 , 2), rlp,/t/t = 0 l/qh;l/t= dH1(p)/?ph, i.e., p = constant. and (with k = 1.21, rlh= (dH1(p)/ipk)t 11:) i.e.. the four-dimensional phase space contains an invariant two-dimensional torus which is characterized by two frequencies w , = dHU(p)l@l and q = 8H'(p)ldpzwhich depend on p. As we have seen, if t he frequencies are incommensurable, the trii-jectories never return to the initial position on the torus (the motion is ciilled quasi-periodic or conditionally periodic). Lct us suppose. for the monicnt. that indeed the freqi~encies(11, and ( 0 2

+

Behavior near elliptic points

147

are incommensurable, i.e., it is not possible to find integers k , , k2 such that k , o , + k 2 0 2= k.w = 0 , k = (k,, k,) and w = ( o , , o,). In order to avoid the possibility k . 0 = 0 it is sufficient to require that no relationship of the form f(w,(p), c ! ) 2 ( ~ ) ) = 0 exists for arbitrary p. Taking the differential of the function /((,),(p),coz(p))it is easy to see that a sufficient condition for incommensurability is to require det(dwi/?pj) = det(t?2H0(p)/dpipj)# 0. Also to avoid problems at the origin we require that 8oi/2pj # 0 at pj = 0. Consider now a perturbation of the form (near integrable system), H(P, q, p ) = HO(p)+ p H 1(P, q ) where H o and HI are analytical (Moser's contribution was to relax this assumption by requiring a finite number of derivatives). If the above conditions are met the phase flow in the four-dimensional space is stable with respect to small changes in H . It can be proved that given any p > 0 there exists 6 > 0 such that if the strength of the perturbation is such that p < 6, the phase space of the perturbed system consists entirely of invariant two-dimensional tori, except for a set of meusure less thun p. Arnold's proof is constructive and requires an infinite number of steps. One of the major dificulties is how to quantify the influence of rationally related frequencies since terms of the form k - w = 0 appear as denominators in series solutions of the equations of motion of the perturbed Hamiltonian system (this is the problem of the 'vanishing denominators'). However, the central question is how important they are to the overall picture. The answer turns out to be that they are not very important and that the previous claim holds; the original tori of the integrable system are not destroyed but merely displaced. The conditions for the abundance of terms kew = 0 can be quantified by means of theorems of Diophantine approximations. It turns out that for almost u11I0 randomly selected frequencies w = (to,, . . ., to,), with oi real, the components are incommensurable for all ks, k = (k,, . . . , k,), with ki integer. Almost every (0 satisfies the inequality /(k.w)l >, K / k / - " (6.10.3) where Ikl = ikll . . . Ik,l, I) = n + I , for all k, where K is a function of (0(the quantity KIA\-"can be interpreted as a measure of the 'irrationality' o f t h e frequencies w). Using these arguments it is possible to estimate the mcnsure of the denominators that vanish. Thc conclusion is that when p is sufficiently small, all tori corresponding t o frequencies satisfying the inequality (6.10.3) d o survive." The great

+

+

8

Chuos in Humiltoniun systems

ijority of the initial conditions in the four-dimensional space are in fact nditionally periodic. Note that in systems with two degrees of freedom e presence of closed invariant curves precludes ergodicity.I2 The general picture obtained by means of uncountable computer nulations (see Chapters 7 and 8) is that in fact tori survive for large :rturbations although they gradually disappear (and eventually reappear) itil no closed curves survive near the original ones.

6.10.3. The twist theovem he KAM theorem tells us that there are many invariant tori (indeed a t of positive Lebesyue measure) that do not disappear (if they satisfy a ven set of conditions they do not disappear). The Poincare-Birkhoff eorems tells us what happens to the rational orbits that disappear. The iist theorem (Moser, 1973) is, in a loose sense, the reverse of the KAM leorem. onsider that a(r) is Cs (s > 5),13 la'(r)l > 0, on an annulus of radius r, < r < b, and that the perturbation is C" close (in the sense of a C h o r m . he C b o r m is defined as the maximum of the absolute values of the kth irtial derivatives, 0 < k < s). Then: :i) a torus survives in u < r < b ii) the perturbation is a twist mapping on the perturbed torus ii) the radius of the torus satisfies (6.10.4) (a(r)- (mln))> Cn-2.5 )r all integers m, n (C is a constant). lote that the inequality (6.10.4) is a consequence in Moser's twist theorem nd an assumption in the KAM theorem. In the KAM theorem we select specific perturbed orbit that satisfies some conditions (e.g. the inequality 5.10.3)) and conclude that it survives. In the twist theorem we focus on n annulus where the perturbation is small and conclude that there is )me torus which satisfies inequality (6.10.4).

6.11. General qualitative picture of near integrable chaotic Hamiltonian systems Ye have discussed what happens to the general picture of an integrable lamiltonian system (Figure 6.4.1) containing two hyperbolic points onnected smoothly by their stable and unstable manifolds encircling one table elliptic point. Under perturbations the stable and unstable manifolds

Near integruble chuotic Humiltoniun systems

149

intersect transversally and a complex picture appears near the hyperbolic points. We have also seen what happens under perturbations near the points. Some orbits disappear, the 'most rational' tori first, each giving rise to a string of elliptic and hyperbolic points with their own stable and unstable manifolds. These points produce in turn a picture similar to the general integrable system (Figure 6.4.1 ) and under perturbation a picture similar to Figure 6.8.1. Simultaneously, within the elliptic points the picture is as Figure 6.10.2 and repeats itself in the newly formed points. K A M curves break up into islands exhibiting the same structure at all length scales (self-similarity) suggesting the applicability of renormalization methods.14 A qualitative picture of the phase space is shown in Figure 6.1 1 . 1 . We expect this picture to he churucteristic of' ull rleur-irltegruhle chuotic Humiltoniun sysrems. Such a prediction is confirmed by numerous computational studies, regardless of the formal details of the models. The Poincare section is a display of the behavior of the system to all possible initial conditions. The picture represents phenomena with multiple time and length scales (e.g., islands corresponding to high periods take a long time to form in a computer experiment). It is important to recognize that not all the phenomena might be displayed in the time scales of interest (for example, in mixing we are interested in low period events).'" Figure 6.1 1 . I . General picture of a near-integrable system.

150

Chaos in Hamiltonian systems

In this context it is interesting to anticipate the effect of such a phase portrait on a line (Berry et al., 1979). Figure 6.1 1.2(a) shows part of the manifold structure of a Hamiltonian system. A line is initially placed horizontally passing by H,, E, and H,. After some time t the line is Figure 6.1 1.2. Sketch of formation of tendrils and whorls; (a) shows part of the manifold structure and orbits near the elliptic point E. ( h ) shows the deformation of a line with an initial condition passing by H I I , , and E (broken lines). In general, the stretching produced by tendrils is much more noticable than the whorl produced by the elliptic point.

,,

Near integrable chaotic Hamiltonian systems

151

&formed as indicated in Figure 6.1 1.2(b) forming both folded and bcortical' structures (named 'tendrils' and 'whorls' respectively by Berry t, ~ l . ) . Note ' ~ that if the 'experiment' is continued for a longer time other higher period points will deform the line as well (for example, the elliptic points, at a rate dependent upon both the winding numbers and the period of the point). These effects will be felt at smaller scales and we can anticipate &hods inside folded structures, whorls inside whorls, folds within folds, and so on. This action is reminiscent of a turbulent flow. problem 6.1 1.1

Elaborate on the differences between the stretching of a line in an actual turbulent flow and Figure 6.1 1.2(h). Problem 6.1 1.2

Criticize the Figure P6.11.2 representing the evolution of a line in a Harniltonian system after the breakup of KAM surfaces. Figure P6.11.2

\ material

line at t=0

\ material line at time r

152

Chaos in Hamiltoniun systems

Bibliography Much of the material of this chapter, especially in the first sections, makes reference to classical mechanics and theory of differential equations and dynamical systems. For an introduction to classical mechanics we suggest Percival and Richards (1982). An excellent but more advanced treatment is given by Arnold (1980). A modern encyclopedic treatment covering quantitative dynamics and the three-body problem is given by Abraham and Marsden (1 985). Traditional treatments are given by Goldstein (1 950) and by Synge (1960). For the general theory of dynamical systems we recommend the books by Hirsch and Smale (1974) and any of the books by Arnold (Arnold. 1983; 1985) and the article by Birkhoff (1920), which is a long but accessible account of the foundations of dynamical systems with two-degrees of freedom written in the language of classical mechanics. Much of the material on chaos reviewed in this section can be found in the recent books by Lichtenberg and Lieberman (1983), Guckenheimer and Holmes (1983) and in the review article by Helleman (1980). The book by Lichtenberg and Lieberman (1983) addresses primarily Hamiltonian systems and is especially useful. Other general articles containing material on dynamical systems and chaos are: Moser (1973) which is the main reference for the twist theorem, and Arnold (1963) which is the main reference for the KAM theorem. This last paper is especially recommended in spite of its difficulty and length. A useful collection of classical papers addressing chaos in Hamiltonian systems is presented by MacKay and Meiss (1987). The Melnikov method is analyzed in detail by Wiggins (1988b). Notes I Newton's second law for a particle of unit mass is: d2x/dt = F where x represents the position of the particle and F the sum of the forces acting on the particle. Using standard techniques (see Hirsch and Smale, 1 9 7 4 ) the second order system can be transformed into a first order system by defining q i = .Y, and pi = d.c,,/dr. I f in addition we define the Hamiltonian H ( p , q ) H ( p . q ) =:lp12 + V ( q ) , which corresponds to total energy of a particle moving in a potential V . Newton's equations can be written as dq,/dt = c'lf/c'p,, dp,/dt = i M / ? q , , which are known as Hamilton's equations (Goldstein, 1 9 5 0 ) . From a more general viewpoint, in classical mechanics it is shown that Hamilton's equations can be derived from Lagrange's equations which in turn are an extension of Newton's equations (see Goldstein. 1950, for a complete discussion). It should be noted that although

Notes

15:

there is no new physics involved. Hamiltonian theory often gives a much more powerful itls~ghtinto the structure of problems in classical mechanics. Hamiltonian systems with the hamiltonian even if the momenta and position are called reversible, i.e., t-+-1ifp-t-p. 1 For example, in an oscillating circuit with no resistance but containing a saturated core, inductance can be described in terms of Hamilton's equations (see Minorsky, l962. p. 62). 7 Note that in a steady two-dimensional isochoric velocity field the streamfunctions ;:~tisfics DGlDr = 0. j A necessary condition is that the Jacobian d(p, q)/ci(l, 8 ) be equal to one. 5 Sincc finding the action-angle variables (often referred to as normal coordinates) i\ ;i way of solving problems in Hamiltonian mechanics, much of the initial effort 111 thib area was focused on finding such transformations. assuminq their existence. Thc general possible ways to accomplish this task are the subject of the HamiltonJacob1 theory (Goldstein, 1950, Chaps. 8 and 9). However, as it became clear in rcccnt ycars it is almost always impossible to find the action-angle variablescorresponding to a glvcn Hamiltonian. 6 Note that the action-angle variable transformation does not work o n the separatrix (see Figure 6.4.1). For example a hyperbolic system dxldl = .u, dyldr = - y cannot hc tr;insformed into a twist map. 7 According to Peixoto's theorem these phase portraits are structurally unstable in the class of 1111 systems (for example if one allows a small amount of dissipation: i.e., volume contraction). Note, however, that this takes usout from the classof Hamiltonian systems. X This has to happen even in dissipative systems and is thecontent of the 'A-lemma'. See Guckenheimer and Holmes, 1983, Section 5.2; Palis, 1969. 0 The '7.-lemma' implies homoclinic tangles. The implications of infinitely many ~ntcrsectionsof the stable and unstable manifolds in Hamiltonian systems were pointed out by Poincare in 1899 (see Arnold, 1963, p. 178) 10 Meaning all, except for a set of Lebesgue measure zero. I I Ilistinguish clearly the assumptions from the conclusions; compare with the twist theorem. I? The effects ofsurviving KAM surfaces on transport in Hamiltonian systems are discussed by MacKay, Meiss, and Percival (1984). A tutorial presentation of these and other matters is given by Salam, Marsden, and Varaiya (1983). 13 The condition C' ( s > 5) is not sharp (see Moser, 1973). This theorem was proved by Moser in 1962 with s 2 333 and later improved by Riissman t o s 2 5; with additional conditions it can be reduced to s 3. 14 A substantial review is given by Escande (1985). An application example is discussed by Greene (1986). 15 Also. it should be emphasized that according to the values of the parameters, the region ofchaos might occupy a very small region of the phase space. In most Hamiltonian systems there is no sharp 'transition to chaos' and chaotic islands and bands surrounding Islands of instability. sometimes too small to be detected by computational means, may exist for any non-zero value of the perturbation. Examples are given in Chapters 7 and 8 . Sometimes the term 'transition to global chaos' is used to refer to the breakup of thc last KAM torus. I6 Note that the whorls can be produced in flows without vorticity and even in llows without circulation (Jones and Aref. 1988).

Captions to color illustrations Figure 1.3.4. Mixing of two immiscible polymers. The image-processed Fourier filtered transmission electron micrograph shows poly(butadiene) (black and light blue) domains in a poly(styrene) matrix (light grey). The black domains correspond to the domains not connected to the boundary. The volume fraction of poly(butadiene) is 0.312 and the average cluster size is 4.8 pm. (Reproduced with permission from Sax (1985).) Figure 1.3.5. Mixing below the mixing transition in a mixing layer, visualization by laser induced fluorescence [the term 'mixing transition' refers to the onset of small scale three-dimensional motion]. The experimental conditions correspond to a speed ratio U,/U, = 0.45, and a Reynolds number based on the local thickness of 1,750. A fluorescent dye is pre-mixed with the low speed free stream. Laser induced fluorescence allows the measurement of the local dye concentration, ( a ) single vortex, (h) pairing vortices. (Reproduced with permission from Koochesfahani and Dimotakis (1986).) Figure 7.3.10. Manifolds corresponding to various values of p, (a) p = 0.3, and (h) p = 0.5. ( c ) Magnified view ofthe stable and unstable manifolds of two period-1 points for p = 0.5 (see 7.3.10(a)).The unstable manifolds of the central point are shown in red, the stable manifold in yellow, the unstable manifolds of the outer point are shown in green. the stable manifold in light blue. The figure shows the results for 15 iterations. More iterations would make the picture hard to visualize; the number of points in the manifolds of the central point is 4.000, the number of points in the manifolds of the outer point is 2,000. ( d ) Magnified view of the manifolds near the transition to global chaos (the figure corresponds to p = 0.38). Note that the manifolds of the central and outer points intersect slightly. Compare with Figure 7.3.2. (Reproduced with permission from Khakhar, Rising, and Ottino (1987).) Figure 7.4.4. Comparison of Poincare sections and experiments (U,,, = 360"): (a) Poincare section corresponding to 8 initial conditions and 1,000 iterations; the symmetry ofthe problem produces a total of 16,000 points. (h) Experiments corresponding to the stretching of a blob initially located in the neighborhood of the period-1 hyperbolic point for 15 periods. Figure 7.4.5. Similar comparison to that of Figure 7.4.4 except that I),,, = 180": ( a ) Poincare section; colored points reveal that the initial conditions 'do not mix' even for a large number of periods, e.g., see magenta points; (h) corresponding experimental result for 10 periods (blob initially located in the neighborhood of the period-1 hyperbolic point). in this case, the blob does not invade the entire chaotic region. Figure 7.4.6. Comparison of two Poincare sections identical in all respects, except that the angular displacement B,,, of the outer cylinder in case ( a ) is 165' whereas in case (h) it is 166'. The value of Oi, is such that !2,,/!2,., = -2. The initial placement of colored points is the same in both cases. Figure 7.4.7. ( u ) Location of periodic points, and (h) corresponding Poincare section (O,,, = 180"). The crosses represent hyperbolic points, the circles elliptic points. The period of the point is indicated by the color; green = 1. red = 2. blue = 3. orange = 4. yellow = 5.. . . White results due to 'superposition' of nearby points of different order.

Color illu.strutions Figure 7.4.10. Poincare sections corresponding to the angular histories of Figure 7.4.9: (a) square (O,,, = 180 ), (h) sin2, (c) sawtooth, and (d) Isin(). The initial placement of colored points is the same in all cases. Note that whereas the large scale features are remarkably similar the distribution of colors reveals some local differences. Figure 7.4.1 1. Stretching map corresponding to square history and 10 periods. The figures were constructed by placing two vectors per pixel and averaging over 10' initial orientations. In (a)U,, = 180" and the white regionscorrespond to stretching of greater than 50; in (h) O,,, = 3 6 0 and the cut-off value is 5,000. Figure ( a )should becompared with 7.4.8 and 7.4.5, (h)with 7.4.4(a,h). Figure 7.5.7. An experiment similar to that of Figure 7.5.5 showing an island at the point of bifurcation ( T = 48.2 s, displacement 970 cm, Re = 1.2, Sr = 0.08). This structure is characteristic of a golden mean rotation speed. Figure 7.5.8. Partial structure of periodic points corresponding to the system of Figure 7.5.2(d), A = hyperbolic period-1; B =elliptic period-2; C = hyperbolic period-2; D = elliptic period-4; and E represents a hole of period-1. Note that the circles represent elliptic points and the squares hyperbolic points. The full white lines connect two elliptic points to a central hyperbolic point; all three points move as a unit. The broken lines connect two period-2 hyperbolic points to a period-1 hyperbolic point; the two period-2 hyperbolic points were born from the period-l hyperbolic point. The two period-2 hyperbolic points interchange their positions after one period. Figure 7.5.9. Illustration of reversibility in regular and chaotic regions. The mixing protocol corresponds to Equation (7.5.la.b) with a period of 30 s. ( a ) is the initial condition at which the vertical line is placed in the chaotic region, while the oblique line is placed in the regular region. ( b ) shows the lines after two periods. Note that the stretching and bending of the line placed in the regular region is very small, and that the line is merely translated. On the other hand, the line placed in the chaotic regions suffers s~gnificantstretching. (0shows the state of the system after being reversed for two periods. Clearly, the line in the regular region managed to return to its initial location successfully, however, the line in the chaotic region loses its identity due to the magnification of errors in the experimental set-up.

Color illustrrrt i o t ~ s

Color illustrations Fig. 7.3.10

d

Fig. 1.3.5

Color il1u.strc1tiorl.s Fig. 7.4.4

Fig. 7.4.6

('olor- illlr.stri~tior~.s

I

Fig. 7.4.7

Fig. 7.4.10

Fig. 7.5.8

Mixing and chaos in two-dimensional time-perjodic flows

In this chapter we study fluid mixing in several two-dimensional timeperiodic flows. The first two flows the tendril-whorl flow and the blinking vortex flow are somewhat idealized and admit analytical treatment; the last two examples - the journal bearing flow and the cavity flow - can be simulated experimentally. -

-

7.1. Introduction This chapter consists of two parts. In the first part (Sections 7.2 and 7.3) we study in some detail two idealized periodic mappings, which can be regarded, in an approximate sense, as building blocks for complex velocity fields: ( 1 ) the tendril-whorl mapping (TW), and (2) the blinking vortex flow (BV).' Both flows can be regarded as prototypical local flow representations of more complex flows and they are simple enough so as to admit some detailed analytical treatment. The analysis concentrates on: (i) the structure of periodic points and their local bifurcations, and (ii) the global bifurcations, or interactions among the stable and unstable manifolds belonging to hyperbolic points. It is found that these simple systems possess a rich and complex mixing behavior. Both mappings are chaotic according to muthemutically accepted definitions. One of the lessons to be extracted from the complexities encountered in the analysis of these mappings is that there are practical limits to the level of probing and understanding of the details of mixing in complex systems. In the second part (Sections 7.4 and 7.5) we focus on two flows which do not admit such a detailed treatment: (3) a Stokes's time-modulated flow in a journal bearing, where we focus on the comparison between experiments and computations, as well as the need for various computational and analytical tools; and (4) several kinds of low Reynolds number cavity flows in this case we compare the performance of steady flows with that of periodic flows and study the evolution and bifurcations of coherent structures. Undoubtedly, many other examples and variations will occur to the reader. -

Tendril-whorl flow

155

should be clear that even though the examples presented here are gc)vcrnd by the theory of Chapters 5 and 6, our problem is not simply one of adaptation and much work remains. Possibly the most important diffcrcncc between conventional dynamical systems studies and the work here is that we are interested in the rate at which phenomena occur. Thercfore attention is focused on low period phenomena since we want to mix both efficiently and quickly. There is also the matter of the area occupied by chaotic and 'non-chaotic' or regular regions. Presently, there are no predictions of the size of the mixed and unmixed regions, no thorough studies regarding the stability of the results with respect to geometrical changes, and no indepth studies of the relationship of the flow to the morphology produced by the m i ~ i n g . ~ An important question to keep in mind throughout this chapter is: What constitutes a complete analysis or understanding of a mixing flow? In the first two examples we follow a program such as the one outlined in Table 7.1. A loose definition as to what constitutes complete understunding is the following: A system can be regarded as 'completely understood', from a practical viewpoint, when the answer to the n + I question in a program, such as the one sketched in the table, can be qualitatively predicted from the previous n answers. In the first two examples, since we have an explicit expression for the mapping, we can accomplish a substantial portion of the program. In the case of the journal bearing flow we have an exact solution to the Navier-Stokes equations, but the Eulerian velocity is not nearly as convenient as having the mapping itself. However, there are number of avenues left for analysis and we exploit many of these. In the case of the cavity flow we do not have a solution for the velocity field and the comparison with computations and analysis has to be of a less direct nature.

7.2. The tendril-whorl flow Flows in two dimensions increase length by forming two basic kinds of structures: tendrils and whorls (see Figure 7.2.1 ) 3 and their combinations. In complex two-dimensional fluid flows we can encounter tendrils within tendrils, whorls within whorls, and all other possible combinations. The mapping we study in this section seems to be the simplest one capable of displaying this kind of behavior. The tendril-whorl ,flow (TW) introduced by Khakhar, Rising, and Ottino (1987) is a discontinuous succession of extensional flows and twist maps. In the simplest case all the flows are identical and the period of alternation is also constant. As we shall see

Table 7.1

Study of symmetries

Analytical

Computational

Experimental

Comments

Entirely analytical

Convenient to produce symmetric Poincare sections

Can be used to verify experimental accuracy

Can be exploited to reduce computational time (e.g.. search of periodic points)

Location of pvr~odicpoints

Only in very simple cases Order one given by conjugate lines Higher order obtained by minimization/symmetry methods

Lowest order points most important in rapid mixing

Stability of periodic points Biiurcation as parameters are changed

Possible in simple cases

Possible (orten only method to obtain eigenvalues)

Regular islands surrounding elliptic points (obstruction to mixing)

PoincurJ sections

Only in very simple cases

Relatively easy

Long time behavior of the system Tor all initial conditions

Only possible in very simple cases

Can be done by placing a blob encircling the point

Possible in simple cases

Relatively easy

Order one. or fixed points. Order two Order n Locul unulysis

Glohul analysrs

Stable and unstable manifolds associated with hyperbolic points (order one, two, n ) . Interactions between manifolds Melnikov method Liapunov exponents Stretching of material lines and blobs

Hard, purely accidental

Relatively easy Hard for large stretchings

Relatively easy Possible, especially in systems with suitable symmetries

Formations of horseshoes

Possible in simple cases

Resolution might be a problem but possible

Pathlines Streamlines Streaklines

Impossible in chaotic flows

Streaklines hardest to obtain

Speed along manirolds is proportional to magnitude of eigenvalueslperiod. Degree of manirold overlapping controls overall rate or dispersion. Generally valid Tor small perturbations only: convenient iranalytical expression for homoclinic/heteroclinic trajectory is available

Relatively little information about mixing (except streaklines)

Mups oj constunr propvrrivs

Stretching ERiciency Striation thickness, etc.

Best route

Possible via image analysis

Useful to make indirect connections

Tendril-whorl jlow

157

the simplest case is complex enough and can be considered as the point of departure for several generalizations (smooth variation, distribution of time periods, etc.). The physical motivation for this flow is that, locally, a velocity field can be decomposed into extension and rotation (see Section 2.8). Alternatively, according to the polar decomposition theorem, a local deformation can be decomposed into stretching and rotation (see Section 2.7). In the simplest case the velocity field over a single period is given by for 0 < t < Text,extensional part (7.2.1) = - CX, 0, = EY, for Text < t < Text Trot, rotational part (7.2.2) uO = o ( r ) [', = 0, where Textdenotes the duration of the extensional component and Trot the duration of the rotational component. Thus, the mapping consists of vortices producing whorls which are periodically squeezed by the hyperbolic flow leading to the formation of tendrils, and so on. The function o ( r ) is a positive quantity that specifies the rate of rotation. Its form is fairly arbitrary and the results are, in a qualitative sense, fairly independent of this choice. The most important aspect is that o ( r ) has a maximum, i.e. d(o(r))/dr= 0 for some r. Nevertheless, independently of the form of w(r), we can integrate the velocity fields (7.2.1) and (7.2.2) over one period to give I?,

+

fext(x, Y) = (xla, a ~ ) , frot(r7 6) = (r, 8 + A$), where a = exp(~T~,,) and A8 = -o(r)Trot/r, i.e., the point (x, y) goes to (.K/z,ay) due to the extensional part, the point (r, 8), which corresponds to (xla, ay), goes to (r, 8 + A8) due to the rotational part, and so on. Figure 7.2.1. Basic structures produced by mixing in two-dimensional flows; ((I) whorl, ( h ) tendril.

158

Mixing und chaos in two-dimensionul time-periodic flows

We consider a rotation given by A8 = - Br exp( - r). We will investigate only a few aspects of this flow. A more thorough analysis is given by Khakhar, Rising, and Ottino (1987), and especially in the Ph.D. thesis of Khakhar (1986). As we shall see, due to the simplicity of the TW map, we can calculate, analytically, up to periodic points of order 2. A similar analysis can be carried out, in theory, for periodic points of order n. The analysis is however, extremely difficult.

7.2.1. Local analysis: location and stability of period-1 and period3 periodic points

The qualitative idea of period-l (or fixed) and period-2 periodic points is indicated in Figure 7.2.2. The composition of the extensional and rotational maps produces, in polar coordinates, f(r, 8 ) = (r', tan - '(a2 tan 8 ) + A6(r1)) where f denotes the composition of (7.2.1) and (7.2.2), f = f,,,.f,,,, and where r' is given by

At the period-l periodic point (see Figure 7.2.2) we have f(r, 8) = (r, 8). Figure 7.2.2. Pictorial representation of period-1 and period-2 points.

us denote the periodic point as p* = (r*, O*). Since r stays constant we require

cos2 O*

(r'2)

+ r 2sin2 O*= I,

which implies 0* = t a n - ' (112). ~ l s o since , the initial and final angles coincide (up to an integer number of rotations, 2nn), 0* = tan- l ( r 2tan O*) + AO(r*) + 2nn. extension

rotation

Simplifying, we obtain Br* exp(-r*) = tan-'[(% - l/r)/2]

+ 2nn

n = 0 , 1 , 2 , . . . ,M

so that depending on the values of a and M we obtain a number of diffcrent solutions. We consider for convenience only the case of M = 0, and make B dimensionless by defining

Be

B = tan - I [(a - ..I/a)/2] so that the radius and angular position of the period-1 periodic points are given by the equations:

O* = t a n - ' ( l l r ) r* exp(l - r*) = 1//3 Thus, thc TW mapping is characterized by two parameters: r and /3. We noticc that the angular position depends only on a, the radial position only on P. It is also easy to see that the origin is a fixed point for all parameter values. For M = 0 (and < 1 + 477) there can be 0 ( p < 1 ), 1 (/I = 0, 1 ) or 2 (b > I ) periodic points. A graphical interpretation of the equations is given in Figure 7.2.3. If M > 0 additional periodic points might appear. The local behavior of the period-1 fixed points is given by the eigenvalues of thc linearized mapping evaluated at the point p* = (r*, O*),

:

= tr(Df)

+ ([:

tr(Df)I2- 1 ) ' 1 2 .

The mapping is f ( s , y) = ((sir)cos A8 - ay sin At), (.u/a) sin AO + rg cos A0) where A0 = AO(r)

and

r = (s2/a2+ a2y2)1'2

160

Mixing und chuos in two-dimensional time-periodic flows

and

' [(') [(*')sin

] [(:) ]

cos AO - rg sin AO , ;

( AO + zy cos AO iu After some manipulations we get

, :

I

cos A ~ I zy sin AO

1

A 8 + ay cos A 8

[(;)sin

This expression is valid for all choices of AO(r).For our choice, we obtain tr[Df(p*)]

=2

+G

(7.2.3)

where G is defined as G = ( r - l / z ) t a n - ' [ ( z - 1/2)/2!(r*- I ) G = g(x)(r*- 1 ). Thus, the character of the eigenvalues depends on the value of G. We have: G >0 + hyperbolic, G =0 + parabolic, 0 > G > - 4 + elliptic, G = - 4 + parabolic, - 4 > G + hyperbolic. Based on the above analysis we can now study the sequence of (local) bifurcations that take place as P is increased from zero for a fixed value

I

Figure 7.2.3. Graphical interpretation of the equation for location of period-l points.

I

I.-

no penod~cpoints

,-

one period~cpoint

Z

r two penodic points r exp ( I - r )

of r (recall that the position of r* depends on

in Figure 7.2.4. If P exceeds 1

p). This is explained

+ 4n the situation is like Figure

7.2.5. At /l= 1 two period-1 points are formed at r* = 1 in the first and third quadrant as shown earlier. Both points are parabolic since G = 0 in this case. When /I becomes greater than 1 , each fixed point splits into two; F ~ g u r e7.2.4. G r a p h ~ c a lrepresentation of bifurcation sequences; the p e r ~ o d i c points are located at a n angle 0 = t a n l ( l / z ) ; the line is mapped by the elongat~onalflow a n d twisted clockwise. T h e intersection(s) correspond to the periodic points material line

I

- ' after one p e r ~ o d

\ material line

H

saddle-node

period doubling

162

Mixing and chaos in two-dimensional time-periodic Jows

the two at radial distance r* > 1 are hyperbolic, while those at r* < 1 are initially elliptic. Thus, at P = 1, r* = 1 there is a 'saddle-node' bifurcation (Guckenheimer and Holmes, 1983, p. 1 56).4 As P is increased further, the inner fixed point, which is initially elliptic, moves closer to the origin and G becomes more negative. If g(cc)> 4, for 1) large enough, G = - 4 and a second bifurcation takes place in which the elliptic point becomes parabolic and then hyperbolic. In addition there are two additional period-2 elliptic points. This bifurcation is called 'flip' or 'period-doubling'. Assuming that cc and fl are large enough to have period-2 fixed points we have f2(p*)= p* (7.2.4) i.e., the following relations hold: f(r7, 0:) = (r;, 0;) f(r;, 0;) = (r?, 0:) where (r:, 0:) and (r:, 0;) are the two period-2 points formed near r*, 0* Figure 7.2.5. Creation of additional periodic points; the whorl formed by rotation intersects the line 11 = tan-'(I/G()more than once in the quadrant x > 0, y > 0 if the rotational strength fi is greater than 1 + 471.

I

A

material line

(by symmetry we can easily obtain the results, r*, O* + n).On substitutin in the mapping, we obtain: r* - r *, (cos2 O:/a2 a' sin2 0:)'l2

+ + a 2 sin2 0;)'"

r* - r2(cos2 * O;/a2 0; = tan- '(a2 tan 0:) 0: = tan- '(a2 tan 0:) which can bc simplified to give: tan 0: tan 0)

+ AO(r;) + AO(r:)

=

I/a2 +

An: = - (n/2 - 20:) An: = - (n/2 - 20)) r) = yr? where

and r , = litan 07, r 2= I/tan 0;. Thc stability of the period-2 periodic points depends on the eigenvalues of the Jacobian cf f 2 ( .). For period-2 periodic points we have +XI, X I = f 2 ( x , )= f(x,) and the Jacobian is calculated as XI +X,

Df2(x

= Df(x,)Df(x )

or

(7.2.6)

Df2(x2)= Df(x, )Df(x2). Aftcr considerable simplification, we obtain: tr[Df(x,)Df(x,)]

=2

+ GIG2+ 2(Gl + G,)

where

A0)(x4 - I ) ( I - r)) z2(x: 1 ) At the birth of the period-' fixcd points r: =r:. GI = Gz=G , and G2 =

-

2 + G'

(7.2.7)

+

r , = r 2= r

so that

G' = G' + 4G. A t thc flip bifurcation. G = - 4 which implies that G ' = O so that thc Period-' points arc parabolic at their point of formation. For slightly

Mixing and chaos in two-dimensional time-periodic flows

64

F~gure7.2.6. (a) Rotat~onalstrength for transition of the periodic point from as a function of the strength of the extensional elliptic to hyperbolic, flow, a, for per~od-land period-2 polnts. (Reproduced with permission from of region in (u). The Khakhar, Rising, and Ottino (1986) ) ( h ) Magn~ficat~on broken line corresponds to u = 10.

a,-,,

5.0

-

(a)

4.0

-

3.0

-

2.0

-

.c

TZ

4 period- l period-2

region shown below 1.o 0

5 .o

10.0

15.0

20.0

Q - - - - ---

1.2

-7 0

1.1

c?.

1 .o

8.O

12.0 (Y

16.0

Tmtlril-whorl flow

165

121-guvalues of /j we find that (;' < 0 and the period-2 points become cIliptic. As /j is increased furthcr, depending on the value of x, a second flip may occur in which the period-2 points become hypcrbolic and two pc1-~od-4elliptic points are formed for each period-2 point. Let us now cc,~~sidcr a few aspects of the problem which are discussed in more detail in Khakhar, Rising, and Ottino (1986). Some aspects of the bifurcation behavior are summarized in Figure 7.2.6, where we have plotted flip bifurcation values ( i t . , the value of /j at which the elliptic point becomes hqpcrbolic) versus x for the period-l and period-2 elliptic points. Figure 7.2.6(h) shows a magnified region with the broken line corresponding to 2 = I0 (as is apparent from this figure the period-1 elliptic points do not exhibit a period doubling cascade of bifurcations). A similar study, much harder from an algebraic viewpoint, can be carried out for period-3 periodic points, etc. 7 . 2 . 2 . Global analysis and inrevactions between manifolds

Hcrc we consider some of the global bifurcations of the TW mapping by stildying the interactions of the manifolds of the hyperbolic period-l periodic points. Obviously, a similar study can be carried out for period-2 periodic points, etc. From the point of view of mixing, the stable and unstable manifolds prov~dcus with broad features of the flow and its ability to mix. As indicated in Chapter 5 we obtain the locus of the manifolds by surrounding the hyperbolic points by a circle of small radius made up of a large number of points and convecting them by the flow. Forward mappings give the unstable manifolds, backward mappings the stable manifolds (actually, the result is a thin filament to the resolution of the graphics output device which encases the manifolds). Obviously, in the case of homoclinic or hetcroclinic behavior larger and larger portions of the manifold become apparent with the number of iterations. However, given the limit of resolution, and the fact that we can work only with a finite number of points. a very large number of iterations is extremely hard to interpret. The manifolds will not appear as continuous lines, which they are, but as a series of disconnected dots.' Far away from the origin (r + x ) the twist mapping acts as a solid b o d rotation. The interesting behavior occurs for r < O i l ) ; the chaotic bellavior is confined to a well defined region shown schematically in Figure '.7.7. The flow drags outside material which enters via conduits around the stable manifolds and leaves via conduits formed around the unstable manifolds. For /I < I the mixing zone disappears and the flow is regular (nlc,stlv) cv~rywhere.~

166

Mixing und chaos in two-dimensionul time-periodic flows

Figure 7.2.8 shows the manifolds of the hyperbolic period-1 periodic points at r* > 1 (P, and P',in Figure 7.2.7) and the origin, 0, for r = 1.5 and two different values of P. The manifolds appear to join smoothly7 and the system has the classical appearance of an integrable Hamiltonian system. At p = 1.075 the stable and unstable manifolds of P I join smoothly with each other (similarly for P',) whereas the manifolds of 0 fly-off to infinity. However, when fi = 1.099 the situation is somehow reversed: the stable and unstable manifolds of 0 join each other and are confined by the region of the stable and unstable manifolds of P I and P',, which also join smoothly with each other. Figure 7.2.9 shows the system at cx = 5.000 and increasing values of fi. In Figure 7.2.9((1)the mixing zone consists of two compartments which are isolated from each other. We expect that K A M surfaces surround the elliptic points in the center of the islands. The picture changes in Figure 7.2.9(h), in which good mixing is expected to take place, due to the presence Figure 7.2.7. Schematic view of mixing zone. Pi and P , are the outer period-] points, P; and P2 are the inner period-1 points The large arrows show the direction of transport along the stable and unstable manifolds.

I.ipurc 7.2.8. Manifolds associated with the central hyperbolic fixed point and period-l points for various values of a and 1.(u) z = 1.500 and P = 1.075. (h) r = 1.500 and /{ = 1.099. Note the change between figures ( ( I ) and (h; in (u) P , forms a homoclinic connection; in ( h ) the central point forms a homoclinic connection. (Reproduced with permission from Khakhar, Rising, and Ottino (IOX6).)

Mixing and chaos in two-dimensional time-periodic flows Figure 7.2.9. Same as Figure 7.2.8, but at larger values of r . In ((1) r = 5.000 and /j = 1.082. (h) r = 5.000 and /j = 1.595. Elliptic islands are expected surrounding P; and P, in case ((1). In ( h ) the islands have been reduced in size (note change of scale). (Reproduced with permission From Khakhar. Ris~ng,and Ottino (19861.)

