Computational Rheology
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Computational Rheology
R. G. Owens Universite de Montreal, Canada
T. N. Phillips Cardiff University, UK
EBB
Imperial College Press
Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
First published 2002 Reprinted 2005
COMPUTATIONAL RHEOLOGY Copyright © 2002 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, US A. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-186-9
Printed in Singapore by Mainland Press
The authors: Tim Phillips (left) and Rob Owens (right). Photograph courtesy of Olivier Burdet. Timothy Phillips obtained his DPhil in numerical analysis from the University of Oxford in 1983. He spent two years as a Research Fellow at the Institute for Computer Applications in Science and Engineering at NASA Langley Research Center in Virginia before taking up a Lectureship at the University of Wales Aberystwyth. He was promoted to a Personal Chair in Mathematics at Aberystwyth in 2000 and was appointed to a Professorship of Mathematics at Cardiff University in 2004. He has held Visiting Professorships at the University of Delaware and the Ecole Polytechnique Federale de Lausanne. His research inter ests include numerical analysis, spectral methods, computational fluid mechanics and rheology.
Robert Owens completed his PhD in viscoelastic flow at the University of Wales Aberystwyth in 1990. He held positions as Assistant Professor at Bogazigi University in Istanbul and Lecturer at Edinburgh University and Napier Uni versity before his nomination in 1998 as Assistant Professor in Non-Newtonian Fluid Mechanics at the Ecole Polytechnique Federale de Lausanne. Since the autumn of 2004 he is an Associate Professor in the Department of Mathemat ics and Statistics at the University of Montreal. His research interests include numerical analysis, spectral methods and computational rheology.
v
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Preface The pace of new developments in numerical techniques for viscoelastic flows has been rapid and there now exists a collection of numerical techniques that can be used to predict the behaviour of model viscoelastic fluids in smooth geometries. Differences between numerical simulations and experimental observations (and there are many) can be attributed, in many cases, to a failure in the modelling process. Thus, we believe, rheological modelling represents the next grand chal lenge for researchers in the field. Indeed, at the start of the 21st century we stand poised at exciting new frontiers: the construction of mathematical models that possess the predictive capability for simulating real fluids through complex geometries remains a relatively unexplored field that is ripe for exploitation. The last major text on the subject, the book by Crochet, Davies and Walters [152], was published in 1984 and concentrated on finite difference methods and the Galerkin finite element method. In the intervening years there has been a number of review articles (see those of Baaijens [25], Crochet [150] and Keunings [324,325,327], for example) and book chapters (see those of Huilgol and PhanThien [296], Tanner [563] and Tanner and Walters [567], for example) that have been devoted to computational techniques for viscoelastic flows. However, the absence of a recent text wholly dedicated to this subject area has provided us with the motivation to write this book. Having said that, we do not claim to provide an exhaustive treatment of the subject of viscoelastic fluid flow within these pages: it is too large and complex an area by far to cover in a book of this size. Neither do we claim to be infallible: in humble recognition of our human frailty any errors in the text brought to our attention will be posted on the world wide web at the URL: http://users.aber.ac.uk/tnp/books/rheolbookerrata.html Our objective in this book is to provide researchers and students in ap plied mathematics and engineering with a flavour of some of the distinctive issues and computational challenges associated with the simulation of viscoelas tic flow problems. We have aimed to make the book fairly self-contained in that the reader will find, in the introductory chapters, details about the derivation of the governing equations, the construction of constitutive models, the mathe matical theory of the systems of partial differential equations governing certain viscoelastic flows, parameter estimation, and the essential elements of numeri cal discretization techniques and numerical algorithms for solving the discrete equations. Readers interested in a more in-depth study of the modelling of poly meric liquids, the mathematics of viscoelasticity, the description of experimental studies of viscoelastic fluids and rheometry are referred to the texts of Bird et al. [72,73], Joseph [315], Boger and Walters [90] and Walters [606], respectively. vn
PREFACE After the introduction the book explores some of the reasons for the high Weissenberg number problem and how innovative techniques have been devised to surmount some of the stumbling blocks. A major part of the book is devoted to four of the benchmark problems that have been associated with the subject over a prolonged period of time and which still have the ability to seduce many a computational rheologist. The benchmark problems are flow through planar and axisymmetric contractions, flow past a sphere, flow past a cylinder and flow between eccentrically rotating cylinders. In addition to comparing the perfor mance of various numerical techniques a discussion of some of the interesting features of the problems from a fluid mechanical perspective is given. The book concludes by looking at some of the recent developments in the modelling of polymeric solutions and melts and at the emergence of stochastic simulation techniques as a powerful numerical tool for solving the equations produced by these models. The authors were both exposed and introduced to the subject of viscoelasticity on arrival in Aberystwyth, the former as a research student and the latter as a university lecturer, when they joined the research group of Ken Walters in the Department of Mathematics. This book reflects the formative influence that this group has had on our research careers and we are more than happy, con sequently, to acknowledge the many stimulating discussions we have had with our colleagues at Aberystwyth which have contributed to our understanding of computational rheology and complex fluids. In particular, we would like to mention David Binding, Russell Davies, and, of course, Ken Walters who has inspired generations of students as well as his colleagues. TNP would also like to acknowledge the fruitful collaboration he has enjoyed over many years with other colleagues within the University of Wales Institute of Non-Newtonian Fluid Mechanics: Peter Townsend, Mike Webster and Rhodri Williams, and the gifted post-doctoral researchers he has supervised: Xavier Escriva, Marc Gerritsma, Rhys Gwynllyw, Andreas Karageorghis, Emmanuel Leriche, Gareth Roberts and Alison Williams. Similarly, RGO would like to acknowledge, with thanks, collaboration over several years with Michel Deville, Peter Monkewitz, Alfio Quarteroni, Cedric Chauviere and Alexei Lozinski. We would like to thank David Boger, Octavio Manero, Gareth McKinley, Nhan Phan-Thien and Ken Walters for allowing us to use figures and pho tographs from their publications. Permission to reproduce these figures and photographs has been obtained from the respective publishers and is acknowl edged at the appropriate places in the book. We would like to thank our departments and institutions for hosting recip rocal visits by both of us over the last four years and for making their facilities available to us. Finally, we would like to thank our wives for their encourage ment, patience and understanding during our partial (and, may we reassure them, temporary) withdrawal from family life. This book is dedicated to them.
Robert G. Owens Lausanne Switzerland
Timothy N. Phillips Aberystwyth United Kingdom January 2002
vm
Preface to the Second Printing We are delighted to have been given this opportunity by the publisher to correct a number of errors that had crept into the original volume of Computational Rheology. We are particularly grateful to a number of colleagues for their con structive criticism of the first edition, made via published reviews or personal communication. Amongst these we should mention Michel Deville, Bruce Finlayson and John Rallison. We have sought, in this edition, to address a number of their concerns and have also now put the derivation of a Fokker-Planck equa tion corresponding to a dilute solution of dumbbells in chapter 2 onto a more solid footing. Of course, there have been a number of exciting developments in computational rheology since the publication of the first edition of the book and we would refer the interested reader to a special issue of the Journal of NonNewtonian Fluid Mechanics (volume 122) for a description of some of these. Robert G. Owens Montreal Canada
Timothy N. Phillips Cardiff United Kingdom October 2004
IX
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To Heather and Gill
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Contents 1
Introduction 1.1 Everything Flows 1.1.1 The Maxwell model and the relaxation time 1.1.2 The Kelvin model and the retardation time 1.2 Non-Newtonian Fluids 1.2.1 Non-Newtonian fluids in shear flow 1.2.2 Non-Newtonian fluids in uniaxial extensionai 1.3 Numerical Simulation of Non-Newtonian Flow 1.3.1 Early work 1.3.2 The high Weissenberg number problem 1.3.3 Numerical work since the mid 1980's
2 Fundamentals 2.1 Some Important Vectors 2.1.1 Fluid velocity and acceleration 2.1.2 Forces and the stress vector 2.2 Conservation Laws and the Stress Tensor 2.2.1 Conservation of mass 2.2.2 Conservation of linear momentum 2.2.3 Conservation of angular momentum 2.2.4 The stress tensor 2.2.5 Conservation of energy 2.3 The Newtonian Fluid 2.4 The Generalized Newtonian Fluid 2.4.1 Derivation of the stress tensor 2.5 The Order Fluids and the CEF Equation 2.5.1 Strain tensors 2.5.2 Order fluids 2.5.3 Limitations of order fluid models 2.5.4 The CEF equation 2.6 More Complicated Constitutive Relations 2.6.1 Differential constitutive models 2.6.2 Integral constitutive models
xiii
flow
1 1 4 7 8 9 14 16 16 17 18 19 19 19 20 21 21 21 21 22 23 . 24 25 26 27 27 28 30 32 33 33 45
CONTENTS 3
Mathematical Theory of Viscoelastic Fluids 3.1 Introduction 3.2 Existence and Uniqueness 3.3 Properties of the Differential Systems 3.3.1 Loss of evolution and Hadamard instability 3.3.2 Classification 3.3.3 Change of type in steady flow 3.3.4 Characteristic variables 3.4 Boundary Conditions 3.5 Singularities 3.5.1 Elliptic problems 3.5.2 Viscoelastic flows 3.5.3 Numerical investigations
49 49 51 54 56 59 61 62 64 65 65 68 71
4
Parameter Estimation in Continuum Models 4.1 Introduction 4.2 Determination of Viscosity 4.2.1 Shear viscosity 4.2.2 Dependence of viscosity on temperature 4.2.3 Dependence of viscosity on pressure 4.2.4 Extensional viscosity 4.3 Determination of the Relaxation Spectrum 4.3.1 Dynamic experiments . 4.3.2 Mathematical problems 4.3.3 Linear regression techniques 4.3.4 Nonlinear regression techniques 4.3.5 Examples 4.3.6 Sampling localization
73 73 76 76 79 81 82 83 84 85 86 88 89 91
5
From the Continuous to the Discrete 5.1 Introduction 5.2 Finite Difference Approximations 5.2.1 Finite differences for viscoelastic flows 5.3 Finite Element Approximations 5.3.1 Finite elements in one dimension 5.3.2 Finite elements in two dimensions 5.3.3 Finite elements for viscoelastic flows 5.4 Finite Volume Methods 5.4.1 Finite volumes for viscoelastic flows 5.5 Spectral Methods 5.5.1 Spectral methods in one dimension 5.5.2 Spectral methods in two dimensions 5.5.3 Spectral methods for viscoelastic flows 5.6 Spectral Element Methods 5.6.1 Spectral elements for viscoelastic flows
xiv
95 95 97 100 104 105 107 110 113 117 121 121 125 127 129 130
CONTENTS 6
Numerical Algorithms for Macroscopic Models 6.1 Introduction 6.2 Prom Picard to Newton 6.3 Differential Models: Steady Flows 6.3.1 Direct methods 6.3.2 Iterative methods 6.4 Differential Models: Transient Flows . 6.4.1 Projection methods 6-4.2 The influence matrix method 6-4.3 Taylor-Galerkin methods 6.4.4 The 9 method 6.4.5 Lagrangian methods 6.5 Computing with Integral Models 6.6 Integral Models: Steady Flows 6.7 Integral Models: Transient Flows 6.7.1 Lagrangian techniques 6.7.2 Eulerian techniques
7
Defeating the High Weissenberg N u m b e r Problem 7.1 Introduction 7.2 Discretization of Differential Constitutive Equations 7.2.1 Streamline upwinding - SU and SUPG 7.2.2 Discontinuous Galerkin methods 7.3 Discretization of the Coupled Governing Equations 7-3.1 Compatible approximation spaces 7.3.2 EVSS-type methods 7.3.3 Change of type and loss of evolution
173 173
Benchmark Problems I: Contraction Flows 8.1 Vortex Growth Dynamics 8.1.1 Boger fluids - observed flow transitions 8.1.2 Boger fluids - effects of changes of geometry 8.1.3 Concentrated polymer solutions and melts - observed flow transitions 8-1.4 Concentrated polymer solutions and melts - effects of change of geometry 8.2 Vortex Growth Mechanisms 8.2.1 Experimental work 8.2.2 Numerical studies 8.3 Numerical Simulation 8.3.1 Comparisons with experiments 8.3.2 Benchmarking numerical methods
201 202 202 206
Benchmark Problems II 9.1 Flow Past a Cylinder in a Channel 9.1.1 Streamline patterns and drag: unbounded flows 9.1.2 Streamline patterns and drag: cylinders in channels . . . 9.1.3 Comparison of numerical and experimental results . . . .
247 247 247 251 252
8
9
xv
135 135 136 137 138 140 149 150 153 155 157 160 • • • 163 164 167 167 170
177 177 182 187 187 190 195
209 213 216 216 222 225 226 231
CONTENTS
9.2
9.3
9.1.4 Purely elastic instabilities 254 9.1.5 Comparison of numerical methods 257 9.1.6 Mesoscopic calculations 263 Flow Past a Sphere in a Tube 267 9.2.1 Drag coefficient 268 9.2.2 Benchmarking numerical methods 274 9.2.3 Negative wakes: steady flow 277 9.2.4 Velocity overshoots - transient flow calculations 281 9.2.5 Comparison between experimental and numerical results . 284 Flow between Eccentrically Rotating Cylinders 287 9.3.1 The Taylor-Couette problem 288 9.3.2 Lubrication theory 291 9.3.3 Statically loaded journal bearing 293 9.3.4 Dynamically loaded journal bearing 297
10 Error Estimation and Adaptive Strategies 10.1 Introduction 10.2 Problem Description 10.2.1 Weak formulation 10.3 Discretization and Error Analysis (Galerkin method) 10.3.1 An error indicator 10.4 Adaptive Strategies 10.4.1 Numerical example: flow past a sphere in a tube
305 305 308 308
11 Contemporary Topics in Computational Rheology 11.1 Advances in Mathematical Modelling 11.2 Dynamics of Dilute Polymer Solutions 11.2.1 The Rouse model 11.2.2 Dumbbell models 11.3 Closure Approximations 11.4 Stochastic Differential Equations 11.4.1 The CONNFFESSIT approach 11.4.2 Variance reduction techniques 11.4.3 Lagrangian particle method 11.4.4 Brownian configuration fields 11.5 Dynamics of Polymer Melts 11.5.1 The Doi-Edwards model 11.5.2 The pom-pom model 11.6 Lattice Boltzmann Methods 11.7 Closing Comments
327 327 330 331 335 335 338 341 343 345 346 347 347 354 357 359
A Some Results about Tensors A.l Existence and Symmetry of the Stress Tensor A.2 Small Displacement Gradient Limit of 7M(x,f,i') A.3 Partial Time Derivative of the Deformation Gradient Tensor F ( x , i , t ' )
361
xvi
311 315 316 317
361 364 364
CONTENTS B Governing Equations in Orthogonal Curvilinear Coordinates B.l Differential Relations and Identities B.2 Differential Operators in Orthogonal Curvilinear Coordinates . . B.2.1 Rectangular coordinates (xi,X2,xs) = (x,y,z) B.2.2 Cylindrical polar coordinates (x\,X2,X3) = (r,6,z) . . . . B.2.3 Spherical polar coordinates (xi, x2,x3) = (r,0,
) B.3 Conservation Equations B.3.1 Rectangular coordinates (x, y, z) B.3.2 Cylindrical polar coordinates (r, 6, z) B.3.3 Spherical polar coordinates (r,0,<j>) . . B.4 Some Important Theorems in Vector and Tensor Calculus . . . . B.4.1 The Reynolds transport theorem B.4.2 The divergence theorem B.4.3 Stokes's theorem
xvn
365 365 366 366 368 369 371 371 371 372 373 373 373 373
Chapter 1
Introduction 1.1
Everything Flows
Figure 1.1: Earthflow on Mission Pass in the California coastal ranges. The lat eral lines on the hillside indicate creep. Photograph by B. Bradley, Department of Geology, University of Colorado, Boulder. Reproduced with permission of the National Geophysical Data Center, Boulder, Colorado, USA. 1
1.1. EVERYTHING FLOWS Rheology is the science dealing with the flow and deformation of materials expe riencing some mechanical force. Professor Eugene Cook Bingham of Lafayette College, Indiana, coined the term "rheology" in the late 1920's after consulting with the professor of classical languages at Lafayette (the Greek ptcx; means "stream" [575]) and took as the motto of the subject the statement by the Greek philosopher Heraclitus "-iravra pet": "everything flows" [333,494]; a motto which has since been adopted by the Society of Rheology. We turn to Judges 5:5 in the Old Testament of the Bible in order to gain an insight into what -navTQ. pu really means. There, the prophetess Deborah stated that The mountains flowed before the Lord. As pointed out by Reiner [494], two facts may be gleaned from this verse. First, the mountains flowed. This may seem surprising: we are used to seeing fluids such as water flowing, but certainly not mountains! Actually though, geological mass wasting processes (occurring under the action of gravity) such as rockslides, mudflows, solifluction and creep, are all examples of the ways in which a mountain, or at least the material lying on a slope, may flow. Shown in Fig. 1.1 is an earthflow on a hillside in the Mission Pass, California. Visible also are lateral lines on the hillside and these indicate creep. Although some geologi cal flow processes may occur rapidly and violently (average speeds of 160km/h have been recorded for some rock and snow avalanches), a great many others are taking place almost imperceptibly. This is what is meant in the second part of Judges 5:5. The mountains flowed before the Lord. The Lord's timescales are not ours. The human timescale may be insufficient to see mountains flowing, but flow they do! Bill Bryson, in his book "A Walk in the Woods" [107] tells us, for example, that the Appalachian Mountains are shrinking on average by 0.03 mm each year, due to erosion. On the rheology web page of the Australian Society of Rheology http://www.pfpc.unimelb.edu.au/asr/rheology.htm may be found the example of the Himalayas: as a result of the collision between the Indian sub-continent with the rest of Asia some 10 million years ago these mountains are continuing to rise (at a rate of approximately 1 mm per year) up from the Indian plains "...in a manner analogous to the piling up of butter (the Himalayas) in front of the spreading knife (the sub-continent of India)...". The fundamental difference between the two flow processes of mountains and butter lies in the viscosities and timescales. Indeed, over a sufficiently long time interval many materials which we classify for all practical purposes as solids could be observed in flow. Glass, for example, flows in window panes under the force of gravity, though admittedly much more slowly than had originally been thought [644,645]. The conclusion of the matter is that the distinction between solids and fluids now seems somewhat more blurred than we may have previously thought. A discussion of material behaviour might be more appropriate. In order to understand the behaviour of materials as fluid-like or solid-like, it is helpful at this point to discuss what is meant by the relaxation time and retardation time of a material. We begin by looking at the two extremes of a Hookean elastic solid and a Newtonian fluid. Consider first a one-dimensional simple shearing motion in a
2
CHAPTER 1. INTRODUCTION Hookean linear elastic solid which occupies at some time t = T < 0, fi = {(x,y,z)
: - o o < x < oo,0 < y < d,-oo < z < oo},
(1-1)
in its equilibrium state, as shown in Fig. 1.2. Suppose that the rectangular crosssection OABC in any plane z = constant now undergoes a small deformation so that at time t — t' > 0 it is a parallelogram OAB'C with angle C'OC denoted by 6(f), independent of the coordinates a;, y and z. Consider a material particle having Eulerian coordinates (x(t'),y(t'),z(t')) = (x{t'),y(T),z{T)). Then we may define a relative deformation gradient ^^(T,^) in the present context by 7xy(T,t')
=
x(t') - x(T) y(t')
ta.n9(t')n9(t'),
(1.2)
for 9 small. Thus we may refer to 6 as a relative deformation gradient or strain, and Hooke's law then states that the relationship between the shear stress axy and the strain 9 is
(1.3)
where G is the so-called rigidity modulus for the material.
Figure 1.2: Deformation of an elastic material OABC. If the strain 6 is a step function in time, given by 0, 9 = 9oH{t) = { 90/2, 0o,
t < 0, t = 0, t > 0,
(1.4)
as shown in Fig. 1.3, then from (1.3) it is clear that the corresponding stress history is cXy — G0oH(t),
(1.5)
and the graph is drawn in Fig. 1.4. Conversely, of course, if a known stress axy is applied then the strain 9 may be determined from (1.3).
1.1. EVERYTHING FLOWS
Figure 1.3: The strain 8 = 90H(t).
Figure 1.4: The stress axy = G90H(t)
1.1.1
The Maxwell model and the relaxation time
Suppose now that a Newtonian fluid occupies the region Q between the same two planes y = 0 and y = d and that the material undergoes the same shearing deformation as before. Then, as required by Newton's hypothesis the shear stress is given by <7xV(t) =Voixy(t),
(1.6)
where % is the constant fluid viscosity, and %y(t) is the rate of strain at time t defined by %y(t) = -~/xy(T,t)
= 6(t).
(1.7)
Eqn. (1.6) leads to the following conclusion: if the fluid experiences a strain history of the type (1.4) then the corresponding shear stress (rxy(t) must be a 4
CHAPTER 1. INTRODUCTION delta function centred at zero, viz., **„(*) = Vo0oS(t).
(1.8)
The relaxation time of a material may be provisionally denned as some appro priate measure of the time required for the shear stress axy in a simple shear flow of the type described to return to zero under constant-strain conditions. We see that the shear stress in the case of a Hookean elastic linear solid will never relax to zero with a non-zero constant strain. Hence we take the relax ation time as infinite. For a Newtonian fluid the relaxation of the stress is immediate and the relaxation time equals zero. Relaxation times which are in finite or zero are never realized in reality: they correspond to the mathematical idealization of Hookean elastic solids and Newtonian liquids. In practice, stress relaxation after the imposition of constant-strain conditions as in (1.4) takes place over some finite non-zero time interval. This is the defining characteristic of so-called viscoelastic materials. To illustrate this behaviour, suppose we have a viscoelastic material undergoing a simple shearing process as in our previous examples where instead of either (1.3) or (1.6) we have the following relationship between shear stress and rate of strain (Maxwell model): 1 d 1 + —0-xy = %y{t)Gdt *y Vo
(1-9)
-^-TTCT.
Then
*v^^(f)) = G W i ) e x p (f axy
exp I — ] ) = G%y{t) exp ( — ] ,
(1.10)
so that
(Gt\
f*
„.G%,(s)exp , (Gs
y
ds + c,
(1.11)
for some constant c. Therefore v*v(f) = / _
^exp
(-\^)
%y(s)ds + cexp ( - £ )
,
where we define the material constant Ai = rfo/G. For finite axy as t -► —oo we require c = 0. Then
hm
lim
£ £ _ „ , exp ( £ ) %y{s) ds ) - ^ , eXp(x)
(1.13)
and using L'Hopital's Rule to evaluate the right-hand side, lim axy(i) = lim rjo7xy(£). t-y—oo
Therefore, if jxy(t) means that
(1-14)
t—f—oo
is bounded as t -> - o o , so is vxy(t),
5
and c = 0 in (1.11)
(1.12)
1.1. EVERYTHING FLOWS The quantity Ai
(t-s)
exp
Ai
is called the relaxation modulus for the Maxwell fluid. If we integrate (1.15) by parts then we obtain 0xy(t)
=
h^exp(
^—\lxy{t,8) (t-s)
-I £"*( ■>s=-oo
A
l
J$=-oo Ai
jxy(t,s)ds,
Ai
\
(t-s)
\
Ai
/
-yxy(t,s)ds.
(1.16)
s)ds,
(1.17)
We may write this integral relationship in the form M
<*■*»(*) = ~ /
(t ~ shxy(t,
J s= — oo
where the function
is called the memory function for the Maxwell fluid. Equations (1.15) and (1.17) show us that the stress at the present time depends upon the strain (or rate-ofstrain) for all past times, but with an exponentially "fading memory". Let us observe that the definition 7a
„(«,«)= f
iXy{t')dt',
(1.18)
JV=t
leads to the result lxV{t,s)
=7zj,(-oo,s) - 7 z y ( - o o , i ) ,
and therefore in (1.16) 0xy(t) = Y7xy(~oo,t)
- -^ /
exp (
— j jXy(-oo,s)ds.
(1.19)
Now suppose that we have a step function strain history (1.4). Then, for t > 0, it follows that a.,(*)
=
- ( f c - ^ y ^ e x p ^ — — jds,
X" e x p V Ai
(1.20)
We now identify the material constant Ai as the relaxation time which, in the present example, is the time taken for the shear stress in a viscoelastic material obeying the Maxwell model to reduce to e _ 1 of its initial value G6o- The relaxation time of water is about 10 _ 1 2 s, that of low density polyethylene about 10s and that of glass in excess of 28 hours! In practice many materials exhibit 6
CHAPTER 1. INTRODUCTION stress relaxation in response to a step strain of the type described in the previous paragraphs. The fact that a given material has a relaxation time associated with it, does not necessarily mean that it will always behave as we may expect, however. Hence the unexpected phenomenon of mountains flowing. What really matters in characterizing material behaviour in experiments is the ratio of a characteristic relaxation time (Ai) of the material to a characteristic time of observation of the flow process (T0, say). Reiner [494] introduced this ratio as the Deborah number De, defined as De=^-.
(1.21)
The Deborah number is zero for a Newtonian fluid and is infinite for a Hookean elastic solid. But it should be clear from the definition (1.21) that, in general, materials will not have one Deborah number associated with them: a material with non-zero characteristic relaxation time could behave in a solid-like way if the characteristic time T0 of the flow process is small (large De). Conversely, a material which may, over short time scales, appear to us as a solid can be observed to flow if To is sufficiently large (small De). An example of a material whose behaviour can be either fluid-like or solid-like is provided by Barnes et al. [43]. "Bouncing putty" is a silicone material which, when left for a sufficient length of time in a container, will find its own level - a fluid characteristic. On the other hand, if a ball of the material is subjected to the sudden deformation involved in being dropped onto a hard surface, it may bounce - a solid-like characteristic.
1.1.2
The Kelvin model and the retardation time
Now suppose that rather than (1.3), (1.6) or (1.9) we have the relationship **»(*) = Gjxy(T,t)+rj0%y(t),
(1.22)
called the Kelvin model. We first of all note that under constant strain the stress remains constant so that the model does not show stress relaxation as in the case of a Maxwell fluid. However, suppose that a stress axy =W is suddenly applied at t = 0, so that
'•*<*> = { £ t>°o,
( L23)
For the model (1.3) the strain 9 ~ -yxy reaches a new value a/G instantaneously. For the Kelvin model the strain takes time to grow to the final value a/G as shown in Fig. 1.5. We may write (1-22) in the form
f+G$ at
=
%
?sM.
(1 .24)
Vo
The solution of this differential equation, for t > 0, is given by
»M-s(i—»(-£)).
V*>
where A2 = Vo/G is called the retardation time of the material and in the 7
1.2. NON-NEWTONIAN FLUIDS
Figure 1.5: Growth in strain 6 according to the Hooke model (1.3) and the Kelvin model (1.22). present context is the time taken for the strain to reach (1 — e _ 1 ) of its final value of a/G after the imposition of a constant shear stress a = a. Alterna tively, and equivalently, it is the time taken for the strain to reduce to e _ 1 of its original value when a shear stress is removed. Viscoelastic materials described by more complicated relations than those given in either the Maxwell model or the Kelvin model typically have non-zero relaxation and retardation times. By linearly superimposing models of the Maxwell and Kelvin types more sophis ticated models exhibiting a spectrum of relaxation and retardation times are possible.
1.2
Non-Newtonian Fluids
Viscoelastic fluids are examples of a broader class of fluids called non-Newtonian fluids. Later, these are denned as fluids for which the extra-stress tensor cannot be expressed as a linear, isotropic function of the components of the velocity gra dient. Examples of non-Newtonian fluids abound in nature and industry: most body fluids, such as blood and mucus are non-Newtonian as are many food prod ucts: egg white, cream cheese, mayonnaise and molten chocolate, for example. The pulp used as part of paper manufacturing is a non-Newtonian material, as are molten plastics and other man-made materials that are formed into a wealth of useful everyday products such as textiles, plastic bags etc. through processes such as extrusion, moulding, film blowing and spinning. Often a material of in dustrial significance is non-Newtonian through the addition of various polymers. The detergent industry adds polymers to shampoos, gels, bleaches and liquid cleansing agents to improve their rheological properties. Inelastic polymers are used, for example, in order to thicken soups and sauces. Paints and toothpaste are thickened in the same way. The particular flow characteristics of ketchup, lipstick and non-drip paint find their explanation in their peculiar rheological 8
CHAPTER 1. INTRODUCTION properties, and these are exploited by their manufacturers. The motor industry also has an interest in non-Newtonian fluids: multi-grade oils have polymer additives (viscosity-index improvers) so as to change the behaviour of their viscosity under extremes of pressure and temperature. As another application area of non-Newtonian fluids, consider one from the world of the oil industry: once the natural pressure which drives oil to the surface from an oil field reduces below a certain point oil companies may use a process known as enhanced oil recovery in order to extract the remaining oil from the reservoir of porous rock. If water is pumped down into the oilfield in order to push the oil to the surface, instabilities at the water/oil interface can occur and a phenomenon known as "fingering" is the result. Polymer solutions find applications in stabilizing the water/oil interface, leading to a greater proportion of the available oil being extracted. We now look at the behaviour of non-Newtonian fluids in two simple types of flows: shear flow and uniaxial extensional flow. The characterization of nonNewtonian fluids based on their behaviour in rheometrical flow situations will be given a more detailed treatment in Chapter 4.
1.2.1
Non-Newtonian fluids in shear flow
The shear viscosity It is generally true of non-Newtonian fluids that the relationship in shear flow between the shear stress and shear-rate is nonlinear. In a simple shear flow the relationship Vxy{t) = Voixy(t),
(1-26)
which holds for a Newtonian fluid, is, in general, replaced by
(1-27)
where rj is termed the apparent viscosity or shear viscosity and 7 = |7 x y | is called the shear-rate. For most non-Newtonian fluids it has been observed experimen tally that the shear viscosity rj in (1.27) is a monotonically decreasing function of shear-rate that tends to some limit ??oo (say) for very high shear-rates. Such fluids are termed shear-thinning or pseudo-plastic. Figs. 1.6 and 1.7 show, respectively, the typical shear viscosity n versus the shear-rate 7 and the shear stress oxy versus 7X!/ behaviour for a shear-thinning fluid. Not all non-Newtonian fluids display shear-thinning behaviour in steady shear flow, however. Boger fluids have more or less constant shear viscosity over a large range of shear-rates and a few fluids (e.g. corn starch suspensions) shear-thicken in steady shear flow, so that the shear viscosity increases with increasing shear-rate. The equation (1.27) has a shear viscosity dependent only on shear-rate and cannot be used to describe the behaviour of many fluids. For example, in a thixotropic (work softening) fluid there is a breakdown of material structure with shear-rate so that constant shear-rate conditions will lead to a decrease in shear viscosity with time. The effect is often reversible, so that if the material is left undisturbed after applying a certain rate of shear a thixotropic fluid will regain its viscosity. The opposite effect is seen in so-called rheopectic (work hardening) fluids where shear forces lead to the formation of structure 9
1.2. NON-NEWTONIAN FLUIDS
Figure 1.6: The shear viscosity i) = ^(7). and the shear viscosity increases over time under constant shear-rate conditions, and then gradually disintegrates when left undisturbed. Normal stresses In order to see a second way in which non-Newtonian fluids (and in particular, viscoelastic fluids) may differ from their Newtonian counterparts we consider a shearing flow with -yxy a constant and associated velocity field u = (u,v,w) = (7z y 2/,0,0).
(1.28)
The pressure and components of stress are then found to be constant,
&XX
= =
&xy
=
C =
=
0,
p
a
ij
=
^yy ~ &zz
-P,
Vojxy,
otherwise,
(1.29) (1.30) (1.31) (1.32)
where u denotes the fluid velocity,
&xx a
yy ~
0Xy
=
vWixy,
(1.33)
&yy
=
^1(7),
(1.34)
zz
=
^2(7),
(1.35)
Oxz
=
a-yz =
a
0
(1.36)
The quantities TJ, NI and AT2 are the so-called viscometric functions. The func tions iVi and N2 are called, respectively, the first and second normal stress differences, often written as iVi(-y) = tfi-f, N2(j) 10
= *272,
CHAPTER 1. INTRODUCTION
Figure 1.7: The shear stress axy =
^(TOT^-
where ^ i and $2 are called, respectively, the first and second normal stress coefficients. All available experimental evidence suggests that non-zero normal stress differences are associated with viscoelastic fluids. So, for example, the normal stress differences are zero for a Newtonian or a generalized Newtonian fluid (see §2.4). One conclusion which we may draw from this observation is that, whereas for a Newtonian fluid a shear stress is all that is required to generate steady simple shear flow, in the viscoelastic case this is not generally so: a normal force would be required on the upper plate. Moreover, non-zero normal stress differences are features of nonlinear models. So, for example, they do not appear in the Maxwell model presented in §1.1.1. Since, when undergoing a shearing flow the restoring force is generally greatest in the direction of the flow, this component of normal stress is larger than the others and it follows that iVi should be expected to be positive. All reliable experimental data confirm this expectation. Experimental evidence for N2 shows it to be small compared to Ni and for polymeric liquids JV2 is non-positive. The presence of normal stresses in a viscoelastic fluid can often manifest themselves in quite spectacular ways. Here we will examine the results of three experiments where normal stress effects are evident. The first two involve free surface flows and the third a swirling flow.
Free surface flows As a first example we look at what happens when a viscoelastic fluid is extruded from a tube. When water is extruded at low Reynolds numbers from a capillary tube having circular cross-section the extrudate may be some 13% greater in diameter than that of the capillary bore, and this will decrease to a diameter some 13% less than that of the capillary bore for high Reynolds number laminar flow. The situation for extrusion of a viscoelastic fluid may be quite different,
11
1.2. NON-NEWTONIAN FLUIDS
Figure 1.8: Die swell for fluids extruded into a neutrally buoyant medium. Left: Extrusion of a Boger fluid of viscosity 11.4 Pa.s. Re = 0.0009. We = 0.272. Right: Extrusion of a Newtonian fluid of viscosity 11.6 Pa.s. Re = 0.001. Photographs by R. J. Binnington, Department of Chemical Engineering, Uni versity of Melbourne. Reprinted from D. V. Boger and K. Walters, Rheological Phenomena in Focus, Copyright (1993), with permission of Elsevier Science. and swellings, at low Reynolds numbers, to diameters some two or three times that of the capillary bore are common. See Fig. 1.8 for photographs showing the significant increases in extrusion diameter that are possible with a viscoelastic fluid being extruded at approximately the same (low) Reynolds number as a Newtonian fluid of comparable viscosity. The so-called die swell (or extrudate swell) effect becomes yet more pronounced as the flow rate increases. Although various theories have been proposed in order to explain the phenomenon of die swell, none is completely satisfactory and there is no one simple explanation available at the present time. A simple model for the swelling behaviour was, however, proposed by Weissenberg. Taking the x—coordinate as being along the axis of the tube it may be seen that the normal stress axx acts along the streamlines within the tube flow region. Once the fluid has passed through the orifice the absence of the tube walls means that the fluid can no longer support this tension along the streamlines and the fluid therefore contracts axially and swells radially. Understanding the mechanism behind die swell is important not least because of the need for accurate prediction of the cross-sectional area and shape of extrudate in industrial flow processes involving polymer melts and solutions. The problem of predicting these extrusion characteristics is particu larly challenging in industrial situations where the cross-section of the bore of the extrusion tube is not circular. In these cases the extrudate may not have a cross-sectional shape that is the same as that of the bore. 12
CHAPTER 1. INTRODUCTION
Figure 1.9: The rod-climbing effect for a viscoelastic liquid. From K. Walters, Rheometry, Chapman and Hall, London, 1975, with permission. As a second example of normal stress effects in free surface flows of vis coelastic fluids we must mention the rod-climbing behaviour of viscoelastic flu ids, commonly known as the Weissenberg effect. If a rod is placed vertically into a beaker of Newtonian fluid and rotated, inertia causes the fluid to dip in profile near the rod and rise at the beaker walls. However, when the same experiment is performed with a viscoelastic fluid the fluid may actually climb the rod. Normal stress effects are thought to be important in explaining this phenomenon too: the hoop stress age produces a tension along the circular streamlines, 'strangulating' the fluid and forcing it up the rod. Swirling flows Normal stress effects manifest themselves also in secondary flow phenomena associated with viscoelastic fluids. Hill [282] observed in 1972, for example, that when a beaker of viscoelastic fluid was swirled by rotating a disc placed on the fluid surface there could be reversal of the flow pattern in the secondary flow when compared with the flow of a Newtonian fluid. Inertia in the Newtonian fluid drives the fluid elements outwards near the disc as shown in Fig.1.10. However, normal stresses can counteract the effect of inertia in a viscoelastic fluid and create a secondary flow in the opposite direction. For further examples of normal stress effects in viscoelastic fluids the inter ested reader is referred to excellent discussions of these in the books of Barnes et al. [43] and Bird et al. [72].
13
1.2. NON-NEWTONIAN FLUIDS
(a)
(b)
Figure 1.10: Secondary flow patterns in (a) Newtonian fluid (b) Viscoelastic fluid.
1.2.2
Non-Newtonian fluids in uniaxial extensional flow-
Consider uniaxial extensional flow, defined as one where the velocity field u = (u, v, w) has components given by CXI
€.Z
u = ex, v = ——, and w = — — ,
(1-37)
where e is the (constant) extensional strain rate. After solving the steady NavierStokes equations (2.33) with the flow field (1.37) we obtain p
=
ff« cryy cjjj
= = =
_£e 2 (4a; 2 + y2 + z2) + constant, 8 -p + 2r)0£, ozz = -p-rjoe, 0, otherwise,
(1.38) (1.39) (1.40) (1-41)
so that axx — ayy — 3%e and ayy — azz = 0. By solving the governing equations for a mathematical model for non-Newtonian flow the stress components are typically of the form Oxx-Vyy
=
«7e(c),
(1-42)
Oxx-Ozz a,
= =
«k(e), 0, i±j,
(1-43) (1.44)
where r}e is called the extensional viscosity. The ratio between the extensional viscosity rje and the shear viscosity is referred to as the Trouton ratio. It is obvious from (1.39)-(1.40) that this value is equal to three for Newtonian fluids. However, it can be very large for many viscoelastic liquids. Polymer melts, for example, can have large Trouton ratios because of the effect of shear-thinning. Dilute polymer solutions may exhibit extensional viscosities that increase greatly with e. We term such fluids tension thickening for obvious reasons.
14
CHAPTER 1. INTRODUCTION Many industrial processes involve extensional flows of non-Newtonian fluids. Stretching flows of polymers are encountered, for example, in coating, calender ing and fibre spinning. In the latter example, tension thickening of the fluid is important to maintain stability in the stretching operation [454]. Of concern to the industrial rheologist may be the reduction or elimination of recirculatory zones in equipment used for a polymer processing operation. Understanding the conditions under which recirculatory regions appear, therefore, is impor tant. In the flow of viscoelastic and Newtonian fluids through abrupt circular entry geometries it is sometimes observed that the vortices in the viscoelastic fluid are significantly larger than those in a Newtonian fluid of the same viscos ity, and that they grow with flow rate. Elasticity per se does not provide the complete answer as to why vortex enhancement can take place, however. For some viscoelastic materials vortex growth or enhancement does not take place. An explanation for these phenomena was provided by White and Baird [625] in 1986. Low density polyethylene showed large vortices and vortex growth whereas polystyrene flowing through the same contraction geometry exhibited almost no secondary flow at all. The difference in behaviour was attributed to the unbounded growth in the extensional stresses with fixed extensional strain rates for the polyethylene, as contrasted with bounded growth of the same in the polystyrene. In work done by Boger and Binnington [87] on tubular en try flow using both a polyacrylamide-based and polyisobutylene-based fluids of constant viscosity a connection was made between the extensibility of the molecules in the fluids and the tendency of such fluids to enhance or reduce vortex size when compared to the vortices generated by Newtonian fluids of comparable viscosity. By rounding the re-entrant corner dramatic differences in the flow field near the corner were observed for the polyacrylamide-based elastic fluid while less dramatic results arose from the same change in the flow geometry for the polyisobutylene-based fluid. An explanation lies in the fact that the sharp corner gave the flow a non-trivial extensional component. The polyacrylamide-based fluid, having a very high Trouton ratio, therefore reacted more dramatically to the rounding of the corner. We will discuss extensional effects in contraction flows in far greater detail in Chapter 8. Boger [85] observed that a similar extensional viscosity component plays a part in determining whether or not the drag experienced by a sphere in creeping flow of an infinite expanse of viscoelastic fluid increases or decreases relative to the Newtonian value. Polyacrylamide corn syrup fluids showed significant drag reduction, whereas polyisobutylene/polybutene fluids of comparable (constant) viscosity can show drag enhancement. An important difference between the two fluids lies in their extensional behaviour. We will have more to say on the influence of the rheological properties of a fluid on the drag experienced by a sphere or cylinder in Chapter 9. One of the most vividly striking of all flow phenomena associated with ex tensional effects in non-Newtonian fluids is the open syphon effect where a fluid having high extensional viscosity may be emptied out of its containing beaker once the flow is initiated by spilling a little over the edge. See Fig. 1.11 for a dramatic sequence of photographs for the phenomenon in the case of a 0.75% aqueous solution of Polyox WSR 301. The high extensional viscosity allows the fluid to sustain the filament so formed in extension and gravity does the rest. Further photographs of the rheological phenomena described in this section, and many others, are available in the fascinating book on the subject by Boger 15
1.3. NUMERICAL SIMULATION OF NON-NEWTONIAN FLOW
Figure 1.11: The open syphon effect with a 0.75% aqueous solution of Polyox WSR 301. A small thread of the fluid has been drawn over the side of the beaker. The sequence of photographs then shows the subsequent part-emptying of the beaker as fluid flows over the edge. Photographs by R. E. Evans, Depart ment of Mathematics, University of Wales, Aberystwyth. Reprinted from D. V. Boger and K. Walters, Rheological Phenomena in Focus, Copyright (1993), with permission of Elsevier Science. and Walters [90].
1.3 1.3.1
Numerical Simulation of Non-Newtonian Flow Early work
Research into non-Newtonian fluid mechanics can be traced back to the late 1940's and early 1950's with the pioneering work of members of the rheological patriarchy who include Rivlin [507], Oldroyd [424] and Lodge [361]. Early work in problem solving focussed on exact solutions and perturbation methods for slow flows of so-called order fluids (see §2.5), valid for small Deborah numbers. Although of little use in polymer processing applications and other real life problems, such flows provided fertile ground in which to plant ideas and later, to develop numerical techniques. A useful list of early work is given by Bird et al. [72] and includes secondary flow in curved tubes [45] and in conduits of non-circular cross-section [339,470,509], flow near a translating sphere [128,233, 604], flow in a converging tube [3,530] and boundary layer flows [50,166,177]. Amongst early papers employing perturbation methods we could also mention work by Tanner and Pipkin [566], Tanner [560] and Pipkin and Tanner [471]. Of course, the validity of analytical solutions is necessarily limited to flows in very simple geometries (for example, the so-called viscometric flows such as Poiseuille flow or Couette flow) or to those cases where the equations for the order fluids are valid perturbations about a Newtonian flow. This would exclude problems with high Deborah number, or flows in geometries where there are sharp corners, for example.
16
CHAPTER 1. INTRODUCTION There seem to us to be two main arguments for using numerical methods for the solution of non-Newtonian flow problems. The first is that non-trivial problems involving highly elastic fluids can be contemplated, thus extending the body of results obtainable by a purely analytical approach. Thus, the com putational rheologist can begin to entertain the thought that the prediction and understanding of the important and (sometimes) dramatic flow phenom ena that are observed experimentally are within his grasp. Secondly, numerical simulation is of great importance in non-Newtonian industrial flow applications. Experiments involving complex industrial fluids can be expensive and time con suming. The costs involved in interrupting a production line operation to make measurements can be prohibitive. The effect of design changes to a flow oper ation can be predicted with comparative ease and convenience using numerical methods, once a fluid has been properly characterized. It is in these contexts that sophisticated packages such as POLYFLOW and FLUENT have proved their worth. The late 1960's and early 1970's saw the start of what has, in the intervening years, been a mushrooming effort in the numerical simulation of non-Newtonian flows. Initially, much effort was expended on the numerical solution of the ordinary differential equations arising from perturbation methods applied to flows at low Deborah numbers. Relevant here are the contributions of Walters and Waters [609], Griffiths et al. [251], Griffiths and Walters [252] and Davies and Walters [165]. From a similar era we find papers appearing in the literature in which attempts were made to solve nearly viscometric flow problems, and in this context we could cite work from the early 1970's by Barnes et al. [44] and Dodson et al. [180] in nearly viscometric pipe flow, by Davies and Walters [165] in journal bearing problems, and by Kramer and Johnson [336] in rotational flows. In the area of inelastic non-Newtonian fluids the generalized Newtonian fluid (see §2.4), useful for modelling flows dominated by shear viscosity, has spawned a body of research literature all of its own. Attempts from the early 1970's to solve flows of such fluids numerically include the papers of Duda and Vrentas [190] and Tanner et al. [565].
1.3.2
T h e high Weissenberg number p r o b l e m
Since the early 1970's the corpus of literature on numerical solutions to nonNewtonian flow problems has seen explosive growth and now includes descrip tions and applications of many sophisticated techniques for the solution of highly non-trivial flows through complex geometries and at high Deborah numbers. It was in 1977 that the first papers involving numerical methods for complex flows in two space dimensions of viscoelastic fluids made their appearance. Perera and Walters [447,448] used a finite difference method for the solution of a number of complex flow problems, and in the same year Kawahara and Takeuchi [319] used a finite element method for analyzing viscoelastic fluid flow. For a list of references of completed work up to 1983 the reader is referred to the review of Crochet and Walters [156] of the same year. Initially, there was considerable optimism about the results which could be achieved by adapting algorithms and techniques that already existed for the solution of Newtonian flow prob lems. Wouldn't standard Galerkin finite element methods, for example, be a tool suitable for the simulation of flows of highly elastic liquids? Such optimism crumbled quickly in the face of disappointing results. Early calculations were 17
1.3. NUMERICAL SIMULATION OF NON-NEWTONIAN FLOW plagued by a loss of convergence of the numerical algorithm at relatively low measures of the fluid elasticity; something which came to be known as the high Weissenberg number problem. (The Weissenberg number is a non-dimensional measure of the fluid elasticity). The problem manifested itself in simulations of every type of flow and using every available numerical method and was present whether differential or integral forms of the constitutive equations were used. By the time Crochet and Walters [156] wrote their comprehensive review of the state of numerical methods for non-Newtonian fluid mechanics in 1983, the high Weissenberg number problem was the outstanding problem in com putational rheology. Numerical methods were unable to handle flows at Weis senberg numbers sufficiently high to make comparisons with many important experimental results, and the numerical solutions were often little more than could be predicted on the basis of the order fluids. In 1984 Crochet, Davies and Walters [152] stated a number of possible contributing factors to numeri cal breakdown including bifurcations in numerical solutions and the possibility of three-dimensional flows arising as bifurcations from two-dimensional flows, failure to deal properly with singularities and change of type of the governing equations. The choice of different rheological models would only change the process of numerical breakdown, not necessarily eliminate it altogether. For example, Crochet and Keunings [154] found that the domain of convergence in Weissenberg number was significantly larger when using an Oldroyd B model rather that an upper-convected Maxwell (UCM) model in die swell problems, but they still encountered a limiting value. By 1989, when Crochet [150] wrote his review of numerical simulation of viscoelastic flow, some progress had been made in overcoming the high Weissenberg number problem, and greater insights gaLied into the underlying mechanisms that trigger numerical breakdown. Prob lems encountered in numerical simulation, including issues relevant to the high Weissenberg number problem, are given a more detailed treatment in Chapter 7.
1.3.3
Numerical work since t h e mid 1980's
Subsequent to the reviews in the 1980's of Crochet and Walters [156], Cro chet et al. [152] and Crochet [150], mentioned in the previous section, several other reviews of work in computational rheology have been published. These are, notably, review articles by Keunings [324,325,327], Baaijens [25] and, at a more popular level, a paper by Crochet and Walters [157]. An enormous (and continuously expanding) bibliography has necessarily limited us in our ambitions to present, at the time of writing, a state-of-the-art description of methods and results in the numerical simulation of isothermal non-Newtonian flow. We confine our review, therefore, primarily to steady rather than transient flows and to the application of numerical methods to four well-established bench mark problems: in Chapter 8 we discuss flow through an abrupt contraction, and in Chapter 9 numerical and experimental work for flow past a sphere, flow past a cylinder and the non-Newtonian journal bearing problem is reviewed. Some of the discretization methods and algorithms used in the viscoelastic literature will be dealt with in Chapters 5 and 6, respectively. In Chapter 11 we present an overview of some recent advances in computational rheology, particularly those at the mesoscale. 18
Chapter 2
Fundamentals This chapter is divided into two main parts. In the first part (Sections 2.1 and 2.2) we will be concerned with the definition of some important kinematic quantities and the derivation of some foundational results in fluid mechanics. In particular, we introduce and use the stress tensor. No attempt is made to relate this tensor to other quantities at this stage. However, from §2.3 onwards we describe a number of models which define a constitutive relationship for the stress tensor. This development begins with the Newtonian model, a simple linear relationship between the stress and the rate of strain, and proceeds to a variety of non-Newtonian models: the generalized Newtonian model (§2.4), the order fluids and the CEF equation (§2.5) and finally, nonlinear constitutive equations of the differential and integral type. As we progress through the chapter the results reviewed become more mathematically demanding and necessarily, therefore, reference is made to other publications for further detail and insights.
2.1
Some Important Vectors
In the mathematical description of the behaviour of fluids it is convenient to use vector (and tensor) valued functions. In particular, we shall introduce here the velocity and stress vectors.
2.1.1
Fluid velocity and acceleration
Adopting a fixed Cartesian frame of reference with orthonormal base vectors ei, e2 and e3 we consider an infinitesimal fluid element (particle) having position vector x(i') = (xi(t'),x2(t'),x3(t')) =xiei + x2e2 + x3e3, at some time t'. Then we define the velocity of the element at time t' = t to be u(x(i),i) =
(2.1)
dt> t'=t
The acceleration of the fluid particle at time t' = t is given by
a(x(t),t)=-^U(x(f),0 Dtf 19
(2.2) t'=t
2.1. SOME IMPORTANT VECTORS We call the operator D/Dt' rule of differentiation, that D _ d
the material derivative and note, using the chain dxi d
ZW ~ dt> + IF'dx'i 2.1.2
7
+ u(x(i'),0"V
(2.3)
Forces and t h e stress vector
Suppose now that an ensemble of infinitesimal fluid elements occupies a volume V(t), the boundary of which is a closed bounded surface S(t), with an outward pointing unit normal n, as shown in Fig. 2.1. If we consider an infinitesimal
Figure 2.1: The material volume V(t). surface area element A having position vector x and area 5S, then we define the stress vector s n (x, t) by requiring that the force exerted at time t on the element A by the fluid into which n is pointing should be sn5S. The total force over the whole of S is then
/
sndS.
(2.4)
S(t)
The fluid in V(t) will, in general, experience forces other than those acting just on the surface S. These are the so-called body forces such as gravity. If p denotes the density of the fluid in V(t) and b the body force per unit mass then the total force experienced by the fluid occupying V(t) will now be / sndS + / phdV. S(t)
V(t)
20
(2.5)
CHAPTER 2. FUNDAMENTALS
2.2 2.2.1
Conservation Laws and the Stress Tensor Conservation of mass
The conservation of mass requires that the rate of change of the mass of the fluid elements occupying V{t) should be zero. That is
dt
I pdV = 0.
(2.6)
v(t) We use the Reynolds transport theorem (see Appendix B.4.1) to write (2.6) as f
(~+pV-u^jdV = 0,
(2.7)
V(t)
so that, since (2.7) is true for all closed bounded volumes V, and assuming a continuous integrand, we deduce that ^ + p V - u = 0.
(2.8)
A fluid is said to be incompressible if jfi = 0 and from (2.8) we therefore conclude that the velocity field must be solenoidal, i.e. V • u = 0.
2.2.2
(2.9)
Conservation of linear m o m e n t u m
The momentum of the fluid occupying the material volume V(t) is easily seen to be / pudV, v(t) so that from the principle of linear momentum (Euler) j
f pudV= V(t)
f sn(x,t)dS+ S(t)
f pbdV.
(2.10)
V(t)
After an application of the Reynolds transport theorem to the left-hand side of (2.10) we deduce, using (2.8), that
V(t)
2.2.3
S(t)
V(t)
Conservation of angular m o m e n t u m
Another law of mechanics, also due to Euler, is the principle of moment of momentum. Applying this law to the material volume V(t) we obtain, on the
21
2.2. CONSERVATION LAWS AND THE STRESS TENSOR assumption that the net torque is due only to the stress s n and the body force b , that j
f (x x pa) dV = f (x x s n ) dS + f (x x pb) dV. V(t)
S(t)
(2.12)
V(t)
Again, using the Reynolds transport theorem to enable us to bring the time derivative inside the integral sign on the left-hand side, and using (2.8) we obtain f [ x x p ~ J dV= V(t)
2.2.4
f (x x s„) dS + I (x x pb) dV. S(t)
(2.13)
V(t)
The stress tensor
Euler's laws of mechanics 1 , applied to an incompressible fluid flow, led to equa tions (2.11) and (2.13). These equations may be used in order to prove first the existence of a stress tensor
sn =n-er, (ii) p ^
= V-a
+ pb,
(2.14)
(Hi) a is symmetric. Proof. See Appendix A.l. In addition to the conservation laws for mass and linear and angular mo mentum, there is another conservation law, not required for isothermal flows. This is the conservation law for energy. throughout this book we work with the definition for the divergence of a second-order Cartesian tensor
22
CHAPTER 2. FUNDAMENTALS
2.2.5
Conservation of energy
The first law of thermodynamics for a system is a statement that the rate of change of the energy of the system is equal to the sum of the rate of work done on the system and the rate at which energy is added to the system. Denoting the internal energy per unit mass by e and by q the heat flow vector per unit area, this leads us to write idt f
Jv(t)
p(lu22+e) V
dV
/
=
f ph-udV+ Jv(t)
f (n-a)-udSJs(t)
[ Js(t)
nqdS. (2.15)
The terms on the left-hand side of (2.15) represent the rate of change of the kinetic and internal energy of the fluid occupying V(t). The first two terms on the right-hand side account for the rate of working on the fluid in V(t) by the body force b and the fluid outside V{t). The last term in (2.15) is the rate at which energy is added to the fluid element due to heat flow. By Fourier's law of heat conduction, we may express q in terms of the gradient of the temperature T thus: q = -kVT,
(2.16)
where k is the thermal conductivity of the fluid. Use of the Reynolds transport theorem on the left-hand side of (2.15) and application of the divergence theorem to the last two terms on the right-hand side of the same equation now enable us to see that
= [ (pb • u + (V • tr) ■ u +
(2.17)
The last steps in our derivation of the energy equation are to assume that all integrands in (2.17) are continuous and to appeal to the equations of continuity (2.8) and of motion (2.14). Eqn. (2.17) thus now simplifies to De p-~=a-:Vu
+ V-(kVT).
(2.18)
It may be shown (see [623], for example) that for incompressible fluids the internal energy equation may be expressed in terms of the temperature T in the form pCp
15t
=
* : V U + V ' {kVT)'
(2 19)
"
where Cp denotes the heat capacity at constant pressure and per unit mass of the fluid. So far in this chapter we have made reference to the Cauchy stress tensor but made no attempt to define it in terms of other variables. The remainder of the chapter will be dedicated to the task of describing the stress tensor for various fluids. We begin by making the Newtonian assumption that the stress tensor is a linear, isotropic function of the components of the velocity gradient. 23
2.3. THE NEWTONIAN FLUID
2.3
The Newtonian Fluid
In order to derive an explicit form for the stress tensor we write the components Oij of
(2.20)
where we define p, the pressure, to be the negative average of the diagonal stress components, i.e. p=~an.
(2.21)
The so-called deviatoric stress or (more generally) extra-stress tensor T is as sumed to be a linear function of the various components of the velocity gradient V u and we may, as a consequence, write the components T^ of T in the form T
v =Aw^,
(2-22)
where A is some fourth-order tensor. A material is said to be isotropic if its physical properties are independent of direction. Since this is true of many fluids it is reasonable to further assume that the dependence of the components of the deviatoric stress tensor on the components of the velocity gradient tensor is invariant under a rotation of axes, i.e. that T is an isotropic tensor function of the components of V u . This further requires that the fourth-order tensor A should be isotropic and it may be shown (see [95] p214, for example) that the components Aijki of A may be written in the form Aijki = ^5ij5ki + ridikdji + vduSjk,
(2.23)
where A, r) and u are constants. Now, from (2.20) we see that since cr is sym metric then so is T and therefore, from (2.22) A must be symmetric in its first two indices. Hence from (2.23) r) = v and so Miki = M>ij6ki + r) {dikSji + 6u6jk),
(2.24)
and from (2.22)
\x
T =
duk
, (dui
,
du
A
^ijjkk+TlJij,
(2.25)
where 7^ denotes the ijth component of the rate-of-strain (or rate-of-deformation) tensor 7, given by 7=(Vu+(Vu)T).
(2.26)
Prom the definition of p, and employing the summation convention, era = -3p+Tu
=> Tu = 0.
24
(2.27)
CHAPTER 2. FUNDAMENTALS Therefore, from (2.25) 3 2A7*fc +mu
=0
3A + 2r? = 0,
(2.28)
a = - -grtkkSij + niij •
( 2 - 2 9)
so that r
We note that ikk = 2^- = 2V ■ u = 0, dxk
(2.30)
for an incompressible fluid (see (2.9)) and therefore aij = -pSij + Tij = -pSij + rrtij.
(2.31)
The final step in the derivation of the equation of motion for a Newtonian fluid is to insert (2.31) into (2.14) to obtain Dui Dt
_
_dp_ dxi dp dxi
_
dTjj. dxj d (dui dxj \dxj
duj\ dxi)
- -^+'M+*'
(2 32)
'
since 3
dxjdxi
-■ — ( V - u ) = 0 ,
dxi
and so, in vector form we have the Navier-Stokes equations for an incompressible fluid: p ^
= -Vp
+ T1oV2u + pb,
(2.33)
where we have written r/o in place of T) for consistency with Chapter 1.
2.4
The Generalized Newtonian Fluid
In §2.3 we saw that the Newtonian fluid is characterized by the assumption that the extra-stress tensor T is a linear isotropic function of the components of the velocity gradient and this led to (2.29) and, in the case of an incompressible fluid, to the derivation of the Navier-Stokes equations (2.33). In a steady simple shear flow with u = (u,«,ti;) = (7 x l ,y,0,0),
25
(2.34)
2.4. THE GENERALIZED NEWTONIAN FLUID the corresponding state of stress is therefore given by /
~P T)ojxy
Vojxy -p
V
0
0
a- =
0 0
\ .
(2.35)
-p )
Many fluids, such as water and air, behave in this way. But as noted in §1.2.1 the Newtonian fluid model is unable to account for the shear-thinning behaviour seen in many fluids and, moreover, the first and second normal stress differences are zero. As a first step towards deriving constitutive relations for non-Newtonian flu ids, and taking account of shear-thinning at least, we could consider abandoning the assumption that the extra-stress tensor is linear in the components of the velocity gradient and instead write the viscosity n as a function of the invariants of 7 , i.e. as a function of quantities associated with 7 which are independent of the coordinate system.
2.4.1
Derivation of the stress tensor
It is well known, as may be confirmed by reference to any elementary textbook on linear algebra, that the eigenvalues Ai, A2 and A3 (say) of a second-order tensor A (say) are invariants of A and, in the case of A being symmetric, are guaranteed to be real. The eigenvalues are defined as the roots of the characteristic equation |A - AI| = 0.
(2.36)
By expanding (2.36) and denoting by {0^} the components of A, we obtain the characteristic equation A3-J1A2 + / 2 A - 7 3 = 0 ,
(2.37)
where, in the case of a symmetric tensor A, h
= =
h
011 + o 22 + a 33 , trA = A1+A2+A3,
(2.38) 2
=
(0,220,33 + 033011 + 011022) - (023 + a
=
AiA2 + A1A3 + A2A3,
l3
+ a\2) ,
(2.39)
and h
=
OH022033 -I-2a230l30l2 - O n a 2 3 - 0220 1 3 — a330 1 2 ,
=
detA = AiA2A3,
(2.40)
are the coefficients of the cubic equation (2.37) and are invariants of A, called the principal invariants of A. If we write, as a generalization of (2.31), that a%i = -P$ij + r)(h,l2,h)iij,
(2.41)
where I\, J 2 and I3 now denote the principal invariants of the rate-of-strain tensor 7, then the stress tensor a with coefficients {u^} is an isotropic function of 7. (See [95], p235, for example). 26
CHAPTER 2. FUNDAMENTALS Instead of working with the principal invariants it is more usual to work with functions 7, II and HI of h, h and J3, defined by I II III
= h = Ax + A2 + A3 = 7 « = tr 7, = I21-2I2 = \l+\22 + \l=7iJiJi=tTi\ = 3I3 + If - Shh = Xf + Xl + Xl =
iaijkiki
= tr-y 3 ,
(2-42) (2.43) (2.44)
where now Ai, A2 and A3 are the eigenvalues of 7. The functions 7, II and III are themselves invariants of 7. It has already been noted in §2.3 that for an incompressible fluid in = 2V • u = 0 so that I vanishes in this case. For the simple shear flow (2.34) III = tr 7 3 = 0,
(2.45)
so that r) is only dependent upon II, and it is usual, as a consequence, to write r] as a function of the shear-rate 7 where
( 2 - 46 )
7 = y ip ■ i = y^iuiii'
when the generalized Newtonian fluid model is used to describe shearing flows. In fact, as noted by Bird et al. [72] (see p233), the generalized Newtonian model should only be used to describe steady shearing flows or shear flows which, if time dependent, are at low Deborah number. Although the normal stresses predicted by the generalized Newtonian model for steady simple shear flows are zero, these are of little importance in determining volume flow rate versus pressure drop relations for flows in straight uniform pipes, and torque versus angular velocity relations for shearing annular flows. Therefore the generalized Newtonian model can give excellent predictions of these relations for shear flows. The same would not necessarily be true for other types of flow, where elastic effects can become important. Specific models expressing the dependence of the shear viscosity on the shear-rate may be found in §4.2.1.
2.5 2.5.1
The Order Fluids and the CEF Equation Strain tensors
Before we are able to progress further we need to say something about strain tensors. Consider a particle moving in a fluid. We suppose that the position vector of the particle is x(i) at the present (reference) time t and that the same particle has position vector x ' at some past time £', so that x(t') = x' as shown in Fig. 2.2. Then by introducing the differentials dx and dx' we define the deformation gradient tensor F(x, t, t') by dx' = F{x,t,t')dx.
(2.47)
In other words, denoting the components of x and x', respectively, by {xj} i = 1 and {x'i]i_l we have, employing the summation convention, dx[ * *
= Fijdxj
(2.48)
-
(2.49)
27
g -
2.5.
THE ORDER FLUIDS AND THE CEF EQUATION
Figure 2.2: Motion of a particle from time t' to time t. As a measure of the relative magnitude of the differentials we can take the scalar product of dx' with itself to get {dx')T dx'
= =
(Fdx)T(Fdx), dxT(FTF)dx.
(2.50) (2.51)
The quantity F T (x,t,t')F(x,t,t') is called the Cauchy-Green strain tensor, which we denote by C(x,t, t') where t' e (—oo,t] so that C(x, t, t) = I, the identity tensor. It is, of course, possible to define a second deformation gradient tensor FJ(X, t,t') (the displacement gradient tensor, referred to again in §6.5) where dx = E(x,t,t')dx',
(2.52)
so that Eij = dxi/dx'j and
EF
< >«=fttH«-
(2 53)
-
The so-called Finger strain tensor B = B(x, t,t') is now given by B = E E T and it may be seen that C B = F T F E E T = F T E T = ( E F ) T = I,
(2.54)
so that B = C - 1 .
2.5.2
Order fluids
Rivlin and Ericksen [510] showed that if the components of the extra-stress tensor T = a + pi for an isotropic fluid are assumed to depend only on the gradients of the velocity, acceleration and higher time derivatives at some time t, then there exists a polynomial equation in T and the so-called Rivlin-Ericksen tensors {A ft }, defined by Qk
Ak(x,t)
= -^C(x,t,t%,=t, 28
(2.55)
CHAPTER 2. FUNDAMENTALS where the partial derivative with respect to t' is taken at fixed x and t (i.e. fol lowing a fluid particle). The coefficients in the polynomial relationship between T and the Rivlin-Ericksen tensors are invariants of the latter. Using the same notation as in §2.5.1, it follows immediately from (2.55) that
AlM
=
^(FTF)L<=i< dFT
-F dt>
FT™ dt
+
T
v=t
r
=
F ( V u ) ' F + F ( V u ) ' J Fl
=
FT((VU)' + (VU)'T)F| , V / \t'=t
=
Vu + (Vu)T.
\t'=t
(2.56)
In deriving (2.56) we have used the following results (see Appendix A.3): 0F(x,t,t') dt' dFT(x,t,t')
at'
=
(yu)'TF{x,t,1?),
(257)
=
FT(x,t,t')(Vu)'.
(2.58)
From (2.56) we conclude that Ay(x,t) = V u -I- V u T = 7. Repeated use of the identities (2.57)-(2.58) and use of induction enables one to deduce the recurrence relation (Eulerian frame of reference) A* + 1 = ^
+ (Vu)A fc + A f c V u T ,
k = 1,2,3,....
(2.59)
If the constitutive relation is a polynomial function of the first TV Rivlin-Ericksen tensors, viz. T = f(Ai,A2,...,Ajv),
(2.60)
for some tensor-valued function f then since the tensor Aj has dimensions t~l where t is time (see the book of Huilgol and Phan-Thien [296] pl51, for exam ple), "order fluids" may be derived by arranging the terms in the polynomial according to orders t~x, t~2, etc. and then terminating the expansion at suc cessive orders to obtain Tx T2 T3
= = =
aiA!, a i A 1 + a 2 A 2 + a 1 iA?, (ai + oi ; iitr(Af)) Ai + a 2 A 2 + o u A j + a 3 A 3
(2.61) (2.62)
+a12(A1A2+A2A1),
(2.63)
where the {a,} are constant coefficients. In order to derive the above we assume that the fluid is incompressible (so trAi = 0), that the stress tensor T is sym metric, and we use the Cayley Hamilton Theorem. These assumptions have the following effects: 1. Since trA x = 0 we ignore a term like (trAi)Ai in the second- and thirdorder models and terms like (trAi)A 2 and (trAi) 2 Ai in the third-order model. From (2.59) we also have trA 2 = trAf, which explains the absence of a term (trA 2 )Ai. in the third-order model. 29
2.5. THE ORDER FLUIDS AND THE CEF EQUATION 2. By the symmetry of the stress tensor T and the Rivlin-Ericksen tensors Ai and A 2 we obtain the result that the expression ai 2 AiA2 + a 2 i A 2 A i in the third-order model can be combined without loss of generality as ai2(AiA2 + A 2 A : ) . 3. The Cayley-Hamilton Theorem states that the Rivlin-Ericksen tensor Ai satisfies its own characteristic equation (see (2.37)) so that A ? - J 1 A ? + / 2 A i - J 3 I = 0, where I1=I
= trAi = 0,
/2 = i ( / 2 - J / ) = -itr(A?), and h = \ (2III + I3-
3(/)(II)) = \lll
= \lv (A?) .
Since the J3I term can be neglected (it can be absorbed into the pressure) we have that Af can be written in terms of tr (Af) A i . This explains the absence of A j in the expansion for the third-order fluid.
2.5.3
Limitations of order fluid models
To relate the parameters in the expansions (2.61) - (2.63) to the viscometric functions we consider steady simple shear flow u = (%yy, 0,0) with %y > 0 so that /
Ai =
0 7*» \ 0
j
/ 0 x y 0 \ 0 0 , A2 = 0 0 0 / \ 0
0 272y 0
0 \ 0 and A 3 = 0, 0 /
(2.64)
and therefore, for this flow,
( =
-'-xx
*xy
"
TXy
Tyy
0
0
0
\
,
Tzz / / 0 (a 1 +272 y a 1 : 1 1 ) %y \ 0
/ i +aii \
°
i
0\ v 0 0 0 0 /
»\
/
i2xy 0 ]+a12\ 0 0 /
0 0
+ a2
0
2j3xy \ 0
/ 0 0 \ 0
2jxy 0 0
0 2j2xy 0
0 \ 0 0/
0\ 0 . 0/
(2.65)
Hence, T
xy
=
=> r? = = Txx-Tyy
T
w
T}(j)%y = (oi 4- 27 2 ai : u + 2a 12 7 2 )7 a: j / a i + 272(ai2+a1:n), Vo + 27 2 (a i 2 + o i : n ) , where ax = r]0 = 2
TJ(0),
(2.66)
2
=
7 *i(7) = - 2 a 2 7
=>a 2
=
-i*i,
(2.67)
-T„ =►011
= =
7 2 * 2 (7) = 2a 2 7 2 + a n 7 2 * 2 - 2a2 = *x + * 2 .
(2.68)
30
CHAPTER 2. FUNDAMENTALS Thus it may be seen that the viscosity is a constant, 770 (say) for the first-order (Newtonian) and second-order fluids. Unless 012 = — a i m the third-order fluid viscosity is a quadratic function of shear-rate, and in order that the model be shear-thinning we then require that 012 + ai : ii < 0. Furthermore, as noted by Bird et al. [72] a shear stress which increases monotonically with shear-rate requires that dT - ^ > 0 ^ 7 OTf
?
+ 47 2 (ai2-f-ai : U ) > 0 ,
(2.69)
i.e. ai+6-y2(ai2+ ai:u) > 0
=> )?o - 67 2 |ai 2 + a i ; n | > 0 =*
; ^ 1>72. 6|oi2 + a i : i i |
(2-70)
We are therefore faced with a limit on the maximum allowable shear-rate if the third-order fluid model is to give realistic shear stress behaviour. No such restrictions for 7 applies for the second-order fluid in this regard, but Tanner [563] has shown that the second-order fluid is unsuitable for describing unsteady flows due to the onset of instabilities. As models for the description of viscoelastic liquids the order fluids are suit able only for • slightly elastic liquids, i.e. where the fluid is only a small departure from the Newtonian fluid, and the Deborah number is low. See Truesdell [582], for example. • 'slow' flows, i.e. flows where the kinematic tensors {A/t} vary slowly. In 1960 Coleman and Noll [146] established a rheological milestone with the introduction of the concept of a 'simple fluid'. The constitutive relation for such a fluid may be represented as a functional relation of the form T(x,t)=^-0(G(x,i,S)),
(2.71)
where the relative finite strain tensor G(x, t, s) = C(x,f, t — s) — I, so that the extra-stress is expressible as a function of the strain history. They then went on to consider "retarded" histories Ga(x,t,s)
= G(x,t,as),
(0
(2.72)
and the corresponding modified Rivlin-Ericksen tensors {A£}, now given by A£(x,i)
=
ak (-l)*_Ga(x,*,S)
=
ak afc(-l)*^G(x,t,p)
_ , ,
(2.73)
where p = as, so that A£(x,i) = afcAfc(x,£). By writing the new stress tensor T a as a multilinear isotropic function of the modified Rivlin-Ericksen tensors
31
2.5. THE ORDER FLUIDS AND THE CEF EQUATION A£ up t o 0 ( a n ) , Coleman and Noll [146] showed that it was possible to expand T a to various orders, viz: TQ TQ
=
r)0A? + O(a),
(2.74) 2
=
7 ? oA«+/3A?Af + 7 A » + 0 ( a ) ,
(2.75)
and so on, so that the order fluids are seen to be approximations to the stresses in a simple fluid when the strain history is "retarded" by a factor a. For this reason (2.74) and (2.75) are known as retarded motion expansions.
2.5.4
The CEF equation.
Let us now assume that the extra-stress tensor T can be expressed as a poly nomial in the Rivlin-Ericksen tensors. For a steady simple shear flow u = (7x»»,0,0),
(2.76)
it can be easily shown that A3 = 0 and therefore, from the recurrence relation ship (2.59), that Afc = 0 , V k > 3. Hence T is now a polynomial in the two tensors A i and A2. Rivlin [508] showed in 1955 that, in this case, there exist etj, i = 1,2,... , 8 , which are polynomials in t r A i , trA 2 , trA 2 , trAf, trA 2 , trA^, t r A i A 2 , t r A 2 A 2 , t r A i A 2 and trAf A 2 , such that T is expressible in the form T
=
a 0 I + a i A x + C 1 2 A 2 + a 3 A i + a 4 A 2 + a 5 (AiA 2 + A 2 A i )
+
a 6 ( A 2 A 2 + A 2 A 2 ) + a 7 ( A 1 A ^ + A^A 1 )
+
a8(A2A2 + A2A2).
(2.77)
The term proportional to the unit tensor can be absorbed into the pressure and a further simplification of the remaining eight term expansion in the case of the shear flow (2.76) was performed by Criminale, Ericksen and Filbey [149] in 1958. The simplification arises from the observation that for the flow (2.76) A2 A2A2 + A2A2 A2A2 + A2A2
= = =
2<72A2, A i A 2 + A 2 A i = 27 2 Ai, 27 2 A 2 , A i A 2 + A 2 Ax = 4 7 ^ , 474A2,
(2.78)
so that the relation (2.77) can be simplified to one involving just A i , A 2 and A 2 . Moreover, incompressibility of the fluid means that trAi = 0 and for the flow (2.76) the coefficients a i , a 2 , . . - , as, become polynomials in 7 2 . The CEF equation can therefore be written as T = /3 1 A 1 + / 3 2 A 2 + / 8 1 1 A 2 ,
(2.79)
and inspection of the components of the tensors in (2.79) leads one easily to see that (2.79) may be written T =
JJ(7)A!
- ^ * i ( 7 ) A 2 + (* 1(7) + *2(7))A?,
(2.80)
where r), $ x and #2 are the (shear-rate dependent) viscosity, first and second normal stress coefficients, respectively, first introduced in §1.2.1. 32
CHAPTER 2. FUNDAMENTALS The importance of the CEF equation (2.80) is that it is exact for flows where Afc = 0, V k > 3, and has found widespread application, for example, in gaining an understanding of the fluid mechanics of rheometers. It may be noted that the first term of (2.80) just gives the generalized Newtonian fluid, considered in §2.4. The remaining terms in (2.80) account for elastic normal stress effects.
2.6
More Complicated Constitutive Relations
The non-Newtonian models which we have presented in §2.4 and §2.5 are only capable of describing certain types of flows or fluids due to the limitations placed upon them. In this section we will outline the derivation of more sophisticated differential and integral models used for the modelling of arbitrary flows of viscoelastic fluids. In describing these models it is convenient to organize them according to two main classes of viscoelastic fluids: differential models of the Oldroyd type will be seen to have utility only for the description of dilute solu tions of polymer molecules, whilst differential models taking account of shearthinning (FENE type, Phan-Thien Tanner, Giesekus models, for example) and integral models of the K-BKZ type may be appropriate for more concentrated polymer solutions or for melts.
2.6.1
Differential constitutive models
The simplest constitutive equations suitable for modelling the behaviour of di lute polymeric solutions under general flow conditions are those of the Oldroyd type. In particular, the Oldroyd B model, derived by Oldroyd [424] in 1950, is an empirical expression generalizing the linear viscoelastic equation by writing the stress/strain relation in tensorial form and satisfying certain admissibility criteria (see Oldroyd [424,426], Crochet et al. [152] or Bird and Weist [77], for example). The Oldroyd B model is also derivable from a molecular model con sisting of a suspension of Hookean dumbbells in a Newtonian solvent, and this is the route which we choose to pursue in the present chapter. It will be seen, however, that a model so crude cannot be expected to capture many features of real rheological fluids. In particular, the Oldroyd B model has a constant shear viscosity and the infinite extensibility of the Hookean connecting spring leads to an extensional viscosity which blows up at a finite extensional rate. A more sophisticated form of the inter-connecting spring force law will there fore be considered afterwards: a Finitely Extensible Nonlinear Elastic (FENE) dumbbell model may be derived, leading to approximations (so-called FENE-P and FENE-CR models) which have bounded extensional viscosities. In the case of the FENE-P model the shear viscosity is a monotonic decreasing function of shear-rate. The elastic dumbbell model We consider an elastic dumbbell immersed in a Newtonian solvent and consisting of two identical beads and an inter-connecting spring, as shown in Fig. 2.3. The two beads each have mass m (say) and have position vectors ri and T2 relative to some fixed coordinate frame. We follow Schieber and Ottinger [661] in writing
33
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS
Figure 2.3: The dumbbell model. the equations of motion for the beads in the dumbbell as
m
| ( ^ "
u ( r i )
)
= _ C
( ^ "
U ( r i )
)
+ F i + B i
'
i = 1 2
' '
(2 81)
'
where u(ri) is the velocity of the solvent at the point with position vector ri, F ; is the force on the ith bead exerted by the spring and Bj is the Brownian force due to the impact of the solvent molecules on the ith bead. The constant £ is the so-called friction coefficient and is the constant of proportionality arising from Stokes's law: the drag force on the ith bead is directly proportional to the difference between the bead velocity and that of the surrounding medium. For spherical beads £ = 6Trnsa where a is the radius of a bead, and ns the solvent viscosity. Note that we have neglected hydrodynamic interactions; that is, any effect which one bead may have on the velocity of the solvent in the neighbourhood of the other bead. We have also neglected external forces such as gravity, for example, on the dumbbell. The ratio m/£ represents a characteristic time scale for velocity fluctuations of the beads due to the Brownian forces and will be denoted henceforth by XB ■ Over time scales large compared to XB we may write the Brownian force Bj in the form Bi(lt= y/2kTCdWi, (2.82) where W ; = Wj(i) is a multi-dimensional Wiener process which, being a Gaus sian stochastic process, is completely characterized by the mean and autocorre lation of its components Wij: E(Witj(t))
= 0, E ( W i j ( t ) W i j ( 0 ) = mrn(t,t').
(2.83)
The coefficient \/2kT(, in (2.82) may be derived from the principle of equipartition of energy (see [296,433], for example) which states, in the present context, that in equilibrium the kinetic energy associated with each physical component of the velocity difference dvijdt — u(rj) for the ith bead is kT/2.
34
CHAPTER 2. FUNDAMENTALS Let Q = V2 — ri denote the end-to-end vector of the dumbbell and suppose, for the sake of simplicity, that the flow is homogeneous so that we may write ufo) = u(0) + KTU
(2.84)
T
where K :— V u is constant in space. Following ideas set out by Schieber and Ottinger [661], we now subtract the two components of (2.81) and introduce the relative velocity V = ^
- KQ,
(2.85)
in order to arrive at the equivalent first-order system of equations mdV = - « V + 2F)dt + dQ = (V + nQ)dt,
2s/kT(dWt, (2.86)
where we use the simpler notation F = F i ( = —F2) and Wt ■= (W2 - W i ) / ^ 2 . Suppose that the probability that a dumbbell has an orientation in the range Q to Q + dQ and a velocity in the range V to V + dV at time t is given by * ( Q , V, t)dQdV, where $ ( Q , V , i) is the dumbbell probability density func tion. Then the Fokker-Planck equation corresponding to the system of equations (2.86) and governing the evolution of * ( Q , V, t) is
£ = -A,(v + ^, 1 + iA, ( c v + „ w + - c | £ . (,87) In this chapter we will consider entropic spring force laws of the form F = Hf(Q)Q,
(2.88)
where H is a spring constant and f(Q) some scalar function of the dumbbell length Q = |Q|. Introducing the time scale Ai = ( / 4 # for the relaxation of the dumbbell configuration we are led to
Taking note that, in general, XB "C AI, we now seek the correct contraction of the Fokker-Planck equation (2.89) into one defined for the marginal probability density function 1>(Q,t)= f *(Q,V,*)rfV. (2.90) iv First, integrating (2.89) throughout with respect to V leads to the continuity equation ^
(2-91)
= - ^ - [ ( « V » + K Q ) V ] ,
where we define the velocity-space average as « • » = \ f ■ $ ( Q , V,«)dV. (2.92) V Jv Secondly, multiplying (2.89) throughout by A B V , integrating throughout with respect to V and letting XB —> 0 leads to XB
OQ
' (<< v
v
» ^) + «
v
35
» ^+
2AT^^)Q^
=
°-
(2-93)
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS Thirdly, multiplying (2.89) throughout by A^VV, integrating throughout with respect to V and letting As —> 0 gives us the Maxwell-Boltzmann relation for the kinetic energy of the dumbbell in equilibrium: ^m
(2.94)
Combining the equations (2.91), (2.93) and (2.94) we finally arrive at the con tracted Fokker-Planck equation
£=-£K5>><»-¥£]-
'-»
To arrive at the equation (2.95) we could, of course, have just naively ne glected the acceleration term on the left-hand side of (2.81) and then writ ten down the equivalent Fokker-Planck equation. However, as pointed out by Schieber [660] this approach is not completely general and, in particular, fails when the prefactor of the stochastic term on the right-hand side of (2.82) is configuration-dependent. Such would be the case, for example, in the encap sulated FENE dumbbell model of Bird and Deaguiar [659] where the isotropic friction tensor £1 is replaced with a more general anisotropic tensor £ = C(Q)Finally, we note (as do Schieber and Ottinger [661]) that equilibration in mo mentum space and the neglect of the acceleration term on the left-hand side of (2.81) (see §13.2 of [73], for example) are both consequences of allowing \B/^I to tend to zero. We may define an ensemble average (•) for any function g of Q by
(9(Q)) = J 9(QMQ,t)dQ.