Tmdril-whorl flow

169

homoclinic and heteroclinic points (the manifolds have been only p;lrtinlly drawn to make the picture understandable). In summary, from the above analysis (which is not complete since the ilnulysis pertains to only period-l periodic points, and even more, those rc5tricted to M = 0 ) we anticipate that if the extensional flow is weak (small x) the mixing will be poor for all /I (strength of the rotation). As x is increased and for a large enough /I there are intersections of the manifolds belonging to the outer periodic points and those of the origin resulting in a single mixing zone. However, based on the results of Chapter 6, we know that the mixing zone is not truly homogeneous, and is expected to contain islands around each elliptic point. Of these the largest are those corresponding to the period-1 elliptic points. The size of the islands decreases with increasing cr and & 7.2.3. Formation of horseshoe maps in the TW map

The above discussion indicates that the TW system is capable of ho~~ioclinic/heteroclinic behavior. We know that a homoclinic intersection implies the existence of horseshoe maps. It is possible to prove that the TW system is capable of producing horseshoe maps of period-I. An analytical proof of existence of horseshoes constitutes a proyf' of chaos. We can show the existence of period-l horseshoes by the construction sketched in Figure 7.2.10 (Rising and Ottino, 1985). We select a rectangle with sides in a ratio x:l and with two corners being periodic points of period-I. By selecting suitable values of P, and mapping forward, i.e., first the extensional flow with p being the stretching axis and then the twist mapping acting clockwise, we can produce vertical stripes. Then, by the inverse mapping. i.e., first the twist mapping counter-clockwise and then the extensional flow with x being the axis of extension, we can produce hori/ontal stripes. Actual examples are given by Khakhar, Rising, and Ottino ( 1 986). Problem 7.2.1 Show that f,,,(r, y) = (sly, rp) in rectangular coordinates, and

-

f,,,(r, 0 ) = (r[(cos2 O/r2)+ r 2 sin2 OIL", t a n 1 ( r 2tan 0)) polar coordinates. Show that fr,,,(r,0) (r, O + AO) in polar coordinates, and fr,,(r,0 ) = (rcos A 0 - y sin AO, .Y sin AO + jt cos AO) 'I1 rectangular coordin:~tes. Pr~hlem7.2.2 Prmc that the point at r = 0 is hyperbolic

Figure 7.2.10. Horseshoe formation in the tendril-whorl flow. In ( a ) the elongational flow acts first, followed by a twist clockwise. In ( h ) the tw~stacts lirst, counter-clockwise, followed by the elongational flow. (c) Actual example corresponding t o r = 10.000 and = 2.180.

\

' I

I

,

I

\

./

.

out of rectangle

twist, ciockwise \,

/ /

\

\

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I

twist, counter-clockwise backward

I

I i

A' goes to A"

stretching,~,,< 0

Blinking vortex jlow

171

problem 7.2.3*

Inject particles in one of the conduits and study the distribution of times in the 'chaotic' region. Interpret in terms of the behavior of manifolds. p~.ohlem7.2.4

consider the flow o, = Gx,, o2 = KGx,. Can a function K = K(t) be specified such that this system will give horseshoes?

7.3. The blinking vortex flow (BV) This flow, introduced by Aref (1984), consists of two co-rotating point vortices, separated by a fixed distance 2u, that blink on and off periodically with a constant period T (Figure 7.3.1).8 At any given time only one of the vortices is on, so that the motion is made up of consecutive twist maps about different centers and is qualitatively similar to the cavity flows studied in Section 7.5.9 When the two vortices act simultaneously, the system is integrable and the streamlines are similar to Figure 7.3.2(a). Figure 7.2.10 continued

T

O""

72

Mixing and chaos in two-dimensional time-periodic flows

The velocity field due to a single point vortex at the origin is given by: o,, = T/2nr or = 0, where T is the strength of the vortex. The mapping consists of two parts; each of the form G(r, 8) = (r, 8 + A8) where r is measured with respect to the center of the vortex and where the angle A8 is given by A0 = TT/2nr2. Placing two vortices at distances ( - a , 0) and (a, 0) in a cartesian co-ordinate system (Figure 7.3.1), the complete mapping, in dimensionless form, is given by: fi(x, y) = (ti+ (x - t i )cos A8 - y sin AH, (x - ti)sin A8 + y cos A8) where ti denotes the position of the vortex i (i = A , B), A0 = p/r2, with p = TT/2na2,and r = ((x- ti)2+ y2)"2 (distances are made dimensionless with respect to u so that the vortices A and B are placed at ti = +_ 1). In the following analysis we assume that the vortex at ti=,= 1 is switched on first and that rotation is counter-clockwise. In this case perturbations are introduced by varying the period of the flow, i.e., by starting from the integrable case, p = 0, and increasing the value of p. As we shall see there are several important differences with the TW mapping: (i) in the BV both flows are weak'' (i.e., the length of filaments increases linearly in time) instead of having a sequence of strong and weak flows as in the TW; and (ii) the BV flow is bounded, i.e., material is not attracted into the flow from large distances; and (iii) the flow is a one-parameter system rather than two as for the TW map." 7.3.1. Poincari sections

Let us start the analysis by considering Poincare sections of the flow. In the process of doing so, the virtues and limitations of such an approach will become apparent: Poincare sections provide some of the information about the limits of possible mixing, but in order to truly understand the inner workings of chaotic systems, we are forced to examine stable and

Figure 7.3.1. Schematic diagram of the blinking vortex system.

Blinking vortex jlow

173

manifolds and to investigate the existence of horseshoes, etc. (see Table 7.1). Figure 7.3.2 shows Poincare sections, t = n T with n = 1 , 2, 3, . . . . for different values of the perturbation p (Doherty and Ottino, 1988). For ~ 1 = 0 ,the system is integrable (see Example 7.3.1). As p is increased, regions of chaos form first near the vortices, then in the center region, until for p = 0 . 5 they occupy the entire region. Figure 7.3.3 shows a magnified view of the Poincare section characterizing the motion near the boundary for p =0.38, for a number of different particles starting at different positions. Near the boundary between the 'regular' and 'chaotic' regions we can observe a chain of islands of regular flow, each island F~gure7.3.2.Poincare sections corresponding to 12 different initial conditions for various values of the flow strength p : (a)11 = 0.01, ( b )p = 0.15, (c) p = 0.25, ( d ) jc = 0.3, ( e ) p = 0.4, and (j')jc = 0.5. A transition to global chaos occurs approximately at p =0.36. (Reproduced with permission From Doherty and Ottino (1988).)

174

Mixing und chuos in two-dimensional time-periodic flows

containing an elliptic periodic point. Between islands there is a hyperbolic point, of the same period as the elliptic points, whose manifolds generate heteroclinic behavior. In addition, in each island the elliptic point is surrounded by a chain of higher period islands and, in theory, the picture repeats itself ud infiniturn. The restriction to mixing imposed by the small Figure 7.3.2 continued

175

Blinking vortex flow

islands appears only for long times (0(104)iterations) since they are of high period. With this as a basis, we explore some of the above behavior by focusing on aspects of local and global bifurcations. The analysis is augmented by computations of Liapunov exponents, 'irreversibility', and mixing Figure 7.3.2 continued

Figure 7.3.3. Magnified view of boundary region for p with permission from Doherty and Ottino (1988).)

= 0.38.

(Reproduced

176

Mixing und chaos in rwo-dimensionul rime-periodic flows

efficiencies. In the last part of the analysis we use the mapping to compute the evolution of the intensity of segregation which might be used to mimic mixing times. Several other possible numerical experiments are suggested and many others might occur to the reader.

7.3.2. Stability o f period-1 periodic points and conjugate lines

Figure 7.3.4 shows two candidates for period-I periodic points (or fixed points). The simplest one is A ' . First, vortex A moves A ' to B' and then vortex B moves B' back to A ' . There are many other ones: For example, the point A " corresponds to more than one rotation due to vortex A , to position B", and then less than one rotation due to vortex B back to position A". I t is clear that if (s*, j*)is a period-I periodic point of the BV flow, then it is necessary that

To find the period-I points, first we map the x-axis by vortex A c-lockwise und with hulf' tile trnyle. Then we do the same thing with vortex B but c~ourtrer-c~loc~kwisr, i.e., one line is the mirror image of the other about J = 0. The points at the intersection of the mapped lines, c~or!jugurelines,

Figure 7.3.4. Schematic representallon o f period-1 points. period-] points

Blinking vortex flow

177

;ire the location of the fixed points (Figure 7.3.5(u)). The reader should vcrjfy that this is indeed the case." Figures 7.3.5(h)-(d) show the location of the period-l periodic points for different values of the flow strength p . The periodic points are labelled by integers n , , 11, which are the number of complete rotations of right and left vortices respectively. At low values of ,u (Figure 7.3.5(u)) there ;ire many periodic points, all of which lie above the x-axis. However, as ,u is increased (Figure 7.3.5(h))all but three intersections disappear, and finally, ;it high strengths (Figure 7.3.5(c))new periodic points are created, some of w h ~ c hlie below the x-axis. question to ask next is which of the periodic points are hyperbolic, which are elliptic, etc. This entails the computation of the Jacobian of the mapping and an analysis of the eigenvalues at each periodic point. As with the T W mapping, we can write the trace of the Jacobian as

In this case G is defined as C; 4 sin(O, + f12)[(A0,A0,

-

-

I ) sin(O, + 0 2 )+ (A0, + A0,) cos(0, + 02)]

Hence. the same analysis applies, as G is varied, but in this case the points hove to be tested one by one. The analysis becomes cumbersome almost immediately (which limits the anlysis to period-l points) and here we describe qualitatively just a few of the major results.'" For example, consider the outermost periodic point located at the centerline, .u = 0 and j.> 0 (for additional details, see Khakhar, 1986). At low flow strengths, ji 0 ', the point is hyperbolic. As / i is increased the point becomes parabolic at 11 2 15 and then elliptic (this kind of bifurcation is characterized by ;in cxchange of stability and the birth of two fixed points of the same period: the bifurcation is called 'pitchfork', Guckenheimer and Holmes, 19x3. p. 156). An analysis for the remaining period-l points yields the follo\ving conclusions: If the graphs of the segments ni and ni are just tangent to one another, at the point of formation the periodic point is llli[i;illy p:irabolic, and on decreasing 11, splits into two period-l periodic Pc"n[s. the one closer to the origin is hyperbolic, the other elliptic (see G~lckcnhcimerand Holmes, 1983, p. 146).

-

7.3.3. H o r s e s h o ~mups in rhr B V flon* As have seen in Chapter 5 thcrc is :I relationship between homoclinic.' ll~[~rc)clinic behavior and the existence of horseshoe maps. I n :I simi1;ir

Mixing and chaos in two-dimensional time-periodic flows Figure 7.3.5. ( u ) Schematic representation of conjugate lines; the intersection between the lines corresponds to period-l points; the point P is mapped as shown by the broken lines, (h) case corresponding to p =0.5, (c) case corresponding to p = 3, (d) case corresponding to 11 = 10. (Figures (h) and (c), reproduced with permission from Khakhar, Rising, and Ottino (1986).)

\\

mapping counter-clockwise with A812

(a)

mapping clockwise with A812

Blinking vortex ,flow

180

Mixing and chaos in two-dimensional time-periodic flows

fashion as with the T W mapping it is possible to prove, by construction, the existence of period-I horseshoe functions in the flow. The construction, which is shown in Figure 7.3.6 is based on the existence of two nearby period-l periodic points. These points and the streamlines of the flow define a quadrilateral which is deformed as indicated in the figure forming horizontal striations by the forward mapping and vertical striations by the backward mapping. The superposition of the figures yields a period-1 horseshoe, and hence, the system is chaotic.14 A similar idea, discussed in some detail in Section 7.5 can be used to examine experimental data.

7.3.4. Liapunov exponents, average ejfieieney, and 'irreversibility'

The Liapunov exponent calculated based on an infinitesimal segment initially placed at X = x, and with orientation M = m, is given by1'

Computationally, we proceed in the following way: The length stretch of an infinitesimal segment with initial orientation mi is calculated as (see Chapter 4)

where f is the mapping x, = f(x,- 1 ) and mi = D f ( x i ~ l ) ~ m i ~ l / i . , ~ , where i i - ,is the length stretch in the i - l to i cycle. The total length stretch, i.'"'. from i = I to i = 11 is

and since the period is constant. the Liapunov exponent is calculated as the limit

F ~ g u r e7 . 3 . 6 . Horseshoe construction. The points A a n d B are periodic points. In ( t i ) the right vortex ( r l acts counter-clucku~scfor t h o turns. then the left \ o r t e \ . not shown. act:, c o u n t e r - c l o c h ~ i s c .in (11) thc lcl't \ o r t c \ acts c l o c h ~ i s c lor two turns. then the r ~ g h tvortex ( r ) acts clockwise for two turns.

\\,

returns to A

I

period- l point

182

Mi.uing trnrl chtros in two-rli~~~c~n,siot~rrl tir~c~-pc~riotlic ,flo~s

For the calculation of the efficiency we need the magnitude of D along the path of the particle. In t h ~ scase it is constant for each half cycle and is given by ( D : D ) ' = 2 ' '/r/r2. ('ompi~tntional results for both the Liapunov exponent (the positive one) and the average efficiency arc shown in Figure 7.3.7. Both quantities tend to a positive limit value for points initially located in the chaotic reglon. the values being independent of the initial location and orientation of the material clement."' As we saw in Chapter 5, positive Liapunov exponents imply exponential rate of stretching of material elements, and hence, good mixing. The computations in Figure 7.3.7 werc carried out using over 50,000 cycles of the flow. The points were located close to one of the vortices, so that prior to the transition to global chaos, the particle is restricted to the chaotic region around the vortex. The behavior of the Liapunov exponent is quite complex (Figure 7.3.7(rr));initially i t decreases with increasing 11, and after the transition to global chaos (11 z 0.36) increases with increasing flow strength. At higher values of , ~ i there is another dip in the graph. Beyond / I 2 5 the calculations, up to a value of /I = 15, show that the exponent increases slowly. O n the other hand, the average efficiency, shown in Figure 7.3.7(h), is better behaved and presents a single maximum. as might have been expected based on the results of Section 4.5, with a value of ( e ) , = 0.16 at 11 = 0.8. The average efficiency seems to level off at a value (e), % 0.07 beyond / I 2 3 and the calculations indicate that it remains almost constant up to /I = 15. I t is important to place the previous results in perspective. From a practical viewpoint, 50,000 cycles is a very impractical way to mix. The values obtained should be considered as a bound for the performance of the flow. Let us reconsider now the question posed in Chapter 2 about kinematical rcvcrsibility of flows. Figures 7.3.8(rr) ( c ) show a line composed of 1 0 , o ~ points. The system is operated at , ~ i= I (the Liapunov exponent is approximately 0.62) and the line is stretched by the flow for different 17 number ofcyclcs forward and then the flow reversed to its original state. The computations werc done with a computational accuracy of lo-'. Figures 7.3.8(rr) ((.) show the rcsults of forward and backward cycles (17 = 5. 10, and 20, respectively). Each plot shows the initial line plus all the points that fail to return it. A point is considered to have returned to its initial location if it fi~llswithin a given radius of its initial location (in this case 0.01 ). These resi~ltsarc not surprising: note that the round-off error grows approximately as ( 10- ' exp[0.61(217)]). Thus, 17 = 5, 10, and 70, give errors of the order of 5 x I O F 5 , 2.4 x 1 0 ', and 5.9 x 10'9 respcct~vely.'

'

I lgurc 7.3.7. ( ( 1 ) 1.lapunov exponent as ;I function of flow atrcngth I ( , ( I ) ) flow c[ll~lcncq;I\ a function of flow strength 11. The different symbols corrcapond r,, cl~ffcrcnt~al ~ n ~ t iconditions i~l and prccisions [ + x,, = (0.9, 0).+ x,, = (0.0.0 ) . 0)x,, (0.99,O) D, x x,,= (-0.99,O) D, C] x,, = ( - 0 . 9 9 , 0 ) ] ; D denotes ci,,t~hlcprwclsion. (Reproduced w ~ t hperm~ssionfrom Khakhar. K~sing.and

-

OIIIIIO (

l987).)

184

Mixing and chuos in two-dimensional time-periodic flows ~i~~~~ 7.3.8, Stretching of a line by the blinking vortex operating at P = 1 for various number of iterations, ((0 n = 5. ( b ) n = 10, (c) n = 2°, using seven-digit precision. ._.- ..... __._,_

....

, c..

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-. .

Blinking vortex jlow

185

7.3.5. Macroscopic dispersion of tracer particles

Figure 7.3.9 shows the clear advantage of mixing in the globally chaotic region when compared with regular flows. First, a circular blob made up of a large number of points (9,500) is placed at a given location in the flow. which is operated at p = 0.8 for 30 cycles. In this case the flow is reglllar and the blob is stretched linearly. Second, we place the blob in the same location as the previous experiment but we operate the flow at = 1.0 for 24 cycles. In this case the blob has been stretched in thin filaments which appear disconnected due to insufficient number of points, precisionof the graphics output, etc. Note that the actual time of operation, !(N. is the same in both cases. The drastic difference can be explained in terms of the Poincare sections of Figure 7.3.3. In Figure 7.3.9(b) the blob lies outside the 'globally chaotic' region. When p is increased the region grows, the blob finds itself inside the chaotic region and the mixing is effective. However, placement of blobs in the chaotic region does not guarantee effective dispersion for short times and in fact, blob experiments provide some evidence for the degree of connectivity of manifolds belonging to different points. Consider the manifolds associated with the periodic points corresponding to the intersections of n, = 0 and n, = 0 (see Figure 7.3.5).19 Figure 7.3.10 (see color plates) shows the stable and unstable manifolds for increasing values of p (the manifolds of the periodic point at s < 0 are not drawn but they can be inferred by symmetry). For low flow strength, p = 0.1, the manifolds of the central periodic point seem to join smoothly, i.e., if there is homoclinic behavior it occurs below the resolution of the graphics device. However, the homoclinic behavior of the outer fixed point is apparent. At higher values of the flow strength, P=0.3. the homoclinic behavior of the central periodic point becomes apparent while that of the outer periodic point increases in scale. At 11=0.5 there are heteroclinic intersections. More detailed computations Indicate that the transition to global chaos occurs for p x 0.36 when the K A M surfaces separating the chaotic region around each vortex from the rest of the flow are destroyed and the outer manifolds intersect with the central ones forming heteroclinic points (see Figure 7.3.10(d)). 7.3.1 1 shows the obstruction to dispersion when the system Operates at a value of 11 = 0.5, which is close to the bifurcation value (see manifolds in Figure 7.3.10). A blob is placed in two different locations ""d OPcratcd on for the same amount of time. Dispersion is very effective In region near the vortices but only a few particles wander to the

186

Mixing und chaos in two-dimensional time-periodic flows Figure 7.3.9. Effect of regular and chaotic flows on a blob, (a) initial condition, ( h ) blob after mixing with p =0.8 and 30 cycles, and ( c ) blob after mixing with p = I and 24 cycles. (Reproduced with permission from Khakhar, Rising, and Ottino (1986).)

Blinking vortex flow

187

r c g i ~ noccupied by the other vortex. The Poincare section, Figure 7.3.2(J') shows that, eventually, particles will cover all the region.20 However, the process is very slow. Usually, we cannot wait this long.21 ~ l t h o u g h ,during mixing, there is always a well defined boundary separating the interior and exterior of the blob, it is hopeless to try to mmpute the exact location of the boundary even for a modest number of iterations (0(101-lo2)),and also, probably unnecessary. Having the length of the boundary, L(t), allows the calculation of the intermaterial area as tr,(t) x L(t)/area and an average striation thickness, s(t) x l / ~ , ( t ) . ~ In many applications one might be interested in a map of the distribution of striation thicknesses in space, e.g., Example 9.2.1. A rough idea can be obtained in several ways. For example consider a blob as before, composed of many points. Consider also that we place a uniform grid on the flow region grid size 6. Counting the number of points in each pixel (or more conveniently color coding the results) gives an idea about the uniformity of the distribution (this was done by Khakhar, 1986). Note also that if the number of points in the blob tends to infinity and the pixel size tends to zero we should see a lamellar structure.23 Some of the above concepts can be quantified by the following parameter:

where C is the concentration of points in the pixel and the angular brackets represent a volume average. Since I is reminiscent of the intensity of segregation (Danckwerts, 1952) we will use this name. Figure 7.3.12 shows an example of such a computation where the value of the intensity of segregation is plotted as a function of pN. Even though the results of the computations are specific to the grid size used and the initial location of the blob. in most cases studied there is a rapid decrease of the intensity of segregation. I t is possible however, for the particles to be 'demixed' at short times. The curve corresponding to 11 = 0.5, which shows an apparently anonlalous behavior, can be explained in terms of the restrictions imposed the degree of overlapping of man~folds.Recall that in this case during the first stage of mixing the particles are dispersed in the region near the "Orticcs ;111dI decliys quickly. In the second stage, however, the dispersion Of the ~;~l.ticlrs is controlled by transport through the cantori. and 1 slowly. as marc and more particles leak through the gaps of the 1.egio11s.At higher flow strengths (1, > 0.5) the restriction is hardly

F ~ g u r e7.3.1 1 . Obstruction t o dispersion of blob d u e t o poor communication of m a n ~ f o l d sa t 11 = 0.5 after 25 cycles of the flow. Placement a s in ( a )a n d ((,) results in dispersion indicated in ( h ) a n d (d). respectively (see Figure 7.3.10 a n d compare with 7.3.2. (Reproduced with permission from Khakhar. Rising. a n d Ottino (1986).)

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190

Mixing und chuos in two-dimensionul time-periodic flows

noticable. In the same spirit as standard experimental determinations we can define a mixing time as the time required for I to be within 5% of the asymptotic value.

Example 7.3.1 Rather than perturbing the integrable system of Figure 7.3.2(a)by turning vortices on and off in a discontinuous manner, consider the situation of Figure E7.3.1 where the system of two co-rotating vortices is perturbed by a linear flow

where K and/or G are time-periodic. Let us consider a few aspects of this system using the Melnikov technique (Section 6.9). Consider two types of perturbations: ( a ) flow strength (G), and (b) flow type ( K ) .

Figure 7.3.12. Decay of 'intensity of segregation'. (Reproduced with permission from Khakhar, Rising, and Ottino (1986).) 4.0

19 1

Blinking rortex flow I Igurc 7.3.1 3. Schematic rcpre\ent:ition of Gibbs's argument regarding mixing

rc\cr\~biliy(ace Endnote 23).

-4resolutionI-

base flow perturbation

192

Mixing and chuos in two-dimensional time-periodic flows

The streamfunction corresponding to the integrable case is *=-

(ln[(x - 1)'

+ y2] + In[(.u + 1)' + y2])

and the corresponding velocities are:

+ +

+

If we denote (x - I ) 2 y2 = Dl, (x 1 )' y2 = Dl, then DID, = 1 on the homoclinic orbit t,h0 associated with the central hyperbolic point. The evolution o f a point (r, 0 )belonging to the homoclinic orbit isgoverned by drldt = - r sin 20,

dOldt = r2 - cos 20,

which can be integrated to give

0 = tan- '(tanh t),

r = (2/cosh t)'12,

or, in x,y co-ordinates .u = 2''' cosh tlcosh 2t,

y = 2'" sinh tlcosh 2t

These are the parametric equations of the homoclinic orbit *O. In order to set-up the Melnikov function, we need to evaluate f = (f,, f 2 ) on $O[f($O(t)]means the velocity of a point belonging to 4'). In this case this happens to be sinh t(l 2 cosh2 r ) , f l = - 2112 cosh2 2r

+

1. - 2112 cosh t(l - 2 sinh2 t )

. 2 -

cosh2 2t We need now to account for the effect of the perturbation 8, = Gx2, g2 = GK.yI on the homoclinic orbit, i.e., (,flu2- f 2 y l ) evaluated on $O. sinh r cosh t ( l + 2 cosh2 t ) coshqt sinh t cosh t ( l - 2 sinh2 r ) f i g , = Gf,y = 2G cosh" 2r

f 1 q 2= GKf1.y = -2GK

Consider now perturbations of the type G ( t ) = Go(', + hhl(r)) when 11, ( t )= c,, exp(ico,r). In this case the Melnikov's function is MI ( t o ) = -Gob exp(icot,,) x

J+' -

exp(itot) il

2(K + I ) sin 2r cosh3 2t

+ (K

-

I ) sinh 2r

cosh2 2r

1

Mixing in u journul beuring flow

wt,ercLls hz(l) =

193

for a perturbation of the type K ( t ) = K,(u + d h 2 ( t ) ) where

1c,, exp(itu,,t) the Melnikov's function is

~ ~ ( l= , , -) GK,d

exp(itut,)

2 sinh 2t exp(itur) [ w s h 3 2r

sinh 2t +

cosh2 2r

T ~ integrations C are cumbersome but it is possible to show that the system of forming homoclinic points (Franjione, 1987). is problem 7.3.1

Interpret p h y 4 ~ a l l ythe parameter p prohlern 7.3.2

Examine geometrically the flow in the neighborhood of A and B in Figure 7.3.6. Is the flow hyperbolic'?

Problem 7.3.3* Consider that a point u,, y, has not returned to its original location after 1:'. Study n forward and 11 backward iterations if (x, - x,)' + (y,, the fraction of points that d o not return for different precisions and different value4 of i . -

Prohlem 7.3.4*

Study numerically the assumption of random reorientation used in the examples of Section 4.5. Introduce a blob of infinitesimal vectors and compute the orientation distribution function as a function of time. Prohlem 7.3.5*

AS a simple model for the decrease in striation thickness in the BV flow assume that s = .so exp(-a(/i)n) where a ( p ) is given by Figure 7.3.7(u) and % is or the order of 2tr. Assume that the intensity of segregation levels off when s = c 5 l K where iiis the pixel size and K is a number of order 10'-lo2. Calculate the values estimated in this way with the ones obtained in Figure 7.3.12. What is the best fit of K ? Develop a more sophisticated model.

7.4. Mixing in a journal bearing flow In thih section we consider mixing in the flow region shown in Figure

7.4.1'' This case differs from the previous two sections in various respects. 111 this case we h;~vea n iinalytical solution for the streamfunction, "d the system can be reali~edin the laboratory, emphasis will be placed On comp:~rison o f numerical and experimental results. Also, we will Concentr:~teon understanding only a rew operating conditions rather than ;I ~ : ~ r i or e tflows ~ in parameter space. The geometrical parameters Of the \.stclll arc given in Figure 7.4.1 ; there are two gcomctricnl ratios:

194

Mixing untl chuos in two-dimmsionul time-periodic i1ow.s

Kill/Kouland ii/R,,,,,; under creeping flow conditions the streamline portrait is determined by the ratio of angular velocities of the inner and outer cylinders, Ri,l/R,u,.The numerical simulations are based on the solution corresponding to the creeping flow case obtained by Wannier2%nd the streamlines corresponding to various cases of interest are shown in Figure 7.4.2(a) (ti).Thus, according to the operating conditions, the flow might display one or two saddle points,2h and time-periodic operation might give rise to homoclinic and heteroclinic trajectories. In what follows we will focus, exclusively, on the case in Figure 7.4.2(~)."The parameter values are R,,/R,,, = cS/R,,, = 0.3, and Oin/R,,,,= -2. The solution corresponding to the creeping flow problem is given by V4$ = 0. Since the problem is linear, the solution can be written as a linear combination of the forcings of the contributions corresponding to the inner and outer cylinders, i.e.,

:,

$ = in(.^. ~ ) f l i+ , , $",l(.~>~ ) f i < , ~ l . This suggests the adoption of a pseudo-steady state viewpoint for the case in which Rin and R,,, are time-periodic: 4(x9 y , t ) = $i,,(-~, y)Rin(O+ $0u1(.y3 .Y)fiout(O> (7.4.1) i.e. $I(\-,y, t ) is determined by the instantaneous ratio of Rin/ROut. In order Figure 7.4.1. Flow region in journal bearing flow and definition of parameters. f i P denotes the approximate location of the period-l hyperbolic point.

Mixing in u journul bearing jlow

195

;\chieve this condition in the laboratory the acceleration terms in the Nnvicr Stokes equation should be negligible. For a viscous dominated f l , ~ ; ,i t . , a flow with characteristic pressure = O ( p V / L ) ,the dimensionless version of the Navier-Stokes equation is to

where Kc = pVL/p is the Reynolds number, and Sr = o L / V is the Strouhal ( V is characteristic velocity, V = 1 vinl+ vOut1, with V,, = R i n R i n and V,,,,,= RoutRout; L is a characteristic length, say L = R,, - Rout;and 0 ) is a characteristic frequency of the motion of the boundaries). Obviously, if both S r < < 1 and Re << 1 the flow can be regarded as quasi-static, and time enters only as a parameter. From Re<<1 we get the condition ( V ; L ) <
1

Figure 7.4.2.Streamlines corresponding to various cases of interest. The vaues of the parameters are: (a) R,,, = 0 , Ri,/R,,, = f , 6/R,,, = 0.5; ( h ) R i , = 0 , K,,, K ,,,, = j. 6,R,,, =0.5: ( c ) Ri,,Q,,, = - 2 , R,,,R,,, = 6/R,,, =0.3; ( d ) (I,, Q,,,, = 3.0, Ri,/R,,, = i, 6,R0,,=0.5.

4,

Mixing unrl chaos in two-dimensionrrl time-periodic jlows

196

conditions requires that the time scale of the perturbation ( T = I bc much larger than the time-scale of the diffusion of momentum, i.c,, T >> L ' s . ' ~ The computations are based on numerical solutions of dy:dt = - ? $ i ? s , dsldt = ?lC/i?y, where R i n ( t and ) R,,,(t) act as an input to the system via Equation (7.4.1 1. The first observation to make is that the streamfunction is i r n l o p c ~ ~ r t i ~ 01' the uc~ruulspeed of the boundary. This implies that as long as t h e velocity histories d o not overlap, i.e., R i n ( t = ) 0 whenever R,,,(t) # 0 2nd vice wrsu. the results for different histories, R i n ( t )and R,,,(r), will be identical provided that the angular displrrcements and oin= Qi,(t) dt ~,,, = R,,,(t) dt are kept the same. This point is indicated graphically in Figure 7.4.3; all discontinuous histories are equivalent to Figure 7.4.3(h) with suitable R i n= const. and R,,, = const. Thus, for discontinuous histories the displacement Hi, or H,,, becomes an additional parameter (note that the ratio R i n'R,,, determines flin0,,,; in what follows we specify the value of I),,,,,). The second observation is that it is convenient to start operatio11 of the system by moving the first cylinder only one-half of its total angular displacement (see Figure 7.4.3). This results in symmetric Poincare sections and savings of computational time. Figure 7.4.4 (see color plates) shows a comparison of a Poincare section and an experiment produced by placing a blob near the location of the period-l hyperbolic point (see Figure 7.4.1 1. In this case the agreement. after just 10 periods, is reasonably good and the blob invades nearly all the chaotic region with the exception of the regular island. (r))

1

1

%M~: , , ,+ Figure 7.4.3. Angular velocity histories: R,,(t), R,,,,(t). As long as the motion5 of the inner and outer cylinders d o not overlap and the displacements arc kept the same, histories such a s ( a ) and ( h ) are indistinguishable and produce identical results (e.g.. identical Poincare sections), Note also that the inlt~al angular displacement is one-half of the total displacement. This results In symmetric Poincare sections.'"

c0

-

-

h L.

h

i-----T-----i

time

time

However, this kind of agreement is the exception nither than the rule. nit h ally, Poincari. sections do not convey a sense of the structure existent within the chaotic region and give no indication of the mixing rate. Figure 7.4.5 (see color plates) shows a similar comparison to that of Figure 7.4.4. In this case however, after the same number of periods, the blob invades only a small part of the chaotic region. A more careful analysis of the poincare sections reveals part of the reason. Initial conditions, identified by color, 'do not mix' due to poor communication of manifolds of low order hyperbolic points, even for a large number of periods (see Section 7.3). However, color-coding does not solve all the problems and Figure 7.4.6 (see color plates) shows an example where this is indeed the case. The figure shows two Poincare sections, identical in all respects except that in Figure 7.4.6(u) the angular displacement, O,,,,, is 165 , and in Figure 7.4.6(b) it is 166'. A naive interpretation of these results, especially after the discussion on tendrils' and whorls of Section 6.1 1, might suggest that the mixing structure in both systems should be very different since ( h ) should produce whorls within whorls whereas ( t r ) should produce just one large whorl (not taking into account the smallest islands invisible in the photographs). Furthermore, if this were the case the results would also indicate that extreme care should be exercised in experimental work since an error of 0.6% could determine the outcome of the result. However. a deeper analysis reveals that this is not so and these variations are unlikely to be of any consequence in the experiments. Indeed it can be shown that the rate of rotation within the large island is nearly a solid body rotation and 'whorls' are not noticeable even for a large number of iterations. The rate of rotation in the smaller islands is even slower and for all practical Purposes they can be ignored in the stretching of lines placed in this region. In most cases the motion within the regular holes does not produce significant stretching (see experiments on reversibility in Section 7.5). It is apparent that a thorough understanding of this system requires a detailed analysis of rate processes. A first step towards this goal is the location of manifolds associated with the hyperbolic points of various Orders. In order to accomplish this we first need to locate the periodic Points of the system. An important consequence of the symmetry of Poincare sections (Figure 's4.3) is that the search for periodic points is o n r c l i r ~ ~ n ~ r i orather t ~ t ~ l than two-dimensional. Figure 7.4.7 (see color plates) shows the location of Periodic points up to order 8 for the case of a discontinuous velocity along with the corresponding Poincare ~ e c t i o n . ~The ' character Of the points can be determined by examining the eigenvalues of the

linearized flow; the crosses represent hyperbolic points whereas the crl-clcs represent elliptic points. Note that, as expected, the elliptic point5 kill within the islands, and there is a heavy concentration of points near the inner cylinder since the motion in the immediate neighborhood of the cylinders has (nearly) circular streamlines and therefore, points of orders are possible. The rate of spreading of a passive tracer is controlled by the 1111\tnbl~ manifolds of the hyperbolic points and is roughly proportional to the value of the eigenvalues and inversely proportional to the period of the point. Figure 7.4.8(u) shows part of the manifolds associated with perrod-I points after five periods of the flow. Figure 7.4.8(h) shows the manrfolds associated with four period-4 hyperbolic points and sixteen periods. Note that the spreading is significantly less and that the period-l manifolds act as a templurp for the structure of the system. A blob placed in the neighborhood of the point is 'captured' by the low-period manifolds. Indeed, Figure 7.4.5(h)(see color plates) can be regarded as an experimental manifestation of this effect. All the previous results, including those of the examples of Sections 7.2 and 7.4, were for the case of a discontinuous velocity history of the boundaries. However, it might be argued that such a history is hard to achieve under laboratory conditions and that smoother histories might result in different phenomena; that is, the relevant question seems to be: How different are the results if the histories are different'? Figure 7.4.9 pcr. pc~r.iorl. shows several limit cases protluc,itly rltr s~lrnrwtrll displricet~~etzt The main result insofar as Poincare sections are concerned is that the results are remarkably similar and that the most important aspect of the flow prescription is the period rather than its shape40 (see comparison in Figure 7.4.10, see color plates). Similar results hold for other systems such as the blinking vortex of Section 7.3. The previous discussion suggests that most of the stretching, and therefore mixing, occurs in the neighborhood of the unstable manifolds of low period points. In order to investigate the regions of significant stretching we adopt the following procedure: We integrate the equation with F(X, 0) = 1 DFIDt = ( V V ) ~F. for various initial conditions X and compute the stretching of a vector with initial orientation M as i. = (F.M(. Figure 7.4.8. ( t r ) Manifolds c~ssociatedwith the pcriod-l hyperbolic point, thc stretching corresponds t o four periods. ( I ) ) Manifolds associated with four periocl-4 points. thc stretching corresponds t o sixteen pcriods (O,,,, = IXO ).

Mixing in u journul heuritzq flow

199

200

tinlr-prriollic flows Mixing unti chaos in two-din~m.siot~ul

The ti~c~rri(lc) .strctc,hirl(lis computed by averaging the stretchings produced by all orientations. Figure 7.4. I 1 ( t ~ ) , ( h(see ) color plates) show the result, o f such compiltations for a small number of periods. The white region corresponds to large stretching (the cut-off value in ( a ) is 50, in ( b ) 5.0001, The remarkable agreement of Figure 7.4.1 l ( u ) with the computations of manifolds shown in Figure 7.4.8, and the experimental results of Figure 7.4.5, is further evidence that the manifolds of the period- I point pro1 ~ d e a template for the stretching occurring in the flow (similarly, Figure 7.4.1 l ( h ) should be compared wth 7.4.4(a)). It is important to point ollt that for a feu periods, say 5 or 10, it is possible to find initial conditions such that the stretching in the chaotic regions is less than or comparable to, that occurring in the regular regions. The results presented in this section give an idea of the possible comparisons between experiments a n d computations and should not be regarded as a complete analysis of this system.

Problem 7.4.1 Discuss the possibility of using the method of conjugate lines of Sect~on Figure 7 3.9. V a r ~ o u sangular histories producing equal displacements. square ( h )sin2. ( ( ) sawtooth. and (ti) s i n . (a) square

( b ) sin2 I

1 I

----

( d ) 1 sin I

(c) sawtooth

t

t

(11)

7.3 to locate period-] periodic points in the alternating cavity flow and the

journal bearing flow.

problem 7.4.2* study the possibility and consequences of having periodic points with tr& < - 2 in the journal bearing flow (Swanson, 1987). problem 7.4.3 Consider that the angular velocities of the inner and outer cylinder are of the form R,,,(r) where

E

= Rg,,(l

+ i: cos t u t )

is small. Show that the Melnikov function takes the form M(t,) = 4 ) cos c f ~ t ,+ F,(tu. 4 ) sin (or,. &((I),

Compute the functions F, and E, (Swanson. 1987).

7.5. Mixing in cavity flows In this section we present experimental results of mixing in several classes of steady and time-periodic cavity flows. However, a number of general remarks will be made regarding the comparison of analysis, computations, and experiments. The experimental system consists of a computer controlled rectangular region capable of producing a two-dimensional velocity field in the x-y plane. The flow region of the experimental system is rectangular with width W a n d height H (see Figure E4.2.3); the experiments described here were conducted in an improved computer-controlled version of the system described by Chien, Rising, and Ottino (1986) (see Leong, 1990). The system consists of two sets of roller-pairs connected by timing belts driven independently by reversible motors, and two neoprene bands that act as moving walls. The entire system is immersed In a plexiglass tank which has a depth of approximately one foot and is entir,ely filled with glycerine." By means of slots accompanying the rollers and suitable partition blocks made of acrylic, the flow regions (Figure 'e5.1) can be adjusted in size to a maximum area of 5 inches by 5.5 inches. (For details the reader is referred to Chien, 1986. and Leong. 1990.) The tracer consists of a mixture of glycerine and a fluorescent dye and is 2-5 mm below the free surface of the fluid by means of a syringe (the diffusion coefficient of the tracer in glycerine is of the order of cm2 s ) . The Reynolds number used in the experiments is the highest compatible with two competing effects: creeping flow and two-dimensionality (i.e., absence of inertia and secondary flows)and minimum dye diffusion during

the time of the experiment. An order of magnitude calculation based on diffusional effects gives a Reynolds number of order one."' The cavity flow apparatus is used to study three classes of steady-state flows and two classes of time-periodic flows. The steady flows are the following: ( i ) the standard cavity flow (only one wall moving); ( i i ) a Ilow in which the walls are moved in the same direction; and ( i i i ) a flow in which the walls are movcd in opposite directions. The velocity of thc lop wall is denoted as I.,,,, and the velocity of the lower wall is denotcd as I~,,,,.Figures 7.5.2(h) (11) show the result of mixing a line of tracer pl:rced vertically in flows (i) (iii) for equal amounts of time. In this case. thc (lows are steady and thereforc, integrable. I t is apparent that the line is trapped by the streamlines and, thereforc, the mixing is poor. Indeed, this is a good technique for generating pictures of the streamlines of the steady flows as shown in Figure 7.5.3(11) (d). The two classes of periodic flows are: (iv) discontinuous operation of the boundaries in a co-rotating sense, i.e., jumping between Figures 7.5.3(11) (c.) with I,,,,, = - I * ~ , ,=, U; the top and bottom walls move for a time (1/2)T "; :and finally, (v). a periodic flow corresponding to wall motions of the form (7.5.l a ) I.,,,, = U,,,, s i n 2 ( ~ t l T , ,+, 2 ) L'h,,l= - Uh,,ls ~ ~ ~ ( ~ ~ / ~ ~ , (7.5. , ~ 1 )b) with equal amplitudes, U,,,, = U,,,, = U , periods TI,,, = T,,,, = T. and a phase angle, r = ~ r / 2 . Thus " ~ the system evolves smoothly in the direction

1.-igurc 7.5.1. f l o w region. The dimensions of thc cavity are W = 10.3 cm ancl I 1 = 6.2 cm.

n

n

glycerine

ters

plexiglass tank

alum

Figum 7.5.2. Stretching of a line placed vertically. a s shown in b).in v a r i o ~ steady flows ( b x d ) .The fluid is glyarim and the t r a a r is a fIUOXuamt dye. The expcrimental.conditions arc the following: Re = 1.0 and the vetocitk~of the top and bottom walls are ,o., =, o = 1.58 cm/s. In (b) both walls move in the same direction, in (c) both walls move in opposite directions, and (d) only the top wall moves. The total time of the experiment is 5 min.

204

M i x i n g und chaos in two-dimmsionul time-periodic flows Figure 7.5.3. Instantaneous picture of the streamlines corresponding to the prescription (7.5.I(a.b)). The experimental conditions arc: (11) r,,,, = U and I.,,,, = 0. ( h ) r,,,, = -r, ,,,. ( c ) r,,,, = 0 and I.,,,, = - U . and (11) - 1 . ,,,, = r, ,,,. The photographs correspond to the case U = 2.69 cm/s. The dimensions of the cavity. constant in all the experiments. arc W = 10.3 cm and tI = 6.2 cm. The Reynolds number is 1.7.

Mixing in cavity jlows

205

1,'igurcs7.5.3(t1), ( h ) , ((.I, (ti) and returns to (u) at the end of the period. T as the governing parameter. ~'igurcs7.5.4 show the result of placing a blob as initial condition when s?stcm operates in a time-periodic mode. It is clear that the periodic ,pcr.;~tionimproves the mixing (otherwise the blob would remain trapped

wc ,xgard the period

1,'igurc 7.5.4. I!volution of u blob of initial conditions originallq plucctl at th, position \- -: 2.2 cm. !. = 3.1 cm (diitnictcr = 0.5 cm). Visualization is providcc. by a fluorcsccnt dye dissoltcd in glycerine. excited by long-wuvc ~~ltritviole light of 365 nm. The top and hottoni walls move according to the prcscriptior of equation (7.5.1) with C ' = 2.60 cni s (Kc, = 1.7 and 0.05 < St. < 0.091. Th. t insti~ntwhen the top wall and hottoni wit11 arc moving pictl~rcs;ire ti~kcni ~ the at m a s i m ~ ~and n i minimum (tcrol spccd. rcspcctivcly. The pictures corrcspont to time period5 7': (111 15 s (a second blob. near the ccntcr of the cavity wa:. added in this cube). (1)) 20 a. ( 1 . ) 25 s. and ( 1 1 ) 30 s: and to wall displuccnicn-(the distance travelled by the walls in one period). ( a ) 1612cm. ( h ) 752c. I',) 951 cm. and ( t i ) 1452 cni. Note the symmetry in Figure (t0.

b! ,,,,~ll'ininpstrcanilincs). I t appears that. as compared with the journal hcal-lllg 110%'. the cavity flow !nixes better and fewer periods arc necessary to ;rchicvc significant mixing. However, this system is a bit more susceptible to experimental error due to corners and the flexibility of the belts. Another significant difference is that there are no fixed hyperbolic points

208

Mixing tlntl chtros in two-rlit71et1,siontrItimr-periodic t1ow.s

in the velocity field and that we cannot properly speak of small perturbations with respect to steady flow. An important difference with the previous sections is that in this case there is no analytical solution for the velocity field. However, computations based on a discretization of the velocity field provide only a crude picture of the flow (e.g., exact location of periodic points, manifolds) and we are therefore left with experiments as the main investigative tool. Therefore, the experimental observations are limited to macroscopic structures, such as islands, produced by mixing, and the question is how d o these structures behave under change of the governing parameter (in this case T ) . Also, any phenomena which are relatively long term (asymptotic)are not directly observable, unless they form structures which are more o r less clearly delineated from the start. As we have seen in the previous section, the objects which can be detected are those with macroscopic spatial extent and low periods. The comparison between analysis, computations, and actual experiments is not trivial and is often misunderstood. In theory there are always three ct~l a cotnputer sitnultrted system, and an systems; a t ~ ~ t r t h r n ~ t i t i system, e.uperinlc~nta1system. However, it is naive to expect that all the information extracted from experiments is easily obtainable from computations, and ric8r w r s a . For example, asymptotic phenomena and sets with measure zero are mathematical objects, and clearly a set with measure zero is not observable by means of computations. The requirements for computational observability are also different from those of experimental observability. In general an object will be observable in computer experiments if it is robust, i.e., relatively insensitive to round-off error, and if it occurs with high probability, i.e., if its presence can be ascertained regardless of the discretization choice (pixel size in the computer screen). O n the other hand, an experimental system is fundamentally analog so that non-robust, low probability events can be seen, though perhaps not easily repeated.35 . 2 Consider now a few experiments comparing the discontinuous and sin histories (Equations 7.5.1 (a,b)). The comparisons reinforce the need for careful experimental interpretation and point out some of the pitfalls of improper and incomplete comparison. Figure 7.5.4 shows the state of the system operating for various values of T under the prescription of Equation 7.5.I(a,b) (i.e. sin". The initial condition in all the experiments is a small blob of t r a ~ e r . "In~ this case the system evolves from no holes to a rather large hole for the highest value of T. Note that the large scale structures of all cases are similar. By analogy with the previous section we might conjecture that this is bec:iuse the initial stretching and folding is

cj(,mill;~tcdby low period hyperbolic points a n d their manifolds. T h e additional folds which occur a s t h e n u m b e r of periods is increased a r e indcc.d contained within the already striated region a n d very quickly the l I I I ~ I c Lfolding ~ serves as a template for further stretching, a n d the structure ,,I,~ ~ shown ~,lc in the photographs is very similar t o the o n e observed for l(,llg (illlcs. T h e holes ( a n d also the large folds) form 'coherent regions' ,\lllch c;Ln be followed in space a n d time: they a r e translated a n d deformed b) tllc Ilow but conserve identity a n d d o not disappear u p o n further ,,1lk11lg.In terms of qualitative appearance. there seems t o be a n o p t i m u m mi\.lllg at T = 20 s. Beyond that, a large regular region grows with illcre;~s~ng 7: r h c mixing structures produced by the discontinuous protocol are s u b ~ t : ~ ~ i t i adifferent lly (Figure 7.5.5). A large hole is present at the lowest c.aluc of T used in the experiments a n d the results of experiments placing blobs inside a n d outside the island offer a nice demonstration of the speed of mixing within the islands a n d in chaotic regions. In Figure 7.5.5(u) the blob was placed inside the island; in ( h ) the blob was placed outside the island: it is clear that the interior of the island does not communicate uith the rest of the flow. An increase in the value of T decreases the size of thc island and the best mixing is obtained a t approximately T = 58 s . It thus appears that the t w o prescriptions, the discontinuous protocol a n d the sin' protocol, give opposite results, a n d that the conclusions of the prcLious section, robustness of Poincare sections (Figure 7.4.10) a n d s o on. cannot possibly hold, since in o n e case we get hole-opening with increasing T whereas in the other we get hole-closing with increasing T. Houever. this is only partially correct. Indeed if the pictures a r e compared at cjc~rrtrltlispltrc~rmentsper period, there is a region (values of displacements In both flows) which produce similar results. Figure 7.5.6 shows a comparison a t equal displacements for both prescriptions. G o o d agreement. in terms of similarity of large scale structures, is seen, especially a t Ion \;dues o f T. However. both systems behave quite differently a t large c.alues o f T. T h e most important consequences of the coherence present in these P1ctllrcs is that careful observations (video recording) allow the detection of pcrlodic points. a n d that the evolution of the system through a series of changes in the parameter T gives information regarding the bifurcations Occurring in the system. Figure 7.5.7 (see color plates) shows a n example ofabifurcation which would be hard t o capture in computer simulations. It " l ~ \ \ . s the behavior of the island boundary at the time of island collapse. result is a consequence of the b r e a k u p of the last K A M tori (rotation

210

Mixing trncl c-htios in two-ilirnm.siontrI time-prriotlic ,flows Figure 7.5.5. Similar to I.'igure 7.5.4 ( K r = 1.2. 0.07 < Sr < 0.1 l ) except that now the motion of the upper itnd lower walls is discontinuous: the top wall moves from lcft to right for half a period and then stops. then the bottom wall moves from right to lcft for half a period and then stops. and so on. There is a five sccond pause between each half period to rcducc inertial cffects. The pictures are taken at the end of the desired period. The pcriods T arc: ( 1 1 ) 34 s. ( h ) 34 s. ( 1 . ) 48 s. and ( t l ) 58 s: the corresponding displacements arc: ( ( I ) 1290 em. ( h ) 650 cm. ( 1 . ) 760 cm. and ( t l ) 1 100 em. Note that the structure shown in ( u ) fits within the 'holes' left in ( h ) and that relatively little stretching is observed in the interior of the island.

Mixing in cuoity jlows

21 1

of the island equal to o r multiple of golden mean). This type of smc,oth visualization is consistent with what is mathematically predicted, but i t is something which might be missed by computer simulations. Figure 7.5.8 (see color plates) shows part of the structure of periodic points corresponding to the system of Figure 7.5.4(d) and part of the biful-cntions that took place up to this point. The entire system might be v i s u u l i ~ ~asda sort of planetary system with the planets (hyperbolic points)

212

Mixing and chaos in two-dimensional time-periodic flows Figure 7.5.6. Comparison of chaotic mixing between the discontinuous (Figure 7.5.5) and the sin2 (Figure 7.5.4) prescriptions, at equal wall displacement per period. The initial condition is a blob of fluorescent dye located at x = 2.2 cm, y = 3.1 cm. The conditions for the sin2 flow are the same as Figure 7.5.4 with periods T: (hl) 20 s, (b2) 35 s, (b3)40 s; and corresponding wall displacements: (hl) 752 cm, (h2) 1129 cm, and (b3) 1075 cm. For comparison with the sin2 flow, the discontinuous flow is started in the following way: the top wall moves for (1/4)Tand the bottom wall moves for (1/2)T, and then the top wall moves for (1/2)T, and so on. The other conditions are the same as in Figure 7.5.5 with T of: ( a l ) 29.2 s, (a2) 51.2 s, and (a3) 59.2 s; and wall displacements: ( a l ) 555 cm, (a2) 778 cm, and (a3) 1125 cm. In general, the results show some similarity of macroscopic structures at low T (up to T = 25 s). Compare ( a l ) and (hl).

(a 3)

discontinuous

Mixing in cavity flows

213

and their moons (elliptic points) returning to their initial locations after the periods characterizing the points. wc have seen that the Smale horseshoe map involves the stretching and folding of a blob onto itself. In the context of mixing in periodic two-dimensional flows such a map has a clear physical significance: A fine subdivision of the blob in the region initially occupied by the blob. Obviously i t is desirable that the flow forms many horseshoes, possibly interacting, in such a way that they influence a large region of the flow. From the point of view of mixing it is desirable that they be of low (number of transformations needed to produce the horseshoe) since we want to achieve mixing as quickly as possible, and also since the domain of a high period horseshoe is usually small. The investigation of horseshoes involves the superposition of forward and backward transformations with the initial location of the blob. The construction is similar to thc one given in Sections 7.2 and 7.3. There are several properties which must be verified in order to deduce the presence of a horseshoe (Moser, 1973). All the conditions must be carefully examined. For example, if the placement of the blob is not exact (and it actually never is) the striations will be partially filled with the tracer. In such a case the system might appear to violate some of the conditions (Chien, Rising, and Ottino, 1986). Chaos magnifies errors and one might wonder to what extent it is possible to unscramble pictures such as those of Figures 7.5.4 and 7.5.5. We anticipate that the configurations in Figures 7.5.2(~1)-(c), which are integrable, can be reversed within experimental error, and there have been experiments showing that this is indeed possible in creeping flows (see Bibliography, Chapter 2). However, we have already seen that this might be impossible, even numerically, if the flow is chaotic (Section 7.3.4). Figure 7.5.9 (see color plates) shows what happens to initial conditions placed in the interior of an island and in a chaotic region (see Figure 7.S.4(tl)).Figure 7.5.9(h)shows the result of forward mappings and Figure 7.5.9((')shows the results of backward mappings. I t is apparent that the line pl;lced in the hole undergoes relatively little stretching" and returns its initial location. However, the line placed in the chaotic region ( ~ ' ~ ~ u m a bnear l y a horseshoe) does not return to its initial location; every 'Irne the line passes by the horseshoe it is stretched exponentially and the experimental errors are magnified. Apparently, two periods are enough to significantly the errors of this particular experimental system. . w e have emphasized that the degree of smoothness achieved in these 'lctures would be hard to achieve in straightforward numeric,I' 1 simulations bee Examples and 9.2.4). Consider now the experimental determination

2 14

Mixing und cthrios in two-rlimensionul time-periodic) ,flows

of a quantity that presents problems from a numerical viewpoint: the estimation of the interfacial area generation (see Example 9.2.3). At the same time, we will consider a feature of the experiments that we have avoided mentioning so far. Even though the flow is two-dimensional, since the camera is not a t infinity, the area of the initial blob appears to Increase in the photographs until it occupies the entire chaotic region.38 Figure 7.5.10 shows the measurements of area and perimeter growth utilizing image analysis for three different cases: a steady flow with one moving wall, and the two time-periodic flows. Both time-periodic flows produce exponential growth of the area, A , and the perimeter. P. Both A and P follow approximately the same growth law, exp(Pt), with similar values of 1: for the discontinuous flow /I%0.022 s - ' , for the sin2 flow /j -, 0.01 9 s I. O n the other hand, in steady flow the growth of both A and P is linear in time. It is then clear that the actual fluid system provides a n alternative visual tool, less controllable perhaps than a mapping, but with the advantage that much of the behavior of the system can be directly observed, particularly in the regions of hyperbolic sets. T h e smoothness and regularity shown in the experiments a r e hard t o achieve in computer simulations where finite pixel size (resolution)a n d round-off errors govern the process a n d the regions often degenerate into pixel clouds by which the underlying structure is o b s c ~ r e d . ~ ' Pvohlem 7.5.1 Using order of magnitude arguments obtain the optimal Rr for the above experiments. Consider the lines stretch exponentially a n d use a n efficiency of order 1 0 - ' 10-" Assume that the striations are unrecognizable if the diffusion distances are of the same order of magnitude a s the striation thickness itself. Pvohlem 7.5.2 Justify the exponential growth of dye area in the experiments. Pvohlem 7.5.3 Estimate the average mixing efficiency in the cavity flow Problem 7.5.4* Rationalize the similarity of the results obtained in 7.5.6 a 1 and bl.40

Bibliography The first two sections are tin abbreviated version of the presentation given in Khakhar, Rising. and O t t i n o (1986). T h e or~ginillpaper treating the

215

Bihliogruphy I I ~ ( I I . C7.5.10. Measurement of area growth and perimeter growth by image ;Inalysis The resolution of the image is 512 x 512 pixels and 256 grey levels. .l.lls nlcasurcments are for three classes of flows: steady, with one wall moving; (ll\contl~luous(Figure 7.5.5(d)):and sin2 (Figure 7 . 5 . 4 ( ~ time-periodic )) flows. ~{,,rll tlmc-periodic flows produce an exponential growth (exp(Pt))in area. A , ,Illtl pcrinictcr. P . with approximately the same exponent P. For the discont l l l , l ~ ) flow ~ ~ ~ fl 20.022 s - I . whereas for the sin2 flow P % 0 . 0 1 9 s ' . O n the ,,thcr hand. in the steady state flow the growth is linear in time.

sin2 flow

0-

-8 m

I

I

100

+steady

200 time (s)

1

flow

2 16

Mixing untl chuos in two-rlimen.sionu1 time-periodic ,flows

blinking vortex system is Aref (1984): the first treatment of the tendrilwhorl map is given in the paper by Khakhar, Rising, and Ottino (1986). Preliminary results for both systems were given in various presentations: The horseshoe construction of the tendril whorl system was given in Rising and Ottino (1985, abstract only): several results regarding efficiency of stretching were given in Khakhar and Ottino (1985, abstract only). The material of Section 7.4, journal bearing flow, is based on the presentation of Swanson and Ottino (1985, abstract only). A substantially morecomplete presentation isgiveninSwansonandOttino( 1990).Thefirst study of the journal bearing flow is the computational study of Aref and Balachandar ( 1 986). Several nice experimental results are presented by Chaiken cJt (11. (1986), however, most studies involve a very large number of periods (order 10' and higher). Section 7.5, cavity flow, is based on experimental results by Leong (Ph.D. thesis, University of Massachusetts, Amherst); the original paper is by Chien, Rising, and Ottino (1986). The relationship between analysis, computations, and experiments is discussed by Ottino rt 01. (1988) in terms of the journal bearing flow and the cavity flow. A related topic is dispersion in Rayleigh Benard flows: even though there is a number of recent papers in this area, the treatments of mixing are rather scarce (e.g., Solomon and Gollub, 1988). A paper addressing mixing in the context of geophysics is given in Hoffman and McKenrie (1985). Notes

I Note that d u e to widespread usage the word 'flow' IS used in two d~ffcrcntways throughout thls work: the first one in the mathcm;~ticnlsense consldcrcd in Chapter 6, the second in the accepted fluid mechanical way. as for example, in the terms 'flow field' o r 'fluid flow'. 2 F o r example. fractal d i m e n s ~ o n sin the chaotic regions. c o r r c l a t ~ o nfunct~ons.etc 3 These terms were defined by Berry 1.1 (11. (1979) In the context of quantum maps. In practice. whorls arc hard t o obtain in most chaotic flows (see Section 7.4) 4 In a saddle-node hifurcation for area preserving systems. as the parameter of the system increases the flow goes from n o fixcd points to two fixed points. one hyperbolic and one elliptic. 5 This kind of limitation is common to moht of the computationi~lstildics described here. 6 Notice that there is efficient mixing in a rcglon where the flow is i~pproximatelylineaf due to the ~ntcractionw ~ t hthe ol~tsldcflow. T h ~ should s be useful in understanding mixing at the nl~croscnlcof turhulcncc. ~ h c r cthe f l o ~ arc. s at some scale. linear and Ianiini~r. 7 Statements like this must alwi~ysbe qualified by saylllg ' w ~ t h i nthe rcsolut~onof the grnphlca o l ~ t p l ~ctc.'. t. distribution of pcrlods, et' X A g a ~ n .scvcl-al possible gcllcrali,;~t~onaarc posbihlc such 0 However. a n important diffcrcncc is that. near the origln. the cakltj flow hchavior 1s I,,, : r wlicrcas it1 the hlinking vortex I.,, :I r .

A, \\c have seen In Section 4.6 this 1s a n indication that this flow might havc a n

operating period for which the efficiency 1s maximized. I I ,lllolllcr diffcrcncc is that D: D diverges at the center of the vortices and the integral ,,('1). I) i\ ilnboundcd. By contrast. the energy d~ssipationin the T W mapping is hchi~vcd.Nevertheless. the efficiency in the BV 1s well behaved. 13 A ,1111ilarconstruction can he used t o find the periodic points in the T W map. and l l l , ~ ~generally. -c of generalized bllnking flows (Rising and Ottino. 1985). I 3 .rile ss:itlcr m ~ g h gtun t a n appreciation of the complexi:y ofchaotic systems by considering rl1.11 ;I \iniilar t~nalysisapplies t o period-11 points, where 11 can be a very large number! 14 .l,lll\ idea was used in Rising and O t t ~ n o(1985). 15 .rile ~ - c ; ~ dshould er notice that in this case. since we d o havc a n expllcit mapplng. r l l ~calculat~onis free of numerical errors which might arise d u e to renormalization ,,('r~-:~~cctories (see Lichtcnberg a n d Lieberman. 1983%p. 2x0). 16 Tllcrc arc still some questions regarding the computation of long tlme averages d u e 10 rhc finite precision nature of the computations. However, when the motion is ~ \ L I ~ I ' I C I C I I chaotlc' II~ and numerical prec~sionhigh (see Figure 7.3.10) the time averages ;ll.c c\lxctcd t o be reliable. 17 N ~ r cthat the accurate computation of length is not trivial: we can label points at~tli l ~ ~ r o d u cadditional e ones if they become too separated and join nearby points. s:~! I and i + I . by short segments to approximate the length. However. a s the number of cvclcs increases we need to a d d polnts at a n exponential rate (see Example 9.2.3) I X Thc.\c numbers are only approximate. Regular reglons a n d chaotic regions are intricately c o ~ i ~ ~ e c tlcf d; ~.point gets 'trapped'in a regular region it remains there for a longtime. I9 Althoi~phthe polnts arc elliptic at the point of formation they are hyperbolic for ~ h craligc of 11-value5 considered here 20 This is one of the reasons why 11 is convenient t o construct Poincare sections using part~clcslabelled with a color representing the initial condition. 71 Thl\ suggests the need for transport studies in Hamiltonian systems. possibly using rcnorlnitlizntion methods. a s indicated earlier. o r in terms of thedynamics of manifolds Iscc I.1g. S.X.3.). 22 111 rhc nilmcrical experiments of Section 7.2.6 we can consider the line separating two difkrent matcrlnls. Then if A is the area of the globally chaotic region and L(O) the in~tiallength of the line. we can write A = L(O)s,, = L(/).\(/).where .\ is a distance of the order of magnitude between the vortices. Thus. s ( t ) 2 I,(/) '. 23 It i h in~crcstingt o point out that J. W. Gibbs used a fluid mixing analogy, a regular flo~ :lclually. to explain the origin of irreversibility in statistical mechanics (Gibbs. 1948). His argument was re-interpreted by Welander (1955) and is closely related to the content of the previous sections. Although the argument is not entirely rigorous it brings u p s o m e important issues a n d it is convenient t o repeat it here. ('("l\idcr home two-dimensional region with area H where we have placed a blob "f 11011-diffusive tracer with density 111=d,,l:drr). We consider that p(x. 1 ) = I if bclOll@ t o the tracer region and p ( x . I ) = 0 anywhere else. The total mass of the trilccr i l l K and the integral of the square density are both equal. i.c.. O 1 , ~ ~ ~ l ; ~ l

L i . d i 1 - M.

1

/I2

dil = M

"here .A1 is less than H. Given these choices. no matter how erective the mixing. both thcsc quulitities remain equal ( t h e particles of the tracer always carry the same 1 . I 1) I). However. what happens if the resolution of the integration is tlii111 the striation thickness:' (see Figure 7.3.13). At this scale the density is (il\ ,TI,, (5.4. nlld the square density. = (i5ttt;ciA)'. Therefore the integral of

-

2 18

Mixing trnd chaos in two-tlimm.siont11 tinlr-periodic ,flows

but since iinl,'iiA cannot be equal t o one everywhere (since hf < K ) we conclude that r

In other words. the density distribution becomes more honiogcncoi~sd u e t o poor resolution. In this context mixing con be viewed a s the minimization of ( 1 1 ) ~In the domain K . 24 This system has been experimentally built by C'haiken rr i l l . (1086) a n d Swnnson a n d Leong, Ph.D. theses (in progress). University of Massachusetts. Amherst, 1986 25 Wannier (1950). O t h e r solutions were presented by Jeffery (1922) a n d Duffing ( 1924). Wannier's method is based o n a complex v:~riobletechnique and gives possibly, l Rlvlln (1977): the most man:~goblc solution. A solution also is presented by B a l l ~ ~and this article considers also :I first-order inertial perturbation and indicates that the streamlines are only distorted even for relatively large Reynolds numbers (0150)). A related article focusing o n the some problem is Kazakin and Rivlin (1978) 26 Note that in creeping flow, the streamline portrait is independent ofthc actual direction of rotation ofthecylinders (thus, ifthe b o u n d o r i e s ~ ~moved re in the oppositedirectlon, all we need t o d o is reverse the direction of all the arrows). Note also that a s long a s the flow is creeping. only the ratio R,,,!R,,,,, is important. not the actual value of the velocities. 27 T h e experiments reported here have been obtained in n computer controlled apparatus (Swanson. 1988). The most important design feature of the apparatus is that it allowsforan unobstructed view of the flow region since the outer cylinder is rotated from the 'outside' with 21 large bearing enclosing the outer cylinder. This is particularly important in the case under study since the period-l hyperbolic point would not be visible in conventional designs (see Figure 7.4.1). Observations are made from the bottom of the apparatus. which is made of glass. The working fluid is glycerine floated o n carbon tetrachloride. 2X This condition can be easily s~~tisfied in laboratory experiments. see also comments in Section 7.5. 29 Note however. that we can not prove that t i l l the points u p to this order have been found; for example, it is possible that there might be more period-X points than those shown in the figure. 30 T h e closest experimental studies are those of Ryu. Chang. a n d Lee 11986). They focused exclusively o n steady flows. O t h e r related studies were conducted by Pan and Acrivos (1967); a n d Bigg a n d Middleman (1974). Bigg a n d Middleman studied laminar mixingoftwo fluids with different viscosities in the standard cavity flow configuration, i t . , with only o n e wall moving. Pan a n d Acrivos focused on tall cavities. f l l W > I . a n d streamline visualization for high Reynolds number flows. Some of the results presented in this section. a n d n comparison with the system of Section 7.4. can be found in Ottino er crl. (1988). 31 As a working fluid. glycerine presents some difliculties; its viscosity is not a s high as it should be t o accomplish comfortably the condition Rc~ccI. However, by using a transparent fluid we can check o n the two-dimensionality of the flow. 32 Both thc Reynolds number ( R c ) and the Strouhnl number (Sr). which are a measure of the inertial efl'ects in the flow. ore kept low enough in the experiments as t o be of no importance (in the computation of Kr ;uid S r the characteristic veloc~tyis then a s 0 and thc chi~rncteristiclength a s ti' CV).

,: i.l l

l]lc c\pcrin~cntsi t takes some time for the motion to set in. (I([{' 1.) = 5 x I ( ) 2 ,,~tlcl.to n?irlinii/c trarl\ierlt effects uc uait for approximately 5 s betucen the 111,,11,~~1 or tllc upper and lohcr bands. ;J ( ) I , , I ~ I L I \ ~ >thcrc . arc infinitely many other po\sibilitics for the motion of the ualls. l O l ~\,iinplc.c\erl within the restricted (:lass of motions. Figures (7.5.1) and (7.5.2). .,ill \ ; l r the relative amplitudes and direction of motion of the boundaries. the ,,ll;,,c ;lrlgle. time periods.etc.: only a few of these possibilities h a i c been explored to date. ~poirltsarc discussed in Ottino r f (11. (1988). Note that there is a factor of 35 .I I,,,, error in the periods T reported in this paper. 7 h h,,rc ll1'1t if the flow were not two-dimensional thc \ellmenth might cross in the lp~l,!~op ~ph. 7 7 ssc,olllnlcnts in the pre\ioussection: i t is Indeed \cry hard to a c h i e ~ e situation a leading I,, I ~ I Cfo~.rnationof iisiblc uhorls in just a few periods. 38 -1 , , l l i ~ i l ; ~indicate\ ~ ~ ~ ) n that for our system the camera & i l l be unable to detect striations ,,llcl1 lilt> arc closer than 10 jlm. Alw. glowing effects and the (small) molecular i l l l j , ~ \ ~ prcsent c~~l durln? the time of the experiments contribute to the area increase. I ~ ;II I I I \ I order approximation. these effects are proportional to the interfacial area bcl\iccn IIIC trilccr and the clear fluid. 3') , ticiallcd ,tudy of breakup of islands in terms of area preserving maps. such a\ those presented by Ottino elul. (1988), is given In the Ph.D. thesis by Rising (1989). 40 Boll1 flc)ws have the same symmetries provided that they are examined at suitable time,. For our purposes a map M with inverse M - ' is said to be symmetric if there exists a map S.with S? = 1. such that M and its inverse are related by M = SM-IS. Thc \ct of points {x} such that {x) = M{x) is called thefixed lineof M. The symmetries of time-periodic Hows can be easily deduced using rules from map algebra. Consider for cxi~mplethe discontinuous history of Figure 7.4.9(a). Let the motion induced by mo\ing thc top wall during a time I be denoted TI (i.e.. a set of particles {x) is mapped to T,{x) at time 1). Similarly. denote by B, the motion induced in the cavlty by moving the bottom wall. Sincc the How 15 a stokes flow and the streamlines are symmetric with respect to the y-axis it follows that TI = S, TI-IS, and B,. = S, B,.-'S,, where S, denote\ the map (x, y) -(-x, y). In this case the fixed line of S , IS the y-axis ~tsell.Furthermore ~f 1 = 1' then the top and bottom motions are related by TI = S,B,-'S, and TI = R B, R , where R = S,.S, = S,S,, denotes at 180" rotation (x. y ) (-x. - y ) . The symmetries of composition of Hows can be deduced by using these rulcs For example. the How Fz, BIT,: 1.e.. a series of top and bottom wall motion\, has symmetry with respect to the x-axis. F,, = S, F ~ , ' s whereas , the flow G,,= H,, TI B,,; i.e.. a series of top and bottom wall motlons. but where observations are made half-way through the bottom wall displacement. has y-symmetry instead. Thi\ means that the Islands ~nthe flow are located in p a r s across the y-axis or on the axis ~tsclf Note that a How can have more than one symmetry. For example. the F,, flow can be written also as F,, = S * Fz; 'S* where the symmetry S * 1s given S * S,T, = T l - l S , .The fixed line of S* 1s not obvious; ~tcorresponds to the mapping of the xis wlth T , , ' ; that is. the inverse of the top wall flow wlth half the displacement. In t h ~ case s the symmetry line is a curve (these ideas are developed In the ;~rtlclcsby J . G Franjlone. C W . Leong. and J . M . O t t ~ n o'Symmetries . with~n chaos' ;I route to effcctlvc mixing'. Phys. Fluids A . 1. 1772-83. 1989 and C. W. Leon?. and J . M Ottino. 'Experiments on mixing due to chaotic advectlon in a cavity'. ,' F l l i l ( j Ihlech.. 209. 46.V99. 1989).

., -

I!,

,,,

-+

-

Mixing and chaos in three-dimensional and open flows

In this chapter we study mlxing in seheral three-dimensional fluid flou, a n d open flows with varying degrees of complexity. W e stress qualitati\c differences with respect t o the time-periodic two-dimensional case a~irl focus o n the topology of the flows a n d in their mixing ability.

8.1. Introduction S o far we have studied mixing in two-dimensional time-perlodic flows a n d have seen that considerable theoretical guidance exists for this case. \2'ith the exception of the tendril-whorl flow. in which fluid entered a n d left through conduits. the systems were closed. W e now move t o the casc 01' three-dimensional a n d o p e n flows. Seheral cases a r e discussed. T h e chaptcr starts by analyzing mixing under creeping flow conditions in two continuous mixing systems. In the first example the partitioned-pipe mixer the helocity field is periodic in space: in the second example the eccentric helical annular mixer - the velocity field is time-periodic. These two flc)\+s a r e analyzed in some detail since they have many potential applications. As we shall see, intuition based o n the two-dimensional case can bc somewhat misleading in the understanding of three-dimensional flous in general. T h e next example, discussed in less detail. includes inertial effcith a n d corresponds t o mixing in a channel with sinusoidal walls. 1.11~. treatment of this case, largely based o n the work by Sobey (1985). is computational in nature a n d relatihely little analytical work is possible. T h e next examples a r e of relevance t o turbulent flou,s. T h e first one of this class is a perturbation o f the Kelhin's cat eyes flow, which is important. as a first approximation. t o mixing in shear layers. T h e next exampic considers the class of admissible flows obtained by constructing solutions of the Navier Stokes equations by expanding the velocity field in a T q l o r series expansion near solid walls. This example is of relevance to p e r t ~ ~ r b c d flows near walls and has obvious r e l e ~ a n c eto turbulence. Finally, the la.;[ e x i ~ m p l e originally . d u e t o H e n o n (1966). is the analysis of mixing in a n

inviscid fluid by Dombre rr (11. (1986)and our presentation is largely based on their analysis.

8.2. Mixing in the partitioned-pipe mixer The partitioned-pipe mixer consists of a pipe partitioned into a sequence of semi-circular ducts by means of rectangular plates placed orthogonally to each other (Figure 8.2.1).The fluid is forced through the pipe by means ofan axial pressure gradient while the pipe is rotated about its axis relative to the assembly of plates, thus resulting in a cross-sectional flow in the ( r , 0) plane in each semi-circular element. We study the mechanical mixing of a Newtonian fluid in the mixer when it operates under creeping flow conditions. The expectation that the flow in the partitioned-pipe mixer is chaotic is based on its similarity to the Kenics" static mixer which, as was mentioned in Chapter 5 (see Endnote 24), resembles the baker's transform in terms of its cross-sectional mixing (Middleman, 1977). Under ideal conditions, each stream is divided into two in each element (see Figure 8.2.2). The main difference between the two mixers is that the sense of rotation of the cross-sectional flow is the same in adjacent elements of the partitioned-pipe mixer while it is opposite in the static mixer. 8.2.1. Approximate velocity field We consider an approximate solution to the fully developed Stokes flow of a Newtonian fluid in the semi-circular compartment of an element of the mixer under creeping flow. In this case the axial and cross-sectional components of the velocity field are independent of each other and can be considered separately. The axial flow is simply a pressure driven flow in a semi-circular duct and in cylindrical coordinates is given by

where R is the radius of the pipe, and the average axial velocity (u,)is 8 - n2 C7p R2 4nZp ?z where p is the viscosity, and p the pressure.' The cross-sectional component is a two-dimensional flow in a semi-circular cavity, and is given as a solution of (U,)

where

=

-

v4* = 0

Figure 8.2.1. Schematic view of the partitioned-pipe mixer; the length of the plate is L.

Figure 8.2.2. Kenics" static mixer, idealized view of the system. Two streams, A and B. enter segregated. After the first twist (clockwise looking from the

entrance) each stream is halved and joined by half of the other stream in the second twist (counterclockwise); the process is repeated periodically. The actual system is more complicated due to flows in the r , 0 plane, produced by the twisting of the planes, and developing flows.

with the velocity field given by

The boundary conditions are I ?* $ = r ,,=o; dr

w = uR -

for r = R , and O E [ O , ~ ] for 8 = 0, n, and r E [0, R]

where 11, = u,(R). An approximate solution to the above problem can be obtained by [he method of weighted residuals (Finlayson, 1972). The streamfunction is taken to be of the form N

$=

Z <m(rbrn(())

m= 1

where {a,) is a set of known trial functions which satisfy the boundary conditions, and (5,) is a set of unknown functions to be found by solving the differential equations, obtained by minimizing the residual over rhe interior as in the Kantorovich-Galerkin method (Kantorovichand Krylov, 1964). The weighting functions in this case are the trial functions themselves, and are taken to be The one-term solution for the stream function for the cross-sectional flow is

where v = ( 1 1/3)'12- 1. The streamlines for the different values of the normalized stream function $* = ll//[4vRR/3v] are shown in Figure 8.2.3. The motion is obtained by integrating the following set of equations which represent the approximate velocity field dr u = - - = /rl( I - rtJ) sin 20 dt

The above equations are dimensionless, with radial distances madc dimensionless with respect to R, axial distances with respect to the length

Mixing in the partitionetl-pipe ntixer

225

of an element L, and time with respect to L / ( u , ) . The dimensionless parameter /J. which we refer to as the tni.uittg strength. is defined as

and is essentially a measure of the cross-sectional stretching per element (as opposed to axial stretching). The analogous parameter in the static mixer is related to the pitch and aspect ratio of each helical unit. Neglecting developing flows, a fluid particle 'jumps' from streamsurface to streamsurface in the manner shown in Figure 8.2.4. The trajectory can be obtained by repeatedly carrying out the following steps: integration from the beginning to the end of an element, turning the co-ordinate system by 90", carrying out the integration until the end of the next element, and returning the co-ordinate system to its original orientation.

Figure 8.2.3. Streamlines for the cross-sectional flow in the partitioned-pipe mixer for different values of the normalized stream function +*. Starting from the innermost streamlines, the values of +* are 0.009, 0.031, 0.060, 0.091, 0.117, and 0.134.