(2.96)
R3
Both Larson [341] and Bird et al. [77] relate the extra-stress tensor T to the ensemble average of the dyadic product Q F , viz. T = -nkTI
+ ns^ + nH(QQf(Q)),
(2.97)
where n is the number density of the dumbbells (see the book of Larson [341] or of Bird et al. [73] for details of the derivation). This is known as the Kramers expression for the stress tensor. As an important step to establishing a con stitutive relation for the extra-stress T we multiply (2.95) throughout by Q Q , integrate over R 3 , and use the divergence theorem and the fact that ip —> 0 as |Q| tends to its maximum permissible length to obtain 8 AkT 1 ^ - «(QQ> - (QQ>« T = — I - Yi(QQf(Q)).
(2.98)
The left-hand side of (2.98) is called the upper-convected derivative of (QQ), v denoted by (QQ), so that (2.98) now reads v 4kT = - ^ 1 -
36
1
TL(QQ/(Q)),
(2.99)
CHAPTER 2. FUNDAMENTALS and by substituting for (QF) into the Kramers expression (2.97) for T we get the Giesekus expression for the stress tensor: T = Vsi-1f(QQ)-
(2.100)
The Hookean spring law. The Oldroyd B equation Suppose now we have a Hookean spring (f(Q) = 1) so that the force law for F is F = HQ. (2.101) Then (2.97) and (2.100) read as follows: T
= -nkTI
+ risi + nHiQQ),
T = VsJ-^iQQ)-
(2.102) (2-103)
We can now eliminate (QQ) between (2.102) and (2.103) by taking the upperconvected derivative of (2.102) and observing that f = - V u - (Vu)T = - 7 ,
(2.104)
so that T-rjsi=-~
T-nkTj-rts^f .
(2.105)
We define the polymeric viscosity, r\v and the characteristic retardation time for the fluid, A2, in terms of the parameters appearing in (2.102) and (2.103) by r,p-
^ 4-ff
andA2_4(7?p + r?s)JT-(??p+??s). A{r]p + r}s)H (rjp + r}a)
Then Eqn. (2.105) can be written in the form T + Ai T = Vo % ( 7 + A2 7T )J ,
(2.106)
where 7]0 = r\v + rjs is the total viscosity. Eqn. (2.106) is the Oldroyd B constitutive equation. By separating out the solvent and polymeric contributions to the stress, viz. T = 17.7 + 1-,
(2.107)
where r is the elastic stress, and substituting into (2.106) we obtain r + Ai T= riPj.
(2.108)
The constitutive equation (2.108) for the elastic stress is called the UpperConvected Maxwell equation for the extra-stress T as r)s —> 0 (T = r ) . From the above results it may be seen that it is unnecessary to solve the Fokker-Planck equation for the configuration pdf in order to determine the extra-stress T. To gain an understanding of the types of rheological fluids for which the Oldroyd B fluid has, historically at least, been considered an appropriate model, we will now investigate its shear and extensional flow properties. 37
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS
S t e a d y s i m p l e s h e a r flow o f a n O l d r o y d B For the steady simple shear flow u = (jxyy, becomes / Txx Txy [ Txy Tyy
\
0
0 \ 0
-
0,0) the constitutive relation (2.106)
r / 0 jxy Ax i 0 0
0 Tzz j
[ V0 +
0 \ / Txx Txy 0 \ 0 \ \ Txy Tyy 0
0 0 / V 0
0 Tzz J
f Txx Txy 0 \ / 0 0 0 \ } \ Txy Tyy 0 7x„ 0 0 ) \
\
0
0 Tzz J \ 0 2
=
fluid
IJO
/ -2A27 >» \ 0
0 0 / J
7«v 0 \ 0 0 , 0 0 /
(2.109)
so t h a t , solving this system yields Txx
= 2r?0(A1-A2)72,
(2.110)
Tyy
= 0,
(2.111)
Txy
= rjojxy,
(2.112)
Tzz
= 0,
(2.113)
from which we have JVi(7) = 2r/ 0 (Ai - A 2 ) 7 2 , iV 2 ( 7 ) = 0 and 7?(7) = %■
(2.114)
Therefore we see t h a t the model predicts a quadratic first normal stress dif ference, a zero second normal stress difference and a constant viscosity. T h e Oldroyd B model has been seen to be useful in providing qualitative predic tions of the flow of Boger fluids, which have a constant viscosity over a range of shear-rates and a quadratic first normal stress difference. But Boger fluids are the exception in having a constant viscosity and as a consequence the Oldroyd B model is of limited use in industrial applications. T h e most serious objection to the Oldroyd B model, however, lies in its behaviour in extensional flows. E x t e n s i o n a l flow o f a n O l d r o y d B fluid Consider, for example, the extensional flow n=(ix,~y,~z),
(2.115)
where e is the (constant) extensional rate. T h e n (2.115) leads to t h e system f Txx Txy 0 \ \ Txy Tyy 0 V 0 0 Tzz J
-
J" / e 0 Ai ( 0 -e/2 [ \ 0 0
+
/ T„ Txy 0 \ / e 0 Txy Tyy 0 0 -e/2 V 0 0 T„ / \ 0 0
=
/ 2e-4A2e2 rjo ( 0 \ 0
38
0 \ / Txx Txy 0 \ 0 Txy Tvy 0 -e/2 / V 0 0 T„ /
0 - e - A2e2 0
0 \) 0 \ -e/2 / J
0 \ 0 -e-A2e2 /
CHAPTER 2. FUNDAMENTALS which has solution .1 - 2A2e 2?7oe l-2A1e' 1 + A2e — Tzz = —T)oe 1 + Aic'
Txx
=
Tyy -*• xy
—
(2.116) (2.117) (2.118)
U;
and means that the extensional viscosity, 7je, is given by ...
%(€)
=
2
l-2A2e ,
*T^2A^
+??0
1 + A2e
TTM'
37? 0 {1- A2e - 2AiA2e2} ( l - 2 A 1 e ) ( l + A1e)
(2.119)
A graph of T]e versus Aie is given in Fig. 2.4. It may be seen that the extensional
Figure 2.4: Extensional viscosity r)e vs. Aie for the Oldroyd B model. viscosity blows up at a finite extensional rate e = 5X" ■ The problem arises from the fact that the dumbbells are infinitely extensible and in fact become infinitely extended in this flow at e = ^-. We can see this by considering the Kramers expression (2.97) for the stress tensor and taking traces throughout so as to obtain (2.120) «?2> = ^ t r T + 3fcT H '
39
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS Using the values for Txx, Tyy and Tzz just obtained in (2.116) - (2.117) for uniaxial extensional flow, we see that /n2.
1 /_ =
.l-2A2e
„
6^oe2(Ai - A2) n H ( l - 2 A i e ) ( l + Aic)
.1 + A 2 e\ 3fcT H '
ZkT
{
'
so that (Q2) -> oo as e -> l/2Ai. It is hardly surprising then that the Oldroyd B model is unable to adequately predict extensional flow of dilute polymer solu tions, even in the case of Boger fluids. Faced with the problematic consequences of infinitely extensible inter-connecting springs, it is natural to progress onto the question of whether something tractable and useful can come out of a finitely extensible spring model. The FENE models As an alternative to the Hookean spring consider the connector force law
F=
T^mwr
(2122)
where Q2 = tr QQ, as before, and QQ is some finite constant. The force law (2.122) was originally proposed by Warner [620] in 1972. It may be seen that with a force law of the type (2.122) the spring cannot be extended beyond a length QQ. The first problem which one encounters in using the Warner force law is that, except for the limiting case Qo -> oo (back to Hookean dumbbells), it is not possible to go to a constitutive equation directly from the diffusion equation for the configuration pdf, as we did in the previous subsection. There fore approximations need to be made. We will consider two approximate FENE models: the so-called FENE-P and the FENE-CR models. The FENE-P model Instead of using the Warner connector force (2.122) Bird et al. [74] followed the ideas of Peterlin [449] (hence the 'P') and used F =
i - ( Q W
(2 123)
'
so that, when F is inserted into the Kramers expression (2.97) for the stress tensor, we have
r
=
-nkTI
+ n(QF),
40
CHAPTER 2. FUNDAMENTALS Then from (2.124), assuming that the flow is three-dimensional and taking traces throughout,
trr ^(l-(Q2/Q2))trr
=
nH(Q2)
r^wm-3nkT'
=
Q2nH{Q2/Q2)
- 3nkT (l - (Q 2 /Qo)) ,
=
nHQl {(Q2IQ2) - 1) + nHQl ~3nkT(l~{Q2/Q2)),
(2.125)
so that
Y^mw) = l+l[l+^f] where b = HQ2/kT.
= z^^
<2126)
Therefore the spring force law becomes F = HQZ(r),
(2.127)
and so the Kramers and Giesekus form of the elastic stress tensor are, respec tively, T = nHZ(T)(QQ) - nkTI, (2.128) and T
= -!j
(2-129)
•
We eliminate (QQ) between (2.128) and (2.129) and obtain, after a few straight forward manipulations, the constitutive equation for the FENE-P model:
=> ZT + XIT - A I —lnZ(r + nkTI)
=
nkTXii, (2.130)
where, as before,
For a steady simple shear flow Bird et al. [74] showed that the model was shearthinning and that as j -¥ oo, rj — r)s -» nfcTAi ( | ) (Xij)~2//3. It was shown by Bird et al. [74] that the zero shear-rate limit of the polymeric viscosity was nfcTAi(o/(6 4-3)). The non-dimensional ratio (rj -r]3)/(r)o-T]s) is shown in Fig. 2.5 on a log-log scale as a function of a non-dimensionalized shear-rate A = A17. As b increases shear-thinning becomes less pronounced and, of course, in the limit b —> 00 the Oldroyd B fluid and a constant shear viscosity are recovered. For steady uniaxial extensional flow it was shown by Bird et al. [74] that for all values of b the extensional viscosity is a continuous function of the extensional rate e and that as e -> 00 r)e - 3T]S ->• 2nkTXib.
(2.131)
Since for e = 0 the polymeric contribution to the extensional viscosity may be shown to be r]e-3r1s = 3nkTXl-^—, (2.132) 0+ 3 41
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS
Figure 2.5: FENE-P model. Non-dimensionalized polymeric contribution to the shear viscosity vs. A = A17. it then follows that
Ve - 3r?s
f(& + 3), ' 31
Hvo-Vs)
(2.133)
as e —> 00. The non-dimensional viscosity ratio appearing on the left-hand of (2.133) is shown as a function of a non-dimensional extensional rate A = Aie in Fig. 2.6 on a log-log scale. The limiting case of b —> 00 is shown in Fig. 2.4. As noted by Bird et al. [74] the model (2.130) was first developed by Tanner [561] who assumed that the configuration pdf was a delta function of the form iKQ,t) = a ( Q - S ( t ) ) ,
(2.134)
for some unknown vector S. Tanner used the Warner spring force law (2.122) and by eliminating S between the Kramers and Giesekus form of the stress tensor, obtained the same constitutive relation (2.130) for the polymeric stress. Bird et al. [74] performed a comparison in the case of a chain model having N beads connected by N — 1 FENE-P springs with experimental data for vari ous rheological fluids, although, as pointed out by Wiest and Tanner [630] the equations after and including Eq. (58) of the paper of Bird et al. are incor rect. See too [75,76] and the paper by Fan et al. [200]. In their paper Wiest and Tanner [630] determined the rheological properties of an infinitely dilute polymer solution modelled by bead-nonlinear spring chains under the Peterlin approximation.
42
CHAPTER 2. FUNDAMENTALS
Figure 2.6: FENE-P model. Non-dimensionalized polymeric contribution to the extensional viscosity vs. A = Aie. The FENE-CR model Let us write the ensemble average (QQ) in terms of a non-dimensional stress tensor A (say) as kT (QQ) = — A . (2.135) H We use the force law (2.123) so that
f(Q) = -
{QVQIY
(2.136)
Then, inserting the expression (2.136) into the Kramers and Giesekus form for the stress tensor and equating the forms we obtain kT v A
4kTT
AH „_,fcT . l m A
-W = — -T lf ' => A = f (l-f(Q)A).
(2.137)
The extra-stress may be recovered from A by using the Kramers expression for the stress tensor, viz. T = -nkTl
+ 77,7 +
nkTf(Q)A.
(2.138)
In 1988 Chilcott and Rallison [134] altered the FENE dumbbell model in an empirical way by replacing the term (I - f(Q)A) on the right-hand side of (2.137) by / ( g ) (I - A). The stress relation (2.138) remained unchanged. For 43
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS the steady simple shear flow u = {^fxyV: 0,0) the FENE-CR constitutive relation leads to 2%yAxy
= — {f(Q)(Axx
- 1)} ,
IxyAyy
= -T- {f(Q)Axy} , M 0 = Y {f(Q)(l-Ayy)},
0 =
f {f(Q)(l-Azz)},
(2.139)
which have solutions
A
=
A„ Azz
= i, 1,Aty A,,, = = -^7^-. = ^ .
"
T^F +1: Ayy = *'
(2140)
(2.U1) (2.141)
Therefore, the viscometric functions assume the form 17(7)
=
fa.+nfcTAi),
(2.142)
iVi(7) = J(§nkT'
( 2143 )
^2(7)
(2.144)
= 0.
The first normal stress difference was shown by the authors to be quadratic with Deborah number for low Deborah numbers and the second normal stress difference predicted by the model is zero. The shear viscosity is a constant and the model was accordingly proposed as suitable for modelling Boger-type fluids. A good quantitative fit with the experimental shear data of Chhabra et al. [133] for Boger fluids was obtained. Furthermore, the model features bounded and continuous extensional viscosities. No direct comparison with experimental re sults for extensional flows seems to have been made by the authors but in a paper by Sizaire and Legat [535] good quantitative agreement was found be tween the extensional viscosities for a polyisobutylene-polybutene Boger fluid measured using the filament stretching device of Tirtaatmadja and Sridhar [577] and numerical simulations involving the FENE-CR model. A detailed comparison of the FENE model and approximations represented by the FENE-P and the FENE-CR models has been done by Herrchen and Ottinger [280] and by Keunings [326]. Steady state flows revealed little difference between the model predictions, with the exception of the FENE-CR model in shear flow, where, as observed previously, there is a constant shear viscosity. Large differences appeared between predictions with the FENE model and its approximations in the case of transient flows, at high shear and extension rates, however, and the interested reader is referred to the papers cited above for further details. A second-order closure model, the so-called FENE-L model, has more re cently been developed by Lielens et al. [354], and has been shown to be able to reproduce the hysteretic behaviour of the FENE model in transient exten sional flows. A description of this model and of a simplified version, the FENELS [354,355], will be supplied in §11.3. 44
CHAPTER 2. FUNDAMENTALS We follow Larson [342] in observing that there is a large number of consti tutive equations which involve the addition of an extra term to the Oldroyd B constitutive equation (2.108) for the polymeric stress so as to produce shearthinning, a bounded extensional viscosity and a (possibly) non-zero second nor mal stress difference. The general form of all these equations is r + A i ? +f (r, 7) = ^ 7 .
(2.145)
For example, among the many constitutive equations of the type described by (2.145) is the Phan-Thien Tanner (PTT) model [456,458] T
+ Xi T + [ exp e ^ t r ( r )
1 r + fAi (JT + rj)
= r)pj,
(2.146)
or + Ai T + £ A — t r ( r ) r + £Ai(-yr + 1-7) = vPi,
(2.147)
T]p
and the Giesekus model [234,235] r + Ai T +a—T2
= r)pj.
(2.148)
T)p
Both of the above models have bounded extensional viscosities and exhibit shearthinning. The interested reader is referred to [77] or [152] for details of other differential constitutive equations suitable for describing melts or concentrated polymer solutions which fall into the general form given by (2.145).
2.6.2
Integral constitutive models
So far in this section we have considered differential constitutive equations in acknowledgement of their widespread use in computational rheology. We now proceed to consider briefly integral constitutive equations which are nonlinear generalizations of the linear integral model in Eqn. (1.17) of §1.1.1, viz. t
r ( x , t) = -
I M(t-
s)7(x, *, t')dt',
(2.149)
—00
where 7 is the second-order tensor having (i,j)th
component
and Zi = x\ — a;, is the ith component of the position vector of a particle at time t' relative to that of the same particle at time t. As in Eqn. (1.17) of §1.1.1, M is a memory function. Before we can consider nonlinear integral models we need to know something about relative finite strain tensors.
45
2.6. MORE COMPLICATED CONSTITUTIVE RELATIONS Relative finite s t r a i n t e n s o r s Consider now the so-called relative finite strain tensor ^°\n.,t,t') 7 [ 0 ] (x,M') = C ( x , i , i ' ) - I ,
defined by (2.151)
and the same as G(x,t,s) in (2.71). This is a generalization of 7(x,t,t') which collapses to 7(x, t, t') in the limit of small displacement gradients. The reader is referred to Appendix A.2 for a demonstration of this result. Hence we see that C(x,t,t') = 7 ( x , t,t')+I, (2.152) in the small displacement gradient limit. By a similar line of reasoning it may be shown that C~1(x,t,t') = I--r(x,t,t'), (2.153) in the same limit, so that the other relative finite strain tensor 7 [ 0 ](x,M') = I - C T 1 ( x , i , t ' ) ,
(2.154)
also equals 7(x, t, t') in the small displacement gradient limit. T h e Rivlin-Sawyers e q u a t i o n The Rivlin-Sawyers equation is an important and popular integral constitutive equation (or, more correctly, class of equations) for the modelling of melts and concentrated polymer solutions, and is a nonlinear generalization of (2.149). The model was proposed by Rivlin and Sawyers in 1971 [511] and has con stitutive equation t
T(x,t) = - J {Vl(Jl,/2,*-<')7[0](x,*,f)+^(/l,/2,t-f)7 [ ° 1 (x»*.O}*'. — oo
(2.155) where I± and 1% are, respectively, the first invariants of the Finger strain tensor C _ 1 and the Cauchy-Green strain tensor C, and V'l and ip2 are scalar functions of I\ and I2. The tensors 71°) and 7r0i are the relative finite strain tensors introduced earlier. If we assume that the functions tpi may be written in the separable form rl>i(Ii,l2,t-t')
= M{t-t?)ipi(I1,I2),
» = 1,2,
(2.156)
it follows that the Rivlin-Sawyers constitutive equation may be written in the factorized form t
r(x,i) = - j M(t - t') { m [ 0 ] + ^27[01} dt'.
(2.157)
—00
Various suggestions have been made in the literature for the strain-dependent factors ipi and i
= exp (-fly/ah
+ (1 - a)I2 - 3) , <^2 = 0, 46
(2.158)
CHAPTER 2. FUNDAMENTALS and Papanastasiou et al. [441]:
" - ( a - « ) + W+p-flV*-°-
(2 159)
'
Both models have zero second normal-stress differences since y>2 = 0 in both cases, and were proposed for general flows. As stated previously, the RAvlin-Sawyers equation is a nonlinear generaliza tion of (2.149). In the limit of small displacement gradients, however, we expect (2.157) to collapse to (2.149). In the infinitesimal displacement gradient limit we have I\ = Ii = 3 and for the correct behaviour in the linear limit we therefore require that
(2.160)
A quick glance at the functions (pi and 2 in (2.158)-(2.159) will be sufficient to convince one that condition (2.160) is satisfied in these cases. Apart from the requirement (2.160), however, 2 are sufficiently general that (2.157) describes an entire class of constitutive relations. As further special cases of (2.160), mention should be made of factorized K-BKZ-type models (see [64, 320,562]) such as the Lodge network model [362,363] and the Doi-Edwards model [184]; the latter being appropriate for polymer melts and is derived from the reptation theory of de Gennes [168]. Further details may be obtained from the book of Bird et al. [72] concerning constitutive equations belonging to the Rivlin-Sawyers family. The Doi-Edwards model is considered in greater detail in §11.5.1 of this book. For details of some techniques for solving both steady and transient viscoelastic flows using constitutive equations of the integral type the reader is referred to §6.5-§6.7.
47
Chapter 3
M a t h e m a t i c a l Theory of Viscoelastic Fluids 3.1
Introduction
The addition of a constitutive equation to the conservation equations of mass and momentum considerably complicates the mathematical analysis and the numerical simulation of viscoelastic flow problems. A complete mathematical theory on the existence and uniqueness of viscoelastic flow problems remains elusive due to the nonlinear nature of the governing set of equations. Despite this bleak picture some progress has been made in understanding the underlying structure and properties of the systems of partial differential equations that govern the flow of viscoelastic fluids. A grasp of these properties is important in order to construct appropriate solution techniques and to impose the boundary and/or initial conditions that will guarantee a well-posed problem. In the early development of computational rheology proponents of the fi nite element method used the Galerkin method for discretizing the constitutive equation. Although the Galerkin method is well suited for elliptic problems it can produce inaccurate approximations to solutions of linear hyperbolic prob lems. When the Galerkin method was applied to viscoelastic flow problems a loss of convergence at quite modest values of the Weissenberg number was ob served. The phrase 'high Weissenberg number problem' (HWNP) was coined to describe the common situation experienced by researchers in which numerical simulations failed beyond some limiting value of the Weissenberg number. We now know that the standard Galerkin method is not appropriate for equations of this type. Initially, computational limitations in terms of attainable Weis senberg number were due to inappropriate choices of numerical methods in the sense that the methods used were not appropriate for the type of the governing equations. This problem has been overcome, to a large extent, thanks to Joseph and his co-workers [316,317] who investigated the type of the systems of partial differential equations that are used to model viscoelastic flows. Today the onset of the HWNP has been extended to higher values of the Weissenberg number. In many problems the breakdown of numerical simulations can be attributed to their inability to resolve adequately stress boundary layers. Of course, the prob lem can be surmounted, in theory, with recourse to more powerful computers 49
3.1. INTRODUCTION but this would just delay the onset of the problem to some higher value of the Weissenberg number. A mathematical understanding of the behaviour of the solution in regions of the flow such as stress boundary layers and singularities would greatly reduce the computing power that is necessary and perhaps lay to rest once and for all numerical manifestations of the HWNP. In this chapter we summarize the analysis which has been performed on viscoelastic models of Maxwell-Oldroyd type. The equations of motion and continuity for unsteady incompressible flow are p(^+(u-V)u) =b-Vp+V-T,
(3.1)
and V -u = 0,
(3.2)
respectively, where u is the velocity vector, p is the isotropic pressure, T is the extra-stress tensor, b is the body force and p is the density of the fluid. The extra-stress tensor is related to the rate-of-strain tensor, 7, by a constitutive equation. For example, the Oldroyd B model is
T + Ai T= f)o ( 7 + M 7 J ,
(3-3)
where Aj and A2 are characteristic relaxation and retardation times of the fluid, respectively, and rjo is the constant shear viscosity. The upper-convected deriva tive of T is defined by v
r)T
T = ^ - + (u-V)T-(Vu)TT-T(Vu).
(3.4)
If the extra-stress tensor is expressed in terms of its solvent and polymeric contributions, viz. T = 1J.7 + r ,
(3.5)
then r , the elastic stress, satisfies T + Ai r = 7 j p 7 .
(3.6)
The constants r}s and r\v are the solvent and polymeric viscosities, respectively, where T}3 + r)p = 170, and 2
A
Vo, T)p= ( l - yj
T)0-
(3.7)
The equation of motion (3.1) may be written in terms of the elastic stress by taking the divergence of (3.5): p ( -^- + (u • V ) u j = b - V p +
T)S V
2
u + V •r
(3.8)
Note that when TJS — 0 the constitutive equation (3.6) is the UCM model with r]p = 770 and r = T. 50
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS
3.2
Existence and Uniqueness
The development of existence and uniqueness theory for solutions of differential equations is an important area of research. If a differential problem possesses these properties then the search for a numerical solution makes sense. The problem is that for many viscoelastic flows, particularly those of practical and industrial interest, existence and uniqueness results are difficult, if not impossi ble, to establish. Although unsatisfactory from the mathematical viewpoint one is forced to adopt a pragmatic approach in many situations and move forward in the faith that, for a given problem, an existence and uniqueness result exists but has not yet been established. The mathematical theory of existence and uniqueness of solutions for steady Newtonian flows is well developed [239,570]. However, the highly nonlinear nature of the governing equations for steady viscoelastic flows has meant that the theory here is still very much in its formative stage of development. The obvious approach in which one performs a perturbation about a Newtonian flow cannot be used for two principal reasons. First, the equations governing the flow of viscoelastic fluids are singular perturbations of the Navier-Stokes equations with higher-order derivatives being introduced into the equations. Secondly, because viscoelastic fluids have memory extra information is required at inflow boundaries to provide information about what happened to the fluid before it entered the domain. Results only exist for special models in special subcritical flows and for small perturbations of supercritical flows of fluids of the Maxwell type. Here the per turbations are about the state of rest or about some uniform motion. The usual technique for establishing the existence of a solution is to construct one that is sufficiently smooth and which satisfies all the conditions imposed on it. The first theoretical results were established for steady flows of UCM fluids. Renardy [495] proved an existence result (see Theorem 3.1) for small perturbations about rest of slow steady flows described by a differential constitutive equation. The flows were defined in a bounded domain with Dirichlet boundary conditions on the velocity, i.e. u = uo on dfi. The absence of an inflow boundary meant that additional boundary conditions for the extra-stress were not needed. Theorem 3.1 Suppose that 1. fi C R 3 is a bounded domain with boundary dfl of class C3, 2. b € H2(fl), uo E H5/2(dfl) small in these spaces,
and the norms of h and u 0 are sufficiently
3. uo is parallel to 9 0 . Then there is a solution of the equations governing the flow of a UCM fluid. The solution satisfies u £ H3(£l), p,r € H2(fl) and it is the only solution for which the norms in these spaces are small apart from an arbitrary constant in the pressure solution. The proof is based on an iterative approach in which one alternates between the solution of an elliptic problem of generalized Stokes type and a hyperbolic system that has the streamlines as characteristics.
51
3.2. EXISTENCE AND UNIQUENESS The divergence operator is applied to the constitutive equation to obtain Ai(u ■ V ) ( V • r ) - A i ( V u ) T ( V ■ T ) + (V ■ r ) = AIT : d2u + T)pV2u,
(3.9)
where the operator r : d2 is denned by
We use the momentum equation to eliminate V • r from (3.9) to obtain V([Ai(u - V) + \]p) - Ai(Vu + ( V u ) T ) V p - [Ai(u • V) - A x (Vu) T + l]b = XIT : d2u + r)pV2u - p[Ai(u • V ) ( u • V) - Ai(Vu) T (u • V) + (u - V)]u. (3.10) Introducing a new variable q defined by q=[Xl(u-V)
+ l]p,
we can write down the iterative scheme which is used to prove the existence of a solution under certain conditions. First of all the initialization u° = 0, p°=q°
= 0, r ° = 0,
(3.11)
is performed. The iteration proceeds by alternating between the solution of a generalized Stokes problem 77 P V 2 u" +1 - Vqn+1
V-u"+1
=
- [ A i ( u n - V ) - A ! ( V u " ) T + l]b
=
-A1(Vun + (Vu")T)Vpn - A i r " : d2un+p[X1(un ■ V ) ( u " • V) n T - A ! ( V u ) ( u " • V) + (u n - V)]u n , (3.12) 0, (3.13)
with u " + 1 = u 0 , on dfl,
[ qn+1dCt = 0,
(3.14)
and the solution of the hyperbolic equations Ai(un+1-V)i)n+1+p"+1=g"+1, Ai(u n + 1 • V ) r " + 1 + rn+l
(3.15)
- Ai[(Vu"+1)Trn+1 + rn+1(Vu"+1)]
= TIP ( V u " + 1 + ( V u n + f ) .
(3.16)
This last step is performed by integrating along the streamlines. No boundary conditions are required for this last step since the streamlines do not cross the boundary because of the confined nature of the flow. In the steady inertialess case the reformulation of the governing equations introduced in the analysis of Renardy [495] has been used as the basis of the EEME approach [332] which 52
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS ensures that the momentum equation remains elliptic whatever the value of the Deborah number (see §7.3.3). This analysis was extended by Renardy [498] to include inflow boundaries. In this case extra boundary conditions need to be imposed to provide inflow information for the extra-stress. The problem considered was the transverse flow through the strip 0 < x < 1. Small perturbations about a uniform flow uo = (U,0,0)T,
po = 0, TO = 0,
in the absence of a body force were sought. Periodic boundary conditions are imposed in the y and z directions with periods Ly and Lz, respectively. The analysis is performed under the assumption that U2 < rjp/pXi in which case the system for u " + 1 and qn+1 is elliptic. The analysis for the situation in which one allows for change of type has not yet been performed. In the analysis of Renardy [498] a small body force is prescribed along with boundary conditions that are small perturbations about the uniform velocity, u(0,y,z)
= (U,0,0)T + ul(y,z),
u(l,y,z)
= (U,0,0)T + u2(y,z).
(3.17)
The boundary conditions must be compatible with the incompressibility condi tion which implies rLz
/■Ly
pLz
rLv
/
/
n • UJL dy dz = /
/
Jo
Jo
Jo
Jo
n ■ u 2 dy dz,
where n = (1,0,0) T is the unit normal in the x direction. The iterative procedure used to prove existence is similar to (3.12)-(3.16) except that here we have uo = (C/,0,0) T .
(3.18)
However, unlike the situation for a confined flow one also needs to impose con ditions on p and T. The solution of the iterative process will satisfy the con stitutive equation and (3.10) but not necessarily the momentum equation (3.1). We need to determine what further conditions are required to ensure that (3.1) is also satisfied by the solution. Now (3.10) was obtained by substituting V ■ r from (3.1) into (3.9). If we set w = p(u • V ) u + Vp - b - V • T ,
(3.19)
then the solution obtained as a result of the iteration process satisfies A i ( u - V ) w - A i ( V u ) T w + w = 0.
(3.20)
If the streamlines do not cross the boundary then (3.20) leads to w = 0. How ever, when there is an inflow boundary the condition w = 0 needs to be imposed as a constraint on the possible inflow data for p and r to ensure that w = 0 throughout the domain. In three dimensions there are ten variables: six extra-stress components, three velocity components and pressure. Renardy [498] proved that four condi tions must be added to the prescribed values of the velocity at the inflow bound ary. The inflow conditions for the seven hyperbolic equations (3.15)-(3.16) are 53
3.3. PROPERTIES OF THE DIFFERENTIAL SYSTEMS subject to the three constraints w = 0. Therefore, four independent condi tions can be imposed at the inflow boundary. The corresponding calculation in two dimensions suggests that two inflow boundary conditions are required. Renardy [498] suggested prescribing the normal components of the extra-stress and solving for the shear stress and pressure. This has to be done for every iter ation. With inflow conditions that satisfy the constraint w = 0, Renardy [498] showed that the iteration converges provided that the data are sufficiently small. The number of inflow conditions that are required to solve the problem can also be deduced from the multiplicity of the streamlines. In steady flow, information at inflow is carried into the domain along the streamlines. This suggests that the number of extra conditions to be added to the prescription of the velocity at inflow is equal to the multiplicity of the streamlines (see §3.3.2). Renardy [498] also considered the Oldroyd B model and proved existence of solutions which are small perturbations about the uniform flow just described. For this model there is no need to restrict the inflow conditions on r in order to satisfy a constraint. All components of the extra-stress can be imposed at inflow boundaries. Renardy [500, 504] has also proved existence of solutions to initial value problems for Maxwell fluids. The analysis of steady problems given above can not be extended in a straightforward fashion to the solution of time-dependent problems. Again, small perturbations about rest and uniform flow are con sidered in order to investigate the effect of an inflow boundary. Guillope and Saut [258] used fixed point theory to prove local existence of solutions of the initial-boundary value problem for the Oldroyd B model in some small time interval. They also showed global existence of solutions for sufficiently small data, which are perturbations about the state of rest. Well-posedness is dis cussed further in §10.2.1.
3.3
Properties of the Differential Systems
The governing equations for viscoelastic models of Maxwell-Oldroyd type can be written as a system of partial differential equations. For planar two-dimensional flows this is a system of six partial differential equations for the six unknowns p, u, v, TXX, Tyy and rxy. These systems may be analyzed using the theory of partial differential equations to give important information about the type of the system, well-posedness and loss of evolution, for example. This information can then be used to construct and develop appropriate numerical methods for solving the governing equations systems that preserve some of these properties. An interesting feature of these systems is that they are of mixed type. For example, the system describing the unsteady flow of an Oldroyd B fluid is neither strictly elliptic nor strictly hyperbolic. In fact, it is of mixed type. The hyperbolic character of these rather innocent-looking differential systems is responsible for the unique set of challenges awaiting practitioners uninitiated in the art of computational rheology. The quasilinear system of equations governing the unsteady flow of an 01-
54
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS droyd B fluid in two dimensions is UX +Vy
=
0,
(3.21) (3.22)
p(Ut + UUX + VUy) +PX~
TXXiX - TXyty ~ T)S(UXX + Uyy)
=
0,
p(Vt + UVX + VVy) +Py
- TXy>X - Tyy^y - r)g(VXX + Vyy)
=
0,
(3.23)
—
0,
(3.24)
=
0,
(3.25)
=
0.
(3.26)
Xi(Txx,t + UTXX<X + vrXXty) - 2(Ai7-xx + r)p)ux ZAiTxylly
+ Txx
Al (TyyJ + UTyytX + VTyy>y) ~ 2XXTXyV -2(XlTyy Xl(rXy,t
+ UTxy,x
+ VTxy>y)
X
+ T)p)Vy + Tyy - {\\TXX
+ T)p)vx
— (X-lTyy + Tjp)Uy + TXy
In order to investigate the type of the Oldroyd B model we need to determine the determinant of the symbol for the governing system of partial differential equations. The symbol of a system is the response of the system to one Fourier mode, both in space and time. The system is hyperbolic if and only if the six roots of the principal part of the determinant of the symbol of the differential operator are real for all wave numbers (see [315] and §3.3.2). If some of the roots are complex then the system is of mixed type. The set of partial differential equations denning the two-dimensional Oldroyd B model may be written as the following quasilinear system Aa
lft+A
dx
where = (p,u,v,Txx,Tyy,Txy)T.
A0
( 0 0 0 = 0 0
\o
(°1 Ai
0 0 0
lo /o A2 =
0 1 0 0
\o
(3.27)
+A
1 pu 0 ~2(XlTxx + Tjp) 0 0 0 pv 0 — 2XlTXy 0
-(Air„„ ^-VP)
The matrices appearing in this system are 0 p 0 0 0 0
0 0 p 0 0 0
0 0 0 Ai 0 0
0 0 0 0 Ai 0
0 0 pu 0 2AiTiy "(A T x i + nP) 1 0 pv 0 -2(A iTyy + VP) 0 55
0 \ 0 0 0 0 Ai )
■)
0 0 Ai u 0 0
0 0 0 0 Xiu 0
0 \ 0 -1 0 0 Aiw /
0 0 0 Ai v 0 0
0 0 -1 0 Xiv 0
0 -1 0 0 0
1
\
Ai» J
3.3. PROPERTIES OF THE DIFFERENTIAL SYSTEMS
f°0
0
0 0 0
\o
0 0 0 0
/ 0 0 0 S = 0 0 \ 0
0 0 0 0 0 0
A3 =
0 0
-1)s
-v* 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
°\
0 0 0 0 0/
and 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 1 0 0
\ 0 0 0 0 1 )
There are two stress tensors which play a role in the analysis of systems of this type, viz. (3.28)
TA TB
=
pUU -
TA-
(3.29)
For the UCM model it can be shown that the tensor TA must be positive definite. The tensor TA satisfies the following differential constitutive equation rA + Ai
TA=
TW. AI
(3.30)
To prove that TA is positive definite we express this differential model in terms of its integral equivalent, i.e. TA(x,t)
= ^Jt
exp(-£^)c-1(x,t,0
(3-31)
where C _ 1 is the Finger strain tensor. This tensor is positive definite and therefore so is TA since, for all vectors £, tTTAt=
I / *
exp(-^-^)€rC-1(x,t,t')€dt'>a|KII2,
(3. 32)
where a > 0.
3.3.1
Loss of evolution and H a d a m a r d instability
The unsteady quasilinear problem comprising (3.27) and prescribed initial and boundary conditions is said to be well-posed if it possesses a unique solution that depends continuously on the data. In this case the mathematical problem is complete and consistent. This means that precisely the right type of data, initial conditions and boundary conditions are prescribed to determine exactly the solution of the problem. The problem is said to be evolutionary if the Cauchy problem is well-posed. If the problem is evolutionary then the system (3.27) is stable to short wave disturbances. The loss of evolution is an instability 56
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS of Hadamard type in which the growth rates of short waves increase without bound as the wavelength tends to zero. The loss of evolution poses a severe problem as far as numerical simulations are concerned since it is no longer true that convergence of the discrete solution can be achieved as the mesh is refined. Consider a plane wave solution of (3.27) propagating in the ^-direction of the form 0(x, t) = 0 O exp (-iwt + ifax + &«/)),
(3-33)
where u is a frequency, and £i and £2 are wave numbers. We define a wave vector £ - f i e s + 6e»> with |£| = \/£i + £2 • Since exp(-iu)t) = exp(3(u)t)exp(—ijft(w)t) the growth rate of the plane wave (3.33) is defined to be 3(w). If S(w) < 0 for all directions £ then all waves of the form (3.33) damp exponentially in time. The satisfaction of this condition prevents the onset of Hadamard instability. Oldroyd B model Consider the quasilinear system of partial differential equations (3.27) for the Oldroyd B model. Inserting (3.33) into (3.27) yields i
(-LOA0
+ a At + 6 A 2 + i(£ 2 + g)Az)
0 = 0.
(3.34)
There exists a non-trivial solution (j) of (3.34) for all £ if det[-uA0
+ frAi + &A2 + »(£ + g)Aa
- iS] = 0.
(3.35)
Note that at present we are not restricting ourselves to very small waves, i.e. for |£| —> 00. We shall consider this important case in due course. The requirement (3.35) leads to 0 £1 £1 pv - iv.(£ + &) £2 0 0 mi 0 0 0
£2 0 P*-«/.(# + $) 0 m2
7713
m.4
0
0 0 -6 0 -£2 \v — i 0 0 A,yV — 0 0
0
-6 -£1 0 0
i
AyV
= 0,
— %
(3.36) where we define v = — UJ + £ • u, and mi
-
m2
=
m3
=
m4
=
-2£i(\1Txx+r)p)-2£2^iTxy, -^2(XiTyy
+ T)p) - 2fiA1Ta:„,
-£2(\iTyy+r)p), -£i{\iTxx
+ rip).
In terms of v the condition for stability is 3(f) > 0. Evaluating the determinant (3.36) gives - fe2 + f2) ( A ^ - i)2 [{pu - iris{& + £22)) (\iu - i) -((Air x x +» ? p )£ 1 2 + 2A 1 rx ! / a6 + (AiTI/1/ + r ? P K|)] 57
=
0.
(3.37)
3.3. PROPERTIES OF THE DIFFERENTIAL SYSTEMS This equation has a double root v — i/Xx, which corresponds to a wave solution with exponential rate of decay — 1/Ai. These are present whether or not the model has a non-zero solvent viscosity. The other roots of (3.37) are given by the solutions of the quadratic equation
^(K&fc+'V^-^-OL p Aiy p p\i
(3-38)
where, for the Oldroyd B model, the tensor TA is denned by
rx=r+^I. Solving the quadratic equation for v gives
To investigate short wave instabilities we look at the behaviour of these roots as |£| -> oo. Since S is independent of £, ^|°->0,
as|€|->oo.
We can divide (3.34) by |£| and let |£| -> oo so that when |£| is very large we may solve (3.34) if det[-wA0
+ 6 ^ 1 + 6 A 2 + i(4 2 + $)A3]
= 0.
(3.40)
In the limit of |£| -> oo the roots are v = 0 and v — ioo when r)s ^ 0. Therefore, the Oldroyd B model cannot be Hadamard unstable. In this analysis we have assumed that the coefficients in the set of partial differential equations (3.27) are constant. This procedure is known as the frozen coefficient approach and is valid in the study of infinitely short waves. Finite wave numbers One can also look at the behaviour of solutions for finite wave numbers. By inspecting (3.39) we see that if TA is positive definite then 5s(v) > 0 for all £. This means that all waves damp exponentially in time. However, if TA loses its positive definiteness then $s{v) may become less than zero. The consequence of this is that the solution grows exponentially in time. So although we did not require the positive definiteness of the tensor TA to show that the Oldroyd B model cannot be Hadamard unstable this condition is necessary for stability for finite wave numbers. U C M model In the case of the UCM model the evaluation of the determinant (3.36) with T)a = 0 gives 2
A?(tf+ £ ) ( « ' - £
£TTA£-PV(V-
58
—)
= 0.
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS The roots of this equation are given by
!/=
i
i
i
±
$TTA£
_
1
vv2Ar v^ ~iAf
,„41,
( }
To investigate short wave instabilities of the UCM model we look at the be haviour of these roots in the limit of |£| -» oo. In this case the roots are given by
u = 0, 0, ± J ^ 1
(3.42)
Since TA is positive definite it follows that 9(i/) > 0 for all directions £ and therefore the waves damp exponentially in time. This means that, like the Oldroyd B model, the UCM model cannot be Hadamard unstable. Note that although the continuous system possesses the property that TA is positive definite this may not be carried over to the corresponding discrete problem. Therefore numerical schemes may become unstable beyond some value of the Weissenberg number even though the continuous problem is well-posed.
3.3.2
Classification
An important step in the mathematical investigation of a system of partial differential equations is classification, i.e. are the equations elliptic, parabolic, hyperbolic or of some more complex form? This has implications concerning the boundary conditions that can be imposed, i.e. how many and what type of boundary data are needed to guarantee a well-posed problem. The classification of a system will also form an important consideration in the choice of numerical algorithm capable of solving the associated discrete problem. Different methods are applicable for boundary value problems and initial value problems. If the system is of mixed type then it is necessary to determine the characteristics and the variables that are propagated along them. One way of classifying a system of partial differential equations is to use the method of discontinuous derivatives [315,316] to determine the characteristic surfaces. The system (3.27) is written as a first-order system by introducing the components of the velocity gradient as additional variables. In the twodimensional case considered in this chapter these are surfaces ip{x,y,t) = 0 across which the vector of dependent variables, 0 , is continuous with bounded jumps in the first derivatives. An alternative approach to the classification of systems of this form is to study the stability of short waves as we have done in §3.3.1. This line of action leads to (3.34). The procedure is to determine the symbol of the differential operator, to compute its determinant and to examine the factors of the principal part of the symbol. Oldroyd B model The symbol of the differential operator defined by (3.27) is the response of the system to a solution of the form (3.33). Therefore, the symbol of the system of 59
3.3. PROPERTIES OF THE DIFFERENTIAL SYSTEMS equations, P(,i£,uj), for the Oldroyd B model is P(4>, it w) = i (-LoA0 + £1A1+
&A 2 + i(& + g)A3 - iS) .
(3.43)
The requirement that detP(4>,i$,u})=0, leads to the polynomial equation (3.37). The principal part of this polynomial is
-i\yv*(8
+ &)2.
(3.44)
The system (3.27) is of mixed type. This may be established by examining the factors comprising the principal part of the symbol (3.44) and identifying the differential operators to which they correspond. There are two parts of the principal part of the symbol (3.44) for the Oldroyd B model: 1. The factor ( ^ + £f) 2 represents a parabolic part and an elliptic part. The parabolic part corresponds to the principal part of the symbol of the equation for vorticity, which is parabolic for time-dependent flows. There is a single real characteristic associated with this part of the system. The elliptic part corresponds to the symbol of the Laplace operator. It arises from the Poisson equationforthe stream function. There are two complex characteristics associated with this operator. 2. There are three real characteristics corresponding to the factor v3. This means that the particle trajectories or streamlines are triply characteristic. There are three quantities, viz. the components of the extra-stress tensor, that are convected along the streamlines. Note that even though this contribution to the principal part of the symbol is not strictly hyperbolic since these real characteristics have multiplicity three, in practice, the solutions behave as if the system were hyperbolic. Therefore, the system of partial differential equations for the Oldroyd B model is of mixed type and portrays traits of elliptic, parabolic and hyperbolic character. U C M model In the case of the UCM model the principal part of the symbol is given by {£+e2)*lA£TTAi-pv2].
(3.45)
The real characteristics are associated with the real zeros of this symbol, i.e. v = 0, 0, ±\r
60
TA
^.
(3.46)
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS Since TA is positive definite there are four real eigenvalues for w for every di rection £ given by w = u • £ (twice) , u • £ ± \JfrA$/p.
(3.47)
The symbol (3.45) for the UCM model comprises three factors: 1. The factor (£2 + £f) corresponds to the symbol of the Poisson equation for the stream function, i.e. V2V = - C , where ( is the vorticity. Since this is an elliptic equation there are two complex characteristics associated with this factor in the symbol. 2. There are two real characteristics associated with the factor v2. There fore, there are two characteristic variables that are convected along the streamlines. 3. Joseph et al. [316] associated the factor (£TTA£ — pv2) with the symbol of the unsteady vorticity equation, the higher-order terms of which are d2(
n
,
„,d(
.
,
2
+ [pv2 -
+[2puv - {TA)xy}^~-
.
.d2( fa)™]—|.
(3.48)
This equation is hyperbolic and, as we have already shown, there are two real characteristics associated with it. Therefore, the system of partial differential equations for the UCM model is of mixed type and portrays traits of elliptic and hyperbolic character since there exist two linearly independent eigenvectors associated with the repeated root v = 0. Note that although there are three convective derivatives in the consti tutive equation there are only two characteristic variables that are transported along the streamlines for the UCM model. These variables have been determined by Gerritsma and Phillips [231] (see §3.3.4).
3.3.3
Change of t y p e in steady flow
Now we set the time derivatives in (3.27) to zero and determine the conditions for change of type for steady flow. A change of type can occur in the steady form of the equations for the UCM model. The product of the last two eigenvalues (3.47),
W
= e - u ± W ^ i ,
(3.49)
multipled by p is given by P « • u) 2 -
£TTAZ
= e(pnn
-
TA)Z
= erai,
(3-50)
where TB is defined by (3.29). This is the symbol of the steady vorticity equa tion. This equation may have real or complex characteristics in contrast to the corresponding unsteady vorticity equation. Furthermore, the situation can arise 61
3.3. PROPERTIES OP THE DIFFERENTIAL SYSTEMS in which this equation is elliptic in one region of the flow and hyperbolic in other regions, as in the case of transonic flow in aerodynamics. If £ T T B £ is negative for all choices of the vectors £ then the two eigenvalues have different signs for all £. This is the case when TB is negative definite. The flow is subcritical in this case. If TB is not negative definite then the flow is supercritical for at least some directions £ since the characteristic variables associated with these eigenvalues are influenced from one direction only. There is no information coming from the opposite direction. If TB is positive definite then the flow is supercritical for all directions £. Note that inertialess steady flow corresponds to p = 0. In this case we have /^•u)2-£
T
r^ = -£Tr^-
(3.51)
Therefore, it follows that TB is negative definite since r A is positive definite. So the inertialess steady flow of a UCM fluid cannot change type. Joseph et al. [316] have also shown that a change of type cannot occur for the system of equations for the Oldroyd B model irrespective of whether the flow is inertialess or not. For the Oldroyd B model the real characteristics are the streamlines. For the UCM model, however, there are situations in which the hyperbolic character of the system of equations for unsteady flow, i.e. p ^ 0, increases in the sense that there are additional real characteristics besides the streamlines.
3.3.4
Characteristic variables
The characteristic variables have been identified by Gerritsma and Phillips [231] for the case of inertialess flow (p = 0) of a UCM fluid. The process begins by computing the left eigenvectors 1 = (h, h, ■ ■ ■ , h)T of the matrix -CJAO + fi Ai + & A 2
-
iS,
for v — 0. The equations for the components of these left eigenvectors are given by £i/i + mih + m3l6 = 0 ,
(3.52)
f2Zi + rn2h + mih = 0 .
(3.53)
and
For both left eigenvectors we have I2 = h — 0. These left eigenvectors will, in general, be a function of x, the point in the computational domain where the characteristics are evaluated, the solution vector (j> (since the coefficients m^ i = 1 , . . . 4, depend on the solution) and the direction vector £. Once the left eigenvectors 1 are known the characteristic variables are com puted using 6Wi = l(i)A05(f>,
t = l,2,
in which S denotes a variation in the variables and W, and l^\ i = 1,2, is an eigenvector satisfying (3.52) and (3.53). Since the left eigenvectors are generally functions of the flow variables, the components of 1 are not constant. Therefore, the variable W will only exist if 62
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS the Pfaff conditions of integrability of a differential form are satisfied. Generally, these conditions cannot be satisfied and the variable W cannot be defined, but 6W always exists and it is only the variation 5W which is meaningful. For a more thorough discussion of the conversion of hyperbolic equations to characteristic form the reader is referred to the book by Hirsch [285]. Premultiplying the system of partial differential equations (3.27) with the left eigenvectors gives the so-called compatibility equations. In general these com patibility equations will not be pure convection equations. Instead the equations will comprise a convection part together with off-diagonal terms. This is not only the case for the hyperbolic part in a system of mixed type, but also for totally hyperbolic systems (see, for instance, pl92 of Hirsch [285]). However, the UCM model contains one left eigenvector which does not de pend on the direction £, and produces a compatibility equation which contains no off-diagonal terms. Furthermore, the quantity SW can be integrated to give a nonlinear characteristic variable W. The left eigenvector with these properties is given by 1 (1) = {2(\1TXX + T)p)(\lTyy + T)p),0, 0, (XlTyy + T)p) , (XlTX X + T)p) , ~ 2Txy ) T . (3.54) It is clear from (3.54) that this eigenvector is independent of the vector £. Premultiplying the system of partial differential equations (3.27) by l^1) gives .DX^Tyy
,-PAIT X X
)- p
r
D\iTxy
- + ( W + Vp) - ^ - - 2rxy -
w
+ {XlTyy + T)p)TXX + (AlTzx + TjpJTyy ~ ^X^y
= 0.
(3-55)
The characteristic variable for this particular left eigenvalue can be integrated and (3.55) can be written in the form Xi D{d^A)
+ 2detrA = r,ptrrA.
(3.56)
So the first characteristic variable is given by W\ = detr^t- The highest order part in (3.56) constitutes pure convection along the streamlines and therefore contains no off-diagonal terms. At inflow the quantity d e t r ^ needs to be pre scribed in order to solve this characteristic equation. The other left eigenvector, l^2), will depend on the direction £ and produces a characteristic equation which contains off-diagonal terms. A particularly elegant left eigenvector is given by 1<2> = (2tf
(AITM
+
Vp)
+
2£I6AITIV
+ 2£22(AXTra + ,,„), 0,0, £ 2 , g, 2^2)T
. (3.57)
This eigenvector yields a linear combination of the extra-stress components and therefore can be integrated easily, provided £ is independent of t and x. The compatibility equation for this left eigenvector is given by DWo Wn n Rll = ~W + li ^ ( ^ + ^ ) 2 + 2 a ( a A i r X 3 ; + 6(A 1 r J/s/ + % ) ) g
dv + 2 6 (6AirXJ/ + £i(Airxx + riP)) — + [£2(AiTxa; + rip) - giXxTyy + rip)]
du
dvs
dx
dyj (3.58)
63
3.4. BOUNDARY CONDITIONS in which W2 = £I(\ITXX + r)p) + 2£i&\iTxy + ^(^iTyy + VP) is the second characteristic variable in the direction £• The velocity derivatives appearing on the right-hand side of (3.58) are the off-diagonal terms. Taking £ to be a unit vector and letting it range over all possible directions we see that the convective part of (3.58) describes the convection of the quadratic form W2 = £TTA£ along the streamlines. Both characteristic variables are closely related to the conditional wellposedness of the system of partial differential equations. The condition that TA is positive definite implies that W\ > 0 and is equivalent to the requirement that W2 > 0 for all directions £. Gerritsma and Phillips [231] show that these left eigenvectors are linearly independent. Knowledge of the characteristic variables is of vital importance, not only to prescribe the proper boundary conditions in numerical calculations, but also to discretize the governing equations in an appropriate manner. For finite volume methods, for instance, one has to select proper inflow and outflow fluxes for the hyperbolic part. Without knowledge of what is convected along a streamline it is not possible to determine these fluxes properly. In weighted residual methods, such as finite element or spectral element methods, one has to match the solution between elements. One way of doing this is to assume that all primary variables are continuous between elements. Another way of matching the solution between elements is to treat every spectral or finite element as an independent domain in which the differential equation has to be satisfied. For elliptic variables, such as the velocity, it is appropriate to impose continuity along element boundaries. For the hyperbolic part only the flux of the characteristic variables need be prescribed at inflow points for a given element. The values to be prescribed are determined by the values of the outgoing characteristic variables in neighbouring elements, whereas, conversely, characteristic variables at an outflow point are not influenced by any changes in the variables in neighbouring elements, thus reflecting the directional dependence in hyperbolic problems.
3.4
Boundary Conditions
Equipped with a knowledge of the type of the system of partial differential equa tions governing viscoelastic flow we are in a position to make an informed choice of appropriate boundary conditions. For Newtonian fluids one can eliminate the extra-stress tensor from the momentum equation to obtain the Navier-Stokes equations. Appropriate boundary conditions in this situation are specification of either the velocity components or the surface traction. However, steady flows of viscoelastic flows cannot be determined uniquely by imposing these conditions alone. For unconfined flows further additional conditions need to be imposed at inflow boundaries to reflect the fact that viscoelastic fluids have memory and therefore the flow within the domain is affected by its history upstream of the inlet boundary. The form that these inflow boundary conditions take depends on the choice of constitutive model. In steady flow, information at an inflow boundary is carried into the domain along the streamlines. When we investi gated the classification of models of Maxwell-Oldroyd type we showed that the streamlines are characteristics. These were of multiplicities two and three, re spectively, for the UCM and Oldroyd B models. This suggests that the number of extra conditions which needs to be imposed at an inflow boundary is equal 64
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS to the multiplicity of the streamlines. Therefore, whereas one can impose all three components of the extra-stress tensor at inflow for the Oldroyd B model one cannot do this for the UCM model since the problem is then overdetermined. This can lead to errors unless the inflow boundary is chosen far enough upstream.
3.5
Singularities
If the boundary of the flow domain is not smooth then the solution or its deriva tives will, in general, have a singularity at certain boundary points. This means that some derivatives of the solution will fail to be integrable. For elliptic prob lems the error in the numerical approximation is bounded by some power of the mesh size or polynomial order multiplied by some norm of the exact solution which is a measure of its regularity. Thus, the presence of singularities degrades the accuracy of numerical approximations.
3.5.1
Elliptic p r o b l e m s
A characteristic property of elliptic partial differential equations is that the regularity of their solutions is dependent on the regularity of the given data. For example, consider the Poisson problem -V20
0
=
/,
in
n,
,„ - q s
= 0,
on
an,
(3 5y)
-
where fi is some bounded domain in K 2 with boundary dil and / is a given function. Suppose that 5 0 is smooth then we have <> / e HS+2(Q) if / € H"(Q,). A consequence of this result is that if / is smooth then so is 0. However, if dfl is not smooth the position is entirely different. If 3 0 has a corner then the solution or derivatives of the solution of (3.59) will have singularities at the corner even if / is smooth. Suppose that $7 is a polygonal domain in which the maximal angle of a corner of c*n is 7 = 7r/2a. A standard finite element approximation h to the solution of (3.59) in this domain satisfies an error estimate of the form
II 0 - 0 / i HffM")- Ch1~t || 0 Hm+i-^n),
(3.60)
for any e > 0, where h is a typical length associated with an element. It is evident, therefore, that boundary singularities cause a deterioration in the error estimates for elliptic problems. For some derivative of the solution the error behaves like 0{h~s) for some s > 0. For example, in the case when a = 37r/4 Grisvard [253] showed that the solution local to that corner has the form 0(r, 6) = dX(r, 0)r 2 / 3 sin f y ) + w(r, 6),
(3.61)
where % is a smooth cut-off function and w € H2(Q). The constant d is known as the stress intensity factor. One can show that the error in the second-order central difference represen tation of the Laplacian near the corner is 0(h~4/3) when a = 3n/4. Therefore, 65
3.5. SINGULARITIES as h —> 0 the error becomes unbounded and no amount of mesh refinement is able to resolve the singular behaviour. In fact, in this situation mesh refinement results in the opposite of the usual, desired effect. The compensating feature of singularities in elliptic problems, however, is that their effect is confined to regions around the singular points. If the form of a singularity in an elliptic problem is known then one can apply one of a number of special techniques to treat the singularity in order to improve the accuracy of numerical solutions. The most successful of these use knowledge of the local behaviour of the solution near the singularity. In finite difference methods the singularity may be subtracted out of the problem in order t o obtain a 'nonsingular' problem for which finite difference methods have a much sounder basis. In finite element methods it is possible to supplement the basis of trial functions by appropriate singular functions [208]. In spectral element methods one can isolate the singularity within a single element and use an expansion in terms of the known singular functions. Consider the plane sector —a < 8 < a, 0 < r < R, of angle 2a with solid stationary walls corresponding to 6 = ±a (see Fig. 3.1). There is a countable set of values of the sector angle for which the following analysis is not applicable. However, the important cases of the salient (2a = 7r/2) and re-entrant (2a = 37r/2) corners are not numbered among the membership of this set. r=R 0=a
0=0
9=-a
Figure 3.1: The plane sector —a <6
0
For the Stokes problem, one can show that the biharmonic equation for the stream function, ip, can be written as the following second-order system in polar coordinates:
r^A
* - A * = 0,
(3.62)
where A =
(&+') <w->) I 66
(3.63)
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS and -\T
*
=
(3.64)
r \ d r
One can show that the solution of (3.62) in a sector local to a corner admits a separation of variables solution in terms of the eigenfunctions of the non selfadjoint operator A, i.e. V(r,6)
=
(3.65)
f^akr^-vk(0), *=i
where (3.66)
Avk = AfcVfc.
Thus, Vfc is of the form (vk, \kvk)J'. The no-slip velocity boundary conditions on the walls determine the form of the eigenfunctions and eigenvalues. Additionally, these depend on the sector angle 2a. The eigenfunctions are given by cos a{\f\k
- 1) cos(v/AT + 1)0 - cos a(\f\k
+ 1) cos(\/A^ - 1)0, if k is odd,
«*(*) sina(v^Afc - l)sin(\Aife + 1)0 — sma(y/Xk~ + l)sin(\/A/fc — 1)0, if k is even, (3.67) and the eigenvalues are the roots of the transcendental equation sin(2oVA) = ±(sin2a)y/\,
Re(V\)
> 0, A £ 1.
(3.68)
The eigenfunctions of the adjoint problem A*wk=\kwk,
(3.69)
where A* is the adjoint of A, are given by wA = {wk,vk)T
where
wk = 2v'k' + (A* - 2)vk. In addition the sets of eigenfunctions {v^}, {w&} satisfy the biorthogonality condition
I w?Avkd9 = 0,
k^l.
J—a
Since A is real the complex eigenvalues occur in conjugate pairs. In the case of a re-entrant corner (a = 37r/4) the first few eigenvalues are shown in Table 3.1 in order of increasing real part. The stream function takes the form f^akr1+^vk(e),
1>{r,9) =
(3.70)
k=i
from which we deduce that there are algebraic singularities in the second deriva tives of the stream function, i.e. in pressure and stress. In particular, the stress 67
3.5. SINGULARITIES
1 2 3 4
0.544484 0.908529 1.629527 ± 0.231251i 2.301327 ± 0.315837i
Table 3.1: Roots of the transcendental equation (3.68) behaves like r - 0 - 4 5 5 5 near to the corner. The asymptotic behaviour of the solu tion of the Navier-Stokes equations is Stokes-like near the singularity. The form of the singularity has also been determined for certain inelastic fluids without memory. For these fluids similarity solutions for the stream func tion of the form r1+af(6) have been sought. Henricksen and Hassager [279] have considered power law fluids with a viscosity r\ = -y™-1. They found that the velocity behaves like ra near to the corner where a = 0.37 for n — 0.5 (shear-thinning fluid) and a = 0.64 for n = 1.5 (shear-thickening fluid). The corresponding asymptotic behaviour of the stress was r~031 and r~054, respec tively. Keiller and Hinch [322] considered different effective concentrations, 4>, of suspensions of rigid rods aligned with the flow. They found that a = 0.58 for (j) = 5 and a = 0.62 for <j> = 20. The corresponding asymptotic behaviour of the stress was shown to be r ~ 0 4 2 and r ~ 0 3 8 , respectively. The analysis of the singularity in the case of viscoelastic fluids is more difficult.
3.5.2
Viscoelastic flows
The governing equations for steady flows of viscoelastic fluids are of mixed elliptic-hyperbolic type unlike their inelastic counterparts. The presence of a hyperbolic part in the governing equations is responsible for some of the prob lems associated with the numerical solution of viscoelastic flows near singular ities. This is because the pollutive effect of discretization error is more serious for hyperbolic problems since errors can be propagated along characteristics. This cannot happen in elliptic problems since there are no real characteristics. The nonlinear nature of the governing equations means that the analysis of the viscoelastic singularity is extremely complicated. Although there have been some developments in the analysis of this problem they have yet to be incor porated into numerical methods. A marriage of analytical and numerical tech niques is essential in order to obtain globally accurate solutions to viscoelastic problems in nonsmooth domains. The common experience of many researchers in the field is that, for problems with singularities, numerical schemes fail to con verge with mesh refinement and that, for increasing values of the Weissenberg number, numerical solutions can only be obtained on ever coarser meshes. The extension of the Stokesian analysis to the viscoelastic situation is dif ficult since the nonlinearity introduced by the convected derivative represents a singular perturbation in the Weissenberg number. Davies [159] proposed an extension of the local Newtonian analysis for the corotational Maxwell model. The governing equations were written in terms of the stream function and Airy stress function, and biorthogonal series expansions for these dependent variables
68
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS were sought of the form oo
9(r,9)
oo
= X>(r)vt(0),
$(r,0)
= 5>*(r)v*(0),
(3.71)
where $ is related to in the same way that * is related to ip (see (3.64)). Un fortunately, this work remains largely incomplete since these series expansions do not yield any information about the asymptotic behaviour of the stream function or stress as r —> 0. Lipscomb et al. [358] have shown that under certain conditions the form of the singularity can be obtained for a second-order fluid. Near a re-entrant corner they showed that the stress singularity behaves like r - 0 - 4 5 5 5 + (We)r - 0 - 9 1 1 . Keunings [325] noted that this result establishes the singular character of the Newtonian limit since there always exists some neighbourhood r 0 .
(3.72)
r ~ % ^ 7 , asr-^0,
(3.73)
This leads to
Ai
which demonstrates that the behaviour is Newtonian-like near the corner with effective viscosity J70A2/A1. This argument breaks down, however, because the condition (3.73) is incompatible with the viscometric conditions on the walls forming the corner unless A2 = Ai in which case the flow is Newtonian. In the region of a corner one can identify several flow regions: a core region away from the walls, a viscometric region near the walls and, possibly, a region containing one or more lip vortices. Renardy [501] investigated the behaviour of the stresses near the re-entrant corner for the UCM fluid. The analysis is based on a fixed Newtonian velocity field and predicts the presence of stress boundary layers near the walls in which there is a transition from viscometric behaviour near the walls to a core region where the convected derivative dominates. For a fixed radial line he found that the stresses behave like r - 0 7 4 . Hinch [283] argued that the Newtonian-like behaviour given by (3.73) is not valid anywhere in the re-entrant corner. Instead, Hinch [283] assumed a similarity solution for the stress near a re-entrant corner but away from the walls of the form ffWOuu, 69
(3.74)
3.5. SINGULARITIES where g is an unknown function. The analysis is based on the assumption (in the absence of lip vortices) that the Oldroyd B fluid behaves asymptotically like the UCM fluid and so it is the elastic stress which dominates the viscous stress everywhere within the sector. The argument is that, in this region of the flow, the elastic stress must be proportional to the square of the velocity since on entering the region it is dominated by the streamwise normal stress term. The analysis neglects the contribution of the polymer relaxation in the core region and fails to satisfy the momentum equation in the proximity of the walls. The behaviour of the stream function predicted by the analysis is proportional to
(3.75)
while away from the walls the solution of Hinch [283] is assumed to hold, i.e.
^«r 1 4 / 9 sinW|0y
(3.76)
The momentum equation is satisfied in this analysis. The value of a is estimated from an order of magnitude argument by determining when the core solution dominates over the viscometric solution. Suppose this happens when 6 sa r1 for small 8. Then the exponents of r in (3.75) and (3.76) match when 7 = 3a - 14/3. To determine a, Renardy [503] assumed that the same boundary marked the transition from viscometric behaviour to the behaviour in the core region. This occurs when the stretch rate is of order unity, i.e. when r~xi]jrg = 0(1). There fore, from (3.75) we have r-1^
« ra~2e «
ra~2+^.
We require that a - 2 + 7 = 0, which along with the above expression for 7 furnishes a — 5/3. To obtain a solution in the viscometric region near the upstream wall, Renardy [503] con structed a matched asymptotic expansion in which the outer solution in the core region is that derived by Hinch [283]. Summarizing, if T and D denote the largest components of the stress and velocity gradient tensors, respectively, then Tsar-2/3, T ss r~ 2 / 3 ,
D a r"4/9, D « r-1/3,
in the core region, in the upstream viscometric region.
f* 77^
This is where the analysis comes to a halt and to proceed further and obtain the inner solution near the downstream wall one has to resort to integrating a system of nonlinear differential equations. 70
CHAPTER 3. MATHEMATICAL THEORY OF VISCOELASTIC FLUIDS Davies and Devlin [161] developed exact singular perturbation series solu tions of the governing equations for an Oldroyd B fluid in planar sectors. For re-entrant corners they showed that solutions which are Newtonian-like away from the walls develop non-integrable stresses at the walls when the no-slip boundary conditions are imposed. Such solutions are physically inadmissible. However, they also showed that the stress can be Newtonian-like away from the walls if the flow separates at the corner into three regions. In the core re gion they showed that the stress behaves like r" where (3 lies in the interval —1 < / 3 < —| for a re-entrant corner. In the region of weak flow they showed that the stress behaves like 3?(r/3), where 9?(/3) > 0, and /3 = - | where S is the angle that the separating streamline makes with the wall. In a recent paper, Fontelos and Friedman [211] have examined the behaviour of general Oldroyd models in the vicinity of a general re-entrant corner. Under certain conditions on the coefficients of the model they showed that near the singularity the stress can be expressed as a sum of a Newtonian stress and a lower-order non-Newtonian stress, i.e. ^2 . . „,
r =T?OT-7 + T .
The singular behaviour of the non-Newtonian part has a complex structure in that it is not self-similar and is highly oscillatory near the downstream wall in the neighbourhood of the corner. Restrictions on the model parameters are necessary to ensure that the singular behaviour of T is sub-dominant to the singular behaviour of the Newtonian contribution. Unfortunately, these restric tions exclude the Oldroyd A and B models.
3.5.3
Numerical investigations
Several numerical investigations have been undertaken to determine the singular behaviour of the solution near the re-entrant corner in contraction flows. The numerical results of Lipscomb et al. [358] suggested that, for sufficiently fine discretizations, the critical Weissenberg number beyond which solutions can no longer be generated decreases with increasing mesh refinement near the singu larity. This finding has been confirmed by many researchers in computational rheology using a wide range of numerical tools. Xue et al. [637] simulated an Oldroyd B fluid (/5 = 0.875) in a planar 4:1 contraction using a finite volume method. Their results indicate the existence of a similarity solution of the form r"a for the stress components where —2/3 < a < —0.545, i.e. the asymptotic behaviour near the corner singularity lies somewhere between the Newtonian solution and the solution derived by Hinch [283]. Sasmal [526], using a finite volume method and the stream function vorticity formulation of the governing equations, also predicted Newtonian-like asymptotic behaviour near the re-entrant corner for most of the flow variables for a UCM fluid in an axisymmetric 4:1 contraction. These finite volume simulations [526,637] suggest that there exist integrable stresses for the UCM and Oldroyd B fluids near the re-entrant corner. This prediction is in contrast with the numerical results of Lipscomb et al. [358] and Coates et al. [142] who obtained results indicating a stress singularity between r - 0 - 9 and r"1. Lipscomb et al. [358], using a Galerkin finite element method for the 4:1 planar contraction, showed that a w 0.91 albeit for low Weissenberg 71
3.5. SINGULARITIES numbers, thus emulating the singular behaviour of a second-order fluid. The corresponding velocity field displays almost Newtonian behaviour. Coates et al. [142] using an EEME/SUPG formulation in an axisymmetric 4:1 contraction geometry suggested that the stress components behave like r~a for We = 0.65 where a is close to unity. This is in agreement with one of the inadmissible solutions (Q = 0.985) of Davies and Devlin [161] for planar flow. These findings demonstrate that inadmissible mathematical solutions may manifest themselves in numerical solutions. Baaijens [24] using a DEVSS/DG formulation for the UCM model in an axisymmetric 4:1 contraction showed that the order of the stress singularity is approximately —2/3 almost independent of 9. This agrees with the analysis of Hinch [283] for the Oldroyd B model. However, on the downstream wall the order of the singularity initially starts at —1/3 before increasing to —2. Con vergence with mesh refinement is unlikely to be attained along the downstream wall due to the influence of the singularity at the corner. Since the analysis of Hinch [283] was based on the Oldroyd B model, Baaijens [24] repeated the numerical experiments for this model with (3 = 0.89 and demonstrated that the order of the singularity was unaffected by the presence of a small amount of solvent viscosity. More recent computations by Aboubacar and Webster [2] for Oldroyd B flow through a planar contraction using a hybrid finite element/finite volume scheme have also corroborated the theoretical results of Hinch along two radial lines towards the re-entrant corner.
72
Chapter 4
Parameter Estimation in Continuum Models 4.1
Introduction
An important issue in computational rheology is the choice of constitutive equa tion. The capability of a particular computation to give agreement with the experimentally observed behaviour of a fluid is predominantly dependent on the choice of model and its capacity to describe the flow under consideration. Within each model there is further flexibility since constitutive models contain a number of material functions and parameters that need to be quantified. The characterization of a fluid in terms of these basic material functions and param eters is therefore essential before computations, if they are to have any physical significance whatsoever, can be performed. Simple representations for the important rheometrical functions exist for most of the standard constitutive equations. The traditional approach to de termining the parameters in these functions has been to perform rheometrical experiments on samples of the material, such as steady simple-shear, smallamplitude oscillatory shear and extensional flow, and to use some form of data analysis, such as least squares, to estimate their values. The choice of a constitutive equation is often guided by its ability to simulate the behaviour of a real fluid in one or more simple rheometrical flows. The de cision as to which rheometrical flow to give greater prominence depends on the application being studied. However, in many situations a combination of effects will need to be modelled. Having completed this process the constitutive equa tion is used to predict the behaviour of the fluid in a more complex configuration in which both shear and extensional effects may be present simultaneously. An excellent case study illustrating the role of parameter estimation in con tinuum models using data generated from rheometrical experiments can be found in the paper of Satrape and Crochet [528]. In this paper the authors demonstrated the difficulties encountered when the derived model is used to simulate a complex flow. Satrape and Crochet [528] sought to determine the parameters in a FENE-CR model to represent the behaviour of a Boger Type I fluid which comprises 0.02 % Separan and 3.8 % water in glucose syrup. Their ultimate goal in this paper was to simulate numerically the flow of this Boger 73
4.1. INTRODUCTION fluid past a sphere and to make comparisons with experimental observations for the same flow configuration. The FENE-CR model was chosen as a candi date for simulating Boger fluids since it possesses a constant viscosity and the first normal stress difference shifts from quadratic to linear behaviour when the shear-rate 7 increases (see §2.6.1). This example demonstrated the difficulties in obtaining agreement between experiments and numerical simulations. These pitfalls are due, perhaps, to the characterization process in which the parame ter estimation is performed using simple rheometrical experiments and then the derived model used in a complex flow configuration. In the case of a steady, simple shearing flow given by u
= (7s»2/>0,0),
the shear stress and normal stress differences for the FENE-CR model (see, too §2.6.1) can be expressed in the form a
xy
=
[Vp > 'Qsllxyi
°**-°™ = I A 7 ( - 1 + [ 1 + ®yy
® zz
=
8(A
^)2^]1/2)'
" ?
where, as in §2.6.1 and with the same notation as used there, b = HQ2/kT. The four material parameters (r)p, r)s, \i,b) occur in only three groups. The first group, the shear viscosity, can be determined directly from the shear viscosity measurements, while the other two o x 2 ( & - 3 ) VPb 62 > 2A!5
8Al
are determined by fitting the first normal stress difference data obtained from a steady shearing flow. Therefore, this experiment can only determine three of the four parameters in the model and thus there is an infinite number of ways of choosing the parameters consistent with this experiment all of which have identical shear viscosities and normal stress differences. To uniquely characterize the fluid another experiment needs to be performed in which the differences between the fluids within a given class can be measured. One can consider either steady, uniaxial extensional flow or small-amplitude oscillatory shear in an attempt to determine the remaining free parameter in the model. Unfortunately, data from these experiments for this fluid were not available when Satrape and Crochet [528] carried out their study. In the case of uniaxial extensional flow for which the flow field is given by
u=
(ex,-ey/2,-ez/2),
where e is a constant extensional rate, the extensional viscosity, Tje, is defined by Vxx—Vyy
=
«7e(e),
Oxx-Ozz
=
«7e(e)>
where, for the FENE-CR model,
»w = '{» + (i +a) a _ 2a)} ■ 74
'«>
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS and a = Aie(l-<2 2 /&), is the solution of the equation 2a 3 + (1 - 2A!e)a2 - [1 + Aic(l - 3/b)]a + Aie(l - 3/6) = 0, Q now denoting a non-dimensionalized dumbbell configuration vector. For finite values of 6 the extensional viscosity tends to a fixed finite value when Aic —> oo, provided a remains finite. The non-dimensionalized polymeric contribution to the extensional viscosity behaves like b as e -> oo (see (2.130)) and therefore fluids that cannot be differentiated in simple shear can be set apart in exten sional flow. Alternatively, in a small-amplitude oscillatory shear flow one can compute the dynamic moduli, which for a FENE-CR model are given by G(w)
= i + AMi-W
G
"^ = "hi + 4-w]-
(4 2)
"
(4 3)
-
Experimental results for Boger fluids [314] show that, when the ratio of sphere to cylinder radii is 0.5 and 0.2 < We < 0.6, there is almost no deviation of the drag on the sphere from its Newtonian value. The numerical simulations of Satrape and Crochet [528] predicted a decrease in drag with increasing Weissenberg number irrespective of the choice of the fourth parameter in the FENE-CR model. Four model fluids with identical shear viscosities and normal stress differences which differed essentially through the ratio T]8/T]O were unable to give qualitative agreement with experimental observations. Therefore, even if extensional data were available for this fluid it is unlikely that the FENE-CR model would be able to model the flow of a Boger fluid past a sphere. The above example serves to highlight some of the difficulties encountered by classical constitutive models such as those of Maxwell, Oldroyd, Giesekus and Phan-Thien and Tanner in modelling complex flows. Experience has shown that it is almost impossible to describe both shear-dominated and extensional flows for a large range of Weissenberg numbers with a universal set of parameters using constitutive equations of this type. The failure of constitutive equations to model complex flows, demonstrated by the quantitative discrepancies be tween experimental observations and numerical simulations, therefore lies, for the most part, in their inability to control independently the shear and exten sional properties of the fluid. An interesting development, which represents an almost quantum shift from the traditional approach to the characterization of fluids, is the work of Peters et al. [451,452] on enhanced constitutive models. They proposed a new class of models that has the ability to control independently the shear and extensional properties of a fluid. The starting point in the construction of these so-called enhanced constitutive models is to write the viscosity, TJ, as V = GXU
(4.4)
where G is the modulus and Ai is the relaxation time. The relaxation time and modulus are then chosen to be functions of the invariants of the extra-stress 75
4.2. DETERMINATION OF VISCOSITY tensor whilst maintaining a fixed shear viscosity. This provides extra freedom to predict extensional properties. For example, if the UCM model is written in the form then a generalized version is \JT)
+ ? =
G(T)
^
where both the relaxation time and modulus are functions of the extra-stress tensor. The functional dependence of Ai(r) and G(T) is arbitrary provided the Maxwell model is recovered in the limit of infinitesimal strains and the product Ai (T)G(T) accurately describes the experimental shear viscosity data for the fluid under consideration. This latter constraint is treated as being absolutely sacrosanct. Thus, the generalized model provides greater flexibility than the standard UCM model in independently characterizing the extensional and shear properties of a material. Peters et al. [451,452] provided numerical simulations of a polymer melt (LDPE) that had been characterized in shear and uniaxial extension using the P T T and Giesekus models. For three problems (4:1 contraction, flow past a cylinder and the cross-slot geometry) these models fail to predict the maximum stress levels and stress relaxation in areas with high planar elongational strain that are observed in experiments. In an attempt to obtain better predictions of the experimental results, Peters et al. [451,452] used their enhanced constitutive models. The viscosity was fixed and the model parameters were determined by fitting data obtained from a cross-slot experiment in which the velocities and stresses were measured using particle tracking velocimetry and the fieldwise birefringence method, respectively. The enhanced models were shown to give better predictions for the flow in a 4:1 contraction and flow past a cylinder. This novel approach to fluid characterization through the use of enhanced con stitutive equations seems to offer the prospect of yielding material functions and parameters that possess the capability of predicting fluid behaviour in complex geometries in a qualitative sense.
4.2
Determination of Viscosity
The viscosity of materials can be influenced by variables such as shear-rate, temperature, pressure and strain-rate, for example. Although the variation of viscosity with shear-rate is traditionally regarded as being the most important from a rheological point of view, there are a number of applications in which the dependence of viscosity on the other variables is more important. For example, the effect of pressure on lubricants and drilling fluids has a tremendous effect on viscosity at elevated pressure, e.g. a viscosity rise of four orders of magnitude can occur for a pressure rise from atmospheric to 0.5 GPa.
4.2.1
Shear viscosity
At the outset it is important to differentiate between Newtonian and nonNewtonian behaviour in the context of shear viscosity. Barnes et al. [43] state 76
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS that Newtonian behaviour in experiments conducted at constant temperature and pressure has the following characteristics: 1. The only stress generated in simple shear flow is the shear stress a, the two normal stresses being zero. 2. The shear viscosity does not vary with shear-rate. 3. The viscosity is constant with respect to the time of shearing and the stress in the liquid falls to zero immediately the shearing is stopped. In any subsequent shearing, however long the period of resting between mea surements, the viscosity is as previously measured. 4. The viscosities measured in different types of deformation are always in a simple proportion to one another, so, for example, the viscosity measured in a uniaxial extensional flow is always three times the value measured in simple shear flow. Any liquid that deviates from the above behaviour is said to be non-Newtonian. The influence of shear-rate on the viscosity of a liquid is determined exper imentally using an instrument called a rheometer. Rheometry is a subject in its own right and therefore only the basic elements can be covered here. The interested reader is directed to the books on rheometry by Walters [606] and Whorlow [629] for further details. When a typical non-Newtonian fluid experiences a shear flow the viscosity appears to be Newtonian at low shear rates with constant viscosity, 770, say. After this initial Newtonian 'plateau' the viscosity is found to decrease with in creasing shear-rate for the vast majority of materials. This behaviour is known as shear-thinning. However, there are some materials, e.g. corn starch suspen sions, for which the reverse behaviour is observed, i.e. the viscosity increases with shear-rate giving rise to what is known as shear-thickening behaviour. The general shape of the curve representing the variation of viscosity (77) with shearrate (7) is shown in Fig. 1.6. For a shear-thinning fluid in the limit of high shearrates the viscosity is again constant, Jjoo? but at a lower level. These constant viscosity extremes in the viscosity versus shear-rate behaviour are sometimes known as the lower and upper Newtonian regions, respectively. The constant value approached as 7 -¥ 0 is known as the 'zero-shear-rate viscosity' and in this chapter it is denoted by 770. In order to model the shear-thinning behaviour described in §1.2.1 and cap ture the main features of the typical viscosity versus shear-rate plot in Fig. 1.6, various viscosity models have been proposed in the literature. The parameters in these models are chosen to give the best least-squares fit, for example, to the discrete viscosity/shear rate data. The Cross model [158] is commonly used to model the shear-thinning behaviour of non-Newtonian fluids due to its ability to predict the two Newtonian plateaux. The Cross model is given by y-rjoo _ % - Voo
(1 +
1 (Kj)™)'
or, equivalently,
^^L
= (K^r,
r]-rjoo
77
(4-5)
4.2. DETERMINATION OF VISCOSITY where K is a constant with the dimensions of time and m is a dimensionless constant. This model has four parameters that can be used to fit the experimen tal data. A typical log-log graph modelling the shear viscosity behaviour of an aqueous solution of polyacrylamide (TJQ = 1.82Pa.s, 7700 = 2.6mPa.s, K = 1.5s and m = 0.6 - see Boger [83] and Barnes et al. [43]) is shown in Fig. 4.1. Another popular model is that due to Carreau [126]: 1 (l + ( t f i 7 ) 2 ) m i / 2 ' where K\ and TOI are constants. The Carreau model is also able to predict the two Newtonian plateaux and the intermediate shear-thinning region.
Figure 4.1: Typical viscosity versus shear-rate behaviour for the Cross model (4.5) Other classes of models can be used with confidence only for restricted ranges of shear-rate. Typical of log-log viscosity versus shear-rate graphs is the linear region away from very small and very large shear-rates. This "power law" region can be modelled by the expression
r) = K2Y
(4.6)
where Ki and n are constants, known as the consistency and power-law index, respectively. Shear-thinning behaviour is described by values of n less than one. The power law model given by (4.6) is a useful empiricism in many engineering applications. We note that, by a simple rearrangement of (4.5) defining the Cross model, Vo (1 ~ V/Vo)
78
= my
(4.7)
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS so that for 77/770 « 7700/77 (77 the geometric mean of 7700 and 770) we have the approximation (4.8) which is (4.6) with K2 = r)oK~m and m = 1 - n. A graph of the power law model shear viscosity superimposed on that of the Cross model for the same parameters as in Fig. 4.1, is shown in Fig. 4.2.
Figure 4.2: Typical viscosity against shear-rate behaviour for the Cross (full line) and power law (dashed line) models plotted on a logarithmic scale.
4.2.2
Dependence of viscosity on temperature
Variations in temperature can have a significant effect on the viscosity of a fluid. Clearly, this has important consequences for many industrial applications. The viscosity of Newtonian liquids decreases as the temperature increases, a phe nomenon known as temperature-thinning. This phenomenon has been known for some time. For example, in the eighteenth century it was known that the reliability of mechanical clocks depended on changes in the viscosity of the lu bricating oil caused by fluctuations in temperature. Variations in temperature caused the clocks to run either fast or slow depending on whether the tempera ture soared or dipped. The clockmaker John Harrison (1693-1776) ingeniously circumvented this problem by dispensing with oil altogether and carving the parts that would normally require lubrication out of a tropical hardwood that exudes its own grease [538]The functioning of lubricants in a car engine is a contemporary example that illustrates the importance in gaining an understanding of the temperature 79
4.2. DETERMINATION OF VISCOSITY dependence of viscosity. The choice of lubricant can have a significant impact on the performance of an engine. If the lubricant thins too easily with increasing temperature then there is a greater risk of engine wear whereas if it is too thick then there is a reduction in efficiency. The strong temperature dependence of viscosity means that great care has to be taken to ensure that isothermal conditions prevail when viscometric exper iments are performed. However, simply maintaining control of the thermostat temperature may not be enough to ensure that the flow is isothermal. In a capillary viscometer, for example, high shear-rates near the capillary wall can give rise to viscous dissipation. Some of the heat generated is conducted to the surroundings and some remains in the fluid where it is transported by conduc tion and convection. This can provide a source of error in the determination of viscosity unless steps are taken to remove the heat generated since the analysis assumes that the flow in the capillary is at a constant temperature. In practice a series of viscosity versus shear-rate curves is determined from experiments performed for a number of thermostat settings, i.e. 77(7, T) is plot ted against 7 for T = Tt, i = 1 , . . . , K. An example of this process is shown in Fig. 4.3 where a series of viscosity versus shear-rate curves is shown for the poly carbonate melt Makrolon for the thermostat settings T = Ti = 260+(i — 1)10°C, i — 1 , . . . ,5, where the data are taken from the M. Phil, thesis of Walters [612].
Figure 4.3: Variation of viscosity with shear-rate for the polycarbonate melt Makrolon for T=260°C (top curve), 270°C, 280°C, 290°C, 300°C (bottom curve). The variation of viscosity with shear-rate and temperature, for this polymer melt, can be described by a so-called master curve, 17(7, T), by plotting the data in terms of the reduced variables [599]: ft
r]R = — Vo
and
7 f l = 7707,
(4-9)
(see Fig. 4.4), where 770 is the relevant zero-shear-rate viscosity. The master
80
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS
Figure 4.4: Makrolon master curve. curve of Fig. 4.4 can be described by the Cross model:
•» = T+WW
(4 10)
'
The temperature dependence is included in the zero-shear viscosity using the Arrhenius law: r)o(T) = A exp
©■
(4.11)
where T is the absolute temperature of the fluid and A and B are constants. Therefore, we may write r)o(T) 7,(71 T )
l + (tf % C07)"
(4.12)
The parameters used to generate the Makrolon master curve are A = 1.67 x 1CT6 Pa.s, B = 11174°K, K = 8.55 x K T 7 P a " 1 and n = 0.94.
4.2.3
Dependence of viscosity on pressure
It is known that the viscosity of a wide range of fluids increases exponentially with increasing pressure. Therefore, the pressure dependence of viscosity can have a profound influence on many industrial processes. At low pressures this effect is small but at elevated pressures the effect can be appreciable and give rise to important effects. For example, at the extreme pressures experienced within a journal bearing at high eccentricity ratios, piezoviscous lubricants can reduce wear by increasing the viscosity. The effect of pressure on viscosity is sometimes modelled by the Barus law ■nip) =r)oexp(ap).
(4.13)
This may be combined with a Cross model for the shear-rate dependence of viscosity to give a viscosity law of the form
(m - noo) ■nii,p)= yio + [1 + (#7)"*] x exp(ccp), 81
(4.14)
4.2. DETERMINATION OF VISCOSITY where K is a function of pressure K = K{p) = exp(ap + E), and 770, rjao, m, a, a and E are material parameters that are estimated by bestfitting experimental data. The parameter a in (4.14) controls the dependence of the low shear-rate plateau on pressure. It is assumed that the dependence of the high shear-rate plateau on pressure is governed by the same exponential law. The parameter a controls the rate of shear-thinning on pressure. Realistically, it may be assumed that this behaviour is independent of the low and high plateau levels reached. It is important to note that the viscosity law (4.14) is consistent with experiments [48,301] that span only limited ranges of the pressures which the lubricants experience under general operating conditions. The variation of viscosity due to shear-rate for the lubricants considered in the paper of Hutton et al. [301] is no more than 50%. However, corresponding changes due to pressure can be as much as tenfold.
4.2.4
Extensional viscosity
The behaviour of the extensional viscosity function (see §1.2.2) is very often qualitatively different from that of the shear viscosity function. For example, highly elastic polymer solutions that possess a viscosity which decreases monotonically in shear often exhibit an extensional viscosity that increases dramati cally with strain-rate. A fluid for which r)e increases with increasing e is said to be tension-thickening, whilst, if ne decreases with increasing e, it is said to be tension-thinning. Continuum mechanics is able to provide some insights into the extensionalviscosity behaviour of non-Newtonian fluids. For example, the following result holds [453]: lim ne(e) = 3 lim 77(7). £->0
7-fO
In the case of Newtonian liquids this relationship is true for all values of e and 7, i.e. Ve = 3%, a result due to Trouton in 1906. Following on from this rheologists have intro duced what is known as the Trouton ratio to measure the departure of the ratio of extensional to shear viscosity from its Newtonian behaviour: T
=
Ve(e)
Elastic liquids are noted for having high Trouton ratios. In order to remove the dependence of TR on both e and 7, Jones et al. [313] introduced the modified definition:
TR(e) = - * < 2 - . Furthermore, Jones et al. [313] have shown that if the fluid is inelastic and isotropic then TR assumes the value three for all values of e. This ratio provides a convenient means of estimating viscoelastic response since any departure from the value three can be associated with viscoelasticity. 82
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS Having determined the shear viscosity using a steady simple shear flow there is very little flexibility in most models when it comes to fitting extensional data. For example, the extensional viscosity for the UCM model (see (2.116)) is given by c\ nVe(e)
2r
-
>°
i
%
~ (l-2A1e)+(l+A1e)There are no free parameters in this expression so the extensional viscosity is prescribed. Note that for the UCM model r}e becomes infinite in the limit Ai<= ->• \. In some situations this property is an advantage. For example, the model is able to simulate the high extensional viscosity levels found in some dilute polymer solutions. Alternative models such as the P T T model [458] allow the extensional viscosity to go through a maximum rather than take infinite values. In the FENE-CR model the extensional viscosity remains bounded when Aic -¥ oo for finite values of a (see (4.1)). However, as shown earlier, there is a free parameter in this model that may be used to fit extensional data.
4.3
Determination of the Relaxation Spectrum
An important quantity in the characterization of viscoelastic polymeric fluids is the relaxation spectrum. In the study of molecular theories of polymeric materials the relaxation spectrum is used to examine the relationship between the molecular weight distribution and viscoelastic properties of a material [197]. The relaxation spectrum can also be used to predict the behaviour of a linear viscoelastic fluid in any other standard experiment and it is a fundamentally important quantity in the analysis of processing experiments such as extrusion. The difficulty is that the relaxation spectrum is not a directly measurable quan tity and the process of extracting it from experimental data frequently leads to an ill-posed problem. In a linear viscoelastic fluid, the relaxation modulus, G(t), relates the stress tensor cr(t) to the rate of strain tensor -y(t) in the constitutive equation a(t)=
I
G(t - s)j(s)ds.
(4.15)
J — oo
For a Maxwell model with N relaxation modes one can write down an expression for the relaxation modulus in the form N
G{t) = YJhie-t'Xi,
(4.16)
j=i
where A{, i = 1 , . . . ,N, are the discrete relaxation times of the material and /ij, i = 1 , . . . ,N, are the weights associated with the relaxation times. These weights are sometimes referred to as partial viscosities. If one of the modes has an infinitely large relaxation time then the material is a solid. In the limit as N —>• oo one can define a continuous spectrum of relaxation times oo
/
H(X)e-^x d(\n\),
(4.17)
-oo
where H{\) is the distribution function of relaxation times, or continuous relax ation spectrum, that weights the importance of the infinity of relaxation times available to the material. 83
4.3. DETERMINATION OF THE RELAXATION SPECTRUM
4.3.1
Dynamic experiments
Dynamic mechanical experiments over a wide frequency range, using highly sophisticated instruments, are performed to provide data from which the relax ation spectrum of polymeric liquids may be determined. In small amplitude oscillatory shear flow with frequency u and strain given by 7*v = 1oeiut,
(4.18)
where the magnitude of 70 is small enough to ensure that the linearity constraint is satisfied, the shear stress is generated using (4.15). If we write axy(t)=G*(u)jxy(t),
(4.19)
where G* is the complex shear modulus, then rOO
G' •(w)=iu>
G(()exp(-iuj(;)d(.
(4.20)
Jo We write G*{u) = G'(w) +iG"(u>), where G' and G" are known collectively as the dynamic moduli and separately as the storage and loss moduli, respectively. In dynamic experiments it is the storage and loss moduli that are measured at specific frequencies. If we insert (4.17) into (4.20) and equate real and imaginary parts we obtain the following pair of Fredholm integral equations
^-f^i^fG
» = f^>OTT-
(4 21
-» < 4 ' 22 '
If the material is a viscoelastic solid there is an additional constant term Ge > 0 in the expression for the storage modulus (4.21), where Ge is known as the equi librium modulus. The inversion of either (4.21) or (4.22) or both as a pair is ill-posed [255]. This ill-posedness is in the sense of Hadamard, i.e. small pertur bations in the data can lead to arbitrarily large perturbations in the solution. As far as the determination of the relaxation spectrum is concerned small inac curacies in the measurement of G'(LO) or G"(u) can lead to the determination of a meaningless relaxation spectrum, H(X), unless the inversion is performed carefully. The discrete relaxation spectrum, H?f(t), has the form N
ffjv(*) = 2 M ( * - A i ) .
(4.23)
The partial viscosity, h,, associated with the ith Maxwell mode is related to the elastic modulus
=
9i*i-
If we insert the discrete relaxation modulus (4.16) into (4.20) and equate the
84
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS real and imaginary parts we obtain the discrete equivalents of (4.21) and (4.22)
SV)=i>7^2,
(4-24)
l
8=1
In a dynamic experiment the dynamic moduli are calculated at a discrete set of frequencies {w* : 1 < i < M}, where typically M > N. The discrete relaxation times are not directly available from (4.24) and (4.25). To obtain them one has to solve the following system of equations N
G'(w4) = X X f c = G ' 4 , l
(4.26)
N
G"{u>i) = Y,Kv9i=G'l,
l
(4.27)
3=1
where Ki
*«
_
w
i^3
Kn
(l+w?A5)'
_
«
^'A?
(l + u;?A|)'
and GJ and G" are, respectively, the experimentally measured storage and loss moduli at a frequency Wj. The number of relaxation modes, N, is also an unknown parameter since it is not known a priori how many modes one needs to fit the data adequately. The set of equations (4.26) and (4.27) is a system of 2M equations for the 2N unknowns {
4.3.2
Mathematical problems
There are practical difficulties associated with the use of methods that fit mod els of the dynamic moduli (4.21) and (4.22) to experimental measurements of the storage and loss moduli. The main difficulty is assessing the extent to which information about the relaxation spectrum is not recoverable because of limi tations in the measurements of the oscillatory shear data. The following are examples of the limitations [160]: 1. the storage and loss moduli are measured over a limited range of frequen cies wmin < u < u}max, say; 2. only discrete sampling at specific frequencies {WJ : 1 < i < M} is per formed; 3. observational errors occur as a natural consequence of the measurement process. 85
4.3. DETERMINATION OF THE RELAXATION SPECTRUM The limitations imposed by the first of these are considered in a recent pa per [160]. It is part of popular rheological folklore that the storage and loss moduli over the frequency range wTOin < u) < oJmax yield information about the relaxation spectrum over the reciprocal frequency range, i.e. (bJmax (w m j n ) _ 1 . However, Davies and Anderssen [160] shattered this myth by showing that the relaxation spectrum is determined on a shorter interval of relaxation < A < e 2ur times defined by e 2 w ~ A key question concerns the number of relaxation times that can be calcu lated from the data provided. There is a serious danger that the overfitting of the data by an excessive number of parameters represents the noise in the data only and not the material behaviour itself. It is commonly known that as one increases the number of modes beyond some critical value, say Nc, there is no discernible improvement in the least squares error. For values of N > Nc there may not be a unique solution to the least squares problem. The manifestation of this is that there may be a number of parameter sets that minimize the least squares error, not all of which are physically meaningful. Therefore, robust al gorithms for determining the relaxation spectrum should terminate before the number of modes exceeds this value. However, the value of Nc is not known a priori. We now look at a number of techniques that have been proposed for the determination of the relaxation spectrum.
4.3.3
Linear regression techniques
In this method the discrete relaxation times are prescribed and the least squares error between the measurements and the expressions for G' and G" is minimized with respect to the unknown elastic moduli. Given the data and the matrices K' and K" (i.e. the relaxation times are specified) one tries to find the best set of values {#; : 1 < i < N} for which the objective function
2
M
N
N
M
Gi-EK^J
* = i£ =l
(4.28)
i=l
is minimized. This problem can also be formulated as 2M
x2 =
N
Gi~22
K
ij9j
(4.29)
= K%,i = 1,. .M,
(4.30)
with Kij - K'itj,i = 1 , . . . M, Ki+MJ
and Gi=G'i,i
= l,...M,
Gi+M=G'!,i
= l,...M.
(4.31)
In practice, of course, the quantities G% (1 < i < 2M) are subject to errors. Suppose that the quantity G, is subject to the standard error o-j, then the mea sured value is denoted by G". The linear regression method seeks to minimize 86
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS the objective function
2
2M
N
n
J
* = Es
G
i - z J ^v'5J
(4.32)
Honerkamp and Weese [286] have derived an elegant solution of this minimiza tion problem by performing a singular value decomposition (SVD) of the matrix K. Let K = UWVT,
(4.33)
where U is the matrix of orthonormal eigenvectors of KKT corresponding to the N largest eigenvalues, V is the matrix of orthonormal eigenvectors of KTK, and W is the diagonal matrix containing the non-zero singular values, Wi, i = 1 , . . . ,N, of K. Then the solution of the minimization problem (4.32) can be written as
B = f>*«/ fc - 1 (u2 , G"),
(4.34)
fc=i
where g and Ga are the vectors of unknowns and measurements, respectively. The standard error for g is given by
*2(g) = ^ X X 2 ( v * ) 2 .
(4-35)
k=i
Note that if some singular value becomes too small the errors become large. Furthermore, because the expression for g contains reciprocals of the singular values, small variations in Ga may cause large changes in g. Therefore, the calculated values of g may be quite different from the true values. It only takes one small singular value or a large condition number of the matrix K, i.e. the ratio of the largest to smallest singular values, for the problem to be ill-posed. This means that the minimum of the objective function may be very flat yielding many solutions that are compatible with the given data. Some of these solutions may be physically meaningless giving, for example, negative values for some of the elastic moduli. Increasing the number of modes merely exacerbates the situation rather than producing an improved approximation. Therefore, there are serious shortcomings in the use of the linear regression method to determine the relaxation spectrum. The source of the difficulty in the method of linear regression lies with the appearance of small denominators in (4.34). Regularization techniques may be applied to the solution of this problem in order to reduce the size of the large standard errors which occur when a singular value becomes small. Honerkamp and Weese [286] introduced a regularized version of (4.34), viz.
e" = E v * ^ ( u ^ ) ,
(4-36)
that converges to g if the regularization parameter /x converges to zero. The form of gM ensures that no small denominators are present for // ^ 0. Note that 87
4.3. DETERMINATION OF THE RELAXATION SPECTRUM g*1 is going to differ from g for /i ^ 0 and therefore cannot minimize the same objective function. For w\ ^> fj, we have Wk
w\+ n
wk''
(4.37)
and for w\ C /x we have Wk
wl+fj,
~Wk
fi
(4.38)
It can be verified that gM is the solution of the following minimization problem: mm*;;, where N
xl=X2+^9l
(4-39)
fc=i
and x 2 is given by (4.32). The search for a minimum of xt leads to a well-defined solution in which the difficulties encountered previously have been overcome provided JJ, has an appropriate value. The extra term in (4.39) is the square of the norm of g. So the minimization procedure takes into account not only the mean-square deviation of Gff from Kg but also the norm of g. The parameter n weights the relative importance given to these two terms. There are several strategies in the theory of Tikhonov regularization to de termine (i. Honerkamp and Weese [286] examined several ways of doing this. These include the discrepancy principle, a method based on the minimization of the predictive mean-square signal error and a self-consistence method. This latter method was shown to be more robust and reliable than the others, as far as the determination of the discrete relaxation spectrum is concerned, in the sense that the estimated regularization parameter is nearly constant and unsuccessful determinations of the spectrum do not seem to appear.
4.3.4
Nonlinear regression techniques
One of the drawbacks of linear regression methods, apart from the obvious ill-posedness, is that the relaxation times are specified a priori. If some of these are placed too close together then the neighbouring columns of K will become almost linearly dependent. This leads to small singular values. Note that if two columns of a matrix are linearly dependent then this will produce a zero singular value. In nonlinear regression methods the relaxation times themselves are allowed to be fully adjustable as well. The objective function is now nonlinear with respect to the relaxation times and therefore nonlinear regression techniques need to be used to determine the solution. Additionally, the number of modes, N, can be chosen to be a freely adjustable parameter. The choice of N is crucial for the success of methods of this type. With too few relaxation modes the least squares error is large. However, when the number of modes, N, is increased the least squares error decays rapidly during the initial 88
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS phase before reaching a terminal value. Winter et al. [633] minimized the x2 function defined by (4.32) with an initial number of modes between 1 and 2 per decade. They found that it was advantageous to start with a large number of relaxation modes and to allow their numerical procedure to merge or eliminate unnecessary ones. Beyond some critical value of N, say Nc, the least squares error will not improve significantly and negative values of the elastic moduli may begin to appear. Winter et al. [634] used the disappearance of negative elastic moduli from the least squares solution as their terminating criterion. Syed Mustapha and Phillips [553] suggested an alternative methodology for maintaining the well-posedness of the problem. They proposed beginning with a single mode and increasing the number of modes gradually to ensure that the problem remains well-posed throughout the process. The procedure in which modes are successively introduced into the spectrum continued until estimates of the standard errors of the unknown parameters were dominated by the cal culated values of these parameters. The final spectrum was then given by the values obtained in the penultimate iteration. Interestingly, the stage at which the calculated values of the parameters were dominated by the standard errors coincided with the appearance of negative elastic moduli in many cases. The nonlinear least squares problem at each stage was solved using the MarquardtLevenberg method. The discrete relaxation times and elastic moduli were de termined in the following order 9i<9i+i(i=
4.3.5
1,...,N-1),
Xi > Xi+1(i = 1 , . . .
,N-1).
Examples
The process of determining the discrete relaxation spectrum of a material using the nonlinear regression method [553] is illustrated by means of the double Gaussian spectrum of Honerkamp and Weese [286] given by 1 9k
■
e(-l/2(lnAfc-lnA,)
2
)
+
e(-l/2(ln
A fc -In A v ) 2 )l
(
(4
4{))
2^/W) (*-l)/(JC-l)
Xk
=
*a(y)
,
k = l,...,K,
(4.41)
where A* = 5 x 10" 2 s, Xy = 5s, Aa = 1 x 10" 3 s, Xb = 1 x 102s and K = 100. The corresponding simulated experimental results for the storage and loss moduli are given by
fc=l
v
*
K'
O.) = f>
(443)
'
where (i-l)/(JV-l)
0Ji = wJ^-)
, « = l,...,JV,
J 89
(4.44)
4.3. DETERMINATION OF THE RELAXATION SPECTRUM gk=9khx(jf\/{K-l),
k=l,...,K,
(4.45)
and u>a = 1 x 1 0 _ 3 s _ 1 , W(, = l x 10 3 s _ 1 and N = 30. The simulated experimental results were assumed to be affected by a relative error of size a and were obtained using G'(u)i)(l+(TN(0,1)),
G"(wi)(l + aN(0,l)),
i=
l,...,N,
where N(0,1) denotes a normally distributed random number. The data listed in Table I of Honerkamp and Weese [286] is used in which a = 0.04. In Fig. 4.5 the exact dynamic moduli given by (4.42) and (4.43) and the numerically simulated fit to the 'noisy' data are compared. An excellent fit to the exact data is obtained by a six mode spectrum determined by the nonlinear regression method. A more sensitive test of the method, however, is a comparison between the relaxation spectrum given by (4.40) and (4.41) and the six mode spectrum generated from the noisy storage and loss moduli. A comparison between the two is provided in Fig. 4.6.
Figure 4.5: The dynamic moduli G' (A) and G" (o) given by (4.42) and (4.43), respectively, and the simulated fit (continuous curves) to the perturbed data with a = 0.04. From S. M. F. D. Sped Mustapha and T. N. Phillips, Determi nation of the relaxation spectrum using dynamic nonlinear regression, J. Phys. D: Appl. Phys., 33:1219-1229, 2000, with permission. In a second example the relaxation spectrum of the model fluid SI (see Davies and Jones [164], for example) is determined. This is a solution of 2.5% weight polyisobutylene in a solvent of 19 parts decalin to 20 parts polybutene. The discrete relaxation times and corresponding elastic moduli computed using nonlinear regression for this material are given in Table 4.1. Also shown in these tables are the estimates of the standard errors used to determine the number of modes that are appropriate. Adding an extra relaxation mode to this spectrum 90
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS
Figure 4.6: Original spectrum (o) and the normalized spectrum (A) obtained using the nonlinear regression technique. From S. M. F. D. Sued Mustapha and T. N. Phillips, Determination of the relaxation spectrum using dynamic nonlin ear regression, J. Phys. D: Appl. Phys., 33:1219-1229, 2000, with permission. produces parameter values smaller than the estimated variances and therefore would be meaningless. In Fig. 4.7, the experimental data are compared with the model predictions based on the determination of the discrete relaxation spectrum.
4.3.6
Sampling localization
An alternative to the regularization and nonlinear regression strategies described earlier are methods based on the sampling localization ideas proposed by Davies and Anderssen [160]. Sampling localization is defined to be any situation where information about the relaxation spectrum on the time interval a < A < b is mode 1 2 3 4 5
Ai
9i
5.044203 0.720174 0.096664 0.009799 0.000432
1.565409 13.796343 39.650799 94.152547 943.854733
°~\i
2.419 6.154 1.434 1.372 3.259
xHT4 xlO" 6 xlO"7 xlO-9 xHT12
°-9i
6.964 2.056 9.088 4.269 1.105
xlO-5 xl0~3 xlO" 3 xlO" 2 xlO 1
Table 4.1: Discrete relaxation spectrum for fluid SI calculated using nonlinear regression and estimates of the variances of the calculated quantities. From S. M. F. D. Syed Mustapha and T. N. Phillips, Determination of the relaxation spectrum using dynamic nonlinear regression, J. Phys. D: Appl. Phys., 33:12191229, 2000, with permission
91
4.3. DETERMINATION OF THE RELAXATION SPECTRUM
Figure 4.7: Comparison of the experimental G' (A) and G" (o) data with the simulated dynamic moduli (continuous curves) versus frequency for the model fluid SI. From S. M. F. D. Syed Mustapha and T. N. Phillips, Determination of the relaxation spectrum using dynamic nonlinear regression, J. Phys. D: Appl. Phys., 33:1219-1229, 2000, with permission. determined completely by the values of the storage or loss moduli in some finite frequency interval or intervals. The aim in this approach is to compute mean values of the relaxation spectrum over small intervals of relaxation times. To achieve this Davies and Anderssen [160] defined nab and gat, to be the partial viscosity and elastic modulus, respectively, over the range a<X
= [
H(X) dX,
(4.46)
Ja
gab = / H(X) ^
= ^
A
Ja
H(X)d(\nX).
(4.47)
Vina
The following two results provide bounds on the range of relaxation spectrum that can be determined from storage and loss moduli information within a pre scribed frequency domain. The proofs may be found in [160]. Theorem 4.1 The partial viscosity in the range a < X < b is determined by the storage modulus over the range e
-T/2
g^/2
—:—• < w <
(4.48)
Theorem 4.2 If 1 < b/a < e", the partial viscosity, r)ai,, over the range a < X
CHAPTER 4. PARAMETER ESTIMATION IN CONTINUUM MODELS (4-4$)- But, ifb/a > e", then 7?a& is determined from the loss modulus in the two distinct ranges e-7r/2
e 7r/2
e-n/2
e 7r/2
—;— k = wo exp(kh),
k =
0,...,K,
where h = TT/(2L — 1), for L > 2, and then centering the intervals (a, b) in (4.47) geometrically on a reduced reciprocal grid of relaxation times denned by x
i = vj1,
j = L,...
,K
-L.
Discrete elastic moduli are determined using moving-average formulae to ap proximate the integral in (4.47). Point estimates of the relaxation spectrum are then given in terms of the discrete elastic moduli. The construction of the moving-average formulae is based on the following data-functional representations for gab, for ln(6/a) < w, derived by Davies and Anderssen [160]:
9ab
n
= g'ab = - f
{ £ " [ m ( M ] - E"[ln(au>)}} G'(w)d(ln w),
J-lnb-w/2
gab = g"ah = f
{ £ ' [ l n ( M ] - E'Maw)]}G"(w)d(lnw), b-iv/2
(4.51) with E'(x) = MmE'Jx),
E"(x) = \imE"(x),
(4.52)
where the functions E't{x) and E"(x) are the real and imaginary components of the elementary sampling function 1 ./x+i7rA -erf ^ — , e > 0. Using the transformation x = l n w + |ln(a&), A = ln(&/a), 0 < A < n, Anderssen and Davies [11] showed that the representations (4.50) and (4.51) for the elastic moduli could be expressed in terms of the continuous moving-average
93
4.3. DETERMINATION OF THE RELAXATION SPECTRUM formulae: 9ab = 9ab = - /
J±(n+A)
{E>< (x + | A) - E" (x - iA)} G' M = \vab/
9ab = 9ab
/
{E> {x + l&)-E'{x-
i A ) } G"
- =
dx.
(4.54) The discrete moving-average formulae, constructed from (4.53) and (4.54), are of the form L
9'j =
- J ] aiG'(uJ+l),
j = L,L + l,...,K-L,
(4.55)
l=-L L
9'j
=
E
A G ' W O . j = L,L + l,...,K-L.
(4.56)
Exponential sampling of the measurement frequencies is required in order for it to be possible to match the arguments of G' and G" in (4.53) and (4.54) and in (4.55) and (4.56). The coefficients in these formulae are determined by appealing to the properties of the elementary sampling function. Note that the coefficients on and /3; depend only on the half-length, L, of the formulae and not on the measurements of the dynamic moduli. Therefore, they are independent of the material being analyzed. Once the coefficients have been determined for a given value of L it is an inexpensive exercise to compute estimates of the relaxation spectrum. Anderssen and Davies [11] presented results showing that good estimates of the spectrum could be recovered if the moving-average formulae were used iteratively. However, a couple of important open questions remain. The first question concerns the mechanics of data sampling. Due to the phenomenon of sampling localization the most effective use of the data requires it to be exponentially sampled with a sampling interval that is an integer divisor of 7T. It remains to be seen how restrictive this limitation on data sampling is in practice. The second question concerns the optimal number of iterations since a balance has to be achieved between a sufficient number to ensure that the amplified noise is filtered and a minimum number to ensure that there is no loss of resolution in the recovered spectrum. An automatic procedure for achieving this balance remains to be determined.
94
Chapter 5
From the Continuous to the Discrete 5.1
Introduction
The numerical solution to a system of partial differential equations involves two stages. In the first stage the governing equations are discretized in a consistent fashion on some mesh using a convergent discrete representation of the solution, i.e. the discrete approximation converges to the exact solution of the problem as the mesh is refined. Note that, for the systems of equations governing the flow of viscoelastic materials, it is not known a priori whether this property is satisfied by a given discretization. In the second stage the discrete problem, which may be nonlinear, needs to be solved accurately and efficiently. These topics are the subject of the next two chapters. Particular difficulties associated with the discretization of the governing equations for viscoelastic flow are related to the mixed type of the system of partial differential equations and the treatment of the convection term. Care must be taken when this system is discretized to ensure that the various oper ators in the equations are treated in an appropriate manner. Convection must be treated very differently from diffusion. Conventional discretization methods such as the finite difference and finite element methods were originally developed for solving elliptic partial differen tial equations. The theoretical basis of the finite element method, for example, for solving elliptic problems stands on a firm mathematical foundation. How ever, without modification these methods may become unstable for convectiondiffusion problems unless the computational mesh is sufficiently fine. In the case of viscoelastic flows traditional discretization methods may become unstable as the elasticity parameter is increased. A natural way of resolving this problem would seem to be to integrate the components of the extra-stress along the streamlines, which are real characteristics of the system of partial differential equations. This is the methodology behind the discretization of integral type constitutive models. However, as Crochet et al. [152] state in this context, 'dif ficult problems are seldom endowed with miracle solutions.' The difficulty in discretizing the convective terms in the constitutive equation is replaced with the difficulty of calculating accurately the strain history. An additional draw95
5.1. INTRODUCTION back is that, since integration is along streamlines, this approach cannot be used for flows with recirculation. Therefore, for flows with recirculation one has to use a differential constitutive model and to use some form of stabilization to retain accuracy and stability of the underlying numerical method for high values of the Weissenberg number. This may involve, for example, the use of artificial diffusivity or the reformulation of the governing equations. For the finite element method the former is achieved by using test functions biased towards the upstream direction [103]. An example of the latter is the EEME formulation in which the momentum equation is reformulated so that it is elliptic in nature [332]. There are other techniques available to the computational rheologist to stabilize calculations for high values of the Deborah number and most robust numerical schemes use a combination of different techniques. We introduce various discretization methods on model convection-diffusion problems in one and two dimensions before going on to describe their application to viscoelastic flow problems. We consider the model problems Lu = -eu" + bu' = f, on I,
(5.1)
where I = (—1,1) with boundary conditions u(-l)=uL,
u(l)=uR,
(5.2)
and Lu = - e V 2 H + b • Vu = / , in Q, with boundary conditions u = 0, on dn,
(5.3) (5.4)
where Q is an open bounded domain in M2 with a Lipschitz-continuous boundary dSl. In (5.1) and (5.3), e and b are positive constants, and b is a constant vector. The basic methods for viscoelastic flows will be described for the Oldroyd B fluid which is representative of many of the differential constitutive relationships in use. The dimensionless form of the governing equations for an Oldroyd B fluid (see §2.6) is - / ? V 2 u + V p - V r = 0, V ■ u = 0, r + We T = (1 - 0)j.
(5.5) (5.6) (5.7)
In these equations r is the viscoelastic part of the extra-stress tensor T where T = r + /?7,
(5.8)
and We = \\UjL is the Weissenberg number, where U is a characteristic ve locity and L is a characteristic length. In these equations time has been nondimensionalized with respect to L/U and pressure and extra-stress with respect to r]oU/L. The parameter (3 is defined by P=Y-,
(5-9)
i.e. the ratio of the retardation and relaxation times of the fluid. In this chapter we describe some of the methods of discretization that have been employed in the solution of viscoelastic problems, viz., finite differences,
96
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE finite elements, finite volumes and spectral methods. The first three methods are examples of local methods. They utilize local approximations to derivatives to form the discrete problem. The discretization parameter for these local methods is h, an element diameter, the distance between neighbouring grid points on a uniform mesh or the maximum distance between grid points on a nonuniform mesh. On the other hand, spectral methods utilize information over the whole interval or domain (or at least, a sub-domain) in approximating derivatives to form the discrete problem and are therefore sometimes referred to as global methods. The discretization parameter in a spectral approximation is N, the order of the polynomial approximation. It is the local or global property of the method that is responsible for the structure of the algebraic systems which arise from the discretization of the problem. Local approximations give rise to banded systems whereas global approximations yield non-sparse systems. For smooth problems spectral methods possess superior approximation properties over their local counterparts. The accuracy of local approximations can be increased by refining the mesh, i.e. decreasing h or by using higher-order approximations to derivatives. The former option increases the size of the system whereas the latter increases the bandwidth.
5.2
Finite Difference Approximations
Finite difference approximations are usually derived by approximating deriva tives using Taylor series expansions of the solution. On the interval / we lay down a uniform mesh Xj =x0+jh,
j = 0 , . . . ,N,
with Nh — \I\. Assuming that u € C 4 (/) we can show that «'(*,)
=
„{xi)
=
u
t , ( 8 j + l ) 2
~U(a!j-l)-^a>^),
u{xj+1)-2u{xj)+u{xj„l)
_h?
ft
(5-10) (i){^
{ 5 n )
!■£
where Xj-i < £,-, TJJ < #j+i. If we neglect the terms of 0(h?) in (5.10) and (5.11) a second-order central difference approximation to (5.1) can be derived LhUj
s
_ e (S+i-^-r)
+ b
(te__^i)
= /(a
,),
(5.12)
in which we have replaced the exact solution, u(x), to (5.1) at x = Xj by its numerical approximation Uj. Collecting terms this can be written in the more convenient form - ( l - \p)
uj+l + 2Uj - ( l + ±p) «,-_! = (jj
where
is the mesh Peclet number. 97
f(Xj),
(5.13)
5.2. FINITE DIFFERENCE APPROXIMATIONS The characteristic equation associated with the finite difference approxima tion (5.12) is Qy3-l)A2 + 2 A - ( ^ + l)=0)
(5.14)
from which the roots may be deduced 1 + -B A = 1, - — * § , 0?2.
(5.15)
It is clear that the numerical solution is oscillatory when /? > 2 irrespective of the nature of the true solution of (5.1). Therefore, the use of this scheme can produce solutions that are physically unrealistic for values of 3 > 2. Two approximations were made in deriving the central difference approx imation (5.12) to (5.1), each of which contribute to the global error denned by ej = U(XJ) —Uj.
(5.16)
In the first, the differential operator L was approximated by Lh by neglecting terms of 0(h2) in (5.10) and (5.11). This approximation is quantified by means of the local truncation error, Tj, which is defined to be the residual obtained as a result of replacing the numerical approximation Uj by U(XJ) in the difference equation (5.12). Thus, Tj = Lhu(xj)
- f(xj)
= Lhu(Xj) - LU(XJ).
(5-17)
We can easily deduce that the local truncation error of the second-order approx imation (5.12) is Tj = ~
[eu^(r,j)
- 2buM(tj)]
= 0{h2).
(5.18)
In the second, the exact solution of the differential equation (5.1) at the grid points is approximated by the discrete solution. The central difference scheme (5.12) was derived by replacing Lhu{xj) with LhUj. Thus the global error satisfies the equation Lhej = Tj,
(5.19)
with eo = eN = 0. Therefore, in order to prove that the global error is of the same order as the local truncation error, we require that the finite differ ence operator Lh, is stable. This operator is stable if it has a bounded inverse, independent of h and e, i.e. \\L?\\
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE equations is diagonally dominant and that the scheme satisfies a discrete max imum principle. Both of these properties are guaranteed for the scheme (5.12) if /3 < 2. This is precisely the condition that is necessary to avoid oscillatory solutions to the discrete problem. To show that the operator L h has a bounded inverse consider the test func tion v(x) = (3 + x)/2. We have Lhv = 6/2 > 0. Next we choose a constant D so that Lh(±u — Dv) < 0. This condition is satisfied at x = Xj, j — 1 , . . . , N — 1, if D > 2\Lhu\/b. Also, for D > \UL\ and D > ||WA| we have (±u — Dv)o < 0 and (±u — DV)N < 0. Hence, by the maximum principle, we can deduce that |w(a;j)| < DV(XJ). In particular, H o c < -DIMU = max{4||i: h u|| 0 0 /6,2|« L |, | u B | } .
(5.20)
This proves that L^ is stable. E x a m p l e 5.1 Consider the boundary value problem -eu"+u'
= 0, 0 < x < l ,
(5.21)
u(0) = 1, «(1) = 0.
(5.22)
The exact solution of this boundary value problem is („\
1 eX
-
P(-(l-X)/e)
u(x) =
(z OOX
.
(5.23)
1 -exp(-l/e) The discrete solution found by solving the difference equation (5.12) is
For small values of e, the solution is close to 1 except for a boundary layer of thickness O(e) where it decays to 0. Note that for given values of b and e we can always choose the mesh size h so that the stability condition ft < 2 is satisfied. While this is feasible for the one-dimensional boundary value problem being considered, it will not always be the case for multi-dimensional problems requiring the solution of a set of partial differential equations. An alternative method, which maintains stability, uses an upwind approximation to the first derivative in (5.1). This corresponds to a backward difference approximation. Thus, an upwind difference approximation to (5.1) is LhU
= -e ^ + i - 2 u j + m - i ^
+b
(-LZ-^i) =
f(Xj).
(5.25)
This can be written in the form -Uj+1 + (2 + )9)Uj - (1 + P)Uj-! = f M f(Xj),
99
(5.26)
5.2. FINITE DIFFERENCE APPROXIMATIONS for which the characteristic equation is \2-(2
+ p)\ + {l + p)=0.
(5.27)
The roots of this equation are A = 1, 1 + / ? .
(5.28)
Thus, this scheme is stable for all values of J3 > 0. The upwind approximation to the solution of (5.21) - (5.22) can be written in the closed form u
i=
1 - \i~N i _ X-N '
X = 1+
P-
< 5 - 29 )
The local truncation error of this scheme is
Tj = 4 " ( 4 ) f e ) + y u " ^ ) = °C0-
(5-3°)
We can show that the scheme is equivalent to
- (« - y ) u" + bu' = /(*) + 0(h2).
(5-31)
Since the upwind difference scheme has introduced a nonphysical coefficient of it" we say that the method introduces numerical diffusion or artificial viscosity [512]. Note that when ft > 2 the 0(h) term in (5.31) dominates the true diffusion term. In Figs. 5.1 and 5.2 we compare the central and upwind difference ap proximations with the exact solution of (5.21) - (5.22) for @ = 10 and /3 = 2, respectively. Here we take e = 1 0 - 2 . In the first case (see Fig. 5.1) /? = 10 and so the stability condition is violated. As anticipated the central difference approximation is highly oscillatory and the upwind approximation exhibits nu merical diffusion in the neighbourhood of the boundary layer. In the second case (see Fig. 5.2) /3 = 2 and both the central and upwind difference approximations exhibit monotone behaviour. Finite difference approximations can be defined on a nonuniform mesh. If hj = Xj+i —Xj, j — 0 , . . . ,N—1, then an approximation to the second derivative of u at x — Xj is
In more than one dimension finite difference approximations are cumbersome to generate, particularly on an irregular domain. Therefore, in many situations the finite element method is to be preferred due to its greater flexibility, which facilitates the treatment of complex geometries.
5.2.1
Finite differences for viscoelastic flows
The formulation of the governing equations in terms of stream function, vorticity and extra-stress has been used as the basis for most finite difference tech niques for simulating viscoelastic flows. Even for Newtonian flows the stream 100
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE
Figure 5.1: Comparison of the solutions obtained with the central differ ence (5.12) (A) and upwind (5.25) (o) approximations with the exact solution (smooth curve) of (5.21) for e = 1CT2, h = 0.1 0 = 10). function/vorticity formulation was dominant until the emergence of the MAC (marker and cell) method of Harlow and Welch [270] in the 1960s. Roache [513] in his treatise on CFD commented that, although it is possible to obtain numer ical solutions using the primitive variable formulation, most successful finite dif ference techniques have been based on the stream function/vorticity approach. In two dimensions, one can introduce a stream function, tp, by 'dx'
(5.32)
and the vorticity, w, by U)
dv dx
=
du dy'
(5.33)
so that U)
=
-V2ip.
(5.34)
The continuity equation is now satisfied identically. Furthermore, the pressure may be eliminated by taking the curl of the momentum equation (5.5): -/3V 2 u> - V x (V ■ T ) = 0, 101
(5.35)
5.2. FINITE DIFFERENCE APPROXIMATIONS
Figure 5.2: Comparison of the solutions obtained with the central differ ence (5.12) (A) and upwind (5.25) (o) approximations with the exact solution (smooth curve) of (5.21) for e = 10" 2 , h = 0.02 0 = 2). where u> = (0,0, u)T. The system of equations is completed by the constitu tive equation (5.7) in which all occurrences of the velocity are replaced by the corresponding expressions in terms of the stream function using (5.32). The stream function and vorticity equations (5.34) and (5.35) are elliptic in character and therefore can be discretized using a second-order finite difference approximation. For example, on a uniform mesh with mesh spacing h a central difference approximation of (5.34) at an interior grid point is h2u>jik = — (V'i-i.fc + ipj+i,k + ipj,k-i + V'j.fc+i - 4V»j,fc).
(5.36)
The constitutive equation (5.7) is hyperbolic and therefore to ensure that the resulting system of discrete equations is diagonally dominant the convective terms in this equation are discretized using an upwind difference approximation. The remaining terms in this equation are discretized using central difference approximations. A major difficulty associated with the ip — to — r formulation is the lack of an explicit boundary condition for the vorticity. On a no-slip boundary there are two conditions for the stream function and these may be used to generate a condition for the vorticity in terms of unknown values of the stream function. At a mesh point on a horizontal wall, for example, we may generate the first-order 102
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE approximation [571] ^
=
_2(^-^,fc)_
( 5 3 7 )
Given that the discretization of the vorticity equation (5.35) is second-order one might be tempted to use a second-order boundary approximation for the vorticity such as the one suggested by Woods [635]. However, Webster [622] argues that without an 0(h2) approximation for the constitutive equation there is little point in using such an approximation since the overall accuracy of the scheme is unlikely to be better than first order. A detailed discussion of the intricacies of this formulation may be found in the book of Crochet et al. [152]. In the primitive variable formulation a Poisson equation for the pressure may be derived. However, care must be taken in the derivation of this equation and the choice of boundary condition in order to ensure that the velocity solution remains divergence free. An equation for the pressure is derived by taking the divergence of (5.5): V 2 p = V • (V • r ) + /3V 2 (V • u).
(5.38)
If in addition the incompressibility constraint (5.6) is imposed a simplified pres sure Poisson equation is obtained V2/> = V - ( V - T ) .
(5.39)
Gresho and Sani [250] show that the solution of (5.39) in conjunction with the momentum equation (5.5) cannot guarantee that V - u = 0. Paradoxically, however, if we do not assume that V - u = 0 and use (5.38) instead then provided suitable boundary conditions are imposed we can be assured that the velocity remains divergence free. The well-posedness of the pressure problem (5.38) requires the specification of a boundary condition - Dirichlet conditions, for example - on each part of the boundary of the domain. On a boundary the gradient of the pressure is known from (5.5). In two dimensions this implies that two derivatives of p are known on the boundary. Despite this apparent over-specification the pressure problem comprising the Poisson equation (5.38) and boundary conditions provided by the imposition of (5.5) on the boundary does have a unique solution, up to a constant pressure level, since the source term on the right-hand side of (5.38) is not arbitrary. Gresho and Sani [250] show that if velocity boundary conditions are prescribed then the proper choice of boundary condition for (5.38) is to apply the normal component of (5.5) since this is the only condition that can ensure that V • u = 0 everywhere in the domain. A finite difference approach for solving the primitive variable formulation may be based on the MAC scheme due to Harlow and Welch [270]. The scheme is characterized by the use of a staggered grid (see Fig. 5.3) and the solution of a Poisson equation for the pressure. The original form of the MAC method was based on an explicit temporal discretization f the time-dependent form of the governing equations. Later developments of the MAC method incorporated an implicit treatment of the diffusion term in the momentum equation [178]. Fur ther details of this method may be found in the books of Peyret and Taylor [455] and Fletcher [210]. However, by the time the MAC method had become estab lished for Newtonian flows finite element methods had become the dominant 103
5.3.
FINITE ELEMENT APPROXIMATIONS
Figure 5.3: A staggered grid force in the solution of non-Newtonian flow problems and so non-Newtonian flow problems have never been targetted in anger by the MAC method.
5.3
Finite Element Approximations
The finite element discretization of an elliptic partial differential equation or certain ordinary differential equations is based on its equivalent variational for mulation. In simple cases the variational problem is a minimization problem of the form: Find u£V such that F{u) < F(v),
Vw e V,
(5.40)
where F : V —► 1R is a functional defined on some function space V. In the finite element method it is the variational or weak formulation of the differential equation that is discretized as opposed to the strong form. The variational statement of (5.1)-(5.2) is: Find u €V such that a{u,v) = (f,v),
Vv£W,
(5.41)
where (•, ■) denotes the usual inner product on L2. Here V and W represent the trial and test spaces, respectively, and the bilinear form a(-, •) is defined by a(u, v) =
(eu'v1 + bu'v) dx.
(5.42)
We take V = Hl(I) and choose a conforming approximation space Vh, i.e. Vh C V. Typically Vh will contain polynomials of a certain order in each element with a given degree of continuity between elements. Define the following: V£
=
V0h
=
{veVh:v(-l)=uL,v(l)=uR}, {))£ V f t : t ; ( - l ) = 0 ; u(l) = 0}. 104
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE In the Galerkin method the basis functions for the test space are chosen from the trial space, i.e. Wh C Vh. Therefore, the Galerkin approximation to (5.41) is: Find uh £ V£ such that o(ufc,«fc) = (/,«*), W 1 6 V0h.
(5.43)
Since the approximation is conforming we can substitute for vh in (5.41) and by subtracting (5.43) we obtain the error projection property o(u - uh,vh)
= 0, Vvh G V0h.
(5.44)
Now, using integration by parts, we can show that the bilinear form o(-, •) is coercive on V$ x V^, i.e. a(v,v) = e(v',v') + \b[v2]l_x = t\v\2, Vv E V0h,
(5.45)
where | • | is the Hl semi-norm defined by
\v\2 = I (v')2dx. We can establish an abstract error estimate for the finite element solution of (5.43). Let wh be any element of V^ and define vh = uh — wh. Then using the properties of the bilinear form
=
a{u-uh,u-uh)
=
e((ehy,(u-wh)') h
<
=
h
+ h
e\e \\u - w \ + b\e \\\u -
a(u-uh,u-wh) b((ehy,u-wh) wh\\L*{I).
Therefore, we have e
<
b inf \u — wh\ + - inf \\u - wh\\L2^y w evg ewhevg h
(5.46)
If b = 0 we obtain an optimal error estimate for the Galerkin approximation in the sense that uh is the best approximation in the space V^ to the solution u in the Hl norm. However, the Galerkin approach only performs well for b ^ 0 if e is sufficiently large. For 6 ^ 0 , the approximation error in the H1 norm is no longer optimal. An estimate of the error in terms of a characteristic mesh spacing h requires the use of results from approximation theory. This involves the determination of an estimate of the error in the best approximation, by functions in the discrete space Vg, to the true solution of (5.1) with respect to a given norm.
5.3.1
Finite elements in one dimension
Consider a piecewise linear approximation to the solution of (5.1)-(5.2). Divide the interval 7 into N elements [ j], j = 1,...,N, with xo — —1 and XN = 1. In the jth element there are two basis functions:
105
5.3.
FINITE ELEMENT APPROXIMATIONS
from which we can construct global basis functions associated with each interior node Xj, j = 1 , . . . , N — 1:
{
^ j W , Xj-i <X<Xj, i,j+i(x)> Xj<x<xj+U 0,
(5.48)
otherwise,
and with the boundary nodes: 0o(a;) ^jvfc)
= =
\tl(x), xa<x<xu X
2,N( )I
X
N-I <X< xN.
(5-49) (5.50)
The functions j, j = 1 , . . . ,iV— 1, have compact support on [XJ-I,XJ+I]. The global basis functions are the so-called 'hat' functions. We can then write the piecewise linear approximation in terms of the nodal values Uj at x = Xj and the global basis functions: N
uh{x) = 5>J^0»0-
(5.51)
i=o In this example a basis for VQ is provided by the piecewise linear polynomials {4>j(x) : j = 1 , . . . ,N — 1}. Therefore, the Galerkin approximation defined by (5.43) generates the following system of equations AM = Mf + g,
(5.52)
where Aij = a((pj,(j>i), Mij = (j,i),
(5.53)
9i = -a((f>0, N,
(5.54)
and
if we also approximate f(x) in terms of the basis functions, i.e. JV
i=0
The matrix A, which is known as the stiffness matrix, is a tridiagonal matrix. This property arises from the support of the basis functions. An important feature of finite element methods is that we can evaluate contributions to the stiffness matrix over each element before combining them to form the global stiffness matrix. In this one-dimensional example this means that we have only to consider the interactions of the two element basis functions. In particular, a simple computation in the j t h element produces the element stiffness and mass matrices
In evaluating the global stiffness and mass matrices we need to remember that the corresponding elemental matrices multiply the vectors of nodal values of 106
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE u and / , respectively, in the element, i.e. ( W J _ I , U J ) T and (fj-i,fj)TThe process of generating A and M is known as the assembly stage. If b = 0 then A is symmetric and positive definite. For this simple problem the left-hand side of the system (5.52) is of the same form as that obtained using the central difference approximation. The matrix M is known as the mass matrix. To obtain a bound on the error for the finite element method based on a piecewise linear approximation we introduce the piecewise linear interpolant, 7TftW, of u, i.e. (u - irhu){xj) = 0, j=0,...,N. Then, if h = max^_ 1 hj and u € H2{I) there exist positive constants C\ and Ci such that (see [410], for example) \\u ~ Khu\\L-z(i) < Cih2\u\H-2{i),
\u - Tr/jultfi^) < C2h\u\H2(I).
(5.56)
Therefore, using the estimate (5.46) we are able to derive the error bound
l« - tiV(j) < Ch(l + p)\u\HHl),
(5.57)
where C is a constant and /3 is the mesh Peclet number. Note that the constant in the error bound is multiplied by the mesh Peclet number so the optimal approximation property of the Galerkin approximation on each finite element mesh is lost when convection terms are present. If the solution is approximated by piecewise polynomials of degree r > 1 on each subinterval and u £ HT+1(I) then the approximation results may be generalized (see [120], for example) \\u-TThu\\L2{I)
< Ci/l r + 1 |u|#"+'(')' l u-7I "h u lif 1 (J r ) < C '2/l''|w|if+i(/),
(5-58)
where TT^U is the interpolant of u at the nodes of the finite element mesh.
5.3.2
Finite elements in two dimensions
The main advantage that finite element methods possess over finite difference methods is that they are extremely versatile - irregular geometries, complicated boundary conditions and variable coefficients can all be accommodated rela tively easily within the general framework. Here we give only a flavour of some of the issues involved in the implementation of the technique in two dimensions. Further details may be found in the texts of Strang and Fix [545], Ciarlet [140] and Johnson [310], for example, which are dedicated entirely to the theory and applications of finite elements. In two dimensions triangles or quadrilaterals are usually used to discretize the computational domain. The procedure for setting up a matrix problem for the unknowns follows closely what has been described for the one-dimensional problem. An important feature of the finite element method is that contri butions to the stiffness and mass matrices can be computed at the elemental level using a local coordinate system. The assembly of the global system then requires information about the mapping from local to global coordinates to be held in an appropriate data structure. Let us consider the case of a triangular partition of the domain. Suppose the vertices of a general triangle x i , x 2 and X3 are to be mapped onto the vertices (0,0), (1,0) and (0,1), respectively, of the parent triangle by an affine 107
5.3.
FINITE ELEMENT APPROXIMATIONS
transformation. This situation is depicted in Fig. 5.4. If (£, rf) are the local coordinates then the mapping is given by x
( f > V) = (1 - S - v)*i + £ x 2 + J?x3.
(5.59)
This mapping defines element basis functions (5.60) so within each element we may write the finite element approximation in the form
uh(x(Z,v)MZ,ri)) = i,Uii(Z,Ti)-
(5.61)
The Jacobian, J, of this transformation is given by J =
d(x,y)
X2 -Xi X3 ~Xi
1/2 - 2/1 2/3 - 3/1
(5.62)
The determinant in (5.62) is constant with a value of twice the area of the physical triangle.
Figure 5.4: Transformation of a general triangle onto the parent triangle To obtain a quadratic approximation within each triangular element one has to add further parameters. It is usual to add the mid-points of the edges in the case of triangles. In this case we need six element basis functions. These are given by 0?(fc V) = (1 - f - V)0- ~ 2£ - 2T/), 0 ? ( £ rf) = 4fr, (5.63) 0?(f,'?)=»j(2iJ-l),
<^=4£(l-£-7/).
In the /i-p version of the finite element method it is preferable to work with a hierarchical basis (see the paper of Szabo and Babuska [554], for example) since this greatly simplifies the implementation of p adaptivity. In this case the quadratic basis is formed by adding the last three basis functions in (5.63) to 108
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE the linear basis functions given by (5.60). Note that in the hierarchical basis so formed the coefficients in the elemental representation of the solution do not correspond to nodal values but are linear combinations of them instead. Let us now consider the case of a quadrilateral partition of the domain. We can map each general quadrilateral onto the parent square {(£,??) : — 1 < 0 r) < 1} (see Fig. 5.5). This square is suitable for computing approximations to the integrals that occur in the definition of the elements of the stiffness matrix using Gaussian quadrature rules. The bilinear element basis functions are now given by
4>l(Z,ri) = \(l - 0 ( 1 -V),
m,v)
= ^(1 + 0(1 + V),
0i(e,»?) = i(i+o(i-»7), <m,*) = |(i-o(i+»?)Note that these basis functions are just tensor products of the one dimensional linear basis functions. The Jacobian of this transformation is given by (23 -- x 4 ) ( l
J
+ »?)
1 + (*2-- * i ) ( l - » 7 ) 4 (*3 "-* )(l + 0 2
+
(Xi-- 0 : 0 ( 1 - 0
(2/3-
-y*)0- + v)
+(2/2- - 2 / i ) ( l
~V)
(5.64)
(ya - 2 / 2 ) ( l + 0 +(2/4-- j / i ) ( i - 0
which is not constant across the element. However, if the quadrilateral is convex one can show that the Jacobian is nonsingular throughout the element. Since the determinant of the Jacobian is not constant the integrals that define the elements of the stiffness matrix cannot, in general, be computed exactly. Therefore, one has no alternative but to resort to numerical quadrature rules for their evaluation.
Figure 5.5: Transformation of general quadrilateral onto the parent quadrilateral element Let us now introduce some notation which will define a finite element ap-
109
5.3.
FINITE ELEMENT APPROXIMATIONS
Figure 5.6: Some low-order finite elements with their nodes proximation. We define the following spaces P*
=
{v:v{x,y)
> J aijx%y-' on each element},
(5.65)
i+j
Qf
=
{v:v(x,y)
V j a,ijXlyi on each element},
(5.66)
i,j=0
where # is the number of degrees of freedom per element. A finite element approximation has order of continuity C if the approximation and its first r derivatives are continuous in fi. For example, a piecewise linear approximation on triangles which is C°, is defined by Ff — C°, and a continuous bilinear approximation on quadrilaterals is defined by Qf — C°. The Hermitian quadrilateral element is defined by Q\G — Cl. This is a bicubic finite element approximation that is C 1 continuous. The 16 degrees of freedom associated with this element are the values of u, du/dx, du/dy, d2u/dxdy, at each corner of the quadrilateral. This element has been used by Marchal and Crochet [383] in viscoelastic flow simulations. Error estimates of the form (5.57) and (5.58) can be generalized to twodimensional problems. See the book of Johnson [310], for example, for the corresponding estimates when fi is partitioned into triangular elements.
5.3.3
Finite elements for viscoelastic flows
The finite element approximation of the equations governing viscoelastic flows is based on a discretization of the weak formulation of (5.5)-(5.7). This is sometimes referred to as the mixed formulation. There are two important issues that need to be addressed in the finite ele ment approximation of viscoelastic flow problems. The first is concerned with the choice of approximation spaces for velocity, pressure and extra-stress (see §7.3.1). Certain compatibility conditions need to be satisfied between these spaces in order to obtain a discrete problem that is well-posed. Although this issue has been studied extensively from the theoretical viewpoint for the velocity-pressure formulation of the Stokes problem the position is not so clear 110
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE for viscoelastic problems. The second issue is concerned with the treatment of the convective term u • V r , particularly for large values of the Weissenberg number. Standard Galerkin methods are well-suited for solving elliptic equa tions for which they possess the best approximation property. However, for convection-dominated convection-diffusion problems, which assume the charac ter of hyperbolic problems, this property is lost. Furthermore, if the solution to the problem is nonsmooth, the Galerkin approximation is polluted by spurious oscillations that are not present in the true solution of the problem. The os cillations appear when the finite element discretization is too coarse to resolve boundary layers and result from a lack of stability of the standard Galerkin fi nite element approach when used to solve convection-dominated problems. The Galerkin finite element may be stabilized using techniques such as SU or SUPG, for example. However, it should be noted that the price of additional stability is often decreased accuracy. Standard Galerkin finite element methods behave in a similar way when the convective terms in the constitutive equation dom inate. Spurious oscillations are generated in the solution resulting in a failure to provide meaningful solutions beyond a certain range of We. Failure to ad dress these issues in the past has had serious repercussions on the robustness of numerical methods. Let V, Q and E denote appropriate function spaces, as yet undefined, for the velocity, pressure and extra-stress, respectively. Then the classical weak formulation of (5.5)-(5.7) is: Find (u,p, r) eVxQxH such that f (/?Vu r : V v - p V - v + r : V v ) dil Jo, [ V uqdn f (T + We¥)
:SdQ
=
0,
(5.67)
=
0,
(5.68)
=
f (1 - 0 ) 7 : S d n , (5.69)
for all v 6 V, q G Q and S £ XL T r e a t m e n t of t h e convective t e r m s In the solution of convection-dominated convection-diffusion problems stability may be accomplished by means of upwinding or SUPG. In viscoelastic flows there is an additional tool at our disposal, viz. a reformulation of the governing equations using a decomposition of the extra-stress tensor and introducing the rate-of-strain tensor or the velocity gradient as an extra unknown. The former can have the effect of introducing a stabilizing elliptic operator into the momen tum equation or increasing the size of the existing elliptic operator. This will be discussed in detail in §7.3.2 and §7.3.3. The most popular approach for stabilizing finite element calculations of vis coelastic flow problems is to use the streamline upwind Petrov-Galerkin (SUPG) method of Brooks and Hughes [103], also known as the streamline diffusion method, in which a different test space is used for the constitutive equation. In the classical weak formulation (5.69) is replaced by /
(T + Wer
- ( 1 - (3)j) : (S + cm • VS) dti = 0,
111
(5.70)
5.3.
FINITE ELEMENT APPROXIMATIONS
for all S e £. The parameter a is typically of the form h/U where U is a characteristic velocity and h is a characteristic length scale associated with the finite element discretization. A number of choices for U have been proposed in the literature, among which are: 1. The magnitude of the velocity at the centre of a biquadratic quadrilateral element [384]. 2. The norm of the velocity [332]. 3. A characteristic speed of the flow [365]. In regions of the flow near singularities or stress boundary layers SUPG may produce oscillatory stress fields. Marchal and Crochet [384] suppressed these oscillations and, in doing so, obtained solutions for higher values of the Weissenberg number using a streamline upwind (SU) method in which the upwind test function is applied to the convective term in the constitutive equation only. In this approach the constitutive equation is augmented by an artificial diffusion term T + Wer
-WeV
■ (emu • V r ) = (1 - (3)j,
(5.71)
the weak form of which is f (r
+ Werj
:Sdn
+
f We(u ■ V r ) : (oca • VS) dfi = f (1 - p)j : S dfl, VS G E.
(5.72)
The added diffusion introduces artificial diffusion in the direction of the stream lines. Although the method appears to be more reliable it is at the expense of accuracy. Crochet and Legat [155] have shown that the extra-stress approxima tion is only first-order accurate. The most serious drawback of this formulation, however, is that substitution of the exact solution into (5.72) will leave a resid ual. Further discussion of this topic will be given in §7.2.1. C o m p a t i b l e a p p r o x i m a t i o n spaces It is well-known that for the classical velocity-pressure formulation of the Stokes problem the corresponding discrete approximation spaces must satisfy a compat ibility condition to ensure that the problem is well-posed. This is the so-called Ladyzhenskaya-Babuska-Brezzi (LBB) condition [239]. For finite elements on a quadrilateral mesh the choice of a quadratic approximation (Q2 — C°) for the velocity and a linear approximation (Qi — C°) for the pressure, for example, satisfies this condition. Theoretical results on existence and uniqueness of the solution to the discrete forms of the weak formulation of the viscoelastic problem (5.5)-(5.7) depend on the approximation spaces satisfying certain compatibility conditions. The general framework for analyzing mixed finite element methods was in augurated by Brezzi [98]. If the approximation spaces satisfy the LBB condition then the underlying discretization is stable. However, for the three fields for mulation of the Stokes problem in terms of velocity, pressure and extra-stress, 112
CHAPTER 5. FROM THE CONTINUOUS T O THE DISCRETE
an additional condition may need to be satisfied between the discrete velocity and extra-stress spaces to ensure a well-posed discrete problem. Evidence for this was first discovered in some numerical results of Marchal and Crochet [384]. They showed that the velocity-pressure and velocity-pressure-stress formulations of the Stokes problem gave different results unless the stress approximation was chosen correctly within the Galerkin formulation. In particular, for the stickslip problem, which possesses a velocity field having an r 1 / 2 singularity on the boundary [506], the latter formulation in which a Qi — C° approximation was used for the stress produced spurious oscillations in the velocity field. The oscillations were reduced by using 2 x 2 linear subelements for the stress and eliminated by using 3 x 3 or 4 x 4 stress subelements (see Fig. 5.7). On the subelements the extra-stress approximation is Q\ — C°.
Figure 5.7: The 4 x 4 subelement of Marchal and Crochet [384] Fortin and Pierre [222] have shown that for the UCM model (/? = 0 in (5.5)(5.7)) a second compatibility or inf-sup condition has to be satisfied to ensure stability of the discretization. A sufficient condition for compatibility is that the extra-stress space contains the gradient of the velocity space. This may be satisfied by choosing a discontinuous extra-stress space when the velocity is continuous. They also showed that the 4 x 4 subelement approximation for the extra-stress [384] resulted in a stable mixed finite element method. However, Baranger and Sandri [42] have shown that if the Stokes problem is derived from (5.5)-(5.7) by setting We = 0 then for any value of /? ^ 0 (in which case the momentum equation retains a diffusion term), the second compatibility condition is not required in order to obtain a stable method. A more exhaustive discussion of compatible approximation spaces is given in §7.3.1.
5.4
Finite Volume Methods
The finite volume method is generally applied to a differential equation or system of differential equations written in conservation form. For example, the twodimensional convection-diffusion equation (5.3) may be written in conservation form as V • (-eVit + bu) = / ,
113
(5.73)
5.4. FINITE VOLUME METHODS provided V • b = 0 or,
du t \ d_ ( e dx {- di + hU) +
a (
du
d-y{-%
\
t
(5.74)
+ b2U
)
The property of conservation of physical quantities, which is preserved by the discrete system, is one of the attractions of the finite volume method. A grid is placed on the computational domain and a control or finite volume is associated with each unknown on the grid as shown in Fig. 5.8. This is the basis of the cell centred finite volume approach. The alternative cell vertex finite volume method will not be described here but interested readers can find details in the book of Morton [410]. Equation (5-74) is integrated over a control volume and finite difference type approximations are used to approximate line integrals over each side of the control volume.
Figure 5.8: A general control volume Integrating (5.74) over the control volume shown in Fig. 5.8 and using the divergence theorem yields
t
du dx
hu
-*Tx),
Each of the integral terms on the left-hand side of (5.75) represents a transport by convection and diffusion through the relevant control-volume face. Central differences may be used to approximate the convective and diffusive fluxes across each face. For example, across the vertical face passing through the point r we have UR
ur —
+ up
du\ • ( dx),
114
UR
-Up
h
'
(5.76)
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE where h is the length of the side of the control volume. This gives rise to the following discretization: Apup -
+ ALuL
ARUR
+ ATuT
+ ABuB
h2
H
fp,
(5-77)
where AR = 1 - l-pu AL = l+
l
-pu AT = l - l-fa, AB = l + ^fo,
AP = AR + A
+A
L
T
+AB,
(5.78) (5.79)
with /?i = bih/e and /?2 = b^h/e. As with the corresponding situation for finite differences the system (5.77) needs to be diagonally dominant to guarantee a bounded solution, i.e. Anb>0,
AP>AR
+ AL
+ AT
(5.80)
+ AB,
where the subscript nb denotes the four nearest neighbouring points to P. This property is satisfied if max{|/Ji | , | f t | } < 2 . If this condition is not satisfied the scheme may still yield a solution but it may not be a physically realistic one. For convection-dominated problems upwind differencing [542] may be used to ensure that the boundedness criterion (5.80) is satisfied. However, a major drawback of the simple first-order upwind difference scheme is that it causes the distribution of transported properties to become smeared, particularly when the flow is not aligned to the grid lines. This lack of accuracy may be remedied using high-order upwind approximations for the convective fluxes. For example, the skew upwind differencing scheme of Raithby [482] and the QUICK (Quadratic Upwind Interpolation for Convection Kinematics) scheme of Leonard [349] were developed with this in mind. It is instructive here to outline the derivation of a high-order upwind scheme such as QUICK. The QUICK scheme is based on the use of a second degree polynomial u(x) = C0 + Cvx + C2x2, biased towards the upstream direction to interpolate the value of the dependent variable at each face of the control volume. If 6i > 0 then, with reference to Fig. 5.9, the values of u at the nodes L, P and R are used to evaluate Co, Ci and GV For the dependent variable ur on the face passing through the point r this gives 'UR
Up +
+
-uP\
fuP-uL\
(xr - Xp) -, _ ('UR -Up
\Xp-XL)
\XR \
XR-Xp
-Xp
(ay -xp)(xT
-XR)
(XL -
(5.81)
XR)
If &i < 0 the values of u at the nodes P , R and RR are used, giving _
=
, / up — UR \ .
UR+ [ \XP-XRJ UR-URR\ XR-XRRJ
.
(Xr ~ Xp) _ (uP-. \X-'. \xp (UP-UR P-XR 115
(ay -xP)(xr (XRR
-xR) -
xP)
(5.82)
5.4. FINITE VOLUME METHODS Thus when the grid is uniform we have „+
=
-(UR+UP)
=
^{UR + UP) - -(uRR-2uR
- -(UR -2UP
+ UL),
+ up).
(5.83) (5.84)
If we define a flow direction velocity switch using (5.85) then the value of the velocity component at the cell interface, ur, may be ap proximated by (5.86) Similar expressions are obtained for ui, ut and u^,. TT
LL
RR
BB Figure 5.9: A control volume for use with the QUICK scheme This approximation leads to a larger stencil (see Fig. 5.9) involving, addi tionally, unknowns at the points TT, RR, BB and LL, and giving rise to the discrete equation ApUp = ^2AnbUnb
H
h2
fp,
(5.87)
nb
where, for example,
AR = l-^(W AL iRR
=
+ \dT + \dr
i+/M^++^r++|r -—
(5.88)
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE Similar expressions may be derived for the other coefficients AT, AB,ATT, ABBThis approximation is third order. A drawback of the scheme is that the main coefficients AR, AL, AT, AB (see (5.88)) are not guaranteed to be posi tive and some of the coefficients ARR, ALL, ATT, ABB are negative. Thus the QUICK scheme is only conditionally stable and therefore may produce oscil latory solutions. Furthermore, special formulae are required near boundaries to maintain the global accuracy of the scheme. There is a tendency for the QUICK scheme to produce overshoot and undershoot values in the presence of large gradients. Gaskell and Lau [227] suggested a modification to this scheme to overcome the unbounded behaviour. The new scheme, which they termed SMART (Sharp and Monotonic Algorithm for Realistic Transport), involves adding a variable parameter to the curvature terms in (5.83) and (5.84).
5.4.1
Finite volumes for viscoelastic flows
Choice of m e s h In Newtonian flow problems a staggered mesh arrangement is generally used (see Fig. 5.3) in which the dependent variables are located at different mesh points - pressure at cell centres and velocity components at the mid-points of cell edges. This type of mesh ensures that the solution is not polluted by spurious modes since the familiar chequerboard mode may be present on a non-staggered mesh. For viscoelastic flow problems the components of the extra-stress tensor must be located on this grid. There are two grids which have been used in viscoelastic flow calculations. These are illustrated in Figs. 5.10 and 5.11. Both configurations possess advantages and disadvantages.
Figure 5.10: Location of the variables on the finite volume grid. All stress components are located at the cell centres. Yoo and Na [642], Sasmal [526], and Xue et al. [636] have used the configu ration shown in Fig. 5.10, in which all the components of the extra-stress tensor are located at the cell centres. This arrangement has the advantage of not hav ing an unknown located at a cell vertex that may happen to be at a geometric 117
5.4. FINITE VOLUME METHODS singularity. For example, in the planar contraction problem some of the stress components are singular at the re-entrant corner. However, the configuration of the unknowns introduces the need for interpolation of the shear stress values in the momentum equations.
Figure 5.11: Location of the variables on the finite volume grid. The shear stress is located at the cell vertices. An alternative configuration is shown in Fig. 5.11 in which the normal stresses are located at the same position as the pressure variables, with the shear stresses located at the corners of the mesh cells. This arrangement has been used by Bevis et al. [65], Gerritsma [228], Mompean and Deville [409] and Bschorer and Brunn [108]. It avoids the need for interpolation of the shear stresses in the momentum equations but risks introducing a shear stress value at geometric singularities. Formulation of the equations If we consider the Oldroyd B model then each of the governing equations (5.5)(5.7) can be cast into the general conservative form
dt
dx \
dx
+ £(**■
dy
S,
(5.89)
where is one of the dependent variables. The coefficients S, 9 and T are defined according to the particular equation under consideration (see Table 5.1), and S$ is the source term which includes all the terms that cannot be accommodated in the convective and diffusion terms. The general equation (5.89) is discretized using the finite volume method ology already described. The discretization yields a set of algebraic equations which, when augmented with the boundary conditions, can be solved to furnish the finite volume approximation.
118
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE Equation w-momentum u-momentum rxx normal stress Tyy normal stress shear stress continuity
u V Txx T
yy
T
xy
1
6 Re Re We We We 0
e
r
Re Re We We We 1
P P 0 0 0 0
Table 5.1: Definition of the coefficients in (5.89) for each of the governing equa tions.
Stabilization techniques In situations where the convective term in the constitutive equation dominates, alternative strategies to upwinding may be used to stabilize the calculations. Xue et al. [636] proposed adding an artificial diffusion term V • {aVr) to both sides of the constitutive equation where a is an artificial diffusion coefficient which has to be chosen so that the boundedness criterion (5.80) for the consti tutive equation is satisfied. The constitutive equation is discretized using the finite volume method in the usual way with the term on the left-hand side being treated at the new time or iteration level and that on the right-hand side at the previous level. Some authors [2,527,616,617], mindful of the computational cost of imple menting stabilization techniques within the finite element context, have exam ined the option of employing hybrid finite element/ finite volume methods for viscoelastic flow problems. Although stabilization techniques are still required they can be implemented far more efficiently than their finite element counter parts. The hybrid method of Sato and Richardson [527] combines a finite ele ment treatment for the momentum equation with a finite volume treatment for the continuity and constitutive equations. A total variation diminishing (TVD) flux-corrected transport (FCT) scheme is applied to the convection terms in the constitutive equation. This scheme eliminates spurious oscillations that arise from the failure to resolve steep gradients which may be present in the flow. The FCT scheme corrects for the excessive dissipation of the first-order upwind scheme by adding a proportion of the difference between the first-order and second-order fluxes to the first-order solution. A flux limiter is used in conjunc tion with the FCT algorithm to ensure that artificial diffusion is only applied locally where necessary. This removes any unwanted overshoots or oscillations that are sometimes associated with the use of second-order methods. Wapperom and Webster [616] proposed a second-order hybrid approach in which the finite volume discretization was applied to the constitutive equation only. Contrary to most other applications of finite volume methods to viscoelas tic flow problems they used a cell-vertex rather than cell-centred approach. They pointed to the recent development of finite volume methods for advection equa tions [374,411,546] as their motivation for adopting this approach. The work by Struijs et al. [546] on advection equations demonstrated that fluctuation distribution can capture steep gradients accurately. Fluctuation distribution is the term used to describe the non-uniform distribution of the flux over a finite volume cell to its vertices prior to updating the approximation. There are a 119
5.4. FINITE VOLUME METHODS number of ways of distributing the flux, some of which give rise to finite volume approximations satisfying certain desirable properties such as conservation, positivity and linearity preservation. Wapperom and Webster [616] found that the last property, which requires that the scheme yields the steady state solution exactly whenever this is a linear function in space for an arbitrary triangulation of the domain, is critical to achieving the goal of second-order accuracy. The semi-Lagrangian method of Phillips and Williams [462] is an alternative approach to the treatment of convection. In this approach the treatments of con vection and diffusion are decoupled and appropriate methods of discretization are applied to each subproblem. At each time step the convection problem dt
+ u • V(f> = 0,
(5.90)
where (j> is either a velocity or stress component, is solved using a particle tracking method. The idea is to determine the position at the previous time step of the particle paths which pass through the vertices of a control volume at the current time. The values of the velocity (u*) and extra-stress (r*) in these deformed control volumes are determined using interpolation from known values on a reference grid at the previous time step. The convection problem (5.90) is solved using a particle-following transfor mation that satisfies
dm
dr(t) dt)(t) = u(£(t),T1(t),T), = v(at),T)(t),T), (5.91) = 1, dt dt dt where £ and r\ are the spatial variables and T is the temporal variable. This transformation is used to determine the departure points at time tn of the vertices of a control volume C»j at time tn+i. The departure points form a distorted cell C?J (see Fig. 5.12). Runge-Kutta methods can be used to solve (5.91) to determine the departure points.
Figure 5.12: The formation of the departure cell C\j using the particle-following transformation (5.91) to determine the vertices of this cell. The next step in the semi-Lagrangian process is to determine the values of the unknowns in the cells Cfj from the values in the cells Cij at time tn. This 120
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE is accomplished using an area-weighting technique (see the papers of Morton et al. [412] and Scroggs and Semazzi [531], for example). For example, 0*™ may be determined using the first-order area weighting scheme
+% = £ -i^f'
(5 92)
-
where ui^j is the area that is in both C8*" and Cr,j, Ax and Ay are the lengths of the horizontal and vertical sides of the cells, respectively, and Z is the set of indices of all control volumes associated with <j> in the computational domain. The delicate part of this calculation is determining how the cell Q"J intersects with the control volumes on the underlying reference grid at time tn and then to perform an area weighting based on the amount of overlap. Full details of this procedure can be found in the thesis of Williams [631]. This scheme possesses the property that the conservation properties of the pure convection problem are preserved exactly if the boundaries of each control volume CIJ remain inside the computational domain from time tn to £n+i> since
£ w / / = area of C / , j .
(5-93)
i,j€Z
Higher-order area-weighting schemes have been derived by Phillips and Williams [463] for linear advection equations.
5.5 5.5.1
Spectral Methods Spectral methods in one dimension
The methods of discretization we have described so far are examples of local methods. The discrete equations generated involve relationships between local grid unknowns. Spectral methods, on the other hand, are global methods and the discrete equations generated involve relationships between the unknowns that are not necessarily local. The major advantage of spectral methods over other methods of discretization is that they can generate high-order approxima tions to the solution of differential equations. If a function u and its derivatives are periodic on the interval (0,2ir) then it is natural to expand it using a Fourier series. Formally, we write oo
Su(x)
=
y ^ Ukexpiikx),
(5.94)
k=—oo
where the Fourier coefficients, Uk, are given by 1 f2* Uk = — / u(x) exp(—ikx) dx, k = 0 , ± 1 , ± 2 , . . . . 2^ JO
(5.95)
If, in addition, the function u is smooth, i.e. infinitely differentiable, then the fcth coefficient Uk decreases exponentially, i.e. faster than any finite power of fc_1 as fc -> oo (see the monograph of Gottlieb and Orszag [246]). If the function to be expanded is not periodic then the eigenfunctions of a singular Sturm-Liouville problem are appropriate as basis functions since the 121
5.5. SPECTRAL METHODS rate of decay of the expansion coefficients is determined solely by the smooth ness of the function and not by any special conditions satisfied by the function at the boundary. The Chebyshev and Legendre polynomials are examples of eigenfunctions of singular Sturm-Liouville problems. The general Sturm-Liouville problem on the interval (—1,1) is -(p(x)u'{x))'
+ q{x)u{x) = \w(x)u(x),
x 6 (-1,1) ,
(5.96)
together with suitable boundary conditions. The coefficients p{x), q(x) and w(x) are real-valued functions such that: p(x) is strictly positive, continuously differentiable on (—1,1) and continuous at x = ± 1 , q(x) is non-negative, continuous and bounded on (—1,1) and w(x) is non-negative, continuous and integrable over (—1,1). The Sturm-Liouville problem is singular if p(x) vanishes at x = ± 1 . This property ensures that the expansion of an infinitely differentiable function in terms of the eigenfunctions of (5.96) converges with spectral accuracy, i.e. the coefficients with respect to this basis of eigenfunctions decay faster than alge braically. The Jacobi polynomials Pj? (x) are the only polynomial eigenfunc tions of the singular Sturm-Liouville problem [120]. These are the eigenfunctions of (5.96) with p(x) = (l-x)l+a(l+x)1+/3,
for
a,/3>-l,
q{x) = 0, w{x) = (1 - x)a(l + xf
.
Important special cases are obtained for a = f3 = 0 (the Legendre polynomials) and for a = 0 = —1/2 (the Chebyshev polynomials). The Jacobi polynomials satisfy the orthogonality property f
P<£'0)(x)Pia>p)(x)w(x)dx
= O,
form^n,
and may be generated using a three-term recurrence relation. We now consider the application of the spectral approach to the solution of the problem (5.1)-(5.2) using expansions based on Legendre polynomials L*k(x) = P^ ' (x) for which the weight function is w(x) — 1. The solution may be approximated by Legendre polynomials in two ways, using expansions based on either Lagrange interpolants or orthogonal polynomials. Lagrange interpolant expansions Here the solution is expanded in terms of the Lagrange interpolants based on the Gauss-Lobatto Legendre points: N
UN(X) = ^Uihiix),
(5.97)
i=0
where hi{x)
= -
W
(l-x2)L'N(x) + i)JM*,)(*-*i)' 122
(5 98)
-
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE are Lagrange basis functions. The points Xi, i = 0 , . . . , TV, known as the GaussLobatto Legendre points, are the zeros of (1 —x2)L'N(x), and Ui is the approx imation to u(xi). The unknown nodal values ttj, i = 0 , . . . , TV, are determined using either collocation or Galerkin methods. In the collocation method satis faction of the differential equation is enforced at certain points in the interval (—1,1). For example, in order to solve (5.1) by the collocation method the unknown nodal values, Ui, i = 1,... ,N — 1, are determined by solving -eu'j,(xj)
+ bu!N(xj) = f(xj),
j = l,...,N-l,
(5.99)
where u0 = ui and «jv = UR. Thus, we arrive at the matrix problem An = b ,
(5.100)
where A is the (TV - 1) x (TV — 1) matrix with entries
A j t k = -eDfl
+ bDi>k,
and bj = f(xj) - A,,oWo - -AJ.JVUJV- The entries of the (TV +1) x (N +1) matrices D and D^ are given by Diti
D\2] =
= h'jixi),
h'jixi).
To construct the Galerkin approximation for this problem we define
Vg = {v e TN(I) : v(-l) = «i,u(l) = uR}, where P ^ ( 7 ) is the space of polynomials of degree at most TV on I. The Galerkin approximation is then determined by solving the discrete weak problem: Find UN 6 Vg such that e(u'N,v'N)N
+b(u'N,vN)N
= (f,vN)N,
V%ePw(/)nH5(7).
(5.101)
The discrete inner product (•, -)JV is defined by N
(,-0(ar 3 -Ms;).