Figure 8.2.4. Qualitative picture of a particle trajectory in the flow field of the partitioned-pipe mixer; the particle jumps from streamsurface to streamsurface of adjacent elements. The streaklines and particle paths coincide; in particular they can not go backwards as in the system of Figure 8.3.3.

Mixing in the partitioned-pipe mixer

227

1t is clear from this procedure that though the particle path is continuous, its derivatives are not, and this results in infinite stresses. However, including the developing flows would significantly complicate the problem (a flow involving smooth trajectories is analyzed in Section 8.3).

8.2.2. Poincare sections and three-dimensional structure

The flow in this case is periodic in axial distance rather than time (as for example, all cases of Chapter 7), so that the most convenient choice for the surfaces of section is the cross-sectional planes at the end of each unit (consisting of two adjacent elements). Poincare maps are then generated by recording every intersection of a trajectory with the surfaces of section in a very long (ideally, infinitely long) mixer, and projecting all the intersections onto a plane parallel to the surfaces. Every trajectory intersects with each surface of section, and the Poincare map should capture some of the mixing in the cross-sectional flow. Quite conceivably, any set of cross-sectional planes separated from each other by the length of a periodic unit could be used to generate a Poincare section by this criterion. However, in all other cases it would not be possible to exploit the symmetries of the mapping (which reduces the computational effort by a factor of 8, Khakhar, Franjione, and Ottino, 1987). Figure 8.2.5 shows the Poincare sections for the partitioned-pipe mixer for various values of the mixing strength (1) = 2, 4, 8, 10). The velocity field in the partitioned-pipe mixer is three-dimensional and it is important to study the relation of the Poincare sections to the overall flow. We visualize the three-dimensional flow by plotting Poincare sections at intermediate lengths, which enables us to follow the progress of the KAM curves through the mixer. In Figure 8.2.6 we have plotted these for /3 = 2, for increasing values of the axial distance, z = 0.2,0.8, 1.4, 2.0 (notice the asymmetry of intermediate Poincare sections). Each KAM curve then the intersecton of a tube with a surface of section, so that the tubes can be reconstructed by joining the KAM curves with their images in neighboring Poincare sections by smooth surfaces (e.g., curve A in Figure 8.2.6). The last figure in the series (8.2.6(d))gives the periodicity various islands; apparently, the smaller islands have higher periods. Notice also in Figure 8.2.6 that the cross-sectional area of the tubes is "Ot constant. This results from changes in the average axial velocity in the tube as it winds through the mixer; the total flow in the tube (average velocity in the tube multiplied by the cross-sectional area of the tube),

Mixing unrl chuos in three-dimen.sionu1 and opetz jlows Figure 8.2.5. Poincare sections for the partitioned-pipe mixer for various values of the mixing strength P : ( a ) /I = 2.0, ( h )/I = 4. ( c , ) /I = 8, and ( d )/I = 10, Initial positions for the different trajectories were placed on an (r-11) grid: eight r-values uniformly spaced between 0 and I . and five 11-values uniformly spaced between 45 and 135 . Each point was mapped for 300 iterations. (Reproduced with permission from Khakhar. Franjione, and Ottino (1987).)

Mixing in the partitioned-pipe mixer

Figure 8.2.5 cotiri~~lrurl

Figure 8.2.5 corrtirrrred

232

Mi.uinq trntl chtros in three-tlinrensiontrl trntl open flows 1-igurc 8.2.6. Poincari. sections for P = 2.0 at intermediate lengths along n pcr~odicunit of length 2. Notice the asymmetry of the PoincnrL; sections. (11) z = 0.2. ( h ) := O.X. (c) := 1.4. ((1) := 2.0. The letter A in (11) through (11) follows the same island down the length of the mixer (the reader might try to follow the evolution of some of the other islands). The numbers in ((1) refer to the periodicity of the islands. (Reproduced with permission from Khakhnr. Frnnjionc. and Ottino (19x71.)

Mixing in the purtitionetl-pipe mi.uer

Figure 8.2.6 cotllitllrc~t/

234

Mixing anti chtros in three-tiimmsiontrl rrnd open flows

Figure 8.2.6 coniinuril

Figure 8.2.7. Schematic view of invariant tubes winding through the mixer. According to Figure 8.2.6(1/) the smaller islands have higher periods (note that axial length scale is different from that of Figures 8.2.1 and 8.2.4). The cross sectional area of the tubes is not constant.

however, remains constant. A qualitative three-dimensional view of the tubes is shown in Figure 8.2.7. Obviously the tubes corresponding to the KAM curves are invariant surfaces, and cannot be crossed by fluid particles. Consequently, the fluid flowing in a particular tube remains in the tube and cannot mix with the rest. Chaotic trajectories, on the other hand, wander in the regions left free by the tubes on two-dimensional homoclinic manifolds. Chaotic trajectories come close to r = R ; such a device should be efficient in mass transfer operations between the bulk of the fluid and the wall of the tube. We should note that the axial flow has a major effect on the Poincare sections, and thus on the cross-sectional mixing. For example, the Poincare section corresponding to plug axial flow (Figure 8.2.8(u))and the same average axial velocity is quite different from the corresponding Poiseuille flow (p = 2, Figure 8.2.5).Another parameter that has a considerable effect on the Poincare section, and thus the mixing, is the sense of rotation in the adjacent elements. The Poincare section (Figure 8.2.8(h)) for the counter-rotating case, which corresponds to the configuration in the Kenics-ixer, indicates that the flow is chaotic over most of the cross-section, and seems to mix better than the co-rotating case (/I = 2). It is not completely clear why this is so and a deeper study is required.

8.2.3. Exit time distributions A pertinent question in open systems is whether or not the phenomena described in the previous section can be captured by various types of exit time distributions. In order to obtain an exit age distribution, we calculate the trajectories and residence times for a large number of particles, initially distributed uniformly (on a square grid) over the crosssection at the entrance of the mixer.Vhe particles then represent a pulse at the entrance of the mixer, and their exit age is found by integrating

along with the equations of motion.' The exit age distributions for a mixer with I0 elements, and two different values of P(2, 8) are shown in Figure 8.2.9, along with the corresponding contour plots of iso-residence times showing the residence times of particles based on their initial location in the cross-section. At low values of /I (Figure 8.2.9(tr))when the rcgular islands are abundant and occupy much Of the cross-section, wc see that the exit agc distribution has two peaks.

238

Mixing and chaos in three-dimensional and open flows

In some sense, the exit age distribution gives an indication of the inhomogeneity of the cross-sectional mixing: fluid streams in some tubes emerge faster than others, and there is no mixing between the streams in the different tubes. This is confirmed by a contour plot of iso-exit times and it is possible to identify the regions responsible for the two peaks. At higher values of the mixing strength, /? = 8 (Figure 8.2.9(/?)), the second peak disappears, in spite of the rather non-uniform cross-sectional mixing reflected in the Poincare section and the contour plot of iso-exit times. In this case, some particles of the peak belong to the regular island, others to the 'chaotic' region outside. In factjhe lowest residence time corresponds Figure 8.2.8. (a) Poincare section for the partitioned-pipe mixer with plug flow, (b) Poincare section for the partitioned-pipe mixer for counter-rotating cross-sectional flows in adjacent elements. The initial positions of the trajectories are the same as in the Poincare sections of Figure 8.2.5. In both cases, \\ = 2. (Reproduced with permission from Khakhar, Franjione. and Ottino (1987).)

(a)

0 = 2.0

t l ~ cri1
8.2.4. Local stretching of' muteriul lines

Let us examine now if the Poincari: sections arc able to capture the details of the stretching produced in the flow (only a few details of the an~~lysis are given here: for a more complete treatment the reader should consult Khakhar, Franjione, and Ottino (1987)).The total length stretch (In i.), average specific rate of stretching ( x ) , and average efficiency were calculated for a mixer consisting of I0 elements, for 21 number of material ( [ I ) ,

Figure 8.2.8 continued

Figure 8.2.9. ( a l . h l ) Exit age distributions for the partitioned-pipe mixer. and (02, b2) the corresponding contour maps showing iso-residence time curves based on the initial positions of the particles in the cross-section. Of 5.000 particles on a rectangular grid in the top half of the mixer, those which fell inside the cross-section were used in the calculations. In ( a t , 02) /{ = 2. T h e initial location of the particles corresponding to each peak is shown in the contour plot by the numbers 1 a n d 2. In ( h l , b2) p = 8 (reproduced with permission from K h a k h a r , Franjione, a n d Ottino, 1987).

0

10

30

20

time (a I )

40

50

242

Mixing and chrros in three-tlimensiontrl und open Jows

elements distributed on the same grid used for the residence t i n c calculations. The results for two values of /j(2, 8 ) are shown in Figilrc 8.2.10, which indicates the initial positions of elements with the highc\t and lowest stretching. The patterns are reminiscent of the Poincar: sections. The material element is represented by a dot if the quantity being displayed is less than the specified value, and by a slash if it is greater than the specified value. Our expectation, mostly based on results of tht Figure 8.2.10. In~tiallocation in the cross-section of material elements with the largest and smallest: length stretch (01, h l ) , and average efficiency (02. hZ). The grid is the same as used for the residence time distribution, the initial orientation olall material elements is m, = (1.0.0) in cylindrical co-ordinates. Dot, quantity is smaller than the specified value: slash. quantity IS larger than the specified value. In (ul. u2) P = 2. ( u l ) log i :dot, < 3 , slash. > 4 : (u2) ( e ) : (h2) (t):dot, 10.16. slash. >0.17 (reproduced with permission from Khakhar. Franjione, and Ottino. 1987).

Mixing in the ptrrtitiotierl-pipe t?ii.uer

243

two-dimensional time-periodic flows of Chapter 7 , is that the fastest stretching takes place in the chaotic regions of the two-dimensional poincart sections. while the stretching in the regular regions is slow and inefficient. However, these calculations indicate that this is not true, in general, in three-dimensional flows. In Figure 8.2.10(tr) the stretching is larger in the chaotic regions (as expected) but the efficiency is lower. Figure 8.2.10(h) is even more dramatic; both the stretching and the efficiency are larger in the regular regions. These seemingly contradictory results are easily rationalized: The jumping from streamsurface to streamsurface results in a continual reorientation of material elements in the regular regions (see Section 4.7). It thus appears that Poincare sections alone cannot give a qualitative description of the entire mixing process, and must be augmented by other analyses. Figure 8.2.10 continued

244

Mixing and chaos in three-rlimmsional trnrl open ,flows

Problem 8.2.1 *

Consider a mixer of field dimensions (L, R ) . What is the optimal value of 1,' to maximize the mixing efficiency'? Attempt a n analysis.

8.3. Mixing in the eccentric helical annular mixer The eccentric helical annular mixer ( E H A M ) provides a n illuminating counterpart for comparison of the results obtained in the previous section The cross-section of this system corresponds to the journal bearing no\\ of Section 7.4 and the axial flow is a pressure driven Poiseuille flow (Figi~rc 8.3.1).4 In this case the system is time-periodic rather than spatiall! periodic and the solution for the flow field, under creeping flow, invol\cs Figure 8.2.10 continued

Mixing in the eccentric helicul unnulur mixer

245

no additional approximations. Furthermore, there is no discontinuous jumping of particles between streamsurface and streamsurface, and the prticle paths have continuous derivatives. It is therefore of importance to investigate the analogies and differences that might exist with respect to the partitioned-pipe mixer. In this particular example we will consider that the inner and outer cylinder rotate with a period T as V0,outer = U c o ~ ~ ( n / T ) VO,inncr = U sin2(nt/T) and the relevant parameter is UT/ROuter(the only effect of the Poisseuille flow is to compress or stretch the particle paths in the axial direction). Figure 8.2.10 continued

246

Mi.uitig rrnrl clttros in thrrr-tlirnrti.siortrrl rrnrl open t1ow.s

In this particular case a literal 'surface of section', i t . . the marking or intersections of trajectories of initially designated initial conditions periodically spaced planes perpendicular to the flow, results in an 'out-offocus' picture (see Figure 8.3.2; in some cases 'islands' might disappear completely). A section in which intersections are recorded at designated time intervals, i.e., a Poincare section, however, reveals no information about the axial structure of the flow since it is identical to those obtained for the journal bearing flow (Section 7.4). It is thus apparent that, in contrast with the case of the partitioned-pipe mixer, conventional Poincare sections are not useful in investigating the structure of the flow and we have to resort to other diagnostic tools. The most revealing visualization in this case is provided by strcaklines (Figure 8.3.3). The calculations however, are considerably more difficult than those of the Poincare sections, since the storage and computational requirements for smoothness increase exponentially with flow time. According to the location of injection the streaklines can undergo complex trajectories reminiscent of the Reynolds's experimcn~.~ or 'shoot through' the mixer undergoing relatively little stretching. The dynamics of the streaklines have been relatively unexplored and more Figure 8.3.1. Schematic of the eccentric helical annular mixer. The length of the mixer is L. In the case of the ligure the cross flow corresponds to 7.4.2(c).

Mixing in the eccentric heliccil utztzul~rmixer

247

work is necessary. I t is significant that even though the axial flow always moves forward, the streaklines can 'go backward' since they can wander into regions of low axial velocity, whereas other parts of the streaklines can bulge forward. Figure 8.3.3(u) shows an instantaneous snapshot of a lateral projection of a streakline injected in a chaotic region, as well as a view from the end of the mixer. It is significant to notice that in the *end view' the streaklines crosses itself at various points, and that these streaklines would be the ones observed for a system with plug flow, or equivalently the journal bearing flow (see also Example 8.5.1). Figure 8.3.3(b) shows a similar computation but now the streakline is injected in a regular region. (The situation however is considerably more complex; the 'regular regions' are not static but move in a time periodic fashion;

Figure 8.3.2. The marking of intersections of trajectories of initially designated initial conditions with periodically spaced planes results in an 'out-of-focus' picture. Close observation reveals that the picture is not symmetric.

248

Mixing unrl chaos in three-rlimensionul unrl open ,flows

thus the dye injection apparatus can inject material in both regular and chaotic regions depending upon the injection time.) Various other diagnostic tests used for the partitioned-pipe mixer in t l l c previous section are valid also for the EHAM. In particular, it is possihlc to inject a dense grid of passive particles or vectors with various orientations and to compute contours of constant age, stretching, efficicncv. and so forth. Multiple peaks in the exit distribution are possible in t h i s mixer. Figure 8.3.4 shows an instantaneous picture of the axial concentration distribution created by the injection of a square array of particles at the entrance of the mixer (the concentration has been averaged with respect to the cross section). I t thus appears that the peaks present in the partitioned-pipe mixer are not due to any artificiality present in the miser and that similar behavior is indeed possible in other systems. Contours of constant stretch reveal that in this case, as opposed to the partitioned-pipe

Figure 8.3.3. Instantaneous p~ctureof a streakline injected in: (a) chaotic region, and ( b ) regular region. In both cases the flow moves from left to right (the axial scale is different in both figures).

Mixing rind rlisprrsion in u furrowed chunnrl

249

mixer, the stretching in the chaotic regions is exponential and linear in the regular regions.

8.4. Mixing and dispersion in a furrowed channel The example described here was considered by Sobey (1985). The system consists of a channel with a minimum gap 2h and with boundaries given by .u, = 1 + t ' ( s , ) and s, = - 1 -y(.u,) where f ( x l ) and g ( x , ) are sinusoidal functions given by ./'(.x,) = (D/2)[1 - cos(2n.ul/L)] g(s, ) = (K/2){! - cos[(2nx1/L) + 41). Thus, the case D = K and d, = 0 is a symmetric channel; if K = 0 one of the walls is flat. Sobey solved the Navier-Stokes equations in this geometry and focused on the case of no net oscillatory flow with a frequency o . The volumetric flow rate, Q ( t ) is given byh Q(t) = 2hU sin(27cwt). For a fixed geometry, the flow is described by two parameters: the Reynolds number, Re = Uhlv, and the Strouhal number, Sr = o h / U , which

250

Mixing trnrl chuos in three-tlimrnsionul unrl open .flows

can be thought of as hj(average distance travelled axially by a fluid particle). Computations indicate that there is a region in the Sr-Kc plane whcl-c flow separation occurs. At very small values of S r the flow is quasi-ste:~d~ and if separation is present (due to R ) the vortices grow and decoy ill phase with Q(t). A[ intermediate values of Sr the vortices d o not decrcaw in s i ~ ewhen Q ( t ) decreases, but instead continue to grow and enrri~in further fluid. However, if Sr is very large, the flow is viscous dominated and vortices d o not form. In this case the distance travelled by a typical particle, between cycle and cycle of the flow, tends to zero and there is 110 lime for separation to occur. The basic flow patterns obtained by Sobey by solving the Navier Stol\cs equations, using a vorticity-streamfunction formulation (Gillani and Swanson, 1976), are given in Figure 8.4.1 (a)-( f ' )which show the instan[aneous streamlines for the flow at Re = 75 and Sr = 0.01 as a function of time. Figure 8.4.1 (u) represents the beginning of the cycle, 8.4.1 ( h ) thc

Figure 8.3.4. Instantaneous picture of the axial concentration (averaged over the cross-section) created by the injection of a square array of particles at thc entrance of thc mixer (U'I.K,,,,= 2). In this case the particles in the regular rcgions rnovc faster than the avcrage speed.

0

5

10 axial distance

15

20

M i x i n g ~itlrlrli.sper.sion in u ,furrower1 chuntlel

25 1

middle of it. The same flow was studied by Ralph (1986) for Reynolds up to 300 and a range of Strouhal numbers of to 1. As we have seen in previous examplcs (notably, Section 7.4) the present only part of the picture insofar as mixing is concerned. Nevertheless, a superficial analysis indicates that the flow will be able to well since it consists of two interacting co-rotating vortices as the blinking vortex flow (Section 7.3) and the cavity flows (Section 7.5). In a rough sense we might interpret Sr as 1/p, if the circulation of the vortex is taken to be UII.Also in the symmetric case we expect the mixing to be rather poor since it is obvious that the centerline is an invariant surface ofthe flow, and prevents interaction between the regions .u, > Oand .u, < 0. As we have seen in previous examples, two-dimensional time-periodic flows are far richer in structure than what might be revealed by a cursory analysis and an in-depth analysis requires an entire arsenal of tools. It is apparent that this flow contains periodic points and is capable of homoclinic and heteroclinic behavior. However, an analysis on the level of that presented for the TW and BV flows is tremendously complicated and only a few aspects of the program of Table 7.1 can be completed. Here we present the results of Sobey for macroscopic dispersion of tracer Figure 8.4.1. Instantaneous streamlines at various points in time for periodic flow at Rr = 75 and Sr = 0.01. for various valuesof ( o t : ( u )0.1. ( h )0.25. ((,) 0.4, ( d ) 0 . 5 ,( r ) 0 . 5 5 ,and ( / )0.75 (reproduced with permission from Sobey. 1985).

p;~rticles.The equations to be .;olved (numerically) arc: SI. t l . ~ tlr , = I . , ( . Y .s2, , . r: I<(,) Sl. (I.\-: 111 = I . . s 2 .1 : KC'). Note that, in general, I., and I.? arc periodic but the period need not bc equal to w - I . Also, the identification of periodic points of the velocity field is far from trivial. Figures 8.4.2 4 show the dispersion, and initial condition, of a plug of 3,000 passive particles for Sr = 0.07 and KC,= 10, 75, and 100, t r f r c v . o ~ ~ (,j.(,/(j of' rlic. ,/low. Qui~litatively,it appears that there is a n optimum /<,, for a given S r . Figures 8.4.5 6 show similar processes occurring in a chi1nncl with n phase angle (1) = i~ and equal amplitudes and another case corresponding to one wall with amplitudc k' = 0. A measure of the ;i\ial dispersion is given by the variance of the axial particle positions ( Y , I,, i = I . . . .. N.

Figure 8.4.2. Evolution of a n initial condition consisting of 3.000 p a w b e particles. for Sr = 0.02 a n d K c = 10. after onc cycle of the flow. The l'ra~ne\ correhpond t o : ( ( 1 ) (or = 0. ( h )ot = 0.2, ( 1 . ) c,,r = 0.5. ((1) = 0.6. I c ) r t i / = 0.75. and ( / I cor = I .O. (Rcproduccd with permih\ion from Sobcy. 19x5.)

M i x i n g und dispersion in a Jurrowrd channel

where ( x , ) is given by

The value of a2 depends on the initial placement of the particles ( . Y , ) ~ and the time at which they are placed in the periodic flow (release time, t o ) . computations indicate that there can be some contraction of the cloud ofparticles, i.e., the growth of a2 is not monotonic, but that if we integrate over to the variance grows almost linearly in time.' It is unclear however, if this behavior is general or if other types of dispersion laws are possible in this system as well as the systems of Sections 8.2 and 8.3.

Problem 8.4.1 Consider that the particles are not material particles but rather actual tracer particles. Discuss the effect of the particle size on the process, i.e., account for Brownian diffusion and/or inertial effects.

Figure 8.4.3. Evolution of an initial condition consisting of 3.000 passive particles for Sr =0.02 and Rr = 25, after one cycle of the flow. The frames correspond to (a) : wt = 0, ( h )cot = 0.25, ( c )ot = 0.5, ( d )c ~ ) t= 0.6, ( e )wr = 0.75, and ( f ) = 1.0. (Reproduced with permission from Sobey, 1985.)

254

M i x i n g lint1 chaos in three-dimension~il~inrlopen ,flows

8.5. Mixing in the Kelvin cat eyes flow Consider again the flow of Example 6.7.1. As we hake seen, the strcanlfunction with respect to a fixed frame is of the form

$ ( x l . x2, t ) = u.u2 + I I I [ C O S ~ x 2 + A c o s ( x l u t ) ] (8.5.1 1 (here we hake made the streamfunction dimensionless with respect ( 0 A C h , 2 , the coordinates .u, a n d .uz with respect t o 11, the velocity L' ~ i ~ respect t o A C , 2 , a n d the time with respect t o 211'AL'). In frame .xi. y, moving with the mean flow the streamfunction $ ( x l . x,. t ) can be reduced to $ ' ( s ' , ,Y;) = In(cosh .u> + A cos s',) (8.5.21 which has a streamline portrait known a s the 'cat eyes' (Figure 8.5.1 1 T h e ~ e l o c i t yo f a fluid particle with respect t o the moving frame is giben b j ,f', = d.u', 'rlt = sinh .u>'(cosh .uL + A cos s',) , f > = d x ; 'tit = A sin .u', '(cosh .u; + A cos .u', ). -

Since the velocity field has heteroclinic trajectories we expect t h a ~ F~gure8.4.4. Evolution of an initial c o n d ~ t ~ ocons~stlng n of 3.000 passite particles, for Sr = 0.02 and He = 10, after one cqcle of the flow. The frame5 correspond to: ( t i ) tor = 0. ( h )tc)t = 0.25. ( c ) oJt = 0.5. ( d )o ~ = t 0.6. ( e )cc)t = 0.75. and (,/ ) o ~ = t 1.0. (Reproduced wlth permiss~onfrom Sobcq. 1985.)

Mixing in the Kelt3in c ~ i ryes t jlow

255

perturbations would lead to transversal intersections of stable and unstable manifolds. As we have seen, a useful technique to determine the existence of transversal intersections is the Melnikov method (Section 6.9). As a simple area preserving perturbationXwe take $1, = i: sin(tor) .'IZ= 0. which gives a Melnikov integral of the form M(t,) = [ - ( : A cos(tut,,)/(l+ A ) ] F ( w ) , where F ( w ) is given by F(w)=

Sr.

sin(.xl)l,,,,,,,,, sin(wr) dl.

The function F(w) is plotted in Figure 8.5.2. Since M(t,) represents the distance between the perturbed manifolds, we expect that an extreme in F(w) should maximize the 'extent of chaos'. In this particular case, the optimal frequency, (11, appears to be in the neighborhood of 0.3. I t is important to check whether or not such a prediction is substantiated by Figure 8.4.5. Similar initial conditions as Figures 8.4.2-4, but in a channel with a phase angle 4 = n and equal amplitudes. Dispersion after one cycle of the flow. The frames correspond to: ( 1 1 ) (01 = 0. ( h ) wt = 0.25, ((,) (ot = 0.5, ( d ) wr = 0.6, ( r ) cut = 0.75, and ( / ) (ot = l .O. (Reproduced with permission from Sobey, 1985.)

other techniques, such as Poincare sections. A Poincare section for tlli, problem consists of recording intersections of initially designated Iluid particles with the r; x i plane every 2n time units. Also, taking advantage of the spatial periodicity of the flow in the xi-direction, we record tllc intersections making .xi mod 2n. Visual inspection of the results shown 111 Figure 8.5.3 indicates that, indeed, chaos is maximized for (,) 2 0.3 (substantially higher values of the frequency, ( 1 ) = 4, indicate a reduction in the degree of chaos and mixing). Since streaklines would be the tool of choice in experimental studies. it is important toexamine how they look in the perturbed case. Experimentall!. Figure 8.4.6. Similar initial conditions as Figures 8.4.2-4, but In a channel with a phase angle 4 = rr and one wall with K = 0. Dispersion after one cycle of the flow. The frames correspond to: ( a ) (or = 0. ( h ) ('11 = 0.25. ( c ) V I = ~ 0.5. ( d ) (or = 0.6. ( r ) ( f ~ = r 0.75. and ( , f )= 1.0. (Reproduced with permission from Sobey. 1985.)

Figure 8.5.1. Streamline portrait corresponding to the streamfunction of Equation (8.5.2)for A = 0.8 (the spacing between vortices is 271);A$ is constant between streamlines.

Mixing in the Kelvin cut eyes flow

257

the streaklines would be injected with respect to the fixed frame, but c~mputationallyit is possible also to investigate injection in a frame moving with the cat eyes (Figure 8.5.4). A comparison between the streaklines and the corresponding Poincare section reveals that the streakline injected within the chaotic region undergoes significant stretching and folding whereas the streaklines injected within the 'cat eyes' and outside 'cat eyes' undergo little stretching. Figure 8.5.5 shows the stable and unstable manifolds associated with the hyperbolic points. The streakline injected with respect to the fixed frame reveals also significant stretching and folding (Figure 8.5.6). However, in this case, as opposed to the Hama flow of Example 2.5.1, the streaklines serve as a good indicator of the position of the vortices (there are none in the Hama flow). It is important to notice that given the time-periodicity of the flow, it is possible to simplify substantially the problem of streakline tracking Figure 8.5.2. Function F(tu) in the Melnikov integral.

258

Mixing und chaos in three-dimensionul and open jlows

(obviously, a similar simplification is possible in the Hama flow); the points defining the streakline injected during the second period undergo the same stretching history as the line formed during injection in the first period, and so on. A more realistic simulation of this flow might involve a sequence of vortices interacting in the fashion described in Example 6.7.2. In this case, however, the attack has to be necessarily computational and the utility of the Melnikov method is somewhat lost. Example 8.5.1

A careful examination of Figure 8.5.4(b) shows the crossing of two streaklines injected with the chaotic region. Is this only possible in a chaotic region or could it happen in an integrable system? Examine the system d . ~ , / d= t - .u2 + sin w t ,

dx2/dt = x , ,

Figure8.5.3. Poincaresections for the perturbed cat eyes flow ( A = 0.8. i:= 0.1 ). The values of cv are as follows: ( u )0. I , ( h )0.2, (c) 0.3, (d)0.4. ( r )0.5, and ( f ) 0.6. The maps consist of eight initial conditions and 100 iterations.

(a)

(b)

M i x i n g in the K e l v i n cut eyes Jow

with solution x, = ( . u , ) cos ~ t - ( . Y ~ sin ) ~ t x 2 = ( . Y , ) ~sin t ( . Y , ) , cos t which is of the form

+

+ [to/(l + [1/(1

x =Q(t)-xo

to2)](costot - cos t ) - w2)](sin tot - t o sin t ) , -

+f(t).

The streakline passing through the point is given by

x'

at time

t,

parametrized by

t',

+

x = Q ( t - t ' ) - x 0- Q ( t - t ' ) - f ( t f ) f ( t ) .

It is possible to show that this flow possesses streaklines which cross and return to their points of injection. For example, for w = 3.1 a streaklines injected at t = 0 at ( 0 . 5 , 0 . 5 ) returns to its initial location after Figure 8.5.4. Snapshots of streaklines injected in a moving frame in the cat eyes flow: (a) outside the eyes, ( h ) within the chaotic region, and (c) within the eyes. The initial conditions (xi, x;)are: (a)(0.0, 2.0). (0.0, 1.8),( h )(0.0, 1.0), (0.0, 0.5), (0.5,0.0), and ( c ) (2.0,O.O). For comparison, a Poincare section is given in (d). The parameters are: A = 0.8, I: = 0.1, co = 0.1. The total time of injection is four periods of the perturbation, 8nlw. /

260

Mixing and chuos in three-dimensional and open flows

t = 207~.Compute some typical streaklines (Franjione and Danielson,

1987). Propose other examples.

8.6. Flows near walls In this section we consider a few examples of structurally unstable flow fields near walls. The importance of this class of flows lies in the conjecture

-

Figure 8.5.5. Stable and unstable manifolds corresponding to the case A 0.8, o = 0.3, for various injection times (the 'most chaotic system', see Figure 8.5.3(c)).

E = 0.1,

Figure 8.5.6. Snapshot of streaklines injected in a fixed frame in the cat eyes flow. The time of injection corresponds to two periods (spanning two cat 'eyes'). The position of the vortices is indicated by crosses. The parameters are: A = 0 . 8 , ~ = 0 . l ,tu=O.l

Flow near wulls

26 1

that chaotic particle tn~jectories might be a precursor to Eulerian turbulence. Also, since many of the flows presented here involve flow separation, they serve to highlight the role of separation and recirculation bubbles in creating favorable conditions for chaos. Even though the development of this section is not as complete as some of the preceding sections and Chapter 7, it should be clear that the possiblities are many and that some of the ideas presented here have a clear connection with the concepts of Section 5.10. The flows considered are asymptotically exact solutions of the Navier-Stokes and continuity equations in two- and three-dimensions.The presentation is based on recent work of Perry et The starting point is to expand the Eulerian velocity field in a Taylor series around a point p, i.e., u(x) = u(p) ( X - p).Vu(x)l,=, i(x - p)(x - p): v v ~ ( x ) l ~.=. ..~ Each term V'"'u(x)l,=, in the expansion represents a tensor of order n + 1 . Denoting Aijk,,,= ( lln!) d"ui/dxj?xk. . ., the expansion can be written as

+

+

+

where the variables associated with the (n + 1 ) order tensor Aijk,,, are of the form (X,)"(.Y,)~(.K,)' with n = (1 + b + c, where u , b, c are integers or zero (c = 0 for a two-dimensional flow).'' It is clear that the tensors Aijk,,, are symmetric in all indices but the first one due to the fact that the order of differentiation is immaterial. These tensors constitute the unknowns and are to be found by forcing the series to satisfy the continuity and Navier-Stokes equations as well as the boundary conditions of the problem in question. The coefficients may have time dependence or not, depending on whether the flow is steady or time varying (due to some external perturbation). The number of independent coefficients, N,, generated by an Nth order expansion of a three-dimensional flow field is given by

Substituting the velocity expansion into the continuity and Navier-Stokes equations and equating coefficients of equal power generates a series of independent relationships between the coefficients Aijk,,,. For example, the continuity equation generates just one relationship for n = 1, three for n = 2, and six for n = 3, and so on. For n = 4 the relationship between cofficientsis given by

262

Mixing unrl chnos in three-rlimmsionul und open ,flows

Table 8.6.1 Two-dimensional

Three-dimension:il

-

and the number of independent relationships is 20. The corresponding expression for a two-dimensional flow is lijk + A22ijk = O The general rule for the number of independent relations betwccn coefficients generated by the continuity equation, E,, is

Similarly the number of independent relations generated by the NavicrStokes equations, EN,, is given by

The corresponding relations for two-dimensional flow are:

A few tabulated values, taken from the work of Perry and Chong (1986)are given in Table 8.6.1. An examination of Table 8.6.1 shows that, in general, several boundary conditions should be specified in order to obtain the numerical value of the remaining coefficients. For example, for n = 5 We need a number of conditions to obtain the unspecified 42 - (15 + 6) = 11 coefficients. Boundary conditions such as no-slip at the wall and a specification of the surface vorticity are commonly used and generate a

Flow rlrtrr wcills

263

series of non-linear equations in terms of the unknown tensor components. For time varying velocity fields the equations become a series of non-lincar differential equations. As an example, consider the flows displayed in Figure 8.6.1. In this case the velocity field was expanded to 5th order in x, the boundary conditions are no-slip at the wall, and the vorticity at u, = 0 is specified to be t o , = K (.uI - u,Z) where K is a constant which determines the strength of the vorticity, and x, determines the points of flow separation and reattachment of the flow. Figure 8.6.1. Two-dimensional flow gcnerated by a fifth ordcr perturbation with K = 0.5, 0, = 45 Y, = I . R e = 50. In ( u )the position of the elliptic point, (x;, 1;) IS (0,0.5)and a hornochnic orblt is produced; in ( h )the point is closer to the wall (0,0.25)and a hcteroclinic orbit appears. connecting the stagnation flows at the walls.

.

264

Mi.uitlg ut~richrlos in three-rlimensionril clnd o p e n flows

Since the vorticity changes sign at &.u, there are two separation point,, We also specify the angle of separation and reattachment of the streamlines O . The relationship between the angle at the point of separation, measurcd in the clockwise direction from the .:all, and the derivatives of pressure and vorticity evaluated at u2 = 0 x1 = f I (separation point), is given by tan 0 = 3p(i(o,/?.~,)/(?p/?xl), which specifies a relationship between the angle, the separation points, and the gradient of pressure. However, these conditions are not sufficient to specify all the coefficients. In order to close the equations the location of a critical point (for example an elliptic point) also must be specified. In this case we can specify the location of an elliptic point; the second critical point (i.e.,the hyperbolic point in Figure 8.6.I ( u ) )arises as a result of solving the problem with the boundary conditions as stated above. Thus, in this problem, a fifth order expansion, the total number of unknowns is 42, and according to Table 8.6.1, the number of independent relations between coefficients provided by the Navier--Stokes equation is 6, whereas the number of relations provided by the continuity equation is 15. Therefore the number of coefficients to be determined is 2 1 . I t can be shown that for a two-dimensional flow the no-slip condition provides 2(N + 1 ) relations, the specification of the surface vorticity N relations, and the angle relations and the location of the elliptic point (x;, x;), two conditions each. For N = 5, this provides an additional 21 relations, from which all coefficients can be determined." In this particular example the length scale L is selected as x,, and thc time scale T as ~ / [ K ( x , ) ~ ]The . resultant velocity field is given by

+

+

v1 = A12x2+ A12,x: + 3All12x:x2 3Al122xlx: A,,,,x: +6All122.~:.~: A12222.~:+ I O A l l l 1 2 2 ~ ~ . ~ :

+

+ 5Ai 12222x1.~;+ A 1 2 2 2 2 2 ~5 2 q = -3All12xlx:- A l 1 2 2 x ~ - 4 A l l 1 2 2 x l x1~ - 0 ~2 3 ~ ~ ~ where the coefficients, and relationships between coefficients, is given by A,,= -1 A i l , , = 113 A 1 2 2 2 = -213 A l l 1 2 2 =(3-A1222)/6 = I / ( x ; ) ~ + 2 / ( 3 ~ ; )+ (1/IO)RAll22x; -A122/(x;12 A111122=-(3/10)A1122 A112222 = -~1122/(xS)~ A 1 2 2 2 2 2 = -(lllO)RA, 1 2 2

and where the values of A , system

,,

and A , , , are given by the solution of the

where R is the Reynolds number of the flow R = ( L ~ ; T ) ;=YK ( x , ) ~ / Y , which is taken equal to one in the computational results shown in Figure 8.6.l(a). The flow constructed has a homoclinic orbit. It appears that the flow is structurally unstable in the class of all perturbations and that a simple time-periodic perturbation will most likely give rise to Lagrangian turbulence. Another example is shown in Figure 8.6.l(h).In this case the flow is subject to the same boundary conditions as Figure 8.6.l(a), with the exception that the arbitrarily designated elliptic point has been moved closer to the wall. The topology of the flow is radically altered. A streamline separates from the wall and reattaches again after some distance. In this case the connection is heteroclinic and the most likely place for chaos is in the region near the end of the unstable manifold (as it 'reattaches' to the wall). The range of applicability of these truncated velocity field equations can be ascertained by means of various checks. For example, the pressure gradient field can be generated in two different ways. One possibility is to expand Vp in terms of a Taylor series; the coefficients of the expansion can be obtained in terms of the known Aijk,..s Another possibility is to write Vp in terms of the Navies-Stokes equation and then to replace the truncated velocity field into v-Vv and V2v. This generates a second expansion for Vp containing more terms. The percent difference at an arbitrary point between the two Vp fields is taken to be a measure of the accuracy, or more properly, consistency, of the solution at that point. Another way to check the accuracy of a solution is to expand about a different point in the field and compare the percent difference in the magnitude of the velocity from both points. In fact, if great accuracy were desired in any particular point in the fluid, an expansion could be done at that point.12 For instance, it may be advantageous to expand about a homoclinic or heteroclinic point in the flow if time dependent perturbations Were being carried out. Figure 8.6.2 shows an example generated by this method in the case of a three-dimensional flow.

266

Mixing und chuos in three-dimensional cind open flows

Time dependent perturbations can produce chaotic trajectories i l l two-dimensional flows with homoclinic and heteroclinic trajectories. Figure 8.6.3 shows an example generated by perturbing the location or the elliptic point according to x; = 0,

u; = d

+ r: sin(2rctot).

The time evolution of the coefficients A , , , and A , , , , is given by

The system of equations can be investigated with an array of techniques suitable for ordinary differential equations. (See, for example, Guckenheimer and Holmes, 1983; Arnold, 1985, 1983.) This is one of thc few instances within the work presented here that the systems can be volunie Figure 8.6.2. Three-dimensional flow field generated by a fifth order perturbation: (0)surface flow pattern, ( h )three-dimensional view. (Reproduced with permission from Perry and Chong (1986).)