(5-102)
j'=o
where the weights, given by
^ = TO)WJ' = °
"•
<5 103
' >
are such that the quadrature rule -1
/
N
4>(x) dx = ^Wj^jxj),
(5.104)
is exact whenever cj> is a polynomial of degree 2TV — 1 or less. If we use the interpolants hj(x),j = 1 , . . . , TV- 1, as test functions in (5.101) we obtain the matrix problem (5.100) where now A is the (TV - 1) x (TV - 1) matrix with entries N A
i,k
= ^i»iflijA,* + i=0
123
K % i
5.5. SPECTRAL METHODS and bj = Wjf(xj) — Aj^uo — AJ^UN- The matrix D is known as the Legendre pseudospectral differentiation matrix. Explicit expressions for the entries of D can be found for the important sets of orthogonal polynomials. For the Legendre polynomials we have 1
LN(XJ)
(XJ -xk)
LN(xk)
3 ± k,
>
0, D
j,k
=
<
l<j N(N + l) 4 '
=
k
j = k = 0,
N(N + 1)
j = k = N.
See Solomonoff and Turkel [539] for the derivation of these entries. Orthogonal polynomial expansions An alternative representation of the solution is in terms of an expansion in Legendre polynomials: N
uN(x)
=
y^UjLj(x),
(5.105)
i=0
where in are the Legendre coefficients of UN- Similarly, we may write N-l
N-2
fl 1)l ar
U
"Wo*) = Yl * '*( )»
X
'N( )
fc=0
= J2 fi*2)L*(a:)»
(5.106)
fc=0
in which coefficients outside the range of the summation are set to zero. The Legendre coefficients of u'N are related to those of WJV by i(1)
u
k-l
(2k - 1)
(2k + 3)
= Uk, k = 1 , . . . ,N.
(5.107)
This recurrence relation has the solution N
4 1} = (2k+i) 52 up-
(5.108)
P=k+i p+k odd
2 ) of u'^ are related to Similarly, one can show that the Legendre coefficients u„-,( k those of UN by
N
4
2)
= (* + i/2) Yl
b(p+1)-*(* + ! ) ] v
p=k+2 p+k even
124
(5.109)
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE Substituting (5.106) into (5.1) and using the orthogonality properties of the Legendre polynomials we obtain -eu£ 2 ) + bu(k1] = fk,k = 0,...,N-2,
(5.110)
where fk =
f(x)Lk(x)
dx.
This system is augmented by the boundary conditions and can be written in terms of the Legendre coefficients Uj, i = 0 , . . . , N, of ujy. As for the finite element method we can derive a bound on the error using ap proximation theory. Let TTNU be the interpolating polynomial of degree N that matches u at the N + 1 Gauss-Lobatto Legendre points Xi, i = 0 , . . . ,N. Then Bernardi and Maday [60] have derived the following bounds on the interpolation error. For u 6 H"(I) for some s > 1, II" - 7rjv«|lz,2(j) < CiN-s\\u\\H>{I),
\u - irNu\Hi{i)
<
C2Nl-a\\u\\H,{i), (5.111)
where C\ and Ci are positive constants independent of N. Therefore, using the estimate (5.46) we are able to derive the following error bound for the Galerkin approximation \u - uN\m{I)
< CM1'* (l +h-N-1) \\u\\Hs{I),
(5.112)
where C is a constant. This bound shows that the rate of convergence of the spectral Galerkin approximation is only limited by the regularity of the solu tion. As the order, N, of the approximation is increased the approximation converges exponentially. The same comments can be made about this bound in the presence of convective terms as for the finite element estimate (5.57). If the convective term in (5.1) is dominant then, as one would expect, oscil lations are present in the spectral approximation since a Gibbs phenomenon oc curs at x = 1 if e/b is very small compared with 1/N. Techniques for stabilizing spectral methods will be discussed shortly in the context of the two-dimensional problem (5.3)-(5.4).
5.5.2
Spectral methods in two dimensions
Grid generation In two-dimensional rectangular domains tensor product expansions may be used to represent the solution. For example, the two-dimensional analogues of (5.97) and (5.105) in the domain [—1,1] x [—1,1] are JV
UN(X,V)
N
= ^2^2ui,jhi(x)hj(y),
(5.113)
i=0 j = 0 N N
uN(x,y) = ^^UijLi^Ljiy), i=0 j=0
125
(5.114)
5.5. SPECTRAL METHODS respectively. In order to apply spectral methods to computational domains which depart from simple rectangular geometries one can map the physical domain onto a more standard computational domain using a suitable coordinate transformation. However, this is not always possible or indeed advisable in which case one must resort to domain decomposition techniques before following the mapping strategy. The problem is to find a transformation which maps an arbitrary physical domain onto the reference square on which expansions of the form (5.113) and (5.114) may be defined. Specifically, we are concerned with continuous trans formations F which map the parent or reference square D = [—1,1] x [—1,1] one-to-one onto a simply connected, bounded region Q in H 2 . Such mappings D —> R 2 are equivalent to the introduction of a curvilinear coordinate system on fi. We assume that the determinant of the Jacobian, \J\, of F is finite and non-zero and that F maps the boundary of D onto the boundary of il. The generation of a grid on a curved quadrilateral may be accomplished by means of a number of techniques. However, they are all based on establishing a relationship between points (x, y) in the physical domain and points (f, r)) in the reference square. If the computational domain is relatively simple an orthogonal or conformal transformation can be used. For coordinate systems which satisfy this property the transformed equations simplify and the grid does not suffer from distortion. The transformation is obtained by solving a system of elliptic partial differential equations in which the generalized coordinates (£, rj) are the dependent variables and the physical coordinates (x,y) are the independent variables (see [576], for example). An elegant alternative is to use the transfinite mapping techniques intro duced by Gordon and Hall [245]. This mapping is expressed in terms of the parametric representation of the boundaries. This technique was first used in the context of spectral methods by Schneidesch and Deville [529] in conjunction with domain decomposition methods. The idea is to construct a univalent map ping U : D -> fi which matches F on the boundary. Suppose that the four sides of 0 can be defined parametrically by F(— 1,TJ), F(1,T?), F(£, - 1 ) and F(£, 1) for — 1 < £, r) < 1. A function U which interpolates F in this way is known as a transfinite interpolant of F . To obtain an exact fit to the boundary segments of ii the approximate mapping takes the form
V&n)
=
<M£)F(-M)+<M£)F(M) +*ifa)F(&-l)+&fo)F(£>l) -MQMTIM-I, -i) - MOMvM-h -h(OMri)F(l,
-1) -
fc(Ofcfa)F(l,
where <j>i and <£2 are so-called blending functions. interpolant they are Ms)
= \(l-s),
i) 1),
(5.115)
In the case of a bilinear
0 2 ( s ) = i ( l + «).
(5.116)
It is impossible to prove analytically that mappings of this form are univalent. However, this property can be checked computationally by showing graphically that grid lines in Q corresponding to lines £ = constant in D do not intersect anywhere and that the same is true for lines r) = constant in D. Alternatively, the determinant of the Jacobian, J, can be evaluated at a set of discrete points. For the mapping to be univalent, | J | must be finite and non-zero. 126
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE In the situation that the four sides of Cl are exactly parametrizable by poly nomials of the same order as the spectral approximation we obtain what is known as an isoparametric mapping. This approach is used widely in the finite element method and has been used in conjunction with the spectral element method by Korczak and Patera [335]. Stabilization techniques Spectral methods for solving the convection-diffusion equation (5.3) may be stabilized in a number of ways. For example, Pasquarelli and Quarteroni [444] proposed a version of the SUPG technique for spectral collocation or Galerkin methods in which the upwinding parameter a assumes the form C & — T77-
Here C is a constant independent of e and N. The value of C may be fixed or allowed to vary at each node on the grid. Although the SUPG scheme improves the stability of spectral approximations the available evidence seems to suggest that the improvement may not be as great as the corresponding situation for finite elements. The likely reason for this is the oscillatory nature of the highorder trial functions that are used in spectral calculations.
5.5.3
Spectral m e t h o d s for viscoelastic flows
Spectral methods have been particularly successful in solving viscoelastic flow problems defined in smooth geometries, such as flow between eccentrically rotat ing cylinders and flow in an undulating tube, for which one would expect them to yield accurate approximations despite the lack of theoretical results available for this class of problems in CFD. The flow between eccentrically rotating cylin ders is considered in §9.3. However, since the flow through an undulating, or wavy walled, tube is not considered later in the chapters on benchmark prob lems, a brief summary of the application of spectral methods to this problem is provided here. The natural coordinate system for the undulating tube problem is cylindrical polars (r, 6, z). The radius of the tube is given by rw{z)
= E U - a c o s ^ J ,
(5.117)
where R is the average radius, a is the amplitude of undulation, and L is the wavelength of the undulation. Pilitsis and Beris [467] solved this problem using two spectral techniques. In the first, the pseudospectral-streamline/ finite difference (PSFD) method, the equations were solved in a protean coordinate system provided by the stream lines and the lines orthogonal to them. In this formulation the stream function, ip, defined by 1 d*l> u = — - w - , v=
I dtp - —,
r oz r or and the coordinate £, which varies along the streamlines, are the independent variables and the cylindrical coordinates (r, z) together with the stress compo nents in the (ip, 9, £) coordinate system are the dependent variables. The approx imation is based on truncated Fourier expansions in the periodic ^-direction and 127
5.5. SPECTRAL METHODS second-order finite differences in the ^-direction. The discrete equations were generated using a collocation procedure. The success of this scheme, which was applied to the UCM model, is due to the technique used to integrate the stresses along the real characteristics. However, this also means that its use is restricted to flows without recirculation since the characteristics of the system of equations are the streamlines. In their second technique, the pseudospectral/finite difference (PCFD) method, Pilitsis and Beris [467,468] solved the equations in a stretched cylindrical coordinate system (r, 6, z) where f = r/rw and so 0 < f < 1. Second-order upwind differencing was used to stabilize the calculations at high De. Fully pseudospectral schemes were developed for this problem by Pilitsis et al. [469] for the ip — w - r formulation and Owens and Phillips [438] for the primitive variable formulation. Both schemes have approximations based on mixed Fourier/Chebyshev representations. The computational domain is the rectangle [0,2n] x [—1,1] formed by the change of variable
L
\rw{z)J
In the computational domain a typical dependent variable, <j>, has the represen tation JV/2-1
M
^2 j=-N/2
^^,fcTfc(i)exp(ijs), k=0
where the coefficients (j>j,k are defined by
+* = j
^ E ~cos ( ^ ) E
^neM-im).
In this representation, 7\ (t) is the Chebyshev polynomial of degree k in t, and CQ = CM = 2, c„ = 1, n = 1 , . . . ,M — 1. Convergence could be obtained by Pilitsis et al. [469] for a large range of Deborah number and the limiting value was higher than that achieved by low-order methods. However, the boundary layers eventually became so steep that even spectral methods were unable to resolve them. Influence of the convective terms A feature of these numerical simulations is that beyond some value of the Weis senberg number, as the flow becomes increasingly convection-dominated, numer ical difficulties are encountered in the form of spurious oscillations. Therefore, care must be exercised in formulating a spectral approximation to viscoelastic flow problems to avoid these numerical instabilities. However, what is observed in many situations is that as the order of the polynomial approximation in creases, the oscillations subside indicating that spectral methods can give rise to stable approximations for convection-dominated problems [558]. In many problems boundary layers may develop in the flow which become increasingly thin as the Weissenberg number increases causing the rate of convergence of the spectral approximation to decrease. Consequently, to obtain stable and exponentially convergent approximations at elevated values of the Weissenberg 128
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE number one has either to increase further the order of polynomial approxima tion or to use some form of domain decomposition technique in an attempt to resolve the boundary layers. In a study of time-dependent Poiseuille flow Sureshkumar and Beris [551] stabilized their spectral method by introducing a stress diffusion term into the constitutive equation. However, in doing so the constitutive equation was effec tively modified.
5.6
Spectral Element Methods
Until relatively recently spectral methods were restricted to the solution of prob lems defined in simple geometries. However, over the last two decades there have been considerable advances in the development of spectral methods in complex geometries. Orszag [431] was the first to suggest ways of extending their range of applicability. His ideas were based essentially on multi-domain techniques in which the computational domain, Q, is partitioned into a number of nonoverlapping regions, fifc,l < k < K, such that U%=1£lk = fl and ftk n fi* = 0 for all k 7^ /. In the majority of applications the partition is assumed to be geometrically conforming in the sense that the intersection of two adjacent subdomains or elements is either a common vertex or an entire edge. However, the development of the mortar element method [62] has provided a framework within which nonconforming discretizations are possible within the spectral ele ment context. Each of the subdomains may be mapped onto the reference square D = [-1,1] x [—1,1] using, for example, the transfinite mapping technique [244]. Spectral multi-domain methods are based on a discretization of the strong form of the differential equation using the collocation technique. Approxima tions are developed on each of the subdomains separately and then patched together using continuity of the solution and possibly derivatives of the solution depending on the order of the differential equation. The continuity conditions along the internal interfaces are enforced pointwise using collocation. This tech nique has been used in the viscoelastic context by Souvaliotis and Beris [541] for flow through an undulating tube and Owens and Phillips [437] for the stick-slip problem. The spectral element method, however, provides greater flexibility. This technique was pioneered by Patera [445] for elliptic problems and the NavierStokes equations. Early theoretical and algorithmic developments of the spectral element method applied to the Stokes and Navier-Stokes problems are due to Maday and Patera [377] and Maday et al. [378]. The method is very similar to the finite element method in that it is the weak formulation of the govern ing equations that is discretized. The method also bears semblance to the p version of the finite element method. The major difference between the two techniques is the choice of trial functions. In the former they are based on Lagrange interpolants whereas in the latter a hierarchical basis is used. For second-order differential equations continuity of the solution is imposed across element boundaries. A popular stabilization technique for spectral element methods is that of bubble stabilization. This technique was devised by Brezzi et al. [99] for stabi lizing finite element approximations for problems of the form (5.3). Canuto [119] showed that a similar technique could be used to stabilize spectral schemes by 129
5.6. SPECTRAL ELEMENT METHODS adding bubbles to the trial and test spaces. Although the method suffers from the problem of choosing appropriate stabilization parameters a variant of the method, employing adjoint residual-free bubbles (see [101,223], for example), overcomes this difficulty. The development of residual-free bubbles provides guidelines for the choice of an optimal bubble space and generates the bubble functions as the solution of local boundary value problems over each element, these problems being related to the original problem to be solved.
5.6.1
Spectral elements for viscoelastic flows
Spectral element methods are more flexible than standard spectral methods since by a judicious positioning of the elements one can, to a certain extent, resolve steep stress gradients that may develop in the flow with increasing elas ticity. This is achieved by concentrating elements in these regions of the flow. Using this methodology the spectral element method [439,590,591] and the p-version of the finite element method [202,557,558,619] have both extended the range of Weissenberg numbers over which converged solutions are obtained on a number of standard benchmark problems by diminishing the influence of numerical errors introduced by stress boundary layers, singularities and the dis cretization of the convective terms. Further efforts to stabilize spectral viscoelastic flow calculations using SUPG have proved to be rather difficult because of the highly oscillatory nature of the high-order trial functions [360]. However, recent work by Chauviere and Owens [130,131] has shown that enhanced stability may be obtained with spectral element methods using SUPG. The test functions for the constitutive equation assume the form hU SAT + JJ N
■ VSjv,
where h is an upwinding factor and U is a characteristic velocity. In the absence of any theoretical results about the optimal choice of upwinding factor for highorder methods, Chauviere and Owens [130,131] suggested choosing h = I/AT2. This choice ensures that the upwinding factor diminishes with p enrichment. Further stability was obtained by solving the constitutive equation elementby-element in which the elements are ordered according to the flow direction. Owens and Chauviere [436] further improved upon their SUPG scheme with the introduction in 2001 of a locally-upwinded spectral technique (LUST). Here, the upwinding factors were allowed to vary within each spectral element. The results of using LUST for the benchmark problem of viscoelastic flow past a single constrained cylinder are presented in §9.1.5 and show that the method is extremely competitive in terms of cost, accuracy and stability with lower-order finite element and finite volume methods for this problem. Compatible approximation spaces For smooth problems the spectral element method provides a velocity approxi mation that is weakly C 1 continuous. This means that the jump in the normal derivative across elements tends to zero as the order of the approximation tends to infinity. Therefore, the use of a continuous extra-stress approximation does not adversely affect the global accuracy. However, for nonsmooth problems such as those containing singularities or stress boundary layers it is wrong to assume 130
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE that the spectral element method will provide a weakly C 1 approximation due to the loss of regularity of the numerical solution. Thus the gradient of the velocity approximation may possess jumps. A continuous extra-stress approxi mation must be able to reproduce them without generating spurious oscillations in the whole computational domain. This cannot be achieved in practice. A source of potential instability is therefore associated with the choice of ap proximation spaces. Here the question is how to choose the finite dimensional spaces Ejv C E, VN C V and QN C Q SO that the resulting discretization is stable. Although we can only address this question for the Stokes problem due to mathematical tractability it seems reasonable to assume that if the approxi mation spaces are not compatible for the Stokes problem then they are unlikely to be compatible for the viscoelastic problem. In order to study the compatibility of approximation spaces for the velocitypressure-stress, or three field, formulation of the Stokes problem we set We = 0 and /3 = 0 in (5.5)-(5.7). Suppose that the velocity, pressure and extra-stress are approximated by polynomials of degree Nu, Np and NT, respectively. The discrete weak form of this problem is: find (TN,IIN,PN) 6 EJV x VN X QN such that O(TJV, SJV) + b(SN, UJV) = 0
&(TJV,VJV) +d(vN,pN)
d{uN,qN)
VSJVEEJV,
= 0 V vN € VN,
,g
l l g
.
= 0 V qN G QN,
where a(; ■) is the symmetric continuous bilinear form on E x E defined by o(r,S) = f r:S(ffi, Ju
(5.119)
and £>(•, •) and d(-, •) are continuous bilinear forms on E x V and V xQ, respec tively, defined by b{r,u) = [ T-.VudQ,
d{u,q) = [ V-ugdfi.
(5.120)
The compatibility between the discrete velocity and pressure spaces has been examined by Maday et al. [379]. They show that if the velocity is represented by polynomials in Pjv(H), then the pressure may be represented in Pjy-2(^) in order to avoid spurious pressure solutions, i.e. Np — Nu — 2. When applied to the velocity-pressure formulation of the Stokes problem this method is known as the Pjv — P J V - 2 spectral method. Recently, Bernardi and Maday [61] have shown that a better error estimate on the pressure may be obtained if the pressure is represented in Pjv-2 H P A N , for a real number A, 0 < A < 1 (see §7-3.1). An important aspect of the discretization of viscoelastic flow problems by spectral methods is the choice of approximation space for the extra-stress tensor. Since the extra-stress fields for a viscoelastic fluid are generally more complex than their corresponding velocity counterparts one would expect that a higherorder approximation is required. Several empirical numerical studies have been performed to ascertain the relationship between NT and Nu. It is clear from 131
5.6. SPECTRAL ELEMENT METHODS these numerical studies that an under-discretization of the extra-stress can re sult in spurious oscillations in the velocity field. Talwar and Khomami [558] have shown for the flow past a square array of cylinders that NT > Nu - 1 is required to avoid these severe oscillations. In fact, they suggested NT = Nu and this has obvious advantages from the point of view of implementation since no interpolation is required between the velocity and extra-stress grids. For the flow through an undulating tube Van Kemenade and Deville [590] have shown that Nu < NT < Nu + 2. They also show that a slightly larger limiting Deborah number is achieved for NT = Nu + 2. However, the small gain hardly seems worth the extra effort. Gerritsma and Phillips [230] examined this question theoretically using the theory of mixed problems developed by Brezzi and Fortin [100]. They showed that for a single spectral element a necessary condition for the discrete problem to be well-posed is NT > Nu for the Stokes problem. This condition ensures that there are no spurious modes in the velocity approximation. Furthermore, they showed that if NT = Nu then optimal error estimates could be derived for the velocity and extra-stress. The convergence analysis requires the satisfaction of a second compatibility condition, this time between the discrete velocity and extra-stress spaces. In particular, there exists a constant C independent of N such that inf
sup
vwGVjvSjveEjv
M*N'*N)1 l|S;v||£||VAr||v7.fCerB*
>C>0,
(5.121)
where KerB* = { v £ F : b(tr, v) = 0 V a € £ } . Gerritsma and Phillips [229] have also examined the compatibility conditions between the Newtonian extra-stress and velocity approximation spaces within a spectral element context. This is a necessary and important prerequisite before examining the impact of constitutive models more complex than the Newtonian one on the compatibility requirements between the discrete spaces. There are a number of reasons why the analysis of constitutive models of Maxwell-Oldroyd type is more difficult than for the Stokes problem. The major reason, of course, is that for these models the systems of governing equations are of mixed el liptic/hyperbolic type rather than totally elliptic as in the Stokes case. The hyperbolic part of these systems requires careful treatment to ensure that the right quantities are convected along the streamlines in an appropriate manner. The choice of the discrete stress space in the spectral element context is motivated in two ways. First of all, by consideration of the properties of the solution of the variational statement of the velocity-pressure-stress formulation of the Stokes problem obtained by setting j3 = 0 and We = 0 in (5.5)-(5.7). Secondly, by looking at an appropriate way of matching the discrete solution between elements to ensure that the global compatibility condition is not vio lated. Since the components of the extra-stress tensor are functions in L2(£l) it seems reasonable to allow the stress to be discontinuous between elements. It turns out that this is also a sufficient condition for the global compatibil ity between the discrete velocity and extra-stress spaces. In fact, suppose that TJV € EJV and u ^ € VN , then since TN
=
VKUJV),
we must have for compatibility that 7(u;v) £ S ^ , 132
\/UN E VN.
CHAPTER 5. FROM THE CONTINUOUS TO THE DISCRETE One way of making this possible is if the discrete stress space is allowed to be discontinuous between elements. It should be noted that the choice of Hermitian elements (giving a continuous 7(ujv)) would also ensure the satisfaction of the compatibility condition but these have only been constructed for rectangular elements. A further discussion of compatibility of approximation spaces may be found in §7.3.1.
133
Chapter 6
Numerical Algorithms for Macroscopic Models 6.1
Introduction
The last twenty or so years have witnessed a veritable explosion of activity in the development of numerical algorithms for solving the algebraic equations arising from a discretization of the governing equations for viscoelastic flow. Most of the endeavour has been devoted to techniques for solving steady and transient problems using differential models and a discussion of these methods is given priority in the present chapter. Differential constitutive equations possess the important property that the evolution of the stress at a given instant in time depends only on the current velocity and stress fields. In this respect differential constitutive equations have a distinct advantage over integral constitutive equations, which require a knowl edge of the entire deformation history of the flow to calculate the stress at a given moment in time. For this reason the study of numerical methods for solving viscoelastic flow problems using integral models has, to a large extent, played a subservient role on the computational rheology stage. However, de spite this disadvantage the integral equation approach to constitutive modelling has some redeeming features. Many of the advances in constitutive modelling over the last several years have yielded stress-strain relationships that are not of differential type. These include the micro-rheological models based on kinetic theory, as well as integral models. Amongst those in the latter class, for exam ple, are the constitutive equations derived from reptation theory that describe the behaviour of polymer melts. There have also been significant developments in the design of accurate and efficient techniques for solving viscoelastic flow problems using integral models. Therefore, the renewed interest in the devel opment of constitutive equations of integral type and the general confidence in their predictive capabilities merit their serious consideration. A discussion of stochastic simulation and Brownian dynamics methods for solving viscoelastic problems using kinetic theory models is delayed until Chapter 11.
135
6.2. FROM PICARD TO NEWTON
6.2
From Picard to Newton
The discretization of the governing equations of viscoelastic flow using a differ ential constitutive model yields a system of nonlinear algebraic equations of the form x " + 1 - x" m
alv ^
j + A(x ) = b,
—
(6.1)
where x = (r, p, u ) T is the vector of unknowns,
I„ =
I
IT 0 0
0 0 0
0 0 Iu
is the diagonal matrix partitioned with respect to the blocks of unknowns r , p and u, and 0 if the problem is steady, 1 if the problem is transient. If m = n then the temporal discretization is said to be fully explicit and if m = n + 1 it is said to be fully implicit. Of course, there are other possible schemes between these extremes in which part of the operator A is discretized implicitly and the remaining part explicitly. Some examples of these will be discussed later in this chapter. The nonlinear operator A may be partitioned with respect to the unknowns associated with each dependent variable. Therefore, in two dimensions it has the following structure A(x)
= \
0 A31
0 A32
A 23 p . A33 ) \ u /
(6.2)
The partition of A corresponds to the discretization of the constitutive, conti nuity and momentum equations, respectively. If m = n then the solution of (6.1) is straightforward requiring a simple computation of terms involving the components of x at the previous time level. However, if m = n +1 the system of nonlinear algebraic equations (6.1) must be solved iteratively. This is achieved by linearizing the nonlinear system and solv ing the resulting linearized equations using either direct or iterative methods. The linearization is usually performed using a Picard type iteration or Newton's method. In the former the system of governing equations is decoupled and the computation of the extra-stress is performed separately from velocity and pres sure. Starting from a known velocity field one calculates the extra-stress using the constitutive equation. The velocity and pressure unknowns are then updated by solving the conservation equations in which the extra-stress contributes to the source term in the momentum equation. This process is repeated to form the basis of an iterative method: ) +Aii(un+1'r-1)Tn+1,r
At
- ) + A32pn+1'r
=
bi-i4l3Un+1'r-1,
A23un+1'
=
b2,
+ A33v.n+1*
=
b3 -
136
A3lTn+l'\
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS where the iteration is performed over r = 1,2,3,... , for each value of n until convergence has been reached. The rate of convergence of Picard-type methods is at best linear. In Newton's method the variables are updated simultaneously by solving the full set of equations. The asymptotic rate of convergence of Newton's method is quadratic. However, convergence is only guaranteed if the initial guess is sufficiently close to the solution. In practice, this is achieved using continuation in the Deborah number in which, starting from the Stokes solution, problems are solved for a succession of increasing values of the Deborah number. This approach often works provided the increments in Deborah number are not too large. In general, Newton's method is more robust than methods based on a Picard iteration, possessing a larger radius of convergence with respect to the Deborah number.
6.3
Differential Models: Steady Flows
Here we are interested in algorithms for solving the nonlinear system of algebraic equations F(x) = A(x) - b = 0.
(6.3)
In the case of a finite element or spectral element discretization of the governing equations for the creeping flow (Re = 0) of an Oldroyd B fluid A(x.) takes the form: ( Eu 0 \ C
0 0 B-r
- D \ B
(
r \ p
,
(6.4)
A ) \ U )
where the subscript u denotes that the matrix E depends on u. Since the inertial term, u • V u , does not appear in the momentum equation when creeping flow is considered, Eu represents the only part of the system (6.4) that is nonlinear. In the present context r , p and u in (6.4) denote the vectors of extra-stress, pressure and velocity unknowns, respectively. Since (6.3) is nonlinear there is no alternative to using iterative methods for its solution. Many solution methods for solving the nonlinear algebraic systems of equations arising from a finite element discretization are based on Newton's method. The quadratic asymptotic rate of convergence is largely responsible for its popularity. The nonlinear system is linearized to obtain J(x)<5x = - F ( x ) ,
(6.5)
where J = ( V F ) T is the Jacobian, i.e. j.
=9Fi(x)
Newton's method for solving (6.3) can be expressed algorithmically as follows: 1. Let r = 0. Initialize the vectors x r and b .
137
6.3. DIFFERENTIAL MODELS: STEADY FLOWS 2. Compute the entries of the Jacobian JT = J ( x r ) . Solve the linear system Jr6xr
= -F(xr).
3. Update the solution Xr_j_i = X r -\~ O X r .
4. Let r = r + 1. 5. Repeat steps (2)-(4) until convergence is obtained. Note that F ( x r ) is the residual vector at the r t h iteration. For large problems Newton's method can be extremely expensive. The most expensive part of the calculation is the construction of the Jacobian and the solution of the linearized system at each iteration (step 2). An economical solution to this problem is to use a modified or approximate Newton method in which the entries in the Jacobian matrix are approximated using finite differences. For example, Burdette et al. [109] approximate the Jacobian at each iteration by one-sided finite differences using Tr
„ ( F i ( x r + Aa: ; ,-e j )-F i (x r ))
where ej is the j t h column of the unit matrix and AXJ is the step length. It remains to choose a solution technique for solving the linearized system (6.5). There are two alternatives. First, direct methods based on an LU decom position of the Jacobian. Secondly, iterative techniques based on methods of conjugate gradient type. Classical iterative techniques such as Jacobi, GaussSeidel and SOR (successive over-relaxation) converge much too slowly to be useful for any practical purposes since their slow convergence would eradicate the benefit of using the quadratically convergent Newton's method.
6.3.1
Direct methods
These methods have been shown to be very robust. Fully coupled methods for solving the linearized equations have generated solutions for large values of the Deborah number. The direct methods that have generally been used are based on a direct LU factorization of the Jacobian and a frontal solver. In this way the full set of equations is solved simultaneously; an approach generally referred to as coupled. Their drawback though is related to the question of efficiency. Even utilizing current computing technology the solution of three-dimensional problems and even two-dimensional problems on refined meshes is extremely expensive. This approach requires substantial memory requirements. This in turn places a restriction on the level of uniform mesh refinement that can be performed. A finite element calculation usually involves the following three phases: 1. calculation of the elemental stiffness and mass matrices; 2. assembly of the global problem; 3. solution of the global problem by Gaussian elimination. 138
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS Direct methods based on Gaussian elimination make use of forward and back ward substitution. If the Jacobian matrix can be decomposed into lower and upper triangular factors, L and £/, respectively, i.e. J = LU,
(6.7)
J6x = - F ,
(6.8)
then the solution of
where F is the residual, may be obtained by solving the triangular systems Ls = F, USx = s.
(6.9)
The matrix J is sparse and the main problem in devising efficient direct methods for large sparse systems is to devise orderings of the equations so that the amount of fill-in is minimized. This is usually accomplished by ensuring that the sparsity patterns of L and U mirror the sparsity patterns of the lower and upper triangular portions of J, respectively. The presence of additional non-zero entries in L and U is known as fill-in. Care needs to be taken to ensure that diagonal entries do not become small during the elimination process. Small pivots result in accumulation of rounding error. This problem can be avoided using pivoting in which the equations are reordered dynamically as the elimination proceeds. Irons [304] proposed a technique for combining phases 2 and 3 of the finite element process known as frontal elimination. The elimination steps for Jitj involves subtracting the quantities T ( 0 T(«)
J
J
j,i
U
r(0 J i,i
'
for I = 1,2,... , min(i — 1, j - 1) and the assembly operations involve adding the quantities for all the finite elements k that depend on the variables i and j . These opera tions can be performed in any order. What is important is that these operations must be completed before row i and column j becomes pivotal. The idea un derlying the frontal elimination method is to eliminate each variable as soon as its row and column is fully assembled. Variables not yet eliminated that are involved in elements that have been assembled constitute the 'front', which separates the region comprising assembled elements from the rest. A full ma trix can be used at each stage to hold rows and columns that correspond to the front. On assembly its order increases to accommodate any new variables. On an elimination, the pivotal row is written to memory and the size of the active matrix is reduced by one. The ordering problem is now associated with the elements and not the variables. Greater efficiency is gained by ordering the elements so as to keep as small a front as possible. In viscoelastic finite element calculations it is found that pivoting is not usually required provided the variables are eliminated in the right order. At the element level, the variables are eliminated in the order - stresses, velocities and 139
6.3. DIFFERENTIAL MODELS: STEADY FLOWS pressures. Furthermore, variables associated with vertices should be eliminated before those associated with mid-side nodes. Failure to follow these guidelines may result in vanishing diagonal terms in the elimination process. One way of reducing the computational requirement is to use a decoupled approach in which the constitutive equation is solved separately from the con servation equations. Although this sort of approach is less expensive (in com putational terms) than the fully coupled approach, experience has shown it to be less robust. Typically, the rate of convergence of the method is slow and a lower limiting value of the Deborah number is found.
6.3.2
Iterative methods
Although Newton's method is robust and has formed the bedrock for many finite element computations of viscoelastic flow problems it is extremely expensive when the system of equations is large. Even for two-dimensional problems direct solvers are prohibitively expensive on the very fine meshes that are sometimes required to resolve the steep stress boundary layers and singularities exhibited by many of the constitutive equations used in numerical simulations. This is due to the cost of assembling and factorizing the global matrix. The resolution of boundary layers requires a fine computational mesh with a corresponding increase in the size of the algebraic system which needs to be solved. In these circumstances iterative methods are essential. Iterative methods do not require the storage or inversion of a large matrix. Instead, the basic computational process is based on matrix-vector multiplications. Many iterative methods exist for solving large systems of linear equations. However, the crucial point is to find the method which is most effective for the problem at hand. A poor choice may lead to slow convergence or even divergence. Let us consider the iterative solution of linear systems of equations of the form Ax = b ,
(6.10)
where A i s a n m x m matrix, and x and b are m-dimensional column vectors. There are two classes of iterative methods: stationary and nonstationary. Clas sical iterative methods such as Jacobi, Gauss-Seidel and SOR are examples of stationary iterative methods. These are based on a decomposition of the matrix A into additive factors. Suppose A — B — C then the general stationary iterative method can be expressed in the form J 3 x f c = b + Cx fc _i,
(6.11)
where B, C and b are independent of the iteration number, k. The convergence of the method depends on the spectral radius, p{B~lC) = max; \Xi(B'1C)\, of the iteration matrix B~1C. If B is nonsingular and p{B~lC) < 1 then it can be shown that the iterative method (6.11) converges to x = j4 _ 1 b for any starting vector xoIterative methods such as SOR have a relaxation parameter at their disposal which may be chosen to enhance the rate of convergence of the method. Al though optimum values of the relaxation parameter can be determined for some elliptic problems such as the Poisson problem, a fairly sophisticated eigenvalue
140
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS analysis of the iteration matrix is necessary in order to determine an appropri ate relaxation parameter for more complicated problems. Therefore, although classical iterative methods are fairly straightforward to implement, more robust techniques are required for many practical problems, including those which arise in fluid mechanics. The conjugate gradient method The conjugate gradient (CG) method is the oldest and best known member of the class of nonstationary iterative methods. In a nonstationary iterative method there is no element of choice in the determination of the iteration pa rameters. They are chosen dynamically at each iteration in such a way as to minimize the error in a certain norm. The CG method was the brainchild of Hestenes and Stiefel [281]. The method was developed for solving symmetric, positive definite systems of linear equations. It converges extremely rapidly when the eigenvalues of A are clustered or lie within distinct clustered groups. The method is based on the idea of projecting an m-dimensional problem into a lower-dimensional Krylov subspace. For a given matrix A and vector b, the associated Krylov sequence is the set of vectors b , Ah, A2h,... The correspond ing Krylov subspaces are the spaces spanned by successively larger collections of these vectors. Let fCn = span{b, Ah,... ,An~lh} be a Krylov subspace of dimension n. The CG method generates a sequence of vectors x n £ fCn with the property that at each step ||e n ||^i is minimized, where ||e||^ =
eTAe.
The CG algorithm is given below. The Conjugate Gradient (CG) Algorithm
1. Choose an initial guess xo and compute r 0 = b — Axo- Set po = r 0 . 2. For n = 1,2,3,... , compute an
=
r£_ 1 r n _ 1 /p£_ 1 J 4p„_i
Xn
— Xn—i -r Q n p n _i
*n
=
r„_i - a „ A p n _ i
Pn Pn
=
r
nrn/rn-lrn-l r„ + PnPn-1
until a convergence stopping criterion is satisfied. The value of the parameter an (the step length) is chosen so that r„ is orthogonal to r n _].. Similarly, the value of /3 n is chosen so that p„ (the search direction) is orthogonal to Apn_i. One can show, by induction, that successive residuals generated by the CG method are orthogonal to all previous residuals in the sequence, i.e. r
n r j = °» 3 < n» and the search directions are A-conjugate, i.e. P n ^ P j = °. 3 < n. 141
6.3. DIFFERENTIAL MODELS: STEADY FLOWS The CG method can be interpreted as a nonlinear optimization algorithm since the solution of the linear system (6.10) is equivalent to the minimization of the functional G(x) defined by G(x) = \-KTAx
-
xTb.
The CG method possesses the property that it converges in at most m steps if exact arithmetic is used. Although this observation is no longer true in floating point arithmetic, the linear systems to which the CG method is applied in practice have coefficient matrices that possess desirable spectra. This means that convergence to a desired accuracy is achieved in substantially fewer than m iterations in many practical situations. If we define the error ek = x — x/. then the following estimate for the error in the energy norm can be derived
||et|U < 2 (^jUJ) ||eo|U,
(6-12)
where re is the condition number of A (see [364], for example). For a sym metric positive definite matrix re is just the ratio of the largest to the smallest eigenvalue. The rate of convergence can be seen to depend on re. Although the condition number is a useful indicator of convergence it does not provide the complete picture since the right-hand side of (6.12) only provides an upper bound for the error. There are two situations in which the estimate (6.12) is pessimistic. The first occurs when the initial guess Xo is A-conjugate to some of the eigenvectors of A. However, in practice components of these eigenvectors may be reintroduced into subsequent iterates through floating point rounding error. The second, which is more common, occurs when some of the eigenvalues of A have multiplicity greater then unity. In these situations convergence of the CG method is rapid, a phenomenon known as superconvergence. Preconditioners As we have seen, the rate of convergence for the CG method depends on the condition number of the coefficient matrix. If the eigenvalues of A are not clustered then the convergence of the CG method is slow. This situation may be improved upon by preconditioning the system with a suitable nonsingular matrix P. The idea underlying the preconditioning philosophy is to transform the original system into an equivalent and better conditioned system P~lAx
= P~lb,
(6.13)
where P is known as the preconditioner. The preconditioner, P, is chosen to be an approximation to A, in some sense, which is easier and cheaper to invert than A with the eigenvalues of P~lA clustered near unity. Ideally the preconditioner should have similar properties to the original matrix and also be sparse so that it is efficient to construct and to store. The trade-off between the cost of constructing and applying the preconditioner, and the expected gain in convergence speed of the iterative method must be borne in mind. The inverse P _ 1 is not constructed explicitly. Instead, systems of equations of the form Py = c, 142
(6.14)
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS are solved. There are, of course, two extreme cases. If P = A then (6.14) is the same (except for a possible change of right-hand side vector) as (6.10), so applying the preconditioner is as difficult as solving the original problem. If P = I then (6.13) is the same as (6.10), so the preconditioning is trivial to apply, but accomplishes nothing since A still needs to be inverted. The transformation (6.13) of the linear system (6.10) is not what is actually used in the computations since P~1A is not necessarily symmetric and positive definite. Instead the preconditioner is decomposed in the form P = QQT and the transformed system written as Q-lAQ-T(QTx)
= Q~lb.
(6.15)
The convergence behaviour of the preconditioned method depends on the spec trum of the matrix R = Q~lAQ~T. A given iterative method is preconditioned in the following manner: 1. Transform the right-hand side vector according to Q~lh
b =
2. Apply the iterative method to the system Rk = b , where x =
QTx.
3. Compute Q~TZ
x =
The decomposition of P given in (6.15) is not needed in practice. The steps of the conjugate gradient method can be rewritten so that the preconditioner is applied in its entirety [242]: The Preconditioned Conjugate Gradient (PCG) Algorithm
1. Choose an initial guess xo and compute r 0 = b — Axo2. Solve P z 0 = r 0 . Set po = z 0 . 3. For n — 1,2,... , compute Oin
=
Xn
=
n-lzn-l/pLl^Pn-l Xn_i+ a„pn_i
rn
=
rn-i
n
=
P-lvn
Pn
=
Pn
=
z
r
T z
-
ftn^Pn-i
/ T
n +
PnPn-1
until convergence stopping criterion is satisfied. 143
6.3. DIFFERENTIAL MODELS: STEADY FLOWS For the PCG method the estimate (6.12) holds for the error but now K is the condition number of R. Since R is spectrally equivalent to P~lA it is sufficient to examine the eigenvalues of P~lA to investigate convergence of the preconditioned version of the CG method. For viscoelastic flow problems the CG method has been used to solve a generalized Stokes problem within a decoupled solution process (see [459], for example). For a specified extra-stress field a finite element or spectral element discretization of the momentum and mass conservation equations leads to a system of algebraic equations of the form (cf. (6.4)) AU +
P = f' (6.16) B u = g, where B is the discrete divergence operator, its transpose is the discrete gradient operator and A is the diffusion operator. The vectors of unknowns now refer to their values at the discretization points or nodes. Block Gaussian elimination yields a symmetric semi-positive definite system for the pressure unknowns
^
Sp = c,
(6.17)
where S = BA~lBT and c = BA~1f—g. The matrix S is made positive definite by imposing a zero volume integral condition on the pressure field. Once the pressure is known the velocity is computed using Au = -BTp
+ f.
(6.18)
The discrete pressure operator S is full due to the presence of A~l. This ne cessitates the use of an iterative method such as CG to solve the system. For the CG method to be effective we need to construct good preconditioners. Maday et al. [376] presented an heuristic argument indicating that the continuous pressure operator is close to the identity operator. Therefore, in the discrete setting S should be close to the variational equivalent of the identity operator, i.e. the pressure mass matrix. Hence, the pressure mass matrix or its diagonal should furnish effective preconditioners for S provided that the element aspect ratios are not too large [261]. In fact, the pressure must be computed using a nested CG iterative method. The inner iteration is associated with the eval uation of the matrix-vector products of the form 5 w in the outer conjugate gradient iteration. To evaluate Sw for a given vector w the following steps are performed: 1. Evaluate t =
BTw.
2. Solve Az = t. 3. Evaluate Sw = Bz. Essentially, this requires the solution of d standard Laplace problems in d di mensions (Step 2). These in turn can be solved using a CG method for which low-order finite element approximations are known to provide effective precon ditioners. An alternative to the Picard approach which decouples the solution of the constitutive equation from the conservation equations is to use a preconditioned GMR.ES iterative method to enforce the coupling. This is a Krylov subspace method for solving nonsymmetric systems of equations. 144
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS The generalized minimum residual method The Generalized Minimum Residual (GMRES) Method, developed by Saad and Schultz [523], was designed to solve nonsymmetric linear systems of equations of the form Ax = b . The principal advantage of the method is that it does not require the assembly or factorization of the coefficient matrix of the system. All that needs to be computed and stored is the residual. The CG method possesses two important properties: the residuals satisfy a minimization property in the A~x norm and the iterates are computed efficiently using a three-term recurrence relation. For general nonsymmetric problems only one of these properties can be satisfied by a given Krylov subspace method. The GMRES method satisfies the minimum residual property. In the GMRES method, the residuals are used to construct a Krylov subspace £ „ , defined by K,n = span{r 0 ,Ar 0 ,>l 2 ro,... , . 4 " - 1 r o } , of dimension n where n is much smaller than the dimension, m, of the system of equations. An orthogonal basis, { q i , . . . , q „ } , for this space is generated using a modified Gram-Schmidt orthogonalization procedure. When this process is applied to the Krylov sequence {vlfcro} this procedure is known as the Arnoldi method [18]. The Arnoldi process is an iteration that transforms a given matrix A to Hessenberg form using orthogonal transformations, i.e. AQ = QH, where Q is an orthogonal matrix and if is a Hessenberg matrix. Since m is large the complete reduction of A to Hessenberg form is not feasible. Instead we consider the first n columns of AQ = QH. The Arnoldi iteration constructs a sequence of Krylov matrices Qn whose columns, the first n columns of Q, span the successive Krylov subspaces fCn. Let Hn be the (n + 1) x n upper left portion of H. Then we have AQn
= Q„+iHn.
(6.19)
The nth column of this equation is Aqn = /i l i n qi -I
1- hn^n
+ hn+i>nqn+i.
(6.20)
This is an (n+l)-term recurrence relation involving all the previously calculated Krylov vectors for determining q n +i- Therefore, we find that, in the absence of symmetry, the generation of an orthogonal basis cannot be done with short recurrences. Instead, all previously computed vectors in the orthogonal sequence have to be retained. In practice, however, the method has to be 'restarted' after a prescribed number of iterations in order to keep storage requirements under control. Once a basis, { q i , . . . , q „ } , for Kn is constructed our problem is to find a vector x n € Kn that minimizes the norm of the residual r n = b — Ax.n. If we write x n = Qny then the least squares problem is to find a vector y that minimizes
\\AQny-b\\. This least squares problem is further transformed using (6.19) to one which seeks the minimum of IIQn+iffny-b||. 145
6.3. DIFFERENTIAL MODELS: STEADY FLOWS Using the properties of the Krylov matrices we can show that Qn+iHny
- b = Q„+i(Hny - ||r 0 ||Q£ + 1 qi) = Qn+i(Hny
- ||r 0 ||ei),
and therefore \\Qn+iHny - b|| = \\Hny - ||r 0 ||ei||, where ei is the first column of the (n + 1) x (n + 1) identity matrix. Therefore, we arrive at the least squares problem which is to find the minimum of ||H„y-|kol|ei||,
(6.21)
for y € R . This least squares problem is solved using a QR factorization algorithm. Computational economies_can be made by using an updating process to compute the QR factorization of Hn from i? n _i instead of constructing the QR factorizations of the successive matrices Hi,H2, H3,... , respectively. The iterates generated by the GMRES method converge monotonically, i.e.
I|r„ + i|| < IknllFurthermore, in common with other Krylov subspace methods, the GMRES method will converge in a number of steps which is no greater than the dimension of the system provided exact arithmetic is used in the computations. This result, interesting as it is, is of no practical value for two reasons: the inability to perform exact arithmetic on finite precision machines, and the desire to obtain convergence in substantially fewer than m iterations when the dimension, m, of the system is large. The computational and storage requirements associated with the GMRES method per iteration increase linearly with the number of iterations. This growth is controlled by restarting the procedure after a chosen number of iterations, k. Once the minimization has been performed over /Cjt, the accumulated data can be cleared, the process restarted with the new residual and the procedure continued until convergence is achieved. The steps of the GMRES(k) algorithm are given below. The GMRES(k) Algorithm
1. Choose xo and compute ro = b — Ax0 and qi = ro/||r 0 ||. 2. For j = 1 , . . . ,k, do: hi,j
=
(4qj,q»),« = 1,2,... ,j,
3
hj+l,j
=
l|qj+l||;
3. Form the approximate solution Xfc =
QkYk,
where y* minimizes (6.21) for y e R*. 146
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS 4. Restart. Compute r^ = b - Ax.k, if satisfied then stop else compute x 0 = x*, qi = rfc/||rfc|| and goto 2. In viscoelastic flow computations Fortin and Fortin [219] have used a precon ditioned GMRES method to enforce the coupling between the computation of the extra-stress from the constitutive equation and the velocity/pressure from the momentum and continuity equations. For a given tensor r , the momentum and continuity equations Au + BTp
=
f-Cr,
(6.22)
Bu
=
g,
(6.23)
represent a steady Stokes problem. The solution of this problem may then be regarded as a function of r , i.e. the velocity u and the pressure p depend on r . Therefore, the discrete form of the constitutive equation -DVL + EUT = 0,
(6.24)
£u(T)T = b u ( T ) ,
(6.25)
may be written in the form
where the right-hand side vector b also depends upon u. This is a nonlinear system of equations for r which can be written in the form G ( r ) = 0. Nonlinear systems of this form can be solved using Newton's method (cf. (6.3)). Within the Newton iteration the GMRES method is applied to the solution of the linearized system J»6T
=
-G(T„),
(6.26)
where J™ = ( V G ) T ( r „ ) is the Jacobian. Rather than compute and store the full Jacobian J " the finite difference approximation jn6r
w
G(T n + MT) - G(T„) h
is used, where h is some small number (typically 1 0 - 3 ) . The iterative process consists of an inner and an outer iteration. The outer iteration is the Newton process while the inner iteration requires the solution of (6.26) using GMRES. The Jacobian J " does not need to be constructed explicitly since only its product with a vector ST is required by the GMRES method. It has to be noted that the solution to a Stokes problem is needed for every inner iteration since the velocity, u ( r ) , corresponding to r = r „ 4- hSrn is required for the computation of J"ST. In order for the method to be efficient and to reduce the number of GM RES iterations an efficient preconditioner needs to be constructed to solve the linear system (6.26). Note, that whereas for some problems the absence of a preconditioner results in slow convergence, here it can result in a lack of con vergence altogether [212]. The theoretical result concerning the convergence of 147
6.3. DIFFERENTIAL MODELS: STEADY FLOWS the GMRES algorithm is dependent on the ability to perform exact arithmetic. In the absence of an infinite precision machine rounding errors are generated that may eventually cause the method to diverge. The choice of preconditioner is delicate and can make the difference between convergence and divergence. Fortin and Zine [212] suggested P — Euo where uo is the best approximation of the velocity field that is available. For example, it may be chosen to be the velocity field obtained at a lower value of the Deborah number. Chauviere and Owens [131] proposed an efficient implementation of this preconditioner for the spectral element method based on an ordering of the elements that allows for the computation of the components of the extra-stress tensor element-byelement. For flows without recirculation Lesaint and Raviart [350] have shown that it is always possible to order the elements according to the propagation of information through the computational domain. As a result of this particular ordering of elements the structure of the resulting global coefficient matrix, Eu, is block lower triangular. By performing and storing LU decompositions of the diagonal blocks the preconditioner, EUn_i: where u„_i is the velocity field from the previous Newton step, may be inverted extremely efficiently. Baaijens [24] has implemented GMRES within a finite element DEVSS/DG framework. The Jacobian matrix for this particular formulation possesses a structure that allows for the construction of an effective preconditioner. The Newton equations (6.5) take the form / Qrr 0 0 ^ Qur
0 0
0 0 0
Qud
Vup
Qdd
( Qdu
f
ST\
fr\
Sd Sp
(6.27)
\SuJ
Vw
for this formulation. The unknowns ST and Sd can be eliminated from this system to obtain a reduced system in the same way that the velocity was elim inated from (6.16) to obtain the pressure equation for the generalized Stokes problem. In particular, we have ST
=
-Q-*(fr
+ QrJvL),
(6.28)
Sd
=
-QddVd + QduSu).
(6.29)
The reduced system formed by inserting (6.28) and (6.29) into (6.27) is o tyup
sv
Vpu Wuu
where fu = fu +
V u r V r r ^cru ~ Wud^idd QUTQTT^T
^idu
Su
(6.30)
+ QudQd^d- We write this more succinctly as J(x)<5x = - F ( x ) ,
(6.31)
where S5£ = (Sp,Su)T is the reduced vector of unknowns. Elimination of the unknowns associated with the extra-stress and rate-of-deformation tensors has reduced the size of the problem enormously. However, the reduced Jacobian, J, is full. Note that the entries in (6.30) do not have to be evaluated explicitly within GMRES since only matrix-vector products of the form J<5x are required within the iterative process. 148
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS To construct an effective preconditioner Baaijens [24] considered the contri butions to the matrix QTT. This matrix, which originates from the linearization of (7.20) with a = 1, comprises two parts: the first corresponding to the jump terms in (7.20) across element boundaries and the second, H say, corresponding to the remaining terms in (7.20). Since a discontinuous approximation is used for the extra-stress tensor, H is a block diagonal matrix with each block associated with a particular finite element. Therefore, contributions to the preconditioning matrix / 0 Qpu \ can be computed on the elemental level and assembled in the same way as the stiffness matrix. Other Krylov subspace methods such as BiCGStab [587] have been used in viscoelastic flow calculations [24,121,583]. The results of Caola et al. [121] for viscoelastic flow past a single confined cylinder are discussed further in §9.1.5. The interested reader will find the work of Tsai and Liu [583] referenced again in §8.3.2. The development of Krylov subspace methods for solving fully coupled viscoelastic flow problems promises to be an active area of research in the future since these matrix-free iterative processes offer tremendous computational sav ings, particularly for three dimensional simulations, over the more traditional direct methods that pervade finite element techniques.
6.4
Differential Models: Transient Flows
A fully implicit discretization of the governing equations yields a temporal scheme that possesses unconditional numerical stability. However, algorithms based on a fully implicit approach require the construction and factorization of large coefficient matrices at every time step if they are to be robust with respect to the elasticity parameter. Thus, compared with conditionally stable explicit techniques, implicit schemes are computationally expensive. Time splitting schemes allow for the possibility of discretizing various parts of the governing equations either explicitly or implicitly at each time step. This introduces a certain amount of decoupling between the equations and indeed be tween the separate terms in the equations. In this way the fully implicit problem at each time step is divided into a number of subproblems, which are solved in sequential fractional steps. Each of the subproblems is a fraction of the size of the fully coupled set of equations. Therefore, this approach leads to a substan tial improvement in computational efficiency, manifested in a reduction in CPU time per time step. The splitting also facilitates the use of specialized solution methods, which may be appropriate for each of the subproblems separately but not for the fully coupled problem. For example, the splitting introduced in the ^-method [524] is based on the character of the momentum and continu ity equations as a generalized Stokes problem for the velocity/pressure and the differential constitutive equation as a hyperbolic equation for the extra-stress. Techniques developed for the solution of transient problems can also be used for solving steady problems. In this case the time step At can be viewed as an iteration parameter and chosen to accelerate convergence towards a steady state solution. The advantage of this sort of approach is that the different operators may be treated in a decoupled approach quite naturally. 149
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS
6.4.1
Projection methods
Projection methods, developed independently by Chorin [139] and Temam [569], are a class of time-splitting methods for solving incompressible flow problems. In this method the governing equations are solved in a sequence of steps by first ignoring the incompressibility constraint and then projecting the result onto the space of incompressible flows. The original method was developed for solving the unsteady Navier-Stokes equations which we write in the dimensionless form Re^
+ Vp
=
F(u),
(6.32)
V u
=
0,
(6.33)
where F(u) = V 2 u - Reu - V u . The continuity equation (6.33) can be differentiated with respect to time to yield V • (du/dt)
= 0.
(6.34)
The construction of the projection method involves the following key steps. First, time is discretized with time step At. Then at every time step F(u) is evaluated and then decomposed into the sum of a vector with zero divergence (du/dt) and a vector with zero curl (Vp). This decomposition exists and is uniquely determined if the initial value problem for the Navier-Stokes equations is well-posed [225]. Finally, the component with zero divergence is used to obtain the velocity field at the end of the time step which, by construction, is divergence-free. In this final step the momentum equation is written in the form Re—
= P(F(u)),
(6.35)
where V is an orthogonal projection operator which projects vectors in i 2 (J7) onto the subspace of vectors with zero divergence in fi satisfying an appropriate boundary condition on 5Q. An explicit version of the projection method was proposed by Fortin et al. [221] in which the convection and diffusion terms in the Navier-Stokes equations are treated explicitly and the pressure is determined so that the velocity at the end of the time step is divergence-free. The scheme may be written as the following fractional step method: Step 1 Re ( U
U
) = Vzun - Reun - V u n .
(6.36)
Step 2 Be(
n A f
j
V-un+1 150
=
-Vp"+\
(6.37)
=
0.
(6.38)
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS A Poisson equation for the pressure may be formed by taking the divergence of (6.37) and enforcing (6.38), i.e. V V
+ 1
= ( g ) v - u \
(6.39)
The velocity boundary conditions are imposed on u n + 1 in step 2. The boundary condition for this Poisson problem is obtained by taking the projection of (6.37) in the direction of the unit outward normal n to the boundary <9ft. This yields the Neumann condition r)nn+1
Tip
if= ~(un+1-u*)-n, on 3ft. (6.40) on /\t The compatibility condition for the Neumann problem (6.39)-(6.40) to possess a unique solution is / V - u* dft = - f ( u n + 1 - u*) • n ds. (6.41) JQ Jan If ft is a bounded domain on which velocity boundary conditions are specified then u n + 1 - n ds = 0,
(6.42) / .an and so the compatibility condition (6.41) is satisfied identically. An important feature of this projection method, which is first-order accurate in time, is that the numerical solution is independent of the choice of boundary condition for the intermediate velocity u*. This is a consequence of the following observations. First, the value of u* at interior points is independent of the value of u* on the boundary due to the explicit nature of the calculation. Secondly, there is a serendipitous cancellation of the value of u* on the boundary, which occurs on the right-hand side of the Poisson problem (6.39) and in the Neumann boundary condition (6.40). Since the first-order projection method (6.36)-(6.38) is explicit the compu tational cost per time step is relatively low. However, there is a price to pay in that the method suffers from a restriction on the maximum time step that can be used to preserve numerical stability. The condition for stability depends on the spatial discretization parameters and the value of the Reynolds number, and is particularly stringent for viscous flows. The time step restriction may be alleviated by the use of implicit schemes. For example, the intermediate velocity u* may be computed implicitly. However, in this case the appropriate choice of pressure boundary condition is not clear. Furthermore, the value of u* on the boundary is not necessarily independent of the value of u* in the interior of ft. Thus, some of the attractive features of the first-order method (6.36)-(6.38) are destroyed. The difficulty associated with the determination of the value of u* on 9ft can be avoided by discretizing the momentum equation implicitly. For example, a first-order implicit projection scheme is: Step 1 /u* - u n \ Re I — ] = -Reun
151
• Vu".
(6.43)
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS
Step 2
V
At
F
J
V •u
(644)
n+1
=
0.
In this scheme the velocity and pressure computations in (6.44) may be decou pled after the spatial discretization using the Uzawa algorithm described earlier. An alternative is the following implicit second-order accurate scheme based on an Adams-Bashforth method for the convection term and a Crank-Nicolson method for the diffusion term. Step 1 Re (
U
* 7 j = -Re ( j u n • Vu" - ^ u " " 1 - V u " " 1 J + 1 V 2 u " .
(6.45)
Step 2
fle(J"+^
U
*)
= -Vp"+i + i v 2 u " + \ (6.46)
,n+l
—
0.
Several projection schemes have been devised which improve on the temporal accuracy of the original Chorin algorithm. Van Kan [589], for example, de veloped a pressure correction method for viscous incompressible flow which is second-order in time and space. The second-order projection methods of Bell et al. [53] were constructed for solving incompressible flow problems in regular domains. A survey of developments in projection methods up to 1990 can be found in the paper of Gresho [249]. The application of projection methods to viscoelastic flow problems follows closely the development of the corresponding techniques for Newtonian flows. However, for viscoelastic flow problems the constitutive equation must also be advanced in time. A generalization of the first-order explicit projection method (6.36)-(6.38) to the governing equations for an Oldroyd B fluid is given by the fractional step method given below. Step 1 J = (1 - j3)jn - r
n
- We(u ■ V r - V u T r -
TVH)".
(6.47) Step 2 Re ( U
7 " " J = /?V 2 u n - Reun ■ V u " + V • r n .
(6.48)
Step 3 Rei
-Vpn+\
=
V
At
F
J
V ■ u"
+1
152
=
0.
(6 . 49)
CHAPTER S. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS In equations (6.47)-(6.49) We denotes the Weissenberg number and the nondimensional viscosity ratio /? is as defined in (5.9). This scheme has been used in conjunction with a spectral spatial discretization for the eccentrically rotat ing cylinder problem [460]. The limitations of explicit methods with respect to stability have motivated the development of a number of mixed explicit/implicit time integration techniques for solving time-dependent viscoelastic flow prob lems. These combine the enhanced stability properties of implicit methods with the computational efficiency of explicit methods. For example, an implicit ver sion of the projection method (6.43)-(6.44) was constructed for viscoelastic flow past a sphere by Owens and Phillips [439]. The ideas underlying projection methods have been employed in the development of a range of time-splitting or fractional-staged schemes for transient viscoelastic flow problems as a means of enforcing incompressibility. This will become obvious as this chapter develops.
6.4.2
The influence matrix method
The influence, or Green's function, technique is a method that has been devised to enforce the incompressibility constraint in a discrete sense when the primi tive variable formulation of the governing equations is used. The technique has been developed for both the finite element method [240] and the tau method [334]. Although it was first introduced for viscoelastic problems by Phillips and Soliman [461], it has formed the foundation of methods developed by Beris and Sureshkumar [58, 551] for the purpose of studying the stability of threedimensional unsteady viscoelastic flows at high Reynolds numbers. In this ap proach the rotation form of the momentum equation is favoured because the spectral element velocity approximation, ujv, arising from a weak formulation semi-conserves kinetic energy for inviscid flows, i.e.
UM
<m = o.
Furthermore, the rotation form is essential for maintaining stability for large Reynolds numbers within a standard spectral collocation scheme [120]. The following time-splitting scheme incorporates an Adams-Bashforth treat ment of the nonlinear terms with a backward Euler treatment of the viscous terms. jfi+l/3 _
u
n
-j^xur-^wxu)"-1)!
A*
+ i
^2B?
un+2/3
_
{(V
"r
r + 1 + (V
" T)"} '
(6 50)
-
un+l/3
=
At 2 3
-VTT,
U«+l - u"+ /
0
At
2Re
(6.51)
{(V2u)"+1 + (V2u)"}.
Here TT is the effective pressure, defined by
1
ftn+"
153
Pdt,
(6.52)
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS
and P = p + | u - u i s a generalized or dynamic pressure that includes a kinetic energy contribution. This pressure is determined by solving a Poisson equation that is derived by taking the divergence of (6.51) and requiring that u™+2//3 is divergence-free. This leads to the equation T7 . . , 7 1 + 1 / 3
The difficulty with the solution of this equation concerns the choice of boundary conditions. This is linked to the continuity equation. It can be shown that V • u n + 1 satisfies the equation (V-u)"+!-
V2(V-u)"+1=0.
^
(6.54)
The solution of this Helmholtz problem is identically zero, i.e. the continuity equation is satisfied identically if and only if V - u " + 1 = 0 at the boundary of the computational domain. Therefore, the appropriate conditions for the pressure are constructed so that V • u " + 1 = 0 on the boundary. In practice this is achieved by solving a number, Nb, of generalized Stokes problems in a pre processing step where Nb corresponds to the number of discretization points, x j , j = 1 , . . . ,Nb, on the boundary. An influence or capacitance matrix, G, is formed from the solutions (UJ,TTJ), j = 1 , . . . ,Nb, of the generalized Stokes problems. The entries of G are given by GiJ=V-uJ-(xJ).
(6.55)
The velocity field at the end of the time step is determined using un+l =
fln+l
+
^
ctjUj,
(6.56)
where a is the solution of the linear system Get = f,
fj = V • u" + 1 (x5).
(6.57)
The construction of the influence matrix can be quite expensive particularly in three dimensions when Nb is large since the method requires the solution and storage of Nb Stokes problems. However, for computationally intensive viscoelastic flow problems the pre-processing step may form just a fraction of the overall effort required for the simulation. It remains to outline the time discretization scheme for the constitutive equa tion in order to complete the temporal scheme. Beris and Sureshkumar [58] used an explicit second-order time integration technique for the time evolution of the Oldroyd B or FENE-CR constitutive models. An alternative method introduces a stress diffusive term into the Oldroyd B model [551]. In this paper it is shown that if the spatial discretization is sufficiently fine then a small enough value of the stress diffusivity, K, can be introduced with the effect of considerably improving the stability of the numerical calculations without distorting appre ciably the flow dynamics. In term of the conformation tensor c, which is related to T by r=
Wi(c_I)' 154
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS the constitutive equation for the modified Oldroyd B model may be written as 1
<9r
j - = - ( u • V ) c + ( V u T c + c V u ) - — ( c - I) + K V 2 C .
(6.58)
If we write F = - ( u • V)c + (VuTc + cVu) - — I , then the temporal discretization may be written as cn+l/2
_cn
At cn+l
_ cn+l/2
At
6.4.3
x
2
(3F n - F " " 1 ) ,
(6.59)
V (K c n + c" + 1 ) + J ( V 2 c " + V 2 c n + 1 ) . 2We ' 2
(6.60)
Taylor-Galerkin methods
Taylor-Galerkin schemes were originally developed for solving problems in com pressible fluid dynamics [185] for which the governing equations are predomi nantly of evolutionary hyperbolic type. The ideology behind the technique is to generate high-order accurate time-stepping schemes that can be coupled to spatial discretizations based on the Galerkin formulation. The derivation of Taylor-Galerkin schemes follows the approach used in the development of LaxWendroff methods in which Taylor series expansions in time are used to generate accurate approximations to the time derivative. Successive time derivatives in this expansion are replaced by spatial derivatives using the original continuous partial differential equations. The resulting semi-discrete equations are then discretized spatially using Galerkin's method. To illustrate these ideas consider the simple linear conservation law
where u is a constant. A first-order approximation in time to (6.61) is —£—
+u-^
=0.
(6.62)
A Taylor series expansion in time provides a more accurate approximation to the time derivative, viz.,
Time derivatives in this approximation are replaced by spatial derivatives using the differential equation. Therefore, a more accurate temporal discretization of (6.61) than (6.62) is
155
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS The second-order explicit scheme (6.64) may be written, equivalently, as the predictor-corrector pair of equations:
=
-At/2-
~Udx
(6 65)
'
-
^ \ ~ ^ = -ud/n+\ (6.66) K At dx ' This formulation is easier to implement than (6.64) for nonlinear conservation laws since it avoids the explicit evaluation of spatial derivatives of the nonlinear flux. Furthermore, the spatial character of these equations is the same as (6.61), unlike (6.64). The discretization is completed by approximating <\> using a finite element representation and applying Galerkin's method to each of equations (6.65) and (6.66). For incompressible flow problems the Taylor-Galerkin approach may be com bined with projection or pressure correction methods to construct efficient com putational schemes that enforce the incompressibility constraint. Van Kan [589] has generalized the projection method of Chorin [139] to second-order for the Navier-Stokes equations. The semi-discrete equations are
eu +
* ( " ir") = (v'u-jteu.vur* -\{Vpn+l V-u
n+1
=
+ Vpn),
0.
(6.67) (6.68)
In this scheme the pressure term has been discretized using the Crank-Nicolson scheme. The treatment of the viscous and convection terms, which are eval uated at the half time step, fits into the Taylor-Galerkin framework we have just described if u n + 5 results from an explicit discretization of the momentum equation from u™ over a time step At/2. This scheme may be written in terms of the following fractional steps: Step la Re\
y
= ( V 2 u - Ren - V u - V p ) " ,
At/2
(6.69)
Step lb
Re[' u * At- u K
n
\ = ( V 2 u - Ren ■ V u ) ) -
T5
- Vpn,
(6.70)
Step 2: V V
+ 1
Tip = | V - U * ,
(6.71)
'- = - V ( p " + 1 - pn).
(6.72)
- P " )
Step 3: Re{-
^
156
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS
The incompressibility of the velocity field at time tn+i is enforced through steps 2 and 3 since u " + 1 determined in step 3 is divergence-free if the pressure differ ence satisfies the Poisson equation (6.71) in step 2. Hawken et al. [277] have generalized this scheme to transient viscoelastic flow problems. The above scheme is modified for viscoelastic flow problems by replacing V 2 u by /3V 2 u + V • r in steps la and lb and including a semidiscretization of the constitutive equation. In the case of the Oldroyd B model, for example, step 1 is augmented by the equations Step lc: We
=
^~K~JT~^'
t(1 -P)i-T~
We u
(
■V
T
-
r V u
- VuTr)]",
(6.73)
Step Id: Wey-
;
= [(1 - 0)
Again this predictor-corrector pair is of the form (6.65) and (6.66). Each of these fractional steps is then discretized spatially using the Galerkin method with the exception of the equations in steps lc and Id which determine r n + 2 and rn+1. These are discretized using the Petrov-Galerkin method [124]. At high Deborah numbers further stabilization techniques are required to obtain converged solutions. Since the velocity gradient appears in the constitutive equation and the finite element approximation of velocity is only C°, discontin uous velocity gradients can cause problems in the computation of the stress in steps lc and Id. For the Stokes problem it can be shown that P2 — C° repre sentations of the velocity and stress are incompatible (see §5.3.3). Matallah et al. [387] proposed a technique that generates a continuous representation of the velocity gradient prior to the computation of the stress in steps la and lb. A continuous velocity gradient was recovered locally by averaging finite element gradient values at the nodes of a triangular element and using quadratic interpo lation on the six recovered nodal quantities on each element. It was found [387] that the scheme that incorporated the recovery of velocity gradients possessed enhanced stability properties in that converged solutions were obtained over a larger range of the Weissenberg number and larger time steps could be used. In steps l a and lc predictions of the velocity and stress are computed at time tn+i from their values at time tn. Steps lb and Id represent an approximation of the constitutive equation based on the mid-point rule and the computation of an intermediate approximation of the velocity. The determination of u* is an explicit computation and results in a velocity field that is not necessarily divergence-free. The pressure difference is computed in step 2 from a Poisson equation with a source term that is essentially the divergence of the intermediate velocity field. In the final step the pressure difference is used to compute a divergence-free velocity field.
6.4.4
The 9 method
Glowinski and Pironneau [241] developed a three-step operator splitting method that decouples the two main difficulties associated with Newtonian flows: the 157
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS nonlinear terms in the momentum equation and the incompressibility constraint. Saramito [524] extended these ideas to the computation of unsteady flows of viscoelastic fluids. The scheme developed by Saramito [524] is second-order accurate in time and decouples the computation of the extra-stress from veloc ity/pressure. The operator A in (6.1) is written in terms of two operators: A{x)=A1(x)
+ A2(x),
(6.75)
which, in the case of the Oldroyd B model, are defined by — «7 V U ], V • r + (1 - a) V 2 u - Vp WT
A1(r,p,u)=\
We
+(1 — o
T
A2(r,p,u)=\
(6.76)
W)T
|,
(6-77)
where w G (0,1) is a parameter associated with the splitting. The temporal approximation is generated using the following scheme from a given initial ap proximation to the velocity and extra-stress at time t = 0: M „n+l-9
M
QM
=
-A2(xn),
(6.78)
=
-Ati***9),
(6.79)
=
-A2(xn+1-e),
(6.80)
_ „n+9
+A2(xn+1-°)
{l_w)At vn+l
M-
+A1(x"+9)
„n+l — 9
^
+ Ax(x" + 1 )
where M is the diagonal matrix defined by M = diag(We, 0, —Re) and 6 € (0,1/2) is a parameter associated with the time scheme. Step 1 comprises the solution of a generalized Stokes problem and an explicit evaluation of the extra-stress. To demonstrate this we write out equation (6.78) in component form: —n+9 _ —n
+ WTn+e - ajn+e
We
= -We r n - (1 - w)rn,
(6.81)
V - un+e = 0,
(6.82)
- Vpn+e
(6.83)
, . 7 1 + 0 _ ,,71
+ V • rn+e + (1 - a)V2un+e
-Re
We can obtain an explicit expression for rn+e from (6.81):
where n x
= (We - (1 - w)GAt)rn 158
- (WeOAt)
rn.
= 0.
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS Therefore, once un+9 is known we can compute r " + 8 using the above equation. Taking the divergence of Tn+e in (6.84) and inserting the result in (6.83) we obtain ~Mn+e 9/\t
- sV2Mn+e + Vpn+e
= g",
(6-85)
where [We-9At(l-w)] ,s = 1 — a{We + wQM)
„ Re n 1 — e = u A , s 9M (We + w6At)
n
V■ v X '
which together with (6.82) comprises a generalized Stokes problem for un+e and In step 2 we determine rn+1~0
and un+1~e
such that
_ n + l - 0 _ _n+0
W&
^ 7 7 - + We Tn+1~e + (1 - w)rn+x-9 = -wrn+e + a7 n+ *,(6.86) (1 — Inj/Xt ..n+1-9 _ „n -Re ( 1 _ 2 g ) A t = - V • r " + * - (1 - a ) V 2 u » + " + V F " + 9 .(6.87) Prom the temporal discretization of the momentum equation in steps 1 and 2 we obtain the velocity field u" + 1 ~ f l explicitly: (6.88) un+i-e = ( ! z i ) u n + * _ i i ^ l u " . u 9 Once un+1~e is known (6.86) becomes a linear convection problem for Tn+1~8. This problem is solved subject to the specification of the extra-stress at inflow, i.e. rn+1-e = T((n + l)M). ^T,
Step 3 follows the same procedure as step 1. The parameters 9 and w are chosen to enhance the convergence properties of the scheme. The optimal values of these parameters can be determined for a linear system of differential equations of the form — = Ax. dt For this model linear problem it can be shown that the choice 0 = 1
7=5 W =
—,
V2 1+ 0' is optimal. This value of 9 minimizes the global error of the linear problem over a time step. These values have been used in the viscoelastic context even though there are no theoretical justifications for doing so. Saramito and Piau [525] have used the 0-method to investigate the influence of extensional properties on flow characteristics in the case of shear-thinning highly elastic fluids in an abrupt contraction. The P T T model was used as the basis of their investigation. The only complication for the P T T model as far as the 0-method is concerned is that the convection problem for Tn+1~9 in step 2 is no longer linear. Sureshkumar et al. [552] used the method to explore the linear and nonlinear dynamics of plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel, both in two dimensions. The Oldroyd B model was used in conjunction with the EVSS-G formulation. in which the velocity gradient is introduced as an additional variable. 159
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS
6.4.5
Lagrangian m e t h o d s
The methods we have described so far in this chapter are based on the solu tion of the governing equations in an Eulerian frame in which the dependent variables are defined on a mesh that is fixed in space and time. An alternative, and arguably a more natural and appropriate approach, for solving the timedependent equations for viscoelastic flow is to use a Lagrangian description of the kinematics of the flow. In this approach the basic independent variables are the material particles and time while the dependent kinematic variables are the spatial coordinates of the particles and the time derivative of the coordinates. Lagrangian techniques for computing time-dependent viscoelastic flows using differential constitutive equations decouple the Lagrangian computation of the polymeric contribution to the extra-stress tensor from an Eulerian treatment of the conservation equations. The components of the polymeric stress tensor are determined by integrating the constitutive equation forward in time along particle trajectories. The polymeric stress tensor is then used to construct the source term in the conservation equations, which are solved on a finite element grid. This grid can either remain fixed in space for all time or move with the material particles. In the latter case the nodes of the finite element mesh are determined by the positions of the material particles. Let {x™ : 1 < i < N} denote the positions of a set of material particles at time tn = nAt. The trajectory, Xj(i), tn < t < i n + i , of each material particle (see Fig. 6.1) is determined using its current velocity, Ui(tn) by solving the initial value problem ^
= m ( t n ) , *(*„) = *?■
(6.89)
This problem can be solved using the forward Euler method x ? + 1 = x? + ufAt,
(6.90)
for example. Alternatively, higher-order methods such as fourth-order RungeKutta methods [226] may be used to solve the initial value problem (6.89) with an intermediate time step Ati„t = At/K, where K > 1 is an integer.
Figure 6.1: Trajectory, Xj(i), of the ith particle. Along the particle trajectory the material derivative reduces to a simple time derivative and therefore the constitutive equation becomes an ordinary differential equation. For example, in terms of the conformation or structure tensor, c, defined by c = (QQ),
160
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS where Q denotes the end-to-end vector of an elastic dumbbell (see §2.6), we have for an Oldroyd B or FENE-CR fluid ft
= -iM(c
- I) + ( ( V u f c + c V u ) .
(6.91)
Recall that /(Q) = 1 for the Oldroyd B model while for the FENE-CR model f(Q) is given by the expression
f(Q) =
1
-(QVQIY
The differential equation (6.91) is solved along each particle trajectory by inte grating forward in time using the value, c n , from the previous time step as the initial condition. Again, the numerical integration may be performed using a fourth-order Runge-Kutta method. For the FENE-CR model the initial value problem (6.91) is stiff due to the presence of the nonlinear spring function / and therefore an implicit method must be used for its solution. In the Lagrangian Particle Method (LPM) [264] the finite element mesh remains fixed in space for all time. The solution of the conservation equations on this finite element mesh provides the approximations u n + 1 and pn+1 using the current values of the conformation tensor c". The conformation tensor is then updated by solving (6.91) along the particle trajectories given by the solution of (6.89). The updated values of the conformation tensor are then used to compute the source term in the momentum equation. These values are defined at material particle points at time tn+i rather than the nodes of the finite element mesh. Therefore, this computation is performed by constructing a linear least squares polynomial in each finite element that fits the available data on the basis of the particles that reside in that element at time i„+i [264]. Obviously, to do this at least four particles need to be located in each element for all time. The method of Harlen at al. [269] also decouples a Lagrangian treatment of the constitutive equation from an Eulerian treatment of the conservation equations. However, the method differs from the LPM since the vertices of the triangles that comprise the finite element mesh are treated as the material particles. Therefore, the nodes of the finite element mesh move in time using their current velocity. Linear approximations are used for the velocity and pressure in each element. Although these give rise to incompatible velocity and pressure approximation spaces stabilization is retrieved by augmenting the continuity equation with a penalty term, i.e. V u
= eV 2 p,
(6.92)
where e is some small parameter. The conformation tensor is updated by integrating the constitutive equa tion forward in time in a frame that deforms with the fluid. Since the upperconvected derivative is the time derivative in this frame the constitutive equation is replaced by the ordinary differential equation
|
= -^(«-D. 161
("»)
6.4. DIFFERENTIAL MODELS: TRANSIENT FLOWS Note that in this frame of reference lower-order terms in c are absent (cf. (6.91)). Piecewise constant approximations are used for the components of the conforma tion tensor over each element. Since a linear approximation is used for velocity each triangle is a material volume. Within a triangular element the vectors p , q along two sides are chosen as the base vectors in the co-deforming frame (see Fig. 6.2). In terms of this basis the components of c and the identity tensor are given by c = R-1c(R-1)T,
I =
J R-
1
I(i?- 1 ) T ,
(6.94)
respectively, where R is the transformation matrix from the co-deforming frame to the Eulerian frame, which in a planar domain is given by R
_
i Px
Qx
Py
Qy
In the co-deforming frame, (6.93) becomes f(RcRT) (c-I). We
dc dt
(6.95)
After solving (6.95) the components of c with respect to the Eulerian frame of reference are determined using c =
R'c(R')T,
where R' is the transformation matrix corresponding to p ' and q', vectors in the deformed material volume as shown in Fig. 6.2. Having determined c n + 1 the conservation equations are solved using standard finite element techniques, for example, in which V • c n + 1 appears as a source term in the momentum equation.
Figure 6.2: Deformation of a material volume over a time step An essential element of Lagrangian techniques is the use of particle tracking techniques for following the trajectories of a fixed set of particles throughout the flow. One of the main difficulties associated with Lagrangian methods concerns the migration of material particles. A set of material particles that is initially well distributed throughout the flow may become concentrated in certain regions of the flow after some period of time leaving other regions bereft of material 162
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS particles. This causes the finite element mesh to become highly distorted and is thus rendered unsuitable for numerical integration. Additionally, for flows with inflow/outflow boundaries particles must be added/deleted as fluid enters/leaves the domain. One method for dealing with the problem of mesh distortion involves remeshing the computational domain from time to time. This method requires the interpolation of variables between the old and the new meshes in which it is necessary to find the position on the old mesh of the new integration points. An alternative method, advocated by Harlen et al. [269], is to retain the nodes as material points but reconnect them in such a way that produces the best triangulation in some sense. For isotropic differential operators the Delaunay triangulation [174], in which the minimum angle in any triangle is maxi mized, is the optimal choice. For a fluid with anisotropic viscosity a modification of the algorithm is necessary in which the triangulation is obtained using a co ordinate stretch to allow the triangles to distort partially with the structure tensor. Gallez et al. [226] modified the LPM of Halin et al. [264] by introducing an adaptive procedure that creates or deletes Lagrangian particles when required. In this Adaptive LPM the number of Lagrangian particles in each element of the mesh is kept within user-specified bounds during the entire duration of the simulation. The creation of particles is achieved by defining a set of reference locations within each finite element. If a Lagrangian particle is not sited close enough to a reference location then a new particle is created at that location. In elements where new particles are created the components of the stress or con formation tensor need to be initialized in order to start the tracking procedure from those points. This is accomplished by computing a P1 — C"1 least squares approximation to the available data within that element. The least squares polynomial is then interpolated in order to determine the required values a t the new particle locations. When the number of particles within an element exceeds the specified maximum number allowed the excess number of particles are deleted. Within an element the closest pair of particles are identified and replaced by a single particle at the mid-point between them with a prescribed average polymer stress.
6.5
Computing with Integral Models
We conclude this chapter by reviewing some of the techniques for solving viscoelastic flow problems using constitutive equations of the integral type. The governing equations are the conservation equations and an integral equation that expresses the extra-stress tensor as a function of strain history. An example of such a relationship is the integral form of the UCM model r(x, t) = - I
M(t~
t')7[0] (x. t, f) dt',
(6.96)
J—oo
where M(t)=(-\^-exp(-t/We),
163
(6.97)
6.6. INTEGRAL MODELS: STEADY FLOWS is the memory function, 7[0](x,t,t')=I-C-
1
(x,t,i'),
is the deformation tensor introduced in (2.151) and C - 1 ( x , t, t') is the Finger strain tensor (see §2.5.1) given explicitly for two-dimensional planar flows by
C-l(x,t,t')
^ / dx \ dx')
/ dx \ \dy')
dx dy dx1 dx'
dx dy V dx'dx'+
dx dy dy'dy'
( dy \ \dx')
dx dy dy'dy' +
( dyx \dy' J
\ (6.98) j
The integral in (6.96) is taken along the particle trajectory that passes through the point x at time t and x' = (x',y') is the position at time t' of the particle that is instantaneously at the point x = (x, y) at time t. We also recall the dis placement gradient tensor E(x, t,t') (see (2.52)), which is related to the Finger strain tensor by CT^M')
= E(x,M')E(x,M')T-
(6-99)
Again, for two-dimensional planar flows we have
E(x,M')
/ dx
dx
dx' dy
dy' dy
V dx'
dy'
\ (6.100) }
More general constitutive equations may be described by modifying the mem ory function in (6.96) and/or replacing the deformation tensor by a prescribed function of the deformation tensor. The Lagrangian treatment of integral constitutive equations such as (6.96) requires the computation of particle trajectories, given a known velocity field. However, the major difference between this approach and a Lagrangian treat ment of differential constitutive equations is that the solution of (6.96) requires the trajectories throughout the past history of the particle instead of over a limited number of time steps. In addition, the strain history of each material particle needs to be computed accurately and stored in order to evaluate the memory integral (6.96). This represents a considerable computational challenge. For transient flows Eulerian and Lagrangian formulations of the governing equations have been used as the basis for numerical simulations. For steady flows an alternative approach can be adopted which uses a mixed Lagrangian/Eulerian formulation. We discuss this further in the next section.
6.6
Integral Models: Steady Flows
In the early development of numerical methods for solving viscoelastic flows us ing integral models a mixed Lagrangian/Eulerian formulation of the governing equations was used. In this formulation the conservation equations and the con stitutive equation are expressed with respect to Eulerian and Lagrangian frames of reference, respectively [63,600]. However, the difficulty with this approach is 164
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS that the velocity, which is a primary variable in the Eulerian representation of the problem, only appears explicitly in the continuity equation. It appears im plicitly in the expression for the deformation tensor 7[o](x, t,t'). Furthermore, the right-hand side of (6.96) must be calculated along particle paths, which are unknown a priori. For steady flows x ' depends on x and time lapse s = t — t' only. Therefore, the integral equation (6.96) may be recast in the form /•OO
r(x,t) = - /
M(*)7 [ 0 ] (x,M-*)<**,
(6-101)
Jo
where 7r0](x, t,t — s) is the deformation tensor at time s relative to the present time t. A number of techniques for solving the steady problem have been proposed (see Viriyayuthakorn and Caswell [600] and Bernstein et al. [63], for example) based on a finite element discretization of the conservation equations coupled with a tracking procedure for determining the stress tensor at the nodes of the finite element mesh on the basis of (6.101). An iterative technique based on the solution for, what is now, an elusive velocity field is employed. The basic method begins with a Newtonian velocity field. This velocity field is used to determine the streamlines by tracking the material particles, x'(xi,s), i = 1 , . . . ,7V, for s > 0. These particles are placed at the nodes of the finite element mesh. The particles are tracked by solving the differential equations dx1 — ^ = u(xj), s > 0 , x{(0) = Xi, i = l,...,N,
(6.102)
where s is the time lapse. The components of the deformation tensor are calcu lated along the streamlines and the components of the stress tensor are estimated at the material points using a quadrature rule to evaluate (6.101). With the change of variable z = s/We, the memory integral (6.101) becomes r(x,i) = - ^ p . J°° exp(-z)y[0](x,t,t-{We)z)dz.
(6.103)
An estimate of this integral may be obtained using the Gauss-Laguerre quadra ture rule, which is appropriate for evaluating integrals on a semi-infinite interval. Thus, we have r(x,t) « _ l i ^ ^ 7 [ 0 ] ( x , i , f - S i V i ,
(6.104)
where zi = Si/We, I = 0 , . . . ,L, are the zeros of the Laguerre polynomial of degree L+l and wj, / = 0 , . . . , L, are the weights. The weights are chosen so that the quadrature rule is exact whenever 7™ (x, t,t — s) is a polynomial of degree 2L+1 in s. To evaluate the stress using (6.104) we need to know the deformation tensor 7™ (x, t, t — s) for s = s/, I = 0 , 1 , . . . , L. This is determined by solving the initial value problems (6.102) numerically by dividing each interval [s;, sj+i]> I = 0,1,... ,L— 1, into an equal number, M say, of subintervals each of length As. The solution of the initial-value problems (6.102) and the determination of the strain history for each material particle are crucial to the success of this approach. The stress is used as a body force in the momentum equation, which 165
6.6. INTEGRAL MODELS: STEADY FLOWS is then solved in conjunction with the continuity equation to obtain an updated velocity field. The process is continued until convergence is obtained. Viriyayuthakorn and Caswell [600] calculated the strain history using the displacement of the nodes of a finite element. Consider the trajectory of a particle that is at the nodal point x, at the current time t. Suppose we have obtained approximations to the position of this particle at times t — (s - As) and t — s. Then its position at time t — (s + As) is calculated using a second-order predictor-corrector scheme (see Fig. 6.3). The position of the particle along the streamline is predicted using x^(s + As) = xj(«) - u;(x'(s))As + iii(x;(s))
As 2 2 '
(6.105)
where the acceleration is given by
As These positions are then corrected using xj(s + As) = x'i(s) -
U i (x'(s))As + Uj(xJ(*))
As 2
(6.106)
where the acceleration is now given by u*(xj(*)) =
Ui(xJ(*))-Ui(xJ(* + A*)) As
The corrector may be applied repeatedly to obtain the position x^(s -I- As) within a specified tolerance. The position of the nodes at a time lapse s = S[ defines a deformed element by means of an isoparametric mapping from which it is a fairly easy task to compute the deformation tensor within an element. The limitation of this approach is that for large displacements, which occur for large values of s, even an exact tracking of the streamlines may not provide an accurate description of the strain history.
X.(s+A S)
Figure 6.3: Determination of a streamline passing through the node Xj of a finite element at the current time. 166
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS Bernstein et al. [63] proposed an alternative approach to the determination of the strain history. Their approach is based on the solution of a differential equation for the deformation gradient tensor, F(x, t, t — s) (see §2.5.1), relative to the present time t: DF{x,t,t-s) JJs
=
_W(B)F(
M
_
g )
F ( x , M - S ) | s = 0 = I,
(6-107)
where K(S) is the transpose of the velocity gradient tensor. Recall from §2.5.1 that the deformation gradient tensor is related to the Cauehy-Green strain ten sor by C ( x , t , t - s ) = FT(x,t,t - s ) F ( x , t , t - s ) . Using a finite element representation in which the velocity gradient is uniform within an element this differential equation can be integrated exactly. In this formulation it is important to keep track of when a streamline leaves one ele ment and enters another. The drawback of this approach is that it is not im mediately applicable to arbitrary finite elements. Luo and Mitsoulis [368] also pursued the approach of integrating (6.107) along streamlines. They used a third-order predictor-corrector method for the determination of the streamlines and a second-order predictor-corrector method for the integration of (6.107). To avoid the accumulation of error due to an inaccurate initial value for F they proposed taking the configuration at the current time as the reference configura tion for every time step. In this way the initial value of the deformation tensor, the unit tensor, is always known exactly. Note that the global deformation gra dient tensor, F(x, t,t'), can be computed using the chain rule to connect local deformation gradient tensors, i.e. F(x, t, t') = F(x, t", i')F(x, t, t"),
(6.108)
where t" is some time evaluation. Goublomme et al. [247] used a parametric equation for the streamlines in terms of the coordinates £, r) in the parent element. A fourth-order RungeKutta method was used to determine the streamlines. If the calculation of a point on the streamline placed it outside the parent element the computation was repeated with a smaller time step until the point lay on the boundary of the parent element corresponding to £,r) = ± 1 . This procedure was used by Goublomme et al. [247] for simulating die-swell using the K-BKZ model.
6.7 6.7.1
Integral Models: Transient Flows Lagrangian techniques
Rasmussen and Hassager [492] have used Lagrangian techniques for comput ing time-dependent viscoelastic flows in axisymmetric geometries using integral constitutive equations such as the UCM model. In their technique, Rasmussen and Hassager [492] followed a set of material particles in time and stored the entire deformation history of these material particles at a discrete set of times. At each time step the nodal positions (xi,yi), 1 < i < N, were determined together with the values of pressure in each finite element. Under the assump tion of creeping flow the motion of the particles is governed by the equation of 167
6.7. INTEGRAL MODELS: TRANSIENT FLOWS motion - Vp + V ■ r
= o,
(6.109)
which in component form is dp dx dp dy
OTXX
!
(jTyx
dx
dy
OTXy
dTyy
dx
'
dy
=
0,
=
0.
Suppose now that a particle that was at position (xo,yo) a t time to is located at position (x(x0,y0,t),y(xo,yQ,t)) at time t. Then the statement of conservation of mass with respect to a Lagrangian frame of reference can be expressed in the form [338]: f dx0 dx det dyo
ctoo ^ dy = 1. dy0
\ dx
(6.110)
dy J
At any given time the positions of the material particles define the nodes of the finite element mesh. This approach is similar to the method of Harlen et al. [269] in this respect. The conservation equations are discretized by introducing finite element approximations for the particle positions and pressure, i.e. JV
M
x(i) = ^ X j ( * ) 0 i , p(t) =
Y^PiXi,
(6.111)
»=i
where fa, 1 = 1,...,N, are the basis functions associated with the material particles and \ii * = 1,■ • • ,M, are the basis functions associated with the M pressure nodes. Within each element a bilinear approximation is used for x and a constant approximation for the pressure. Therefore, Xi is the characteristic function on the iih element. The Jacobian of the transformation from the parent element to the physical element is dy dx J = dx dy d£ dr} <9£ d-q' Similarly, we can define Jo by replacing x and y in the expression for J by Xo and yo, respectively. It can easily be shown that dxp dy0 dx dy
dx0 dy0 dy dx
Jo J'
(6.112)
Thus, the weak form of the continuity equation can be written in the form j
j
( J - Jo)Xk didri = 0,
(6.113)
for k = 1 , . . . ,M. This equation represents mass conservation for each of the finite elements for all time. 168
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS The weak formulation of the momentum equation is obtained by multiplying the equation of motion (6.109) with a bilinear test function 4>j and performing an integration by parts over the domain. This process leads to / pV4>j dxdy — I Jn Jn
TV'4>J
dxdy + / (n-ar)(pjds JdQn
= 0,
(6.114)
where d£ln is the part of the boundary on which a natural boundary condition, in the form of specified tractions, is imposed and n is the unit outward normal to the boundary. The stress from the constitutive equation is now introduced into the second term in this equation to give TV4>J
dxdy = -
Jn
/ M(t-t')f[0](x,t,t')dt'V4>jdxdy. Jn J-oo
(6.115)
The time discretization is performed by keeping the present time t > 0 fixed and choosing discrete points in the interval (—oo,t) according to i_i = — oo, t0 = 0, tT = t and ti-i < U for / = 1 , . . . , T. Therefore, using a backward Euler scheme to evaluate the time integral we have / TV
« - /V]
/
M(t-t')-y[0](x,t,ti)dt'V4>j
dxdy.
(6.116)
This approximation is first-order in time. A second-order version of the method may be obtained by approximating 7r 0 i(x, t, t') by a Taylor expansion about ti to first-order in (t1 - tt) [491,493]. Using the definition of the Finger strain tensor in terms of the displacement gradient tensor we have E(x,M,)TV^- =
(6.117)
VKPJ,
where
„
f
a
a '
MtiYdyiti))'
Therefore, if we define the symmetric matrix C, which is time-dependent, by
then the weak form of the equation of motion is r
r
ptT—i
/ pV4>j dxdy + Jn Jn J-oo
-ffxi Ja
i=i
M(t - t')dt'Vj dxdy
V Cf [' M(t- t') dt') dxdy \i=o
•'''-i
/
+ f
(n • *)4>j ds
=
0,
(6.118)
for j = 1,... ,N, since 7r 0 ](x,i,^T) = 0. Note that this equation is valid for any admissible memory function. It may be simplified for the memory function given by (6.97) for the UCM model. 169
6.7. INTEGRAL MODELS: TRANSIENT FLOWS At each time step there are 2N + M equations (6.113) and (6.118) for the 2N + M unknowns x,, i = 1 , . . . ,N, and pk, k = 1 , . . . , M. The entire defor mation history is stored in the nodal coordinates (xi(ti),yi(ti)), i = 1 , . . . ,N, for all times ti, I = 0 , 1 , . . . , T. At each new time the discrete equations (6.113) and (6.118) are solved using Newton's method in which the computation of the Jacobian is straightforward.
6.7.2
Eulerian techniques
Peters et al. [450] and Hulsen et al. [298] have developed an Eulerian technique for solving integral constitutive equations of the general Rivlin-Sawyers type (2.154) that circumvents the problems associated with mesh deformation. In their approach, which has become known as the method of deformation fields, the deformation history is represented by a finite number of deformation fields that are convected and deformed as the flow evolves. These fields are used to approximate the integral in (2.154) using a finite weighted sum. In the original method of deformation fields [450] the past time t' is regarded as the reference time and B(x, t, t') (the Finger strain tensor of §2.5.1) is the field that measures the deformation of a fluid element currently at position x with respect to the reference time t' in the past. By definition, at the moment of creation the deformation field B(x, t, t') satisfies B (x, *',*')= IThe Finger strain tensor evolves in time according to B ( x , t , r ' ) = 0,
(6.119)
or — B ( x , t, t') = V u T ( x , i)B(x, t, t') + B(x, t, t') Vu(x, t).
(6.120)
The key to the success of the method is that the evolution of B(x, t, f) only requires a knowledge of the velocity field at the current time. This means that one does not have to store velocity fields from past times as one would have to do in a Lagrangian calculation. Instead the information about the flow history resides in the deformation tensors B(x, t, t') with t' < t. Peters et al. [450] proceeded by introducing a cut-off time TC and approximat ing the integral in (2.154) using a quadrature rule involving deformation fields B(x, t, t'j) at a finite number of reference times t't = t — r;, i = 0 , . . . , Nn - 1, in the interval t — TC < t' < t. In terms of the time lapse, r, we have To - 0 < TX < ■ ■ ■ < T i V o - l =
Tc.
Since the memory function usually decreases rapidly for small times the deformation fields for which r is large contribute less to the evaluation of the stress than those for which r is small. Therefore, in the computation of the stress using a time discretization of the right-hand side of (2.154) it makes sense to use small time steps over the recent past and larger time increments over the distant past. To avoid unnecessary interpolation of fields between the discrete time intervals the small time step is taken to be the same as that used in the conservation equations and the larger time step just a multiple of this. At each time t the deformation fields are updated in the following way: 170
CHAPTER 6. NUMERICAL ALGORITHMS FOR MACROSCOPIC MODELS 1. Convect and deform all those deformation fields with reference times t' < t over a time step by solving the evolution equation (6.120) using the current velocity. 2. Destroy the oldest deformation field, i.e. the one with reference time t — TC. 3. Create a new deformation field with reference time t + At. The integral on the right-hand side of (2.154) may be evaluated by approx imating B(x, t,t') using a linear finite element approximation with respect to the time lapse with nodes n, i = 0 , . . . , No — 1, i-e. ND-1
B(x, t, t - T) = Y,
B x
( ; *. * ~ TMT),
(6-121)
i=0
where fa, i = 0 , . . . , Np — 2, are the standard linear finite element basis func tions. The last basis function (J)ND-I is defined to be unity for r > r c to ensure that there is no cut-off error for t < r c . Substitution of (6.121) into (2.154) yields ND-1
T(X,*) = -
Yl
w
i{tPn[o](^t,t~n)
+ V>2j[0](^t,t-Ti)},
(6.122)
i=0
where the weights are given by /-OO
wt =
Af(r)0i(r) dr.
(6.123)
JQ
Note that the weights only need to be computed once and then stored for use at all subsequent time steps. Once the stress is calculated using (6.122) the velocity and pressure at the new time t + At are determined by solving the conservation equations using standard numerical techniques. The evolution equation (6.120) is discretized using the forward Euler method in time and the discontinuous Galerkin finite element method in space in which Py — C _ 1 approximations are used for the deformation fields. Peters et al. [450] found that typically 100 deformation fields were sufficient to represent the deformation history accurately. In a more recent implementation of the deformation fields method, Hulsen et al. [298] abandoned the use of absolute reference times t' in the deformation fields and replaced these with age T = t — t' as the independent variable instead of t'. The authors showed that various drawbacks of the original deformation fields method were overcome and that the new implementation was more stable. The deformation fields technique represents a major breakthrough in the simulation of time-dependent integral constitutive equations. The method has been validated on the benchmark problem of flow past a sphere and excellent agreement is obtained with other methods with respect to the determination of the drag factor. The method can be extended to the wide range of new models that are under development such as the Mead-Larson-Doi model for linear polymers, which includes convective constraint release and tube stretch.
171
Chapter 7
Defeating t h e High Weissenberg N u m b e r Problem 7.1
Introduction
Prom the outset attempts at numerically simulating flows of viscoelastic fluids have been hampered by a breakdown in convergence of the algorithms employed at critical values of the Weissenberg or Deborah numbers. The first manifesta tion of the so-called high Weissenberg number problem was in the late 1970's for calculations of viscoelastic flow using finite difference methods and Galerkin finite element methods. Despite the success that such methods had enjoyed for various Newtonian flows the limitations of their usefulness in the viscoelastic context became clear: converged solutions up to Deborah numbers of only 0(1) proved possible. Thus the successful computation of highly elastic flows exhibit ing interesting experimentally observed phenomena remained elusive. The quest for more sophisticated methods had begun. Decades later much effort is still be ing expended in the search for suitable methods of integration of the constitutive equations and appropriate choices of approximation spaces. Our understanding of the difficult issues in computational rheology and how they may be overcome is now much better than in the late 1970's and it is the purpose of this chapter to summarize some of the significant contributions published since then. The loss of convergence of iterative methods at limiting values of the Weisenberg number is due either to limitations of the model or to numerical ap proximation errors. As mentioned in §3.2 the dearth of existence and unique ness results for viscoelastic flow problems means that one may sometimes un knowingly be attempting to compute an inexistent or non-unique solution to a problem. The sources of numerical error are numerous but include the follow ing. First, those incurred by integrating the coupled nonlinear mixed elliptichyperbolic system of equations governing certain viscoelastic flows using insuf ficiently accurate numerical techniques, such as the Galerkin method, for ex ample (see §7.2). Secondly, numerical oscillations which propagate into the flow domain and ultimately destroy the quality of the solution will also often
173
7.1. INTRODUCTION occur if the approximation spaces for the primary flow variables are not chosen so as to ensure that the discrete problem is well-posed. Typically the requirements for a well-posed problem will include the satisfaction of compatibility conditions between the approximation spaces for velocity and pressure (the "inf-sup" or LBB condition [98]) and velocity and stress [222] (see §7.3.1). Thirdly, inad equate resolution of steep boundary layers due to discretizations that are too coarse, or inaccurate representation of the solution of a flow problem in the re gion of singularities (see §3.5), can be sources of error. Details of computations for benchmark problems possessing singularities and/or steep boundary layers may be found in Chapters 8 and 9. As early as 1982, Mendelson et a!. [401] addressed the need to understand the cause of the De limit. Beginning with a steady two-dimensional family of solutions emanating from De — 0 they classified possible variations in a solution variable X (e.g. pressure drop or drag factor). The first possibility, illustrated in Fig. 7.1(a), is that X is a single valued function of De for all De. The remaining cases deal with so-called irregular points. The first of these (Fig. 7.1(b)) is that at a certain value Decru of De at least two families of solutions cross. Beyond this bifurcation point a solution branch may still be two-dimensional, steady and stable. On the other hand, other branches may exist which are temporally unstable of fully three-dimensional. Although the Jacobian in a Newton's method for the coupled system of equations vanishes at Decrit it may still be possible to continue calculations to higher values of De. However, as remarked by Mendelson et al. [401], what is possibly being calculated by a steady two-dimensional algorithm is a temporally unstable solution, making the computed results meaningless. The next possibility (Fig. 7.1(c)) identified by Mendelson et al. [401] is a limit point (or turning point). Again, the Jacobian in a Newton's method vanishes, although it may still be possible to return along a different branch of solutions at lower Deborah number. Brown et al. [104] and Keunings [324] identified another possible scenario at a limiting value Decru of the Deborah number: a termination point (Fig. 7.1(d)). Beyond Decrn no connected real solution exists. Semi-analytical evidence for limit points in the strong formulation of viscoelastic models exists [340, 402]. Nevertheless, it is very often the case that what is observed at the discrete level as an irregular point at some critical value Decrit of De is, in fact, just a numerical artifact. A true limit point would be expected to manifest itself by occurring at a value Decru to which one can demonstrate convergence with mesh refinement. Critical values of the Deborah number have, however, often been seen to be mesh-dependent. By changing the discrete formulation spurious irregular points can be made to occur at a higher value of De and can even be removed altogether. To substantiate some of the statements made above, we begin by returning to the reference of Mendelson et al. [401]. The authors used a mixed Galerkin finite element method to compute steady flow of a UCM and a second-order fluid (SOF) through a planar contraction, and driven cavity flow of an SOF. Existence of these SOF flows may be shown by the plane flow theorems of Giesekus [233] and Tanner and Pipkin [566]. Huilgol's theorem [295] establishes additionally, uniqueness for the flow of an SOF in the driven cavity problem. Thus, no gen uine limit or termination points were to be anticipated for any De with the SOF in the numerical solutions. For computations with the UCM model, the De limit achieved depended upon the mesh spacing and "spikyness" of the first normal 174
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM
D%ri.
Ehj* D e
De
rxfcri, D e
Figure 7.1: Possible variations in a solution X.
stress difference on the downstream wall, and ultimately resulted in numeri cal breakdown, although no change in sign of the determinant of the Jacobian matrix could be detected. Although the Jacobian matrix in the calculations of Mendelson and co-workers remained non-singular for the SOF calculations through the contraction, oscillations prevented solutions for this problem be yond De = 2.3 being obtained. Since Huilgol's theorem guarantees that the Stokesian velocity field is the unique velocity field for the SOF driven cavity problem, the (unique) elastic stress is determinable explicitly in terms of this velocity field. The numerical calculations of Mendelson et al. [401] could not progress beyond a Deborah number of 0.18, however. Knowledge of the exact solution field enabled the authors to show that several flow features were, in fact, attributable to numerical error. A later paper by Yeh et al. [639] found a limit point at a Weissenberg num ber between 0.6 and 0.8 with associated change of sign in the determinant of the Jacobian matrix in the authors' numerical solution for flow of a UCM fluid through a 4:1 axisymmetric contraction. The location of this limit point was not sensitive to mesh refinement and was conjectured, on this basis, to be an intrinsic property of the UCM model. More recent computations with the UCM model for the 4:1 axisymmetric contraction have been performed by Baaijens [24] up to a Weissenberg number of 5, however, and away from the downstream wall convergence with mesh refinement was observed, thereby casting doubt upon the earlier conjectures of a true limit point at the values seen by Yeh et al. [639]. A couple of years after the work of Yeh et al. Keunings [323] reported calcula tions using the Giesekus and UCM models for viscoelastic flow through a planar contraction. A standard mixed Galerkin finite element method was used. Limit points with both the Giesekus and UCM models manifested themselves. The 175
7.1. INTRODUCTION crucial difference between the two cases was that, whereas the limit point for the Giesekus model calculations was highly sensitive to mesh refinement and thus could be disregarded as spurious, a limit point for the UCM model appeared at a Weissenberg number of about 0.6 on all meshes except for the finest. Although in his 1986 paper Keunings [323] dismissed the UCM limit point as spurious, an alternative explanation was offered by him a little later [324]: perhaps the stress singularity overwhelmed the true turning point of the UCM model for the finest mesh used. Hence the turning points revealed with the coarser mesh calculations were mooted as evidence of a plausible true limit point. However - as one of several examples that we could cite - solutions up to a Weissenberg number of 3 obtained for the 4:1 UCM planar contraction problem by Alves et al. [9] (using a high-resolution MINMOD [271] finite volume scheme), have laid to rest all doubts about the origins of the limit points seen by Keunings [323]. In steady finite element calculations involving viscoelastic flow in a journal bearing having small (e = 0.1) and moderate (e = 0.4) eccentricities, Beris and co-workers [55,56] encountered irregular points. The location of bifurcation points for the UCM model was very sensitive to the choice of mesh and these points were therefore dismissed as spurious. Limit points were also found by the authors in the calculation of SOF flow when the eccentricity was moderate, in contradiction to the theorems cited above for existence and uniqueness for this model. Beris et al. [55] claimed that, since the locations of limit points for the UCM fluid calculations in the journal bearing with small eccentricity were relatively stable to mesh refinement, these limit points were therefore fea tures of the strong formulation of this model. The same authors were obliged to backtrack on this conclusion three years later, however, when their highly accurate spectral/finite element calculations [57] converged to oscillation-free solutions at Deborah numbers 30 times larger than had proved possible in [55]. The Deborah number limit which the authors encountered in [56] in the case of the moderate eccentricity journal bearing flow of a UCM fluid was similarly considered an artifact of the discretization. As a final example where limit points have been demonstrated conclusively to be spurious we refer to a study of circular Poiseuille flow of a Johnson-Segalman fluid [312] by Van Schaftingen and Crochet [593] in 1985. An analytical solu tion showed that, for a solvent-to-polymeric viscosity ratio T)S/TIP < 1/8 and We > 0.3497, infinite stresses and pressure gradients were possible inside the tube. These were expected to cause problems for the numerical simulations. At We = 0.3497 the authors showed analytically that two of the stress compo nents possessed infinite gradients with respect to We at the wall. Despite these analytical observations only one (method A) of the three mixed finite element formulations utilized gave rise to a limit point on the meshes used, even though the same approximation spaces were employed for velocity, pressure and extrastress in all three cases. The limiting values for the Weissenberg number were 0.6707 with 4 finite elements, 0.3451 with 16 finite elements and 0.3285 with 64 elements, and these were evidently spurious. When the pressure was inter polated with discontinuous linear elements in method A instead of continuous first-order polynomials then the limit points were removed altogether. References to these and other papers in the literature, establishing the con nection between approximation error and the onset of oscillations and spurious irregular points may be found in the reviews by Keunings [324], Crochet [150] and Baaijens [25]. 176
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM Having delineated some of the catastrophic consequences of numerical error in the pursuit of stable and accurate solutions to the flow of highly elastic liquids we now proceed to examine some of the sources of these errors in more detail.
7.2
Discretization of Differential Constitutive Equations
A family of constitutive models of the differential type for an elastic stress T may be defined by r + Ai r + f ( r , 7) = 7?P7,
(7.1)
where r\v is a polymeric viscosity, 7 is the rate-of-strain tensor and f is some (possibly nonlinear) function of r and 7 (but not their derivatives), as men tioned in §2.6.1. The upper-convected derivative V has been defined already in Eqn. (3.4). The family described by (7.1) is very broad and includes models of the Maxwell and Oldroyd type, the Phan-Thien Tanner models [456,458] and the Giesekus model [234,235]. We begin our discussion of appropriate numerical methods for viscoelastic flows by noting here that given some velocity field u, (7.1) constitutes a system of first-order hyperbolic equations. As observed by Johnson and co-workers [311] and King et al. [332], amongst others, although centred finite difference methods or Galerkin finite element methods converge for convection-dominated convection/diffusion problems and linear hyperbolic problems, the error estimates are not optimal [311] and these methods may produce highly oscillatory numerical solutions for such problems unless the ex act solution is globally smooth. Similar comments [246] apply in the case of spectral-type methods. We are thus led to consider numerical methods bet ter suited from the point of view of accuracy and stability, for the solution of hyperbolic equations. Two important classes of methods which have enjoyed widespread use in the computational rheology community and of which we now proceed to provide general descriptions, are streamline upwinding methods and discontinuous Galerkin or Lesaint-Raviart [350] methods.
7.2.1
Streamline upwinding - SU a n d S U P G
The unwelcome presence of spurious oscillations generated by use of central dif ference methods or Galerkin finite element methods for convection-dominated convection/diffusion problems has been demonstrated by Brooks and Hughes [103,292]. In these two papers the authors demonstrated, using a one-dimension al advection/diffusion problem, that nodally exact solutions to the central dif ference discretization of the equation could be realized by adding the correct amount of artificial diffusion. Since a Galerkin finite element discretization of the same problem involving piecewise linear interpolations leads to the same cen tral difference approximations it could thus be seen that this Galerkin method under diffused. The problem in extending the idea of adding artificial diffusion to multi-dimensional convection-dominated problems is that spurious crosswind diffusion may result, thus corrupting the solution and destroying accuracy. It was this consideration that led Brooks and Hughes to introduce the so-called streamline-upwind/Petrov-Galerkin (SUPG) finite element formulation. 177
7.2. DISCRETIZATION OF CONSTITUTIVE EQUATIONS Consider, for example, the convection-diffusion problem dp = u ■ Vtp - V ■ (KVtp) = / , infi,
. {
'
for some bounded Lipschitzian domain fi C R d having a boundary dfl. In (7.2) JK" is a constant diffusion tensor, u is a constant velocity vector and / is some source term. The Galerkin weak formulation of (7.2) is: Find ip £ HQ(Q) such that f (u • V
dQ 4- f fw dft, Vw £ H^(Cl).
(7.3)
JQ
Then, integrating the second term by parts and using the summation convention we have,
/ ui^w JQ
oxi
do = - [ Kij^^tm+ JQ
[ fw dfi, Vw G fl£(n). (7.4)
dxj dxi
JQ
If we now add an artificial diffusion term ~
/ '„
JQ
dip dw ~Br~'dx~-
ii
'
onto the second term in (7.4) we may avoid the problem of crosswind diffusion by denning, as do Brooks and Hughes [103], K to be a streamline-upwind artificial diffusivity tensor with components Kij = * T l f ,
(7-5)
for some scalar k. The problem to be solved is then: Find ip € HQ(Q) such that
/ uip-wdsi JQ
oxi
= - f Ki:i^^JQ
dn+ f fw dn, Vw e H^(n),
dxj dxi
(7.6)
JQ
where w is the perturbed weighting function , Uj dw W =W + k TT^TT 7T—. ||U|| OXj
._ _. (7.7)
This is the SU formulation of the convection-diffusion problem (7.2). The SUPG formulation for the solution of (7.2) weights the residual of (7.2) with the perturbed weighting function w so that / (&P - f)w dO = 0.
(7.8)
JQ
Note that this weak formulation is not a Galerkin method since w $ HQ(CI). Furthermore, although there is a streamline diffusion term in the equations the method is consistent in the sense that each term in the residual is multiplied by the weight function w in the weak formulation (7.8). The first application of streamline-upwind methods to the computation of viscoelastic flows was by Marchal and Crochet [384] in 1987. The authors used 178
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM nine-node Lagrangian elements with biquadratic shape functions for the inter polation of the velocity field whilst bilinear elements were used for the pressure and 4 x 4 bilinear subelements for the stresses. The authors used both the SUPG method and an (inconsistent) streamline upwind (SU) method for the UCM constitutive equations. Denoting the approximation space for the stress by T,h the SU formulation used for the constitutive equation was: Find rh £ T,h such that
[ I Th + Xi rh -VPi(uh) J : Sh dn +Ai / (u h ■ V r f t ) : (kwh ■ VSh)
dfl = 0,
VSft e Efc,
(7.9)
where uft is the discrete velocity, w' 1 = u f t /||u / , || 2 and k is a scalar quantity derived as an ad hoc generalization of the formula of Brooks and Hughes [103, 292] for one dimension and proportional in each element to the element size along the direction of flow. Thus the SU method of Marchal and Crochet [384] was similar in spirit to (7.6). The authors performed an integration by parts on (7.9) and neglected the surface integral to obtain f (rh + \x Th -T]pj(uh)
):Shdfl-X1fv-^h-
V T ' 1 ) : Sh dfi = 0, (7.10)
where the artificial diffusivity tensor (3h is given by
u U
k k
Marchal and Crochet [384] found that the consistent SUPG integration of the constitutive equation produced oscillations in the numerical calculation of a stick-slip flow and flow through an abrupt contraction. The reason for the disappointing result was attributed, at the time, to the coupling between the momentum and constitutive equations since SUPG could be expected to give reasonable results when one calculated the stresses on the basis of a fixed velocity field. Hence the authors rejected SUPG as a suitable vehicle for the stabilization of viscoelastic calculations involving differential constitutive models. This was unfortunate since, as pointed out by Crochet and Legat [155] five years later, the real culprits in the poor performance of the SUPG method in the earlier paper were the singularities in the stick-slip and contraction flow problems. Crochet and Legat [155] substantiated their claim by illustrating that the SUPG method was stable and accurate for solving flows of a Maxwell fluid around a sphere in a tube and through a corrugated tube; both smooth geometries. A similarly poor performance (no better than Galerkin finite element elements) of the SUPG method had also been observed by Luo and Tanner [371] in their attempts to solve flow of a UCM and Oldroyd B fluid through a circular contraction using a decoupled finite element scheme. Rosenberg and Keunings [516] too, in a paper dealing with various numerical techniques for the integration of the UCM constitutive model with given kinematics, found that SUPG and SU were 179
7.2. DISCRETIZATION OF CONSTITUTIVE EQUATIONS superior to a Galerkin method for a stick-slip problem but remained inaccurate near the stick-slip singularity. In contrast to the spurious numerical oscillations encountered by Marchal and Crochet [384] with the SUPG method, the SU method (7.9) allowed the au thors to perform calculations at Deborah numbers as high as 27 for the stick-slip problem, for example, with no loss of convergence. Deborah numbers (based upon the upstream wall shear-rate) in excess of 20 and 60 could be reached for the 4:1 planar contraction problem and the 4:1 circular contraction prob lems, respectively. However, the introduction of the SU method as a stable integration scheme for differential viscoelastic constitutive equations has met with strong criticism. Luo and Tanner [371] argued that since the surface in tegral which arises in the integration by parts of (7.9) (and which generally does not vanish when assembled over all elements) is of the same order as the retained volume integral (the artificial diffusion term in (7.10)) it was wrong to neglect it and then argue, as did Crochet et al. [153], that the SU scheme differed from the standard Galerkin finite element method only in the addition of a diffusion term which vanished as the element size reduced to zero. In fact the use of an augmented test function for the treatment of the convective term leads to a modified problem [371,564] and also changes the dominant balance at singularities [332]. Although by using Richardson extrapolation on the results from several meshes Crochet et al. [153] showed that in the limit h — ► 0 good predictions could be obtained with the SU scheme for the flow resistance for vis coelastic flow through an undulating tube, the fact remains that the SU scheme cannot be more than first-order accurate. Using a one-dimensional simplified version of an Oldroyd/Maxwell type constitutive equation, Tanner and Jin [564] showed that the error in the stress gradient could be 0(1) for high Weissenberg numbers and solution errors were predicted to be O (h1/2) and 0(h) for meshes with unequal and equal elements, respectively. In contrast, Johnson et al. [311] showed that the global error for the SUPG method of Brooks and Hughes [103] for their advection/diffusion problem was O (/i fc+1 / 2 ) for degree k polynomial basis functions and that the error incorporated a streamline derivative term, whereas the error for the Galerkin method was half an order lower at O (hk) with no control over the derivatives, except for smooth problems. The results were confirmed numerically for viscoelastic flows by King et al. [332]. In summary, the implications of the above discussion for stabilizing finite element calculations for differential constitutive models of the type (7.1) are clear: both high-order accuracy and stability may be realized for problems in smooth geometries using the SUPG method of Brooks and Hughes [103] and its multi-dimensional extensions [293,294]. For problems with singularities SUPG may smooth the stress in the bulk flow but because the augmented test function collapses to the test function in the Galerkin formulation near a solid boundary there will be oscillations near boundary singularities [332]. In the absence of a complete theory accounting for the asymptotic forms of stress in fluids of 01droyd and Maxwell type throughout a small sector containing the singular point (see §3.5) it may be appropriate to use some modified constitutive equation such as the modified UCM (MUCM) [13,332] or modified Chilcott-Rallison (MCR) models [203] which have Newtonian-like singular behaviour at singularities on non-deformable boundaries. The SUPG finite element method features strongly in the literature and its performance relative to other methods of integration has been the subject of 180
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM several expositions (see [169,564], for example). The SUPG formulation has been particularly successful when applied to the EVSS/AVSS (see §7.3.2) or EEME (see §7.3.3) formulations of smooth problems. Further to the study by King et al. [332], Burdette et al. [109] combined SUPG with the EEME formulation to good effect in order to investigate the flow of a UCM fluid through an axisymmetric corrugated tube. Excellent agreement with the spectral/finite difference calculations of Pilitsis and Beris [467] up to We = 15 were reported. This contrasts with the findings of Debae et al. [169] who, using a combination of SUPG with the 4 x 4 element of Maxchal and Crochet [384], reported numerical breakdown for the same problem for We > 8. Debae et al. [169] found that the combination of the EVSS method of Rajagopalan et al. [485] with SUPG did not, somewhat unexpectedly, yield accurate results beyond We = 5, although by using higher-order interpolations for the stress and rate-of-strain tensor the inaccuracies were overcome. The moral of the story seems to be that the use of SUPG does not guarantee in itself, even for smooth problems, accurate and stable calculations for all mixed methods. Particular mention should be made of the great success of EVSS/SUPG and AVSS/SUPG calculations for the benchmark problem of flow past a sphere in a tube. In 1995 Fan and Crochet [202] experimented with three different SUPG formulations in combination with a high-order (p < 7) EVSS finite element method. The first choice of SUPG formulation considered by Fan and Crochet [202] was similar to that proposed by Brooks and Hughes [103]: the test tensor S^ was augmented to S/i + / i ^ - V S h ,
(7.12)
where h was the element size in the flow direction. However, the magnitude of the upwinding factor may be seen to be non-vanishing in general on sta tionary walls, where the differential equations degenerate into a set of algebraic equations for the stress. Fan and Crochet [202] could not obtain p-convergent results with this choice of upwinding factor. A second possibility, which col lapsed properly to the Galerkin formulation on stationary walls was proposed by Lunsmann et al. [365] in their EVSS finite element solution for viscoelastic flow past a sphere in a tube. Here the normalizing factor in the upwind term was not \uh\ as in (7.12) but some characteristic velocity of the flow U. The difficulty with this formulation was that results depended upon the choice of U. So Fan and Crochet [202] chose a scaling factor l / | u h | m instead, computed as the reciprocal of the mean value of uh in an element. The scheme reverted to a Galerkin method on stationary walls and there was none of the arbitrariness inherent in the scheme of Lunsmann et al. [365]. This third choice of SUPG method enabled the authors to extend the limiting Deborah number from the 1.6 of Lunsmann et al. [365] to 2.0. A recent adaptive AVSS/SUPG code devel oped by Warichet and Legat [619] has further extended the limiting Deborah number for p-convergent solutions to 2.5. High-order methods have enjoyed good success when compared with SUPG finite element formulations for smooth problems. In a study in 1994, Khomami et al. [329] found that high (p < 16) order h-p type Galerkin finite element meth ods produced stable and accurate discretizations for flow of a UCM fluid past square arrays of cylinders and through a corrugated pipe. Of the techniques 181
7.2. DISCRETIZATION OF CONSTITUTIVE EQUATIONS investigated EVSS/SU and EVSS/SUPG were also stable for these flows. How ever, the h-p finite element method gave rise to exponential convergence rates to the exact solution whilst all the lower-order methods were linear in their convergence rate. Moreover, it was found that the high-order h-p finite element method was much more cost efficient than the lower-order schemes when used to calculate a solution at a prescribed level of accuracy, as measured by global deviation from mass conservation. We say more about high-order spectral-type methods and their stabilization in §5.5.
7.2.2
Discontinuous Galerkin m e t h o d s
The discontinuous Galerkin (DG) method was first introduced by Lesaint and Raviart [350] in 1974 for the solution of the neutron transport equation and applied in the viscoelastic context for the first time by Fortin and Fortin [218] some 15 years later. In particular, Fortin and Fortin [218] approximated the components of stress using discontinuous elements. In addition to allowing a solution of the constitutive equation on an element-by-element basis (see, for example [213]), two of the main advantages of the DG method over continu ous finite element stress interpolations, as remarked by Baaijens [23], are that satisfaction of the velocity-stress compatibility condition, shown to be required in the standard mixed finite element approximation of the three fields Stokes problem (see [257]) may be satisfied easily (see §7.3.1). Moreover, when itera tive methods (such as GMRES [523]) are used, preconditioning can be achieved efficiently at the elemental level. Another possible benefit, as noted by Fan [203] is that there may be greater robustness in dealing with problems with discon tinuous boundary conditions because the approximations allow for jumps in the stress. Error estimates for the DG finite element approximations to the solutions of the Oldroyd B equations on quadrilateral meshes were derived by Bahhar et al. [33] and it was shown that with Qk+i continuous finite element approx imations for the velocities, Qk discontinuous elements for the stresses and P* discontinuous elements for the pressure, that error estimates of O (hk+1/2) were valid in the energy norm. This is half an order higher than for the continuous Galerkin method (see §7.2.1). Derivation of mathematical description Let fl C R d be a bounded Lipschitzian domain with boundary 9 0 . Then letting n denote an outward pointing normal vector to dil the inflow boundary d£l~ of dtt with respect to a given velocity field u is defined as dn~ = {x G an ■. u ( x ) - n ( x ) < 0 } .
(7.13)
With obvious notation we define the outflow boundary dfl+ = 9fi\dfi~, and denote a suitable function space for the elastic extra-stresses by E. Suppose now we partition fi into K (say) non-overlapping subdomains {fik}Then weak forms of the governing equations for the isothermal flow of a Maxwelltype fluid in 0 will include those for the momentum and continuity equations,
182
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM and for the constitutive equation: Find r E S such that V J / (r + Xir) : S d n f c - 7 7 P / j:SdQk\=Q, lJak \ / Jnk )
V S e E . (7.14)
k=1
In order to derive a DG formulation of (7.14) we begin by writing the convected derivative term in (7.14) (that appears as part of the upper-convected derivative) as Y,
(u-VT):Sdnk = J2
fc=l^a*
k=lJSlk
V-(u(T:S))dflfc
K
-J2
(u-VS):TdQk.
(7.15)
fc=1-/nfc
Suppose now that wefixa value k and that a point x on the interior of an edge of flk is shared between Qk and one other neighbouring element fig (say), as shown in Fig. 7.2.
Figure 7.2: Neighbouring subdomains Qk and f^. Then we denote by r e ( x ) the (external) stress tensor evaluated a,t x & fig and by r'(x) the (internal) stress tensor evaluated at x € Qk. By applying the divergence theorem to the first term on the right-hand side of (7.15) we get K K r r ]T / {u-VT):Sdnk = Y, fc=1
"'fifc
fc=1
K
r (u ' V S )
(n-u)T:Sdrk-Y,
-/aOfc
fc=1
: T dfi
*•
A}fc
(7.16)
183
7.2. DISCRETIZATION OF CONSTITUTIVE EQUATIONS
We may then let f a r e + (1 - a ) r ! \ ari + (l-a)T*
on d^ \dSl~, ondn+\dft+,
. [
. '
for some a £ [0,1] to obtain the result K
V
,
X
/" (u • V r ) : S dnk
= V /"
(n • u)r i n f l o w : S dr*
K
+ V / « ■
(n-uJCar' + fl-ayjiSdTfc
-
+
V / (n-u)(aTi + ( l - a ) T e ) : S i T i J=J ■'an+\en+
+
V /"
(n ■ U)T J : S dTfc
fc=1J9n+n9n+
if
V ) / ( u - V S ) :TdQk.
(7.18)
fc=i,/"*
From a second integration by parts and application of the divergence theorem to the last term in (7.18) with r = r 1 on <90fc\<9fi~ we therefore deduce that the left-hand side may be replaced in (7.15) by K
„
K
V
/ (u • V r ) : S dVtk + « V k=1JQk
f
(n • u)[r] : S dTk
J k=1
sa^\ea-
K
+(l-a)T
(n-u)[T]:SdTt,
(7.19)
k=1Jdn+\dn+
where we use the notation [T] = re — T'. That is, the DG formulation for (7.14) is: Find r € £ such that K „ , K N Y\ I (T + \IT) :Sdnk+a\iy/
+(l-a)Aiy/
f
(n-u)[r] : S dTk
{n-M)[r}:SdTk-YVp
k=1JdU+\dQ+
k=1
7
: S dQk = 0,
Juk
VS e E.
(7.20)
D G m e t h o d s for viscoelastic flows The earliest applications and developments of the DG method for viscoelastic flows were undertaken by the research groups of Fortin and Fortin [213,218,219]. In their first paper, in 1989, Fortin and Fortin [218] used the DG scheme with a = 1 to solve the flow of an Oldroyd B fluid through a 4:1 contraction. They approximated the velocities with biquadratic finite elements, the pressure with discontinuous piecewise lineax approximations and used discontinuous incom plete biquadratic approximations for the components of stress. Unfortunately, 184
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM their decoupled scheme for the steady-state problem (employed with Picard it eration) did not converge for any Deborah number and a transient algorithm and the SU formulation of (7.20) were found to be necessary. Results up to De = 12 were presented. A year later Fortin and Fortin [219] obtained solu tions of the steady-state problem using an improved method based on a GMRES iterative method and the same approximation spaces chosen in [218]. Admit tedly though, values of the upwinding parameter a up to 10 had to be employed in order to eliminate oscillations that appeared in the solutions beyond De = 2 for the Oldroyd B stick-slip problem and thus the authors were effectively solv ing a modified problem, similar to that which would have arisen with the SU method [384] (see §7.2.1). Basombrio et al. [47] later showed that the oscilla tions that plagued the Fortin and Fortin [219] calculations for De > 2 could be reduced by using linear stress interpolations rather than Ql • An improved GM RES method (applied to the stress components only) was introduced by Fortin and Zine [212] in 1992. The authors also used the EVSS formulation (see §7.3.2) and (inconsistent) streamline upwinding (see §7.2.1) to obtain solutions for the flow of an Oldroyd B fluid in the stick-slip and 4:1 contraction geometries. The improved GMRES method of Fortin and Zine [212] was then made more af fordable by Fortin et al. [213] who used it for the DG stress formulation, thus allowing the solution of the constitutive equation on an element-by-element ba sis and avoiding the assembly of a large matrix for the stress computation. The method was used for the solution of the stick-slip problem using the Oldroyd B and Phan-Thien Tanner (PTT) models and the approximation spaces used were the same as those which had been employed by Fortin and Fortin [218]. Although the achievable Weissenberg numbers for the Oldroyd B stick-slip prob lem were not great (We = 5 for the finest mesh) the numerical strategy was far more robust for the P T T fluid and the comparative cheapness of the method augured well for three-dimensional applications. Baaijens et al. [26], in the first of a string of publications on DG methods for viscoelastic flows [22-24,26,27,29] applied a DG method with GMRES in order to simulate the flow of a shear-thinning solution of polyisobutylene around a confined cylinder by using a single mode P T T model. The insufficiency of determining model parameters from simple viscometric flows in order to predict complex flow phenomena was highlighted in this paper: dramatic discrepancies between predicted and measured stress profiles were observed in the wake of the cylinder. However, in a paper published the following year, Baaijens et al. [29] used a four mode P T T model, higher characteristic flow rates (larger range of De) and some new experimental equipment to compare results for the same fluid, and agreement between the measured and predicted results was good for the symmetrically arranged cylinder and even excellent for the asymmetric case. Results were also produced with a single mode P T T model, as in [26], but this time showed an improvement over the original results. The difference in the quality of the agreement between the two papers was not fully understood by the authors but the most obvious explanation was that there had been a change in the rheological behaviour of the fluid between the two research reports. Important improvements in the performance of DG methods for viscoelas tic flows were demonstrated by Baaijens [22, 23] in two papers in 1994. In the first [23], the basic DG method for unsteady flows was improved by ex plicitly approximating the r e in (7.20), making possible the elimination of the stress degrees of freedom at the elemental level. The additional enforcement of 185
7.2. DISCRETIZATION OF CONSTITUTIVE EQUATIONS monotonicity (see van Leer [592]) eliminated the oscillations which otherwise affected the numerical results generated by a fully implicit DG method for the stick-slip flow of a PTT fluid. Moreover, implementation of this new, so-called implicit/explicit DG (DGE) method was easier than for the earlier implicit DG method. The only drawback of the DGE method was that convergence with time-stepping to the steady solution could be slow for UCM-like models. In the same year Baaijens [22] used a DG formulation with Newton's method for steady flows and experimented with two different stress approximations - linear stress elements and constant stress elements. In the latter case the governing equa tions were stabilized with a Galerkin least squares stabilization scheme, following ideas of Franca and Stenberg [224]. Remarkably stable finite element methods for the computation of viscoelastic flow resulted, especially with the piecewise constant stress elements. Whereas convergence of the Newton scheme for UCM fluid flow past a sphere in a tube was lost at De = 1.6 for the linear stress elements, calculations continued with the stabilized scheme as far as De = 4.0. However, convergence with mesh refinement could not be demonstrated. It was therefore a further breakthrough by Baaijens and co-workers [27] when the DGE scheme [23], now stabilized using the DEVSS method of Guenette and Fortin [257] (see §7.3.2), yielded mesh-convergent solutions to the same problem up to De — 2.5. Disappointment followed, however, when experimental results for a polymer melt past a confined cylinder could not be predicted accurately by parameter-fitted P T T and Giesekus models in the mixed shear/elongational flow region between the cylinder and the walls. In addition, neither model was able to correctly predict the birefringence distribution in the cylinder wake. As with many papers in the literature, growing confidence in the faithfulness of numerical simulations to accurately reflect the predictions of the models them selves is leading to a more concerted effort to develop more realistic models (see §11.1). In 1998 Baaijens [24] used a preconditioned GMRES method to solve the DEVSS/DG equations for a UCM fluid. Solutions at We = 2.2 for the falling sphere-in-a-tube problem, and at We == 5 for the 4:1 axisymmetric contraction flow were reported. In the latter case, convergence with mesh refinement away from the downstream wall was observed and the asymptotic solution behaviour predicted by Hinch [283] was corroborated numerically. Comparison of S U P G and D G methods In 1997 Fan [203] performed a comparative study of the performance of the EVSS/SUPG formulation and a DG method in both the pure mixed formula tion (see (7.25)) (DG/MIX) as well as in the EVSS formulation (see (7.37)) (EVSS/DG), in which solutions to three benchmark problems were computed: 1. UCM fluid flowing steadily past a sphere in a tube, 2. axisymmetric steady flow of an MCR fluid (a) through a 4:1 contraction, (b) in a stick-slip geometry. The MCR model was chosen because of the Newtonian-like stress singular ity which meant that the stresses for the two non-smooth problems above were 186
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM integrable and convergent solutions could be attained. The DG/MIX formu lation, unsurprisingly, did not perform well in any of the problems considered. For the sphere problem the limiting Weissenberg number decreased with mesh refinement (We = 1.2 for the finest mesh and maximum polynomial order 6 for the stresses) and the method exhibited strong stress oscillations for the two problems having singularities. Performance of the EVSS/DG and EVSS/SUPG methods were comparable for the sphere and stick-slip problems and no limit was found on the Weissenberg number in the case of the stick-slip flow. Al though EVSS/SUPG required fewer degrees of freedom than EVSS/DG on the same mesh, the benefit of the extra robustness of the EVSS/DG method for nonsmooth problems was demonstrated in the case of the 4:1 contraction problem: changing the pressure approximation from being discontinuous to continuous helped the EVSS/SUPG method a little but it could still only be used to at tain a Weissenberg number of 10, compared with the value We = 21 from the EVSS/DG method.
7.3
Discretization of the Coupled Governing Equations
Thus far we have considered discretization methods appropriate for differen tial constitutive models, considered as a set of first-order hyperbolic equations. However, the challenge of designing stable and accurate numerical methods for solving viscoelastic flows goes further than this. Proper account has to be taken of the fact that the constitutive equations are part of a larger set of coupled governing equations of mixed mathematical type. We shall presently see that the multi-field discretization that is required may involve the observation of cer tain compatibility constraints on the choice of approximation spaces in order to ensure that the problem is well-posed. Moreover, change of type and loss of evolution (see Chapter 3), whether in the strong formulation or only in the discrete equations, can have disastrous effects upon the convergence properties of standard numerical schemes.
7.3.1
Compatible approximation spaces
Using the same notation as in §7.2, let us recall the governing equations for the steady inertialess flow of an Oldroyd B fluid in some bounded open domain Cl C Hd with Lipschitzian boundary 9 0 . These are as follows: V u + Vp
= =
0, V - r + b,
r + Ai r
=
vir(u),
-r)sV2u
(7.21)
where b denotes a body force and equations (7.21) are to be solved subject, say, to Dirichlet boundary conditions u = 0,
on dfl,
(7.22)
and elastic stress inflow conditions r = 0 on 3n i n f l o w . 187
(7.23)
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS Then, given suitable function spaces V, Q and E (which we leave unspecified) a Galerkin weak statement of (7.21) may be written: Find (u,p,r) eV x Q xT, such that
I V • u q dfl = 0, Vg € Q, [ r)s V u T : V v dO. + f r : V v dQ, JQ
(7.24)
JQ
- f V • v p dQ. - f b - v dft JQ
=
0, Vv € V,
JQ
f (T + Ax r -Tj p 7(u) J : S dft
=
0,
VS e E.
Given conforming finite/spectral element approximation spaces Qs C Q, Vs C V, Ea C E we then arrive at the classical three fields discrete formulation, equivequiv alent to the so-called MIX1 method of Crochet et al. [152]: Find (us,ps,Ts) £ Vs x Qs x Y,s such that / V us qsdQ. / ?7S V u f : V v { dH+ JQ
-
Jn
rs:
0, Vqs G Qs,
=
0,
Vvd dfl (7.25)
V-vsPsdSlJQ
=
hvsdn
Vv<5 6 Vs,
JQ
f (TS + Ai TS -r)p-y{us) J : S4 dfi = 0, VS5 £ E 4 . On account of the intractability of the full set of equations (7.25), all available analyses (e.g. [42,222,230]) of compatibility conditions between the approxi mation spaces Qs, Vs and E^ for the well-posedness of (7.25) are restricted to linearized versions of the constitutive equation for slow flows or the three fields Stokes problem obtained from (7.25) by setting Ai = 0. From the results of Fortin and Pierre [222] in 1989 and Baranger and Sandri [42] in 1992 it may be shown that whether one considers the mixed method above for the limiting case of the UCM fluid (AJL = 0,rjs = 0) or the Oldroyd B fluid (Ai = 0,t]a ^ 0), the LBB or inf-sup condition [98] applies. This restricts the choice of approximation spaces Vs and Qs and states that any admissible Vs or Qs must be such that there exists a constant /? > 0, independent of the discretization parameter d, such that inf
sup .
.{V'Vf'q^
> /3-
(7.26)
qs€Qs vjeKs | v ,5|/fi(Q)rf||9(S||L2(n)
Fortin [214] showed that the inf-sup condition (7.26) would be satisfied when ever, given a vector field u eV, one can explicitly build a discrete u<5 6 Vs such that / V ■ usqs dQ= j V • uqs dn, JQ
V© e Qs,
(7.27)
JQ
for all ug depending continuously on u, i.e. 3 C independent of S such that \us\H1(Q)d < C|u| H i ( n ) d. 188
(7.28)
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM In 1981 Fortin [215] used (7.27)-(7.28) to identify several two-dimensional and three-dimensional finite elements satisfying the LBB condition, amongst them the quadrilateral element Q% — Pi (9-node biquadratic velocity - linear pressure). In 1985 Fortin and Fortin [217] experimented with six different finite elements and discussed their relative merits. Now suppose that Cl is the union of non-overlapping spectral elements {fl*} and let Pjvfc(njt) denote the space of restrictions to a spectral element Qk of polynomials with d variables and degree < Nk for each variable. Then, taking Vs = {v5 e H^(Si) : v 5 | n , e P j v j n * ) } ,
(7-29)
Maday et a!. [379] showed that the inf-sup condition (7.26) is satisfied with Qs = {qs e Lg(ft) : qs \nk € TNk_2(flk)}
.
(7.30)
This has proved to be a popular choice. More recently, Bernardi and Maday [61] have proposed Qs = {qs E L20(n) : qd \„h e P j v . ^ ^ J n P ^ ] ^ ) } ,
(7.31)
for a real number A, 0 < A < 1 and [XNk] denoting the integer part of XNk. The authors showed that this choice of Qs led to a better a priori error estimate on the pressure than with (7.30). In the same paper the authors also proved an inf-sup condition when the pressure was chosen in the pressure space Qs = {qs € Lg(fi) : qs \a> € VNk-i(ilk)}
,
(7.32)
i.e. the polynomials representing the pressure in each spectral element are of total degree Nk — 1. The inf-sup condition for this space coupled with Vs above is standard in finite element methods. The drawback in its application in the spectral context is that the benefits of tensorized bases are lost. Finally, for the non-conforming mortar element method, Ben Belgacem et al. [52] showed that Vs given by (7.29) and Qs given by (7.30) or (7.31) led to the LBB condition being satisfied. Baranger and Sandri [42] have shown that no compatibility conditions are required on Vs, Qs and Us other than the LBB condition for mixed finite element approximations to the three fields Stokesian limit of the Oldroyd B equations (Ai = 0, T)s ^ 0). The situation is more complicated in the corresponding UCM (Ai = 0, r/s = 0) limit, however. We begin by writing down the standard mixed finite/spectral element formulation for the Stokes problem: Find (us,ps) € Vs x Qs such that
L
V -usqS(Kl
=
0, Vg5 e Qs,
?)Vu| : V v s dfi - / V • v 5 ps dfl
(7.33)
./fi Q
L
hv5dfl
=
0, Vvs e Vs,
where t] = t]v + i)s is the total viscosity. In general, and as observed by Marchal and Crochet [384], the results obtained with the mixed method (7.25) for the
189
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS three fields UCM Stokes limit will not be the same as those found from solving (7.33). This is because the equality T , = vir(ua),
(7.34)
must hold in a pointwise and not merely in a weak sense for there to be equiv alence of (7.33) and the Stokesian limit of (7.25). We see that if 7(u 5 ) G E«,
(7.35)
then such pointwise equality follows (we may take Ai = 0 and S$ = rs — r)pj(u$) in the last equation of (7.25)). Therefore (7.35) is a sufficient condition for the equivalence of the Stokesian limit of (7.25) and (7.33) and hence we may further state that (7.26) and (7.35) are sufficient compatibility conditions for the wellposedness of the Stokesian limit of (7.25). Different ways of satisfying (7.35) may be found in the literature. For example, Marchal and Crochet [383] pro posed the use of Hermitian elements for the approximation of the velocities, thus ensuring that the components of the discrete rate-of-strain tensor were continuous and allowing the use of continuous stress approximations as well. However, these Hermitian elements were restricted to rectangular shape. On the basis of (7.35) it is more practical to seek discontinuous stress approxima tions. The Lesaint-Raviart method [350] and its applications in the realm of viscoelastic computations, has already been described in §7.2.2. For a single spectral element Gerritsma and Phillips [230] have shown that for the three fields Stokesian limit of the UCM equations (7.25) well-posedness of the dis crete problem is guaranteed if, in addition to satisfying the inf-sup condition (7.26) for the velocity/pressure approximation spaces, the degree of polynomial approximation in the tensorized basis for the stresses is at least as great as that used for the components of velocity. In [229] the same authors used discon tinuous spectral elements for the stress in order to solve the three fields Stokes equations for the stick-slip problem. The same degree approximations were used for the representation of the stress and velocity components. The condition (7.35) is not necessary in order that the Stokesian limit of the UCM equations (7.25) is well-posed, however. Marchal and Crochet [384], for example, only satisfied the condition (7.35) approximately by using a Q% — Q\ velocity-pressure element and a stress finite element consisting of 4 x 4 bilinear sub-elements. Two years later Fortin and Pierre [222] showed that the contin uous mixed finite element method of Marchal and Crochet led to a well-posed discrete system of equations and that an a priori error estimate for the nu merical solution was of optimal order. As stated by Baranger and Sandri [42], satisfaction of the compatibility condition on the velocity and stress approxi mation spaces in the case of continuous stress approximations demands that, in each element, the number of interior degrees of freedom for each component of Tg is greater than or equal to the number of all degrees of freedom of each component of v^.
7.3.2
EVSS-type m e t h o d s
EVSS It has already been noted that if T}s ^ 0 in the Stokesian limit of (7.25) then the compatibility condition between Vs and E$ may be dropped, thus greatly 190
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM widening the possible choice of approximating spaces for TS- In fact, even if r)s = 0 it is possible, by means of a suitable change of stress variable, to ensure that the discrete system of equations governing the Stokesian limit be wellposed, provided that the LBB condition is still satisfied. An example of such a change of variables is provided in the Elastic Viscous Split Stress (EVSS) formulation, first used by Perera and Walters [448] and Mendelson et al. [401] for flows of second-order fluids (see §2.5.2 and §7.1). In the EVSS formulation we begin by changing the stress variable to as (say) where this is defined in the present context as
(7.36)
Then substituting this into (7.25) gives us the system of equations equivalent to the MIX4 method of Crochet et al. [152]: Find (us,ps,as) 6 V$ x Qs x £<j such that
/ V ■ usqs dfl
=
0,
\ % 6 Qs,
JQQ
l T)S7uJ : Vvs dfl 4- / as : Vvs dfi . JQ
JQ
Q
(7.37)
f
V • vsps dflbvs dfl = 0, Vva e Vs, n JQ as + Ai av5 +\1r)p V 7 (u*) : Ss dU = 0, \/S5 € £«. n \ /
l(
It is, of course, obvious that, in general, 7 will not be continuous across element interfaces so that strictly speaking it is not possible to have both T$ and as be longing to continuous approximation spaces. It may be seen that the Stokesian limit of (7.37) leads trivially to a problem having exactly the same solution as the classical (u,p) formulation (7.33) since Ai = 0 =>• a$ = 0. As a consequence, no compatibility conditions other than the usual LBB condition are required for the well-posedness of the Stokesian limiting equations. One disadvantage of the formulation of the last equation in (7.37) is that second-order derivatives in the velocity are present in the third term and cannot be handled for a C° continuous velocity field. One way round the problem is to follow Crochet et al. [152] and use integration by parts on this term. Using the summation convention and with obvious notation for the stress and velocity components, we have
f ?:ss dn = / ukp^Sij dsi- ! ^-ikiSn da- [ iikp-Sij dn, Jn
dxk - / Ukjij~-
JQ
=
JQ
+ /
dxk
Jn dxk d£l- / -^ikjSij
JQdxk
nkuk'yijSijdT.
JQ dxk dfl - / jik-~Sij
Jn
dxk
dfl (7.38)
JdQ
The drawback of integrating by parts as above is that boundary conditions are required on the rate-of-strain tensor 7 for those parts of dfl which are not fluxfree, although this may not be a problem if it is valid to assume fully developed flow there. An additional problem with integrating by parts is that it may lead to a numerically unstable scheme: Rajagopalan et al. [485] attributed the poor results of Beris et al. [55] and Van Schaftingen and Crochet [593], at least in 191
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS part, to their treatment of the convected derivative of the rate-of-strain tensor in MIX4 in this way. Therefore Rajagopalan et al. [485] proposed treating the rateof-strain tensor 7 5 as a separate unknown and computing it by performing an L2 projection of the symmetric part of the velocity gradient onto some continuous approximation space Tg (say). Thus the EVSS scheme may be written: Find (ns,ps,trs,js) 6 Vs x Qs x T,s x r 5 such that
L V-\i q dn s s
/
Q JQ
= 0,
VqseQs,
?)Vuj : Vv<5 dU + as ■ V v 5 dfl „ Jn / V -vspsdflhvsdfl = 0, Vv* e Vs, Jn Ja
f
{is - ( V u j + V i t f ) ) : Gs dn
=
0,
(7.39)
VG a e Ts.
JQ
In a survey of different mixed finite elements Debae et al. [169] found that the EVSS formulation of Rajagopalan et al. [485] (when used with biquadratic continuous elements for ua and bilinear continuous elements for the remaining variables) was inexpensive in computer time and remarkably stable and accurate for the four benchmark problems considered. Furthermore, it is comparatively cheap for simulating fluids possessing multiple relaxation times and, therefore, was proposed as a good candidate for three-dimensional flows. Khomami et al. [329] investigated the performance of lower-order Galerkin finite element methods, with and without EVSS, as well as EVSS/SU and EVSS/SUPG and h-p finite elements for steady flow of a UCM fluid past a square array of cylin ders and through a corrugated tube. Only the h-p, EVSS/SU and EVSS/SUPG methods produced stable and accurate discretizations for the flows considered. In a further study a year later, Talwar and Khomami [559] showed that although higher-order h-p Galerkin methods without stress splitting were stable and pro duced exponentially convergent solutions for steady inertial flows of viscoelastic fluids under change-of-type conditions, an EVSS formulation was needed in or der to avoid a limit on the elasticity number. Upwinding was needed with lower-order EVSS/Galerkin discretizations to stabilize them. For comments on the study by Fan [203] on the relative performance of EVSS/SUPG methods and DG methods for three benchmark problems, see §7.2.2. Several other pa pers [202,365,619] utilizing the EVSS formulation have been referred to already in this chapter. With the EVSS formulation Luo [366] used a transient and de coupled operator splitting/SUPG method for flow of a UCM fluid past a sphere in a tube. Although convergent solutions up to a Weissenberg number of 2.8 proved possible for this problem the values for the drag factor diverged consider ably from those obtained by Jin et al. [308], Warichet and Legat [619], Fan and Crochet [202] and Baaijens et al. [27] beyond a Weissenberg number of approx imately 1.5. Baaijens [25] suggested that this was possibly due to insufficient mesh refinement by Luo. The EVSS formulation has also seen several applications in finite volume methods for viscoelastic flows, notably in work by Sasmal [526] in 1995, by Dou and Phan-Thien [186] in parallel finite volume methods in 1998 and by Xue et al. [636] in three-dimensional finite volume methods in the same year. 192
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM As stated already, the available analyses for the compatibility conditions for the approximation spaces for velocity, pressure and stress are for linearized forms of the constitutive equations and the Stokesian limit of viscoelastic models. In this limit the constitutive equation becomes an algebraic relation between the components of the stress and velocity gradient tensors. It was observed by Szady et al. [556], for example, that on solid stationary boundaries the constitutive equation again becomes an algebraic relation between stress and rate-of-strain since the convection term vanishes in this instance. On the conjecture that, in this case, similar compatibility requirements may apply to the approximation spaces as had been established by Fortin and Pierre [222], for example, for the three fields Stokes equations, Brown et al. [106] and Szady et al. [556] developed an EVSS-G method, similar to the EVSS method of Rajagopalan et al. [485] but a continuous L2 projection G^ of Vu^ and not of 7 was taken as the new variable in the problem. The compatibility condition required by the algebraic system of equations to which the Oldroyd B model degenerates on a solid boundary were stated by the authors to imply that as and G<$ should consist of the same degree polynomials. Since in [556] the velocity field was approximated by Lagrangian biquadratic elements the components of G^ and as were approximated by bilinear finite elements. The same elements were used for the pressure. One advantage that EVSS-G has over the EVSS method is that the velocity gradient terms appearing in the upper-convected derivative of the stress erg may be replaced by their continuous projections, thus regularizing the operator in some sense. In order to test the EVSS-G algorithm against the EVSS formulation Szady et al. [556] considered the stability of inertialess planar Couette flow of a UCM fluid when undergoing two-dimensional infinitesimal disturbances. The earlier work by Brown et al. [106] showed that the strong formulation was stable for all De. No numerical instability was found for the EVSS-G/SUPG method. A mesh-dependent limit on the achievable De for the EVSS formulation had been observed by Brown et al. [106], however. By using bilinear approximations for the components of stress a considerable saving was possible over the cost incurred by the use of the biquadratic stress elements used in the EVSS formulation. AVSS The change of variables involved in stress splitting does not, of course, have to be done in the manner described in (7.36). Sun et al. [548] proposed an Adaptive Viscoelastic Stress Splitting (AVSS) scheme whereby rather than (7.36) a change of variables trs = TS-0i,
(7.40)
was carried out, the parameter (3 being allowed to change adaptively in such a way that the viscous contribution to &$ was of at least the same order as that of the elastic contribution. Letting hk denote the characteristic size of the finite element fi& and by suitably non-dimensionalizing the coordinates and field variables, Sun et al. [548] showed that if (3 was chosen according to
fi = ahk.^max, Nlmax 193
(7.41)
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS with a parameter a — 0(1), then the viscous and elastic terms in the momen tum equation would be balanced, the kinematics would not be overly sensitive to changes in the elastic stress and a decoupled algorithm could be used. By com bining AVSS first with a streamline integration algorithm and then an SUPG method the authors achieved convergence to high Weissenberg numbers (3.2 and 1.55, respectively) for the flow of a UCM fluid past a sphere in a tube. The results showed that the decoupled AVSS was more stable than the correspond ing EVSS formulation. However, as with the results of Luo [366], an obvious departure from the majority consensus [27,308,619] for the values of the drag factor beyond a Weissenberg number of approximately 1.5 was evident, pointing to insufficient mesh refinement. The experiences of Warichet and Legat [619] with an AVSS/SUPG formulation for the computation of UCM fluid flow past a sphere in a tube, have been reported already in §7.2.1. The benchmark problem of flow past a sphere in a tube is discussed further in §9.2.2. DEVSS Popular though they have proved to be, the EVSS and EVSS-G methods re quire a change of variables which may be impossible to perform for some con stitutive equations. An example is the Grmela model [254]. Motivated by the concern to apply EVSS-type methods to a wider class of rheological models, Guenette and Fortin [257] introduced in 1995 the Discrete EVSS (DEVSS) method which requires no change of variables. Under this formulation the discrete UCM equations given by (7.25) with rjs = 0 are rewritten as: Find (us,ps,Ts,is) eVs x Qs xT,s xTs such that / V us qs dil
=
0, V^ € Qs,
=
0, \/vseVs,
=
0, VS* £ E«,
/ a (j(us) - 7<s) : V v a dQ. + / TS : Vv,s dft
JQ
JQ
- I V-vspgdnJQ
[ b-vsdtl
(7.42)
JQ
f (TS + Ai T5 -TiPi(us)\
:Ssdtt
[ (js-j(u5)):G5dn
= o, vc 5 er 5 ,
JQ
where a is any positive parameter. It should be noted that since 7,5 is the con tinuous L2 projection of the discrete rate-of-strain tensor 7(11,5) that the viscous term in (7.42) typically does not vanish. As remarked a little earlier, no explicit change of variables is required and no derivative of the rate-of-strain tensor (or its continuous projection) is required in (7.42). Numerical experiments per formed by the authors for the 4:1 contraction problem and the stick-slip problem with the P T T and Grmela models demonstrated the stability of the method: using the P T T model in the stick-slip problem no limiting Weissenberg number could be found. In a later paper Fortin et al. [220] showed that no extra com patibility conditions beyond the usual LBB condition [98] were required for the stability of the formulation in the UCM Stokesian limit and for some linearized versions of the Oldroyd B model. This fact allowed the authors in [257] and [220] to take Qj "~ C° elements for the velocity, Qi~ C° elements for the extra-stress and rate-of-strain, and linear elements for the pressure. Thus lower-order finite 194
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM elements were permissible for Tg with the consequent saving in computational cost. The DEVSS method has seen applications in combination with the DG method (see §7.2.2) by Baaijens and co-workers [24,27] and in the modified form DEVSS-G by Liu et al. [359], for example, where the projection of the velocity gradient and not of the rate-of-strain tensor is considered as a separate variable. For a description of the use of a DAVSS-G/DG method and a DEVSSG/DG method for solving viscoelastic flow past a confined cylinder, the reader is referred to the papers of Sun et al. [549] and Caola et al. [121] and to our discussion in §9.1.5.
7.3.3
Change of t y p e and loss of evolution
Some discussion has taken place already in Chapter 3 of the possibility of a change of type and loss of evolution in the flow of viscoelastic fluids. Here, we want to consider the phenomena further and explore the connection with the high Weissenberg number problem. Joseph and Saut [317] and Joseph et al. [316] considered a class of fluids having zero Newtonian viscosity and in particular those having constitutive equations of the form A i ^ = % > 7 + S,
(7.43)
where r is the extra-stress tensor, Ai a characteristic relaxation time, r)p the vis cosity and S a tensor of lower-order terms possibly involving r and the velocity u but not their derivatives. The derivative V/Vt in (7.43) is the corotational or Jaumann derivative defined by VT
Vt
1
BT
— + (u • dt
V)T
+ -r ((1 - a) V u T - (1 + a) V u ) 2
(744)
T
+-((l-a)Vu-(l+a)Vu )r, for some parameter a 6 [—1,1]- When S = — r and a = — 1 and 1 we recover the lower-convected Maxwell (LCM) and UCM models, respectively. Inspired by the original analysis of Rutkevich [521,522] (who dealt with the three cases a = —1,0,1) Joseph et al. [316] considered a small perturbation about a constant base flow solution to the momentum equation, equation of continuity and (7.43) and showed that the system would be of evolution type, i.e. stable to short wave disturbances, if the eigenvalues T\ >T2 > T 3 of the stress disturbance f satisfied ^ +^a(r1+r3)-i(r1-f3)>0.
(7.45)
In particular, for a = 1 (UCM) the flow is evolutionary if T3 > -rjp/M and for a = - 1 if Tx < ffo/Ax- To see that these conditions always hold and that, therefore, the UCM and LCM models are always evolutionary we begin by writing out the integral representations of the UCM and LCM models, T(X,*) = - ^ I + / Ai Jo
r1pXi2exp(-s/Xi)C-1(x,t,t-s)ds,
195
(7.46)
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS and r ( x , t)=7^-IAi
/
T}p\l2 exp(-s/Ai)C(x, t, t - s)ds,
(7.47)
Jo
respectively. We then follow Dupret and Marchal [195] and Crochet [150] by observing that since the Cauchy-Green and Finger strain tensors (see §2.5.1) are always positive definite then so are the tensors T\ = r + r)p/\il and r _ i = r — r/p/AiI for the cases a = 1 and o = —1 respectively, provided, of course that they are positive definite initially on the upstream boundary. Likewise, the Oldroyd B fluid will be stable to short wave disturbances. For other models which may be written in the general form (7.43) (PTT [456,458], Johnson-Segalman [312], Giesekus [234,235] and corotational Maxwell models, for example) the condition (7.45) may not be satisfied and these models may show loss of evolution in certain flows [194,317]. In general, loss of evolution may occur when certain stresses exceed critical values. The reason why the loss of evolution in certain models is serious for numerical simulations is that time-dependent solutions to certain flows with such models may explode once the Deborah number reaches a sufficiently high value. The problem will be exacerbated with mesh refinement [324] and, as remarked by Joseph and Saut [317], Hadamard instabilities are much stronger than those studied in bifurcation theory. Change of Type Joseph and Saut [317] demonstrated that for the class of models (7.43) in plane flow (six unknowns) two of the characteristics of the full steady quasilinear system are always imaginary, two are real and correspond to the streamlines and the remaining two can be real or complex. These last two are associated with the steady vorticity equation. The authors showed that this equation could exhibit change of type and could be hyperbolic in some parts of the flow and elliptic elsewhere. An important result proved by the authors was that if the vorticity equation of an inertialess steady flow becomes hyperbolic then there is loss of evolution and the onset of instabilities of the Hadamard type for the corresponding time-dependent problem. We have already remarked that the UCM, LCM and Oldroyd B models never lose evolution. This means that the vorticity equation for steady inertialess flows of such fluids must be elliptic everywhere. To make the connection between change of type and loss of evolution yet clearer we cite Dupret and Marchal [195] in stating that the UCM model changes type when the tensor T\ — pun is not positive definite. It has been seen already that T\ is positive definite so that for steady inertialess flows it is indeed the case that the UCM model cannot change type. Dupret and Marchal [195] also showed that whether the Reynolds number vanishes or not the Oldroyd B fluid never manifests a change of type. Just as with loss of evolution, there are potentially severe consequences of change of type in numerical calculations of viscoelastic flows: Galerkin methods, for example, can be very inaccurate for nonlinear hyperbolic problems. In the light of all the comments above it may be tempting to think that by us ing the LCM, UCM or a model with a Newtonian component in the extra-stress such as the Oldroyd B model, any difficulties with short wave disturbances in time-dependent calculations or the appearance of hyperbolic regions of vorticity 196
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM will be overcome. The depressing reality is that numerical errors can result in a loss of positive definiteness of the computed T\ in the case of a UCM fluid, for example, with consequent loss of evolution for a time dependent calculation and/or change of type, at the discrete level. Dupret et al. [196] used the stan dard mixed Galerkin formulation to solve the flow of a UCM and Oldroyd B fluid past a sphere in a tube and through an abrupt contraction. The result was that the tensor r% lost its positive definiteness in regions of high stress gradients at a Weissenberg number lower than critical limiting value and the loss of positive definiteness of T\, was marked by the onset of large oscillations in the computed velocity field. In a study that demonstrated the same phenomena at the discrete level, Brown et al. [104] attempted to solve flow of a UCM fluid through two different contraction geometries and between eccentrically rotating cylinders us ing a Galerkin method with an elastic stress splitting (7.36). The tensor n lost positive-definiteness at a Deborah number of just 0.1 for the abrupt contraction flow, whilst the same phenomenon was observed for De > 1.5 near the outer cylinder for the eccentrically rotating cylinders calculations. Oscillations in the velocity field and stress fields resulted. In order to demonstrate that the onset of oscillations was related to a change of type, Brown et al. [104] introduced inertia into the equations and thus allowed a genuine change of type to occur. For Re = 1000, De = 0.22 and eccentricity e = 0.1 the hyperbolic region almost filled the gap and large oscillations resulted. Refining the mesh only amplified the oscillations and showed that the Galerkin method was unstable for this flow.
The EEME method The Explicitly Elliptic Momentum Equation (EEME) for differential models having no solvent viscosity was introduced by Renardy [495] as part of an ex istence proof for steady inertialess flows of viscoelastic fluids. We illustrate its derivation in the case of the governing equations (7.21) for the flow of a UCM fluid (T]S = 0) in the absence of body forces (b = 0). Let us compute the diver gence of the elastic stress tensor r from the constitutive equation in (7.21). We therefore get, using the summation convention, drjk dxj
, _d_ / dxj \
l
dTjk\ _ . _<9_ (duj_ dxi J dxj \dxi
- A — (T — ^ dxj \ 3 dxt)
— (^dxj \dxk
P
\ J
+ ^^]=0 dxj)
(748)
Computing the derivatives above, using the continuity equation and simplifying leads to V • r + A z (u • V ) V • r - A! ( V u T ) (V • r ) - \
l T j l
^ ^
-
T? P V 2 U
= 0. (7.49)
Use of the momentum equation V • r = V p and the identity (u • V ) V p = V [(u ■ V)p] - ( V u ) V p ,
197
(7.50)
7.3. DISCRETIZATION OF THE COUPLED EQUATIONS enables us to write (7.49) as Vp + XiV ((u • V)p) - A a (Vu)(V • r ) - A x ( V u T ) ( V - r ) - A X T ^ ^ J - r? p V 2 u = 0.
(7.51)
That is, writing a modified pressure q = p + Ai(u - V)p we have Vq
=
Ai (V • T ) ( V U + V u T ) + \lTjt
= =
V-[(rkI + AiT)Vu]+Ai(V-T)(Vu)T, A1V-[X(Vu)] + A 1 ( V - x ) V u T ,
^
^
+ r;pV2U,
(7.52)
where the tensor x = T + ??p/AiI. The importance of the EEME formulation (7.52) is that, as seen earlier, the tensor T + %/AiI is always positive definite for the strong formulation of the equations for flow of a UCM fluid. Hence, the momentum equation in the form (7.52) is a second-order elliptic equation in the velocity for all values of the Deborah number. The EEME formulation was first used for the numerical simulation of viscoelastic flow by King et al. [332] in 1988, who used it in the computation of steady UCM flow between rotating cylinders and for the flow of a modified UCM fluid [13] in the stick-slip geometry. Since then the accuracy and stability of the EEME formulation has been demonstrated for a number of different problems and constitutive models. Burdette et al. [109] used the EEME formulation for calculations up to a Weissenberg number of 15 for flow of a UCM fluid through an axisymmetric corrugated tube with sinusoidally varying crosssection. Excellent agreement with the calculations of Pilitsis and Beris [467] was reported. Shortly afterwards the EEME formulation was extended to timedependent flows by Northey et al. [418,419]. The authors computed flows of the UCM fluid between concentric and eccentric rotating cylinders and also used the EEME finite element method for calculation of time-periodic states in the UCM viscoelastic Taylor-Couette flow. Rajagopalan et al. [484] demonstrated that accurate and stable solutions with multi-mode UCM models could be ob tained using the EEME formulation. Sample calculations were described by the authors using a two mode model for flow between eccentrically rotating cylin ders. On account of their Newtonian-like structure near re-entrant corners the modified UCM model and a modified form of the Chilcott-Rallison dumbbell model were used by Coates et al. [142] in 1992 with the EEME/SUPG method to examine steady-state viscoelastic flow through axisymmetric contractions. Unlike calculations in the literature performed with the UCM model, the au thors could achieve higher Deborah numbers with increasing mesh refinement. The EEME method has been used by a number of different research groups (see, for example, [308,365]) to obtain solutions at high Deborah number for the flow of a UCM fluid past a sphere in a tube and also for flow of a P T T fluid through the same geometry. The results of Lunsmann et al. [365] were convergent for a ratio of the sphere radius to that of the tube of 1/2 up to a Deborah number of 1.4. Slightly higher Deborah numbers could be achieved using the EVSS formulation. Jin et al. [308] obtained good agreement with Lunsmann et al. [365] up to a Deborah number of 1.2 and reported reliable and accurate results up to a Deborah number of about 2.0. 198
CHAPTER 7. DEFEATING THE HIGH WEISSENBERG NUMBER PROBLEM The EEME method, despite its obvious successes as exemplified in the dis cussion above, does not now seem to be as popular as it once was and has given way to (D-)EVSS type methods combined with some form of upwind stabiliza tion.