Figure 8.6.2continued

contracting and strange attractors appear to be possible (of course t I 1 1 \ depends upon the number of dimensions and the exact details of tl~c systems). This brings up the possibility of complex Eulerian signals. The computations correspond to the following values of the parametel-\: ti = 0.3, i: = 0.05, (o = $271, R = I . Figure 8 . 6 . 3 ( ~ 1shows ) a PoincarL: s e c t i o ~ ~ corresponding to the integrable system ( i : = 0 ) . whereas Figure 8.6.3(/,) shows a Poincarc section corresponding to similar conditions but for 1 1 1 ~ time-dependent perturbation of the elliptic point. Note that the point\ leak as the unstable manifold 'reattaches' to the wall which is shown in Figure 8 . 6 . 3 ( ~ ) Problem 8.6.1

Study the restrictions imposed by non-slip on a wall with normal 1,. Prove that in a third-order expansion of the velocity field, the no-h11p condition implies that the following coefficients are zero: A , , A ? , . I , ,, A 2 l . A l l , , A z l l , A l l , , , A2111Problem 8.6.2

Show that the restriction imposed by the Navier Stokes equ a t ~ ' o non a third order expansion of a two-dimensional velocity field takes the f o ~ m ~ ( A ~ A ~ ~ ~ + A ~ A ~ 2 ~+) A+ ~( zAA l1 ~2 A) 6l ~l ~ ( A ,Al l1z 2 2+) + t l A 1 2rii

+

-

~ ( A I A Z+I AI z A z I z ) + ( A l l A z I + A 2 1 A z 2 ) 6 ~ 1 ( A z I IAzI27)+dALi I ~ I ~ . t-igurc 8.6.3. Behavior of the bubble under a time-periodic perturbation o I the elliptic point. of the form Y; = 0 . .Y; = d I: sin(2ncur)with d = 0.2. I. = 0.05. ( o = X/2n, a n d Kr = I . t l g u r c X.6.3(u) shows a Poincart: section c o r r c h p o n d ~ t ~ g to the integrable system (I: = 0). whereas ( h ) shows a Po~ncart: section corresponding t o similar conditions but for the time-dependent perturbation of the elliptlc point. Note that the points leak as the unstable n1anllr)ld 'reattaches' t o the wall, which 1s shown in ((,).

+

Streamlines in an inviscid

flow

269

Problem 8.6.3 Show that the equations describing the time evolution of the coefficients Ai22

anc

* ^1222

a r e

volume contracting.

Problem 8.6.4 Show that the angle of separation and attachment depends only on the viscous terms, and that the non-linear and time dependent terms are identically zero for all expansion orders. 8.7. Streamlines in an inviscid flow As seen in Chapter 3, a steady isochoric flow of an inviscid fluid is governed by

v x a) = V(\jj + p/p + \q ) and therefore ф + p/p + \q2 is constant over streamlines (and also pathlines and streaklines). The simplest case corresponds to ф + p/p + \q2 uniform over some region of space and this implies that the vorticity and velocity are co-linear at every possible x, i.e., Furthermore, since V'O> = 0, we have and the streamlines are constrained to belong to surfaces /?(x) = constant, 13 which act as a constant of the motion. However, if fi is uniform, i.e., Vj8 = 0, this requirement disappears and v(x) is not constrained to belong to any surface. Arnold conjectured that such flows might have a complex topology and Henon, in a short note (Henon, 1966), took on his suggestion and examined the case dxx/dt = A sin x 3 + С cos x2 dx2/dt = В sin x{ + A cos x3 dx3/dt = С sin x2 + В cos x{ for 0 < xt < 2л, / = 1,2,3, which corresponds to the case л = + 1. Note that this is a steady flow and that the portrait of iso-vorticity (surfaces \w\ = 1/2 constant), iso-dissipation ((D: D ) = constant), helicity (o-v = constant), etc. are all very simple looking and can be computed easily. The structure of fixed points of the flow, however, and their associated manifolds can be extremely complicated and a complete analysis is not attempted here. Henon integrated the equations numerically and recorded the intersections with the plane (0 < x < 2n, 0 < у < 2n, z = 0, Poincare section). Figure 8.7.1 shows Henon's results for the case A = 3 1 / 2 , В = 2 1 / 2 , С = 1. The points joined by curves correspond to the same pathline (i.e., they

2 70

Mixing trnd c/zuo.s itz three-rlinzm.siontr1 tint1 open jlows

all originate from the same initial condition); all the isolated point, originate also from one initial condition (this behavior was called 'semi-ergodic' by Henon). The behavior of A = B = C = I is similar except that the 'semi-crgodic' region occupies more space. The same flow was analyzed, in a more complete study, by Dombre cJt (11. (1986). I t is easy to check that the velocity field has eight fixed points corresponding to - C sin u 2 = B cos s, = -B

+ [(B' + C'

-

AL)/2]'

+ B'

-

C2)/2]'/'

-

B')/2]'i'.

sin u , = A cos .u, = + [ ( A 2

- A sin u ,

=

C cos .u2 =

+ [(C' + A'

The condition that x,, .uz, and .u,, be real is that a triangle be formed with sides A ' , B 2 , and C ' . The condition dxldt = 0 implies o = 0 and therefore at the fixed point Vv is symmetric (since Q = 0 ) . I t follows that the eigenvalues of Vv are real and the flow is hyperbolic (the sum of thc eigenvalues is zero since V . v = 0 ) . Furthermore they are all different except Figure X 7 1 . Poincark section in the ABC flow for the case A C'= I.

=

3",

B = 2'".

in the case A = B = C = 0. (For C' = 0 the equations are integrable and the streamlines belong to the surfaces B sin .Y, + A cos .Y, = const.). The stable and unstable manifolds of the fixed points (either twodimensional surfaces or lines) have either to join smoothly or intersect transversally and complex possibilities of the type depicted in Section 5.10 are possible. Such a scenario is indicated by Dombre et trl. Extensive numerical simulations indicate that the system displays chaos." I f the flow were a truly effective mixing flow every filament of length i, placed in any orientation whatever in the chaotic regions will increase its length exponentially. However, even though this is suggested by the numerical calculations, i t is not true in general. Given the character of the flow we know that if the filament of length j., is placed initially coinciding with a vortex line with vorticity wo its length, when it moves to a region of vorticity o is given by i = (1o1/1o0l)i.,. However, o is bounded (in this case its value is exactly equal to v ) and we have to conclude that the stretching i. cunnot incrrclse esponentiully as is expected in the chaotic region of a two-dimensional time-periodic flow no matter what the initial orientation. Note however, that the stretching history can be extremely complicated, and that filaments placed in any position, most likely, will be stretched exponentially.

Bibliography Most of the work reported in this chapter is currently in progress and the presentation should be regarded as a sketch of the current state of the field and as an outline of possibilities. Some of the original material on the partitioned-pipe mixer appeared in the Ph.D. thesis by Khakhar (1986). A more thorough presentation appears in Khakhar, Franjione, and Ottino (1987). T h e work is continued and considerably extended in the Ph.D. thesis of Franjione. The treatment of Section 8.4 follows closely the presentation by Franjione and Ottino (1987, abstract only). The material in Section 8.4 follows entirely the article by Sobey (1985). Additional material can be found in Ralph (1986). The treatment of the Kelvin cat eyes flows (Section 8.5) is based on the Ph.D. thesis of Danielson. The treatment of flows near walls is based on the work by Perry et al. (see references in Section 8.6) and recent work by DanielSon. T h e material in Section 8.7 is based on the short article by Henon (1966) and Dombre et a[. (1986) with some interjections of our own. A relevant paper for the analysis of three-dimensional flows is Crawford

and Omohundro (1984) since it e m p h a s i ~ e sdynamical aspects which are not visible in a PoincarC map.

Notes

I In our calculation\. we u\c :t 3-term approximation of the scrics I ,lr. 0 ) : \uch .tppro\im:ttion nac h u n d to give a rcawnahly accurate dcxription of the flon a \ comparcd to a 100 term approximation, scc Khahhar. 1086. 2 -l-wo other po\\~hilitic\come to m ~ n dwith regard to the calculation I h c first I \ 'solving' the partial dlffcrcntial equation\ for the tracer concentration. a task \\hicIl would require con\idcrahlc computational effort. I h c cccond i\ atlding \tocha\ticit! at each htcp of the mapplnp to m i m ~ cmolecular tl~ffusion. 3 There are xeveral other standard poss~bilities,for example injecting particles w ~ t h:I speed proportional to the axial speed. However, this 1s hard to do experimentally. 4 An analytical solut~onis glven by Snyder ant1 Goldstem (1965) I h c croshsectional flow is based on the solution by Wannicr (see Scction 7 4 ) 5 See photograph 103 in van Dyke (1082). The EHAM has been an,tly/ed experimentally hy Kusch ( 1988). 6 Note that the equation defines the speed U . 7 A related flow. with thc same geometry whlch should produce similar results corrchpoli~l.. to the case in which one of the \vtrll.\ IS moved in a time-period~cmanner. Anothclexample of a spatially p e r ~ o d ~system c produc~ngefficient mixing due to sccondal-! vortices is a succession of tw~stetlpipe xclions (elbows). See Jones. Thomab, and Arcl ( 1990). X Obviously, a r e a l ~ s t ~pcrturbat~on c and n f ~ ~ analy\i.; ll of the prohlcm would gi\c rlse to three-d~mcnc~onal effect\ w h ~ c harc not con\itlcrcd here 9 For a dchcription of the method xcc Perry and Chong (19x6). a short introduction i.; given in Perry and Fairlie ( 1074):a general review is given in Perry and Chong (1987). 10 I h c spatial coordinates arc madc dimcncionless with rcxpect to I. and the time bit11 respect to 7'. where Land 7'aresuitahly sclcctcd length and timescales, which depend on the boundary conditions of the prohlcm I I Of courhe since the system i h non-linear this ncetl not he true in general. 12 However, a point worth mcntioningis that wecannot prove that the solution gcncratecl I)! this methotl is unique since the equations arc non-linear. 13 These surfitces can bc very complicated hut are, at least, tlifferentiable. 14 Recent works examining this system arc Gallow;ty and Frisch (1986. 19x7). bh1c.h examine the linear stability in the presence ofdis\ipntion; and Feingold, ~adanoff;111~1 Piro (Ic)X7), which examines a tliscrctirctl version of the flow of the form

+ A sin .Y, + C' cos + H sin \ , + A cos .x, u, + C' sin I, + H cos Y ,

Y', = .Y,

. x i = .Y, Y;

=

.\,

(motl Zn) (mod 271) nod 271)

In this case the emphasis is on invariant \tructurcs and 'diff~~sion'.

Epilogue: dvfusion and reaction in lamellar structures and microstructures in chaotic flows

The chaotic flows described in Chapters 7 and 8 provide a 'fabric' on which it is possible to superimpose several processes of interest such as interdiffusion of fluids and stretching and breakup of microstructures. A brief sketch of possibilities is discussed here. I f the fluids are purely pussitle, the connectedness of material surfaces suggests a way of incorporating the effects of stretching on diffusion and reaction processes. In the first section of this chapter we study diffusion-reaction processes occurring at small scales in lamellar structures; the presentation is general and independent of specific flow fields. Another prototypical situation, discussed in the second part of the chapter, corresponds to active particles; in this case the fluid particles are endowed with some 'structure' in such a way that they can mimic the behavior of small droplets, an interfacial region, or macromolecules (which we shall refer to as microstructures). The microstructure is governed by some evolution equation, for example, a vector equation based on suitable physics, which coupled to the underlying flow, governs the processes of stretching, change of orientation. breakup, etc. The nature of the presentation is speculative and is intended primarily as a catalog of possibilities and as an outline of future problems.

9.1. Transport at striation thickness scales The simplest problem of stretching of microstructures corresponds to the case of passive material elements. For example, in the case of passive mixing, which can be regarded as the case of two fluids of similar viscosity and without interfacial tension, the boundary between the two fluids acts as a marker of the flow; the motion is topological and the interface can be regarded as pussire. In the case of uctioe it~terfucc.s' the interfaces interact with the flow and modify it. Obviously, when describing mixing, it is too complicated to try to attack the problem in full. I t is convenient to describe mixing in terms of passive interfaces and then add, possibly

at small scales, the effect of active interf:ices.' Let us consider first illv purely passive case, the active case is considered in Section 9.3 Within the framework of the motions described so far, x = @,(X),2nd in particular all the flows of Chapters 7 and 8, initially designated material volumes remain connected, i.e., topological features are conserved and cuts of partially mixed materials reveal a lamellar structure. The striation thickness, s, serves as a measure of the mechanical mixing3 (see Figure 1.1.1). We wish to preserve a similar structure for the case of diffusing fluids in such a way that a properly defined striation thickness can he thought to act as the underlying fabric on which the transport processvs and reactions occur. We define a tracer as a hypothetical non-reactive material m that moves everywhere with the mean mass velocity v = (p, I,)", (see Section 2.3). Thus, if to, represents the mass fraction of the traccr (to, = p,/p) we have Dtu,/Dt = 0. Under these conditions, material surfaces of the tracer remain connected and map the average mass motion of the flow. The patterns produced by the tracer are similar to those shown in the examples of Chapters 1,7, and 8: tendrils, whorls, and primarily, stretched and folded structures. Diffusive species traverse the hypothetical surfaces of the tracer, concurrent with stretching and folding of the tracer surfaccs, which is governed by the equations of Chapter 3. The entire process is governed by the convective-diffusion equation

1

where r, represents the rate of reaction of species-s (an important simplification, which we will adopt here, is that the transport processes and the chemical reaction d o not affect the fluid mechanics). Sincc in general it is impossible to predict the stretching of surfaces in complete detail,4 it is clear also that the diffusion and reaction problem cannot be solved in a completely general way, and that we have to search for simple representations that contain the essential physics. We take the structures of Figure 7.2.1 as the morphologic~alhuilrliwl hlocks of arbitrary flows. These material regions (labelled by the material particle X,) stretch and deform, in a Lagrangian sense, as indicated In Figure 9.1.1, aiding the diffusion and overall reaction processes. We assume that the gradient of concentration has (locally) only one non-vanishing component, i.e., (0. I .? 1 V(l),= (?(I),/?.Y.0, 0 )

275

Trtrtlsport trt s t r i t i t i o t ~t l ~ i t ~ l i t ~.sc.tr/c.s c~.~.~

where .\- is a co-ordinate direction which is normal to thc planes in the lamellar structure (Figure 9.1 . I ).' The simplest case corresponds to a periodic structure with striation thickncsses s , and s,: in such a casc we can study a single pair of striations thercby avoiding thc complexities arising from distributions of striation t h i c k n e s ~ c s .Consider ~ now the application of (9.1.1) in a Lagrangian sense, i.e., by focusing our attention on a (small) identifiable material region of fluid labelled by thc material particle X, and denoted S., Figure 9 1.1. D e f o r m a t ~ o n sof structures In flows:

(11)

tendr~ls,( h ) whorl. ; ~ n d

(0a pair of strlatlons. w ~ t hcorresponding concentration fields along a cut Note that the Instantaneous l o c a t ~ o nof the maxlmum d ~ r e c t ~ oofn c o n t r a c t ~ o n not necessar~lynormal t o the striations.

IS

(a)

motion concentration profile

along broken line

maximum direction of contraction

The simplest theories of diffusion arc bnscd on Fick's law, i.c.. rllc relative diffi~sionvelocities are given by u,= ~ - (0~7Ills VW,~. (9.1.3) Since us transforms as ul, = Q - u , and V transforms as V' = Q - V we concludc that the diffusion equation has the same form in any moving framc 1;' provided that the velocity field is referred to the frame F'. An identical argument applies to the energy equation.' In particular we can apply Equation (9.I . I ) to a frame attached to a material particle X, such t hot. at small distances from X,,, in frame F', we have

where c, is the concentration of species-s and v,,, is the relative velocity field of Section 2.8. According to the lamellar assumption (9.1.2) we have

where n is a normal vector in the direction of V C , . ~In general, n does not coincide with the instantaneous (local) maximum direction of contraction (see Figure 9.1.l(c)).Another way of writing Equation (9.1.5) is in terms of the surface area stretch (for V - v = 0)

The stretching function r(X, t ) ( E D In tl/Dt) depends on the fluid mechanics and provides the tie up with the previous chapters.' In the simplest case the problem is represented by one equation of the type (9.1.5) and the elements of Figure 9.1 . I . According to the circumstances the complexity of the problem can be augmented by incorporating additional effects (for example a different stretching history for each microflow Sx,,,? see Section 9.2). However, even in the simplest cases, i.e., simple chemical reactions, the multiplicity of parameters makes the analysis quite complicated and it is hard to obtain results of general validity. Example 9.1.1 The equation

where L is Vv, which is a generalization of (9.1.4).and where c., can be either concentration or temperature, has been cxtcnsivcly analyzed in the context

277

Trcrt~sporrcrt stricrtiot~thic~kt1c~s.s .sc~rlr.s

of various applications. For example. Townsend ( 195 1 )considered the case

where the D,,s are the diagonal components of D. which in general are functions of time. but are considered constant here. A heat spot is released at the origin, and according to the value of the determinant of D, it is converted into a filament (det D > 0 ) or into a flat disk (det D < 0). We take tr D = 0 and consider D l , 3 DZ23 D33; i.e., D l , > 0 and D,, < 0. The temperature evolves as

where the eris are given by

and T,,,(t) = A / L I , u ~ L I ~ , where A is a constant determined by the initial conditions. Thus, the isotherms are ellipsoids, and for long times ( D l ,t >> I ) , T,,,(t) decays exponentially in time. Batchelor (1959)considered a similar problem, but with the initial condition T = A, sin(l-x ) r 2 , s,).In this case where I = (I,, 12, I,) is a constant vector and x = (s,, the solution is of the form T(x, t ) = A(t) sin(m(t).x) where m ( t ) is giver, by dm/dt = -D-m, and therefore (assuming 1, # 0 ) we obtain m2= + tn: + ni: + 15 exp(-2D3,t), and

as t

-+

rd

ast+m T-+ A , , ( ( Y + ~ / ~sin(m(t) D ~ ~ ) x), which shows that the normal to surfaces of constant T (i.e., with normal V T ) approaches the direction of the greatest rate of contraction. Note however, that this local result is valid only for linear velocity fields.'"

Example 9.1 -2 . O n many occasions it is sufficient to characterize the mixing in average or structured terms, e.g., profile of striation thickness across a

278

Epilogue

mixing layer, average striation thickness in the cross-section of an extruder+ etc. Consider a thin sheet of a tracer which at time t = 0 coincides \vith the boundary between materials. As mixing proceeds, and interdiffusion occurs, the thin layer S of the tracer moves with the mean mass velocity, and if the mixing is effective, it is finely distributed throughout a material volume V. We define p*(X, t ) as the mass of tracer per unit volume (p* = 0 for points x not belonging to the thin layer). It follows that

and

Similarly, we define pn(X,t ) as the mass of tracer per unit area of the surface at the location X and with orientation n. We have

and

We define a,(X, t) (and similarly, a,,(x,t)) as a, = p*/pn. The quantity a,. can be interpreted as the area of tracer/volume.ll Furthermore a,. is proportional to the area stretch, a, = ~ : ~ / d eFt and obeys (see Ottino, 1982)

Problem 9.1.1 Show that material planes remain flat in the flow field v,,,

=x-(VV)~.

Problem 9.1.2 Confirm the invariance of the convective-diffusion equation. Problem 9.1.3 Show that the distance 6(t) between two small, nearby, parallel planes with normal n varies as a function of time as (Ottino, Ranz, and Macosko1979) I DS(t) = D: nn. 6(t) Df

Trunsport crt striation thickne.ss .scule.s

279

Therefore. l/O(t) is proportional to the intermaterial area per unit volume, a,, (Example 9.1.2). problem 9.1.4

Build a simple model to interpret the determination of mixing times in terms of the temperature decay of hot spots.12 9.1 . I . Parameters and variables characterizing transport at small scales

It is important to recognize that the simplest problem of stretching with diffusion and reaction at small scales is characterized by, at least, three time scales and many parameters. Such multiplicity of parameters makes a general analysis difficult. There are three time-scales associated with local processes, the time-scale of diffusion, which is related to diffusional distances, O(sZ(t)/Ds),the time scale of the reaction(s) (in the case of multiple reactions there can be several times scales), and the time-scale of the thinning of the striation thicknesses, O(l/a) where a is the stretching function (see Section 2.9). There is also at least one time-scale associated with the macroscopic processes of mixing, which can be taken as the exit age of the element, t,,, (see Figure 9.1.2). Let us consider now the independent variables and parameters characterizing the system. In order to cast the equations in dimensionless form it is convenient to define the following variables: I-' = tit,, reduced time 'warped time'

Figure 9.1.2. U'i-premixed reactor. The reactants A and B enter the system and are mixed by complex motions. The microflow element SXmexits the system after a time t,,,.

where rl is the area stretch associated with the microflow clement .' ;ll,d r,. is a characteristic time. The usual choices are I,. = I,,,, (diffusional I I I ~ , ~ ) and t,. = r, (reaction time). A dimensionless distance is defined as spirc-c1S L - I I ~~L ~I I S Part~ tho i ~ t ~ t r ~ ~ t i r ~I ' L~I t~cI I~Oof' o ~l/,c, rs f = s/s(t), We also define the following fluid mechanical parameters:

(r(t))

=

:1;

r ( t l )dtr,

tinw trcleragrd oul~reof the stretc.hing ,firrlc.r ion

Based on these definitions we define the following characteristic times: characteristic jluid ntecltuniccrl time (stretc~hi~t,~) instantatteous d(ffusionu1 tinte t, = s ( ~ ) ~ / D , ,

t ~ ( t= ) I/(r(t)),

and t ~ =, $IDK, characteristic diffusion time ( K denotes a reference species). The characteristic time of reaction is denoted t,. The variables I-, 7, and the fluid mechanical parameters, s(t), x(t). and ~ ( t )are , associated with the microflow element S,,, the label X, being omitted without fear of confusion. In terms of these variables the convective diffusion Equation (9.1.6)can be transformed into the following two forms (see Chella and Ottino, 1984):

where Ci represents the dimensionless concentration of species-i, Ai is a diffusion coefficient ratio, and Pi is the initial concentration ratio of species-i. R i represents a reaction term (production) of species-i. Note that the contribution of stretching is double; first, an increase in the area available for diffusion accelerates the interdiffusion process; second, the continual stretching does not allow the mass fluxes to dccaY. Equations (9.1.6a, and b) provide two different viewpoints of the infli~ence of local stretching on diffusion and chemical reaction: Equation (9.1.(,a) shows that in the time scale I- and striation thickness based scale f . the apparent diffusion coefficient is augmented by a factor $(r)due to th" local stretching; Equation (9.1.6b) shows that in the warped time s a ~ l ct

T,crnsport

trt

str'itrtiort t/tic.krirss sc.crles

28 1

and striation ~hickness based scale <, the apparent reaction rate is multiplied by a factor 11-q~) due to the local stretching." The usual initial conditions correspond to complete segregation of the streams: the physical situation might correspond to streams being fed to the reactor (in this case the interface is a streaksurface) or two segregated volumes initially placed in a batch reactor. As initial condition we establish that the streams are unmixed. i.e., T or 3 0, c?C,/?<= 0, for = 0, I . where Hi(() is a unit step function discontinuous at ( = 4, where 4 is the volumetric fraction of the one-stream (one,of the streams is arbitrarily designated as 'one'). l5 T h e boundary conditions, in the periodic case, are obtained from symmetry considerations: for ( = 0, I . (?c,/ay= 0, 7 or r 2 0, Consider now the value of the initial striation thickness, so. In establishing this value we face two competing effects: We might argue that if so is too large, and curvature effects are important, the lamellar structure assumption is probably not quite valid and consequently the model a poor approximation. O n the other hand, if we wait longer and select a smaller so, we might miss some of the diffusion-reaction process and not be able to estimate the initial conditions. However, if so is large, as for example during the first stages of mixing in a shear flow (see for example, the Hama flow of Example 2.5.1 ), the interfaces are essentially non-interacting and therefore it really does not matter that the lamellar structure has not been established.

<

Problem 9.1.5 Eliminate the convective term (v = x . D ) in the equation

where c is a vector of concentrations and K is diagonal, by transforming the time t as A = A(() and x as = Q - x , where Q is diagonal. Show that the convective term disappears if we choose

<

This leaves several choices for the selection of A([) (Chella, 1984). 9.1.2. Regimes Equations (9.1.6a and b) form a system of coupled, non-linear partial differential equations, which can be solved, in general, only by numerical

Table 9.1 Average striation thickness

Reactor size

Age of microflow element

Entrance

Kinetic control Slow reactions

-c

.2 0

s

2

u

"

g .L

X

B

t

*Exit

t < t D<< t , N o appreciable reaction until reaching scales at which reaction behaves as homogeneous

t 2 t, System of O D E s with initial conditions (classical reactor analysis)

2

t D<< t R

t << t,, t D N o appreciable reaction

Combination of O D E S and P D E s (some reactions might be in equilibrium whereas others might be diffusion controlled)

t = O(t,, t ~ ) System of coupled ODES

2

Some reaction kinetics controlled, others diffusion controlled Diffusion control Fast reactions f R << f D

t << t o N o appreciable reaction

t
tR2 tD Reaction practically complete

'

0

Stretching at *small scales

Trtrnsport

clt

stricltion thickness sc.tr1r.s

283

means. The limit cases. 'fast'or 'slow' reactions, are more tractable than the general case. The different regimes of reaction are shown. schematically. in Table 9.1 . I . It is evident that the presence of at least three major time scales produces two dimensionless ratios (Damkohler numbers) which characterize the processes occurring at small scales: DtrjL'= tR/tb 1 6 Dull tD/tR. Note that the characterization of a reaction as fast or slow depends, in gneral, on t,, t,, and t, (see Figure 9.1.3).

--

Slow reactions A reaction is called 'slow' if t,/tR << 1 during the course of the reaction, except possibly during the first stages of mixing (this is a conservative criterion since t,,, is defined with respect to the initial segregation - the loosely defined so). Actually, the diffusion time-scale varies in time, since the striation thickness is decreasing in time. Consider a simple example corresponding to exponential thinning. Initially, the diffusional time-scale, t , = s ( t ) ' / ~ ,might be larger than the reaction time-scale, i.e., s(O)'/D > t,, but after some time, we might achieve t, z t,, since s(t) decays as s(0) exp(-xt), and quite rapidly afterwards we might be in a regime such that t, << t,. From this time onwards, intrinsic kinetics controls and local mixing has no influence on the course of the reaction. Under these conditions the mass balance for species-s reduces to: dC,7/dt= /j,,R,. (9.1.7) This limit corresponds to residence time theory (see Chapter 1). Figure 9.1.3. Schematic representation of regimes corresponding to slow and fast reactions.

case corresponding to 'fast reaction'

/to L

0

time (age)

case corresponding 'slow reaction'

-

284

Epilogue

Vtry f i s t reactions

A reaction is called 'very fast' ~f t,)/t, >> 1 during the course of the reactloll, In such a case the reacting species cannot co-ex~stand the reactlng 1011, reduces to a plane. The rate of diffusion to this plane controls the ovcl-;~ll reaction rate, and we lose the influence of the intrinsic kinetics. ' I t l c controlling factor is the rate of stretching and the location of the reaction Lone, which in the simplest case does not move and acts as a m a t ~ r i ; ~ l surface." In addition to the initial and boundary cond~tions,the f l i i \ of reactants to the plane must satisfy the overall reaction stoichiometry. Hcrc the variable T, with I, selected as I,,,, contains all the effects of the local convect~ve mixing (through the stretching function r(X. t ) ) and tllc problem reduces to

Thus, the solution of (9.1.8) for a particular reaction scheme allows thc superpos~tionof any stretching history (cho~ceof r(X. I ) ) . This is thc b a x idea behind the use of fast reactions as tracers of the fluid motion.lx Exumple 9.1.3

Figure E9.1.3 represents the effect of the local flow on a series-parallel reaction (A + B -t P), with a specific rate of reaction k2, R + B -t S. ~ i i t h a speclfic rate of reaction k,). The system is governed by the equatiollr

The parameters are: i: = 0. I , A, = A, = I, /r', = 0.5,4 = 0.5, initial diffuri~n time, t,,, = 2.25 x 10' s, final diffusion time, t,,, = 2.25 x 10-I s ( i t . . the striation th~cknessreduction is 109, the characteristic reactlon time is 1, = 2.25 x 1 0 ' s. Two types of flows are considered: a hyperbolic flow, I., = Eu,, o2 = -E.u2, and a shear flow r l , = jay2. c., = 0; the initial orientation in both cases is N = (0, I ) . The values of i: and $ are selected such tllat they produce the same striation thickness reduction in the total time, 10 s. The value of R in the ordinate in Figures E9.1.3 represents 111" average concentration of species-R in the S,,,, clement. The two I l o al-c ~ compared with that of a premixed rc;ictor. where the initial condition corresponds to A and B mixed molccul;~rly. In this ex;~mplc.tlic t w o

unpremixed systems have the same overall values of DeriLJ when the mechanical time t, is defined as t,,,/ln(.s,,/.s,). Extensive computations indicate that. for simple reactions, this definition of t, gives a reasonable measure of convective mixing; for example, even though the fluid mechanical histories x(t) are substantially different, the final results do not seem to be very sensitive to the form of x(t) for values of DuIL' as convective effects are high as 0.5. On the other hand, when DaILJ < not significant (Chella and Ottino, 1984). Example 9.1.4 The previous example showed a case where the functional form of r ( t ) does not seem to be very important and flows with equal overall values of D L I I ~ ' give similar selectivities. In more complex reactions, however, the fluid mechanical path can produce drastic differences in the final product. Figure E9.1.4 shows the result of a more elaborate simulation (see Fields and Ottino, 1987a.b) using similar stretching paths to those used in Example 9.1.3. The physical situation corresponds to a co-polymerization between two species, A , and A,, with a third one, B, leading to the

Figure E9.1.3. Effect of local kinematics on a series-parallel reaction. The figure shows the evolution of concentration of the intermediate reactant R.

premixed feed

/

/ 2

elongational flow

4

6

time (s)

/ 8

10

286

Epilogue

formation of (A,-B) units (soft segments, denoted P , ) and (A2-B) units (hard segments, denoted P,). The reaction is exothermic and the diffusion coefficients are a function of the concentration. High molecular weight polymer is preferentially formed near the interface. Average propertics (gross concentration profiles, average molecular weight, etc.) are not path dependent whereas other properties such as the maximum molecular weight across the striation are strongly path dependent.

Problem 9.1.6 Calculate the penetration thickness of a product P, 6,, produced by a very fast reaction A + B -+ P. The plane of reaction undergoes area stretch given by q -,exp({ cl dt). Show that for cr constant 6, reaches a steady state proportional to ( D / u ) ' ~Does ~ . this happen for any other functional form of cr(t)?

Figure E9.1.4. Effect of stretching path on a copolymerization. Molecular weight across the interface of reaction for (a) shear and ( h ) elongational histories corresponding to a dimensionless time of 250, (c) evolution of molecular weight for shear and elongational histories over a longer time scale; the initial and final values of the striation thicknesses in the shear and elongational prescriptions are the same. (Reproduced with permission from Fields and Ottino (1987b).)

Transport at striation thickness scales Figure E9.1.4 conr~nurd

288

Epilogue

Problem 9.1.7

As a crude indicator, assume that reactions are diffusion controlled 11 tI,/tR > lo4 and kinetically controlled if !,It, < 1 0 ' . Calculate the lengt/, scales necessary for these two regimes for the following second order reactions: HCI + HONa. k , 2 lo", C O , + H O N a , A , -, 1.3 x lo-'. H C O O C H , H O N a , k2 = 4.7 x lo1( k 2in l/mol. s and at 30°C).

+

9.2. Complications and illustrations 9.2.1. Distortions of lamellar structures and distribution effects

Here we consider briefly some of the additional physics necessary to complete the simplified picture described in Section 9.1. It will become apparent that while many effects can be considered we might get diminishing returns by their introduction. In any case we want to o b t a ~ n results of general validity, and then, if necessary, according to the physical situation, complicate the picture. Striation thickness distribution

In general, the striation thicknesses are distributed rather than spatii~lly periodic. and the distributions can give rise to large scale diffusion effects and 'isolation of reactants' (Ottino. 1982). In some cases, such ;IS combustion, this effect is referred to as flame shortening (Marble and Broadwell, 1977), in other problems such as polymerizations this cflcct produces permanent results due to variations in the diffusion coeffictent and local 'quenching'. Numerical experiments (direct simulations) provide a glimpse into this Changes in topology

By definition. the material surfaces of the tracer d o not change the topology and remain connected throughout time (however, they rnigllt appear disconnected in two-dimensional cuts as in Figure 1.1.2). H o L I ~ v ~ ~ the iso-concenration surfaces ofdiffiusitiy scalars move at a different velocity than the average mass velocity and might change topology (initially connected regions can break and form islands). (See Gibson, 1968.1 'Phis effect distorts the lamellar appearance. Note also that Equations (9.1.6a and b ) apply to only one microflow element; in practice we have the following complications: A distribution of fl~lidmechanical histories, x(X, t ) : Each element might have a different stretching history identified by its label X nltholt!ll 21 given flow might be well represented by a 'typical type'. For cxanlplc.

all steady curvilinear flows (Section 4.4) have r l 2 t . in a globally chaotic flow 'almost all Xs' evolve as r l 2 exp(t). A distribution of initial striation thicknesses, s,,s: This distribution might depend on the feeding or initial placement of the reactants. As an example consider the inlet condition in a multi-jet reactor. A large s, can be used to signify that the reacting surfaces d o not interact. A distribution of ages: The microflow elements identifed by labels X will exit the reactor (or reaction zone) after a time t,,, according to an age distribution function (Figure 9.1.2).20 In spite of these effects, it should be emphasized that the main idea is that the overall rate of reaction might be controlled h y the .smulle.st .sc,clle.s. Therefore, most of the physics is controlled by striation thickness scales and is modified only quantitatively, but not qualitatively, by distribution r rdiffusionally controlled, then the problem effects. If the reactions ~ ~ 1101 becomes simpler and falls within the scope of classical reaction engineering.

Example 9.2.1 Consider a system, a simplification of the co-polymerization of Example 9.1.4 (Fields and Ottino, 1 9 8 7 ~ )The . reaction is of the form A, + B + P , and A , + B + P,. For simplicity the striation thickness is constant in time. Figure E9.2.1 shows two systems with the same mean value of the striation Figure E9.2.1. Instantaneous concentration of reactant in a system with: ( u ) uniform and ( h )distributed str~ationthicknesses. (Reproduced with permission from Fields and Ottino (1987b3.3

ABABABABABABABABABAB x =0

x= 1

290

Epilogue

thickness, one of the systems has a uniform striation thickness the other has a distribution of values. Figure E9.2.1 shows two snapshots of the reacting systems at the same real time; it is clear that thick striations remain unreacted for relatively long times (this effect has been called 'isolation of reactants'). Problem 9.2.1 The rate of variation of the concentration of a scalar c when moving : ~ t speed v in a fluid is given by

Consequently, the velocity, c, =,,,, ~ ( x t ,) = const. is given by

corresponding to the speed of a surfi~cc

Use this result to prove that, at the point x, and at any instant t,

Compute a similar expression for the velocity of regions of zero gradient, i.e., Vc = 0 (Gibson, 1968). 9.2.2. Illustrations The following illustrations serve to reinforce some of the previous concepts and exemplify the limits of current computational efforts and the ability to resolve striation thickness scales. Example 9.2.2 Figure E9.2.2 shows what happens during the wrapping of an interface due to the first stages of instability in a shear flow in the direction .Y. for times such that the layer is constrained to be two-dimensional (Corcos and Sherman, 1984). The initial velocity and concentration profiles are p = - 0.5Ap[(nPr)1'2y/(2(5)1 u, = U erf[n"2y/(26)], i.e., the streamwise velocity changes from + U to - U and the scalar from +Ap to -Ap. The value of 6 is such that 6 = U[((7uX/dy),,,]- ' : the Reynolds number is defined as Re = U6/v and the Schmidt number is \','D. The time-scale is made dimensionless with respect to d/U where d is the length scale of the initial disturbance. Figure E9.2.2(a) shows the wrapping of the interface along with the instantaneous picture of the streamlines, and iso-concentration curvcs of

Cotnplications and illustrations Figure E9.2.2(a).Wrapping of the interface between two fluids due to the first stages of instability in a shear flow: ( a l ) location of the interface at dimensionless times 0.5, l .O, 1.5, and 2.0, with a = (2nii)ld = 0.43. Re = 100: ((12)iso-concentration curves of a passive scalar with Sc = I . a = 0.4, Re = 100, at times 1.5 and 2.0. (Reproduced with permission from Corcos and Sherman (1984).)

29 1

292

Epilogue

the diffusive passive scalar. Note that the stretch of the interface is fiiil-lv small (O(10))and that the lamellar structure assumption is a reasonable one except near the center of the 'cat eye'" (something more complica[cd occurs during vortex pairing). The details of the stretching history of [Ilc elements are highly complicated. With the exception of elements near the hyperbolic points, all other points experience a quasi-periodic s t r e t c h ~ n ~ history where a ( X , t ) becomes negative. Corcos and Sherman (1984) noted. as might have been expected, that the stretching history is highly sensitiic to the initial conditions near the hyperbolic point. Example 9.2.3 As we have seen in Section 9.1.3, the accurate computation of material interfaces is important in problems involving fast reactions producing t h i n reaction zones. The problem also has relevance to computations such 2 s those described above and the direct computations given below. Order of magnitude calculations indicate formidable problems due to stori~gc requirements. Computationally, the material interface is composed of a large number of points connected by small linear segments. However, duc to the flow the interface stretches and folds, and if the distance between points becomes too large, the linear segments connecting adjacent polnts could cross, which is physically inadmissible. In order to maintain consistent resolution, the numbel of particles should be increased in such a way as to be able to resolve the striation thickness accurately. Obviously, such a calculation is a precondition for the resolution of the concentration Figure 9.2.2(u) continued

field within the striations and the computation of concentration gradients without loss of information due to averaging. The storage requirements can be estimated based on the idea of Figure 9.2.2(h).Consider a striation thickness reduction of lo-', maintaining throughout the process a ratio of particle separation (S) to striation thickness ( s ) of lo-", where p is of Figure 9.2.2(u) rontinuvd

294

Epilogue

order one. The simplest possible case, which is a lower bound for nio1.c: complicated situations, corresponds to a two-dimensional (deterministic, time-periodic) mapping in a 'globally chaotic' regime where the length of a line increases, on the average, as Lo exp(crk),where k refers to the number of mappings and a is a suitable Liapunov exponent (in the case of integration of the Eulerian velocity field, we have estimated that tllc number of operations for each dimension, F, is of the order 10 to 100, and that each period requires 10' time steps with t of order 3). An estimate of the number of floating point operations per mapping, taking into account the loss of precision due to growth of errors, for an accuracy of O.OI'!;,. yields the results shown in Table 9.2 (Franjione and Ottino, 1987b). I t should be noted that the bulk of the computational effort is not due to growth of errors. For example for the case of a striation reduction of relaxing the accuracy to 1% and 5 % , reduces the storage requirements by only 1.5 and 2 Gigabytes respectively. Example 9.2.4 There have been a few direct simulations incorporating chemical reactions. Figure E9.2.2(h). Tracking of the interface between two fluids being mixed. The distance between material points is 6, the striation thickness is s. Material points are used to represent the blob itself or the interface between the blob and the surrounding fluid.

Complictrtions trntl i1lu.strtrtio11.s

295

Table 9.2. Cor~~puttrtiontrl esrimtrtcs ,fiw r~ltrtcritrlline rrtrckin~qin u rhuotic floul on tr rornpurcr ctrptrhlc of 10' ,flotrtiny point opertrrions prr second. The c~trl~~~rltrtion.s trssurllc cr = I , p = I . F = 20, trnd t = 3 Striation thickness reduction (r)

-

3 4 5

Computational time Explicit mapping

Integration of Eulerian velocity

Number of required digits of precision

Storage requirements (bytes)

4 minutes 6.5 hours

1.5 weeks 36 motlrhs 300 ycclrs

9 11 13

5.2 x 10' 600 x 1o9 70 x lo1*

1 tnot~th

The article by Riley et ul. (1986) presents one of the most complete. They considered a binary chemical reaction with no heat release (i.e., the reaction does not alter the fluid mechanics) in a temporully growing mixing layer, which is simpler to analyze, and contains some of the same physics as a spatiully developing mixing layer (i.e., their geometry is similar to that in Figure E9.2.2(u)). The initial condition was also qualitatively similar to that of Corcos and Sherman (1984). Riley, Metcalfe, and Orzag (1986) used a pseudo spectral method using 323 and 643 grids; typical runs involved between 500 and I000 time steps and consumed one to two hours of Cray 1 computer time. The Schmidt number was kept of order one so that the resolution requirements for both velocity and concentration were roughly the same, the Reynolds number based on the Taylor microscale was approximately 50, and the diffusivities of all species equal. Figure E9.2.2(c) shows snapshots of iso-vorticity plots and iso-concentration contours, the iso-concentration contours being simpler and more lamellarlike (compare with the cat eyes flow). Note that the limit of resolution is set by the computational grid. Striations created below this scale will be averaged out and the lamellar structure, if present, will be lost. If the reactions are slow the loss of resolution is not serious. However, depending on time-scales, the reaction might be diffusion controlled at these length scales and in this case the resolution problems become serious (see previous example). Similar treatments have included compressibility, for low Mach numbers, and also heat release (McMurtry et al., 1986). Problem 9.2.2 A possible way of connecting the results for microflow elements with Eulerian measurements is by computing the spectrum of concentration of a product or a passive scalar. Lundgren (1985) used an axisymmetric microflow element to obtain the product spectrum corresponding to small

296

Epilogue

length scales (large wave numbers). T o what extent is the spect~-u,,l dependent upon the characteristics of the microflow element'?

Problem 9.2.3* Study the computational problems associated with the determinatio~lof the striation thickness evolution by means of Eulerian sensors. Use the BV flow (Section 7.3) and consider a probe such that it marks 'one' if it is contained within a striation and 'zero' if it is not. Discuss the possibility of using this signal to determine a 'mixing time'. Problem 9.2.4* Consider the analysis of Kerr's direct simulation data (Kerr, 1985) done by Ashurst el al. (1986). Kerr conducted a three-dimensional Figure E9.2.2(c). Direct simulation of a mixing layer with a bimolecular reaction: (cl ) iso-vorticity ;and (c2)iso-concentration, showing a single roll-up. The initial condition is similar to that of 9.2.2(a) and is taken as u ( z ) = (Ul2) tanh(0.55z/zm); the time is made dimensionless with respect to (dul dz)-' at z = 0, whereas distances are made dimensionless with respect to UT; the Reynolds number is defined as Uz,lv and is equal to 50; the second Damkohler number based on a diffusional time defined with respect to the scale z, is 12.5. (Reproduced with permission from Riley, Metcalfe. and Orzag (1986).) The frames show the state of the system at various times.

simulation in a cubic box using up to 128"oints to compute velocity and passive scalar (concentration, c ) fields with periodic boundary conditions in all three directions (the equations were forced to maintain steady-state turbulence by driving the wave numbers less than a specific cut-off value). Ashurst rt t i / . took 'snapshoots' of the flow and at various points x calculated the eigenvalues of the symmetric part of Vv, denoted a, p, and ( r 3 /j3 7 ) and the direction of Vc (they analyzed 8 planes each of which consisted of 12g3 points and in each point they recorded the value of o,, rlo,/dx,, and Vc). They found that 23% of the points analyzed were such that /j < 0 and the eigenvalues were ordered approximately as IfiI x ($4 and H x 1 .2IYI (a sphere becomes a filament). In the remaining 77% of the cases fi > 0 and /j' -,ly1/3 and H x (3/4)Iyl (a sphere becomes a disk). Moreover, in this case Vc was found to be nearly parallel to the maximum direction of compression of the flow ( 7 ) . Attempt a rationalization of these findings in terms of the two-dimensional results of Sections 7.2 (tendril-whorl) and 7.3 (blinking vortex). Note that if the efficiency decreases, the flow becomes less extensional. Compare the results .~~ also with the classical findings of Batchelor and T o w n ~ e n d Investigate ;I

Figure E9.2.2(c) continued

298

Epilogue

the roll of loss of precision in similar computations in light of the Example 9.2.3. (For additional details see Ashurst et rrl., 1987.)

9.3. Passive and active microstructures The simplest problem of deformation and rotation of m i c r o s t r u c t u r ~ ~ corresponds to the case of passive material elements. By definition. a pussive mutrriul filument of length dx and orientation m = dx/(dx)evolves according to the equations of Chapter 3, i.e.,

D(ln E.)/Dt = (Vv):mm (9.3.1a ) DmlDt = ( V V ) ~- .(D:mrn)m. ~ (9.3. l b ) A similar viewpoint can be adopted in the case of trttrire mic~ro.str~rc~rir~.f (both infinitesimal filaments and planes). For example, in the casc of filaments, an active element described by a vector equation obeys a somewhat more complicated expression than (9.3.la,b); if the element hxs some internal resistance, such as surface tension or viscosity in the cast of a droplet or an 'entropic force' in the case of a coil, it stretches and rotates in a different way than a passive element. In the context of mixing the most interesting case corresponds to the case of droplets. As we have seen, for miscible fluids, molecular diffusion becomes the controlling mechanism when the striations are reduced to small enough length scales. In the case of immiscible fluids at the beginning of the process, the blobs or striations fed into the flow are large and the stretching and breakup are dominated by inertial and viscous forces. However, as the length scales of the blobs decrease, surface tension forces might play a dominant role in preventing further stretching and breakup. The deformation of the blobs is related in a complicated way to the velocity field. In practical mixing problems, the velocity field, even in the case of a single fluid, is not known, and in any case it would be perturbed by the presence of a second fluid. If the fluids are immiscible, coalescence and other hydrodynamic interactions further complicate the analysis. as does accounting for blobs of several length scales produced by the mixing process. In practice the fluids are more often than not, rheologicall~ complex. In the same style as Section 9.1, it is desirable to start the analysts from a clearly defined point. T o a first order approximation, the velocity field with respect to a frame fixed on the drop's center of mass, denoted X, and far away from it, denoted by the superscript a,can be approximated by v" = x L + higher order terms, (9.3.2)

-

where the tensor L = L(X. t ) = D + R is ;I function of the fluid mechanical path.'" The central point is to investigate the role of L in the stretching and breakup of the drop. Denote the viscosity of the d r o p a s p , and the viscosity of the surrounding fluid as jc, (the subindices 'i' and 'e' represent the it~tcriorof the d r o p and the crtclrior fluid, respectively). Neglecting any body forces that arise due to the rotation and translation of the moving frame and assuming that the Reynolds numbers p,Sr~'/ic, and p,Str'/~c, are vanishingly small ('tr' is a characteristic length scale, c.g., the radius of the undisturbed drop, and S is characteristic shear rate, IL(), the problem is governed by the creeping flow equations

V. v,,,

,L~,,,V~V,,, = VP,?

= 0,

m=~,e

(9.3.3)

with the boundary conditions v +v'

=x.L

as 1x1 +

7-

v, = vi and n . (T, T,)= 2 H o n , at the surface of the d r o p (denoted S ) , where o is the interfacial tension, and 2 H is the mean curvature at the point with exterior nit normal n. As formulated above the main parameters of the system are'4: (i) the dimensionless strain rate o r cupillury number Cu = u S ~ ~ / u , which is the ratio of the viscous to surface tension forces, (ii) the viscosity ratio p = jii;jc,, (iii) the imposed flow, L = L(X, 1 ), and (iv) the initial shape of the d r o p S,,. Note that the equations contain no explicit time dependence. This fact has an important consequence: the velocity field is instantaneously determined by the d r o p shape and the imposed flow, o r equivalently, the drop shape at time t is uniquely determined by the initial condition S,, and the value of L(X, r ) (Rallison, 1984). There are almost no experimental data studying the deformation of droplets o r microstructures with a prescribed deformation history (i.e., a specification of L(X, 1)). Also, a general theoretical analysis is plainly impossible and simplifications are necessary. Therefore, the bulk of analysis and experimentation has centered on prototypical two-dimensional (steady)flows such as simple shear and plane hyperbolic flow. Theoretically it is possible to s t ~ ~ dalso y the case of axisymmetric extensional flow (Acrivos, 19x3; Rallison, 19x4). However, only recently have studies been conducted with linear two-dimensional flows spanning the range between Pure shear and pure straining (Bentley a n d Leal, 19X6b). (See Problem 2.5.3.) -

300

Epilogue 9.3.1. Experimental studies

The focus of both theoretical and experimental investigations is to find the deformation produced by a given flow and how 'strong' it must be in order to break the droplet. Consider first some of the classical observations corresponding to the limits of simple sltecir and planor esret~siot~~il ,/lo,\.. Experiments2"ndicate that when Cu is small, the deformation of a n initially spherical droplet is small, and if the flow is steady the drop attains a steady shape (the time response to viscous stresses being of order p~i,rr,o ) . When the deviation from sphericity is small the usual measure or deformation is D = (I- b)/(l+ b); when the drops are significantly elongated (slender drops), the preferred measure is lla (see Figure 9.3.1). The most important experimental observations are the foll~wing'~: ( i ) At low Cu (Cu < 0.1 or SO) and in both flows the deformation D increases linearly with Cu. (ii) In both flows, for low values of the viscosity ratio p, and large values of Cu, drops are slender and have pointed ends. (iii) In most cases there is a critical capillary number, Cu,, such that ir the strain rate is increased the extent of deformation increases until surface tension forces are surpassed by the viscous forces and the drop continues to elongate in the steady flow. The critical capillary number corresponds to the value of Cu (the shear rate being increased quasi-statically) such that

dNdt > 0 for Cu > Cu,

and

dNdt = 0 for Cu < Crr,

However, in simple shear there is a value of p, z 4 such that if p > 11, drops would not break no matter how high the value of Cu. There have been many experimental studies for the limit cases of simple shear and planar extensional flows. However, experimental studies in the class of linear flows studied in Example 4.2.3 (see also Figure P2.5.3) which span the limit cases of fi = 0 (pure stretching) and D = 0 (pure rotation) as K is varied from 1 to - I, are relatively more recent. Simple shear corresponds to K = 0 (such flows can be generated in a four roller apparatus by varying the rotational speed of diagonally opposed pairs of cylinders, see Problem 2.5.3). In the first thorough study, Bentley (1985) considered the effect of such a flow by carrying out experiments under quasi-static conditions (dSldt small) and for K in the range 0.2 to 1.'' As might be expected, the curves of Ca, = Spea/a versus p for flows with more vorticity than hyperbolic two-dimensional flow (K < I ) were found to lie between those of simple shear and hyperbolic two-dimensional flow (Figure 9.3.2). For K = 1, Cu, was almost independent of p for p > I . For

Ptrssive und uctioe microstructures

30 1

K < 1, however, Ca,, increased with p, for p > I . For the flow with the most vorticity considered, there was a limiting value ( p z 14.7, based on theory), beyond which breakup was not possible. For high and intermediate viscosity ratios (p > 0.05),the data agreed well with the small deformation theory of Barthks-Biesel and Acrivos (1973), and for low viscosity ratios ( p < 0.01) with an empirically modified version of the result of Hinch and Acrivos (1979) for hyperbolic extensional flow. In a related study, Stone, Bentley, and Leal (1986) focused on the interfacial driven motion which

Figure 9.3.1. ( a )Drop being deformed in a flow; the flow in the neighborhood of the drop is Lax (the center of coordinates corresponds to the center of mass of the drop). (b) Deformation measures for nearly spherical drops, D = ( I - b)l(l+ b ) , and elongated drops, [la.

local velocity field x .L

(a)

near spherical drop (viscous)

(b)

pointed drop (non-viscous)

302

Epilogue

occurs after the flow is stopped abruptly, for various flows with K = I .(). 0.8, 0.6, 0.4, and 0.2 and viscosity ratios in the range 0.01 < p < 12. The! found that for modest extensions d r o p breakup does not occur with the flow but may occur as the result of a new mechanism, which they called 'end pinching'. In this mechanism the ends of the elongated d r o p become spherical, forming a dumbbell-like shape; the ends then proceed to pinch off leaving a cylindrical thread of fluid which can again repeat the proces\ Figure 0.3.2. Critical capillary number a s a function of viscosity ratio p = ~ ( , ' / 1 , . with flow type, K , a s a parameter. Comparison between experimental data and small ((I(<:')) deformation theory (full line) and large deformation theory (broken line). (Reproduced with permission from Bentley and Leal (1986a).) The valuea of K increase from top t o bottom (circles, K = 1.0: triangles; K = 0.8, diamonds. K = 0.6; inverted triangles, K = 0.4: crosses. K = 0.2).

or relax back to a spherical shape. This mechanism was observed only in moderately elongated drops.'Tigure 9.3.3 shows the critical elongation necessary to produce breakup after cessation of the flow. One of the conclusions of this study is that both very viscous and inviscid drops are hard to break; if the drops are very viscous they do not break in any flow, however, if the drops are inviscid, a very large elongation under quasi steady state conditions is needed before breakup can be achieved. 9.3.2. Theoretical studies

There is a strong correspondence between experimental and theoreticall anaytical studies. Analytical studies can be divided into those applicable to high viscosity droplets, where deviation from sphericity is small (small deformation analyses), and those applicable to low viscosity droplets highly elongated, only slightly bent and nearly axisymmetric and where it is possible to use results obtained in the context of slender-body theory [for reviews through 1983 see Acrivos ( 1 983) and Rallison ( 1 984)]. In the -

Figure 9.3.3. Necessary elongation to produce breakup following an abrupt halt of the flow, as a function of viscosity ratio p = /l,/ii,. The triangles denote the smallest elongation for which a drop was observed to break: the squares denote the largest elongation for which the droplet relaxed back to a sphere. (Reproduced with permission from Stone. Bentley. and Leal, (1986).)

304

Epilogue

context of theoretical studies, breakup can be interpreted in various w a ~ , : according to the theory i t might be unbounded growth. absence of ;I steady-state shape. o r reaching a prescribed amplitude in the growth 01' capillary waves. The basis of v , ~ r r l lrl~fi)r.r,~rrtior~ rrr1tr1j~sc.sis the exact general solutions of the creeping flow equations for spherical geometries. due to Lamb (19321. When the drop is only slightly deformed from a spherical shape, the I l o ~ field inside and outside the droplet can be obtained as a regular perturbation series, with the deformation from sphericity as the srn:lll parameter, and the evolution of the d r o p surface can be predicted. This method was first used by Taylor (1934) to study a d r o p in shear f l o ~ t . assuming the drop to be deformed into a n ellipsoid. Several years later Cox (1969) generalized Taylor's first order solution to the case of all lincar flows. The shape of a slightly deformed drop is given by

where i: is a small parameter. The shape of the drop is characterized h! the time evolution of the tensor A . Since the evolution equation is linealin A, a steady shape exists for all Crr and therefore the model is not capable of predicting breakup. However. if some of the higher order terms arc taken into account, the equation is non-linear and beyond a particular Crr no steady solution is possible. Using computer algebra to fi~cilit:\tc the anaysis. Barthes-Biesel and Acrivos (1973) obtained the second ortlcr solution to the problem. Even though the expansions should not be carrictl out to the same order of approximation. a point which was noted by the authors, the model was able to predict the breakup of drops. Their results were verified by the data of Bentley (1985). The /rrr
found that there could be several possible steady-state solutions for the drop shape. but based on physical grounds he selected the one in which the drop was stretched the least as the only realistic solution (this was also the solution found by Taylor (1964)). Several years later. Acrivos and Lo (1978) obtained the same conclusion by means of a stability analysis. Hinch and Acrivos (1980) studied the case of a hyperbolic two-dimensional flow as perturbation of the axisymmetric case. They found that even though the cross-section was not circular, the area in the axisymmetric and two-dimensional cascs were approximately the same (Cu, = 0 . 1 4 5 p 1 "). Thus, it appears that only the area and not the shape is important in the breakup process and suggests the possibility of using the approximation of circular cross-sectional area for other flows. The analysis corresponding to simple shcar flow is substantially more complicated than the case of axisymmetric flow. Hinch and Acrivos (1980) sti~diedthe case in which the d r o p is almost sligned with the streamlines in a shear flow by assuming that the d r o p cross section remains circular due to the slenderness. They found that the d r o p axis bends slightly and that the elongated droplet attains a sigmoidal shape. but that the deviation from a straight line was small for most cases of interest. A steady-state analysis of the d r o p shape indicated that steady shapes were possible for all values of Crr. However, a time dependent simulation indicated that the drops were unstable beyond Crr, = 0 . 0 5 4 1 p 2 ". In numerical calculations involvingjump in the strain rate they found that drops 'broke' (interpreted as failure of the numerical scheme) at the lower strain rate.'" Extensions of these analyses yield models which can be incorporated in the context of the flows of Chapters 7 and 8. For example, for the case p << 1 and Ca >> I , the dynamics of a nearly axisymmetric drop with pointed ends, characterized by a n orientation m ( m = 1 ) and a length I(t), is given by (Khakhar and Ottino, 1 9 8 6 ~ )

+

D In I(t)/Dt = L: mm - (a/2(5)'l'/~,u)[(/(t)/rr)'!~/(l 0.8~(l(t)/~)~)] (9.3.5a) DmlDr = ( G D + R)' em - ( G D : mm)m (9.3.5b) where G(t) = ( I + 1 2 . 5 ~ ~ I ( r ) ~-) /2.5u3/l(t)". (l The underlined term in (9.3.5a) acts as a resistance to the deformation [contrast with (9.3.la,b)]. A very long drop, (I(t)/u)+ r,, G I , rotates and stretches as a passive element since the resistance to stretching is unable to counterbalance the effect of the outer flow. Note also that since G > 1 the droplet 'feels' a flow which is slightly more extensional than the actual flow. Other researchers have sought a unification of results and developed a format which is useful for both low and high viscosity drops. Olbricht,

-

306

Epilog uc.

Rallison. and Lcal (1982) developed two phenomenological models: one in which thc microstructure was characterized as a vector [for example I(t)rn. as in (9.3.5a.b)l. thc other as tensor. The vector model is uscful elongated drops and macromolecules, for example in terms of a dumbbell model; the tensor model for high viscosity drops (see 9.3.4). Both models involve a number of phenomenological coefficients. which can be obtained by comparison of the models to particular cases. For example. in a n empirical scheme for breakup of drops, one of the coefficients M:\\ backed-out from the small deformation results of Barthes-Biesel and Acrivos (1973) and subsequently used to predict the critical strain rate for breakup in other flows. The vector equation, useful for a variety of axisymmetric elements. such as droplets or macromolecules, is

+

DplDt = p.(GD + R ) - G(F/(I F))[(D: pp)/(p.p)lp- [/)/(I + F)]p (9.3.6) where p is the length of the element, and G, F , and /) arc suitably selcctcd parameters." Such models can be incorporated in the context of thc simulations described in Chapters 7 and 8. Example 9.3.1 In the context of the model of Equation (9.3.5a,b) a drop is said to brcak when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with L:mm constant in time (for example, an orthogonal stagnation flow with thc drop axis initially coincident with the maximum direction of stretching). Rearranging Equation (9.3.5a), and defining a characteristic length as tr/pl'-', we obtain

L: mrn/(D: D)"' = efficiency = (1/2(5)'i2E)[(1,"2/(l+ 0.81;')] where I, denotes the steady-state length and E = p'lhCu. A graphical interpretation of the roots I, is given in Figure E9.3.1. The horizontal line represents the asymptotic value of the efficiency (i.e., corresponding to DrnlDt = 0), which in three-dimensions is (2/3)'12, and the value of the resistance is a function of the drop length for various values of the dimensionless strain rate E. For E < E , there are two steady states: one stable and the other unstable. For E > E,. there are no steady states and the drop extends indefinitely (see Khakhar and Ottino, 1 9 8 6 ~ ) . Pvohlem 9.3.1 Estimate the order of magnitude of the additional body forces that arise due to the translation and rotation of the moving frame. How significant Lire they to the analysis based on Equation (9.3.3)?

Problem 9.3.2

Stone. Hentley, and Leal (1986) investigatcd the time evolution of the d r o p Icngth, L(t)ltr, for conditions slightly above the critical capillary number in the flow field of Figure P2.5.3. Thcy plotted their results using a dimensionless time, ( ; K 1 ' r . They found that curves of L(r)/u versus G K ' ' r for a wide range of K frill approximately on the same curve. Justify. Problem 9.3.3 Cast the results of Hinch and Acrivos (1980) and Huckmaster (1973) in the form of Equations (9.3.5a.b).

9.4. Active microstructures as prototypes The interaction of active microstructures and a chaotic flow field might provide a prototype for a variety of physical problems. We will mention a few possible situations. For example, such a picture might help to understand why small amounts of high polymers ( o r needle-like particles), when added to laminar o r turbulent flows, produce (sometimes) consequences at a macroscopic level which are out of proportion to their c ~ n c e n t r a t i o n . ~Manifestation ' of these effects might be the drag reduction produced by polymers in turbulent flows and the increase of viscosity and non-Newtonian effects displayed by polymer solutions injected into porous media. Even though the two problems are very different one is a low -

Figure E9.3.1. Graphical interpretation of breakup of pointed drops. The ligure corresponds to the case in which the droplet does not rotate with the flow. 11

resistance to stretching

b

stable

unstable

drop length

308

Epilogue

Reynolds number flow, the other is high Reynolds flow there seems to be agreement that the explanation of the phenomena in the case of' polymers is due to flow-induced conformational changes and that much benefit can be obtained by focusing o n the behavior of a single microstructure in a linear, possibly time dependent. flow. Macromolecular stretching can modify the surrounding flow: in turbulence the smullc\r eddies might be suppressed; in porous media the hydrodynamic interactionh might result in non-Newtonian effects. Even though considerable experimental and theoretical work has bccn done in both areas, the suggestion here is that microstructures in chaotic flows might have a bearing o n these kinds of problems. However. since o u r primary focus is o n a few c o m m o n aspects of both problems, our review of the literature is rather superficial." A review of experimental studies of flow induced conformation studies is given by Leal (1984). M o h ~ of the work focusing o n conformational changes has been o n well controlled linear flows. for example, opposed jets, two-roll mill, four-roll mill. six-roll mill (see. e.g., Berry and MacKley, 1977); only a few studit\ have focused o n spatially periodic flows (Nollert and Olbricht, 19851. There should be n o question that such flows contain some of the r e l e ~ a n ~ aspects of both problems (i.e., porous media, turbulent flows) and thar the conceptual and experimental advantages are significant. T h e flows arc relatively simple to characterize completely and measurement of conform;^tional changes by light scattering are r e l a t ~ ~ e lsimple y t o perform ('I\ opposed t o measurements in a n actual turbulent flow o r the interior of a real porous media). What is missing, however is the unsretrtij nature of the flow experienced by microstructures in actual turbulent flows ancl topologically disordered porous media. O u r thesis here is that there i \ some opportunity for a n unexplored mid-ground. O n e such possible mid-ground is t o focus o n the behavior of microstructures in two-dimensional chaotic flows (either experimentnlly or theoretically). Carefully designed low Reynolds number chaotic flo\\\ provide several key ingredients missing in the studies listed above. Let L I \ consider a few

( i ) Chaotic flows provide unsteady (chaotic) Lagrangian histories. The flows are capable of generating regions where the stretching is mild and time-periodic and chaotic regions where. o l e r time. the stretching of material elements is nearly exponential (efficient flows)." In the chaotic regions the microstructure is neLer in steady extension: the maximum directions of stretching constantly change. ( i i ) T h e unsteadiness of the maximum directions of stretching, prcscnl

in turbulent flows, is also present in chaotic two-dimensional flows, such as the journal bearing flow (Chapter 8); the randomness associated with porous media, usually ascribed to the porous media itself, is present in the partitioned-pipe mixer, which is spatially periodic, or in the eccentric helical annular mixer, which is time periodic. (iii) Chaotic flows allow the investigation of spatial co-operative effects (e.g., percolation of chaotic regions, percolation of significantly stretched regions, see Figure 7.4.1 1 , color plates). (iv) Practical mixing problems start from an unmixed state; for example, polymer is injected into a pipe line and the drag reducing effects take place during mixing of the polymer with the flow. This effect cannot be incorporated into the conventional picture; however, chaotic flows can mimic this effect. The central question is what happens to microstructures, such as those described by Equation (9.3.6), when placed in a chaotic flow (and also, but this is a harder question since it involves hydrodynamic interactions, what happens to the ,flow as a result of the interaction between the microstructure and the flow). The simplest studies can proceed under the assumption that it is possible to decouple the global and local aspects of the problem (i.e., there is no hydrodynamic interactions). Even with this idealization, several questions come to mind. For example, where is the region of significantly stretched elements'? The answer to this question might have relevance in several co-operative phenomena. Besides stretching, several other phenomena can be incorporated at the microstructural level; breakup and coagulation/coalescence came to mind. A whole range of phenomena can be anticipated; for example it seems obvious that the process can generate fractal-like structures as obtained in diffusion limited aggregation (see, e.g., Witten and Sander, 1981, 1983; Meakin, 1985).

Example 9.4.1 * Consider stretching and breakup of pointed drops (with volume (4/3)7-rtr3) in the flow described in Section 8.2 (partitioned-pipe mixer). The objective is to determine if it is possible to predict (qualitatively) the behavior of the droplets based on the results obtained for the stretching of (passive) material filaments. In an admittedly oversimplified model (Franjione and Ottino, 1987, Unpublished), we assume that the drops are characterized by a vector p 1 = lpl have a viscosity ratio p , and obey an evolution equation of the type (9.3.5), i.e. DplDt = p - ( V v ) * - (((3 - 1 ) [ D : P P / ( P - P )+~ . f ) p

where

(Vv)* = GD + R, is the velocity gradient experienced by the droplet a s it moves in the no\\. and G a n d ,I' are functions defined a s G = [ I + 12.5(t1.1)" [ I - 2.5(ti 1)". f = [ ~ / 2 ( 5 ) " ~ ~ , a ] ( '12/[1 a / f ) + 0.8p(a/1)3].

Further, we will arbitrarily assume that the drops break when / , t i > 10. T h e droplets, somewhat stretched, are placed a t the entrance of tllc partitioned-pipe mixer and arranged in a square lattice. T h e initial orientation is purely radial and the initial number of droplets is 3.14 x I oJ. We will consider two operating conditions, P = 2 a n d /) = 8. T h e valucs of the parameters for the results presented below are p = 3 x and

Figure E9.4.1. Stretching and breakup of microstructures in the flow lield of the partitioned-pipe mixer for two operating conditions ( ( u ) /I = 2. ( h ) /I = 8 ) . The dots represent the region where the mlcrostructurei have broken after 10 sections. Note that in ( b ) a significant amount of breakup occurs near the center of the channel.

Active microstructures us prototypes

31 1

= 1. where L is the length of the plate and Vz the average (L/V,)a/2(5)' axial velocity. O n physical grounds, we restrict G to be positive. If an) of the droplets, as it moves through the mixer, produces G
Figure E9.4.1 continued

3 12

Epilogue

efficient regions are the large regular islands where no breakup occurs all. The results for P = 8 are no more encouraging. Figure E9.4.1 ( h )~ h o \ \ , ~ that breakup occurs near the walls and in two large central regions. whereas Figure 8.2.10(ul) and (02) although roughly consistent u.itll each other do not predict such a pattern. It thus appears that these results could hardly have been predicted based on the maps of stretching of Section 8.3. Obviously, extensive studies are needed in order to make qualitative predictions of breakup in this and other complex flows.

Example 9.4.2* The chaotic flows of Chapters 7 and 8 can be used to study variouh aggregation processes. Possibly the simplest example, which can bc Figure E9.4.2. Coagulation in the blinking vortex flow. Figure 9.4.2(11) shows the initial location of particles with capture diameter 0.003 which did not coagulate in the flow field of the blinking vortex flow operating at 11 = 0.8 after 25 cycles; the approximate location of the bounding K A M surface is shown by a full line; ( h ) example of a non-monotonic histogram of cluster order; the operating conditions correspond to 11 = 0.4 and a capture diameter 0.001 after 212 cycles of the flow. The initial arrangement was achieved in the following way: 3,600 particles were placed in the region - I < .u < 1, -0.01 < y < 0.01 and mixed for 5,000 cycles without coagulation; (c) time evolution of cluster order for a well mixed condition (11= 1.0, 6 = 0.001, 5,000 cycles). The symbols represent the computational results (circles, order-I: triangles. order-2; diamond, order-3, squares, the total number of clusters); the curves are the solution predicted by a mean field kinetic model.

regarded as model of gradient coagulation (Levich, 1962), corresponds to irreversible binding of particles with capture diameter 6 (Muzzio and Ottino, 1988). More complex examples might consider aggregation and breakup processes, growth of fractal-like structures, Brownian diffusion, etc. In this example, when two particles coagulate, the size of the cluster does not change. The number of particles forming part of a cluster is denoted as the order of the cluster. Figure 9.4.2((1)shows the initial location of particles with capture diameter 0.003 which did not coagulate in the flow field of the blinking vortex flow (Section 7.3) operating at p = 0.8 after 25 cycles. It is apparent that the regular regions are not nearly as effective as the chaotic regions and that most of the coagulation takes place within the bounding KAM surface. The dynamics of this simple coagulation process is quite complicated and it is possible to find conditions leading to non-monotonic histograms of cluster order; this seems to be due to the presence of relatively large regular and chaotic regions (Figure 9.4.2(b) shows a case at a value of p slightly above the transition to global chaos). Under 'well mixed' conditions, i.e., when the stirring is efficient enough as to destroy large scale concentration variations, Figure E9.4.2 continued 10,000.0 T

cluster order (b)

Figure E9.4.2 continued 10,000

time

(c)

the kinetics of cluster formation can be described by simple laws, such as Smoluchowski's mean field kinetics (even though there is no diffusion), and the dependency of the kinetic constant upon parameters such as p and 6, can be predicted based on simple models. Figure 9.4.2(c) shows the type of agreement observed between computations and a kinetic model for clusters of order one (single particles), two (dimers), and the total number of particles.

Bibliography The content of this chapter is more indicative of future possibilities than of the current state of the art. The common thread is the role played by local and global flows on diffusion and reaction processes and in stretching, breakup, and aggregation processes. For example, the fluid mechanical path is not very important in conversion and selectivity of simple reactions but appears to be very important in complex reactions such as polymerizations; small variations in the coefficients of a dynamical equation appear to determine whether a microstructure breaks or not. There is a substantial amount of literature on the foundations of both topics and the review presented here is a rather biased account suited to our needs. The sections on diffusion and reaction rely on transport across stretching interfaces. The main ideas and a brief historical review can be found in Ottino (1982). The importance of stretching of material surfaces is addressed in many works; one of the earliest is Batchelor (1952) and various other articles referred to in the chapter. In the context of chemical engineering, the idea of a local flow was used by Fisher (1968) and by Chan and Scriven (1970); in mechanical engineering, the idea has been used in the context of combustion, e.g., Spalding (1978a). The transformations of Section 9.1 can be found in Ranz (1979); Ottino (1982); and Chella and Ottino (1984) (many of the examples are taken from this work). The basic ideas of the lamellar viewpoint are discussed by Ranz ('1985). The review of stretching and breakup does not d o justice to the state of the art and the reader should consult the reviews by Rallison (1984); and Acrivos (1983). The most complete summary of experimental work is given by Grace (1982). Probably the most complete experimental studies are those from Leal et al. A partial listing includes Bentley and Leal (1986a); and Stone, Bentley, and Leal (1986). Complete details are given by Bentley (1985).

Epilogue Notes I These terms wcre coined by Arcf and Tryggvason (1984). 3 This viewpoint was suggested by Khakar. Chclln. and Ottino (19x4). 3 This is a good approximation if the distribution o f s is narrow. In chaotic flows the distribution can be very broad, especially if there are large regular regions. 4 See Example 9.2.3. 5 If the structure is nearly axisymmetric, as the whorl of Problem 4.3.5, we have. approximately.

where .Y is the radial distance. 6 In a few simple cases we might be able to solve some problems taking into account the distribution. One case is the vortex wrapping of Problem 4.2.5. 7 Throughout this chapter we will use an approximate form of the continuity cqunliol~. namely.

Similarly. the energy equation is written as

where T is the temperature, r,- is an average thermal diffusivity, and r,. is a gcncratic term that includes energy generation due to chemical reaction, viscous dissipation. and volume change (see Bird, Stewart, and Lightfoot, 1960, Chap. 18). We d o not consider multi-component effects. 8 This equation is valid for a material region S,", of size d where a linear velocity field approximation is valid. For turbulent flow thisscale can be taken as the Kolmogoro scale, (~'/O~I:)~:'. where E is the viscous dissipation per unit volume. Note, however. that there can be significant stretching and folding below these scales as seen in thc examples of Sections 7.1. 9 For example, we know that if the flow is a steady curvilineal flow, the stretching history s( decays as I/(; in chaotic flows, on the average, s(, is nearly constant, etc. 10 A brief list of other works using similar equations are: Saffman, 1963; Tennekes and Lumley. 1980; Foister and van de Ven. 1980; and Batchelor. 1979. l l The application of u,. to mixing was suggested by Corrsin (1954). 12 See Ottino and Macosko (1980). and references therein, and Townsend's hot spot model, Example 9.1.1. 13 This delinition of 'warped time' coincides with that of Ranz (1979) iff,. = t,,,,. 14 The choice of the scaling time t,. is also important in improving the accuracy of numerical solutions (it should be selected in such a way that each term (e.g.. i 2 C ,i v in Equations (9.1.6a.b) is of the order one. The coellicient (e.g.. r,/r,,,) then reflects the importance of the term). Thus, 1,. is chosen to be r, for 'slow' reactions and it,,, for 'hst' reactions. 15 Also, note that the streams themselves can be partially premixed and that everything can be generalized to the case of three or more streams. I6 Damkohler (19361 defined what is now called the first Damkiihler number, Du,. as (length/velocity )/characteristic reaction time which can be interpreted ns the riitio of the characteristic Eulcrian time t o a characteristic reaction time. O u r Du~'.'isdefined in an analogous fashion. the superscript ( L )refers to the use ofa Lagrangian time-scale.