199
Chapter 8
Benchmark Problems I: Contraction Flows
Figure 8.1: The 2:1 planar contraction. The flow geometry for circular or planar entry flows is shown in Fig. 8.1. Fluid passes from one channel or tube into another of smaller radius and in the pro cess generates a complex flow exhibiting regions of strong shearing near the walls and uniaxial extension along the centreline. As a consequence of the shear and extensional regions the attraction of the problem to the rheological commu nity is that, despite the simple flow geometry, non-Newtonian fluids manifest significantly different behaviour to Newtonian fluids in flowing through the con traction. In particular, interest in the literature has focussed on (i) vortex behaviour, both near the re-entrant corner leading into the smaller section (socalled lip vortices) and in the salient corners, (ii) the pressure drop across the contraction (as a function of the Deborah number) required to maintain a given flow rate, (iii) the particle paths upstream of the contraction and (iv) velocity overshoots along the axis of symmetry. In acknowledgement of the importance of contraction flows and their ready accessibility to numerical simulation, the 201
8.1. VORTEX GROWTH DYNAMICS problem was chosen as a benchmark problem for the computation of viscoelastic flows as far back as 1988 [275]. Numerical simulations for the contraction problem have actually been around a lot longer, however: the finite difference calculations of Newtonian flow by Vrentes et al. [601] were published in 1966 and viscoelastic simulations first appeared in the literature in 1981 with the paper by Cochrane et al. [143]. Interest in the entry-flow problem itself dates back even further, to 1860 with the work by Hagenbach [263], and some 30 years later by Couette [147] and Boussinesq [96], whose primary interest lay in the development of a capillary rheometer to measure the viscosity of Newtonian fluids. Excellent review articles describing the experimental, theoretical and numerical work for the contraction up to the late 1980's have been written by Boger [84] and White et al. [628] and further details of historical developments in the analysis of viscoelastic entry flows may be found there. A keynote paper by Walters and Webster [611] in 2001 provided a brief review of some landmark achievements in computational rheology (confined, regrettably, however to sim ulations with models of the PTT and Oldroyd-Maxwell type) and a description of some new attempts to obtain agreement between numerical and experimental work for contraction-flow problems.
8.1
Vortex Growth Dynamics
Undoubtedly one of the main topics of interest in non-Newtonian entry flows is understanding the mechanisms underlying the various flow transitions that occur as the Deborah number is increased. In particular, there has been much discussion in the literature attempting to identify the relevant parameters dic tating the different ways in which vortices grow (or diminish) in non-Newtonian fluids. In the review that follows, and as indicated in Fig. 8.1, we denote by Du, D and L\ the upstream tube diameter, the downstream tube diameter and the vortex attachment length, respectively. Throughout the discussion fi refers to the contraction ratio Du/D and 6 the opening angle of the corner vortex, as also shown in Fig. 8.1. In the following two subsections we will summarize what has been observed experimentally for constant-viscosity Boger fluids before moving on to describe the corresponding results for shear-thinning fluids.
8.1.1
Boger fluids - observed flow transitions
It was not until 1979 and the paper by Nguyen and Boger [417], that the flow of constant viscosity viscoelastic fluids through a contraction was studied. By taking a dilute solution of polyacrylamide (PAA) in a highly viscous glucose syrup the authors were able to eliminate both inertial and shear-thinning effects in their circular contraction flow experiments. The authors performed their experiments for a number of different contraction ratios in the range [7.67,14.83] and found that, beyond the shear-rate giving rise to a quadratic normal stress behaviour, the salient corner vortices grew dramatically in size. Indeed, defining a Weissenberg number We based on the downstream average velocity and a Maxwell relaxation time computed from the downstream wall shear-rate, it was found that increasing We from 0.218 to 0.263 led to a 100% increase in the vortex attachment length. The dimensionless vortex length x = Lx/Du for each fluid tested was found to be a unique function of We, independent of 0, for 202
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS this vortex growth regime. It was found, furthermore, that by plotting j3We/x against an elasticity number El = •q^Xi/pD2, where p is the fluid density, that all the data for different fluids (including a shear-thinning fluid that was tested) fell on the same curve. Nguyen and Boger [417] concluded that elasticity was the sole factor responsible for the formation and growth of the observed vortices. Increasing the Weissenberg number further led to continued growth of the vortices and to a breaking of the axisymmetry of the flow. Larger shear-rates still (c. 50s" 1 ) resulted in rotation of the axisymmetric vortices around the curved surface of the larger tube until at wall shear-rates of 3 0 0 s - 1 and above helical flow was observed, the material from the vortex region spurting periodically into the downstream tube. All the flow transitions observed by Nguyen and Boger occurred at very low (< 2.9 x 10~2) Reynolds numbers. The work of Muller [413] and Lawler et al. [347] represent the first quanti tative measurements of the velocity field in viscoelastic entry flow. The authors considered flow of a polyisobutylene/polybutene (PIB/PB) Boger-type fluid through a 4:1 axisymmetric contraction. Lawler et al. [347] defined a Deborah number De in terms of the downstream velocity, radius and viscometric func tions, and showed that the flow became time-dependent and three-dimensional near the contraction lip for De « 0.8. At a second critical Deborah number De « 1 . 2 a vortex that was steady and two-dimensional formed at the lip. Boger et al. [89] performed experiments with two different Boger-type flu ids: a PAA/corn syrup solution and a PIB/PB solution, through circular con tractions and a variety of contraction ratios (3 in the range 4 to 16. The two test fluids were chosen to have very similar dynamic and steady shear properties. However, the vortex growth dynamics observed, for example, for the 4:1 circular contraction were very different: before the PAA/corn syrup fluid vortex showed any significant signs of growth the cell boundary passed from being concave (at lower shear-rates) to convex (at higher shear-rates) and the centre of rotation moved nearer the contraction lip. For values of A17 > 1 the cell reattachment length increased. A photographic sequence showing the principal vortex growth transitions for the PAA/corn syrup solution is shown in Fig. 8.2. In the case of the PIB/PB fluid the cell reattachment length decreased from the Newtonian value for A17 > 1.6 to approximately zero at A17 = 2.37. As the cell length decreased a second vortex, emanating from the lip, grew and eventually took over the salient corner vortex and the reattachment length of the new vortex was seen to grow with A17. For (3 = 16 the entry flow fields developed in the same way and no lip vortices were observed. Lip vortices could be seen for the PAA/CS fluid only for f3 < 2 [84]. Boger et al. [89] concluded from the results above that clearly a knowledge of the steady and dynamic shear prop erties of a fluid were insufficient for predicting its behaviour in circular entry flows. Boger [84], a year later, stated that the probable explanation for the difference in the behaviour of the two fluids lay in their different extensional viscosities. Consistent with the observations of Lawler et al. [347], Boger and co-workers [84,89] noticed for lower contraction ratios that when lip vortices be came dominant for either fluids these vortices rotates with increasing frequency as A17 increased until at some critical value of A17 the flow again became twodimensional and steady. Both the results of Lawler et al. [347] and Boger et al. [89] then showed a transition back to three-dimensional time-dependent flow at significantly higher Weissenberg numbers. The flow transitions of a highly elastic PIB/PB Boger fluid through axisym203
8.1. VORTEX GROWTH DYNAMICS
(a)
(b)
(c)
(d)
Figure 8.2: Vortex transitions for a 0.04% PAA/corn syrup-water solution in a 4:1 circular contraction, (a) 7 = 1.1s""1, Re = 5.7 x 10~ 4 , We = 0.079, (b) 7 = 3.4s" 1 , Re = 1.76 x lO" 3 , We = 0.120, (c) 7 = 9.3s" 1 , Re = 4.8 x 10" 3 , We = 0.179, (d) 7 = 24.2s" 1 , Re = 1.25 x 10~ 2 , We = 0.204. Reprinted from D. V. Boger, D. U. Hur and R. J. Binnington, Further observations of elastic effects in tubular entry flows, J. Non-Newtonian Fluid Mech., 20:31-49, Copyright (1986), with permission from Elsevier Science.
204
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS metric contractions for a wide variety of contraction ratios (/3 = 2,3,4,5,6 and 8) were the subject of an experimental study by McKinley et al. [389] in 1991. The authors defined a shear-rate dependent Deborah number in the same way as Lawler et al. [347], calculating this from a Maxwell relaxation time [84] which in turn was derived from the viscoelastic properties of the fluid in the downstream tube. For contraction ratios 2 < /3 < 5, McKinley et al. [389] found that at a critical Deborah number (De = 1.5 for the 4:1 contraction) the flow near the lip underwent a Hopf bifurcation to a three-dimensional, time-dependent motion, the velocity previously being steady and two-dimensional. Although a similar Hopf bifurcation had been observed by Lawler et al. [347] for a 4:1 axisymmetric contraction flow, the flow was not observed by McKinley et al. [389] to subse quently revert to a steady two-dimensional state at higher shear-rates. No Hopf bifurcation was seen by the authors at higher contraction ratios. For 3 < /9 < 5 and higher Deborah numbers McKinley et al. [389] saw a period-doubling tran sition in the velocity components near the lip. Subsequently the development of a quasi-periodic state for 2 < ft < 5, occurring at a Deborah number De w 3.0 and almost independent of the contraction ratio, was recorded. It was at this stage that the formation of a lip vortex could be observed. As seen by Boger and co-workers [84,89] the lip vortex then grew upstream and radially outwards as the Deborah number continued to increase. Once the lip vortex reached the corner of the upstream tube it grew rapidly to form a large convex elastic vortex, as described previously by other workers in the field. At the same time as the flow passed through the transitions just described, changes in other flow features could be seen: for 2 < /3 < 5 and at Deborah numbers comparable to those at which the lip vortex became evident, the axial velocity was seen to decrease as the fluid approached the contraction throat, before accelerating again closer to the contraction plane. This resulted in the now well-documented phenomenon of divergent flow, shown by McKinley et al. [389] to have no direct relation with the appearance of lip vortices. The sensitivity of the diverging flow to the contraction ratio led McKinley et al. [389] to suggest that upstream shear-rates and the dependence of the total Hencky strain /■CO
e=
fU;(co)
i d t = JO
Ju,(-GO)
J
^=ln(/?2),
(8.1)
U
Z
on /? may be important in accounting for the appearance of the diverging flow. Once the reattachment length of the elastic vortex reached its maximum value a further increase in De led to the onset, when /3 equalled 4, of a pulsating axisymmetric vortex. A pulsating vortex was also seen when /3 was equal to 3. However, for larger /? the vortex became asymmetric. Furthermore, it no longer pulsated but rather rotated around the upstream tube. The flow behaviour of a PIB/PB fluid in a 4:1 axisymmetric contraction was further investigated by Yesilata et al. [640] who determined the critical conditions for the onset of elastic instabilities by measuring pressure differences across the contraction plane as the flow rate was increased. The first elastic instability recorded by Yesilata et al. [640] was a transition to a weakly timeperiodic flow, identified as the lip instability of Lawler et al. [347] and McKinley et al. [389]. By using the same definition of the Deborah number as McKinley et al. [389] (based on the Maxwell relaxation time) the flow transition was seen 205
8.1. VORTEX GROWTH DYNAMICS to occur in a critical value range 1.57 < Dent < 1.67 and agreed well with the value of 1.5 given by McKinley et al. [389]. The corresponding De range based on an Oldroyd relaxation time (see [84]) was 4.5 < Deo < 4.8. What was not reported in the literature prior to the paper by Yesilata et al. [640] was that, although this first instability was localized near the lip and decayed rapidly away from the contraction plane in the upstream direction, the instability was convected into the downstream tube. As the Deborah number Deo increased further a sudden decrease was found in the frequency of the oscillations in the pressure measurements. At these high Deborah numbers (8.2 < Deo < 12.0) the upstream flow underwent a bifurcation from axisymmetric steady flow to a three-dimensional time-periodic flow. Between the first instability and this second one the lip vortices were seen to grow rapidly radially and then axially. Finally, at a Deborah number of Deo « 13.5 the flow appeared irregularly time-dependent. Yesilata et al. [640] concluded that the growth of elastic vortices observed by them for Deo > 12 was in qualitative agreement with the explanation put forward by Cogswell [144] and Binding [66], who stated that significant vortex growth was attributable to high extensional viscosities: the PIB/PB fluid used in the experiments of Yesilata et al. [640] was tension-thickening. That instabilities originated near the lip in both the experiments of McKinley et al. [389] and Yesilata et al. [640] was not seen as surprising by the authors in [640]: in the global critical condition (9.3) for the onset of elastic instabilities proposed by McKinley et al. [392], the curvature 1/72. is a maximum in this region and the tensile stress Tu is large.
8.1.2
Boger fluids - effects of changes of geometry
In a further attempt to understand the mechanisms responsible for vortex growth in elastic liquids, several authors [87,198,607,610] have investigated the effect of changing the geometry on the flow of Boger-type fluids: by altering the contraction ratios, rounding the corners of the contraction lip and comparing the flow transitions for planar and circular contractions. Walters and Webster [610] experimented with several different constant vis cosity dilute solutions of PAA in a water/maltose syrup mixture. No significant vortex activity could be found in the flow of these fluids through a 4:1 planar contraction and rounding the corner in such a geometry slightly reduced the already weak vortices present for the sharp-cornered case. The observations contrasted markedly with what could be seen for a 4.4:1 circular contraction. Here, vortex enhancement was clearly visible as the flow rate (and hence both Reynolds number and Weissenberg number) increased, up to the point where fluid inertial effects became important. Rounding half of the re-entrant corner had the effect of reducing vortex activity in one region and increasing it in the other. A change in the lip geometry was thus seen to affect the whole flow field. The difference in the level of vortex activity for Boger fluids flowing through planar and three-dimensional geometries observed by Walters and Webster [610] was corroborated by Walters and Rawlinson [607] in their investigation of the flow of such fluids through a 13.3:1 planar contraction and a 13.3:1 squaresquare contraction geometry. Planar and square-square contraction geometries were again employed for experiments with PAA/water-syrup Boger fluids by Evans and Walters in 1986 [198]. A comparison of the flow characteristics for 206
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS two Boger fluids at the same Reynolds number passing through an 80:1 planar geometry revealed that, whereas in the case of the less elastic of the two there was no vortex activity, two lip vortices were clearly visible for the more elastic fluid. Moreover, there was an asymmetry in the streamlines that could not be attributed to geometrical imperfections. Consistent with the effect observed by Walters and Webster [610] when a corner was rounded, Evans and Walters found that cutting away a corner on one side of the square contraction lip had a profound influence on the global flow structure: the vortex on the side whose corner had been cut becoming smaller and the other (looking at the flow through the plane of symmetry) increasing in size. Boger fluids not only manifest enormous differences in the level of vortex activity, depending on whether the flow is through a planar or axisymmetric contraction, but would also seem to exhibit different mechanisms to bring about the large entry pressures in excess of those for Newtonian fluids having the same viscosity, and which have been observed at high flow rates. By "excess entry pressure" we mean the pressure that is in excess of that required to produce fully developed flow over the sum of the upstream and downstream sections. Binding and Walters [70] found that for an axisymmetric contraction the growth in the entry pressure was accompanied by the development of large recirculating zones. In the planar geometry the salient corner vortices all but disappeared after a certain flow rate. At the point where a significant increase in the excess entry pressure could be measured, however, there was manifested a "bulb" shape (divergent flow) in the flow field (see the paper of Cable and Boger [115]). At yet higher flow rates vortices returned but the flow was unstable. The results cited in the preceding paragraphs from the work of Walters and co-workers were largely presented by the original authors without any attempt being made to explain what was observed. An interesting demonstration of the effect of fluid rheology on the size and structure of recirculation cells in contraction flows was presented by Boger and Binnington in 1994 [87]. Although the two Boger fluids used by the authors (a PIB/PB and PAA/water-syrup solution) each had approximately the same zeroshear Maxwell relaxation time they reacted very differently to the rounding of the corner in a 4:1 circular contraction. Flow patterns for the PIB/PB solution, the so-called Ml fluid, had already been presented for the 4:1 circular abrupt contraction by Boger and Binnington in 1990 [86]. Whereas the vortex in the abrupt contraction grew to a maximum cell size of x = 0-22 as the flow rate was increased, no significant differences could be seen between the vortex in the rounded entry flow and the Newtonian value. A change in shape of the abrupt contraction vortex from concave to convex with increasing flow rate also distinguished it from the rounded entry flow. At a downstream shear-rate of 3 0 2 s - 1 a lip vortex could be seen for the abrupt contraction and was entirely absent from the smoothed contraction at 7 « 473s - 1 . "Bulb flow", observed previously by Binding and Walters [70], was more pronounced for the abrupt entry flow than for the smoothed case. Flow of the PAA/corn syrup-water fluid in the same two geometries led to far more dramatic differences, however. For the abrupt entry the corner vor tices grew rapidly until the flow lost its symmetry and began to rotate. At the rounded entry, in contrast, it was the growth of the lip vortex and subse quent expansion into a corner vortex that primarily accounted for an increase in the vortex attachment length at higher shear-rates. Rounding the corner, 207
8.1. VORTEX GROWTH DYNAMICS argued Boger and Binnington [86], was seen to be effectively the same thing as decreasing the contraction ratio (3. As will be discussed in the next chapter, in the section dealing with the drag on spheres falling through polymeric liquids (§9.2.1), two fluids having the same steady and dynamic shear properties may exhibit very different flow behaviour in the same experimental setup. Boger and Binnington [86], consistent with comments by Chmielewski et al. [135], speculated that the water/corn syrup solution was a better solvent for PAA than PB was for PIB. Hence, at equilib rium PAA is more extended relative to its full length than PIB and thus the PIB may be extended further than the PAA before vortices, conjectured to be a stress-relief mechanism [624], need to grow. Thus the extensional rheological properties of the two fluids were supposed to be significant in explaining the different reactions of the two Boger fluids to rounding the corners of the con traction. Extensional data [389,544] is now available for the two types of Boger fluid used by Boger and Binnington [86] and will be discussed in the paragraphs that follow. The combined influence of changes of geometry, flow rate and fluid rheology on pressure drop and vortex growth dynamics were investigated in a paper by Rothstein and McKinley [519] who considered flow of a solution of polystyrene in oligomeric styrene (PS/PS) through 2:1:2, 4:1:4 and 8:1:8 contraction-expansion geometries. This paper extended earlier work done by the same authors [518] for the 4:1:4 case and, importantly, utilized transient extensional rheology mea surements to explain the huge differences observed in the literature in vortex growth transitions for different Boger fluids in different geometries. A dimensionless pressure drop Ap(De, /3) was defined by Rothstein and McKinley [519] for a sharp axisymmetric contraction-expansion in the same manner as their earlier paper [518] by computing the difference in the pressure measurements upstream and downstream of the contraction, subtracting the pressure difference due to Poiseuille flow in the geometry and finally normaliz ing by dividing the result by the pressure drop for a Newtonian fluid. Monotonic increases in Ap were found for all contraction ratios once a /3-dependent criti cal Deborah number was exceeded. These increases were accompanied by the onset of elastic flow instabilities. Unlike the case of the PAA/CS fluid studied by Boger and Binnington [87], rounding the corner did no more than shift the onset conditions for different flow regimes to higher Deborah numbers. As had been commented in the earlier paper of Rothstein and McKinley [518], no nu merical simulation using simple dumbbell models [142,321,555] had been able to predict the large additional pressure drops observed beyond critical Deborah numbers: probably because of the failure of the models used in [142,321,555] to predict the stress-conformation hysteresis first seen by Doyle et al. [189] in measurements of uniaxial transient extensional flow. By taking account of an approximate residence time for a fluid element in the vicinity of the curved throat region, a shifted Deborah number incorporating the radius of curvature of the rounded corner could be used to collapse the dimensionless pressure drop data on to a single curve, for a given value of /3. This could be important for numerical analysts since it would allow them to compute numerical solutions for contraction problems with rounded corners (thus avoiding a singularity) but deduce pressure drop data for the sharp-cornered case. Results for the 4:1:4 and 8:1:4 contraction-expansions showed similar vortex growth dynamics: the Newtonian-like Moffat-type vortices growing in size and 208
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS strength with Deborah number, until the onset of a global instability, the vortex now precessing in the azimuthal direction in conjunction with a fluid jet into the throat upstream of the contraction plane. Rounding the corners just shunted the vortex growth dynamics to higher Deborah numbers. In contrast with the higher contraction ratios, the flow for the 2:1:2 case showed a decrease in the corner vortex size at Deborah numbers approaching unity and the development of a lip vortex that grew outwards and upwards. Interestingly, and in seeming contradiction to previous experimental measurements [87,389], the lip vortex was found to be steady in time. Two particularly important results in connection with vortex growth dynamics were presented by Rothstein and McKinley [519]. First, the authors provided graphical evidence that it was possible, as described a little earlier, for the pressure drop data to predict a single generic vortex size for a given contraction ratio by plotting the attachment length against a Deborah number shifted to take account of the radius of curvature of the rounded corner. Secondly, and more importantly, Rothstein and McKinley [519] gave the first quantitative evidence in the literature that the flow kinematics for different viscoelastic fluids in entry flows could be rationalized on the basis of rheological data. Specifically, the authors showed how the ratio of normal stresses arising from the wall shear flow and the extensional stresses resulting from the centreline extensional flow could be used to predict whether or not, for a given contraction ratio, vortex growth would originate at the lip or the salient corner. We will discuss this result further in §8.2.1.
8.1.3
Concentrated polymer solutions and melts - observed flow transitions
The first entry flow study conducted with a polymer melt was by Tordella [579] in 1957 who demonstrated the existence of recirculating vortices in a low density polyethylene (LDPE) melt in a circular contraction. A good summary of exper imental work up to 1986 involving both polymer solutions and polymer melts may be found in Tables 1 and 2 of the review of White et al. [628]. Although there exist several other important papers in the literature of the early 1970's dealing with polymer melt entry flows (e.g. [34,176]) we begin our review with the publication of Cogswell [144] in 1972. Our choice of starting point has been motivated by the depth of insight shown by Cogswell [144] in interpreting the die entry flow profiles he obtained for flows of LDPE, polypropylene and acrylic. Cogswell [144] showed that melts having an extensional viscosity that increased with extensional rate exhibited vortex growth. White and Kondo [624], some five years later, undertook a study of melt flow from a reservoir into a capil lary die (j3 fa 5.1) and used polystyrene, LDPE and high density polyethylene (HDPE) in their investigation. Of the three types of melts investigated the LDPE and polystyrene both exhibited vortices but the HDPE did not. White and Kondo [624] noted the connection drawn by Cogswell [144] between exten sional flow properties and the potential for vortex growth in abrupt contractions, and believed his idea to be basically sound. Vortices, they further stated, could be interpreted in terms of the development of large tensile stresses disrupting radial flow and induced by the extensional flow going into the contraction. Vor tices develop as a stress-relief mechanism. The first truly comprehensive experimental investigation of vortex growth dynamics for polymer solutions in contraction flows was done by Cable and 209
8.1. VORTEX GROWTH DYNAMICS Boger in 1978-79 and published in a series of three landmark papers [115-117]. The authors examined the vortex flow patterns for solutions of polyacrylamide in both 4:1 and 2:1 circular contractions. Before the onset of unstable flow they identified two distinct flow regimes: a vortex growth regime at low flow rates and a divergent flow ("bulb flow") regime at moderate flow rates, the latter being marked additionally by a decrease in vortex size as inertia! effects began to be important. The test fluids all had similar shear-thinning properties and characterization of the fluids was restricted to steady shear data. Although Cable and Boger [115-117] did not state the reasons for vortex growth they nevertheless proposed that, in the vortex growth regime, the nature of the entry flow pattern was completely determined by the ratio of elastic to viscous forces (a Weissenberg number) and that the dimensionless attachment length \ = Li jDu was proportional to a Weissenberg number We defined in terms of the Maxwell relaxation time. The authors went on to show that for both the vortex growth and divergent flow regime the We/x data for all the fluids could be collapsed on to a single curve when plotted against a suitably defined Reynolds number. Boger et al. [89] later admitted that knowledge of the extensional properties of the fluid may be required to give an adequate explanation of flow behaviour. Cable and Boger [115-117] proceeded to show that beyond the vortex growth regime, a slow increase in the flow rate led to a steady two-dimensional flow regime which they termed divergent flow and associated with a deceleration of the fluid elements along the centreline as the contraction throat was approached. In the 4:1 contraction and in the divergent flow regime multiple steady flow states were possible at the same volumetric flow rate and this was interpreted by the authors as an instability. By reducing the flow rate from a higher value (associated with random distortions) down to that which was typical of the divergent flow regime a periodic three-dimensional flow was obtained. Further reduction yielded a transition to the vortex growth regime, so that by decel erating the flow rate the divergent flow regime was bypassed altogether. By increasing the flow rate progressively through the vortex growth and divergent regimes a transition to random and unstable flow was seen. Whereas passage from the vortex growth regime to the divergent flow regime was marked by a maximum vortex size, transition from the divergent flow regime to chaotic flow was characterized by the authors as occurring at a critical value (increasing with Reynolds number) of a normal to shear stress ratio. In two important and insightful papers, White and Baird [625,626] at tempted to explain why significant vortex growth could be observed in LDPE flow through 4:1 and 8:1 planar contractions but not for flow of polystyrene through the same geometries. Slightly larger vortices were observable for the LDPE 8:1 experiment than in the 4:1 case. In [625] the authors attributed the vortex growth for LDPE for both contraction ratios to the polymer's extensional stress growth. In [626], that vortex growth could not be attributed to the magni tude of the shear and normal stress difference alone, was confirmed by comparing these quantities for LDPE at 150°C and polystyrene at 165°C and 190°C over several decades of shear-rate. The viscometric function values for LDPE at any fixed shear-rate j were higher than those for PS at 190°C but lower than those for PS at 165°C Yet LDPE had larger vortices for comparable shear-rates than the PS melt at either temperature. Neither could vortex growth be correlated solely with fluid elasticity. By plotting an approximation to the Weissenberg
210
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS number against shear-rate, it was shown by White and Baird [625,626] that LDPE had a higher Weissenberg number than either of the polystyrenes at low shear-rates and the lowest Weissenberg number for shear-rates 7 « 100s""1. Yet for all shear-rates LDPE exhibited larger vortices. Did this result contradict those of Nguyen and Boger [417] and White and Kondo [624] then, who at tempted to correlate vortex growth with fluid elasticity? Not necessarily, since White and Baird [626] were establishing the point that elasticity could not be used to distinguish between the flow behaviour of different polymers. Given the inadequacy of shear properties to explain entry flow behaviour it was natural to turn to the extensional properties of the melts to gain some un derstanding. Examination of the transient extensional viscosity of LDPE over a wide range of extensional rates showed that it exhibited unbounded growth with time, whereas the equivalent quantities for polystyrene at 165°C and 190°C showed bounded growth. The difficulty with correlating vortex growth with ex tensional stress, however, was that the same magnitude of extensional stress as was observed in LDPE could be generated in the PS melt by sufficiently increas ing the flow rate: but still no comparable vortices! Birefringence measurements on the LDPE and polystyrene made it possible to calculate the ratio of the first extensional stress difference Tn — T22 along the centreline to the downstream wall shear stress Ti2W- Although no full explanation could be offered by the authors for why N
_ Tn -T22 1~12w
should be an important factor in determining vortex growth it could be seen clearly that LDPE had an N ratio value more than twice that of PS at 190°C for the 4:1 and 8:1 contraction ratios. Moreover, iV increased for LDPE when the contraction ratio went from 4:1 to 8:1, consistent with the observation of larger vortices in the larger contraction ratio for LDPE. In contrast N did not change significantly between the two contraction ratios for polystyrene. A high molecular weight PIB dissolved in tetradecane (C14) was the test fluid of choice for laser-Doppler velocimetry (LDV) and birefringence (FIB) studies of viscoelastic entry flows by Raiford et al. [481] and Quinzani et al. [478]. The fluid had been thoroughly characterized in linear viscoelastic and shear flows by Quinzani et al. [480] in an earlier paper. Raiford et al. [481] used LDV measurements of the velocity components to describe flow transitions of the (shear-thinning) test fluid in axisymmetric contraction geometries having contraction ratios /3 = 2, 4 and 8. Experiments were performed for each of the three contraction ratios at low, moderate and high flow rates. Inertial forces were negligible for the low flow rates and dominant for the high flow rates. For low flow rates the authors defined a dimensionless axial velocity by scaling it with the average downstream axial velocity. Dimensionless coordinates were defined by dividing them by the downstream tube radius. It was then found that in the near-Newtonian limit the flow near the contraction plane scaled with the flow in the small tube at fixed dimensionless axial positions near the contraction plane. The salient corner vortices were reported to be of Newtonian flow proportions (x w 0.17). At moderate flow rates the corner vortex had increased slightly in size (x & 0.25) and of particular interest now was evidence of an accelerating core in the centre of the upstream tube as the contraction plane was approached, in dicative of growing extension rates and extension-thinning in this region. Scaled 211
8.1. VORTEX GROWTH DYNAMICS radial profiles of the upstream dimensionless axial velocity continued to show an accelerating core as the flow rate was increased yet further, but now the axial velocity near the wall retained its upstream shape, consistent with the observation also made that the corner vortex had decreased substantially due to inertial effects. At this high flow rate (Q = 96cm 3 /s) a small lip vortex was now in evidence for all values of /? but inertial forces suppressed any growth in this new feature with increasing flow rate. The experimental work of Quinzani et al. [478] with the same fluid used both LDV and FIB to measure velocity and stress fields for flow through a 3.97:1 planar contraction. Measurements at six different flow rates (Q = 34.8cm3fs to Q = 360cm 3 /s) were performed and graphs of profiles of the axial velocity, shear stress and first normal stress difference at various scanning locations upstream and downstream of the contraction were supplied by the authors. In contrast to the slight increase in vortex attachment length seen by Raiford et al. [481] as the flow rate was increased to moderate levels, Quinzani et al. [478] observed a shrinking vortex as their (shear-rate dependent) Deborah number De increased to De = 0.66. This was not surprising since as the flow rate increased so did the shear-thinning, inertial and (presumably) tension-thinning effects. No lip vortices were seen by the authors. The different qualitative behaviour of shear-thinning polymeric fluids in axisymmetric and planar contractions will be discussed further in the next subsection. A challenge to providing too simplistic an explanation of the role played by extensional flow properties of viscoelastic fluids in vortex growth dynamics was issued by Byars et al. [114] in the course of their report on entry flow experi ments with the test fluid SI; a shear-thinning solution of PIB in a PB/decalin solvent. Whilst admitting that extensional behaviour plays an important role in determining the vortex size in polymeric fluids, Byars et al. [114] pointed out that the extensional stress growth coefficient 77+ for flow of the fluid SI in a 4:1 axisymmetric geometry only reached about 1% of its steady value and the transient Trouton ratio Tr+ = r}f(e,t)/r)(\/3e) would not be large in value and largely independent of strain rate e. That differences in the steady extensional viscosity between two different polymeric fluids could not be used to explain the differing vortex growth dynamics was made explicit by the authors in con sidering entry flows of PIB and PAA fluids. The vortex growth dynamics are known to be very different but it is evident that if Tr+ is considerably lower in a 4:1 contraction than the steady-state value for PIB one cannot appeal to the differences in the steady-state Trouton ratios for PAA and PIB fluids to explain the differences in the respective vortex growth pathways. Therefore, Byars et al. [114] proposed, it was necessary to consider the extensional stress growth behaviour at low strains. Taking a Weissenberg number We defined in terms of the downstream shearrate dependent properties, Byars et al. [114] showed that for We > 0.65 as the Weissenberg number increased so did the vortex attachment length x, reaching a value in excess of 0.5 for We = 5. This was despite the growing influence of shear-thinning and inertial effects with increasing flow rate. Rounding the corner of the contraction resulted in a simple shift of the Weissenberg number at which the various vortex sizes observed for the abrupt contraction were realized. Unlike the experiments of Raiford et al. [481], no lip vortex was seen at any time in the flow transitions observed by Byars et al. [114] and vortex growth was from the salient corner. 212
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS
8.1.4
Concentrated polymer solutions and melts - effects of change of geometry
Mention has been made already in the previous section of the differences in flow phenomena observed by Quinzani et al. [478] and Raiford et al. [481] in planar and axisymmetric geometries, respectively, for flow of a PIB/C14 solution. It has also been seen that rounding the corner in the 4:1 axisymmetric contraction experiment of Byars et al. [114] did not change the dynamics substantially apart from shifting the observed transitions to a higher Weissenberg number. This shift in Weissenberg number with no associated change of the structure of the transitions is similar then to what was seen by McKinley et al. [389] for their Boger-type fluids. As mentioned in §8.1.2 Boger and Binnington [87] observed dramatic changes when the corner was rounded for entry flow of a PAA/watersyrup Boger fluid, however, so the simple Weissenberg number shift does not, unfortunately, hold for all fluids. Walters and Webster [610] sought to assess the effects of geometry changes on the vortex size of an aqueous solution of PA A. The recirculating regions for both a 4:1 abrupt planar contraction and 4.4:1 circular abrupt contraction were both substantial in size and this was to be contrasted with the virtually non-existent vortices for the flow of their PAA Boger-type solution in the 4:1 planar geometry. The recirculating flows for the aqueous solution in both planar and axisymmetric cases were very weak, however. It is of particular interest to note here that the difference in vortex sizes between the aqueous PAA solution and the Boger-type PAA solution cannot be accounted for just on the basis of elasticity. Rounding the corners of either geometry resulted in a reduction of the size of the vortex region. An important contribution to our understanding of the dependence on con traction ratio of the vortex transitions which a polymer solution undergoes was supplied by Evans and Walters [198] in 1986. In their experiments the same 1% aqueous solution of polyacrylamide was forced through planar abrupt con tractions with /? = 4, 16 and 80, at increasing flow rates. It was found that for the 4:1 case the salient corner vortex grew in size and intensity, the vortex boundary gradually going from concave (Newtonian-like) to convex. See Fig. 8.3. For the higher contraction ratios the salient corner grew towards the re entrant lip until a separate lip vortex developed there too, the re-entrant corner vortex then growing to embrace the salient corner vortex and the attachment length increasing with further increases in the flow rate. In a later paper [199], the same authors considered the question of whether or not a change in the polyacrylamide concentration would lead to a lip vortex even for the 4:1 planar contraction. Although a salient corner vortex enhancement was observed for 0.5% and 0.3% PAA solutions, a lip vortex did indeed appear and grow for a 0.2% PAA solution. The sequence of vortex transitions for this concentration may be seen in Fig. 8.4. Rounding the corner with the 0.2% PAA solution led to no vortex enhancement whatsoever. The results of Evans and Walters [199] for different polymer concentrations were discussed, and an interpretation in terms of the competing effects of elasticity and inertia given, in the paper by Xue et al. [637]. Further details may be found in §8.2.1. White and Baird [626] proposed an interpretation based on the extensional properties of their LDPE and polystyrene melts to account for the fact that whereas larger vortices at comparable flow rates could be observed for flow 213
8.1. VORTEX GROWTH DYNAMICS
Figure 8.3: Sequence of photographs showing the streamlines for a 1% aqueous solution of polyacrylamide in a 4:1 planar contraction. The flow rate increases in moving from (A) through to (F). Reprinted from R. E. Evans and K. Wal ters, Flow characteristics associated with abrupt changes in geometry in the case of highly elastic liquids, J. Non-Newtonian Fluid Mech., 20:11-29, Copyright (1986), with permission from Elsevier Science.
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CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS
Figure 8.4: Sequence of photographs showing the streamlines for a 0.2% aqueous solution of polyacrylamide in a 4:1 planar contraction. The flow rate increases in moving from (A) through to (F). Note, in contrast to Fig. 8.3 that the devel opment of a lip vortex is now clearly visible. Reprinted from R. E. Evans and K. Walters, Further remarks on the lip-vortex mechanism of vortex enhance ment in planar-contraction flows, J. Non-Newtonian Fluid Mech., 32:95-105, Copyright (1989), with permission from Elsevier Science.
215
8.2. VORTEX GROWTH MECHANISMS of LDPE through an 8:1 planar contraction than in a 4:1 contraction, little difference was obvious for the polystyrene melt. White and Baird [626] argued that in the (3 = 8 contraction the change in the fluid velocities going from the upstream channel to the downstream channel would be greater than in the 4:1 case. Hence, the extensional strain rates would be greater, implying that for a tension-thickening fluid such as LDPE larger extensional stresses would develop and, as a consequence, larger vortices could be expected. The polystyrene's extensional viscosity, being only a weak function of extension rate, led to little observable difference in the vortices generated in the two cases. Raiford et al. [481] observed that the growing recirculation region for their 8:1 axisymmetric flow of a PIB/C14 fluid was much larger than in the 4:1 case. Lip vortices were seen to appear for all the contraction ratios (/? = 2, 4 and 8) but unlike in the experiments of Evans and Walters [198], these did not change in size with increased flow rate, due to suppression by inertial forces.
8.2 8.2.1
Vortex Growth Mechanisms Experimental work
It will be clear from the summary given in the preceding paragraphs of some of the experimental results that have been obtained over the years for the nonNewtonian entry flow problem, that the underlying fluid mechanics for vortex growth or suppression involves a rich and complex dependence on flow geometry and fluid rheology. Anyone attempting to undertake the task of giving a con vincing explanation for the greatly differing vortex growth dynamics observed in the laboratory for polymer solutions and melts, at varying flow rates, in either planar or circular contractions (of assorted contraction ratios) with or without rounded corners, does so at their peril! At issue is not only the reattachment length of the secondary cells but also whether or not growth, if present at all, originates principally from the contraction lip or salient corner. Perhaps that is why comparatively few authors have wished to commit themselves to giving reasons for what is seen in the laboratory, preferring instead to give a description of the results followed by (at most) a few vague words of conclusion about the importance of extensional stress effects, before moving swiftly and gratefully on to the acknowledgements and reference list! We cannot hope to do much better here, but feel compelled to say something about what is known (or proposed, at least) at the time of writing. As noted already, Cogswell [144,145] has been honoured as the founding father of the majority school of researchers who have seen clear evidence of the need to acknowledge the role played by the extensional rheology of a fluid in any attempts at reconciling the vortex growth phenomena measured for entry flows of different non-Newtonian fluids. Whilst some authors (e.g. [66,624]) have, since the paper of Cogswell [144] in 1972, openly called into question the correctness of the analysis contained therein, there is broad agreement today that the basic ideas of Cogswell [144,145] were sound. Although there are several authors who have sought correlations between the secondary cell size or type and shear-rate dependent data such as the Weissenberg number or elasticity number E = We/Re (e.g. [114,116,417,624,637]), it has been pointed out [626] that such correlations are of primary importance when considering cell
216
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS
Figure 8.5: Possible vortex mechanisms as functions of the elasticity number E and viscoelastic Mach number M. Reprinted from S. -C. Xue, N. PhanThien and R. I. Tanner, Three dimensional numerical simulations of viscoelastic flows through planar contractions, J. Non-Newtonian Fluid Mech., 74:195-245, Copyright (1998), with permission from Elsevier Science. sizes of a particular fluid. Arguments based solely on fluid elasticity have been shown in a number of papers to be inadequate as explanations accounting for differences in vortex behaviour between different fluids. As has been mentioned already, a striking example is to be found in the paper by White and Baird [626] dealing with the entry flow of an LDPE melt and polystyrene melts at 165°C and 190°C Elasticity could be ruled out as the explanation for the difference between the LDPE and polystyrene behaviour: whereas the LDPE had a higher (shear-rate dependent) Weissenberg number at low shear-rates than either of the polystyrenes, at intermediate shear-rates (7 « 10s - 1 ) its Weissenberg number was less than that of the polystyrene at 165°C but greater than the polystyrene at 190°C. At the highest shear-rates (7 « 100s - 1 ) the Weissenberg number for the LDPE was less than that of either of the two polystyrene melts. Similar conclusions could be drawn from the work of White and Kondo [624] and Boger and Binnington [87], for example. It should be added, to be fair, that the authors cited above [114,116,417, 624,637] still acknowledge the importance of the role played by the extensional behaviour of fluids in determining vortex size, particularly when a comparison is being made between flows of different fluids or through different geometries. For example, Xue et al. [637] observed that if extension plays a role in determining vortex behaviour then it must be the case that it is the transient extensional 217
8.2. VORTEX GROWTH MECHANISMS behaviour that is significant. However, in their finite volume computations with UCM and PTT fluids, Xue et al. [637] felt that, whether or not a salient corner mechanism or a lip vortex mechanism came to dominate, the vortex activity was determined by the competing processes characterized by a zero-shear-rate Weissenberg number We and a zero-shear-rate Reynolds number Re, or more conveniently by an elasticity number E = We/Re and a viscoelastic Mach number M = y/WeRe. In referring to the experimental work of Evans and Walters [199] it was claimed (see Fig. 8.5) that, for low polymer concentrations (lower viscosity), E could be sufficiently high to cause the appearance of a lip vortex mechanism. For yet higher values of E, such as would be the case for higher polymer concentrations or polymer melts, there is no lip vortex and salient corner enhancement is what is observed. Xue et al. [637] also proposed the scenario depicted in Fig. 8.5 as a means of understanding the different features of vortex activity seen in the work of White and Baird [626] with LDPE and PS melts, referred to a little earlier. What is confusing is that the qualitative picture shown in Fig. 8.5 was presented by Xue et al. [637] for a specific fluid in a 4:1 abrupt contraction so that, as has been highlighted already, interpretations based on non-extensional properties may have value in aiding the understanding of vortex mechanisms for a particular fluid in a particular geometry, under different flow conditions, but as a means of differentiating between one fluid and another are, to our minds at least, dubious. In our description of experimental work on non-Newtonian entry flows we have already cautioned, based upon the evidence available, against too simplistic an interpretation of vortex dynamics in terms of extensional properties of the test fluids. In the experiments of White and Baird [625,626] the magnitude of extensional stresses was eliminated as a possible reason for the observed differences in the vortices of LDPE and PS melts. Although the extensional stresses m — T22 along the centreline for LDPE were larger than for the PS melts for a given flow rate, by increasing the flow rate sufficiently in the PS melt experiments extensional stresses of the same magnitude as had been realized for LDPE at the lower flow rate were possible. But still there was no vortex growth in the PS melts. Mention has been made already of the perceptive observations of Byars et al. [114] with regard to the inappropriateness of citing differences in the steady extensional viscosity between fluids as the sole explanation for the differences in vortex growth behaviour observed, for example, for PIB and PAA fluids [87]. The analysis of Binding [66] in 1988 added another voice of support to the recognition amongst rheologists of the important role played by a ratio of exten sional and shear properties in predicting vortex growth mechanisms. Although the strongest arguments - some of which have been presented already in this section - are in favour of considering the transient behaviour of both extensional and shear stresses, Binding's work, dealing as it does with steady-state quan tities, is still a useful addition to the literature, not least because the author derived explicit formulae relating the entry pressure drop and vortex attachment length to the shear and extensional data for flow of a general viscoelastic fluid through both planar and circular contractions. In the following paragraphs we present some details of the planar contraction analysis but the axisymmetric case may be deduced without undue difficulty. In deriving his formulae Bind ing [66] assumed that the only non-zero component of the velocity field in the core region between the vortices was in the axial direction and was fully devel218
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS oped, that the entry angle 6 was small and that the separating curve between the core and recirculatory regions was a slowly varying function of the axial coordinate z. Moreover, Binding [66] expressed the steady-state shear viscosity T) and the extensional viscosity rje as power-law functions of shear-rate 7 and extensional rate e, respectively, to wit ri(i) r,e(e)
= =
*7n_1, U'-1,
(8-2) (8.3)
where k,£,n and t are constants. After minimizing the total power consumption required to maintain flow in the converging region between the vortices, Binding [66] showed that the envelope y = ±H{z) through which the core fluid passed in the planar case ("funnel flow") satisfied the equation (_dH\t+1 = fc(l+n)*+! V dz) £{2n + l)JntnH
iQ(2n + \ 2dH2n
l)\n-t J
^ ' '
where d is the depth of the channel, Jnt is an integral function of n and t, and Q is the volumetric flow rate. From (8.4) Binding [66] deduced that since n and t are both positive, enhancement (corresponding to a smaller value of -(dH/dz)) would occur \it>n, i.e. whenever the Trouton ratio was an increasing function of shear-rate. Hence, materials that tension-thickened and shear-thinned would exhibit vortex growth, as would materials that tension-thinned less rapidly than they shear-thinned. Binding [66] went on to state, in a manner consistent with the "stress-relief mechanism" idea of White and Kondo [624], that vortices grew with increasing flow rate in order to reduce extension rates (and therefore, the tensile stresses). By integrating (8.4) an explicit formula for the vortex attach ment length could be deduced. The theory of Binding [66] was put to the test the same year by Binding and Walters [70] in experiments in both circular and planar contractions with shear-thinning aqueous solutions of polyacrylamide, xanthan gum and a PAA Boger fluid. By minimizing the viscous dissipation in the converging region and performing an energy balance, Binding [66] had derived an expression for the entry pressure drop in the form
pe = f(k,£,n,t)^S^\
(8.5)
where / was some given function of the parameters k, I, n and t, and -yw was the downstream wall shear-rate, proportional to the volumetric flow rate Q. Thus, by measuring pe and Q and plotting logpe against logQ the slope t(n + l)/(t+l) could be read off and t deduced once n had been determined from the shear data. The constant £ could then be determined from a straightforward application of equation (8.5). Agreement between the resulting extensional vis cosity r)e and the extensional viscosity determined from a spinline rheometer for the shear-thinning polymer solutions was very good. Agreement was also good between the measured and predicted vortex attachment length for these polymer solutions for flow rates where the vortices were seen to grow in the laboratory, despite the fact that Binding's definition of the vortex attachment length was given as the distance over which the flow deviated from fully devel oped flow, and was therefore in practice longer than the actual vortex length. 219
8.2. VORTEX GROWTH MECHANISMS Tension-thinning in the xanthan gum experiments was adequately captured by the Binding model. The biggest disappointment was in the application of the Binding theory to entry flow of the Boger fluid. The fact that two completely different mechanisms led to enhancement of entry pressure (see §8.1.2), depend ing on whether flow was through a planar or axisymmetric contraction, with no one-to-one correlation between vortex enhancement and entry pressure, meant that extensional viscosity data from theory and experiment showed a marked discrepancy for the axisymmetric case tested although, somewhat unexpectedly, the Binding theory did well in predicting the vortex attachment length in this case. Despite the good agreement between the predictions of the Binding theory and the experimental measurements of Binding and Walters [70], at least for shear-thinning PAA solutions, Zirnsak and Boger [657] later cast doubt upon the value of such agreement by observing that the extensional viscosity mea surements of Binding and Walters [70] were not the steady state extensional vis cosities, as required by the Binding theory. Therefore Zirnsak and Boger [657] proposed testing the Binding theory from entry flow measurements for semidilute solutions of semi-rigid rod-like xanthan gum where it was possible to measure extensional viscosities more accurately. The authors found that in creasing the flow rate through their 4.4:1 axisymmetric contraction did little to increase the size of the secondary flow vortices and this was consistent with earlier observations of Binnington and Boger [71] for a 0.01% solution of xan than in a syrup/water solvent. Semi-dilute suspensions of rigid rods have been seen to behave similarly in axisymmetric contraction flows [37,357]. Rounding the contraction corner led to little difference in the vortex sizes seen by Zirnsak and Boger [657]. Vortex growth was, however, seen with an increase in gum concentration. The independence of vortex size with flow rate led Zirnsak and Boger [657] to suggest the importance of extensional properties of the fluid on vortex development and the attachment length of the secondary flow could be related well to the Trouton ratio for the solution. Power-law fits were made by Zirnsak and Boger [657] to their shear and extensional data (thus avoiding entry pressure measurements) and a theoretical attachment length was calculated from the Binding formula for different shearrates. Agreement with the experimentally measured vortex attachment length was excellent. Binding's theory has thus been shown to be a useful contribution to our understanding of the mechanisms underlying vortex development, even though steady shear and extensional properties are used. Since the original analysis in 1988 extensions incorporating the effects of fluid elasticity have appeared in the literature [67,69]. Maia and Binding [381] further extended the Binding analysis to deduce the extensional viscosity of the test fluid Si in circular contraction flows. Although the previous developments of the Binding theory were concerned with contraction flows in the high wall shear-rate regime, Maia and Binding [381] showed that the funnel flow (referred to earlier and required by the analysis) occurred at lower shear-rates than had been previously recognized. Therefore, a piecewise series of power-law approxi mations to the entry pressure drop could be taken at different values of the wall shear-rate, and the parameters t and i determined locally for the evaluation of the extensional viscosity r]e. The extensional viscosity coming from this modi fied Binding theory was then compared with that from the uniaxial stretching experiments of Ooi and Sridhar [430] for a circular contraction having a contrac220
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS tion ratio equal to 24.4:1, and showed initial strain-thinning followed by strong strain-hardening until breakdown at a wall shear-rate of 186s - 1 , this breakdown corresponding to the onset of flow instabilities. No obvious agreement could be seen between the theoretical predictions and the experiments. In fact, the mea surements of Ooi and Sridhar [430] showed strong strain-hardening followed at higher extensional rates by slower strain-thinning. This was mirrored in the growth and subsequent decay in the vortex cell sizes measured by Maia and Binding [381] and shown for contraction ratios between 24.4:1 and 100:1. An empirical dependence of the cell size \ o n the contraction ratio /? in the form X oc ft~0A was proposed by the authors. In [380] the opposite route to the one normally followed in the Binding theory was taken: knowing the uniaxial extensional viscosity for various consti tutive models chosen to model the fluid SI, Maia compared predicted pressure drops with those from experimental results. A comparison between the Couette correction predicted by a PTT model and a K-BKZ model (using the original inelastic Binding theory [66]) and the experimentally determined values at vari ous shear-rates showed that the P T T model was better at predicting the correct behaviour at low ( I s - 1 ) and high (> 10 3 s _ 1 ) shear-rates. The incorporation of elasticity into the Binding theory, as in [67], had only a minor influence on the P T T prediction of the Couette correction. A serious attempt at rationalizing the kinematics of flows of different poly meric fluids through contractions on the basis of their rheological properties was presented by Rothstein and McKinley [519] during the course of their experi mental investigation of the flow of a PS/PS Boger fluid through axisymmetric contraction/expansions having varying contraction ratios. Rothstein and McKinley [519] drew attention to the conjecture in the literature [89,389,518] that the difference in flow transitions observed for fluids having approximately the same shear properties must be accounted for on the basis of their differing extensional rheology. This could arise from differing solvent quality and hence differing equilibrium configurations of the dissolved polymer, or the differing stiffness of the polymer molecules themselves. Evolution of properties such as the transient extensional stress would be affected. Focussing now on one poly mer solution: a PS/PS solution, Rothstein and McKinley [519] associated the presence of a lip vortex for ft — 2 and its replacement by a corner vortex for contraction ratios of /3 = 4 and 8, with the fact that the flow was elastically shear-dominated at the lowest contraction ratio, and that the growth of the ex tensional stresses for the higher ratios was sufficient, in these cases, for a corner vortex to supplant the lip vortex. The same transition was not observed for a P I B / P B Boger fluid, where the lip vortex was present for j3 < 8, but it was pos sible that this could be attributed to the PS being dissolved in a better solvent than the PIB. Having argued that the flow kinematics involved in this transition from a lip to a corner vortex could arise from a change from shear-dominated to extension-dominated flow, Rothstein and McKinley [519] proposed that the transition be associated with a normal stress ratio £, given by
C=
Nlh0 ?
l
(TZZ
-Trr)/rj0e
.,
(8-6)
where Nx is the first normal stress difference generated by the shear flow near the walls, and «o is the zero-shear-rate viscosity of the fluid. A plot of £ against contraction ratio is shown in Fig. 8.6 for the PS/PS solution of Rothstein and 221
8.2. VORTEX GROWTH MECHANISMS
Figure 8.6: The normal stress ratio £ as a function of contraction ratio and evaluated at the rate of deformation corresponding to the onset of significant elastic vortex growth. ■: 0.025 wt.% PS/PS solution (7^ = 5.1s - 1 ) from [519], • : 0.31 wt.% PIB/PB solution (jw = 9.2s" 1 ) from [389], and two PAA/CS solutions; A: Fluid C (7™ = 0.14s" 1 ) and T: Fluid E (% = 2.5s" 1 ) from [544]. Reprinted from J. P. Rothstein and G. H. McKinley, The axisymmetric contraction-expansion: the role of extensional rheology on vortex growth dynam ics and the enhanced pressure drop, J. Non-Newtonian Fluid Mech., 98:33-63, Copyright (2001), with permission from Elsevier Science. McKinley [519], the PIB/PB solution of McKinley et al. [389] and two PAA/CS solutions characterized by Stokes [544]. The plots are shown for each fluid at the small tube shear-rate corresponding to the onset of the first significant elastic vortex growth, and readily demonstrate that the ratio C, is significantly larger for the PIB/PB fluid than for the other fluids, leading to different flow transitions. The importance of the work of Rothstein and McKinley [519] is that it represents the first attempt at quantitatively accounting for the difference in vortex transitions between one fluid and another in terms of their rheological properties.
8.2.2
Numerical studies
The importance of extensional effects in entry flows has been demonstrated in several numerical studies. Debbaut and Crochet [171] included dependence of the viscosity function upon both the second and third invariants of the rate-ofstrain tensor 7, for a number of inelastic and viscoelastic (UCM-type) models. In their finite element calculations of flow of a UCM fluid through a 4:1 abrupt circular contraction it was found that the corner vortex grew slowly with Deb222
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS orah number. However, when a generalized Newtonian model was employed that had the same extensional viscosity as the UCM fluid in uniaxial extension, but with zero first normal stress difference Nx in simple shear flow, there was a rapid growth in the vortex size, leading the authors to conclude that a rapidly growing extensional viscosity had a positive effect on vortex growth, but that Nx > 0 had the opposite effect. It is well known that for a viscosity function depending only on shear-rate, the effect of shear-thinning on an inelastic fluid is to decrease the vortex size and intensity [330]. This was confirmed by Deb baut and Crochet [171] with a shear-thinning Bird-Carreau fluid. Debbaut and Crochet [171] then chose a modified Bird-Carreau fluid with r\ now a constant in uniaxial extensional flow and monotonic decreasing with shear-rate in simple shear flow. Vortex size and intensity were now growing functions of Weissenberg number. The obvious difference between the two Bird-Carreau fluids lay in the magnitude of their Trouton ratios: 3 for the first Bird-Carreau fluid, significantly larger for the second. In a finite element study of vortex growth for polyethylene melts, Luo and Mitsoulis [369] in 1990 demonstrated, using a multi-mode K-BKZ model and a circular contraction geometry, that whereas LDPE melts exhibited rapid vortex growth, HDPE melts did not. Luo and Mitsoulis [369] concluded that the crucial factor in the different vortex development with flow rate was that the LDPE had a strain-thickening extensional viscosity whereas the extensional viscosity for HDPE was strain-thinning. Further evidence of the important role played by extensional effects in con traction flows was provided by Keiller [321] in a paper dealing with finite dif ference calculations of /? = oo entry flows of FENE-type [134] fluids. Keiller defined a Weissenberg number We based upon the downstream wall shear-rate and observed that for an Oldroyd B fluid the vortices were weak in both axisymmetric and planar contractions. For the FENE fluid, vortices that grew in both size and strength with We were observed, although the vortices for planar flow were much weaker than in the axisymmetric case. What was interesting about the vortex strength for the FENE model was how it varied with the maximum extensibility parameter Q0 as a function of We. For We < 48 larger vortices were seen with Q% = 25 than for Ql = 100. However, the opposite was true for large We. Keiller [321] interpreted this result by saying that for the shorter spring less extension was required before the nonlinearity of the spring resulted in enhanced extensional stresses. At higher Weissenberg numbers the more ex tensible spring enters its nonlinear regime and the higher extensional viscosities realisable in this case manifest themselves in higher extensional stresses and larger vortices. The PTT model has been a useful tool in the hands of numericists to probe any connection that may exist between vortex growth and the extensional vis cosity r)e. Byars et al. [114], in seeking to establish that the maximum steady value of rje had little effect on calculated entry flow, cited the numerical results of Debbaut et al. [172] who tested the sensitivity of vortex size and intensity on the adjustable parameters £ and e (see Eqns. (2.143) and (2.144)) in the P T T model. By fixing £ and varying e Debbaut et al. [172] were able to change the extensional rheology. However, the numerical results showed little dependence on e. It should be pointed out at this juncture, however, that more recent cal culations [36] than those of Debbaut et al. [172] show that the vortex size in a 4:1 planar contraction can exhibit great sensitivity to variations in e. Clearly, 223
8.2. VORTEX GROWTH MECHANISMS supporting an argument with one set of numerical results can be risky! The importance of the role played by a fluid's extensional rheology in deter mining vortex growth patterns was investigated for the P T T models by Saramito and Piau [525]. Four different fluids were used in their numerical experiments: two being of the exponential type and two of the linear type (see §2.6.1 and Eqns. (2.143) and (2.144)). One of the exponential models (Fluid 1) had an extensional viscosity that was a monotonic decreasing function of the exten sion rate; the other (Fluid 2) had an extensional viscosity that went through a maximum before decaying at higher extension rates. Both the linear PTT models (Fluids 3 and 4) were tension-thickening but plateaued at higher exten sion rates. Computations were performed with upwinded finite elements, the 8 method being employed to march the solution forward to a steady state. Flow patterns for an 8:1 circular contraction plotted for increasing values of a nondimensional number \ijw based upon an averaged downstream wall shear-rate, revealed that for Fluid 1 the vortex intensity and size decreased for \ijw suf ficiently large and only a little growth was reported for lower values of Ai7u,. In contrast, the size and intensity of the recirculating region associated with Fluid 2 both went through a maximum. For both linear models vortex size and activity increased over most of the range of Ai7 w , tending to a plateau for higher values of Xi'jw Whereas vortex activity and extensional effects were seen to be correlated, an attempt at plotting the intensity and reattachment lengths against a non-dimensional ratio of the (steady) normal stress difference in uniaxial extension to the (steady) shear stress, revealed a complex relationship from which it was difficult to draw firm conclusions. Investigators at the University of Delft have, through their finite element/ streamline-integration calculations of viscoelastic entry flows [297,299,614] given further insight into some of the mechanisms driving secondary cell growth. Hulsen and van der Zanden [299] used an eight mode Giesekus model in their finite element simulations of flow through a 4:1 and 5.75:1 axisymmetric con traction. The authors there argued that vortex growth was a mechanism for fulfilling the balance of momentum in the axial direction ld(rarz) r dr
|
dazz dz
_Q
The size of the vortex was thus related to the ratio of the extensional stress and shear stress. This is consistent with the statement of White and Kondo [624] that vortices are a stress-relief mechanism, and White and Baird's conclusion from their study of melts in entry flows [626] that it is the ratio of exten sional stress to shear stress, and not the magnitude of the extensional stresses themselves that determines the presence or absence of vortex growth. In order to further support their statement that it was the ratio of extensional stress to shear stress that was of importance for vortex enhancement in contraction flows, Hulsen and van der Zanden [299] performed a parametric study with a single mode Giesekus model. The extensional to shear viscosity ratio could be increased by either decreasing the mobility parameter a (see (2.145)) from 0.5 or decreasing the solvent viscosity. When the viscosity ratio was increased by changing either parameter in the manner just described, the size of the opening angle 0 also increased, attesting to vortex enhancement. These conclusions are further supported by the paper of Hulsen [297] in 1993 in simulations with a single mode P T T model and the same method used by 224
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS Hulsen and van der Zanden [299] two years previously. It was found that, by taking the PTT parameter e small (e = 0.02) and Re — 0, larger vortices for a 4:1 contraction could be observed than was the case with simulations at a larger value of e (e = 0.25). The ratio of extensional to shear properties was larger for e = 0.02 than when e was taken equal to 0.25, for a fixed value of £. In addition to considering vortex growth mechanisms, Hulsen [297] conjectured on the basis of his observations that divergence of the streamlines ("bulb flow") following vortex growth for e small and fixed elasticity number We/Re was a critical phenomenon associated with a change of type of the vorticity equation. The divergent flow regime began to appear when the hyperbolic vorticity region started to spread upstream of the contraction. The decrease of the vortex size beyond a certain Deborah number followed by the development of divergent flow was qualitatively similar to what had been seen by Cable and Boger [115]. Interestingly, some time later, Purnode and Legat [474] presented numerical evidence for the influence of change of type on the growth and eventual reduction in size of the lip vortex they observed in flow of a FENE-P fluid through a 4.4:1 planar contraction. As the hyperbolic vorticity region grew in influence and extent in the region upstream of the contraction due either to (i) increasing the Weissenberg number at a fixed elasticity number We/Re or (ii) increasing the maximum extensibility QQ at a fixed Weissenberg number, the lip vortex diminished in size. The divergent flow regime referred to by Hulsen [297] was also in evidence as the vortex shrank, so that the results of the two papers were seen to be in harmony.
8.3
Numerical Simulation
The history of the numerical simulation of non-Newtonian entry flows goes back at least as far as the late 1970's [151,319,446] and a good description of the work done up to 1986 may be found on pages 149 to 156 of the review by White et al. [628]. The presence of a singularity in the entry-flow geometry as well as regions of high shear and extension in the flow have presented a grand challenge to numericists over the years, as they have struggled to develop robust and accurate numerical methods. We will not occupy ourselves overly with the magnitude of the Deborah numbers achieved by the computational community since the earliest papers were written, however, believing that the size of the limiting Deborah number encountered using a given method is not necessarily related to the quality of the solution obtained. Therefore we begin the review by looking at the comparatively small number of papers where sincere attempts have been made to simulate experimental results. Here, differences between experimental and numerical results may involve both modelling and numerical errors and an appreciation of the underlying physics is required for reasonable agreement for non-trivial flows. Then we move on to consider comparisons between one set of numerical results and another, or between numerical predictions and analytical results, where these exist. On the whole, these comparisons are useful but, arguably, neither as illuminating nor as demanding as simulating physical flows.
225
8.3. NUMERICAL SIMULATION
8.3.1
Comparisons with experiments
An attempt at predicting the vortex characteristics observed by Walters and Webster [610] and Walters and Rawlinson [607] (see §8.1.2) for constant viscos ity and shear-thinning fluids, was made by Davies et al. [162] in 1984 using both finite difference and finite element methods. For 4:1 abrupt planar contractions the authors used UCM, Oldroyd B, second-order equivalent and White-Metzner models, and for the 4:1 axisymmetric problem the Oldroyd B model was the model of choice. Finite difference and finite element simulations with the UCM model in the planar case led to spurious vortex growth when the meshes were insufficiently fine. Numerical predictions with the finest finite element mesh showed no growth in the corner vortex, consistent with the behaviour of a Boger fluid in such a geometry [610]. It should be pointed out that the finite difference calculations with a UCM model reported by Walters and Webster [610] actually predicted an increase in the vortex size with increasing Weissenberg number; possibly because of too coarse a finite difference mesh. Finite element compu tations by Davies et al. [162] with a White-Metzner model having a power-law viscosity were not successful in predicting the substantial vortices seen by Wal ters and Webster [610] in the flow of an aqueous solution of PAA through a 4:1 planar contraction. No justification was given by the authors for their choice of parameters in the power-law fit of the shear viscosity. Although finite el ement calculations with the UCM model in the 4:1 axisymmetric contraction seemed to show a growth in the intensity and size of the vortex with increase in Weissenberg number, the vortices were smaller than those observed in the constant viscosity PAA/CS-water solution of the experiments of Walters and Webster [610] experiments. Moreover, the Weissenberg numbers attainable in the experiments were an order of magnitude higher. Although models with a single relaxation time have been used in order to simulate flows of LDPE melts in contractions (e.g. [627] with a single mode P T T model) it is generally acknowledged that a spectrum of relaxation times is required for an adequate fit of the material functions. Multi-relaxation time K-BKZ integral models were used, for example, in the finite element codes of Dupont and Crochet [192] and Luo and Mitsoulis [369] for the modelling of polyethylene melts in circular contractions. A little later Hulsen and van der Zanden [299] set out to show that integral models were not necessarily supe rior to multi-mode differential models for simulating contraction flows of LDPE melts and used an 8-mode Giesekus model with viscosity-relaxation time param eters from the work of Laun [364] for both 4:1 and 5.75:1 circular contraction flows. Hulsen and van der Zanden [299] compared their results with those of Dupont and Crochet [192] and Luo and Mitsoulis [369] for 4:1 and 5.75:1 circu lar contractions for a number of flow quantities: entrance pressure corrections, vortex intensity and the opening angle 6, as functions of the recoverable shear SR = NI(JW)/2T(JW) (where j w denotes the downstream tube wall shear-rate). For the opening angle 9 and the 4:1 contraction, results for the model of Hulsen and van der Zanden [299] were on the whole in very close agreement with those of the integral model calculations of the other authors, showing vortex enhance ment with increasing flow rate. Moreover, calculations in this case with the differential model could be continued to De = 256, way in excess of what was achieved in the other papers. The comparisons for the 5.75:1 contraction in cluded the experimentally determined values of White and Kondo [624] for their 226
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS LDPE melt. Computations by Hulsen and van der Zanden [299] on their finest mesh were in reasonable agreement with the integral model calculations, al though those of Luo and Mitsoulis [369] attained a limiting value for 6, not seen in the computations with the Giesekus model. Agreement with the opening angle results of White and Kondo [624] was good even up to the highest Debo rah number although Hulsen and van der Zanden [299] pointed out that White and Kondo [624] used seven different LDPE melts to get the 8 data over the entire range of shear-rates. Since different LDPE melts can have very different extensional properties (and therefore vortex dynamics) it was quite certain that the experimental 6 vs. SR curve with which the numerical simulations were compared was not universal for all LDPE melts. More recently Mitsoulis [408] has used the original method of Luo and Mit soulis [368] in simulations of the IUPAC-LDPE melt [396] with a multi-mode K-BKZ model based both on the original PSM damping function [441] (see Eqn. (2.156)) and its modification by Olley [428]. The IUPAC-LDPE melt had been modelled previously by Luo and Tanner [370] with the PSM/K-BKZ multi-mode model. Simulations were undertaken by Mitsoulis [408] both for a 4:1 axisymmetric and 14:1 planar contraction flow. Results with the PSM/KBKZ model in the axisymmetric geometry showed that the size of the corner vortex grew monotonically with apparent shear-rate before reaching a plateau and finally diminishing at high flow rates. In doing this it followed the trend of the extensional viscosity. For the planar problem, however, the vortex for the PSM/K-BKZ fluid model diminished in size and intensity with increasing flow rate. This was in sharp contrast with the experimental results of Wassner et al. [621] and was explained by the failure of the model to predict any strain hardening of the LDPE melt. To rectify this, calculations were performed with two different choices of parameters in the Olley model. A planar extensional viscosity function as great in magnitude as the uniaxial extensional viscosity could now be realized. The choice of the parameter set leading to a greater amount of strain hardening resulted in a close correspondence in vortex size to the experimental findings of Wassner et al. [621] and showed that far more strain hardening was exhibited by the polymer melt at high strain rates than had been previously thought. An EEME finite element method was used by Coates et al. [142] in 1992 to study the flow of Boger fluids through a 4:1 and 8:1 axisymmetric contraction. The MUCM model of Apelian et al. [13] and a modified form of the dumb bell model (MCR) of Chilcott and Rallison [134] were used in the study and comparisons were made with the PAA data of Boger et al. [89] for the /3 = 4 contraction and the PIB/PB data of McKinley et al. [389] for the 8:1 contrac tion. The choice of experimental data was motivated by the need for results from steady, two-dimensional flows over a sufficiently large range of Deborah numbers for the two contraction ratios, although nothing was done in the cal culations to differentiate between the behaviour of the PAA and PIB Boger fluids. For the 4:1 contraction both the MUCM and MCR models captured the development with Deborah number of the corner vortex as described by Boger et al. [89]. A graphical comparison of the dimensionless attachment length \ as a function of the Deborah number for the PAA Boger fluid and prediction of the MUCM model revealed unexpectedly good agreement: unexpectedly, because, as noted by Coates et al. [142], no attempt had been made to model precisely the rheology of the PAA solution. 227
8.3. NUMERICAL SIMULATION Although all of the main features of the vortex development reported by McKinley et al. [389] for flow of their PIB/PB fluid through an 8:1 contraction at low to moderate Deborah numbers, were reproduced qualitatively by the MUCM and MCR simulations of Coates et al. [142], a quantitative comparison between the measured and predicted values of x revealed that both constitutive equations predicted similar vortex growth over the range of Deborah numbers but that these overpredicted the observed values. Matching vortex size and shape between the MUCM predictions and the experimental measurements was only possible by choosing the laboratory results at (zero shear-rate) Deborah numbers up to eight times greater than for the numerical computations. This, said Coates et al. [142], was not surprising because a proper quantitative fit of the PIB/PB rheology had not been attempted: such an exercise would require a multi-mode model. The numerical simulation of the earlier experimental results of Evans and Walters [198,199] (see §8.1.4) with shear-thinning polyacrylamide solutions in planar geometries having contraction ratios 4, 16 and 80, was the objective of Purnode and Crochet [473]. The authors used a single-mode FENE-P model and the viscosities (solvent and polymeric), relaxation time and maximum ex tensibility parameter QQ were chosen so as to try and match the steady shear and extensional properties of three different concentrations of a PAA solution: 1%, 0.5% and 0.25%. These corresponded to some concentrations chosen by Evans and Walters in [198,199] with the exception of the last, where Evans and Walters [198,199] worked with a 0.2% aqueous polyacrylamide solution. The paper of Purnode and Crochet [473] represented a thorough investigation of the influence of contraction ratio, corner sharpness and PAA concentration on the vortex dynamics. It was encouraging that all the essential vortex features of the investigation of Evans and Walters [198,199] were observed by Purnode and Crochet [473] with the three contraction ratios and three different polymer concentrations. In particular, in the inertial flow of the low viscosity, 0.25% PAA solution lip vortices developed with increasing flow rate for a 4:1 contrac tion, these lip vortices growing to envelop the salient corner vortex before finally inertial effects took over and caused the recirculatory regions to shrink again. Although the results were qualitatively correct, however, the phenomena seen in the computations of Purnode and Crochet [473] did not occur at the same flow rates as in the experiments of Evans and Walters [198,199]. There were at least a couple of reasons for this lack of quantitative agreement. First, the computations of Purnode and Crochet [473] were two-dimensional whereas it was clear that there were three-dimensional effects in the work of Evans and Walters [198,199]. Secondly, so crude a model as the FENE-P could not possi bly hope to capture all of the physics contained in a proper description of the PAA solution. Precise rheological characterization by experimentalists and a detailed de scription of experimental parameters and results may be what are needed on occasions in order to encourage the numerical community to model the fluid and simulate the flow. One good example may be found in the experiments conducted by Quinzani et al. [478] with a concentrated solution of PIB in C14 in a 3.97:1 planar abrupt contraction. Four years previously, Quinzani et al. [480] had characterized PIB solutions of various concentrations and at various tem peratures in linear viscoelastic, steady shear flow and transient shear flow ex periments. This included a four mode fit for a Maxwell model. A later paper 228
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS by Quinzani et al. [479] revealed that the best fit to the transient extensional viscosity could be made with a P T T model. The detailed LDV and FIB mea surements by Quinzani et al. [478] of the velocity and stress fields at six different sets of flow conditions were presented in the form of numerous profiles of these quantities taken along lines in the radial and axial directions, and represented the most complete study up to that time of viscoelastic entry flow. Shear stress and normal stress difference measurements near the re-entrant corner were also given and seemed to indicate that a similarity form T(r,6) = rx-2G(6),
(8.8)
could apply, with an exponent A larger than that for a Newtonian fluid, due to shear-thinning effects. Given the quality of the information made available in [478] and [480] it is not surprising that several attempts have been made to numerically simulate the findings of Quinzani et al. [478]. Mompean and Deville [409] used a finite volume method to compute axial velocity profiles and profiles of the first normal stress difference along various radial coordinate lines upstream and downstream of the contraction plane in a 4:1 planar contraction. Despite the fact that the authors were using an Oldroyd B fluid to model the shear-thinning polymer solution of Quinzani et al. [478] the overall agreement was good for the axial velocity profiles, although the predicted first normal stress difference did not fare so well, and tended to underestimate the measured values near the axis of symmetry. Another finite volume simulation of the Quinzani et al. [478] results was by Xue et al. [637], this time with a single mode P T T model having parameters fitted as in [479], in order to investigate the dynamics upstream and downstream of the contraction plane, as well as along the axis of symmetry. Details of the comparison are somewhat lengthy and the reader is referred to the original paper for these. Overall agreement between predictions and measurements was described by the authors as reasonably good, bearing in mind that no attempt was made to exactly match the viscometric and extensional properties of the test fluid. Numerical experiments near the re-entrant corner confirmed the findings of Quinzani et al. [478] that shear-thinning reduced the strength of the corner singularity. Xue et al. [637] also sought to simulate flow of a Boger fluid through a three-dimensional 4:1 abrupt contraction of rectangular cross-section using a UCM model under the same experimental flow conditions used by Walters and Webster [610]. Although Xue et al. [637] claimed that the trend of corner vortex behaviour was consistent with the earlier experimental observations of Walters and Webster [610], there were discrepancies in the results, including the prediction by the code of Xue et al. [637] of a small lip vortex at higher flow rates. More recently still, finite element simulations in a planar 3.97:1 contraction using a single mode FENE fluid with parameters matched to the shear data in [478,480] have been performed by Bonvin [93]. As with Xue et al. [637] the dynamics upstream, downstream and along the axis of symmetry were probed and a detailed commentary may be found in [93]. An investigation of the corner singularity revealed that overall the FENE model gave a better description than an Oldroyd B fluid, which is not surprising given the influence of shear-thinning effects near the corner. Another good example of experimentalists paving the way for the computa tional community in an effort to entice its members into performing numerical 229
8.3. NUMERICAL SIMULATION simulations is found in the paper of Byars et al. [114]. The authors performed ex periments for the shear-thinning test fluid SI in a 4:1 axisymmetric contraction geometry with abrupt and rounded corners and provided parameter fitting to the shear and extensional data of Ooi and Sridhar [430] for the P T T , Giesekus and K-BKZ models. Mitsoulis [406] allowed himself to be enticed: using a modified K-BKZ model with a spectrum of four relaxation times and the finite element numerical scheme of Luo and Mitsoulis [368], the author computed solutions in a 4:1 contraction with both abrupt and rounded corners. Despite the Weissenberg number limit encountered, the author was able to undertake simulations over the entire range of experimental flow rates realized in [114]. In both experiment and simulation the corner vortices were seen to grow monotonically with Weis senberg number, whether the corner was sharp or rounded. Discrepancies were, however, evident in the graphical comparison of the calculated and measured non-dimensional attachment length x, this becoming as much as a 30% underprediction for the highest flow rates and the rounded corner case. A slightly larger vortex was possible, however, by increasing the steady-state extensional viscosity in the model. In 2001 Mitsoulis [407] performed numerical simulations for the flow of the SI fluid using a K-BKZ integral model with the same choice of parameters as originally used by Byars et al. [114] and Mitsoulis [406]. This time, however, the simulations were done in a 24:1 axisymmetric contraction and the results were compared with the experimental data of Maia and Bind ing [380,381]. The simulations were seen to capture the main features of the experimentally observed vortex dynamics, with a monotonic increase of the vor tex size with flow rate up to a wall shear-rate -yw of about 200s~ 1 . At a wall shear-rate of about 186s - 1 flow instabilities gave rise to larger discrepancies be tween the experimental and simulation results. Results for the overall pressure drop were in close agreement with the experimental measurements up to the onset of flow instabilities. Although comparisons mentioned in this section between numerical predic tions and experimental measurements have focussed mainly on vortex dynamics, mention should be made again of the failure to date of numerical simulations using simple dumbbell models to predict the large additional pressure drop (see §8.1.2) seen in contraction flows of Boger fluids [142,321,555]. Rothstein and McKinley [519] suggested that the reason for the failure was that the constitutive models used in [142,321,555], for example, were unable to describe the exten sional rheology of polymer solutions, and, in particular, to predict the stressconformation hysteresis observed in uniaxial transient extensional flow [189]. However, Rothstein and McKinley [519] pointed out that pressure drop mea surements for their PS Boger fluid through a /3 = 2 contraction-expansion were still greatly enhanced and yet in this flow the dissipative stress suggested by stress-conformation hysteresis was not in evidence. So correctly capturing the transient extensional behaviour of polymer solutions might not be sufficient in all circumstances. If numerical simulations are to stand any chance of predicting accurately (and not just qualitatively) vortex growth dynamics and enhanced pressure drops for contraction flows, the message would therefore seem to be that, at the very least, the models used must adequately describe the fluid's transient extensional properties. The same observation would apply to other flows (such as the flow past a sphere or a cylinder) that have a high extensional component. Mesoscopic-scale modelling in a micro-macro or lattice Boltzmann framework 230
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS (see Chapter 11) would seem, at the time of writing, to hold most promise.
8.3.2
B e n c h m a r k i n g numerical m e t h o d s
Non-isothermal and three-dimensional simulations The vast majority of numerical simulations of non-Newtonian entry flows have dealt with two-dimensional planar or axisymmetric isothermal flow problems. Exceptions are the papers by Wapperom et al. [614], Wachs and Clermont [602], Kunisch and Marduel [337] and Yesilata et al. [641] for non-isothermal fluid flows, and the papers of Mompean and Deville [409] and Xue et al. [637] that solve the three-dimensional rectangular contraction problem using finite volume methods. Baloch et al. [36] had previously used a semi-implicit Taylor PetrovGalerkin pressure correction scheme [124] to solve for three-dimensional flow of a P T T fluid through a 40:3:3 expansion geometry. Given the extra cost involved, the comparative rarity of three-dimensional simulations is hardly surprising. Mompean and Deville [409] presented results for the three-dimensional 4:1:4 contraction problem, using an Oldroyd B fluid and a finite volume technique. The authors used a staggered mesh for the variables, the method in the paper being derived from the MAC method of Harlow and Welch [270]. Nonlinear terms in the momentum and constitutive equation were handled with the QUICK scheme proposed by Leonard [349] and the authors were able to reach a Deborah number of 27.3, although it is not clear that the corner singularity was accurately represented by the numerics. Of interest in the results presented was the observation that the salient corner vortex in the mid-plane of the three-dimensional contraction was always smaller than the two-dimensional one for equivalent flow rates. The three-dimensional simulations of Xue et al. [637] were performed us ing an implicit finite volume scheme based on the SIMPLEST [636] algorithm. Mention has also been made already in this review of the results obtained by the authors using both a UCM and a single mode P T T model to simulate the experimental results of Walters and Webster [610] and Quinzani et al. [478,479]. Entry flow simulations since 1986 A vast literature exists for entry flow simulations, stretching back at least as far as the late 1970's. Given the excellent review which is already available [628] of numerical methods for contraction flows that appeared in the literature up to 1986, we take this year as the starting point of our discussion of some of the more notable contributions to the field. To make the presentation more digestible, and in view of the very different vortex behaviour which is observed at least for Boger fluids - depending on whether flow through a plane or circular contraction is being simulated, the reviewed papers will be divided into those that deal with planar flows and those that are for circular contractions. Those that deal with both are included in the first part.
231
8.3. NUMERICAL SIMULATION Planar contractions Along with most other computational rheologists who, through the years, had been troubled by the loss of convergence of numerical methods for nontrivial viscoelastic flows beyond some critical Weissenberg number (the so-called "high Weissenberg number problem", or H WNP ;see§3.1and§7.1), Marchal and Crochet [383] attempted to choose the approximating subspaces for the velocity and stress fields in such a way as to provide greater accuracy. Specifically, they sought to construct a velocity field approximation based upon Hermitian shape functions in order that an equivalence criterion between the velocitypressure and mixed formulations might be satisfied in the limit of vanishing Deborah number (see §7.3.1). Limiting Deborah numbers of the order of 6 were encountered in their simulations of the flow of an Oldroyd B fluid through 4:1 planar and circular contractions and this extended by a factor of four the range of Deborah numbers over which converged solutions had previously been possible with classical Lagrangian elements. However, the Hermitian finite elements were restricted to rectangular shape and this limited the applicability of the scheme. Significant differences were observed between the vortex dynamics for the planar and axisymmetric cases: in the planar contraction no significant corner vortex could be observed but a lip vortex developed after De = 2.84 and grew in size and strength thereafter. For the axisymmetric contraction, vortex growth was via the salient corner vortex, consistent with experimental observations (see §8.1.2). The first great breakthrough in the battle with the HWNP for contraction flows was realized a year later by the same authors [384]. A choice of biquadratic shape functions for the velocity approximations, bilinear elements for the pres sure and 4 x 4 linear stress subelements, combined with inconsistent SU (see §5.3.3 and §7.2.1) led to a very stable formulation and no loss of convergence of the iterative method for the Oldroyd B problem in either a 4:1 planar or ax isymmetric geometry was encountered. Galerkin and SUPG formulations were found to perform poorly for the planar problem and consequently were not con sidered for simulations in the axisymmetric case. For the planar case, the SU 4 x 4 method predicted small and weak salient corner vortices. A very small lip vortex manifested itself at De = 7.6 but this may have been a numerical arti fact. Large vortices, growing in size with De, were predicted for the Oldroyd B fluid in the axisymmetric contraction. Although the shape was similar to those which had been seen by Boger [84], they occurred at De values differing by an order of magnitude from their experimental counterparts. Slightly larger vor tices than for the Oldroyd B fluid were observed with a UCM fluid through the axisymmetric contraction at similar flow rates. In this case, however, a limiting Deborah number of 7.68 was encountered. It should be pointed out here, however, that the use of SU amounts to a change in the constitutive equation being solved [371] and that in a study in 1991 conducted by Tanner and Jin [564] of Galerkin, SUPG and SU schemes for solving Oldroyd-type constitutive equations, the authors concluded that for the one-dimensional problem being considered the SU scheme delivered the least accurate solution. Computations by Luo and Tanner [371] in 1989 were per formed using a decoupled finite element scheme with Picard iterations. The flow of UCM and Oldroyd B fluids through both planar and circular 4:1 con-
232
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS tractions were calculated and comparisons were made between the results of Galerkin, SUPG and SU formulations. The poor performance of SUPG meth ods for abrupt contractions was confirmed by the authors: the SUPG method providing no more stability for this problem than the Galerkin method, al though doing better in this regard than the 4 x 4 SUPG method of Marchal and Crochet [384]. Weissenberg numbers of 5.5 and 6.4 were obtainable with the 01droyd B model and the inconsistent SU method in the planar and axisymmetric geometries, respectively, but inspection of the stress field revealed the smearing effect of diffusion. The 4 x 4 mixed finite element method introduced by Marchal and Crochet [384] in 1987 has enjoyed several applications in the literature since then, often in conjunction with (inconsistent) SU. One example is the paper of Bodart and Crochet [81] who extended the original 4 x 4 method to calculate time-dependent flows of an Oldroyd B fluid, and in particular to examine their stability. Galerkin and SU formulations were used with their 4 x 4 method and both planar and axisymmetric contractions were considered. Earlier numerical work by Fortin and Esselaoui [216] and El Hadj and Tanguy [262] had suggested that nonsymmetric and periodic flows could develop for the contraction problem if steady flow and symmetry restrictions were dispensed with. This seemed to be in agreement with experimental observations (see [417] and [389], for example, in the case of Boger fluids). The numerical results of Bodart and Crochet [81] showed that the temporal stability of the 4:1 planar contraction solutions depended heavily upon the mesh and the form of boundary conditions, the Deborah number and whether a Galerkin or SU formulation was being used. All numerical solutions with SU were stable for this geometry, however, and for all the methods used the flow remained symmetric. For the circular contraction, no evidence of swirling flow was found. A second example of the use of the 4 x 4 method in entry flow simulations is found in the paper by Purnode and Legat [474] where simulations of a FENE-P fluid through a planar 4:1 contraction were performed. Introduction of the SU method was found, additionally, to be necessary for simulations with a very large extensibility parameter Q0. Mention of the correlation drawn by the authors between the lip vortex mechanism and a change of type has been made already in this review (see §8.2.2). Results obtained by Purnode and Legat [474] for the case Qo —> oo (Oldroyd B fluid) showed that the vortex was greatly reduced in size compared to simulations at the same Weissenberg and Reynolds numbers at finite values of the extensibility parameter. This, the authors claimed, was in line with the results of Song and Yoo [540] in their finite difference simulations with a UCM fluid. Keunings [323] and Rosenberg and Keunings [517] concluded on the basis of numerous mesh refinement studies with the UCM model in a 4:1 planar contraction that, since the limit point for the Weissenberg number was not sensitive to the choice of mesh, nor (in the case of a rounded corner), to whether C° or C1 representations were used, a genuine limit point of the Maxwell fluid was possibly in evidence. Subsequent numerical simulations have, of course, shown that an intrinsic limit point of the Maxwell model, if it exists at all, occurs at a higher value of We than predicted on the basis of the calculations of Keunings [323] and Rosenberg and Keunings [517]. We make further comment to this effect in §7.1. Not many simulations for planar viscoelastic entry flows using finite differ233
8.3. NUMERICAL SIMULATION ence methods have made their appearance in the literature since 1986. Three exceptions are the papers by Choi et al. [137], Keiller [321] and Olsson [429]. In the paper of Choi et al. [137] an upwind corrected finite difference scheme involving an artificial viscosity term was reported, yielding second-order accu racy in their computations of a single mode Giesekus fluid flowing through a planar 4:1 contraction. Although earlier computations by Davies et al. [162] with a White-Metzner model had been unsuccessful in predicting vortex growth in the 4:1 planar contraction, the results of Choi et al. [137] with the Giesekus fluid demonstrated that significant corner vortex enhancement could take place with increasing We. This was stated as being consistent with the experimen tal results of Evans and Walters [198]. The competing effects of elasticity and inertia were also discussed by the authors and showed, unremarkably, that as the viscoelastic Mach number M = y/ReWe increased at low elasticity number E — We/Re, the salient corner vortex decreased whereas the opposite was true at higher values of E. A Giesekus model was also used by Olsson [429] in his study of transient viscoelastic flow through a 2:1 and a 4:1 planar contraction. Simulations were performed with a slightly compressible fluid and although the re-entrant corner was rounded, it was gradually sharpened in order to better approximate the abrupt contraction problem. Olsson [429] used a second-order accurate finite difference method on a composite overlapping grid and in order to avoid oscillations at the interpolation boundaries it was found necessary to introduce second-order artificial viscosity. Computations with the 2:1 contrac tion flow revealed that, as the re-entrant corner was made sharper, the gradient of the axial tensile stress increased significantly. Plots showing vortex formation with time for the 4:1 contraction were particularly interesting; revealing that the vortex growth built up from just below the re-entrant corner rather than in the salient corner. The observations of Keiller [321] on the effect of polymer extensibility on vortex growth for FENE dumbbell models in P = oo entry flows, have been discussed already in §8.2.2. Keiller [321] used a decoupled finite difference time stepping scheme to investigate both planar and axisymmetric entry flows of Oldroyd and FENE fluids. In addition to discussing the mechanisms of vortex growth, Keiller [321] also considered the behaviour of a non-dimensional pressure drop across the contraction for the Oldroyd and FENE models and found that for both planar and circular contractions the pressure drops were less than for a Newtonian fluid. Up to the point where the computational domain became too small to assume with validity that a Poiseuille velocity profile existed at exit, Keiller [321] found that the Oldroyd B pressure drop decreased linearly with We. This contrasted sharply with the earlier findings of Debbaut et al. [172] who showed that an increase in the pressure drop beyond We PS 20 could be anticipated in both planar and axisymmetric 4:1 contraction flows of an Oldroyd B fluid. Keiller [321] justified his results on the basis of the low transient shear viscosity of the Oldroyd B model. A number of computational studies on planar contraction problems [124, 384,387,527,642] were reviewed by Phillips and Williams [462] and their results compared. Phillips and Williams [462] used a finite volume scheme for solving the planar flow of an Oldroyd B fluid through a 4:1 contraction, in which a staggered grid for the dependent variables was employed, and a particle tracking method used in order to evaluate the convective terms appearing in both the momentum and constitutive equations. Included in the comparison of results 234
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS
Figure 8.7: The length L\ of the salient corner vortex. Oldroyd B fluid. 4:1 planar contraction. ■: Phillips and Williams [462] (Re = 0), ▼: Matallah et al. [387] (Re = 0), o: Sato and Richardson [527] (Re = 0.01), +: Yoo and Na [642] (Re = 0) and x: Marchal and Crochet [384] (Re = 0). Reprinted from T. N. Phillips and A. J. Williams, Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. Non-Newtonian Fluid Mech., 86:215-246, Copyright (1999), with permission from Elsevier Science. for the Oldroyd B planar entry problem were the results of Matallah et al. [387], whose simulations used a Taylor-Galerkin finite element method. Such methods are part of a long tradition in the work of the Webster/Townsend group [35,36,124,125,388]. For a summary of the paper of Carew et al. [124] the reader is referred to the paper of Phillips and Williams [462]. We discuss the paper of Matallah et al. [387] in more detail later in this section. Also included in the comparison were the predictions of Sato and Richardson [527] who solved for transient viscoelastic entry flow using an implicit finite volume method for a UCM or Oldroyd B constitutive equation coupled with a Galerkin finite element method for the momentum equation. Time integration was performed with the 9 method. Another finite volume scheme, this time that of Yoo and Na [642], appeared in the list of papers cited for comparison purposes by Phillips and Williams [462]. These authors investigated the planar 4:1 contraction problem with the Oldroyd B model and found a strong lip vortex that grew in size with We. Finally, in the list of results for comparison were those of Marchal and Crochet [384], referred to already at some length in this section. The details of the comparison done by Phillips and Williams are given in Fig. 8.7. Shown are the vortex attachment lengths predicted by [384,387,462, 527,642] for the Oldroyd B 4:1 planar contraction problem at Re = 0, where the upstream channel has a width of eight units. The various definitions of Weissenberg number and Deborah number used by the authors featuring in
235
8.3. NUMERICAL SIMULATION
Figure 8.8: The streamlines for a) We = 0, b) We = 1.0, c) We = 2.0 and d) We = 2.5. Oldroyd B fluid. Re = 0. 4:1 planar contraction. Reprinted from T. N. Phillips and A. J. Williams, Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. Non-Newtonian Fluid Mech., 86:215-246, Copyright (1999), with permission from Elsevier Science. the comparison have been translated, for comparison purposes, into a single Weissenberg number We = 2 \ ^ ,
(8.9)
where Ai is the fluid relaxation time, U the mean velocity in the downstream channel and D is the downstream channel width. A ratio of the solvent to polymeric viscosity equal to 1/9 was taken by all authors for the results shown in Fig. 8.7. The results of the computations of Phillips and Williams [462] are also shown in the figure. The results of Phillips and Williams [462] may be seen to agree very closely with those of Sato and Richardson [527] (whose results were obtained at Re = 0.01) up to a Weissenberg number of about 1, and are smaller than those published in the paper of Yoo and Na [642]. It is quite sobering to note, however, that there is a large variation in the vortex attachment lengths even for Newtonian creeping flow, with L\ for the cited simulations varying for 236
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS
Figure 8.9: The streamlines for a) We = 0, b) We = 1.0, c) We = 2.0 and d) We = 2.5. Oldroyd B fluid. Re = 1. 4:1 planar contraction. Reprinted from T. N. Phillips and A. J. Williams, Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. Non-Newtonian Fluid Mech., 86:215-246, Copyright (1999), with permission from Elsevier Science. We = 0 between approximately 1.4 to 1.65. Although the data of Marchal and Crochet [384] are given in Fig. 8.7, it should be noted here (and Phillips and Williams do the same), that to be fair to these authors their results were not in a form that allowed them to be readily translated onto such a plot. Streamlines for the converged solutions obtained by Phillips and Williams [462] at Re = 0 are given in Fig. 8.8 (a)-(d) for Weissenberg numbers of 0, 1, 2 and 2.5 (the maximum attained by the authors). It may be seen that as We increases Lx decreases slightly and that at We = 2 a lip vortex appears. The behaviour was observed to compare well with that of Matallah et al. [387], although the authors did not have confidence that the lip vortex was anything other than a numerical artifact. Phillips and Williams [462] also investigated the effect of inertia in the pla nar contraction flow by performing calculations at Re = 1. The corner vortices were now about 20% smaller than the equivalent vortices at Re = 0 and the streamlines are shown in Fig. 8.9. Phillips and Williams [462] also showed that 237
8.3. NUMERICAL SIMULATION
Figure 8.10: The length L\ of the salient corner vortex. Oldroyd B fluid. Re = 1. 4:1 planar contraction. ■: Phillips and Williams [462], T: Matallah et al. [387], o: Sato and Richardson [527], +: Carew et al. [124]. Reprinted from T. N. Phillips and A. J. Willliams, Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. Non-Newtonian Fluid Mech., 86:215-246, Copyright (1999), with permission from Elsevier Science. they were weaker. A comparison of the vortex attachment length obtained by Phillips and Williams [462] with those from the published results of Matallah et al. [387], Sato and Richardson [527] and Carew et al. [124], as shown in Fig. 8.10, again reveal good quantitative agreement between those of Phillips and Williams [462] and of Sato and Richardson [527] for low values of the Weissenberg number. Although there is general agreement in the trend of the plot ted data as We increases it may be seen from Fig. 8.10 that Carew et al. [124] predict consistently larger values of L\ than the other authors. A cell-vertex hybrid finite element/finite volume scheme, the key features of which were documented in two papers of Wapperom and Webster [616,617] (see too, §5.4.1) was used by Aboubacar and Webster [2] for the numerical solu tion of the Oldroyd B 4:1 planar contraction problem. The general framework of the method was a Taylor-Galerkin/pressure-correction scheme. A recovery technique for velocity gradients was additionally employed. The use of a discon tinuity capturing technique, achieved via reduced integration within the control volume surrounding the re-entrant corner increased the critical Weissenberg number (denned as in (8.9)) from 2.4 (without reduced integration) to 3.7, on one of the meshes (12,779 degrees of freedom). The lip vortex growth with increasing Weissenberg number observed on the same mesh was seen to disap pear with mesh refinement up to a Weissenberg number of 2. On the finest mesh (32,717 degrees of freedom), however, a miniscule lip vortex was seen at 238
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS We = 2 and the development of a trailing edge vortex could also be detected. Although the calculations of Aboubacar and Webster [2] were for an Oldroyd B fluid, good agreement was demonstrated between the dimensionless vortex at tachment length x and that predicted by the SIMPLER-type MINMOD [271] finite volume procedure of Alves et al. [9] for planar flow of a UCM fluid. Both sets of results revealed a salient corner cell that decreased in size with Weis senberg number. As a further demonstration of the accuracy of their scheme, Aboubacar and Webster [2] corroborated the theoretical results of Hinch [283] for the corner singularity along two radial lines towards the re-entrant corner. In a related article, Aboubacar et al. [1] provided additional results for the planar contraction problem by first rounding the corner in order to assess sta bility compared with the abrupt corner case, and secondly using models of the P T T type in addition to the Oldroyd B model. Further results for planar and axisymmetric contractions with Oldroyd B and P T T models were presented in [611]. Circular contractions We begin our review of the numerical simulation literature on the circular contraction problem with the paper of Debbaut and Crochet [170]. On the basis of computations performed by Keunings and colleagues [323,328,517] questions were being asked about whether the limit points encountered in computations of both planar and circular contraction flows of viscoelastic fluids were of purely numerical origin or were inherent in the models being used for these simulations. Debbaut and Crochet [170] took the three same finite element meshes used by Keunings and Crochet [328] in order to calculate the flow of the same P T T fluid through a 4:1 contraction, but now chose discontinuous representations of the pressure with a stronger enforcement of the continuity equation. The result was that the limiting Weissenberg numbers of 1.22 and 1.54 (defined in terms of the downstream mean axial velocity) found by Keunings and Crochet [328] for their two coarsest meshes were increased by a factor of four. A smoother mesh, with a longer exit section than in the meshes of Keunings and Crochet [328] permitted convergence up to We = 9. The obvious conclusion from the work of Debbaut and Crochet [170] was that the limit points encountered by Keunings and Crochet [328] were of numerical origin. When, two years later, Debbaut et al. [172] used the 4 x 4 SU method of Marchal and Crochet [384] for the same fluid in the same geometry, Weissenberg numbers up to We = 50 were possible and no limit points were encountered. Numerical experiments were performed by the authors to investigate the effects of changing the P T T parameters £ and e (see Eqns. (2.143) and (2.144)) on the vortex size and intensity and the Couette correction factor. For the P T T parameter choice of [170,328] (£ = 0.2 and e = 0.015) the growth of the corner vortex up to a Weissenberg number of We = 10 and gradual decay to Newtonian-like proportions at We = 50 could be related to the steep rise of the extensional viscosity of the PTT fluid for low values of the extension rate followed by a decrease at higher extension rates and rapid shear-thinning. However, in general it proved difficult to predict the macroscopic flow features on the basis of the material parameters £ and e. We have discussed earlier in the chapter (see §8.2.2) the results of Hulsen [297] with a single mode PTT model of the linear type, as he varied e and 239
8.3. NUMERICAL SIMULATION thus altered the extensional characteristics of the model. Here, the relationship between the parametric value and vortex size for differing Weissenberg numbers and Reynolds numbers seemed clearer than for Debbaut et al. [172]. Attempts by Debbaut et al. [172] to match the vortex size from the experimental results of Boger et al. [89] with that predicted for Boger fluids using an Oldroyd B model and the 4 x 4 SU method were seen to have limited success: vortices of the same size for the experiments and computations were only possible at recoverable shear-rates differing by an order of magnitude. The reason for the quantitative discrepancy was not understood at the time but more recent comments (see [88,321]) relating to the appropriate definition of the Weissenberg number in such comparisons, as well as the documented failures of Oldroyd models to adequately model even constant viscosity viscoelastic fluids, may hold the key. Another possible reason relates to the accuracy of the SU formulation, noted already in our discussion of numerical results for the planar contraction problem (see §8.3.2 and [371,564]). Given the important differences in vortex growth behaviour seen in experi mental work with entry flows of different types of melts, numerical simulation in this area is of great significance. We have referred already in this review (see §8.3.1) to the integral formulations of Dupont and Crochet [192] and Luo and Mitsoulis [369] for the K-BKZ model and to the finite element streamline inte gration method of Hulsen and van der Zanden [299] for an eight mode Giesekus model. A comparison of the results of all of these authors for the opening angle 9 and entry pressure correction for the 4:1 and 5.75:1 circular contraction prob lems may be found in [299]. The experimental results of White and Kondo [624] for the opening angle 9 for the 5.75:1 contraction were superimposed on the plot ted data for comparison purposes and showed the results of Hulsen and van der Zanden [299] particularly favourably. As an alternative to the K-BKZ model, Clermont and de la Lande [141] chose to apply their streamtube method to the Goddard-Miller constitutive model [12,141]. The streamtube method used by Clermont and de la Lande [141] involved the mapping of open streamlines onto straight and parallel lines, the flow region covered by the open stream lines thus being mapped in this way onto a rectangular computational domain. The one-to-one coordinate transformation became an unknown to be determined alongside the discrete pressure and extra-stress values. Evaluation of derivatives appearing in the momentum equation was done using finite difference formulae, and a Levenberg-Marquardt algorithm [248] was employed to solve the discrete set of equations. An increase in the vortex zone near the salient corner with Weissenberg number was observed by the authors and claimed to be consistent with the experimental and numerical results reported elsewhere in the literature. Results were presented up to a Weissenberg number of approximately 30. One of the more successful time discretizations in viscoelastic flows has been the 9 method, employed by Saramito and Piau [525] in their PTT simulations for the 8:1 circular contraction (see the earlier reference in §6.4.4 and §8.2.2) and the mixed finite element method of Saramito [524]. In this latter paper Saramito [524] used, as did Saramito and Piau [525], a Lesaint-Raviart scheme [350] for the transport of the normal stresses and a Baba-Tabata scheme [30] for the shear stress components. Different finite element representations were used for the normal stress components and the shear stress. Saramito [524] applied his transient algorithm to the computation of the flow through a 4:1 circular contraction of an exponential-type P T T fluid, having a maximum in the 240
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS extensional viscosity. As with Saramito and Piau [525] before him, the vortex intensity and size both had maxima when plotted against the non-dimensional parameter Ai7„,, based upon an averaged downstream wall shear-rate. Vortex activity could be seen to be related to the extensional properties of the fluid and the results were stated as being consistent with those of Debbaut et al. [172]. For another time discretization technique, that of characteristics, particu larly well suited to solving hyperbolic transport problems, the reader is referred to the papers by Basombrio [46] and Kabanemi et al. [318], where the method has been successfully applied to the solution of 4:1 contraction problems. Consistent with experimental observations of vortex growth behaviour for Boger fluids both in axisymmetric and planar geometries, the recent numeri cal results of Phillips and Williams [464] for an Oldroyd B fluid in a circular contraction, obtained with the same numerical method as in their earlier pa per [462], revealed startling differences in the predicted vortex growth rate. In Fig. 8.11 are shown the results from [462] and [464] at Weissenberg numbers up to 1.5 for planar and circular 4:1 contractions. Vortex growth is vigorous in the circular case, whereas, as has been noted above, only a slight increase of the at tachment length was evidenced in the Phillips and Williams [464] results for the planar contraction. For a given Weissenberg number it was found by Phillips and Williams that the vortex could be some 72% larger in the axisymmetric case than in the planar geometry. We conclude this subsection by looking briefly at the application of special formulations or techniques that have been applied to the circular contraction problem and which have enjoyed particular success in permitting high Deborah numbers to be attained. These include regularization of models used near the corner, finite volume methods, adaptive//i-p methods and DEVSS/DG meth ods. The EEME finite element method of Coates et al. [142] (see §7.3.3) was seen to produce results which, in the case of an MUCM fluid in a 4:1 contrac tion, predicted a vortex attachment length that was in excellent agreement with the PAA Boger fluid results of Boger et al. [89]. By using the MUCM or MCR models [13,134] a Newtonian-like asymptotic behaviour near the re-entrant cor ner was realized and meant that the limiting Deborah number increased with mesh refinement. However, for the UCM model the upper limit on the Deborah number decreased with global mesh refinement. Defining a Deborah number based on the average downstream velocity and tube radius, the maximum Deb orah number for a mesh having 20,756 unknowns was 1.44, and for a finer mesh still this maximum dropped to 0.65. Coates et al. [142] conjectured that the stresses were no longer square-integrable beyond the maximum attainable Deb orah number on the finest mesh. A Deborah number defined in the same way as by Coates et al. [142] of up to 6.25 was reached in the finite volume calculations by Sasmal [526] in 1995 for the UCM fluid in a 4:1 abrupt symmetric contraction. A stream function-vorticitystress formulation was used by the author in the EVSS form of the equations, and a staggered grid arrangement placed stresses at cell centres and the stream function and vorticity unknowns at the corner points. Velocities were defined at cell faces. Remarkably, Sasmal [526] found that the limiting Deborah number value of De — 6.25 was attainable on all three of his meshes, irrespective of how coarse or fine they were. This is at variance with the experience of Coates et al. [142] and therefore reasons need to be elicited to explain the different findings. The explanation is almost certainly found in the differing behaviour of 241
8.3. NUMERICAL SIMULATION
Figure 8.11: The streamlines for a) We = 0, b) We = 0.5, c) We = 1.0 and d) We = 1.5. Oldroyd B fluid. Re = 0. Left: 4:1 planar contraction. Right: 4:1 axisymmetric contraction. From T. N. Phillips and A. J. Williams, Semi-Lagrangian finite volume methods for viscoelastic planar and axisymmetric contraction flows, Proceedings of ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, UK. Reproduced with permission of the Institute of Mathematics and its Applications. the sets of solutions of Coates et al. [142] and Sasmal [526] near the re-entrant corner. Whereas the corner stresses of Coates et al. [142] were predicted to be no longer square-integrable beyond a modest Deborah number, the computations of Sasmal [526] led to a Newtonian-like behaviour near the corner, presumably (bearing in mind the analytical work of Hinch [283] and Renardy [503]) because of poor solution accuracy there. The attainment of high Weissenberg numbers with no modification to the true UCM corner behaviour has been possible with both EVSS/SUPG finite element methods [202] and DEVSS/DG finite element methods [24]. Fan and Crochet [202] found that an SUPG method involving an upwinding term where the velocity vector was scaled with an average value over a finite element, was remarkably robust. This was in contrast with disappointing results for the ax isymmetric contraction obtained by Marchal and Crochet [384] and Luo and 242
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS Tanner [371] with the usual SUPG formulation of Brooks and Hughes [103]. With the new SUPG formulation Weissenberg numbers (based on the down stream wall shear-rate) of up to 20 were possible. However, no convergence studies were done for this problem and it was only for a smooth contraction that the authors demonstrated convergence with p-enrichment (at lower Weis senberg numbers). Arguably the best results for the UCM 4:1 circular contraction problem ob tained to date are those by Baaijens [24] who used a DEVSS/DG finite element formulation and calculated flows on his finest mesh up to De = 5 without loss of convergence. Here the Deborah number was defined in the same way as that of Sasmal [526] and Coates et al. [142]. Corner vortex intensities were in good agreement with those of Coates et al. [142] whereas Sasmal [526] predicted sig nificantly stronger vortex activity. As pointed out by Baaijens [24], although higher Deborah numbers had been achieved by Marchal and Crochet [384] and Sasmal [526] for this problem, the methods used by both groups were known to be only first-order accurate for the stresses, whereas the DEVSS/DG method used by Baaijens [24] had an accuracy of 0(/i 3 / 2 ). Baaijens [24] further demon strated the robustness and accuracy of his method by observing that convergence with mesh refinement could be seen everywhere away from the downstream wall, and moreover that, near the singularity, the analytical results of Hinch [283] were corroborated: the order of the stress singularity in TZZ behaving like r _ 2 / / 3 in a core region away from the walls. Numerical methods for contraction flows: some comparative studies The comparative performance of several numerical methods or formulations used for the solution of the viscoelastic entry flow problem have been investi gated in a number of papers: those of Tsai and Liu [583], Matallah et al. [387], Debae et al. [169] and Fan [203], represent just a selection. Tsai and Liu [583] considered the relative efficiency of three numerical sche mes which could be used to solve the linearized system of equations arising from a finite element discretization of the Oldroyd B 4:1 planar contraction problem and stick-slip problem. The first scheme was a modified frontal solver based on Gaussian elimination and first proposed by Hood [287] and later improved by Duff et al. [191] and Zlatev [658]. A dropping procedure [191] was implemented by Tsai and Liu [583] whereby a modified Jacobian matrix was employed in favour of the full Jacobian and where small-valued elements of the Jacobian matrix were discarded. A drop tolerance e measured the difference between the reduced Jacobian and the full Jacobian. The modified Jacobian matrix was also used as the preconditioner for two iterative methods: the BiCGStab method of Van der Vorst [587] and the GMRES method of Saad and Schultz [523]. The GMRES method had first been utilized in viscoelastic calculations by Fortin and Fortin [219] and was also the solution method used by Baaijens [24] in his finite element simulations of a UCM fluid flow through a 4:1 circular contraction. The finite element discretization used by Tsai and Liu [583] was the SU 4 x 4 method of Marchal and Crochet [384] and allowed them to reach a Deborah number of 34.2 with the iterative methods. Perhaps the most significant result of the paper was the comparison of the CPU time required by the frontal solver and the two iterative methods to solve the Oldroyd B problem at a Deborah number 243
8.3. NUMERICAL SIMULATION De = 6.3 starting with the De = 6 solution as the initial guess. Although it is not stated, the same error tolerance was presumably used in all three cases. Both iterative schemes were shown to be faster than the direct solver with the BiCGStab method outperforming GMRES for all but the largest (e = 0.01) drop tolerances. BiCGStab was also shown to lead to a solution two and a half to three times faster than the direct method. In a comparison of the streamlines for flow up to De = 1.56 the results of Tsai and Liu were seen to be in good agreement with those of Yoo and Na [642] who used a finite volume method, although the vortices of Tsai and Liu [583] were slightly smaller. The publication of Matallah et al. [387] represents one in the long pedigree of papers on the subject of Taylor-Galerkin methods for viscoelastic flows coming from the Townsend and Webster stables, and tracing its ancestry back to 1993 and the original paper of Carew et al. [124]. The authors sought to investigate the performance of various recovery and stress-splitting schemes for simulations with the Oldroyd B model in the framework of a Taylor-Galerkin/pressure cor rection method with consistent streamline upwinding. Flow past a cylinder and through a 4:1 planar contraction were considered. It was found that by using a direct local recovery method (see, for example, [278]) and thereby furnishing the constitutive equation with continuous velocity gradients, that both a DEVSS scheme and the conventional scheme of Carew et al. [124] were provided with greater stability and that the limiting Deborah number (De = 3Ai) for the 4:1 planar abrupt contraction was doubled in the case of the conventional scheme. A lip vortex could be seen for the solution obtained using the conventional scheme with recovery for Deborah numbers beyond 6 until computations were halted at De = 24. A coupled EVSS scheme was tested against the conventional scheme with recovery and was found to give exaggerated vortices beyond De « 6. The problem here was not the EVSS scheme per se but the continuity condition cor rection that was employed in order to maintain stability. To prove the point, a decoupled EVSS method, not requiring continuity correction for stability, was found to give identical results to those predicted by the recovery scheme. When continuity correction was then added the over-exaggerated vortices seen in the solution of the coupled EVSS scheme reappeared. Matallah et al. [387] compared the non-dimensional cell size predicted by the conventional scheme with and without recovery with those of Yoo and Na [642], Basombrio et al. [46,47], Sato and Richardson [527], Marchal and Crochet [384] and the experimental polymer melt results of White and Baird [626]. Basom brio et al. [46,47] used a transient decoupled method with a Lesaint-Raviart discontinuous method for the stresses [47], and a method of characteristics [46]. The results of Matallah et al. [387] at Re = 0 were found to compare well with those of Basombrio et al. [47] and Yoo and Na [642], but the results of Marchal and Crochet [384], as can be seen from Fig. 8.7, were somewhat different. The overall trend seen in the calculations of Matallah et al. [387] was a decrease in the vortex size up to De « 6 and a more or less constant size vortex from that point onwards. The vortex size at Re = 1 for increasing values of De agreed well with the predictions of Sato and Richardson [527] and, as expected, were smaller for comparable Deborah numbers than at Re — 0. Useful light was shed on the issue of appropriate mixed finite element for mulations and methods of integration for the viscoelastic contraction problem by Debae et al. [169]. The authors compared a simple mixed method, the 4 x 4 method [384] and two different interpolations for the EVSS method [485]; each 244
CHAPTER 8. BENCHMARK PROBLEMS I: CONTRACTION FLOWS within a Galerkin, SUPG and SU framework. The two EVSS methods differed in the degree of the approximation spaces that were used: in the one method (EVSS1) the finite elements used for the components of the stress and rateof-strain tensor were linear P 1 — C° elements, of the same degree as for the pressure; in the other (EVSS2) biquadratic polynomials P2 — C° were used. In both EVSS1 and EVSS2 P2 - C° elements were used for the velocity compo nents. The problem solved by the authors was that of steady flow of a UCM fluid in a 4:1 circular contraction and results were presented for the Couette cor rection and the non-dimensional vortex re-attachment length as functions of a Weissenberg number, defined in terms of a downstream wall shear-rate. As has been documented already in this chapter, the use of the SUPG method for the abrupt contraction proved disastrous and no figures of the results were supplied in this case. The SU technique, as is to be expected, proved much more stable than the Galerkin formulation. Results were presented for the problem up to We = 9.5 (although the authors could have gone further) with the simple mixed method being the only one to diverge before this point. The results for both the Couette correction and the vortex attachment length clearly showed the effect of the artificial diffusivity in the case of the two EVSS formulations: the EVSS Couette correction results were slightly lower but the vortex attachment lengths markedly higher than the corresponding quantities predicted by the 4 x 4 method. Reducing the magnitude of the diffusivity parameter for EVSS1 and EVSS2 brought the re-attachment lengths calculated with these methods into closer conformity with that of the 4 x 4 method. Overall, the authors con cluded on the basis of calculations performed additionally for UCM flow past a sphere and through an undulating tube, that the EVSS1 method, because of its cheapness and robustness, as well as the low cost involved in calculating flows of multi-mode fluid models, could be commended as a good candidate for three-dimensional simulations. The final comparative study of our review is that of Fan [203] in 1997. The author considered the performance, for the 4:1 circular contraction problem (amongst others), of a discontinuous Galerkin method in both a mixed formu lation (MIX/DG) and EVSS formulation (EVSS/DG), as well as two slightly different EVSS/SUPG methods. For all the methods considered Legendre poly nomials of degree two were used for the stresses and velocities, and of degree 1 for the remaining variables. The two EVSS/SUPG methods differed in the fact that for one a discontinuous pressure was employed (EVSS/SUPG) and for the other this field was continuous (EVSS/SUPG*). Fan [203] used the same MCR model as had been employed by Coates et al. [142] in order to avoid non-integrable stresses near the corner singularity and to compare results with theirs. Using the same definition of the Weissenberg number as Debae et al. [169], Fan [203] used a mesh whose smallest element (near the corner) was smaller than that of Coates et al. [142] and found that a limiting Weissenberg number of 21 applied in the case of the EVSS/DG method, 10.5 in the case of the MIX/DG method, and 7.5 in the case of the EVSS/SUPG method; this latter value rising to 10 when continuous pressure approximations were used (EVSS/SUPG*). Coates et al. [142] reported a limiting Weissenberg number of 9.4 with their finest mesh and the SUPG/EEME formulation. Vortices pre dicted with the EVSS/DG method of Fan [203] showed expansion upstream with increasing Weissenberg number and were consistent with those of Coates et al. Fan [203] concluded that, although the EVSS/DG method was a little 245
8.3. NUMERICAL SIMULATION
more expensive than the EVSS/SUPG method because of the discontinuous interpolation for the stresses, the additional cost was more than offset by the benefit of an increase in robustness.
246
Chapter 9
Benchmark Problems II 9.1
Flow Past a Cylinder in a Channel
Early interest in viscoelastic flows past cylindrical bodies arose principally be cause of the importance of such flows in instrument technology where hot-wire probes or sensors may be placed in an elastic fluid flow and correlations made between the drag and heat transfer coefficient data (see James and Acosta [306], for example). The problem of flow past a cylinder placed centrally between two plates (see Fig. 9.1) would seem to be well suited as a benchmark problem for gaining a better understanding of viscoelastic liquids in a complex flow: near the cylinder both shearing and extensional effects are present - the flow in the gap between the cylinder and channel walls being primarily a shear flow, and the flow near the axis of symmetry upstream and in the wake of the cylinder has a high extensional component. The geometry is, despite the complexity of the flow through it, comparatively simple to build and is more amenable to birefringence measurements than its close cousin the falling-sphere-in-a-tube problem. From a numerical point of view, the geometry lends itself well to numerical methods, particularly because of the absence of geometrical singularities. Having said this, however, the accurate solution of the stress field near the cylinder surface and in the wake of the cylinder, poses a huge challenge to the computational rheologist. For this reason the problem was chosen as a benchmark problem for the comparison of various codes in terms of stability and accuracy [105]. The origins of the analysis of creeping Newtonian flow past a submerged body go back to the middle of the 19th century and the work of Stokes [543] who produced a solution for slow uniform flow past a sphere and derived a formula for the drag on the sphere. But a similar analysis for two-dimensional flow past a cylinder did not work and it was left to Oseen [432] in 1910 to identify the difficulty as the neglect of the convective term in the Navier-Stokes equations far from the cylinder. It was not until 1957, however, that Proudman and Pearson [472] provided a complete solution to the problem of uniform flow past a cylinder using matched asymptotic expansions.
9.1.1
Streamline patterns and drag: unbounded flows
Early work in viscoelastic flow past a cylinder primarily sought to address the questions of how the streamline patterns and drag were modified by elasticity. 247
9.1. FLOW PAST A CYLINDER IN A CHANNEL
Figure 9.1: Cylinder radius R placed symmetrically in a two-dimensional chan nel of half width H. The earliest theoretical analysis of viscoelastic flow past a cylinder was per formed by Ultman and Denn [585]. Their paper is important because it demon strated the possibility of a change of mathematical type in the vorticity equation, depending upon whether the product Re We of the Reynolds and Weissenberg numbers is greater or less than unity. Only the subcritical case ReWe < 1 was considered. By assuming small deformation rates the constitutive equation was linearized with respect to velocity deviations from the streamwise velocity. Oseen's approximation was used to linearize the momentum equation. However, as pointed out by Mena and Caswell [399] some three years later, Ultman and Denn [585] satisfied the inner boundary conditions in an approximate (numer ical) manner by solving an overdetermined system, rather than using a proper matching procedure. As a consequence, the results of Ultman and Denn are at variance with those in the literature that have appeared since the publication of their paper. In particular, Ultman and Denn predicted an upstream shift in the streamlines even for very low values of the Deborah number. Although this was supported in their paper by dye-streak experiments with aqueous polymer solu tions the validity of their results seems questionable in the light of subsequent papers. Mena and Caswell [399], for example, presented an analysis for flow of an Oldroyd B fluid past an immersed body by solving the Oseen equations in the far field and matching this solution asymptotically with the Stokes solution near the body. For flow past a cylinder Mena and Caswell [399] predicted that the effect of elasticity was to shift the streamlines in the downstream direction. The authors also showed that the Newtonian drag coefficient was unchanged to first order and that the first contribution due to elasticity appeared as a quadratic term in the velocity. The same two results were shown to apply to flow past a sphere. Drag measurements by Broadbent and Mena [102] for flow past cylin ders and spheres of a solution of polyacrylamide in glycerol or water confirmed that, for viscoelastic liquids, the departure from the Newtonian drag appeared as a quadratic function of velocity. However, these authors observed no visible change in the streamline pattern when compared with the purely viscous case. The earliest published numerical solutions of flow of a viscoelastic fluid past a submerged circular cylinder were those arising from the finite difference proce dures of Pilate and Crochet [466] and Townsend [580,581]. Pilate and Crochet considered plane flow of a second-order fluid and found, in qualitative agreement with Mena and Caswell [399] and Broadbent and Mena [102], that viscoelastic248
CHAPTER 9. BENCHMARK PROBLEMS II ity reduced the drag coefficient for very low Reynolds numbers. However, the variation of the drag coefficient was linear in Weissenberg number for values of the Reynolds number up to 40, rather than quadratic. Chilcott and Rallison [134], who observed the same linear dependence some years later, noted that this behaviour was due to the choice of boundary conditions on the stress at the finite upstream boundary. In agreement with the experimental observa tions by James and Acosta [306], drag increases were possible for large values of the Reynolds number. Just as with Broadbent and Mena [102] little could be observed by the authors in elastic effects at low Reynolds numbers on the streamline shift. Townsend [580,581] in his simulations with 2- and 4-constant Oldroyd models saw a small downstream shift of the streamline pattern which was increased by the presence of shear-thinning. His results for the drag coeffi cient agreed qualitatively with those of Pilate and Crochet [466], the small drag decrease at low Reynolds numbers becoming a large increase in drag at higher Reynolds numbers. The question of whether or not elastic effects cause a downstream or up stream shift in the streamlines was shown to be more complex than previously observed or predicted with the publication by Manero and Mena [382] in 1981 of experimental results for slow flow of solutions of polyacrylamide in water or in a water-glycerol mixture. Results at Weissenberg numbers of 0.6 and 2.0 may be seen in Fig. 9.2 (a) and (b), respectively. For values of the Weissenberg number We < 1 a downstream displacement of streamlines was observed consistent with the majority view up to that time and in qualitative agreement with predictions from the small perturbation analysis of Mena and Caswell [399]. Interestingly, however, for We > 1 an upstream displacement of the streamlines appeared. Similar experimental evidence for this behaviour had been published by Zana et al. [643] in 1975 for flow past spheres. Chilcott and Rallison [134] used their finitely extensible dumbbell constitu tive model (see §2.6.1) in finite difference simulations of viscoelastic flows past bubbles, solid cylinders and spheres. By considering the trace, tr(QQ), of the ensemble average of the dyadic product QQ of the dumbbell end-to-end vector Q the authors were able to identify three regions of high polymer extension for flow past a cylinder or sphere. These were (a) a short distance upstream of the cylinder, (b) at the sides of the cylinder where high shear stresses can cause large deformations in the dumbbells and (c) downstream of the rear stagnation point. The wake region (c) was pinpointed as the one exerting the greatest single influence on the dynamics of the flow. As just remarked, Chilcott and Ralli son [134] observed a linear fall-off of the drag coefficient for flow past spheres and cylinders for small (
9.1. FLOW PAST A CYLINDER IN A CHANNEL
(a)
(b)
Figure 9.2: Pictures of flow of a shear-thinning aqueous solution of polyacrylamide past a cylinder in a wide channel. Flow is from left to right, (a) We = 0.6, (b) We = 2. Reprinted from 0. Manero and B. Mena, On the flow of viscoelastic liquids past a circular cylinder, J. Non-Newtonian Fluid Mech., 9:379-387, Copyright (1981), with permission from Elsevier Science. that the perturbation of the velocity field from the Stokes flow was larger down stream than upstream. Chilcott and Rallison [134] showed that the positive gradient of dumbbell extension in the near wake of the sphere influenced the flow field by acting as a body force and accelerating the fluid close to the sphere. Since (slightly downstream of the rear stagnation point) the extension gradients become negative after the point of maximum extension, their influence could be seen as a deceleration there. Thus the acceleration very close to the sphere may be adequate to produce a measurable upstream shift in the streamlines with an overall downstream shift a little further away from the rear stagnation point. The question was further discussed by Bush [110,111] in the context of the stagnation flow behind a sphere. The shear viscosity of the model developed by Chilcott and Rallison [134] is constant and no overshoot in the axial velocity in the wake of the cylinder (or sphere) was observed. However, Chilcott and Rallison [134] conjectured that the presence of shear-thinning would result in 250
CHAPTER 9. BENCHMARK PROBLEMS II a reduction of flow resistance in the shear flow regions near the cylinder and increase the influence of polymer forcing close to the rear stagnation point and possibly result in greater fluid acceleration there. Hassager [274] called the ve locity overshoot which has been observed in flows of shear-thinning fluids past rigid spheres [78,534] a "negative wake" since when the experiment of a falling sphere is observed in the laboratory frame of reference (and not with respect to axes fixed in the sphere) the velocity overshoot manifests itself as a region where the axial velocity is in the opposite direction to that of the sphere. A more recent numerical analysis of unbounded viscoelastic flow past a cylin der has been performed by Matallah et al. [387] who used a recovery scheme in conjunction with a Taylor-Galerkin/pressure correction method with consis tent streamline upwinding in order to obtain greater stability at higher Deborah numbers for flow of an Oldroyd B fluid than proved possible using either con ventional Taylor-Galerkin [124] or EVSS alternatives. By computing flow at Re = 10 past a sphere the authors were able to compare the streamlines of their results with those of Townsend [580] and Pilate and Crochet [466]. Good agree ment appeared to be reached and, in particular, the downstream shift of the streamlines and the elongated wake behind the cylinder when compared with the Newtonian case, were corroborated.
9.1.2
Streamline patterns and drag: cylinders in channels
So far our discussion of viscoelastic flow past a cylinder has concerned unbounded flows. Unlike the case of a sphere falling in a tube (see [127], for example) there is no non-Newtonian equivalent of the Faxen-Bohlin formula [91,206,265] for the correction to the velocity field necessitated by the presence of the chan nel walls. Experiments were performed by Dhahir and Walters [179] in 1989 with Boger fluids, aqueous solutions of polyacrylamide and aqueous solutions of xanthan gum, in order to investigate the effect of viscoelasticity on the flow of non-Newtonian fluids past a cylinder contained in a confined geometry. By placing the cylinder asymmetrically between the channel walls Dhahir and Wal ters [179] showed that, unlike a Newtonian fluid, increasing the flow rate of the Boger fluid resulted in the dividing streamline shifting towards the closer wall indicating that a relatively smaller proportion of the liquid passed through the narrower gap as the flow rate was increased. This is thought to be a mani festation of the extensional viscosity: molecules that enter the narrower gap elongate more strongly than those in the broader gap and therefore in the nar row gap the extensional viscosity and resistance to flow are greater. For all the non-Newtonian flows tested a force was generated at right angles to the mean flow direction when the cylinder was placed in an asymmetric position. This force was directed towards the nearer of the two channel walls. As with the previously cited papers for unbounded flow past spheres and cylinders the pres ence of viscoelasticity resulted in a reduction of the total drag on the cylinder. Numerical simulations with the POLYFLOW package, using generalized New tonian, White-Metzner and UCM models were limited by a severe constraint on the maximum possible Weissenberg number and, although qualitative agree ment with experimental results was possible for the effect of elasticity on the force experienced by the cylinder, the simulations were not able to predict the observed change in streamline patterns referred to above. The preference of a viscoelastic liquid to flow through the wider of the two 251
9.1. FLOW PAST A CYLINDER IN A CHANNEL gaps in the case of an asymmetrically placed cylinder was confirmed numeri cally by Carew and Townsend [123] using a Galerkin finite element code and fluids of the Oldroyd B and exponential P T T type. The wall separation to cylinder diameter ratio was fixed at 12/7. With a ratio of the two gaps between the cylinder and channel walls equal to 1/4 the viscoelastic fluids were almost stagnant in the small gap, for sufficiently large Weissenberg numbers. Interest ingly, the presence of shear-thinning in the P T T model was observed to result in a small upstream shift of the streamlines relative to the Newtonian flow pat tern. A comparison of Newtonian and Oldroyd cases showed no visible change. For the eccentric placement of the cylinder, the demonstration by Dhahir and Walters of a transverse force towards the closer of the two channel walls was confirmed by Carew and Townsend's finite element calculations. Drag reduction with increasing elasticity was also ratified. Huang and Feng [291] used an EVSS finite element method from the pack age POLYFLOW to investigate the dependence of the drag on a cylinder and the velocity profile in its wake as functions of the aspect ratio A = R/H and rheological properties of the fluid. Their results showed that wall proximity shortened the wake and increased the drag and that the presence of elasticity tended to reduce these effects. For small A elasticity tended to increase the drag and lengthen the wake for Reynolds numbers in the range 0.1 < Re < 10.
9.1.3
Comparison of numerical and experimental results
From 1994 onwards papers began to appear in the literature comparing ex perimental and numerical results for confined flow past a cylinder. Encour aged by the good performance of the P T T model in simulations of the flow of a shear-thinning 5.0%wt polyisobutylene/tetradecane (PIB/C14) solution through a planar 4:1 contraction [17,21,477], Baaijens et al. [26] sought to establish whether or not a single mode P T T model would do an equally good job in predicting stress fields for flow of the same fluid around a cylinder placed symmetrically between two parallel plates. Laser Doppler anemometry (LDA) velocity and flow induced birefringence (FIB) stress measurements were com pared with numerical predictions from a discontinuous Galerkin (DG) finite element code (see §7.2.2) at a Deborah number of 0.22. The authors showed that the measured velocities compared quite well with the predicted data and any discrepancies were attributed to slight differences in the experimental and numerical flow rates, a slight eccentricity of the cylinder with respect to the two plane parallel plates and the possibility of a small offset error in the experimen tal streamwise ordinate. However, dramatic discrepancies were found between computed and measured stresses downstream of the cylinder along the axis of symmetry: the measured stresses reached a maximum that was twice as high as the computed value. Moreover, the measured stresses relaxed much more slowly than the computed stresses. In an attempt to improve on these results Baaijens et al. [29] built a new experimental apparatus and this time sought to simulate the same test fluid at Deborah numbers lying between 0.25 and 2.32 using finite elements with a parameter-fitted generalized Newtonian model and with both single and four mode P T T models. A comparison was also made in the case of an asymmetrically confined cylinder. On this occasion good agreement was found between computed and measured stresses at all flow rates, with the four mode P T T model doing an excellent job in predicting the downstream normal 252
CHAPTER 9. BENCHMARK PROBLEMS II stress differences. The discrepancy between these results and those obtained a year earlier was attributed to a change in the rheological behaviour of the test fluid during the former experiments. In a continued effort to investigate the predictive capabilities of existing differential models for multidimensional complex flows, Baaijens et al. [27] chose to assess the worth of the exponential version of the P T T model and the Giesekus model for simulations of flow of low density polyethylene (LDPE) past confined cylinders. The parameters for the models were fitted on steady shear data only, and primarily a four mode fit was used. A modification of the DG method was applied for the computa tions and comparisons were made at Deborah numbers of 4.4 and 8.1 between the experimental and computed birefringence patterns. Although the Giesekus model appeared to do the better job at capturing the birefringent tail in the wake of the cylinder, neither of the models used proved capable of predicting the birefringence distribution correctly. Problems were also encountered with both models in predicting the mixed shear-extensional flow between the cylinder and the walls. The same conclusions had to be drawn for simulations of flow past an asymmetrically confined cylinder. Attention has been given in the literature to the reliability of numerical sim ulations involving integral equations of the Rivlin-Sawyers type in predicting the confined flow past a cylinder of polymer melts. The usefulness of the P T T model and Rivlin-Sawyers equation [511] with the Papanastasiou, Scriven and Macosko (PSM) damping function [441] in describing the rheological behaviour of linear low density polyethylene (LLDPE) and LDPE in steady shear flows and transient extensional flows was shown by Hartt in his PhD thesis [272]. In 1996 Hartt and Baird [273] fitted the parameters of these models to the shear viscosity and extensional viscosity data of both LLDPE and LDPE melts and then compared the experimental birefringence data with the predictions of fi nite element calculations. For the single mode P T T model both Marchal and Crochet's 4 x 4 SUPG and SU formulations were used, whereas a finite element method developed by Dupont et al. [193] was employed for the computations with the Rivlin-Sawyers constitutive equation. Differences were observed be tween the two melts in the stress fields along the centreline downstream of the cylinder. The LDPE melt exhibited large extensional stresses there but these were not present for the LLDPE melt. The difference in the extensional stresses was predicted with both the single mode P T T model and the Rivlin-Sawyers equation. The use of only one relaxation time with the P T T model was thought to explain why only qualitative agreement could be obtained when this model was used to simulate the flow of the two melts. When the finite element results with the Rivlin-Sawyers model were compared with the birefringence patterns for the flow of LDPE at Deborah numbers of 0.62 and 3.2 past the cylinder, the agreement was good in the wake and better overall than with the P T T . The authors also looked at flow past three cylinders placed in a series along the centreline of the planar channel. The birefringence patterns in the wake of the third cylinder were very different from those observed in the wake of a single cylinder and it was thought that this was due to the dependence of fluid elasticity and extensional viscosity on the deformation history. In this situation both the P T T and Rivlin-Sawyers models failed to predict the dependence of the flow behaviour on the deformation history for the LDPE melt. A Rivlin-Sawyers constitutive equation with the PSM damping function was the model of choice for Mitsoulis [405] who compared his numerical solutions 253
9.1. FLOW PAST A CYLINDER IN A CHANNEL with the reported experimental data on polyethylenes by Hartt and Baird [273] and with the LDPE data of Baaijens et al. [27]. Both 2:1 and 16:1 aspect ratios were considered. Numerical results obtained using the data for Baaijens's LDPE melt led to good agreement with the P T T results of Baaijens et al. [27] and their experimental results. However, the simulations for the LDPE/LLDPE melts of Hartt and Baird [273] could not reproduce the sharp wedge-like lines in the birefringence patterns in the wake of the cylinder. The reason for this failure was attributed to the fact that the integral equations did not capture accurately the planar extensional characteristics of the melts. Therefore, Mitsoulis [405] modified the constitutive equation by introducing a strain-memory function associated with every relaxation time in the melt. In this way it was possible to obtain strain-thickening of the planar extensional viscosity. The use of the new constitutive equation to simulate the three melts resulted in an improvement in the representation of the birefringence lines in the wake of the cylinder. However, the Giesekus model of Baaijens et al. [27] still seemed to do better and, in the absence of steady-state planar extensional data for a wide range of extensional rates, further work remained to be done in fitting parameters for Mitsoulis's model.
9.1.4
Purely elastic instabilities
The first experimental investigation into purely elastic (i.e. in the absence of inertia) instabilities in viscoelastic flow past a circular cylinder placed symmet rically in a channel was conducted by McKinley et al. [391] in 1993. The fluid used in their study was a highly elastic fluid of the Boger type and consisted of a very dilute solution of polyisobutylene in a polybutene/tetradecane solvent. The viscosity of this fluid was fairly constant over four decades of shear-rate and the maximum Reynolds number in all the experiments performed was only about 0.04. In this way inertial and shear-thinning effects were eliminated and any instabilities that appeared could be attributed solely to the fluid elasticity. McKinley et al. [391] defined a shear-rate dependent relaxation time based on the viscometric functions in steady shear flows as
M7) = | § ,
(..i)
and then defined a (shear-rate dependent) Deborah number De for the flow by
De = A l ( l H
(9.2)
R where R denotes the cylinder radius and (uz) an average axial velocity in the channel. For most of the results A was taken to be 0.5. McKinley et al. [391] sought to simulate the flow using a single-mode FENE-CR [134] constitutive equation and the finite element code developed by Lunsmann et al. [365]. How ever, computations were only possible up to a Deborah number De = 0.8 before convergence of the method was lost. Since the experimental data showed that the flow was still two-dimensional and steady at this point, no comparisons in the unstable regime were possible. Quantitative measurements of the velocity field were made by the authors using laser Doppler velocimetry (LDV). Results at low (w 1) Deborah numbers before the onset of instabilities revealed that 254
CHAPTER 9. BENCHMARK PROBLEMS II as the Deborah number increased, the axial velocity in the wake of the cylinder crept up to its fully developed value more and more slowly and gave rise to a downstream shift of the wake and streamlines relative to the Newtonian flow. At De = 1 . 3 the onset of a steady three-dimensional flow instability confined to the wake of the cylinder, was observed. Profiles measured in the neutral di rection parallel to the cylinder axis were presented by the authors to show the development of a steady three-dimensional cellular structure which extended down the length of the cylinder. These birefringent bands extended as far as 10 cylinder radii downstream and the wavelength of the disturbances (bands) was approximately equal to the radius of the cylinder. As the flow rate increased further a second flow transition, this time to a time-periodic instability, was observed for De > 1.85, the bands now beginning to move slowly towards the channel walls. In order to inspect the effect of the aspect ratio, A was succes sively reduced to the value 0.17. The results of plotting a stability diagram of Deborah number against a Weissenberg number (this latter based on the shear-rate 7 in the gap) revealed that the onset point of the first instability en countered depended more strongly on the Deborah number and cylinder radius than on the Weissenberg number and channel gap. A precise criterion for the critical conditions required for the onset of purely elastic instabilities in a wide range of different geometries was elucidated by McKinley et al. [392] in 1996. The authors presented experimental and theoret ical evidence to support their claim that the destabilizing mechanism leading to purely elastic instabilities was the combination of streamline curvature and elastic normal stresses which give rise to tension along the streamlines. Specifi cally, a criterion for the onset of purely elastic instabilities was proposed by the authors in the form Aif/ni n 7707
1/2
> Mara,
(9-3)
where U is the characteristic streamwise fluid speed, "R. is a characteristic radius of curvature, m is the stress in the direction of the streamline, % is * n e (zero shear-rate) viscosity, 7 is a characteristic rate-of-strain and Mcrii some critical magnitude. McKinley et al. [392] suggested that the dimensionless group on the left-hand side of (9.3) could be thought of as a viscoelastic Gortler number with the onset of an elastic instability and the generation of streamwise vorticity occurring when this Gortler number exceeded a critical magnitude. The authors interpreted the term XiU as a streamwise length scale over which the perturbations to the stress and velocity fields relax, so that Ai U/1Z is a measure of the distance over which disturbances are advected relative to the streamline curvature. The product XlU Til
n 7707' then gave a measure of the relative magnitude of Ai U/1Z to the elastic stresses in the base flow. The authors also suggested a way in which shear-thinning effects and a spectrum of relaxation times could be incorporated into the dimensionless criterion (9.3). The criterion (9.3) was shown by the authors to reduce to well established results in the literature in the case of geometrically simple flows (circular Couette cell, cone and plate rheometer, etc.) and was also applied to more complex problems. For the problem of viscoelastic flow past a circular 255
9.1. FLOW PAST A CYLINDER IN A CHANNEL cylinder a relationship — = - + —,
(a,b, constants),
(9.4)
between the curvature 7?. -1 and R and H was postulated. The maximum axial velocity gradient e m a i in the wake of the cylinder was taken to be emax « ^ = ^ (a + Ab),
(9.5)
and this was shown to describe the calculated values of emax extremely well for a Newtonian fluid. Hence, calculating the value of TH in (9.3) for the Oldroyd B fluid (with solvent to total viscosity ratio equal to /3) to be approximately m = 2(1 - 0 t o o A i 4 „ ,
(9-6)
and combining all the above results in (9.3) resulted in the condition AXE/N2
R
1/2
{a + Ab)' 2(1-/3)
> M ,criti
In an unpublished paper (but see [440] for a similar analysis) Oztekin et al. performed a numerical linear stability analysis in a narrow wedge-shaped region in the wake of the cylinder. From a linear regression fit to the numerically calculated values at the onset of the elastic instability the constant o, b and Merit could be determined. A stability diagram of 1/De cr jt versus A for onset of elastic instabilities in the wake of the cylinder was presented by McKinley et al. [392] for both the numerical linear stability analysis of Oztekin et al. and experimental results of Byars [113] and McKinley et al. [391]. Straight lines could be drawn convincingly through the various data sets, validating a relationship of the form (9.7), although for all non-zero values of A (confined flows) lines drawn through the experimental points had smaller slopes than the line through the data of Oztekin et al., the flow thus remaining stable to higher Deborah numbers than predicted by the Oldroyd B linear stability analysis. The question of whether unbounded viscoelastic flow past a cylinder at zero Reynolds number was stable or unstable proved impossible to answer with certainty, based upon the analysis presented in the paper by McKinley et al. [392]. If the equation for the maximum velocity gradient (9.5) were still to be valid as A ->■ 0, however, then the analysis predicted a critical Deborah number Decrit « 7.1. In experiments involving particle image velocimetry (PIV) measurements of the velocity field for flow of a PIB-based Boger fluid past a cylinder in a channel with A = 1/16, Shiang et al. [533] were able to obtain steady flow conditions in the region near the rear stagnation point for (zero shear-rate) Deborah numbers of 0.6 and 1.2. However, physical constraints on the time over which an experiment could be performed prevented the experiments attaining similar steady flow conditions for De = 3. As a consequence, the flow transition predicted (in their unpublished paper) by Oztekin et al. to occur at De > 3 could not be observed. 256
CHAPTER 9. BENCHMARK PROBLEMS II One of the reasons for the slow progress in the stability analysis of multi dimensional flows of viscoelastic fluids in complex geometries was identified by Sureshkumar et al. [552] to be the difficulty associated with the potentially very large generalized eigenvalue problem generated by the linear stability analysis. Sureshkumar et al. [552] therefore proposed an alternative time-dependent sim ulation in which numerically accurate methods were employed to compute the linearized equation set resulting from a perturbation about a (numerically com puted) base flow. Their numerical analysis of linear stability to two-dimensional disturbances to a base flow past a linear periodic array of cylinders in a channel was conducted using a 6 method [524,525] based on the DEVSS-G/SUPG finite element method for the inertialess Oldroyd B equations. In this way it was hoped that the flow of a Boger fluid could be simulated and, by varying the separation distance between the cylinders, different levels of interaction in the flow fields around the separate cylinders could be anticipated, in turn affecting the stability of the flows. The results presented by Sureshkumar et al. [552] focussed on the standard A = 1/2 ratio and separation distances L ranging over 2.5 to 30 were chosen for the analysis. In keeping with other simulations of flow of an Oldroyd B fluid past a cylinder in a channel, a viscosity ratio (3 = 0.59 was taken. Computations with L = 6 and 30 on a mesh with approximately 30,000 degrees of freedom were stable to two-dimensional perturbations in the stress for (zero shear-rate) Deborah numbers up to 0.71 and the most dangerous eigenfunction decayed at rates which closely matched the behaviour for plane Couette flow. In the case of closely spaced cylinders with L = 2.5 the critical Deborah number for the onset of an instability was reported to be DeCTit K 0-64 and a Hopf bifurcation from the base flow was suggested. The observed instability persisted even with mesh refinement. Contour plots of the stream function showed that the eigenfunction associated with the most dangerous eigenvalue in the unstable flow exhibited a cellular structure close to the channel wall which was convected past the cylinder by the mean flow. No secondary flow was observed near the cylinder surface. Linear stability computations were also performed for L = 3 and L = 3.5 and the critical Deborah number for the onset of instabilities was seen to scale linearly with L. Assuming that this linear scaling could be extrapolated to the case L = 30 a critical Deborah number of 7.7 was predicted, way outside what could be realized numerically. The nonlinear dynamics of the L = 2.5 separation distance case were explored by solving the fully nonlinear discrete equations for De > Decrit. The results showed that for De > Decru the steady state was linearly unstable and that the flow then evolved into a nonlinear timeperiodic state or limit cycle which occurred as a supercritical Hopf bifurcation. An explanation given for the instability observed for flow past closely spaced cylinders was that it was driven by an interaction between a shear mode and a time-periodic pulsation caused by the closely spaced cylinder geometry.
9.1.5
Comparison of numerical m e t h o d s
Despite the fact that there is no geometrical singularity present for the cylinderin-a-channel problem the presence of steep stress boundary layers and a severe normal stress wake combine to present a great challenge to those who seek to achieve accurate results at high Deborah numbers. This problem has proved to be numerically even more difficult than the related sphere problem because, for 257
9.1. FLOW PAST A CYLINDER IN A CHANNEL the same aspect ratio, the planar flow past a cylinder in a channel undergoes stronger contraction and expansion than axisymmetric flow past a sphere in a tube. Particular difficulty is associated with numerical simulations of the UCM and Oldroyd B fluids in this geometry on account of their strongly tensionthickening properties. All the usual grid-based numerical methods have been tested on the flow past a single cylinder, either in unbounded flow or when constrained to lie in a channel. A representative list of references for each method is as follows: • Finite difference methods [134,466,580,581] • Spectral element methods [131,436] • Finite volume methods [10,186,187,290,427,457] • Finite element methods [26,27,29,41,122,204,273,291,300,307,309,359, 387,388,391,405,443,549,552]. Since reference has already been made in our review of work on this bench mark problem to many of the papers enumerated above, we choose to focus our attention on some more recent results [10,131,187,204,359,436,443,549] where comparisons may be easily made and some conclusions drawn. All Deborah numbers and Weissenberg numbers quoted in the discussion below are defined in the same way:
We = De =
h<£l,
(9.8)
where (uz) denotes the average value of the entry axial velocity component. Liu et al. [359] used numerical simulation to investigate flow of polymer so lutions around a periodic linear array of cylinders where the fluid was modelled by the Giesekus, FENE-P and FENE-CR models. The Oldroyd B model was considered as a limiting case of the FENE-CR model as the maximum dumb bell length Q0 was allowed to go to infinity. The authors used two different finite element formulations for solving the problem: the EVSS-G [106] and the DEVSS-G [257,556] formulations, the 'G' indicating the fact that the velocity gradient V u was approximated by a (continuous) variable G separate from the other fields. Liu et al. [359] succeeded in getting converged solutions up to a Weissenberg number of approximately unity for the problem of a single cylinder and aspect ratio A = 1/2. The meshes used required up to 67,495 degrees of freedom. It was noted that the velocity profiles along the centreline upstream of the cylinder superposed for all three models when normalized with the aver age velocity across the channel. Downstream of the cylinder, however, as has been noted on several occasions already in this review, the solutions exhibited pronounced and differing downstream shifts relative to Stokes flow, the axial ve locities along the centreline recovering more slowly than in the Stokesian case to their fully developed values. One point of particular interest was the observation by Liu et al. [359] that the degree of downstream shift appeared to correlate with the shear-rate-dependent relaxation time Ai(7) = * I ( 7 ) / 2 J 7 ( 7 ) , so that, for fixed 7 and ignoring shear-thinning, the relaxation time of the models was proportional to *x and the ordering of the models in terms of increasing down stream shift should be (and was indeed observed to be): Newtonian, FENE-P 258
CHAPTER 9. BENCHMARK PROBLEMS II and FENE-CR. Since * i increases with Q0 for the FENE-CR model, it there fore came as no surprise to see that the downstream shift of the velocity for the Oldroyd B model was greater than any of the other models. The predicted stress fields at We = 1 . 0 were in qualitative agreement with the FIB measurements by Baaijens et al. [29] for a PIB/ C14 solution. Plots of the axial stress and average molecular extension clearly showed that the regions where the polymer molecules became highly extended were near the stagnation points on the cylinder and adjacent to the cylinder and channel surfaces (cf. the review of comments made by Chilcott and Rallison [134] a little earlier). Maximum polymer extension usually occurred adjacent to the cylinder surface where the shear-rates were large, although for the Giesekus model the maximum extensions shifted to the wake of the sphere for larger We. Liu et al. [359] also observed that the effect of elasticity was to increase the length and intensity of the stress wake. A surprising conclusion from inspecting the drag on the cylinder predicted by the three models was that since the extensional behaviour of all the models was identical, it must therefore be the shear behaviour of the fluids that is crucial in determining the difference in the drag forces, at least for the range of Weissenberg numbers covered in the simulations. A comparison with other references of the drag on the cylinder computed by Liu et al. [359] for flow of an Oldroyd B fluid past a cylinder in a channel with A = 1/2 will be made in Table 9.1. A new stabilized formulation featuring a stabilization term in the momentum equations where this consisted of a square residual of the continuity equation, was introduced by Fan et al. [204] in 1999 and presented a challenge to the widely accepted belief that the momentum equation should be made explicitly elliptic. The constitutive equation (UCM and Oldroyd B) was handled with a consistent SUPG method, and h-p finite elements were used for the discretization of the resulting set of weak equations. The resulting Galerkin least squares formulation (called MIX1 by the authors) was then tested on three benchmark problems: flow of a UCM fluid between eccentric cylinders, flow of a UCM fluid around a sphere in a tube and finally, flow of a UCM and Oldroyd B fluid around a cylinder in a channel with aspect ratio A = 1/2. For the cylinder problem and using a UCM fluid, comparisons were made with two other finite element formulations: MIXO, in which a Galerkin method was used for the momentum and continuity equations and the SUPG technique for the constitutive equation, and the DEVSS method. For the UCM fluid, limiting Deborah numbers of 0.75 and 0.8 were encountered for MIX1 and DEVSS, respectively; the major difficulty in obtaining convergence being identified as the accurate determination of the axial normal stress profile in the wake of the cylinder. Convergence could not be obtained for any Deborah number using MIXO. For flow of an Oldroyd B fluid (with solvent to total viscosity ratio /? = 0.59) through the same geometry the h-p MIXO method performed as well as the MIX1 and DEVSS formulations but convergence was lost after a Deborah number of 1.05. Again, inspecting the axial normal stress in the wake of the cylinder identified the convergence problem: no convergent trend was evident in the stress profiles for those Deborah numbers larger than the value where the predicted drag force attained a minimum, and led Fan et al. [204] to suspect that, as a consequence, numerical solutions for the Oldroyd B fluid at Deborah numbers higher than about 0.8 were numerical artifacts. A comparison with other references for the drag on the cylinder computed by Fan et al. [204] with MIXO is presented in 259
9.1. FLOW PAST A CYLINDER IN A CHANNEL Table 9.1. Chauviere and Owens [131] used a stabilized spectral element method for the Oldroyd B benchmark problem and arrived at the same conclusion as Fan et al. [204]: as shown in Fig. 9.3, convergence with mesh refinement of the axial normal stress solution in the wake of the cylinder was lost somewhere between De = 0.7 and De = 0.8. In fact, taking z as the axial coordinate, as shown in Fig. 9.1, if the zz component of the Oldroyd B constitutive equation is examined along the axis of symmetry in the wake of the cylinder at a point z*, say, where the normal elastic stress component TZZ is assumed to attain a local maximum (or minimum) T*Z, say, then since at z* the streamline derivative will be zero, the equation satisfied by T*Z is
where r)p is the polymeric viscosity and uz the axial velocity component. Thus, if duz/dz approaches l/2Ai for Ai sufficiently large, TZZ becomes unbounded. The presence of the nonlinear terms in the P T T model, for example, ensures that the same axial normal stress growth mechanism does not take place, as was verified by calculations by the same authors [131]. The strong statement on the conjectured Deborah number limit for Oldroydtype fluids by Fan et al. [204] did not, perhaps, cast the best of lights upon some of the numerical results of Sun et al. [549] for the cylinder problem. Sun et al. [549] developed a numerical method with several different components: a combination of the Adaptive Viscous Stress Splitting (AVSS) of earlier work [548] and the DEVSS method of Guenette and Fortin [257] for preserving the elliptic character of the momentum and continuity equation pair, a continuous approximation G for the velocity gradient field, and a discontinuous Galerkin method. The resulting DAVSS-G/DG method was used for solving the system of equations for the Oldroyd B and Giesekus models and applied to the cylinder problem for aspect ratios of A = 1/2 and A = 1/8. Drag calculation results were compared for the Oldroyd B (/? = 0.59) with those from the DEVSS-G method of Liu et al. [359] for the larger of the two ratios, and additionally with a DAVSS/SUPG method for A = 1/8. The DAVSS-G/DG Oldroyd B calculations converged for the narrow channel up to De = 1.85 (cf. the comments of Fan et al. [204]) and significantly, no axial normal stress profiles in the wake of the cylinder were presented by the authors. Hence no assessment of whether the authors achieved convergence in the critical region in the wake of the cylinder can be made. Drag calculation results agreed well with the DEVSS-G calculations of Liu et al. [359] up to a Deborah number of 1.0 (see Table 9.1). However, the drag is not, in general, a sufficiently discerning measure of solution accuracy [130]. For the wider channel, converged solutions up to De — 12.35 were possible compared with the maximum values of De = 6.9 for the DAVSS-G/SUPG method and 3.6 for the DEVSS-G/SUPG method. The maximum value of Deborah number achievable with the DAVSS-G/DG method was relatively insensitive to the choice of which of three meshes (Ml - M3) was used, these meshes varying in the total number of degrees of freedom required from 25,495 (Ml) to 80,760 (M3). Good agreement between the numerical meth-
260
ffl
> '-a •-3 H pi
a
M 2! O
to 05
a s > *d £J O td f H
S
CO
Figure 9.3: Profiles of the non-dimensionalized axial normal stress TZZ on the cylinder surface (—1 < z < 1) and in the wake of the cylinder (z > 1) with mesh refinement, at Deborah numbers of 0.7 (left) and 0.8 (right). Oldroyd B fluid. A = 1/2, (3 = 0.59. iV refers to the polynomial degree used in the stress approximation. Reprinted from C. Chauviere and R. G. Owens, A new spectral element method for the reliable computation of viscoelastic flow, Comput. Meth. Appl. Mech. Engrg., 190:3999-4018, Copyright (2001), with permission from Elsevier Science.
9.1. FLOW PAST A CYLINDER IN A CHANNEL ods could be seen in the predicted axial normal stress profile in the wake of the cylinder for the wide channel. However, although calculations with the three meshes for the DAVSS-G/DG method gave rise to good quantitative agreement for the maximum axial stress value in the wake of the cylinder up to De PS 2 there was wide disagreement between the results with the different meshes beyond that point. The equivalent results for the narrow channel, had they been available, would presumably have shown more dramatically the lack of convergence in the wake, this case being a more severe test of the numerics. Independent calculations by Pasquali [443], also using a DAVSS-G/DG method (with 82,925 degrees of freedom), were in excellent agreement with the drag values computed by Sun et al. [549] on mesh M3 for the A = 1/8 case: there was only a 1% difference in their respective values at De = 2.2. However, at a Deborah number of 2.0, the maximum values of the axial normal stress in the wake of the cylinder computed by the two sets of authors differed by 54%! Shear-thinning in the Giesekus model meant that for a wide channel calculation with the DAVSSG/DG formulation no limiting Deborah number was encountered by Sun et al. [549] with the two finest meshes used and good convergence of the axial stress profiles with mesh refinement could be observed. Channel flow past a cylinder with PTT, UCM and Oldroyd B fluids was the problem solved in three papers by Dou and Phan-Thien [186,187,457], who in each of these papers used an unstructured finite volume method based on the SIMPLER [642] algorithm. In [187] Dou and Phan-Thien utilized both a DEVSS and DAVSS formulation together with an independent interpolation of the vorticity to arrive at, respectively, the DEVSS-w and DAVSS-OJ formulations. Computations done with both methods and four different meshes for Oldroyd B flow past a cylinder in a channel with A = 1/2 revealed that, whereas the DEVSS-w method converged better than the DAVSS-w method, it would only go up to a limiting Deborah number De = 0.9. The DAVSS-w method allowed computations to proceed up to De = 1.8. However, as admitted by the authors, the adaptive viscosity in the DAVSS-w method which is large in high stress regions, tends to amplify discretization errors, calling into question the overall accuracy of the predictions. Caola et al. [121] used the DEVSS-G/DG finite element method presented in [549] for solving the flow of an Oldroyd B fluid past a single confined cylinder. The method involved an operator-splitting time integration method (see the paper of Smith et al. [537]) for the decoupling of the pressure and velocity calculation from the integration of the constitutive equation. The resulting generalized Stokes problem was solved using a BiCGStab iterative method [587]. Up to 751,110 degrees of freedom were used by the authors and a limiting Deborah number of 1.1 was encountered for the 2:1 cylinder problem. Owens and Chauviere [436] developed a locally upwinded spectral technique (LUST) for viscoelastic flow calculations. Unlike previous work by the same authors using stabilized spectral element methods [130,131] the upwinding fac tors were allowed to vary within each spectral element and were related to the Deborah number and the local mesh spacing. Mesh converged solutions up to a Deborah number of 1.2 were presented for the Oldroyd B 2:1 cylinder-in-achannel problem and showed excellent agreement with the drag results of Caola et al. [121] (see Fig. 9.4). A comparison of the drag results for the 2:1 Oldroyd B cylinder problem between the DEVSS-w and DAVSS-w methods of Dou et al. [187], the MIXO 262
CHAPTER 9. BENCHMARK PROBLEMS II results of Fan et al. [204], the DEVSS-G results of Liu et al. [359] and the DAVSS-G/DG results of Sun et al. [549] are presented in Table 9.1. We also show the drag results by Caola et al. [121], Owens and Chauviere [436] and the fine-mesh (M120) finite volume results of Alves et al. [10]. In the latter, the method of Oliveira et al. [427] was employed. For the results shown in Table 9.1 the SMART scheme of Gaskell and Lau [227] was used by Alves et al. in the treatment of the convective terms in the Oldroyd B equations. The following comments may be made: 1. A wide discrepancy is obvious, even for lower Deborah numbers where all the methods converged. This is disturbing and adds to ones anxiety about the worth of some of the solutions presented. Further proof of the poor accuracy realized in the finite volume calculations of Dou and PhanThien [187] was presented by these authors in their plots of axial stress profiles in the wake of the cylinder at a Deborah number of only 0.6. No convergent tendency was clear with mesh refinement. 2. Only by extrapolating the DAVSS-w drag factor data to zero mesh size (column 3 of the results in Table 9.1) could any kind of reasonable agree ment with other published results be seen. The conclusion would seem to be that adaptive viscosity methods, whilst enhancing stability, compro mise the accuracy of the solution and especially in low-order finite volume implementations. The disparity in the computed drag data may be seen yet more clearly in Fig 9.4. There we see that the only results that match well in the range 0.5 < De < 0.7 are those of Alves et al. [10], Fan et al. [204], Caola et al. [121] and Owens and Chauviere [436]. For 0.7 < De < 1.0 two trends are evident: first, the results of Alves et al. [10] and Fan et al. [204] and secondly, the results of Owens and Chauviere [436] and of Caola et al. [121]. The results of the first group are slightly higher than those predicted by the second.
9.1.6
Mesoscopic calculations
As an alternative to numerical methods designed for integrating closed form constitutive equations Hulsen et al. [300,434] have proposed a new approach for calculating viscoelastic flows where the polymer stress is determined from a microscopic model. Previously, Ottinger and Laso [207, 344,345] had in troduced the so-called CONNFFESSIT approach that combined finite element techniques and Brownian dynamics simulations in order to calculate the poly mer contribution to the stress from a large ensemble of model polymers. Hulsen et al. [300,434] circumvented the difficulties in the CONNFFESSIT approach associated with the tracking of individual molecules by working instead with ensembles of configuration fields, continuously defined over the flow domain. In order to test the new approach, Hulsen et al. [300] presented results for the start-up of planar flow of an Oldroyd B fluid past a confined cylinder with aspect ratio A = 1/2. Since both a closed form constitutive equation and stochastic equation for the configuration fields is known for solutions of Hookean dumb-bells, Hulsen et al. [300,434] were able to compare their microscopic cal culations with those from the macroscopic approach. The DEVSS formulation of Guenette and Fortin [257] was used for the discretization of the momentum 263
De
to
0.0 0.025 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.85
DEVSS-w [187] 131.811 131.500 131.079 129.715 126.239 123.155 121.018 120.024 119.884 120.606 122.099 124.206
DAVSS-w [187] 131.811 131.502 131.082 129.723 126.406 123.515 121.562 120.577 120.485 121.130 122.572 124.480 126.885 129.716 133.157 136.942 141.948 147.172 152.820 158.664 165.236
Extrap. FV [187] 131.319 131.028 130.577 129.094 125.197 121.872 119.495 117.966 117.134 116.686 117.232 117.868 118.776 119.687 120.496 121.541 122.858
DEVSS-G/SUPG [359] 132.34
119.48 118.72 118.54
119.47
MLXO [204] 132.36
130.36 126.62 123.19 120.59 118.83 117.78 117.32 117.36 117.80 118.49
DAVSS-G/DG [549] 132.34
130.33 126.63 123.26 120.76 119.11 118.17 117.84 117.98 118.50 119.32 120.38 121.65 123.07 124.66 126.32 128.01 129.67 131.37 132.30
START FV [10] 132.209 131.788 130.343 126.618 123.195 120.596 118.832 117.786 117.328 117.370 117.865 118.56
LUST [436] 132.357
DEVSS-G/DG [121] 132.384
t-1
O
118.827 117.775 117.291 117.237 117.503 118.030 118.786 119.764
118.763
32 i-3
O
117.783 118.031
Table 9.1: Drags computed by various authors for the Oldroyd B confined cylinder problem. /3 = 0.59 and A = 1/2. N.B. The results of Liu et al. [359] are as supplied in the paper of Sun et al. [549]. The results of Sun et al. [549], other than those at De = 0,0.5 - 1.0, have been calculated from the drag correction factors as supplied by Dou and Phan-Thien [187]. The results of Dou and Phan-Thien [187] have been calculated from their drag correction data. All other results are as originally supplied by the authors.
2 d
H I—I
>
a >
H tr1
CHAPTER 9. BENCHMARK PROBLEMS II
Figure 9.4: Comparison of computed drags on a single confined cylinder. 01droyd B model. j3 = 0.59. A = 1/2. Results shown are those of Dou and PhanThien (DAVSS-w) [187], Liu et al. (DEVSS-G) [359], Owens and Chauviere (LUST) [436], Sun et al. (DAVSS-G/DG) [549], Alves et al. (SMART-FV) [10], Fan et al. (MIX0) [204] and Caola et al. (DEVSS-G/DG) [121]. equation and equation of continuity. The DG method was used for the con stitutive equation in the macroscopic case and the stochastic equation in the microscopic case. A solvent to viscosity ratio /3 = 1/9 and Reynolds number Re = 0.01 were taken and results were presented for the Oldroyd B fluid at times t < 7 and De = 0.6. Up to 8,000 configuration fields were used with the microscopic method and good agreement between the two methods was clear in the predicted drag coefficients. A non-dimensionalized conformation tensor b , proportional to the ensemble average of the dyadic product Q Q of the end-to-end vector Q for a Hookean dumbbell was known to have a determinant whose theoretical minimum is unity [613]. However, plots of detb against time for five different meshes at a Deborah number De = 0.6 for the macroscopic method revealed large errors in detb for the coarse meshes, the value of the determinant sometimes falling significantly below the theoretical minimum. An important difference between the microscopic calculations and the macroscopic ones was that when 4,000 con figuration fields were taken with the new approach and the calculations allowed to run up to t = 30 for the same Deborah number, detb became negative for 265
9.1. FLOW PAST A CYLINDER IN A CHANNEL the macroscopic method and eventually blew up. In contrast, the microscopic method remained stable with detb fluctuating near the theoretical minimum. Thus the greater stability of the microscopic method over a macroscopic ap proach was verified. A DEVSS/DG finite element method was used a year later by the same authors in conjunction with the Brownian configuration field technique to solve for the flow of a FENE fluid in the same geometry as [300]. Only the results of one run, at a Deborah number of 0.6, were presented by the authors since the aim of the paper was the determination of the parameters in the FENE-P model in such a way that the main flow features of the FENE fluid were recovered. The favourable time stability properties of the micro-macro approach and the ease with which the essential physics of polymeric liquids can be incorporated into the solution process are encouraging pointers to the future of numerical simulation of viscoelastic fluids.
266
CHAPTER 9. BENCHMARK PROBLEMS II
9.2
Flow Past a Sphere in a Tube
Figure 9.5: Geometry of a falling sphere in a cylindrical tube. The motion of a sphere through an incompressible viscous fluid at low Reynolds number is a classical problem and one of the oldest in theoretical fluid mechanics. The appearance of the problem in the literature dates back to 1851 and the work of G. G. Stokes [543] who developed an analytical solution for the case of creeping flow and an unbounded fluid. The first rigorous solution to the problem of the influence of bounding walls on the drag experienced by a sphere moving at low Reynolds numbers in a fluid was presented by Fax6n [205,206]. For a fascinating account of the development of the theory of the motion of a sphere in an incompressible fluid at low Reynolds numbers by Stokes, Oseen, Lamb and Faxen between 1845 and 1960, the reader is referred to the essay by Lindgren [356]. The problem of viscoelastic flow past a sphere falling in a cylindrical tube has received attention since the 1970's and manifests some important differences with the Newtonian problem, some of which are discussed in the sections that follow. Despite the simplicity of the geometry (see Fig. 9.5) and the absence of geometric singularities, reliable experimental and numerical data for the viscoelastic problem are difficult to obtain because of the complex mixture of shear and extensional flow regions that are present. The practical interest of problems involving flow around obstacles is to be seen, for example in the settling of suspensions, fluidized beds and rheometry [594]. Attempts to use falling-ball viscometers for the measurement of 267
9.2. FLOW PAST A SPHERE IN A TUBE non-Newtonian viscosities have been only partially successful. Although some success has been obtained with inelastic non-Newtonian fluids [112], the use of falling ball viscometers for viscoelastic fluids has been found to be inconvenient, principally because of the time required for the stress field in the fluid to relax between successive drops of the ball [136,442]. In §9.2.1 - §9.2.5 we will attempt to put into context some of the important issues in the benchmark problem of non-Newtonian flow past a sphere. For additional detail and insight the reader is referred to the review on the motion of spherical particles in viscoelastic liquids by McKinley [390].