3 17

Notes

-

17 For examplc. for the reaction A + IJB P. the conditions b r a stationary reacting zone are: (I)(.,,,, s.\~,,(, ,,, s,,) = I . ( I J C ,,#, c,,,)(D, D,)' = I. 18 The case of List reactions has been considered in the context of combustion and has reccivcd considerable attention in applicd mathematics (see for cxample Kapila. 1983): the LISC of fist reactions as tracers is discussed in Ottino (1981 ). 19 An extreme case is given by polymerizations; see Ex;imple 9.2.1. It is possible also to imagine situations where imperfect mixing can lead to 'thermal exlosions' due to imbalance between heat generation and dissipation due to thermal conduction. The presence ofa thermal explosion depends strongly upon the Lewis number (Ottino, 1982). 20 This topic dates from the work of Danckwerts (19581 and Zweitering (1959). See references in Chapter I. A book devoted to these issues is Nauman and Buffham (1983). 21 See Section 8.5. 22 Batchelor and Townsend ( 1956)obtained.for homogeneous turbulence. r = 0.43(1:/~0~,'. /I = o.12(c//O1 ;.= -0.55(1:,'~0' '. 23 The tensor L(X. t ) plays the same role as thestretching function, r(X, 1 ) in Section 9.1. 24 This formulation is largely due to Taylor (see Chapter I ). 25 As indicated in our presentation in Chapter I . Taylor (1934) invented the two basic experimental configurations; to mimic a two-dimensional hyperbolic flow hc invented the four-roller apparatus, to mimic shear flows he constructed a parallel flow apparatus. 26 A few references discussing various aspects of experimental results. prior to the work of Bentley (1985). are the following: Rumscheidt and Mason (1961). repeated some of the same experiments of Taylor (1934); their rcsults matched the theory for small deformations. Karam and Bellingr (1968). carried out experiments in simple shear flow using a Couette apparatus over a relatively wide range of p and obtained a graph of Cu, versus p which had a minimum for p in the range 0.2 to I. They also reported a viscosity ratio of p s 4 beyond which breakup was not possible in shear flow. A few years later an exhaustive experimental study was reported by Grace (1971, published 1982). In the first half of his paper Grace concentrated on shear and hyperbolic flows (as Taylor had done), but on a much wider range of viscosity ratios ( I O - q o 950). whereas in the second half of the paper he concentrated on the applicability ofthe results to static mixers. In addition to Cu,, the time for breakup and the length and deformation at breakup were also recorded. He also investigatcd the breakup under conditions of super-critical strain (Ctr > Cu,). Grace also found a maximum viscosity ratio, slightly lower than Karam and Bellinger's, p s 3.5, beyond which drops could not be broken in shear flow. At low viscosity ratios he found Ca, s p-n.s"or simple shear flow and Ca, s p O . l hfor two-dimensional extensional flow. For drops of higher viscosity ratio, the data are more sparse and indicate an increase of Ca, with p . There have been relatively few studies focusing on the effect of the history of the imposed flow and the dynamics of the drop. One early study is by Torza, Cox, and Mason (1972). They found that the rate of increase of the strain rate (dS/(/t) had a significant impact on breakup. Though data are only reported for relatively low values ofdS/dr. they found that the critical strain rate was a function ofdS/dt, especially at high dS/dt. 27 See Bentley and Leal ( 1986a)for details of the apparatus and Bentley and Leal ( 1986b) for experimental results. 28 It is significant that the breakup mechanism wus rtor due to capillary wave instabilities as studied by various researchers (e.g., Mikami, Cox, and Mason, 1975; Khakhar and Ottino, 1987). A long thread is not equivalent to an infinite cylinder and end effects cannot be neglected under most conditions of interest. Obviously, a significant difference between an infinite and a linite cylinder is that an infinite thread is an equilibrium shape whereas a finite cylinder is not; in the case of an infinite cylinder the only question is whether o r not it is stable (Stone and Leal, 1989). However, if drops are significantly stretched, Na>>l@, as they are in chaotic flows, the prevalent mode of breakup is by capillary waves.

'.

'

Epilog ur 20 Other results for time dependent flows indicate the severity of the flu~dmechanic;~l path. For example. Khakhnr and Ottino (1987) considered the number of f r a g n i c n ~ ~ N produced ny bursting of an ittlittirc, thread by means of a model based o ~ gi r o ~ l h of capillary w;~vcs.The flow field around the thread was taken to be I., = - :E(I)~. I.: = 1(1)I. Two cases were compared; one with constant 1. the other with l ( r ) = S/( l + Sr ) where S is a constant. The number of fragments goes as Cii2~"forconstant 1 and as Cii' for the u s e 1(r)= S, ( I + SI ) (where Crr is the capillary number. Cii = ~r,Sil.o. and il the initial diameter of the thread). This implies that under identical conditions n flow with dccaying efficiency will produce significantly fewer fragments than onc with constant elliciency. These results appear to have interesting implications in thc context ofexperimental studiesof breakup in chaotic flows,such as those ofsections 7.4 and 7.5. Although the experiments are just at the beginning and it is early to draw lirm conclusions, it does appear that there is significantly more stretching and breakup within the chaotic regions as compared with the regular regions, where thc elliciency of the flow is poor. 30 Choices corresponding to two-dimensional extensional flow and simple shear flow are given by Khakhar and Ottino (1986b). 31 This simple picture implicitly considers the possibility of drag reduction withour any wall eNects. Such a viewpoint was advocated by de Gennes (1986). 32 Fornn introduction to the literatureofdrag reduction see Lumley (1973). Virk (1975). Rerman (1978). and many of the articles in Vol. 24(5). J . 1!/' Rhrology. 1980. 33 Recent studies involving experiments in the journal bearing flow of Section 7.4 show that droplets in chaotic regions can stretch by several orders of magnitude and break primarily by capillary wave instabilities whereas drops placed in large islands, such as those of Figure 7.4.5(a), hardly deform at all (Tjahjadi, 1989). 34 Recent theoretical analyses and computations involving fast bimolecular reactions ( A B- P) in systems with initially distributed striations of the reactants - such as Figure E9.2.1- have shown scaling behavior. Owing to the unevenness in the striation distribution the reaction planes move and the average striation thickness increases in time. The remarkable result is that, regardless of the initial distribution, the system evolves in such a way as to produce a self-similar time-dependent striation thickness distribution (F. J . Muzzio and J . M. Ottino, 'Evolution of a lamellar system with diffusion and reaction: a scaling approach', Phys. Rev. Lerr., 63,47-50, 1989).

+

Cartesian vectors and tensors

The objective of this appendix is to cover some indispensable background with minimal mathematical complexity. The discussion is restricted to Cartesian tensors.'

Properties of vector spaces Vectors, in a general sense,2 are denoted as u, v, w, etc. We require them to satisfy the following properties:

I Sum (1) Associative (2) Commutative (3) Existence of zero (0) (4) Existence of negative

u+(v+w)=(u+v)+w u+v=v+u u+O=u u+(-u)=O. 11 Multiplication

( I ) Associative (2) Unit multipication (3) Distributive (4) Distributive Properties I + 11 form a vector

dpu)=( 4 0 ~ lu=u ( a + P)u = r u flu a(u + v) = au + av. space.

+

111 Inner product (1) u - v = v - u (2) ( u + v ) . w = u . w + v . w (3) a ( u - v ) = r u - v (4) u - u > 0 if u # 0 (magnitude of u, lul = + (u.u)'I2, denoted u). (5) u.o=o. Properties I + 11 + I l l form an Inner Product Spuce. No reference is necessary to basis, dimension, etc. Properties I , 11, and 111, are quite general and with proper definitionsof '+', addition, and '.', multiplication,

they might be obeycd by n suitablc class of functions. In particular. the, itre obeyed by Cartcaian vectors and tensors. Some important propertic, arc a direct consequence of 1 I l l . A space obeying 1 111 satisfies tllc inequalities: Cauchy S c h w a r ~ Iu-vI 6 IuI -IvI

IU + VI < IuI + IvI.

Triangle

Operations Co-ordinate system, basis, dual basis

From here on the discussion is restricted to three-dimensional space. In general, a point x in space is located by its co-ordinates, x = x ( s l . r'. .s.'). For example, in cylindrical coordinates, x is specified by r, (I, z . The nrirlr~.ol hllsi.s jei) is defined as ei = ix/isi

-

We require the je,) to be linearly independent, i.e., xiei = 0 implies th~tt xi 0." Consider a vector v located at x, i.e., v = v(x). The components defined with respect to the natural basis, r l ( x )= e, v(x) are called the conrrc~ruriantc8omponent.s.The dual basis, ( e i ) , is defincd such that e i - e J= 1 if i = j e i - e j = 0 if i # j The components defined with respect to the dual husis ui(x) = ei. v(x) are called the corclritrnr componrnts. A coordinate system is callcd orrhogpnul if the natural basis is such that e i . e j # O if i = j e i . e j = 0 if i # j For an orthogonal co-ordinate system the vectors e < i , - ei/leil = ei/c)i

form an orthonormal basis, i.e., e
.

= dij

where Oij is the Kroncker dclta ( h i i = I i f i = j and ijii = 0 if i # , j ) . '1'hc components with respect to the orthonormal basis (e(';] are called ph!,.sic.irl

C ~ O I I I ~ ~ ~ I I OFI oI r~ fut , ~ . ure ' LISC we

record the magnitudes cJi in two co-ordinate

systems Cylindrical : Spherical:

See Figure A . 1 . Figure A . I

or = I , o,, = I.. or =

=

I

I . o,, = r . o4 = I . sin O

322

Appendix: C u r t e s i u ~rectors ~ trr~dterlsors

Cross product - definition

.

The cross product, denoted x is defined in terms ( e i j e , x e2= e , ez x e , = -e, e , x e, = -el and so forth. In general. ei x e, = cijke, (qj, is called the permutation symbol) and is such that I;.. = , ~ k -

+ 1.

ifijk

-1,

ifijk = 321,132,or213 if any two indices are equal

0, Problem Verify that 6,,ijm, Problem Verify that

=

123,23I,or312

=3

bjrijks ijjsSkr

~ ~ ~ ~= l : ~ , , ,-

Problem Using the results given above prove u x (V x w)=v(u-w)-w(u.v). This identity provides a means for proving the Cauchy Schwarz inequalit! Vector representation in terms of bases

Any vector v admits an infinite number of possible representations. -1'hus v can be expressed in terms of the bases ( e , ) and ( e j ) as v = l,.e.= rrei 1

1

1

I.

The are the components of v with respect to { e , ) , and the ri are the components with respect to ( e i ) . Thus. one of the possible ways of identifying v is by means of its components. tii

[ V I E( r , , 2;2,c,) where the brackets, [ 1, denote matrix representation.5 It is important not to confuse the vector itself with its component representation. How to obtain components

Using the inner product and the representation of v (using any subindex. but never repeating them) we write: v . e1. = ~ . . e .I . e . c=. ( e i . e j )= ~ ; .ldJ . . =c1. 1

1

1

1

so v = ( v . e j ) e j ,which is a representation of the vector in terms of itself."

Appendix: Cartesian rectors and tensors

323

Remark: If the basis is not orthogonal we need the dual basis and v is expressed as v = ( V .el)ei= (v.ei)el.

-

Addition and subtraction and multiplication by scalar in component form v

+u

and XV

.v vxw

-

-

(ri+ ui)ei

=( ~ 1 1 : ~ ) ~ ~ .

Scalar product ~I . e .c..e . = uI.t.J.e.. eJ. = u.r .h.. = u.r - u .c. I ~ ' i - 11' J

I

Cross product

ejcj x e,wk = ej x ekcjwk= cjkieicjwk= ~

~

~

The origin of tensors - linear scalar functions and linear vector functions Linear vector functions provide a convenient and general way of introducing tensors. Before discussing them, let us consider linear scalar functions. Linear scalar functions (LSF)

Definition: Mapping f ( ): V

+R

(vectors into reals)

such that f(x + Y ) =.f(x) +!(Y) f(ax) = ?/'(XI. Representation theorem Any L S F has the representation f ( x ) = f - x , where the vector f is unique and independent of x.

( i ) Uniqueness: Assume f and f' such that f - x =f'.x = f(x). Then f - x - f'.x = 0 and using property III(2), (f - f'). x = 0. Since x is arbitrary, using property III(5) we obtain f = f'. (ii) Existence: f(x) = f(xiei) and using linearity f ( x ) = xi f(ei) = x.e,f(e,) = f(ei)ei. x, so f = f(ei)ei. Problem Show that

,/(.)=I 1,

where the bars denote magnitude, is not a LSF.

Linear. vector .frrnctions (1- V F ) I>cfinition: Mapping t ( . ) such that V + V (vectors into t(x T y ) = f ( ~ ) f(y) t(xx) = xt(x).

hectors)

+

Exutnplc 1 The multiplication of a vector by scalar is a LVF. Exanlple 2 I>cfinc (a b).u -a(b.u), where the dot can be interpreted as (a b) actin! on u. Note that the result is parallel to a. The entity (a b) is ailled a d!ad. Notc also that if the order of a and b is reversed the result of the operation is parallel to b. (b a).u = b(a.u). Similarly, u.(a b) =- (u.a)b. A note uhout notution The dyad (a b) is often denoted (a 8b). Customarily. within this notation the product of a vector and a tensor. here denoted as T-v. is denoted a, .l'v. The contraction (or inner product) of two tensors or a tensor a n d dyad. here designated as T: S or T: nn, is designated as T - S and Tan S n . respectively. The main motivation for this notation, which the reader i \ urged to master. is that the inner product of both vectors and tensors is designated with the same symbol.

Representation theorem Any I.VF has the representation

+

+

t(u)= ( e l t l ) - u (e,t,).u (e,t,).u = ( e l t l ezt, e,t,).u = (eiti).u= e,(ti.u) whcrc the ti arc unique.

+

+

( i ) L'rlicluc~rlc~s.~: Consider two representations t(u) = e,(t,. u) = ei(t;.u).

Taking the inner product with ei, ei.t(u) = ( t i - u )= (ti-u) and by the same reasoning as with LSF, ti = tj. ( i i ) E.~I'sIL~II(.~: We write t(u) = e,(t,-u)= ti(u) e, VL'CIO~

component5 basis

It is enough to prove that t j ( u )is a LSF. Write

t(u) = t,(u)e, t ( v ) = ti(v)ei.

Appendix: Cartesian vectors and tensors Then

+

+

+

t(u) t(v) = ti(u)ei ti(v)ei= [ti(u) ri(v)]ei. Also since t( ) is a LVF,

+ v) = ti(u + v)ei, ti(u) + t,(v) = ti@+ v). t(u

then

Also t(ru) = (eiti).ru = ei(ti.ru)= eiti(ru)= r(eiti).u = rei(ti.u) = eiati(u) then ri(ru)= rti(u).

Tensors A LVF is the most general linear transformation of vectors into vectors.

Symbolically, t(u) is written as t(u) = T - U , where T = e,t, e2t2+ e3t3.This sum of dyadics is called a tensor. (More precisely, a second rank tensor. Unless stated explicitly, whenever we use the word tensor we mean a second rank tensor.) Similarly, we can define linear transformations between tensors, and so on.

+

Components of a tensor

Any of the ti, for example, t,, has component representation t , = t l Jej, where t , means the jth component of t, [Remark: Note convention here. We could just as well have written tjl to mean the same thing.] Thus, in general, t 1. = r.*I J J and T = e.t. = e.t..e. = t.e.e. 1 1 I I J J I J L J ' Usually, the components rij are denoted with the same letter as the tensor, 1.e.. T = TiJeiej. As with vectors, the array of Tij,usually displayed in matrix form, is one of the infinitely many possible representations of the tensor T. Thus, we can write T = T.l .e.e. = Tr .ere'. I I J 1 I I J as the representations of T with respect to the bases { e , ; and (ef).

326

Appendix: Cartesian uectors and tensors How to obtain components

The ij component of S, denoted [SIjj, is obtained as [SIij = ei.(S.ej). Note that the vector corresponding to the first subindex is placed on the left. It is easy to see why this works: [SIij = ei.(Sklekel-ej) = Sk,ej.ekel.ej = SklSikblj = SilSlj= SkjSik = Sij. Some properties of tensors Zero tensor (0) Defined such that 0.u = 0 for all u. The components are

[OIij

= ei.(O-ej)= ei.O = 0.

Thus the component representation is

in any basis. Unit tensor ( I ) Defined such that I mu = u for u. The components are [llij = e i . ( l . e j )= ei.ej= Sij. Thus the component representation is

in any basis. Note also that

Product by scalar Defined such that 2T.u

-

T-xu.

-

Addition (and subtraction) Defined such that IT S).u T-u S . U .

+

+

Inverse tensor Givcn T , there exists 'r-'. such that if det T #0, then T - T - = 1. T - . '- T = 1. If det 1' # 0, T is callcd non-singular. I t can be shown that det 'l' docs not depend on the basis and that 'r-I is unique. The rule of computation of T ' is similar to that used in matrices.

'

Appendix: Cartesian Gectors and tensors Transpose Vector function t T ( . )such that tr(u).v= t(v).u. Theorem tT( ) is a LVF: Note tT(u+v).w= t(w).(u+ v ) = t(w).u + t(w).v = tT(u).w+ tT(v).w= (tT(u)+ tT(v)).w. Since w is arbitrary,

+

tT(u v ) = tT(u)+ uT(v).

Also tT(u).rw= atT(u).w= rt(w).u = t(w).ru = tT(ru).w

and since w is arbitrary, rtT(u)= tT(ru). Note: The transpose of T is denoted TT.Thus, we write (TT.v).u= (T-u).v. Component representation of transpose [TTIij= ei. (TT.ej)= ( ~ ~ . e = ~ (T.ei).ej ) - e ~ = Tji. Note: If T = Tijeiej,then TT= Tjieiej.If T = TT, T is called symmetric. If T = -TT, T is called antisymmctric or skew. Problem Show that T = (TT)T. Problem Show that any tensor T can be decomposed, uniquely. into symmetric and antisymmetric parts: T, = (T + TT)/2,symmetric. TA= (T - TT)/2,antisymmetric. Component representation of a dyad = akbl(ei.ek)(el.ej) = u,h,di,dlj = uih,dlj= aibj.

Note:

(a b)' = (b a). Useful relations inuolcing unit vectors ( 1 ) eiej-ek= ei(ej.ek)= eidjk (2) eiej.eke,= ei(ej.e,)el = eiSkjel (3) eieJ x ek= e,[ej x ek] (4) e, x eJek= Ce, x ejle, (5) eiej:eke,= (ej.ek)(ei.el) = 6jkiii,.

328

Appendix: Cartesian vectors and tensors

Warning: definition (5) is not universal. For example, some authors define eiej:ekel= (ej.el)(ei.ek) rather than (ej.e,)(ei.el). Remarks: ck6jk= (i. Tk16jk = Tjl bjk= d k j . Operations in terms of components

Addition (and subtraction)

T + S = TiJeiej+ Sijeiej= (Tij + Sij)eiej, Multiplication by scalar rS = rSijeiej,

which is a tensor.

which is a tensor

Product of vector by tensors

which is a vector. T.V = T,J€iC!j'Ckek = (Tijck)eibjk = (Tikuk)ei, Note: This corresponds to usual matrix multiplication ( [ 3 x 3][3 x I]). Similarly, v.T is interpreted as: v.T = ckek TiJeiej= (c, Tij)Skiej = (ciTij)ej. Note: This corresponds to usual matrix multiplication ([l x 3 ] [ 3 x 31). Inner product (or double dot product or contraction) of two tensors which is a scalar. T :S = T,-Jeiej: Sklekel= TijSklSilbjk = TikSki = TljSjl, Similarly, T : vw = T.IJ.uJ.w. I

uv: wz = U i V j W j Z i . Problem Show that if T = TT and S = -ST then T : S = 0. Thus if ':' is interpreted as an inner product this means that the projection is zero. Product of two tensors which is a tensor. T .S = TiJeiej. &,ekel= TijSklbjkeiel = TikSkleiel, Note: This corresponds to usual matrix multiplication ( [ 3 x 3][3 x 31).

Change of orthonormal basis The problem is how to find the components of vectors and tensors when they are seen in different bases. Consider two such bases: lei} and {ej). Denote a; = cos(ang1e between ei and ei)= e;.ei and arrange the a; as in Table A . l

Appendix: Cartesian vectors and tensors

Table A. 1

Then, for example, from Table A.l

In general, e!1 = a [J eJ . ei = aie'. J J'

:/I.

We denote the matrix of the a j as [Q] (note convention here)

CQI =

[:!:! a3

a3

a3

Q is an orthogonal matrix. Consider now some of the properties of Q. Note that e i . ei = d,,. Replacing the transformation we obtain

Similarly, 01

Note that a x b-c = Volume of prism

=

02

6, CI

b2 c2

Then, e; xe;.e;= 1 .

Since e; = a i e , + u$e, eI, = u i e , + uie,

then detjurj = 1 .

a31

+ a:e, + u:e3

c3I

330

Appendix: Cartesian vectors and tensors

Problem Prove that uiu;

= 6,,.

Problem Prove that detla;] = 1.

Transformations of Cartesian co-ordinates Vectors The same vector v can be written in (ej) and {ei}as 1: = cses = c'e' rr' Since es = aze: we write L' ase' = c'e' s r r r r (csa: - c:)e: = 0. Since all the components of 0 are zero, L"r = r IS) or in matrix form, [v'] = [Q][v]. Similarly, 0, = ci'u' or in matrix form, [v] = [QT][v']. r r7

Further properties of Q Since v transforms as [v'] = [Q][v] and the length of v is invariant, we obtain = Ca'lCa'l= CQICalCQICaI = CQTICQICalCal = Cal [a]= la21

then,

[QTICQI= C11 or

CQTl= CQ-

'I.

Also, det([QT][Q]) = det[l] = 1 det[QT] det[Q] = (det[Q])2 = 1 det[Q] = k 1. Q is called proper orthogonal if det[Q] = 1 .

+

Problem Prove that [Q] preserves angles between vectors.

Appendix: Cartesian uectors und tensors

33 1

Tensors

Consider two vectors v and u and a tensor T such that u = T . v . The question is: What happens when T is seen in terms of {e;) and (e,)'? We know that: U: =aJu. I J and Uj

= Tjqt',

Then u: = uiT.14 6 4 u: = uiT.J4 ~ P4 uP' ' Also, since u'.= T'.IP u'P 1

we get T i , = aja4,Tj, or, in matrix form, with the usual rules of multiplicaton,

Problem Derive the above result using matrix manipulations. Consider

[u'l

= [Ql [ul

and

Since

CQ- 'I = CQTI then

CQTICv'1= Cv1. In {e,),

Cul = CTICvI whereas in {el), [u'] = [T'][v'].

Combining

Cull= CQl [TI Cvl and

[v'l

= [QICvI.

332

Appendix: Cartesian t'ecrors and tensors

Scalar functions of tensor arguments There are two kinds: (i) Those for which the relationship is dependent upon the choice of basis. (ii) Those for which the relationship is independent of the choice of basis. These are called incuriant or isotropic functions. Examples: f(T) = TI, belongs to class (i); as we shall see det(T)and tr(T) belong to class (ii). Trace

Definition: Mapping, tr( ), 8 + R, such that tr(T + S) = tr(T) + tr(S) tr(rT) = r tr(T) (i.e., linear), and tr(a b) = a-b. Calculation: tr(T) = tr(Ti,eiej)= Tj tr(eiej)= 7;.j6ij= Ti. Properties: tr(ST)= tr(S) tr(S.T) = tr(T-S). Magnitude of a tensor

IT/

= /[tr(T-TT)]Ii2I.

The Cauchy-Schwarz inequality, for tensors, reads IT: SI < IT1 ISI. Problem Show that IT1 = I(T:TT)1121. (See earlier comments about notation.) Problem Show that in

1 1=

1 if In1 = 1

Problem Show that T: TT2 0, Problem Prove that tr(A.B.C) is invariant.

= tr(B.C.A).

Problem Prove uv: wz = uw: vz = (u.z)(v.w).

Use this result to prove that the trace

Appendix: Curtesiun rectors und tensors

333

Determinant

The determinant is calculated using the representation of T in any basis. The result is independent of the basis (without proof). det T =

7.11 TI, 7-13, 7i1 T,, TZ3. T3 1

T32

T.33

P~soblem

Prove that dct(A

I) =

[det(A)]-

I.

Problem

Prove that det(T') = det(T).

b:igenvalues and eigenvectors Givcn a tensor T, its eigenvalues and eigenvectors are the solutions of: Tan = i.n gives the right eigenvectors ni and eigenvalues i.i (i = 1 to 3) m - T = fim gives the left eigenvectors mj and eigenvalues fij ( i = 1 to 3). Note that the left eigenvectors are also given by T T - m= [Im. Procedure

Note that ( T - i.l).n = 0. T - n - i.1 .n = 0 or The system admits a non-trivial solution if and only if det(T - - j . l ) = 0 (third-order polynomial) which generates, at the most, three different eigenvalues (real or complex). The corresponding eigenvectors ni are the solutions of (T-j.I).n=O, replacing i. by L,.Normally, the eigenvalues are normalized, i.e., they are reported such that In,l = I . Churacteristic equation

det(T - i i ) = 0 can be written as i."1,i2 + lI,i-1I1,I = O where

334

Appendi.~:Curt esirln rectors unri tensors

I,.. II,.. and Ill, are called the principal invariants of the tensor. As we have sccn I,.. 11-,.and Ill-,. arc independent of the choice of basis. Chyley-Humilton theorem (without proof) Any tensor T satisfies its own characteristic equation, i.e., 1- .3 - l l ~ r 2 + l l , ' l ~ - I l I , l = O . Note: Besides I,, 11,. and 111, the other invariants commonly encountered in the literature are the moments I; = tr('f), II', = tr('f2), 111; = tr(-r3). Representution theorem for symmetric tensor (without proof) Any isotropic scalar function of a symmetric tensor argument can be expressed as a function of the invariants of the argument: ,f(T)= (](IT? 111, 111,) or. alternatively, ,/IT) = gC(l', , I I.;.. 11 I.;.).

Example Show that if T is symmetric, the eigenvectors belonging to different eigenvalues are orthogonal. Ilcnotc n,, ii and nj. il as two sets of eigenvcctors eigenvalues such that ii# /.j. then ,. I . n ., = ;..n. I ! and T.nj = ;.,nj then, cross-multiplying. (T.ni).nj= iini.nj and (T-nj).ni= ijnj-ni. Using the delinition of transpose. since ?' is symmetric. and

. (1.;-

.

/.j)nisnj = 0.

Since. by assumption. ii# i j then ni and nj are orthogonal.

Exumple Obtain the rcprcscntation of the above tensor with respect to its eigenvectors. Construct T..= ( ' r - nI . ) . nI . = ; . . n . . n i = / ...I (.)11' . I

I

So the matrix representation is

,

0

0

0

0

i,

i

Problem

Using the Cayley Hamilton theorem prove that 'T' = Q - T - Q ' and T have the same invariants. Problem

Using the Cayley Hamilton theorem prove that T". in terms of 1, '1'. and T'.

II

> 3, can be written

Problem

Show that if T is invariant, then Tnis invariant. Problem

Show that T and TT have the same cigenvalucs. Prohlcm

Show that if 'l'.n

=in.

thcn T

' an = i

In.

Problem

Show that if T = 'f.''and the 7; are real then the eigenvalues of T are real. Problem

Continuing the problem above, construct the matrix [L] such that its columns, n,, are the eigenvectors of T, i.e.,

rl.1=

rnln2n,l

where the ni are written as ( 3 x 1) vectors. Show that [LT][I'][I.] is diagonal. Problem

Prove that T and T ' = Q . T . V T have the same eigenvalues and eigenvectors. Problem

Show that if W

=

-

W ' thcn i. = 0 is an eigenvalue.

Problem

Show that thc maximum (or minimum) of D : nn, with D given and In1 = I , is given by the solution of the eigenvalue problem D . n = Ln. Identify i. with a 1.agrange multiplier.

Differentiation of scalars, vectors, and tensors Grudicnt of a scalur - definition

of

V f (f(x), sculur function)

d f ' = , f ( x + d x ) - , / ( x ) = dx.vf'.

Appendix: Curtesiun vectors und tensors

336 Denoting ldxl

= ds

V f is a vector. Gradient of a vector - definition of Vv (v(x), vector function)

dv = v(x + dx) - v(x) = dx .Vv. Denoting (dxl = ds

Vv is a tensor (second order). Gradient of a tensor - definition of VT (T(x), scalar function)

d T = T(x + dx) - T(x) EE dx-VT. Denoting ldxl

= ds

VT is a third order tensor. Note: In every case V( ) is independent of dx. This is what makes the definition useful. Example Find the components of V( ). Consider Vf, f(x). Note that the unit vector can be written as

Then, using the chain rule, df - G f dxk - .ds dx, ds Vf is vector, since df is scalar. Then df = dx,(Vf and

therefore

)k

Appendix: Carteslun rectors unrl tensors

and

Thus, the operator V( ), applied to

'.' is:

The symbol '.' denotes a scalar, or tensor- preceded by an operation (no or ' x '). If '.' is a vector or a tensor i t should be inserted in symbol. component form. ( ' a '

Divergence of a vector field Definition: V is followed by '.'. Thus,

If V . v = 0, v is called solenoidal. Divergence of a tensor field Definition: V is followed by

V S T= el

(

'.'. Thus, a?, T

-.Ti.k e . ek = e,.ejek-= aij I

i ui

axi

-

-

e

i~,,

a vector.

-

ix,

Luplacian of a scalar field Definition: divergence of the gradient

; Y ?r'f - ?l?f . e . - - =aij,(.xi ' ? x j c,ui?.uj ?Xi ?.u,

V.qf'=e,,

a scalar

This is often indicated as V2f Gradient of a urctor field Vv

= el

i

(

cj

e,ej, ? u, i u, Note that the ij component of Vv 1s (c't,/c'w,)

Luplacian of a vector field

Note: why not

r,e,

=

a tensor.

338

Appendix: Cartesiun vectors un~Itensors

Curl of a vector field

If V x v = 0, v is called irrotational. Other operations, such as V x T, can be defined similarly. Problem Prove the following results. Both j'and g are scalar functions, v and u are vector functions.

V(f9) = fV9 + 9Vf V(f+g)=Vf+Vg v~(u+v)=v.u+v~v V.(fv) = (Vf).v + f(V-v)

v

.

.

(fv) = f ( V V )+ ~ (Vf) v

Problem Prove the following results V*(vw) = v - v w ~(V'V)

+

v x f ~ = v f ~ v + f ( v ~ ~ ) V2(V.v)= v . v 2 v V.(fl)=Vf V . ( V V )= ~ V(V.v) V(a-x) = a, where a is a constant vector and x = xiei VxVf=O V-VXVEO vv:1 =v . v ~ ( l / l x l= ) -x/1xI3 where x = xiei.

Integral theorems for vectors and tensors Divergence theorem or Gauss's theorem Scalars

Vectors

Appendix: Curtesiun oectors

N I I ~tensors

Tensors

The unit vector n is the outward normal t o c'V. Stokes's theorem Vectors

[n.(V x v) ds]

= fc (t-v)dc

Tensors

The symbols are explained in Figure A.2.

Notes 1 Excellent references for this material in the context of continuum mechanics and fluid mechanics, respectively. are Chadwick (1976)and Aris (1962).Portions of this appendix are based on the appendix of Coleman, Markovitz, and Noll (1966). 2 As we shall see this definition includes both 'vectors' and 'tensors'.

3 Unless explicitly stated we use Einstein's summation convention: rcpcatcd indices are summed from 1 to 3, i.e.. riei = riel + r,e, + r3e3.For example. q j r j r imeans the double sum 11 Tijrjriwith both indims running from I to 3. 4 This applies to components of vectors and tensors. Throughout this work, unless explicitly stated otherwise, we will deal with orthonormal bases. 5 As is usual in fluid mechanics we do not distinguish between column and row vectors to simplify the notation; also, the distinction it is always clear by the context of the operation. Figure A.Z.

n

6 'I'he re~idcrrnighr find i r in\trucIr\c r r ) conlp'irc t l i i h \\irh [lie c\pari\iorl of a f~rnclrori in terms of base functions, a, in Fourier herich for example. and to identify the terms (v'e,) with the Fouriercoelficienta. 7 In the Iltcraturc one~on~e~imcscncouritcrs opcr,~lioli\siich as ~ v . V lO. n e intcrprclarion 15 lo regard V. by irhelf. as a vecror. We prefcr. ho~vcver10 regard ir ,is a n opcraror according to rhe defin~rion\g r \ c ~ lriho\e.

Frequently used symbols

acceleration r;rdius o f a sphcriciil droplet ir~tcrnia~erial arc;i densit! concent r;rtiorl c;ipillary nu111her stretching tensor diffusion cocflicicrlt differential arca clcr~lent(prcbcrlt St;lte) differential arca clement (rcfcrcncc s1:ite I cfficienc! Linetic energ! \xithin matcri;~l \olurnc I, hod? force right hirnd \idc ol' a ~1ilTerenti;rl equ;~tion tbpical ~ n a p p ~ n gas. i n Scctic~rih 7.11 and 7.3 dcfi)rrnat~ongr;~diel~t \ilic:~r rate r p i c a l Li~grangiirnful~ctiorl t ! p i ~ ~I l( u l e r ~ ; ~l un n c r i ~ ) ~ ~ cur\;Iturc Ha~n~lton~;~n action i n Il:rm~lroni;~nsbstemli intensit! o f segregation invariant set III horseshoe map Jacohian tlo\v cl~;~r;~ctcr 111 1111e:rr l 1 t 1 :~ h - I. pure rot;111~1rial tlou. li I. orthogon;tl stapn:itlori f l o ~ Icl~gtlro f a droplet rcli.rc~icc II~:II~~I.II Ic1ig111 III;II~~I;I~ IIII~ ;it ti111e r ~ ~ r ~ e l ~ t i of i t i ;
rcfcrcr~ccstirte oS~~i;tteri:~l fi1;1111ent out\idc normal t o a rrl;~terial \~ILII~I~ o r i c ~ ~ l ; ~ l iof o n:in ilreil ~ICIIICII~. prchcrlt st,ltc r c f c r c ~ ~ ct ei ~ t eo f an area element prc\sure \ixx)\it! r;ttio ( ~ r i c ~ ~ t ; ~of~ ~i onri ic r o s t r ~ ~ e~I ~i ~t rhe I~ll&!ltl 1 - - 1p; ;I \;~riahlci n a l i a ~ n i l t o n i ; ~ >>>tern n I \ \ irh cornp~)llcrlts11, I tllc fi\cd p ~ ~ i rindicated lt bj I~)c:itior~f' Ileal fluv a \ari;~hlei n ;I Hamiltolliitn \!stem ( ~ i t comporlcrlts h (1,)

orthgonnl tensor o r r n i ~ t r i x ~ ~ u l . cterm c 111 clitTu.;ion c q c ~ a t i o l ~ Kc! l ~ o l d s~ ~ u r l ~ h c r ~II.~;II~LIII 111ich11e~s Srr~)uhalrlull~bcr real time 1ractik)n tlrllc period rsmpcraturc \trc\\ tcnwlr \ elocit! rcfcrcncc material \ o l u ~ ~ ~ c ~ i i ~ i r c r\OIIIITIC i ~ ~ l :it t i r i ~ cr ~ I I ~ I ,II~~III~II I I ~ ~t c ~ \tri;~tior~s ) ~ L I \ I ~ I O Iiector ~ t p o \ i t i o l ~< ~ c c ~ ~ p i c h> p;irticIc \ ill t1111e1 1 p:irt~clcS. LI\LI;III! d e h i g ~ ~ i ~ h! tcd il\ l i c ~ \ i t i o;it~ ~time I

List of ji.eyirrnrly irsrrl synlbols

342 Creek symbols

9 @,(. )

4 (

,)

w

R

stretching function parameter in tendril- whorl flow parameter in tendril whorl flow mixing strength in the partitioned pipe mixer strain strain rate reduced time extensional rate perturbation parameter internal energy per unit mass energy dissipation per unit volume per unit time area stretch length stretch viscosity mixing strength in the blinking vortex flow kinematical viscosity dimensionless space with respect to striation thickness density Liapunov exponent interfacial tension warped time viscous part of the stress tensor potential function motion o r flow streamfunction frequency vorticity vorticity o r spin tensor

Subscripts hot c e i I

bottom wall in rhe cavity flow critical value exterior fluid interior fluid species-i

in

inncr cvlindcr in a journal bearing flow I,th term in il vector 11th iteration outer cylinder in the journal bearing flow bpecies-.\ stable manifold steady state top wall in the cavity flow unstable manifold reference state; initial condition quantities referring to an infinitesimal area element angle in Hamiltonian systems quantities referring to an infinitesimal line element quantities referring to infinite time

Superscripts (s) 0

r

species-s initial position quantity referring to the frame F' quantities referring to flow far away from droplet or microstructure

Special symbols D D . Dt

u

n [MI ( )

[ ,]

when applied to a vector f, it denotes the matrlx i f ,i X , (also Dx material derivative set union intersection brackets denote matrix representation of tensor M pointed brackets denote tlme or area average Poisson brackets

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Rhinc\. 1'. H. (11)X3) Vort~cit) d!~i;irilic\ of tlic (iee;i~~ic pc~icr;iI ~ I ~ C U ~ ; IIIIII. I I ~KPI.. II. 111riiI \ l o ~ / i . .18. 433 07. Klle!. J . J.. R. W. Mctc;ilfc. and S .I 0. r t . i ~(10X61 1)irect \irrl~~latiiin\ ofchern~c;~ll~ rccrctlng turbulcnt niix~nglci>er>./ > I I I , \t. 111ti/\.29. 406 2 2 . Rising H . (1089) Applications of chaos and dynamical systems approaches to mixing in fluid. P h . D . 'rhesi5. I k p t of M~ithemnticsand Statistics. 1;niversity of Ma,sachusetts. Amherst. Ki\ing. 11.. ,rnd J M . Oitino (1085) L,'\c of horscshtrc furict~on\In n u ~ d~ n ~ t i n\!\tern\. p U~tii.:IIII. /'/I!.\. SIC .. 30. 1600 (abstract oiil) ). Robert\. f-'. A. (I9X5)T-ITcct\ ofa pcriodicd~>turbancet)11 \tructure and ~lil'iingin I U ~ ~ L I I C I I I \hear I;rer> and \va!ies. PIi.1) . I h c ~ \ (;r.~duatc . A e r o n . ~ u f ~ c aL;rboriitor~c\. l ('~liforni;~ In.;titute of Tcchilolopx. Rom-Kedar. V.. A . 1.eonard. and S. Wiggins (1990) A n an:rlytic;rl study oftransport. mixing. a n d chaos in a n unsteady vortical flow. J. f'luitl Mecll. 214.337-94. Ko\cn\\reig. K. I:. (198.5) t e r r o l ~ ~ t l r o t / ~ t ~ r ~Kew t ~ ~ iYork: i ~ . \ . C'arnbr~dgelin~versit) Prc\s. Ro\hho. A. (10761 Structure of turbulent \hear flows: a new looh. .Al'ltl J . . 14. 1349 57. Roughton. I.. J . W . ant1 H. ('hirnce ( 19631 in /(,I/~tiiyirc~\ I)/ ,ir!l[itiic. c llrt~ii\rri.Vol. X. A. Wc~>sbergcr.cd.. Ch'lp. XIV. NCH York . I~iterscience. Rumscheidt. 1'. D.. and S. In pollmcr blend\: transport-morpI~oIog> relationship>. 1'h.D. Thcais. Dcpt. o f C h c m ~ c a lEngineering. L!ni\crsi~qofMassachusct1s. Amherst. Sax. J . IS.. a n d J . M. Ottlrlo (1985) Influence of morphology on the transport properllcs of polyhutirdiene p o l y s t ~ r e n eblends: experimental results. PoI!.ti~rt.. 26. 1073- 80. Schlichting, 11. (1955) Ho~rtit/tir! Itr,,rr r l i c o r ~ .K c u Yorh: Mc
ccce~itric~ I I I I I ~ I ~.11('/11, U~. .I.. 6-12 467. S o h e . I . J . (I9S.i) I)ispcr\~onc a i ~ \ ~h!d \cp;~rationduring o\c~llator>flo\r. through :I rurrc>\\ed cl1;1111ie1. (./ICJIII. b.~tq.Sc 1,. 40. 2120 34. Solomon. T . H. ant1 J. 1'. (iolluh 119XX)Chaotic particle trun\port in time dependent Kayleigh-Hhnard convection. Pl~j,.\.KIT. :l. to appear. Sp;~l(li~ip. I). I*. I 19771 The l:S('IHO thtor! of turbi~let~t c o ~ ~ i b u s t i o1111pc,ri[rl r~. ('o11~,!11, .\li,<.ll.1;lt.,so.t i 1.s 76 I.?. Sp;~lcling.I). H. (107X;1)The inflilence\ of laminar tr;lnsport ; ~ n dchcn~icalkinetic5 ~ I I the Ilrnc-nie;~rire:ictio~irate 111 :I l ~ ~ r b ~ ~l1;11iic. l e n tin I'ro(,. 1 ?!/I SVIII/I. I/III.)OII (~~III/III\~~O pp. 431 40. The ombu bus ti or^ In\titute. Spalding. I). 13. 11078b1<'liemic;~l reaction5 iri ti~rhulentfluitls. in I'l~t~.\ico-('lrc~~trrctrl I I i i l r i ~ c l ~ ~ ~ \r:r ~IJrr~[.. ~ r i c I.~r.rc11 OOrh hirrl~ilir~. ('or~/c~-r~lc.c,. pp. 321 28. I.ontlon: Adtance l ' ~ ~ h l i c ; ~ t ~ ohnlsa. . Sperlccr. K S.. and K. M . Wile! 11951 ) The mixing of \ c r \iscou\ liqultls. . I . ('olloicl. 5c.i.. 6. 133 45. Stone, 11.. B. J . Hentlcy, and L. <;. Leal (1986) A n experimental study ol'transicnt effects in the breakup of viscous drops. J. Fluid M e c l ~.. 173. 13 1-58. Stone. f 1 . A.. and L. ( i . Ideal (1989) Relaxation and breakupof an initialy extended drop in an otherwise quiescent fluid. J. I.71cid Mrcll. 198.39')-427. Stuart. .I.'1.. (1967) O n finite arr~pl~tude oscillation in laminar mixing I;lycrh. J . Flrrid .\lc~e~h.. 29. 417 40. Swanson, P. I). and J . M. Ottino (1990) A comparative. compuntalional and cxperimcntal study of chaotic mixing of viscous fluids. J. Rttici Mccll. 213. 277-249. Swanson. P. D.. and .I. \I.Ottino (198.5)('hilotic mixing of tiscous liquids b e t ~ e e ~ i ccccntrlc cylinders. Mrrll. 4111. \'IIJ\. Sot... 30. 1702 (abstract orily). S ~ i r ~ n e !H . . L. (1985) Ohscr\ations of complex d n a r n i c s arid ch;ios. in F ~ r r ~ t l i r r ~ ~ ~ ~ ~ ~ r /JVO/J/~,I~I.\ I I I .\r[rri\ri(~trl ~irr,c./~rr~ric.\ 1'1. li. (i. I). Cohen. ed.. pp. 253 80. Amsterdam: lilseiicr (the a p p e n d i of thi5 paper was published in 1083 in IJl~j.\ii~ir. 7D. 3 15). Synge. J . I.. ( IYhO) ('lassical dxnamic5. in t1~111dhrrc.L tier I'l~,r.\iL.Ill I . S. I-liiggc. ed.. pp. I 225. Berlin: Spririgcr-Verlag. Tadmor. Z.. and ('. <;. <;egos (1070) P r ~ ~ ~ c . i p01l rp01?.111ur ~\ proc.c,\vi~lll.Kcu Yorh: Wiley Intcr~cic~ice. Ttlnner. K. I. (19761 A test ptirticlc ;ippro;ich to flo\r classification for ii\coelastic fluids. Al('11t; J.. 22. 910 14. 'Fanner. K. I.. and K . K. lluilgol 119751 O n a elassilication scheme for flou fields. K/rc,ol. A(,rlr. 14. 959 -62. Taylor. G. I. (I9341 The formation of emulsion.; in defirl;~blcfields of flow. I'roc.. Ko!. Soc., Lond.. A146. 501-23. Ta>lor. G . 1. 119641 <'onic;d free surf;~ces; ~ n dflu~ciin1crf;lces. in I'ro~.. llrlr I~rr.C O ~ I O . /lpl~l.:Glcc.l~..pp. 790 6. Munich. Tennckes. I I.. and J. I.. Lumle! ( 1 0 8 0 ) .A fir.\/ co[cr\cJill rtohtrlr,rrc,e. ('iimbridge. Mass. : M1.T l'ress. 6th printing. Thirring. W. ( lc)7X) A c.ordr\c, it1 tirur/~c~~n[iti~~~rl p/~~..\ic\. I : (~lci.~,~iccrI ~ / J I I ~ I I I I.\j~.\roiir\. ~~,LI/ New York. Springer-Vcrlag. I'orra. S.. R . ('ox. :rnd S. (;. Mason (1072) l'article motions in shc;~rcdsuspensions. XXVII. Tronhlenl and steady deformation and burst of liquid drops. J. ('~lloitl. S1.i.. 38. 305 41 I . l'ownscnd. A . A . 11051 1 .She dlflusioll or heat spota in ihotropic turbulcricc. i.'rrJc'. KOJ.

S o . . 1 . t 1 1 4209. 418 30. 'rritton. D. J. (1977)I'l~y.\i~,trl \l~riili/ytrior~r~ 3 . Kcu Yorh: Van Kostrand Re~nhold(rcprintcd 10x21. Truesdell. C. A. (1954) '1'11~~ klt~rtilclri~~s I!/ r~~rric,iry. Bloomington: 1ndi;uiu Univers~tyPress. trues dell.(^. A. ( 1977) (1 /;r,vr cotirv<,in ri~riot~ul c,ot~ri~~tit~tt~ t~~c,(,l~clt~ic,.\. Purr 1: F'~it~di~ri~~,trr~r/ t,otli.oprs.New York: Academic I'ress. Truesdell, <'. A,. and K. l'oupin ( I 960) The cl;rssic;rl field theories. in I l ~ r t ~ d h ider i~~l~ Pl~ysik.111 1. S. Fliigpe. ed.. pp. 226 703. Rerlln: Springer-Verlag. Ulbrecht. J. J.. and G . K. Patterson. eds. ( 1 985) Mixing i~/'liyuid,sh~rr~c~~I~i~n~culu~i~~~ New York: <;ordon and 13re;ich. van Ilyke. M . (1982) At1 trlh~rti~ of fllritl rrlorior~.Stanford: The Parabolic Press. Veronis. G. (1973) Large scale ocean circulation. ..lilr.. Appl. .\lrc.l1.. 13. 1 -92. Veronis. G. (1977) The use of tracer in circulation studies. in 7'11~SPU. ).i)lunlr 6 . .Wtrrirtc. tr~r~tlt.llit~q. F.. D. (ioldberg. I. N.McCave. J. J. 0'Rrien.and J. H. Steele.eds.. pp. 169-88. New York: Wiley. Virk. P. S. (1975) Drag reduction fundamentals. /IIC'kE J.. 21. 625 56. von Mises, R.. and K . 0 . Friedrichs (1971) Fluit1 tnrc.11ut1ic.s.New York. Springer-Verl;~g. Wallerstein. G . (1988) Mixing in stars, Scirtlc,r, 240, 1743. 50. Walters. P. (1982) An irltrodl~c,riot~ ro rr(qodic, rhrory. hew York. Springer-Verlag. Wannier. G . H . (1950) A contribution to the hydrodynamics of lubrication. Qirurr. .4ppl. Murk.. VIII. 1 - 32. Welander. P. (1955) Studies on the general development of motion in a two-dimensional. ideal fluid. 7i~llrt.s,7, 141 56. Wickert. 1'. D., <:.W.. Macosko. and W. E. Kanz (1987) Small-scale mixing phcnonien;t during reaction injection moulding. Polytnrr. 28. l I05 10. Wiggins, S. (1988a) On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class oiardin;rr> diMerential equations. SIAM J . App1::Vurh.. ro appear Wiggins, S. (1988b) Glohul hifurcutiot~.~ und chuos-ut~ulyticulmrlliods. New York: Springer-Vcrlag. Witten. T . A.. and L. M. Sander (1981) Iliffusion-limited aggregation. a kinetic critical phenomenon. P11j.s. Rer.. l.r,rr.. 47. 1400 3. Witten. 'I'. A,, and I.. M. Sander (1983) Diffusion-limited aggregation. Ph!,s. Her... B27. 5686 97. Zweilcring. I'h. N. (1959) The degree of m~xingin continuous flow systems. Chetn. k t ~ y . Sci.. 11. 1 15.

Author index

Italic numbers refer to figures. Abraham. R . 11.. 109, 124, 127. 152 Acrivos. A.. 218. 299. 301. 303-7. 31 5 Allcgrc. C. J.. 8 Andronob. A. A.. 128 Aref. I{.. 1 . 3. 130. 140. 153, 171. 216. 272. 3 16 Aris. R.. 17. 40. 339 Arnold.V.I..18.101 2.127.132.136.146. 152 - 3 Ashurst. W. '1' .. 296. 298 Astarita, Ci.. 67 Balachandar. S., 2 l 6 Balals. N. L.. 150 1 and 216 lin Bern or (11. (I979)J Ballal. B. Y.. 218 Barthbs-Biesel. D.. 301. 304. 306 Batchelor. C;. K.. 6. Y. 51, 57 8. 62. 75. 139 40. 277. 2Y7, 315 17 Bcllingcr. J . C.. 31 7 Bcntlcy. B. J.. 41. 299. 302 3. 304. 307. 315. 317 Berker. R . 95 Rcrman, N. S.. 318 Berry. M. V.. 150 1 . 216. 308 Bigg. D.. 2 1 X Bird. R . B.. 6. 22. 316 Birhhoff. G . I).. 145 6. 152 Howen. R . M.. 22. 41 Brenner. H.. 47 Bro;tdwell. J E.. 288 Brothman. A., 3 Brown. <;. I... X Buckmasrer. J. D.. 304. 307 Buffham. B. A,. 16. 317 Burgers. J . M.. 57 G~ntwell.I%. J.. 62 ("~suell. J. H., 34 ('h;idwich. P.. 20. 40. 62 3. 339 <'h;~ihcn.J.. Y. I I. 21 6. 218 C-han. K . I-..6

Cfian. W. C'.. 315 Chance. B.. 17 <'hang. H.-N.. 218 Chella. R.. 7.72.81 .X6.90.94.28&1.285. 315 16 Chevray. R.. 9. 1 I . 216. 218 Lin haikcn (11. (I986)] Chicn. W.-L.. 9. 201. 213. 216 Chong. M . S.. 261. 206 7. 272 Cohen. E. G . D.. [see Helleman (I9XO)and Swinney (I985)] Coleman. H. I).. 9 4 - 5. 339 Coles. I).. 62 Cooper. A . R.. I 6 Corcos. G . M.. 290. 291 3. 292. 295 Corrsin. S.. 3. 6. 9. 11. 316 Cox. R . Ci.. 304. 317 Crawford. J . D.. 271

f

Damkiitiler. G.. 316 Danckwcrts. 1'. V.. 4. 7. 16. 187. 317 1)anielson. T. J.. 271 I>;t\ies R . M. [see Batchelor and .Townsend ( 1956)J de Gennes. 1'. Ci.. 318 1)ebaney. R. I... 127 Dimotakis, P. I<. Fig. 1.3.5 (see color plates) 12. 13. 60 Iloherty. M. F.. l.(ll. 173 Dombrc. T.. 111. 127 Duffing. Ci.. 218 Echi~rt.(... xi. 4 Eringen. A. ('. [see Howen ( 1976~1 Eisc;~ndc. I). 1. .. 153,, t.airlie. B. D.. 35. 171 1Fcldm;in. S. M.. 3 I-'cigenbaum. M. J.. 127 Fcingold. M.. 272 1'1elds. S. D.. 13. 285. 2x6 7 , 2x9 1;inlnyson. B. A,. 224 Fisher. I>. A,. 31 5 Folstcr. R. T.. 3 16

Franjionc. J. G.. 110. 227.228-3.5. 2.38-4.5. 27 1 . 204 t'rictlrichs. K. 0..22 Frisch. U.. 221 and 271 1 in 1)onibre 1.1 111. (I9X6)I. 272 <;alloway. I).. 272 (iarcia-RcjOn. A. [see Khirkhar. ('hell;r.;~nd Ottino (1984) and I.eal (IOX4)] (iibbs. J . W.. 191. 217 (iihson. ('. H.. 2x8. 3 0 . 296 tin Ashurst 1.1 ttl. (I9XhlJ. 298 [in Ashurst rr ttl. (I9X7)J (iiesckus. 11.. 41 ~;illirni. K. W,. 250 (iogos. C. (i.. 16 (ioldherg. F. D.. 16 (ioldstein. G . A,. 272 Goldstein. H.. 152 3 (iollub. J. P.. 216 Grace. 11. P.. 315. 317 (ireene. J . M.. 128. 153. 221 and 271 [ i n Dombrc t.1 01. (19X6)] (iuckenheimcr. J.. 97. 101. 1 12. 127 9. 141. 152 3. 162. 177, 266 Harna. F. K.. 25. 62 Hawthorne. W . K..3. 5 Hellernirn. R. I{. <;., 132. 145, 152 tleller. J. P.. 40 Ilinon. .M.. 2 2 0 , 221 and 271 [in I>ombre cJr trl. I9X6)I. 269 Ilinch. F. J.. 305. 307 Ilirsch. M. W., 25. 97. 1 0 1 3. 127 X. 136 Hoffman. K . R. A.. 10. 210 Hollirnd. W. R.. I6 Holloway, G . . xi Holmes. l'.. 97. 101. 112. 127 9. 141. I52 3. 162, 177. 266 Hottel. H. ('.. 4. 5 Huilgol. R. R.. 34. 67 Ilyman. I).. xi loos. <;.. 127 Jeffre. G . B.. 2 1 X Jcpson. C. ti.. 4. 7. I h Jones. S. W.. 153. 272 ~ o s ~ p r). h . D.. I 27 Jou. W.-H.. 295 (in McMurtry cJr ~rl.( It)X6)] Kadanoff. L. I>.. 272 Kanterovich. I.. V.. 224 Kupila. A. K.. 317 Karam. H. J.. 71 7 Karweit. M.. 6. 0 . I I Kari~kia.J. Y.. 2IX Kcrr. K . M.. ?)(I. 290 [ i n A\hurst 1.1 '11. (10x6)).208 [in Ashursr rr rtl. llOX7)1

Ksrb~ein.A.. 206 [in Ashursl rr trl. (l986)j. 2')X [in Ashurst rr rrl. (I9X7)I Khaikcn. S. F.. 128 Khahhar. I). V.. 67. X X 03. 155. 158. 104. 165. 167 8 . 169. 177. 178. 183. 18.5 l.'ip. 7.3.10 (set color plates). 187. IXS. /YO. 213. 227. 22s 35. 238 4.i. 271 2. 305 6. 316 I X Kim. J.-S.. I28 Kirstrnannson. S. S.. xi Kol~nogorov.A. N.. 146 Koochesfh;tni. M . M.. 13. Fig. 1.3.5 (see color plates) Krylov. V. 1.. 224 Kusch. H . A,. 272 Lamb. 11.. 63. 139. 304 1.andford. 0. 1.. 127. 129 1 . ~ 1 .I*. Ci.. 41. 67. 85. 299, 301. .
Mctcalfc. K. W..205 and 3 5 [in McMurtr!, P I trl. ( 19X6)] M~iike-Lye.R. C'.. 12. 13 Middleman. S.. 16. 120. 218. 221 Mikami. '1' .. 317 Milne-'l~homaorl. I.. M . 63 Minorak). X . . 153 Moffat. t t . K.. 59 Mohr. W. I)..4. 7. 16 Monson. P. A,. X Moser. J.. 126. 137. 148. 152 3 Muzzio. F. J . , 313 Naum;tn. F . B.. 16. 317 Noll. W . , 82. 84.04,. 339 Sollert. M I:.. 308 O'Brien. J . J.. 16 [in (ioldbergrr ul. (197711 Olbricht. W. I... 67. 85. 3 0 5 . 308 Omohundro. S., 272 Onsagcr. L.. 41 Or7ag. S. A , , 295 Osecn, C. W.. 95 Ostrousky. N. [see Meakin (1985)J Ottino. J . M..3. 7. X. 9. 13 tin 1.w 1.1 ui. (1980)].l3.16 17.75.77.81 X6.XX.XY. 90. 94. 121. 140. 1.55. 158. 164. 165. 167. 168. 109. 173-5. 178 50. 183, 1x5-6. Fig. 7.3.10(seccolor plates). 188 YO. 201. 21.%13.21~17.219.227.228-3.5. 238-45. 271.27X.280.2XS.286.288-9. 289,294.305-6.313, 3 1.5-1 8 Palis. J.. 153 Pan. I-.. 2IX I'iipantoniou. I). A.. 12. 13 I'alterson. A. K.. 29 Patterson. G . K.. xi. 315 jiri Ran/ il9Xj)l Peizoto. M . M..100 l'ercival. I . C.. 102. 127. 157. 153 Perry. A . F... 35. 261 2. 266 -7. 272 Piro. 0..272 Poincark. H.. 125. 153 Pomphrey. K.. 140 Prandtl. I... x i . 6. 22. 40 Rallison. J . M.. 16. 67. 83. 299. 303. 306. 3 15 Kalph. M . E.. 2.51. 271 Riingcl-Nafiiile. ('. 1 acc Kh;~Lh;ir.C'hella. and Ottino (1984) and Leal (1984)l Ranr.. W . I:.. 13. 13 tin Lcc ui.i 1 PXOi]. 278 315. 316 Kcichl. 1.F:.. 7 Rhtncs. P. H.. 16 Richard\. I)..102. 127. 152 K~le).J . J.. 195. 105 [in M c M u r ~ r ycr (11. 119X6)I

Rising. t1..9. 121-3. 155. 158. 164. 165. 167-8. 169.17#40.18.7.INS-6. Fig. 7.3.10 (see color plates). 210. 21.3-14. 216. 216 [in O t t ~ n o e r u l(IOXX)]. . 217 R i ~ l i n K. S.. 218 Kobcrts. F. A , . 138 Roberts. J . M.. 17 Kom-Kedar. V.. 143 Kosenbweig. R E.. 47 Koshko. A,. X Roughton. F. J. W.. 17 Rumacheidt. 1.. D.. 317 Kiiasmann. H.. 140. 153 Kqu. H.-W.. 218 Saffman. 1'. G.. 3 16 Salarn. F. M. A,. 153 Snndcr. L. M.. 309 Sax. J . L . . X. 17. Fig. 1.3.4 (bee color plates) Saxton. R. I... 4. 7. I0 Schlichting. H.. 6 Schowalter, W. R.. 63 Schuster. H . Ci.. 122. 127 S c r i ~ c n .1.. f .. 3 15 Scrrin. J.. 18. 20. 45 6. 52. 54. 57. 62 S c ~ i c k LI. , M.. 8 S h i l ~ C.'. I).. 109. 124. 127 Sherman. F. S.. 200. 291 3. 292. 295 Shinnar. R.. 16 Smale.S..20.97. 101 3. 108. 112. 115. 119. 126 9, 113) S m ~ t hJ. . C.. xi Snyder. W. 1.. 277 Sohey. I . J.. 220. 249. 232 3. 271 Sofia. S.. 6 Solornon. 1.. H.. 216 Souard. A . M..221 and 271 [in Domhre 1.1 (11. i 19x6 11 Spalding. TI. B . 16, 271 Spencer. K. S.. 4. 6. 129 Stanley. H . E . [see Mcakin (19X5)l Steelc. J . 11. 16 [in (hldherg cr (11. (1977)] Stewart. W. E.. 6. 22. 316 Stone. 11.. 303. 307. 315 Stuart. J . T.. I67 Suanaon. 1'. I).. 216. 216 tin Ottino rr (11. Cl9XXiI S ~ a n s o n W. . X.. 250 Swinney. H . L.. 127 Synge. J . L.. 118. IS?

c2r

'I'abor. M.. 19. 11. 116. 218. in Chiliken rr (11. (1986)]. 150 1 and 216 lin Berr) t11. ( I979 I] Tadmor. %.. 16 'l'an. Q . % 19.I11. . 216. 118. in ('haiken 1.1 ctl. (IOX6)I

358

Aurhor int1e.u

Tanner. R . I.. 67 Taylor. G . I.. 3. 4. 7. 40. 304 5. 317 Tict.jens. 0.Ci.. xi. 22. 40 'Tennehcs. H.. 316 Thirring. W.. 127 Thomiis. E. L.. 13 l'homas. 0. M.. 273 Torza. S.. 317 Toupin. R.. 10. 24. 40. 46. 59. 62 3. 03 Townsend. A . A,. 6. 277. 297. 317 Tritton. L). J.. 63 I'ruesdell. C'. A., l X 19. 24. 4 0 I . 46. 53. 59. 62 3. 67. 94 Tryggvason. C;.. 1. 3. 316 'I'urcotte. D. I... 8 1Jlbrcch1, J . J.. xi. 315 [in Ranz (IOX5)l van de Vcn. T . G. M.. 316 van Dyke. M..272 Varniyn. 1'. l'.. 153 Veronis, Ci.. 9. 16

Virk. P. S.. 318 Vitt. F.. A.. I28 von Misses. R.. 21 Voro5. A,. 150 1 and 216 [in (1979)l Wallerstein. C;.. 6 Walters. f'.. 17 Wannier. G . 11.. 194. 218 Wciss. N. 0.. 17 Weissbcrger. A. [see Roughton (1963)l Welander. P.. 4. 5. 5. 9. 217 Wendell. D. S.. 4. 5 WicLerl. 1'. D., 13 Wiggins. S.. 143. 152 Wilev. R. M.. 4. 6. 129 witten. T. A,. 309 Wollan. G . N..3 Zwcitering. -1.h. N.. 4. 7. 317

Subject index

Kote: A subscript following a page number indicates an endnote; for example 16,. indicatcs the second footnote of page 16. An italic number indicates a figure or table. acceleri~tion. 20. 5 1 acceleration and changc of frame. 5 1 acecssible domain. 85. 86 action-angle variables. 132 3. 153,,, active intcrPaccs. 3. 273 activc microstructures. 173. 298 in chaotic flows, 307 additional body forces due to frame transformation. 52. 306 advcction. xi age. 279. 289 agitation. xi analytical tools in the study of mixing systems. 156 antisymmetric tensor. 51. 327 area preserving maps, 101 arca stretch. definition, 31 arca stretch. finite arcas. 36, 37 X astrophysics. 6 asymptotic efficiency, 65 asymptotic stability. 101 average elliciency. 65 definitions, 65 6 in the linear two-dirncnsional flow. 70 in the HV flow. 180 aterage mass velocity. 22 axisymmetric cxtcnsional Ilow.efficiency. 68 Baker's transformation. 6.1 19,119,122,129,, barotropic fluid. 54 basis. 320 behavior near elliptic points. 143 Hcltr;~miflows. 87 I3eri1oulli equation(s). 55 bifi~rcations diagrams. Ihl flip or period-doubling. 162 in the cavity flow. 209 in the tendril whorl map. 161, I64 saddle-node. 162 blending. xi. 7

blinking vortcx flow ( R V ) . 171 horseshoe maps, 177 PoincarC sections, 172 stability of period- l periodic points. 176 body couples. 46 body forces due to framc transformation, 52 breakup and coalescence. 16, experimental studies 300 role in mixing, -7 thcoreticnl studies. 303 bubble, separation. -768 building blocks. 4, 154 RV flow. 171 canc~nicaltransformation. 133 capillary number. 299 ci~pillarywave instabilities. 3 1 72, cat eyes flou. 254. 292 ('auchy's equation of motion. 46 Cauchy Grcen strain tensor. 32 Cauchy- Schwari. inequality, 65, 332 Cauchy Ricmann condit~ons.60 Cauchy's construction, 45 Cauchy's equation of motion. 46 Cauchy's tetrahedron. 45 cavity flow. 201 cabity flou. related studies. ?I#,,, center. 27. 28 centrifugal force. 52 change of framc. 49 change of orthonc)rmal basis. 328 chaotic behavior near elliptic points. 143 chaos in dynan~icalsystems, general rcfercnces. 125 7 i r ~Homiltonian systems. 151 possibilities in higher dinlcnsions. 124 chi~ractcristictimes. 270 XO characterization o l the mixed state. 7 circulation. 53

cI:issiciil 111e:1nsfor ti\~~;iIiziition offlo\\\. 22 clu!,tcr order distributio~i.31 3 ctx~gularionin chac~ticIlo\\s. 312 cc)hcrent strtrctures. 8. 209 combu.\tion. 5. 16,,,. 3 17,, complcs potential. 60. 90 composition of niutions. 32 cc~tiip~ttation nuniericnl requirements. 208. 102 5 of \tretching and cfficict~c!.. 64 computational tc~olsin the srudy of miling sptelnb. 156 conjugale lines. 176. I X; 0. 200 conser\;uion of mask. 42 L;igrangian \crsion. 42 1:ulerinn ~ c r s i o n 42 . cunstant stretch h i s t o r m o ~ i o n s 82 , contact forccs. 43 continuit\. equation {or mass balance). 42 continuoits d n a m i c a l s)stems. 97 continuum mech;t~iics.rolc in thc analysis of [nixing. xi. 6 conxra\ari;irit componentz. 320 con\ cctiie-diff~~aion cqunt ion. 276, 180 copol~meriration.3 6 - 7 Corioli\ 'force'. 52 correlation f ~ ~ n c t i o n129,. cot;iriant colnponcnta. 320 creeping flois. mising in. 11. 194. 21 X,,, critical capillary number. 300 2 curvature or line>. 37 cur\aturc of surf;iccs. 36 cutting and folding. 6 qcles homoclinic co~incction.I I I hcteroclinic connection. I I I h~perbolic.107 tnanifolds of. 107. 1 1 I 1 I4 cjclic ~ a r i a b l c .133 c!lindcr. potential flu\\ o u t s ~ d ca, 60. 61 ,

1)amkiihler number>. 2x3. 316, ,, decay of efficiency. 66. 67. 8 9 definitiotia of bre;ikup in theoretical studies. 304. 307 deformation ;in;~lq.sesfor droplets larpc deformation. 304 small deformation. 304 deform;ttion gradient. 30. 82. O H delinition.; of ch;~os.summiiry. 124. 129>dcgrccb of frccdom. 13 1 2 diffcomorpliism. I8 d 309 diffusion l ~ m ~ t caggregation. diffus~on.rolc In !nixing. I d ~ r c c tsimulatio~i.296 discrctc dynamical system. 97 tlispcrs~on.1x5. 751 3 dih~ipiilivesystem\. OX. 138. 268

distribution effects. 2XX tii\srgc~iccof initial corlditions. 110 drag rcducrion. 207. ! I N , drop breakup e.\pcri~ne~itul studies. 300. 3 17,,, theoretical studies. 30.3 dropleta niotion. g o ~ t r n i n gequ;~tion\.299 t c r j \ ~ s c o u s 303 . 4 sle~idcror lo\v \ iscosir!.. 303 5 dual babis. 320 & n a m ~ c ; ~slstems. l 97

,,,,

earl] works in mixing. 3 Earth. mi.\ing in the mantle of. X l o eccentric helical annular mixer. 244 elliciency of mixing. 64 cfliciencj in linear three-di~ncnsion;ilflow. X4. Xn in sequences of Ilo\bs. $7. SY in steady tuo-dimensional flou. 69. 70 eigcnvaluc5 in linear two-dimensional flou. s4 e~gcnv;tlues.mapping. 159 ellipttc point. 102. 137 energ) equation, 47 cqui\alcnce among iarious definition!, of chaos. 174 crgodicity. 17,-. 12XLZ Fuler.5 accclcratiun. 52 luler's auiom. 43 Eulcr's formul;~.20 Euler~an \icwpoint. 20. 40. 40, 41, vclocit), 45 ex;tmples of stretching and efficiency. 68 exit time. 7 exit timc distribution.\. 237 expansion work. 47 experimental tools in the study of mixing hysletna. 1.56 extra bod! forces. 52 extruders. 90 fiimilies of pcr~odicpoints. 103 fast reactions. 13. 8 2 . 284 I-ick's law. 276 first law of thermodynamics. 48 fixed point, 24 fixed points and periodic points. 99 flow. 14. IS atid motion. terminology. 17, bisuali~at~on, 14. 25 flow class~fication.14. 25 and maps, 97 flow s extensional flow. 68 near walls, 760

hlnlplc illcar. OX \\ ~ t ha \ p c c ~ ; ~ Corm l of. 1'1. 77 u ~ t hl)(Gv) I)! - 0 . 77 a ~ t h111Vv)/ ) I small. 70 w ~ t ha special form of F. X2 fluid mechanical h~>toric>. ?XX f O ~ l l \ . '7. 2X folding. 107. 100 formation of horhcshoe map\ in the -1-W map. 170 1 formation of horscbhoc m;ip\ in thc R V fll)\$.I
~rnnl~\cihlc Ilu~db.15 IIllpIIl&!cIl~c111 11lI\II1~.1.1. 17: ,, irnport;tricc of rcor~cntat~on. 87 improper n o d c ~2:. 28 integrable tnist mapping. 133 i n t c g r a h i l ~ t ~14. . 131 ~ntcgrabilitof llarniltonian 2~btcrns.132. l.l4 ~ntensific;~t~or~ of vortic~t).56 7 Intc'nzltJ of >cgrcgatiori. X. Ih2. IS3 interactions betiwen mnnifolds. 16.5 ~nrerf;~ce.i actlvc. 2. pasbiic. I . 2 intermatcr~alarea per unit \olumc. 3. 278. 31hl, In\;irlant sct. I?X,, subbpascs. 105 ~nr.;rriantsin linear three-dimensional flu\*. X5 inv~scidfluid. 47. 40 irrational o r h ~ t s .143. 134. 147 irre\.ersih~l~t!.180. 184 ~rrot;~tional flair. 55. 90 ~\ochoricflo\vs. 10. 6%. 130 iso-concentration cur\.es and surfaces, 3. 290 isolation of reactant>. 2XX Jacobian. 19. 121. 15.3, jet\. I? Jordan form. 102 journal hearing flow. 193 K a n t o r o ~ i c hGalerkin method. 224 Kelvin cat eye floa. 130. 254 klncrnatical viespoint. ix kinc~natics.xi. 40 kinematics of deformation rate of strain. 35 strain. 30 Kolmogoro\ Arnold -Moscr theorern (KAM theorem). 146 Kolmogorov length scales. 13. 3 16, Lagrange's theorern. 22 I.agr;inp~anand Iiulerian viewpoints. 20. 40. 40, I.agrang~anturbulcncc. I lamcllar structures. 1. 2. 5 complications and ~llustrations.288 initial conditions for conccntration. 281 dlffuaiori and reaction in. 174 85 distortion effects. 288 distribution effects. 289 large deformation analyses. 304 laher iriduccd fluorescence. 13. Fig. 1.3.5 (bee color platcsl. 12

leitking from bubble. 268 length stretch. 31 linear area preserving map. 101. 103 isochoric two-dimensional flow. 29 momentum. 43 scalar funct~ons.323 subspaces. 105 threc-d~mensionalflow. 84 two-dimensional flow. 25. 69. 78 two-dimensional flow. streamlines. 27 vector functions. 324 Liapunov computation in the BV flow, 180, I83 exponents. 1 1 1. 1 2g2,. 138 stability. 101 Liouville's theorem. 98, 128,. 135 Lipschitz constant, 23, 97 local equilibrium. tractions. 45 flow. 298 processes. 4. 15 local stability and linearized maps. 100 macroscopic bztlance of vorticity. 57 macroscopic dispersion of traccr particles, 187 manifolds of a cycle, 107. 111. 114 15 of hyperbolic points, 125 mantle of the Earth, 10 mass flux vector. 45 material description. 20 material integrals. 36 volume, 42 matrix transrormations, relationship with mixing, 7 maximum directions o l stretching. 39 measurement of mixing, 7 mechanical energy equation, 47 Mclnikov's method, 141 application examples 190, 191. 201, 255, 25 7 microflow element. 276 microstructure, 67, 306 microstructures in chaotic flows. 307 mixed cup average, 72 mixing and chaos in three-dimensional and open flows, 220 and chaos in two-dimensional time-periodic flows, 154 cavity flows. 71. 201 mantle of the earth. N 10 improvement. 91 In oceans, 5. 9. 16,,

in the partitioned-pipe mixer. 221 In time-period~cflows. 105 in stars. astrophysics, 6 layer. Fig. 1.3.5 (see color plates) mathemat~cnldelinition. 1 1 7. 118. 124. I28,, of diffusing and reacting fluids. 15 of immiscible fluids. 15. 298 of polymers. Fig. 1.3.4 (see color platcsj of a single fluid. 14 strongly measure-thcoretic. I I8 strongly topologically. I I9 morphological building blocks. 274 motion 14. 18 motion (velocity) around a point. 33 motions in multicomponent rncdia. 22 with constant stretch history. 82 multicomponent media, 22 Navier-Stokes equation, 47. 195. 250, 261 near integrable chaotic Hamiltonian systems, 149 Newtonian fluid. 47 8. 56. 221 Newton's 'second law', 152, Newton's 'third law', 43 node. 27, 28 non-topological motions. 19 non-wandering points, 115 ob~ectivity,49 obscrvability, 208 obstruction to dispersion. 187 9 oceans. mixing In. 5, 9. 16,? orbit. 22, 98 orthogonal matrix. 329 stagnation flow. 29 transformation. 49 parabolic point. 102. 137 pararnetcrs and variables characterizing transport at small scales. 279 particle path, 22 partitioned-pipe mixer. 221 2 p:tssive ~nterfaces.1, 3 microstructures, 298 scalar, 292 pathlines. 22 periodic points, orbits, etc. eigcnvalues in thc neighborhood of. 158 flows. 24 method for location of. 178 NO orbit, 100 pictorial representation. 158,160,161,176 polnts. 25 reorientation, 83. 87

~ ~ r i o d i c i tof ! ~ \ l , ~ n d Fig. s . 7.5.X ( w e color pI;llc~).?>? : perturbatior~, i i p p l ~ c ~ ~ t iofo nMclnlhol's method. I90 3 H;~miltonl;tns)htclns. 141 pcrturhed shear flou. 25. 20. 254 phase ,pace. 97 1l;imiltoniall systems. 13.5 I'oincare section, ;ind tori. 136 physical medning of dcfor~nationtensor. 3X of ( V v ) ' . 38 \elocity gradient. 38 physical picture of mixing. I I'iol;~ tensor. 41 planar hyperbolic flo14. 4 Poincarb HirkhoK theorcm. 144 Poincart sections. 100. 103. 104. 105. 136, 137. 172. 197. ,728 31, 258 and three-dimensional structure. 2'7 in the BV flou. 172 i In the journal bearing flow. F ~ g s .7.4.6. 7.4.10 (.\ee color plates) Poincarc's recurrence. 136 point transformation. I X point \.orriccs in the plane. 139 pointed drops, 301. 304 Poisson's br;icket. 1 32 polar decomposition theorem. 32 polymer mixing. Fig. 1.3.4 (see color plates). 16, polymer molecules in chaotic flows. 308 polymer processing. 7. I3 polymerization. 28.5 7 porou, modla and chaotic flows. 308 possible u a y s to improve mixing. 91. 93 potent~;ilfunction. 59 present statc, I 9 principle conserv;ition of ;ingular momentum, 46 conscr\'ation of energy, 48 conservalion of lincar momentum. 43 conservation of mass. 42 proper orthogonal transformation. 49 properties of 1,; and P,. 66

,

qualitative picture of ncar-integrable chaotic tlamiltonian systems. 149 quasi-periodic. 144 rate of area generation, 277 VdtCS of change of material integral$. 36 rational orbits. 137. 143, 144 reattachment to wall, 268 reference configuration. 19. 21, 41 ,,, regimcs. 28 1, 282 slow reactions. 283

\er> f,ist re;~ction,,284 regular and chaotic flo~ts.m i ~ i n gcflecrs. 186.242-3 rcnormalization methods in Hamiltoliian skstcms. 153,, reprcscntatlon theorcm. 323 reorlentation. period~c,strong. ucak. 83 residence tlme. 237 9. 24(1 1 Reynolds number. 195. 201. 249. 299 typic;ll lulues in Larious processes. 9 Re! nolds's theorem ( o r tranhport ). 20. 128, rotation number. 143 saddle. 20. 27 scal'ir lector function. 326 scnsitirity to ~nltlalconditions and Liapunov exponents. I I I . 110 separ;ition bubble. ,763 sequences of flows. 87 shcar flow strctch~ngin, 29&2, 291 7 time pcrturhation. 75 shear layer. 138 shear stress. 47 signarures of chaos. I I 1 simple shcar flow, 4. 27. 28. 68 singular point, 24 slender body theory. 304 small dcformation analysis. 304 Smoluchowski's kinetics. 3 1.5 hpin tensor. 33 spiral sink. -77, 28 spiral source. 27. 28 stability definitions. 101 of area preserclng two-dimensional maps. 101 of perlod-l and period-? periodic points in the -I'W flow. 158 of periodic points in the B V floe. 158 stable and unstable manifolds, 105 stz~ndardtuo-dimensional cavity flow. 72 stars, mixing in, astrophysics, 6 static mixers. 129?,, 223 statisticiil thcory. 6 steady flows, 24 steady curvllinenl flows. 82. 83 stirred tdnks. 4 stirring. xi stopped flow mcthod, 17,, strange attractors, 129,, strcaklincs. 23, 246. 248 9 crossing of, 258 in fixed frames. 260 in lime dcpendcnt shear flows, 26 with respect to moving frames, ,759 streamfunction, 92, 130