9.2.1
Drag coefficient
By balancing the drag force experienced by a sphere moving at very low Reyn olds numbers through an infinite expanse of Newtonian fluid with its net weight, Stokes was able to deduce the sphere's settling velocity t/jvoo as UNoo = 2a2Apg/9r]o,
(9-10)
where Ap denotes the density difference of the sphere and the fluid and r)0 is the fluid viscosity. When the same sphere falls along the axis of a right circular cylinder filled with the same Newtonian fluid, the formula for the final settling velocity C/jv, say, has to be modified to take account of wall effects and is usually written in the form UN = UNoolKN{a/R),
(9.11)
where n^(a/R), the so-called Newtonian wall correction factor, was stated by Happel and Brenner [97,265] in the form
where f(a/R)
may be represented by the Faxen-Bohlin series [91,206] f(a/R)
= 2.10444(a/i?) - 2.08877(a/i?) 3 + . . . ,
(9.13)
valid for sufficiently small aspect ratios a/R. Since for Newtonian fluids there is a linear relationship between the drag and the velocity of the sphere, either the velocity or the drag may be corrected for wall effects using a relation of the type (9.11). In the case of non-Newtonian fluids the linearity in the drag/velocity rela tionship is lost and it is on the velocity that wall effects have been reported in the literature (e.g. [127,256,400]). In the case of incompressible Rivlin-Ericksen fluids [510], Caswell [127] de duced the non-Newtonian equivalent of the Faxen-Bohlin formula for a sphere experiencing a force of magnitude F on its surface:
U-U„- 2.0444^ (|) (! - | S (DO + ° I*" ™">' (9.14) By dropping spheres of varying densities and sizes in tubes of differing diameters, Caswell [127] plotted the settling speeds against 268
CHAPTER 9. BENCHMARK PROBLEMS II
2.08877 / a \ / a w 2.U88Y7 \R~) \ ~ 2.10444 \RJ and, using the experimental data of Turian [584], was able to determine from (9.14) approximations to both the zero-shear-rate viscosity of Turian's polymer solution and the terminal velocities Uoo that such spheres would attain in an unbounded expanse of the same polymer. Mena et al. [400] systematically investigated the influence of the Theologi cal properties of various fluids (Newtonian, viscoelastic, inelastic and constant viscosity-elastic) on the wall correction factor. The authors began with a gen eralization of the formula for the drag D experienced by a sphere whilst moving with speed U along the central axis of a right circular cylinder filled with a shear-thinning fluid. This was written as
D = 6ffa*j(7)t//s,
(9.15)
where n = re(7, a/R) is a non-Newtonian wall correction factor. In the particular case of very low shear-rates the drag-velocity relationship was written by the authors as D = GiraqoU K0,
(9.16)
with 1 KO
=
1-Ha/RY
the zero-shear-rate wall correction factor. Prom (9.16) one may write D (1 - f(a/R)) &irar)o
= U,
(9.17)
so that
D
=v+DiWK)m
67ra7jo
(jUg)
67rar?o
Supposing now that the left-hand side of equation (9.18) is put approximately equal to U^oo, the authors divided throughout by U to get
« s %»= i + ^m.
„ i+Kmf{a/m,
(9.i9)
thus yielding a non-Newtonian wall correction factor K=
i-f(a/R)(n(i)/voY
(9 20)
-
Thus (9.20) may be considered to be a generalization of the Newtonian wall correction factor and it may be seen that K -> KJV as 7 -> 0. The authors pointed out the similarity in form between the wall correction factor K given in (9.20) and the one proposed by Gu and Tanner [256] K
= 1
1
,\t
/pv
—a(n){a/R) 269
(9 21
-
)
9.2. FLOW PAST A SPHERE IN A TUBE in their analysis of power-law fluids, where a(n) is some function of the powerlaw index n. The authors found that, although for the shear-thinning viscoelastic fluid elastic effects could be important at lower shear-rates, at higher shearrates shear-thinning effects became predominant, and the experimental drag for both the inelastic non-Newtonian fluid, and the shear-thinning viscoelastic fluid could be predicted adequately by using (9.15) and (9.20). Further evidence for the validity of (9.20) was provided by Arigo and McKinley [14] who plotted experimental values for the viscoelastic wall correction against De{y) for flow of a shear-thinning test fluid in a tube with aspect ra tio a/R — 0.121 together with the Mena et al. [400] shear-thinning correlation (9.20). Excellent agreement was obtained at high -De(7) when the flow was dominated by shear-thinning in the viscosity. Mena et al. [400] compared their experimentally determined wall correction factor for their (shear-thinning) viscoelastic fluid with the numerically com puted factors of Hassager and Bisgaard [276] and the results were in good agreement for the (small) range of Weissenberg numbers We for which Has sager and Bisgaard [276] had been able to compute solutions. This agreement was possible for low We despite the fact that Hassager and Bisgaard [276] used a UCM model. Agreement with the experimental wall factor results of Sigli and Coutanceau [534] for a polyox solution was also found. For a Boger fluid (a polyacrylamide in glucose syrup solution) similar results to those of Chhabra et al. [133] were evident when a non-dimensional drag was plotted against the Weissenberg number. Influence of rheological properties on the drag The influence of the rheological properties of non-Newtonian fluids on the drag exerted on a freely falling sphere has been the subject of numerous investiga tions. Early analyses [128,233,351,399] gave rise to small perturbation solu tions valid for small Reynolds numbers and Weissenberg numbers and dealt with spheres in unbounded expanses of fluid. All of them predicted a small downstream shift in the streamlines and a small 0(We2) drag reduction. Early experimental evidence for these theoretical predictions was supplied by Tan ner [351], Broadbent and Mena [102] and Manero and Mena [382]. Of major importance in the investigation of rheological influences on the drag on a sphere falling through constant viscosity (Boger-type) viscoelastic fluids, has been the role played by the extensional properties of the fluid. In 1980 Chhabra et al. [133] measured the drag on a sphere falling slowly through an unbounded expanse of various glucose-Separan solutions. Although for (strainrate-dependent) Weissenberg numbers in the range [0,0.1] no significant devi ation could be observed, for higher Weissenberg numbers once the first nor mal stress difference was outside the quadratic region, the drag dropped to an asymptote 26% less than the Stokesian value. That the Weissenberg number is in itself insufficient in general to characterize the drag force in a viscoelastic fluid was ably demonstrated in experiments by Chmielewski et al. [135] us ing corn-syrup based and polybutene-based Boger fluids. Although the drag behaviour was consistent with the results of Chhabra et al. [133] for the poly acrylamide (PAA)/corn-syrup solution, no decrease was seen in the drag for the polyisobutylene/polybutene (PIB/PB) solutions in the Weissenberg num ber range 0.01 to 0.7. For Weissenberg numbers in excess of 0.3 the drag steadily 270
CHAPTER 9. BENCHMARK PROBLEMS II increased until at 0.7 the drag was some 15% higher than in the Stokes case. Measurements of the drag force on spheres in creeping motions through the Ml (PIB/PB Boger-type) fluid by Tirtaatmadja et al. [578] revealed a small drag reduction compared to the Newtonian value for small We followed by an in crease of more than 20% above the Newtonian value at a Weissenberg number greater than 1.6. A similar difference of behaviour between PIB/PB and PAA/corn-syrup Boger fluids had been observed by Boger et al. [89] in tubular contraction flow experiments (see §8.1.1). The two sets of fluids had similar viscosities and relaxation times. However, whereas the PAA solutions could give rise to vortex-growth behaviour, the PIB/PB solutions produced secondary flow vor tices whose reattachment lengths could decrease with Weissenberg number. Of great importance in the explanation for the very different behaviours observed by Chmielewski et al. [135] and Boger et al. [89] is consideration of the extensional behaviour of the two constant viscosity fluids. Here, the numerical predictions of Chilcott and Rallison [134] for a sphere settling in an unbounded dilute polymer solution, and those of Lunsmann et al. [365] and Satrape and Crochet [528] for a sphere falling along the axis of a cylindrical tube, are able to give us some insight. All the authors performed simulations with the FENE-CR model and found that, whatever the value of the non-dimensionalized maximum extensibility b = HQl/kT, the drag was more or less constant or decreased with Deborah number for small Deborah numbers. However, for Deborah numbers larger than 1.0 the size of 6 was observed by Chilcott and Rallison [134] to govern the drag behaviour: for Vb — 10, for example, an asymptote above the Newtonian drag was seen for large De whereas at -\A = 2.5 the drag fell monotonically for all De. Evidence may be found in the papers of Lunsmann et al. [365] and Satrape and Crochet [528] too, of an upturn in the drag for a sufficiently large b, and for sufficiently small aspect ratio a/R. Satrape and Crochet [528] also document an interesting case (a/R = 0.1, b = 10.48) where the drag decreases initially, then increases above the Newtonian drag and finally decreases below this level again. Hence, amassing the evidence and attempt ing to arrive at a verdict, it would seem that associating a constant viscosity PAA/CS fluid with a FENE-CR model having a small value of b and a PIB/PB model with a FENE-CR model having a larger value of b, that the difference in the observed drags between the two fluids may be attributed to the fact that the PIB molecules can be stretched more in PB solutions than can the PAA molecules in corn syrup/water [16,135]. An explanation for why a drag increase should be associated with larger b is that the more extensible polymer molecules form, at high Deborah numbers, a birefringent strand [268] in the wake of the sphere; this increases the effective size of the sphere and pushes the drag force up [51] . Such an increase in drag with Deborah number above that expected from the Newtonian wall effect was observed experimentally by Rajagopalan et al. [483] in their experiments with a sphere sedimenting through a PIB Boger fluid. Finite element calculations with an Oldroyd B fluid and experimental measurements with a PIB/PB Boger fluid (a/R = 0.243) by Becker et al. [51] showed a small drag decrease and then a pronounced increase at higher Deborah numbers and further corroborated the findings of Tirtaatmadja et al. [578] and Chmielewski et al. [135]. In an attempt to isolate and interpret the effects of extension thickening, first normal stress difference and shear-thinning on the drag on a sphere falling through 271
9.2. FLOW PAST A SPHERE IN A TUBE a non-Newtonian fluid, both Debbaut and Crochet [171] and, ten years later, Rameshwaran et al. [487], performed finite element simulations with various inelastic and viscoelastic models. Amongst the models chosen, both groups of authors used a generalized Newtonian model where the viscosity not only depended upon the shear-rate j but also on an extensional rate e. Thus it would be possible to obtain arbitrary values of the Trouton ratio in uniaxial extensional flow. For this model, the two papers showed that extension thickening with a zero normal stress difference led to drag enhancement with increasing flow rate. For low values of De, Ni and the extensional viscosity were shown to have opposite effects. Neither groups of authors considered the case of a shear-thinning viscoelas tic fluid. From their drag measurements for shear-thinning viscoelastic fluids Mena et al. [400] concluded that whether the viscoelastic medium was infinite or contained in a cylinder, the overall trend was the same: in the low shearrate region elastic effects were the predominant factor in drag reduction, but at higher shear-rates shear-thinning took over as the primary cause for further drag reduction. There is little consideration given in the literature to the effect of the second normal stress difference upon the drag. However, Mitsoulis [404] and Satrape and Crochet [528] modified the integral and differential form, respectively, of the UCM equation so as to incorporate a non-zero second normal stress difference. Their results were shown to be in excellent agreement for the case N2/Ni = —0.2 and the effect was to increase the drag slightly. Experimental investigation by Navez and Walters [416] into spheres falling in tubes filled with two different shear-thinning solutions yielded similar con clusions to those of Mena et al. [400]. Comparing the wall correction factor for an aqueous polyacrylamide solution with that for the numerical simulation of an inelastic generalized Newtonian model it was observed that a close match could be obtained and thus it was concluded that viscoelastic effects were small, and certainly much smaller than shear-thinning effects. The wall correction fac tor for the experimental results decreased monotonically when plotted against the Weissenberg number until inertial forces caused an upturn in the graph. A comparison with the generalized Newtonian model permitted the authors to draw the same conclusion about the dominant influence of shear-thinning in a second test fluid, Si (PIB in PB and decalin). Of interest here, however, was the graph obtained by dividing the wall correction factor for the Si fluid with that for the generalized Newtonian fluid and plotting this against We. Since the Reynolds numbers in the experiment were never greater than 0(1) the up turn in the graph at higher Weissenberg numbers was attributed to viscoelastic effects. This was stated to be "not inconsistent" with the behaviour that had been postulated earlier by Walters and Tanner [608]. Numerical simulations performed with the PTT model by the group at Sydney [308,547,550] have confirmed that the reduction in the drag force on a sphere in a shear-thinning viscoelastic liquid is mainly caused by shear-thinning effects for sufficiently high shear-rates. However, the upturn predicted by Walters and Tanner [608] in the drag force was not seen in their PTT simulations. In experiments performed by Degand and Walters [173] for a sphere falling in a tightly fitting cylindrical container (/? = 0.88) filled with the test fluid SI, an ad hoc calculation for an average shear-rate in the narrow gap and a Newtonian analysis allowed the authors to estimate that 84% of the observed drag decrease 272
CHAPTER 9. BENCHMARK PROBLEMS II was due to shear-thinning. The calculations of Mitsoulis [404] using a K-BKZ model showed reasonable agreement with the Degand and Walters [173] drag data. Influence of aspect ratio on the drag The influence of the cylinder walls on the drag on the sphere for a constant vis cosity Boger fluid has been studied extensively in the literature. Nevertheless, gaining a clear overall picture from the various references is complicated by the use of different fluids and the manner in which the aspect ratios are changed. Jones et al. [314] performed experiments with two different Boger fluids: the one (Type 1) a solution of PAA in maltose syrup/water, and the other (Type 2) a PIB/PB-based Boger fluid. Although drag enhancement beyond the Newtonian level was evident for all Weissenberg numbers, at an aspect ratio /? = a/R — 0.25 for the Type 1 fluid, this had all but disappeared at /3 = 0.5 where the behaviour was essentially observed to be Newtonian. Data for f3 « 0.91 showed a settling velocity considerably lower than in a comparable Newtonian fluid, thus indicat ing a significant drag enhancement. The reason for this drag enhancement was explained on the basis of the dominance of extensional viscosity effects in such a situation. For an aged sample of the Type 2 fluid Jones et al. [314] concluded that the (K/KAT, We) data was essentially independent of (3 for 0 < 0 < 0.15 and showed a steady increase from 1 to 2 over We 6 [0,2]. For 0.15 < (3 < 0.2, the authors observed a small increase with j3 in K/K^ values at each We. For 0.2 < /3 < 0.5, K/KN then decreased with /3 for each We. In a previous paper, Chhabra and Uhlherr [132] had concluded that for /? < 0.2 and 0.2 < We < 10 the elastic wall correction factor did not depend on /?. The experiments by Degand and Walters [173] with a PAA in maltose syrup/ water base for (3 = 0.88 were a natural extension of the work of Jones et al. [314]. Attempts at performing the free-falling sphere experiment at this high aspect ratio had been hampered by a low maximum obtainable Weissenberg number: it just was not possible to find a sphere sufficiently dense! In fact, it is considered unlikely that the drag upturn hypothesized in [314,608] will ever be observed for spheres sedimenting freely in closely fitting tubes (/? -► 1) [483]. Therefore it was decided to fix the sphere and pump the fluid past it. Manifested in the K/KN versus We graph was a Newtonian profile at low We, followed by a drag reduction to another (lower) plateau and finally, at higher Weissenberg numbers still, substantial drag enhancement as a consequence of the dominance of extensional effects. Evidence of drag enhancement at the opposite end of the aspect ratio scale (/3 —> 0) has been found numerically by Bodart and Crochet [82] in their compu tations with UCM and Oldroyd B models. For aspect ratios between 0.02 and 0.2 a minimum at We ?a 1 was discernible in the graphs of K/KJV versus We. For /3 = 0.5 the K/KN graph seemed to be monotonically decreasing. Numerical results by Arigo et al. [16] were able to converge to higher Weissenberg numbers than those detailed in the paper by Bodart and Crochet [82] and showed, even at (3 = 0.5, that a small upturn in the wall correction factor was discernible for Deborah numbers larger than 2. As with the observations of Jones et al. [314] and Chhabra and Uhhler [132], the results of Bodart and Crochet [82] for K/KN seemed little affected by the value of (3 for f3 < 0.1. The discovery by Jones et al. [314] of a decline in the magnitude of the 273
9.2. FLOW PAST A SPHERE IN A TUBE viscoelastic drag enhancement K/KN when ft « 0.5 was corroborated in the experimental findings of Arigo et al. [16] at ft RJ 0.4 and ft « 0.6. For lower values of ft (0.121 and 0.243) a small initial drag reduction was followed, at higher Deborah numbers, by a more significant drag enhancement. Rajagopalan et al. [483] changed the sphere to cylinder radius ratio by chang ing the size of the spheres, thus altering the elasticity number E = De/Re at the same time. Hence although their results for ft = 0.121 followed a familiar pattern of drag reduction at low Deborah numbers followed by enhancement at higher Deborah numbers, they observed that as ft progressively increased (from 0.243 to 0.632) the steady-state response of the system changed from drag enhancement to a drag decrease (settling speed greater than Newtonian). However, it should be noted that, although the terminal settling velocity grew larger relative to the Newtonian value, the Newtonian settling velocity also decreased with growing ft, leading to a reduction in the viscoelastic settling ve locity too. Arigo and McKinley [14] released spheres of the same size in tubes of varying radii, thus keeping the elasticity number constant from one experiment to the next. They found that, as the aspect ratio increased (0.089 to 0.387), the steady drag on the sphere in their shear-thinning aqueous polyacrylamide also increased, due to the greater wall proximity.
9.2.2
Benchmarking numerical methods
In acknowledgement of its importance in providing insight into rheological effects of fluid viscoelasticity in a comparatively simple (but, nevertheless, non-trivial) flow, and because of the inherent difficulties associated with the adequate res olution of stress boundary layers and wakes, the problem of a sphere falling freely in a cylindrical tube filled with a viscoelastic fluid was chosen as a bench mark problem by the computational rheology community back in 1988 [275]. In particular, the quantity chosen for the comparison of one numerical method or algorithm with another has often been taken to be the steady-state drag factor (or, wall correction factor) Drag on sphere 6-K-qoaU
which represents, as in (9.15), the ratio of the drag experienced by the sphere to that which would be experienced by the same sphere settling steadily in an unbounded expanse of a Newtonian fluid having the same viscosity. Although there are numerical results available in the literature for shear-thinning fluids such as those described by the two principal versions of the P T T model [16, 111,122,308,372,415,483,550,648] and for constant viscosity fluids such as the Oldroyd B fluid [82,122,130,232,318,365,372,415,647] and FENE-type fluids [82, 365,483] and these for a variety of aspect ratios, undoubtedly the most popular benchmark configuration has been the one in which the fluid is governed by the UCM model and the aspect ratio ft is taken equal to 0.5. The precise reasons behind the eventual breakdown of all numerical algorithms beyond a certain Deborah number have been the subject of much discussion and debate in the literature. In 1990 Zheng et al. [647] inferred that the loss of convergence of their boundary integral calculations for an Oldroyd B fluid in the ft = 0.5 geometry was attributable to a real limiting Deborah number somewhere between 0.5 and 0.7 and arising from an instability in their similarity solution along the centreline 274
CHAPTER 9. BENCHMARK PROBLEMS II of the tube. However, as pointed out by Lunsmann et al. [365], Zheng et al. [647] arrived at their conclusion by performing time integration of the transient equations, and hence no conclusion about the existence of steady similarity solutions is possible. Moreover, Lunsmann et al. [365] observed that the solution field near the rear stagnation point on the sphere in the work of Zheng et al. [647] began to stray from the similarity form before their calculations started to diverge. Shortly afterwards, Brown et al. [106] performed a numerical linear stability analysis of the flow of a UCM fluid past a sphere in a tube and found for the 0.5 aspect ratio case that the flow approached neutral stability at De = 1.6. Interestingly, this value for the Deborah number was approximately what was encountered as a limiting value in the numerical calculations of steady flow by Crochet and Legat [155], Rasmussen and Hassager [492] and Lunsmann et al. [365], and led to the conjecture of the existence of a physical limit point [105]. Lunsmann et al. [365] conjectured that the reason for the loss of convergence of their scheme was due to the inability of the mesh to resolve the very steep boundary layers around the sphere surface and in the sphere's wake, but were unable to support this conclusion on the basis of their calculations: the max imum attainable Deborah number did not increase with mesh refinement. In 1991 a paper by Jin et al. [308] appeared in which EEME finite element methods were used to good effect to obtain mesh converged solutions at Deborah numbers up to 2.2, thus challenging the assumption of the existence of a physical limit point at De = 1.6. That no physical limit seemed to exist at De = 1.6 was fur ther demonstrated by Arigo et al. [16]. Using an EVSS finite element method and careful mesh refinement, the authors showed, by calculating solutions at De = 2.2, that convergence with mesh refinement was possible up to Deborah numbers higher than 1.6. Lunsmann et al. [365] had also used an EVSS finite element method. Thus the result by Brown et al. [106] was shown to be prob ably due to a spurious numerical instability, arising from an insufficiently fine mesh. As pointed out by Rajagopalan et al. [483] no experiments with Boger fluids have thus far documented the onset of a similar viscoelastic instability. Since 1995, the authors of several published works have developed algorithms and discretizations enabling them to break through the De — 2 barrier for the steady settling of a sphere in a tube filled with a UCM fluid and aspect ratio /? = 0.5. Most of these references to date may be found in Table 9.2. A notable (and deliberate) omission from the table is a reference to the work of Baaijens [22] who used a DG method with constant stress elements, allowing him to proceed to Deborah numbers of at least 4 on all his meshes. However, as admitted by the author in a later review paper [25], convergence with mesh refinement could not be established. Also omitted, on account of the similarity of the method with that of Arigo et al. [16] two years earlier, are the EVSS/SUPG results of Fan [203], who achieved a Weissenberg number of 2.1. Equally, we have omitted any mention in Table 9.2 of a paper of Baaijens [24] where the author used a GMRES method for the solution of equations discretized using a DEVSS/DG method. This is because the DEVSS/DG method used was the same as that in a paper of the previous year by Baaijens et al. [27]. Baaijens [24] reached a Weissenberg number of 2.2 on a mesh having 28,078 degrees of freedom for the UCM confined sphere problem but did not report the computed drag factors. The borderline case of simulations by Hulsen et al. [298] with the deformation fields method, and with which the authors attained a Weissenberg 275
We Wemax 0.4 0.8 1.0 1.4 1.6 1.8 2.0 2.1 2.2 2.4 2.6 2.8 3.0 3.2
EEME [308] 2.2 5.1843 4.5270 4.3420 4.1432 4.0986 4.0800 4.0760 4.0812 4.0903
OS/SUPG [366] 2.8
3.9465 3.8639 3.7403 3.6638 3.5925 3.5142 3.4327
Adaptive h-p [619] 2.5 5.1862 4.5274 4.3405 4.1336 4.0831 4.0557 4.0454 4.0451 4.0476 4.0580
DEVSS/DG [27] 2.5 5.186 4.528 4.341 4.134 4.084 4.057 4.048 4.049 4.061
EVSS/DG [203] 2.1
4.339 4.142 4.084 4.056 4.041 4.041
AVSS/SI [548] 3.2 5.2223 4.5355 4.3291 4.0957 4.0084 3.9505 3.9388
EVSS/SUPG [16] 2.2
4.0492
DEVSS [204] 2.2 5.1928 4.5364 4.3513 4.1343 4.0839 4.0563 4.0455
MIX1 [204] 2.2 5.1870 4.5284 4.3407 4.1337 4.0832 4.0556 4.0450
3.8934 3.8792 3.8961 3.8865 3.8889 3.9307
4.0786
4.0486
4.0474
4.3354
Table 9.2: Drag correction factors K for steady settling of a sphere in a tube. UCM fluid. 8 = 0.5.
CHAPTER 9. BENCHMARK PROBLEMS II number of 2.0 for the UCM sphere benchmark problem, will not be discussed here. Details of the deformation fields method may be found in §6.7.2. It will be noted from Table 9.2 that excellent agreement up to De « 1.6 exists between drag factors predicted by all the works. Significant discrepancy is evident from that point onwards, however, in the case of the operator split ting (OS) method of Luo [366] and the AVSS/SI method of Sun et al. [548]. The results in the OS paper of Luo [366] were criticized by the same author two years later [367] as being inaccurate on account of a completely unstruc tured triangular mesh that was unable to resolve properly the thin elastic stress boundary layer on the sphere. This defect was remedied by the author and mesh-converged solutions up to a limiting Deborah number of 2.8 were possible but at a cost of some 200,000 degrees of freedom. The drag results of Sun et al. [548], although not diverging from the majority consensus as widely as those of Luo [367], would nevertheless seem to be inaccurate near the sphere, most likely on account of an adaptive viscosity which, although contributing greatly to the stability of the algorithm, can introduce large discretization errors in regions of high stress. Only 15, 582 degrees of freedom were required by the authors to reach De = 3.2 with their finest mesh. We conclude this section with two remarks. The first (and this will be discussed further in Chapter 10) is that the drag factor is a poor indicator of the solution quality although it is a convenient quantity for the comparison of numerical results. The same point is made by several authors: see for example, [27,130,202]. Clearly, although good solution accuracy should lead to credible drag factors, the close agreement seen in the results of [16,27,308,619] should not be interpreted as meaning necessarily that equally good agreement exists in the computation of the solution fields for these references. The second remark to make is that even if recent numerical results indicate that De sa 1.6 is not a limit point inherent in the UCM model it is quite plausible that a limit point still exists. No amount of mesh refinement or advances in sophisticated algorithms will be able to continue the process of obtaining accurate and mesh-convergent solutions at higher and higher Deborah numbers forever. Indeed, following a similar analysis to that pursued in §9.1.5 on the benchmark problem of flow past a cylinder in a channel, it may be seen that if duz/dz —> l/2Ai for Ai sufficiently large, exactly the same process operational in the planar flow will apply in the axisymmetric problem, leading to unbounded tensile elastic normal stresses in the wake of the sphere.
9.2.3
Negative wakes: steady flow
Beyond a critical Reynolds number a recirculating toroidal vortex develops in the wake of a sphere moving through a Newtonian fluid. However, whereas for lower Reynolds numbers the fluid velocity in the wake of a sphere falling in an otherwise stationary Newtonian fluid decays monotonically to zero, flow reversal, sometimes several radii downstream, may occur when the same sphere steadily sediments through a viscoelastic fluid. This region of flow reversal, where the fluid may move in a direction opposite to that of the sphere, was first called a "negative wake" by Hassager in 1979 [274] who observed the phe nomenon in a study of bubbles rising in shear-thinning viscoelastic fluids. In 1977 Sigli and Coutanceau [534] performed experiments to investigate the wake structure for a sphere falling in a tube filled with a sheax-thinning viscoelastic 277
9.2. FLOW PAST A SPHERE IN A TUBE fluid at low Deborah and Reynolds numbers. For sphere to tube aspect ratios between 0.25 to 0.75 they found a toroidal vortex in the wake of the sphere. Increasing the aspect ratio increased the elastic effect whereas the presence of inertia resulted in a dampening of elastic effects. Bisgaard [78] found the same effects due to fluid elasticity in experiments involving a shear-thinning solution of PAA in glycerol and aspect ratios between 0.04 and 0.18. LDV measurements also revealed oscillations in the wake velocity for Deborah numbers in excess of 30 and may have pointed to the onset of an elastic instability. A step in the right direction towards identifying the mechanisms that drive elastic negative wakes was taken by Maalouf and Sigli [373] in 1984. Four types of fluid were used by the authors in this study: a Newtonian fluid, an inelastic shear-thinning fluid, an elastic constant-viscosity fluid and various shear-thinning viscoelastic fluids. Then the flows of these fluids past ellipsoids, ovoids and cylinders were measured. Interestingly, only in the case of the shearthinning viscoelastic fluid was a negative wake observed and only when the elasticity number E, measuring the relative importance of elastic and inertial effects, exceeded some critical value. Further evidence that the presence of both elasticity and shear-thinning could be necessary for the presence of a negative wake in steady, low-Reynolds number flows was presented by Bush [110] in 1993. In experiments conducted with constant viscosity PAA/glucose syrup Boger-type viscoelastic fluids and a rigid sphere held on the centreline of a circular cylinder, no axial velocity overshoot (as one would see a negative wake as an observer on a moving sphere) could be observed. However, a downstream shift of the streamlines, correspond ing to a lengthening of the wake compared to a Newtonian fluid and resulting from an enhanced decelerating force on the fluid far behind the sphere, was evident from the results. Numerical evidence for a lengthening of the down stream wake, using boundary-element methods and finite elements, had been previously supplied by Zheng et al. [646,647], Carew and Townsend [122] and Jin et al. [308]. A plausible explanation for the fact that the velocity in the wake behind a moving sphere may decay much more slowly than for a Newtonian fluid (and hence the downstream shift of streamlines) was preferred by Harlen [266,267]: the polymer molecules in steady flow of dilute polymer solutions past solid ob stacles at low Reynolds numbers but high Deborah numbers may become highly extended in a narrow region downstream of flow stagnation points, producing what is called a "birefringent strand". The reason for the high extension in these molecules is that since they pass close to a stagnation point they remain in the flow sufficiently long to experience large strains. It is the birefringent strand that provides the enhanced decelerating force. In addition to the down stream shift of streamlines far from the sphere, Bush [110] also observed that an acceleration effect close to the sphere could be sufficient to cause an upstream shift of the streamlines very near the rear stagnation point. The possibility of a localized upstream shift in the streamlines had been noted previously by Chilcott and Rallison [134]. Arigo et al. [16] were no more successful in their search for a negative wake in the flow of a Boger-type fluid past a sphere in a tube than Bush [110]. The authors studied a sphere sedimenting steadily under gravity through a cylindrical tube filled with a PIB/PB Boger fluid and aspect ratios varying between 0.121 and 0.5 were used in the experiments. However, neither a negative wake nor the onset of a flow instability were detected in 278
CHAPTER 9. BENCHMARK PROBLEMS II any of their measurements. Interestingly enough, the same Boger fluid used by Arigo et al. [16] exhibited a three-dimensional instability at high Deborah numbers in flow past a cylinder [391]. As with Bush [110], elastic effects due to the extensional flow in the wake caused the wake to extend further downstream with increasing Deborah numbers; an effect that became more pronounced as the aspect ratio decreased. However, unlike the results of Bush [110] no up stream shift of streamlines very near the rear stagnation point of the sphere could be detected. The fact that Bush [110] investigated pressure-driven flow past a stationary sphere provides a possible explanation for this difference. More recently, Arigo and McKinley [15] conducted a thorough investigation into the formation of negative wakes for a sphere sedimenting steadily in a shearthinning viscoelastic solution of PAA in a 50/50 water/glycerin mixture. For both aspect ratios of 0.121 and 0.243 it was found that the stagnation point where the flow reversed in the wake of the sphere was equal to three sphere radii away from the sphere centre for all sphere densities and Deborah numbers, contradicting the earlier results of Sigli and Coutanceau [534] and Bisgaard [78], and leading the authors to conclude that the dynamics that control the posi tioning of the stagnation point are independent of De and primarily influenced by the geometry. The distance from the sphere of the point of attainment of the minimum axial velocity and the absolute value of the minimum attained by the axial velocity were both found to be functions of De, however, with both increas ing with increasing De. Differences in the results of Sigli and Coutanceau [534], Bisgaard [78] and Arigo and McKinley [15] could be attributed to rheological differences in the polymer solutions used by the different authors. Arigo and McKinley found that a negative wake occurred for all the aspect ratios (0.06 to 0.243). Increasing the aspect ratio magnified the negative wake and shifted it further downstream. However, a negative wake formed even for the small est aspect ratio and led the authors to conclude that a negative wake was a phenomenon that would form even in an unbounded domain. Thus the driving mechanism for the formation of a negative wake was not the confining cylinder walls but the interaction of shear and extensional flow in the wake of the sphere. Although there is no experimental evidence available to date for negative wakes in steady flows of constant-viscosity viscoelastic fluids, an elastic fluid which is not shear-thinning is present in the numerical simulations of Satrape and Crochet [528]. Previous attempts at finding a negative wake with constantviscosity models had been unsuccessful [134,365]. However, in their numerical study of wake formations in FENE-CR fluids, Satrape and Crochet [528] found that by taking the dimensionless maximum dumbbell length b sufficiently small (b — 10.48) flow reversal could be observed. The smaller the value of 6, the greater was the amplitude of the negative wake. Although negative wakes man ifested themselves for aspect ratios of both 0.1 and 0.5 the negative wake was further downstream and smaller in magnitude for the smaller aspect ratio. Harlen et al. [269] also observed, taking the same value of b as Satrape and Crochet [528], that finite extensibility in dumbbell models could lead to the appearance of steady state negative wakes. Their calculations involving a split Lagrangian-Eulerian finite element method with the FENE-CR model revealed that the magnitude of the flow reversal increased and moved further downstream with increasing Weissenberg number. Increasing the maximum extensibility pa rameter b for fixed values of the polymer concentration and Weissenberg number had the effect of sweeping the negative wake downstream. Qualitatively similar 279
9.2. FLOW PAST A SPHERE IN A TUBE results to those obtained by Bisgaard [78] could be realized for 6 = 5. Arigo and Mckinley [15], referring to an unpublished paper by Harlen (but see [267] since then) and considering the sign of V ■ r on the right-hand side of the z-component of the momentum equation, explained that the formation of a negative wake is intimately connected to the relative magnitudes of the axial gradient of the extensional tensile stress drzz/dz and the radial gradient of the shearing stress {I/r)d{rrrz)/dr in the wake. For large values of b the tensile stresses may become very large and a negative wake is not seen. Smaller values of b imply a dramatic reduction in the tensile stresses: the radial variations in the shear stress may now dominate the axial variations in the tensile stresses and a negative wake may result. Harlen [267] pointed out that in the wake of the sphere, along the down stream axis, the polymers are extended in the axial direction but that the de gree of extension (and hence the polymeric stress) decreases with increasing distance from the sphere. The axial gradient of the tensile stress then drives a flow towards the sphere, leading to a slowly decaying wake. However, away from the downstream axis, the polymers are stretched in a direction at an angle to the axis and the effect of this rotation of the polymer stretch direction is to produce a force away from the sphere. It is this force that gives rise to the negative wake. Chilcott and Rallison [134] did not observe a negative wake in their original paper since they expected to find the velocity overshoot for large values of b [528]. Arigo et al. [391] did not observe any negative wakes in their finite element calculations with the FENE-CR model, either, again because the choice of b appropriate for modelling their Boger fluid was not sufficiently small for shear forces to dominate in the wake. Numerical predictions of negative wakes have been possible with the P T T model and here references to the simulations by Jin et al. [308], Zheng et al. [648], Bush [111], and Sun and Tanner [550] are relevant. Zheng et al. [648] considered four different fluids: Newtonian, Carreau (Generalized Newtonian), UCM and P T T . A negative wake was obtained with the P T T model and Zheng et al. [648] concluded that both shear-thinning and elasticity were needed for there to be a negative wake in this case. Bush [111] performed numerical simulations with four different sets of parameters for the P T T model, giving rise to different shear and extensional properties. An attempt was then made to reproduce the overshoot seen in some of his experiments with a fixed rigid sphere and shear-thinning PA A solutions. For a P T T model having a comparatively weak extensional response, overshoot and an upstream shift of streamlines with Weissenberg number could be observed. However, increasing the Weissenberg number with one of the other P T T models led to a greater tendency for the downstream shift of streamlines. Bush concluded that the Weissenberg number was insufficient to characterize the behaviour of the velocity profile and that the extensional response must be taken into account. Denoting the Trouton ratio by Tr, Bush [111] concluded that it was when the ratio We/Tr was increased that there was an upstream shift of streamlines behind the sphere and the possible formation of a negative wake. An attempt at harmonizing the two criteria for the onset of a negative wake: that of Harlen [267] in the context of the FENE-CR model and of Bush [111] for the onset of a negative wake in his work with P T T models, was suggested by Arigo and McKinley [15] and further details may be found there. The key factor in both criteria is the size of the extensional stress that develops in the 280
CHAPTER 9. BENCHMARK PROBLEMS II wake of the sphere.
9.2.4
Velocity overshoots - transient flow calculations
Amongst the first analytic solutions for the transient motion of a sphere accel erating through a viscoelastic fluid were those of Thomas and Walters [574] and King and Waters [331]. King and Waters [331] used Laplace transform meth ods in order to arrive at an expression for the velocity of a sphere (valid for small Deborah numbers) as it accelerated under the action of a constant force through an unbounded expanse of a fluid governed by the linear Jeffreys model. Their results showed that the sphere velocity could undergo oscillations about the terminal velocity. Both the maximum velocity, occurring at the first over shoot, and the period of the oscillations, were found to be proportional to the square root of the relaxation time of the fluid. Importantly, the solution pre dicts that, for fluids having large retardation times, the solution is overdamped and the velocity goes through a single overshoot. For fluids having a smaller solvent contribution the sphere oscillations may be only lightly damped. Arigo and McKinley [14] identified the mechanism behind the oscillation in the sphere velocity about the terminal velocity as a phase lag between inertial accelera tions and the growth with time of the viscoelastic stresses that contribute to the drag experienced by the sphere. In the same vein, Harlen [266] explained the phenomenon of overshoot by observing that in a flow started from rest, before the birefringent strand in the wake of the sphere has become established, the stress exerted by the polymer is lower than when it is fully extended and so the elastic contribution to the drag on the sphere is less than its ultimate value. As a consequence the sphere may reach a speed which is greater than its final settling speed. Of interest in the more recent experimental and numerical literature on the transient motion of a sphere in a viscoelastic fluid have been the effects of constraining walls, departures from the linear viscoelastic regime and the possible inclusion of shear-thinning effects. In 1992 Zheng and Phan-Thien [646] used a boundary element method in order to simulate the unsteady motion of a sphere falling under gravity along the axis of symmetry of a cylindrical tube filled with a UCM fluid. Observing that the maximum velocity occurred at the first overshoot the authors also noted that increasing the Weissenberg number for a fixed aspect ratio had the effect of increasing the magnitude of the velocity overshoot and the time taken for the sphere to realize its peak speed. Reducing the Weissenberg number resulted in an increase in the frequency of the oscillations but a decrease in their amplitude. This was consistent with the earlier results of Thomas and Walters [574] and King and Waters [331]. Only for moderate (non-zero) values of the Weissenberg number were negative velocities (sphere bouncing) observed. The authors also examined wall effects by varying the aspect ratio from 1/6 to 1/2. The tube walls were found to reduce the effects of the elasticity of the fluid, i.e. increasing the aspect ratio resulted in a decrease of the maximum velocity overshoot. A couple of years later Bodart and Crochet [82] used finite elements to calculate transient flow of an Oldroyd B fluid around a sphere but their method of selecting material parameters was different from that of Zheng and PhanThien [646]: the Weissenberg number was based on the final steady speed of the
281
9.2. FLOW PAST A SPHERE IN A TUBE sphere rather than on a characteristic speed Tj u
HVP
+ Vs)
° ~ —A
'
4Traps
as in [646], where ps denotes the sphere density. Zheng and Phan-Thien [646] varied the Weissenberg number by changing the relaxation time Ai of the fluid so that the elasticity number, E = rjp\i/a2pf (where pf denotes the fluid den sity) measuring the relative importance of fluid viscoelasticity and inertia (and interpreted by Becker et al. [51] as a ratio of an elastic time scale for growth of viscoelastic stresses to a viscous time scale for diffusion of vorticity information), also changed. Bodart and Crochet [82] on the other hand, with the intention of simulating laboratory conditions changed We by varying the sphere density only, thus keeping E constant. These differences in selecting material parame ters are important because some of the results of Bodart and Crochet [82] seem to be in contradiction with those of Zheng and Phan-Thien [646]. In particular, Bodart and Crochet [82] found that increasing the Weissenberg number resulted in a decrease in the ratio of the magnitude of the initial overshoot to the steady state value, although as with Zheng and Phan-Thien [646] the magnitude of the overshoot increased with We. Translating the relative overshoot result of Bodart and Crochet [82] to laboratory language means that for two spheres of the same size, the heavier of the two will have a smaller relative overshoot, as seems intuitively reasonable. The higher the solvent viscosity, the greater was the damping effect found to be on the amplitude of the velocity oscillations. For a given Weissenberg number, Bodart and Crochet [82] found that de creasing the aspect ratio resulted in a decrease of the overshoot and the am plitude of the oscillations, in apparent contradiction with the results of Zheng and Phan-Thien [646]. Happily, Bodart and Crochet [82] were able to claim that when their numerical method was applied to the specific flows of Becker et al. [51] their results were in perfect agreement with theirs. Becker et al. [51] had examined both experimentally and numerically the motion of a sphere acceler ating from rest along the centreline of a tube filled with a PIB Boger fluid. This fluid had previously been extensively characterized by Quinzani et al. [480] in 1990. A Lagrangian finite element method following the basic idea of Rasmussen and Hassager [492] with both single and four mode Oldroyd B models was used for the numerical simulation. A digital imaging system was used in the experi ments. Becker et al. [51] fixed the aspect ratio fi = 0.243 and then took spheres of the same size but increasing density in order to vary the Deborah number, defined in the same way as the Weissenberg number of Bodart and Crochet [82]. It was found that as the sphere density (and therefore the Deborah number) increased both the magnitude of the velocity overshoot and the ultimate steady state velocity increased too. Transient numerical results using the single mode Oldroyd B model revealed the same qualitative behaviour as in the experiments, but with a much larger relative overshoot and a monotonic decay to the steady state. The use of the four mode model capturing approximately the spectrum of relaxation times in the Boger fluid in the experiments, proved to be much more encouraging and closer qualitative agreement with the experimental data resulted. No direct comparisons with experiments were made by Harlen et al. [269] who performed numerical simulations of the motion of a sphere falling in a FENE-CR fluid using a split Lagrangian-Eulerian method. It was interesting, 282
CHAPTER 9. BENCHMARK PROBLEMS II nevertheless, to see what happened when the sphere was allowed to fall from rest under the action of a constant force through an Oldroyd B fluid (b ->■ oo limit). A negative velocity perturbation (velocity overshoot) was observed in the transient wake behind the sphere for a simulation at We = 1 although as the polymer strand developed the perturbation changed size and, in the long time limit, the velocity was seen to decay more slowly throughout the wake than for Stokes flow. Improvements to the numerical results obtained by Becker et al. [51] for the prediction of the transient velocity of a sphere for an aspect ratio of 0.243 were published by Rajagopalan et al. [483] in 1996. Whereas the results of Becker et al. [51] with a four mode Oldroyd B model consistently underpredicted the velocity of the sphere, Rajagopalan et al. [483] obtained good quantitative agreement by using a four mode P T T model. For a larger aspect ratio (0.396) agreement was still excellent up to the point in time where the authors' EVSS finite element method diverged. Agreement was less satisfactory for /? = 0.632. Thus the incorporation of nonlinear variations in the viscometric properties of a rheological model was shown to be valuable. Comparisons between the predictions of different rheological models (FENE-CR, Oldroyd B, PTT) showed that the graphs of the velocity of the sphere versus time superposed and led to the conclusion that it is the inertia of the sphere and linear viscoelastic properties of the fluid that govern the initial motion. For larger time and higher strains (unsurprisingly) the nonlinear fluid rheology became important. The experimental results showed that the ratio of the magnitude of the overshoot to the steady state settling velocity decreased with increasing aspect ratio. This would seem to contradict the results of Bodart and Crochet [82]. However, as pointed out by Rajagopalan et al. [483], Bodart and Crochet [82] changed (3 by modifying the tube radius and keeping the elasticity number E constant, whereas Rajagopalan et al. [483] changed a in order to modify the aspect ratio, and thus E also changed. No damped oscillations were observed experimentally by the authors after the initial overshoot because of the high solvent viscosity of the Boger fluid. Wall effects were found to lower the steady settling velocity and to increase the damping rate of the velocity overshoot. In contrast to the overdamped transient response observed in the experi ments of Rajagopalan et al. [483] with their PIB/PB fluids, Arigo and McKinley [14] reported a series of decaying velocity oscillations about the final velocity for a shear-thinning PAA solution, qualitatively similar to what had been pre dicted by the King and Waters [331] solution. The authors studied successively the effect on the sphere velocity of varying first, the density contrast ps/pf of the sphere and fluid and secondly, changing the aspect ratio whilst keeping the density contrast and fluid elasticity number constant. The magnitude, Umax, of the maximum velocity attained by the sphere was found to increase with sphere density although UmaK/U3, where Us is the ve locity of the sphere, decreased at the same time. The effect of increasing /3 (by choosing tubes of different diameters) was to decrease the velocity value at all times and to increase the frequency of the oscillations and rate of damp ing. However, UmaK/Us remained unchanged. These results, after taking ac count of differing definitions of the Deborah number and the way in which aspect ratios were chosen, were in broad agreement with the numerical re sults of Zheng an Phan-Thien [646] and Satrape and Crochet [528]. For an aspect ratio of 0.243 Arigo and McKinley [14] actually observed the velocity for 283
9.2. FLOW PAST A SPHERE IN A TUBE the sphere become temporarily negative during the first oscillation: the sphere "bounced"! The paper is of particular significance as it represented the first reported quantitative experimental measurements of transient oscillations and rebound in the problem of a sphere accelerating from rest, although rebounding had been predicted in some regions of parameter space by Bodart and Crochet [82] and there is photographic evidence by Walters and Tanner [608] for the phenomenon. A qualitative explanation for the experimental observations was provided by Arigo and McKinley [14] by applying a simple velocity correction to the solution of King and Waters [331] in order to take account of wall effects. It should be pointed out however, as do Arigo and McKinley [14], that whereas an analysis incorporating linear viscoelasticity and the inertia of the sphere describe well the sphere's initial transient response, such an analysis is unable to predict the formation of a negative wake, which is a large strain phenomenon and therefore associated with the nonlinear fluid rheology.
9.2.5
Comparison between experimental and numerical results
Several studies have appeared in the literature where an attempt has been made to compare numerical predictions and experimental results. Becker et al. [51] used a finite element Lagrangian approach with both single and multi-mode integral formulations of the Oldroyd B model to simulate the acceleration of a sphere from rest under the influence of a constant gravitational force. Com parisons were made with experimental data from spheres settling in a tube (/3 = 0.243) filled with a PIB Boger fluid. Transient numerical results for the axial velocity component from the single mode Oldroyd B fluid for four different Deborah numbers revealed that although qualitatively similar to the experimen tal data, the relative velocity overshoot was larger. The experiments and simulation results with the single mode Oldroyd model for the steady state drag showed the same qualitative trends: a small decrease in drag at low Deborah numbers followed by a larger increase in the drag at higher Deborah numbers. As noted already in this chapter the reason for the drag increase is the development, in the wake of the sphere, of a region of highly extended polymer, thus increasing the effective size of the sphere. Simulations performed with a four mode Oldroyd B fluid (where the spectrum of relaxation times was obtained from the linear viscoelastic measurements of Quinzani et al. [480]) predicted a smaller velocity overshoot than had been seen with the single mode calculations and were in closer qualitative agreement with the ex perimental data. The failure of the single mode Oldroyd B model to provide a satisfactory qualitative prediction to elastic effects in complex flows of a Boger fluid is not surprising: as pointed out by Boger [85], the Oldroyd B model does not, in general, do a good job for flows such as that past a sphere where extensional stresses can be very high. The quantitative fit with experimental data obtained by Becker et al. [51] for the sphere velocity with their four mode Oldroyd B model was further im proved upon by Rajagopalan et al. [483] when two different four mode PTT models (having different sets of nonlinear parameters) were used. The conclu sion drawn was that nonlinear viscometric properties were essential, in addition to a spectrum of relaxation times, for obtaining good qualitative descriptions of the transient motion of the sphere. Attempts by Rajagopalan et al. [483] at fit284
CHAPTER 9. BENCHMARK PROBLEMS II ting experimental results to experimental data were not an unqualified success, however. The numerical results obtained by Rajagopalan et al. [483] for the nor malized sphere velocity at smaller density contrasts exhibited far larger velocity overshoots than was observed experimentally, possibly because of experimental error at low Deborah numbers. Also, at a higher aspect ratio (0.632) one of the four mode P T T models (PTTa) predicted a settling velocity some 20% higher than was measured in the experiments. This discrepancy was thought to be due to the development of large shear stresses near the sphere equator which could not be adequately matched with the choice of model parameters used. Computations with the other four mode PTT model (PTTb), whose nonlinear parameters were chosen to match more closely the fluid viscosity over a broader range of Deborah numbers predicted a far more realistic settling velocity. The same PIB Boger fluid as used in the experiments of Becker et al. [51] and Rajagopalan et al. [483] featured in the study by Arigo et al. [16] of the steady settling of a sphere in a cylinder filled with viscoelastic fluid. Numerical simulations were also performed with single mode UCM and FENE-CR models and a four mode P T T model. The parameters for the P T T model were the same as for PTTa in the work of Rajagopalan et al. [483]. Computations with the FENE-CR model (with extensibility parameter b = 12) and the Oldroyd B models provided lower and upper bounds, respectively, for the experimen tally measured steady state drag correction factor. The four mode P T T model PTTa surprisingly, and disappointingly, predicted a drag correction factor sub stantially lower than the experimentally observed values. This showed that a multi-mode discrete relaxation time spectrum with nonlinear viscometric prop erties is not in itself an adequate guarantee of a good fit with experimentally determined quantities. Simulations performed with the FENE-CR model for f3 = 0.243 showed good agreement at low Deborah numbers with the LDV measurements of the centre line axial velocity both upstream and downstream. As the Deborah number in creased the velocity was over-predicted close to the sphere and under-predicted further downstream, however. One of the reasons which could be conjectured for the failure of simple models of the Maxwell, P T T or closed-form FENE-type to quantitatively predict the steady-state drag on a sphere settling in a dilute polymer solution is that these models do not contain enough of the underlying physics of real viscoelastic fluids. Therefore Yang and Khomami [638] sought to simulate the experimental findings of Arigo et al. [16] by using not only FENE-CR, FENE-P and Giesekus models but also the single and multi-mode versions of the constitutive equation of Verhoef et al. [597]. This latter model is a closed form constitutive equation in which the additional dissipative contribution to the polymeric stress arising from a stress-conformation hysteresis in transient extensional flow [189] is modelled as being purely viscous. The predictions of the model in transient extensional flow of Boger fluids were comparable to the FENE model and were shown to give better predictions of the transient extensional viscosity than either the models of Hinch [284] or Rallison [486]. The importance of the inclusion of additional dissipative stresses in the present context is that they are expected to give rise to an increase in the predicted drag on the sphere. The multimode Giesekus and Verhoef models were shown by Yang and Khomami [638] to give the best overall agreement with the steady shear and transient uniaxial extensional viscosities for the fluid as measured by Arigo et al. [16]. However, 285
9.2. FLOW PAST A SPHERE IN A TUBE attempts to fit the steady state drag coefficient for aspect ratios of 0.121 and 0.243 were disappointing: although at an aspect rate of 0.121 the basic trends in the drag correction factor when the FENE-CR model was used with different values of b were consistent with the observations of Chmielewski et al. [135] (see §9.2.1), this model proved unable to quantitatively describe the measured drag coefficient. Neither the FENE-P nor the single mode Verhoef models were able to do much better. On a brighter note, the multi-mode form of the Verhoef and Giesekus models provided a better description of the experimental drag data than the other models at We = 2 but, disappointingly, both overpredicted the drag for larger Weissenberg mumbers. For the larger aspect ratio the multimode models both underpredicted the drag for nearly all values of We. Two conclusions for the constitutive modelling may be drawn. First, the ability of a model to describe the steady state viscometric functions and tran sient uniaxial extensional properties of a given fluid is no measure of how well it will perform in more complex flow situations, such as flow around a sedimenting sphere. Secondly, the inclusion of dissipative stresses should not be assumed to be sufficient an enhancement to the usual closed-form dumbbell-type equations to give a better quantitative match with experimental data. The au thors concluded their study with the hope that Brownian dynamics simulations with bead-rod or bead-spring chains would be sufficient to correctly capture the physics of real polymeric fluids. Brownian dynamics with the configuration field method [300,434] was the approach adopted by Fan et al. [201] for simulating the flow of a suspension of fibres past a sphere in a tube. Predictions of dimensionless drag force on the sphere for a sphere-to-tube aspect ratio of 0.5 and a variety of fibre aspect ratios (measuring the ratio of a fibre's length to its cross-sectional diameter) were in excellent agreement with the experimental data of Milliken et al. [403] and give basis for hope that the numerical solution of problems requiring close quantitative agreement with experimental data may be realized more effectively than ever before by using mesoscopic models.
286
CHAPTER 9. BENCHMARK PROBLEMS II
9.3
Flow between Eccentrically Rotating Cylin ders
The flow between eccentrically rotating cylinders is particularly alluring as a benchmark problem since it is an example of a flow that is enclosed in a domain free from sharp boundaries. In this geometry one does not have to worry about whether the entry and exit lengths are sufficiently long or about the influence of corner singularities, for example. However, at high eccentricities boundary layers may form in the flow and the resolution of these requires a fine level of discretization. The flow between eccentrically rotating cylinders is of particular interest in the study of journal bearing lubrication. The ability to predict and assess the performance of lubricants within a journal bearing under a wide range of engine operating conditions is fundamentally important to lubrication engineers and engine manufacturers alike. Polymers were originally added to mineral oils to minimize the dependence of viscosity on temperature. It is important that lubri cants are mobile enough to aid cold-starting of engines and yet viscous enough to prevent wear under 'hot-running' conditions. When polymers are added to mineral oils to formulate a multigrade oil, the oil becomes non-Newtonian and viscoelastic. A fundamentally important question is what effect, if any, does viscoelasticity have on bearing characteristics when multigrade oils are used in automobile engines? A review of the journal bearing problem forms the major part of this sec tion. However, it would be perverse not to mention the concentric configuration, known as the Taylor-Couette problem, which is one of the classical problems in fluid mechanics. Therefore, this section will begin by reviewing recent develop ments in the viscoelastic Taylor-Couette problem.
Figure 9.6: The eccentric cylinder geometry with eccentricity e and average gap c. Consider a three-dimensional eccentric cylinder configuration comprising two cylinders, each of length L, whose axes are parallel to the z-axis. The region be tween the cylinders is occupied by a fluid. The inner and outer cylinders, known 287
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS as the journal and bearing, respectively, are of radius Rj and i?s, respectively. A parameter of interest, particularly when making comparisons with lubrication theory, is L/D where D — 2RB is the diameter of the bearing. The axes of the journal and bearing are separated by a distance e, and it is customary to define an eccentricity ratio e e = -, c where c = RB — Rj is known as the gap, so that 0 < e < 1. A two-dimensional cross-section (in the (r, 6) plane) of the geometry is shown in Fig. 9.6. In this figure 6 is the azimuthal angle between the line joining the centres of the journal and the bearing (the a;-axis) and a line joining the centre of the journal to a point on the journal surface. The inner and outer cylinders rotate with angular speeds fij and fis, respectively. When the centre of the journal, as well as that of the bearing, is fixed the bearing is said to be statically loaded. In a statically loaded journal bearing problem the eccentricity ratio is specified and the load required to maintain the journal at the equilibrium position is computed. The load on the journal is the force exerted on the journal by the fluid, calculated by integrating the pressure over the surface, Tj, of the journal, i.e. F = /
pndS.
(9.23)
The load, F, can be resolved into components Fx and Fy in the x- and ydirections (see Fig. 9.6). In a dynamically loaded journal bearing the journal centre is no longer stationary but moves freely in response to forces acting on it. This means that the journal moves translationally in space, and so the flow domain is no longer fixed in time. Given forces F c and F p an external load of the form F = F c + F p sin(wt),
(9.24)
has been applied to a dynamically loaded journal in the literature [259,260,352, 353,459] (see further in §9.3.4) and the path of the journal computed. This provides information about the maximum eccentricity ratio attained and hence, in the case of an oil-filled journal bearing, one can deduce the minimum oil film thickness (MOFT) in the bearing.
9.3.1
The Taylor-Couette problem
In this problem a fluid fills the region between two concentric cylinders (e = 0). The flow is driven by the rotation of one or both of the cylinders and the stability of the flow is investigated with increasing inertia and/or elasticity. For a Newtonian liquid, G. I. Taylor [568] showed theoretically and experi mentally that the purely azimuthal shearing flow that occurs in a Couette cell when the inner cylinder rotates at low speeds becomes unstable as the inertial forces are increased and is replaced by a flow with steady toroidal roll cells. The height of each of these cells is approximately equal to the gap. The onset of secondary flow occurs at a critical value, TCTit, of the Taylor number, T, where
r j
~ (£-Mi)288
CHAPTER 9. BENCHMARK PROBLEMS II The Taylor number is a measure of the ratio of the centrifugal to the viscous forces in the system. Most of the early work on the Taylor-Couette problem concentrated on the determination of Tcrn and the corresponding wavelength of the disturbance for which the simple circular Couette flow becomes unstable. Theoretically, one can show that the onset of instabilities in this geometry for Newtonian viscous fluids is a direct consequence of the presence of the nonlinear convective term in the momentum equation. The effect of this term is amplified with increasing Reynolds number. The stability threshold can be determined by performing a linear stability analysis in which a small disturbance is added to a stable primary flow. The equations of motion are then linearized with respect to this disturbance and decomposed into Fourier modes in order to examine the associated growth rates. In the Taylor-Couette flow the disturbance is usually assumed to be axisymmetric and, since the primary flow contains only the azimuthal velocity component u°, we may write ur(r, z,t) ug(r,z,t) uz(r,z,t) p(r, z, t)
= = = =
U(r)exp(iaz + vt), V{r)exp(iaz + vt)+u°e(r,z,t), W(r)exp(iaz + vt), P(r) exp(iaz + vt) + p°(r, z, t).
,g
2g,
In these representations of the disturbance, a is the wave number of the assumed periodicity in the axial direction and v is the frequency. The real and imaginary parts of the frequency describe the growth rate of the disturbance and the frequency of oscillation, respectively. If $s(v) = 0, there is no oscillation and the disturbance is steady. Otherwise it is said to be overstable (Hopf bifurcation). If $l(v) > 0, the disturbance grows in time and the primary flow is unstable to a disturbance of the form (9.25). If 5ft(i/) < 0, the disturbance decays in time and the primary flow is stable to the disturbance. The stability analysis proceeds by determining values of a, v and T for which non-trivial solutions U(r), V(r) and W(r) of the linearized equations and associated boundary conditions exist. This requires the solution of a generalized eigenvalue problem for v. The condition $l(v) = 0 is satisfied by points on the neutral stability curve which forms the boundary between the stable and unstable flow regimes. One can expect viscoelastic instabilities to be qualitatively different from their Newtonian counterparts due to the additional convective terms that are present in the constitutive equation. There are two interesting questions that are pertinent to the study of stability of viscoelastic flows. First, what are the modifying effects of viscoelasticity on the inertial Taylor-Couette instability? Secondly, what happens when inertia is neglected? The quest for an answer to the first question has formed the focus of most research on the viscoelastic Taylor-Couette problem. The second question is interesting since inertialess instabilities do not occur for Newtonian fluids. It is also interesting because purely elastic linear instabilities do not exist in parallel shear flows for the UCM fluid. However, inertialess instabilities have been detected for viscoelastic flows with curved streamlines, the simplest of which is Taylor-Couette flow. Effect of viscoelasticity on the inertial instability In contrast to the Newtonian Taylor-Couette problem, the corresponding vis coelastic problem has received scant attention until relatively recently. An ex ception is the work of Thomas and Walters [572,573] who extended the linear 289
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS stability analysis to include the UCM fluid in the limiting case of a narrow gap between two rotating cylinders. Thomas and Walters [572] demonstrated that, in certain situations, when elasticity in the flow is important the secondary flow is overstable or time-periodic. In addition, when there is a large relative motion between the two cylinders, they found that elasticity is a destabilizing influence on the inertial instability causing the onset of secondary flow to occur at a lower value of TCTit. Beard et al. [49] confirmed these findings. When the Deborah number is increased above a threshold the primary solution bifurcates and a new unstable mode appears. This overstable mode reduces Tcru considerably compared with the corresponding critical value of the Taylor number for New tonian fluids. As elastic effects are increased, Beard et al. [49] showed that an overstable mode eventually becomes more unstable than the stationary mode. Numerical calculations by Avgousti and Beris [20], using a pseudospectral method to solve the axisymmetric linear stability and bifurcation problems, reached the same conclusions as Beard et al. [49]. Avgousti and Beris [20] also examined the pattern and symmetry of the secondary flow after the onset of instability for the Oldroyd B fluid based on the interactions between Hopf bifur cations and symmetry [243]. At the bifurcation point a breaking of symmetries takes place and the primary flow becomes unstable to infinitesimal disturbances. The presence of symmetry is responsible for a degenerate Hopf bifurcation, i.e. the eigenvalues of the Jacobian matrix at the bifurcation point that are respon sible for the onset of instability - those for which 9?(i/) = 0 - have multiplicity greater than unity. Avgousti and Beris [20] used the theory of bifurcations in the presence of symmetries developed by Golubitsky et al. [243] to show the existence of two different time-periodic families of solutions characterized by either spirals (rotating wave) or ribbons (standing wave). This work was generalized by Avgousti and Beris [19] to include the stability of non-axisymmetric disturbances. As for axisymmetric disturbances the onset of instability was accompanied by a Hopf bifurcation. For the UCM model and for a particular set of geometric and kinematic parameters they showed that overstability for the non-axisymmetric case sets in before the correspond ing axisymmetric situation provided the elasticity number De/Re is sufficiently large. Purely elastic instabilities The discovery of purely elastic instabilities, which occur at Re = 0, is responsible for a renaissance in the viscoelastic Taylor-Couette problem. Purely elastic in stabilities were first observed in experiments performed by Muller et al. [414] and Larson et al. [343] for low Reynolds number Taylor-Couette flow. The experi ments were performed using 1000 p.p.m. high-molecular-weight polyisobutylene in a viscous solvent. This is an example of a Boger fluid for which the Oldroyd B model is a reasonable model. A secondary flow is shown to develop due solely to the viscoelastic character of the fluid. Muller et al. [414] and Larson et al. [343] also performed a linear stability analysis in the limit of a small gap. The the oretical predictions were in agreement with the experimental observations. For a sufficiently large value, Decru, of the Deborah number this work revealed the growth of an overstable mode and that as c/Rj decreased the critical value of De increased. Shaqfeh et al. [532] performed a linear stability analysis on a version of 290
CHAPTER 9. BENCHMARK PROBLEMS II the Oldroyd B model that had been modified to include second normal stress differences. The purpose of the work was to examine the influence of the ratio of the first and second normal stress coefficients and the ratio of solvent to polymeric viscosities on the onset of flow instability. Previous studies had shown that JV2 can play an important role in the stability of elastic flows even if the magnitude of N2 is significantly less than that of Ni. Since the Oldroyd B model has N2 = 0, Shaqfeh et al. [532] considered a simple version of the Oldroyd 8 constant model in order to predict the effects of N% in highly elastic flows. The linear stability theory predicted that two possible flow structures are possible near the onset of instability - a standing wave structure characterized by radially propagating vortices and a travelling wave structure characterized by vortices propagating up or down the coaxial cylinders. The strength and dimensions of these vortices were shown to depend strongly on the gap. They also showed that a positive second normal stress coefficient destabilizes the flow while a negative value of the second normal stress coefficient strongly stabilizes the flow. Shaqfeh et al. [532] also found that, for a dilute viscous polystyrene solution for which the second normal stress coefficient is approximately zero, DeCTn decreased as the polymer concentration increased in qualitative agree ment with theoretical predictions.
9.3.2
Lubrication theory
The traditional approach to the study of journal bearing lubrication has been via the lubrication approximation introduced by Reynolds [505]. This enables an equation for the pressure within the thin film region of the geometry to be written separately from the kinematic and constitutive equations describing the flow of the lubricant. This simplifies greatly the task of calculating the reaction forces engendered by the lubricant. The effectiveness of the lubrication approximation has been supported by experimental evidence in a very wide range of lubrication studies. The reliability of the approximation is responsible for its pervasive appeal and usage. However, there are at least two contexts in which the approximation may be open to question. The first is in predicting the fine details of the nonlinear dynamics of the journal bearing under dynamic loading conditions. Here, the precise pressure boundary conditions exploited in Reynolds's equation can have a profound effect on the dynamics of the journal. The second context is in studying the role of viscoelasticity in journal bearing lubrication. If the lubrication approximation is not invoked, there is no option but to solve the full set of coupled equations governing the flow of the lubricant, taking proper account of the moving parts of the geometry. Until recently this task has proved too formidable a calculation, but with current computing power combined with efficient and accurate numerical methods, the calculation may be attempted. In the remainder of this chapter we set CIB = 0 and fij = w in order to be consistent with the notation used in the lubrication literature. Reynolds's equation A full film Newtonian fluid will impart a load on a static journal in the direction orthogonal to the line joining the centres of the journal and the bearing. When the gap c is small compared with other dimensions, it is possible to perform an 291
9.3.
FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS
order of magnitude analysis on the full Navier-Stokes equations. This results in Reynolds's equation [118], which, in the absence of inertia, is
|
(d + <™e?%) + ( # ) 2 | (d + <-*)»!) = -6,^± si „ 9 , (9.26)
where 6 is the azimuthal angle between the line joining the centres of the journal and the bearing, and the radial line from the centre of the journal to a point on its surface. In many physical situations the long or short bearing approximations to Reynolds's equation (9.26) are valid. When either of these approximations is applied to the above equation a closed form expression for the pressure may be derived. Note that the expression for the load (9.23) does not include the extra-stress contribution to the Cauchy stress tensor. Long bearing theory The long bearing approximation assumes that the pressure field is constant along the ^-direction and thus the effects of the side boundary conditions are negligible. When dp/dz is neglected, Reynolds's equation (9.26) can be solved analytically to obtain the pressure, 6uwR2 e sin 0(2 + e cos 8) c2 (2 + e )(1 + ecosfl)2
.„ „„.
Using (9.23), the load and torque for a long bearing are given by Fx = 0, Fy = V
' = , c V ( l - e 2 )(2 + e2)
(9.28)
and C =iH^L + f. 2 2
(9.29)
Short bearing theory The short bearing approximation assumes that the contribution of the side boundary conditions gives rise to a much larger pressure gradient than that due to the rotation of the journal. With dp/69 neglected, Reynolds's equation (9.26) can again be solved analytically for the pressure, viz. Znutsm.6 I'L2 3 c (l+ecos0) V 4 2
,\ )
/„ ™s
In this case the load and torque for a short bearing are given by
and C =^* + Sf.
Cy^l - e2) 292
2.
(9.32)
CHAPTER 9. BENCHMARK PROBLEMS II
e 0.70 0.80 0.90 0.95 0.99
No Approximation C Fy 0.11x10° 0.54xl0- 2 0.24x10° 0.64xl0- 2 0.80x10° 0.89x10-2 0.23X101 0.12X10- 1 0.17xl0 2 0.30 x l O - 1
Short Bearing Theory F C 0.18x10° 0.53x10-2 0.35x10° 0.63x10-2 O.lOxlO1 0.88x10-2 0.29X101 0.13X10- 1 0.33 xlO 2 0.33X10- 1
Table 9.3: Comparison of the load and torque on the journal calculated with and without using the lubrication approximation for L/D = 1/10.
e 0.70 0.80 0.90 0.95 0.99
No Approximation F C r 4 y 0.82 xl0° 0.21xl0 0.27 xlO 4 0.11 xlO 1 0.40 xlO 4 0.16 xlO 1 0.56 xlO 4 0.23 xlO 1 0.13 xlO 5 0.51 xlO 1
Long Bearing Theory C Fy 0.22 xlO 4 0.84 xl0° 0.28 xlO 4 0.11 xlO 1 0.41 xlO 4 0.16 xlO 1 0.59 xlO 4 0.23 xlO 1 0.13 xlO 5 0.53 xlO 1
Table 9.4: Comparison of the load and torque on the journal calculated with and without using the lubrication approximation for L/D = 10/1.
Numerical comparisons Li et al. [352] compared the component of force Fv and the torque C predicted by the lubrication approximation with those computed using a spectral element technique on the full set of governing equations for a constant viscosity lubri cant. In Tables 9.3 and 9.4 comparisons are presented in which the short bearing approximation theory for L/D = 1/10 and the long bearing approximation the ory for L/D = 10/1 are used, respectively, for a range of eccentricity ratios. The parameters used in these comparisons are provided in Table 9.5. As expected the computed force and torque are in agreement with long bearing lubrication approximation theory for L/D = 10/1 for all eccentricities (see Table 9.4). When L/D = 1 / 1 0 one would expect agreement between the computed values and the prediction of short bearing lubrication theory. However, an inspection of Table 9.3 clearly reveals that such agreement is lacking. The explanation for this rests in the way the component of force Fy is calculated. In short bearing lubrication theory the contribution of the extra-stress to the calculation of the force is omitted. This has negligible effect for L/D — 10/1, but has increasing influence for short bearings. If the contribution of the extra-stress is neglected in the computation of Fy when L/D = 1/10 then excellent agreement with short bearing theory is obtained [352].
9.3.3
Statically loaded journal bearing
In the eccentric cylinder geometry it is natural to formulate the governing equa tions in terms of cylindrical bipolar coordinates. This enables the region be tween the cylinders to be mapped conformally onto a concentric geometry. The 293
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS Journal radius (Rj) Bearing radius (RB) Constant viscosity (?j) Density (p)
0.03125m 0.03129m 5 x 10 _ 3 Pa.s 820kg/m 3
Table 9.5: The geometric and lubricant parameters for the journal bearing
boundaries of the eccentric cylinder geometry are therefore represented exactly. Not only does this allow for a Fourier basis in the azimuthal direction, which means that the solution satisfies implicitly the periodicity conditions, but it also avoids the introduction of corner singularities arising from an approximation of the boundary. The cylindrical bipolar coordinates (£, rj) are given by: o sinh £ a sin n - , y= ', X X where x = cosh£ + COST? and a is a constant depending on e, Rj and RB- The Stokes problem written in bipolar coordinates is given in the paper of Roberts et al. [514]. Beris et aJ. [56,57] have calculated the viscoelastic flow between eccentri cally rotating cylinders for a variety of constitutive relations using both finite element and spectral/finite element methods. The use of a Fourier expansion in the azimuthal direction for numerical simulations of the UCM model alleviated numerical oscillations that were present in the earlier finite element work [54,55] and resolved the stress boundary layers that exist for high elasticity, as mea sured by the Deborah number. Their work was based on a formulation of the governing equations in terms of the stream function and extra-stress tensor. In their spectral/finite element work Fourier expansion functions were used in the azimuthal direction and cubic Hermite and quadratic Lagrangian approxima tions were used for the stream function and extra-stress, respectively, in the radial direction. The use of Fourier expansion functions in the azimuthal direc tion enabled converged solutions to be determined for higher values of De than were possible using the finite element method. For an eccentricity ratio of 0.1 solutions were obtained up to De = 90, a factor of 30 increase over that realized using finite elements. However, the method was less successful at predicting the flow and stress fields for high values of De at large eccentricity ratios. Roberts et al. [514] developed a spectral approximation for Stokes flow for the three-dimensional journal bearing problem based on a tensor product of Chebyshev and Fourier approximations. This was used as the basis for vis coelastic flow calculations in two dimensions by Phillips and Roberts [460] and in three dimensions by Roberts and Walters [515] using the UCM and Oldroyd B models. The Deborah and Weissenberg numbers for this flow are defined by x=
De
=
Xiu,
We
=
\\LO—,
c
where, as usual, Ai denotes the relaxation time and u is the angular speed of the journal. The bearing is at rest. The fully spectral discretization enabled solutions to be obtained for high values of the Deborah number at large eccen tricity ratios. Note that although the values of the relaxation time are small, the 294
CHAPTER 9. BENCHMARK PROBLEMS II Ai
0.0 1.0 1.0 1.0 1.0 1.5 2.0
xl0~6 xHT5 xlO-4 xHT3 xHT3 xlO-3
Fx 0.00 4.48 xlO 1 4.48 xlO 2 4.48 xlO 3 4.18 xlO 4 5.90 xlO 4 7.41 xlO 4
Fy
2.30 2.30 2.31 2.31 2.24 2.18 2.11
xlO b xlO 5 xlO 5 xlO 5 xlO 5 xlO 5 xlO 5
C 9.25 9.24 9.24 9.24 9.18 9.12 9.07
Table 9.6: UCM results for eccentric cylinder on a 12 x 12 mesh with D = 0.03125m, c = 0.00004m and e = 0.5. high shear-rates generated in the journal bearing in the region of smallest gap due to the rotation of the journal give rise to large local Weissenberg numbers. Numerical results are presented in Table 9.6 for e = 0.5 on a 12 x 12 FourierChebyshev grid [460] showing the variation of the couple and components of the force per unit length on the journal with relaxation time. The method of false transients was used to solve the governing time-dependent equations. This allowed for the use of a time step for the constitutive equation that is at least 104 times the size of the time step required for the momentum equation and enabled greater efficiency when only the steady state solution was required. The maximum local Weissenberg number occurred in the region of smallest gap and was approximately 780 for the situations simulated. These physical choices of the parameters are a severe test for any numerical algorithm. For non-zero values of Ai the horizontal load increases linearly with Ai and the resultant force no longer acts normally to the line joining the centres of the journal and the bearing. The physical relaxation times for lubricating oils under isothermal, isobaric conditions are typically in the range [10 _ 6 s, 10 _ 5 s]. These studies have shown that, for constant viscosity lubricants operating un der isothermal conditions, relaxation times of O(10~ 4 )s are required before viscoelasticity can be expected to increase load-bearing capacity. Increased loadbearing capacity occurs when Fx becomes of a comparable size to Fv and acts in a direction to move the journal away from the region of smallest gap. These initial studies [460,515] neglected the effects of temperature and pres sure that are clearly important in journal bearing lubrication. Viscosity in creases under pressure (p) but decreases as the temperature (T) increases. Re laxation times behave in a similar fashion. Davies and Li [163] incorporated these effects into the journal bearing model by employing the White-Metzner constitutive equation, i.e. T + Ax ( p , r , 7 ) T = J ? ( p , T , 7)7,
(9.33)
where 7 is the scalar invariant
7 = y 27:7The White-Metzner model is one of the simplest coordinate invariant viscoelastic fluid models that allows for a single mode relaxation time Ai. A possible functional form for Xi(p,T,Af) is to make it proportional to the viscosity [163],
295
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS i.e.
Ai(p,r,7) =
fcA»j(P,T,7),
(9.34)
where k\ is a constant. The viscosity of lubricating oils is, in general, a function of shear-rate, tem perature and pressure. In the journal bearing, changes in pressure contribute to the largest variation in viscosity. The variations of viscosity due to shearthinning for lubricants of industrial interest under normal operating conditions are no more than 50%, whereas corresponding changes due to pressure can be as much as tenfold. Davies and Li [163] expressed viscosity as a combination of the well-known Cross, Barus and Vogel laws, which, neglecting extensional viscosity effects, led to a relation of the form
»,(p,T,7) = (^oo + JT-M^L^} [l + (K(j,,T)j)*
«P («P+ pj ,
(9.35)
where K(p,T)
=
K0exp(ap+^
Here TJOQ is the asymptotic high shear-rate viscosity, a is the pressure coeffi cient of viscosity, /3 is a reference temperature, and KQ, a, ft and m are further material constants. The material parameters are estimated by best-fitting ex perimental data. The parameter a in (9.35) controls the dependence of the low shear-rate plateau on pressure. It is assumed that the dependence of the high shear-rate plateau on pressure is governed by the same exponential law. The parameter a controls the rate of shear-thinning on pressure. Realistically, it may be assumed that this behaviour is independent of the low and high plateau levels reached. It is important to note that the viscosity law (9.35) is consis tent with experiments [48,301] that span only limited ranges of the pressures which the lubricants experience under general operating conditions. Recent ex perimental work provides evidence that the characteristic relaxation times of multigrade oils are strong functions of pressure [68]. These findings furnish a possible mechanism for the improved performance of multigrade oils in journal bearings. Under nonisothermal conditions, the governing equations are augmented by the energy equation with appropriate boundary conditions. Davies and Li [163] showed that in order to obtain equilibrium temperature fields which have max ima in the small gap, Dirichlet and Biot conditions should be imposed on the journal and bearing, respectively, i.e. T = Tj,
onTj,
f^ = X
( T r _ T B )
'
°
n I V
In these boundary conditions Bi denotes the Biot number, h is a characteristic thickness, and Tr is a reference temperature on TBDavies and Li [163] provided evidence which suggests that, at high eccentric ities, pressure-thickening dominates the viscosity behaviour rather than shearthinning or temperature-thinning. The particular form of the White-Metzner model used in the numerical simulations predicted that this can increase the 296
CHAPTER 9. BENCHMARK PROBLEMS II normal stresses by at least two orders of magnitude, in comparison with a con stant viscosity model, resulting in increases of 20 % and higher in load-bearing capacity. Therefore, this study demonstrates that viscoelasticity can have a ben eficial effect on the load bearing characteristics of a statically loaded bearing operating under nonisothermal conditions.
9.3.4
Dynamically loaded j o u r n a l bearing
A dynamically loaded journal bearing consists of a journal whose axis is not stationary but which moves in response to forces acting upon it. Such forces consist of the reaction force of the fluid on the journal and also an applied load, such as the force of a piston. Computationally, the dynamically loaded journal bearing presents a major computational challenge because of the need to track the position of the journal in time and to construct a new mesh at each time step. Experimental work undertaken by Williamson et al. [632] using a journal bearing simulator, an experimental rig in which the journal locus is prescribed and the bearing load is derived, has demonstrated that the viscoelastic proper ties of multigrade oils can have a measurable effect on journal bearing charac teristics in a realistic configuration. A journal bearing simulator is a practical piece of apparatus that permits extreme eccentricity ratios, up to around 0.98, to be reached. It is at these extreme values that major effects are observed. In this experimental study the viscous and viscoelastic properties of lubricants at shear-rates of practical importance were measured using a Weissenberg Rheogoniometer and a Lodge Stressmeter. It may seem rather surprising that oils which possess such low relaxation times can produce such measurable effects. How ever, it seems that the first normal stress difference, for multigrade oils, is a stronger function of pressure than the shear stress. This may explain why the beneficial effects of viscoelasticity only manifest themselves at higher eccentric ity ratios when the pressures are likely to be high enough for the normal stress amplification effect to be relevant. The numerical simulations of Phillips et al. [459] seem to support the exper imental observations of Williamson et al. [632] in that the beneficial effects of viscoelasticity only materialize when the journal is literally up against the wall, i.e. at high eccentricity ratios, oils with larger relaxation times can produce larger MOFTs. The numerical computations are based on the White-Metzner constitutive equation with the viscosity law given by (9.35). Cavitation model It is generally accepted that, under many operating conditions, a complete lu bricating film is not maintained in a journal bearing. Under such conditions the lubricant is unable to sustain the very low pressures that are generated within the journal bearing mechanism. The result of this is that the fluid ruptures and a cavitation region is formed. The occurrence of cavitation in journal bear ings has been shown to result in reduced power loss, bearing torque and load capacity [305]. However, Dowson and Taylor [188], in an excellent review of cavitation, point out that cavitation need not have detrimental effects on the load carrying capacity of bearings. Although cavitation damage may occur in journal bearings there is significant evidence to suggest that cavitation may re297
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS duce wear by damping out the self-excited instability that occurs in dynamically loaded journal bearings. This instability is known as whirl instability. The suggestion that whirl instability occurs in the full-film dynamically loaded journal bearing (with zero applied load) has been made by many in vestigators. When this instability occurs, the journal moves from an unstable equilibrium position and gradually tends towards the journal following a spi ral path characterized by an eccentricity ratio that is monotonieally increasing. The period of this orbit is twice that of the angular velocity of the journal about its own axis resulting in the term half-speed whirl. In this situation the bearing is deemed to have failed since the journal will eventually touch the bearing. In practice the full-film condition is not realistic, from the physical point of view, since in many journal bearing models the large negative pressures produced in the oil film cause the oil to vaporize leading to cavitation. A comprehensive treatment of cavitation involves the solution of a two-phase flow problem in which the phases are separated by a permeable free surface or interface. Such a treatment requires the solution of the conservation equations in each of the lubricant and vapour phases together with appropriate conditions on the interface between the phases. The position of the interface moves in time since the journal is subject to dynamic loading. This presents a formidable computational undertaking. The inclusion of cavitation modelling in journal bearings has been inves tigated in numerous papers and a large number of different approaches have been proposed. The use of a cavitation model circumvents the problems as sociated with the solution of a complicated two-phase flow problem in which the fluid/vapour interface is a moving boundary. The most popular model is the so-called 7r-film, or half-Sommerfeld cavitation model in which it is assumed that the cavitation region occupies the divergent half of the region between the journal and the bearing. The lubricant is assumed to occupy the convergent part of this region. Thus, for the purposes of calculating the force exerted on the journal by the fluid, the pressure boundary condition on the journal has been set as follows: p = 0, for
n < 9 < 2n.
If the short bearing approximation is used, a more sophisticated model is the oscillating Tr-film model. This allows the cavitation region to change dynamically in response to the behaviour of the journal. In this model the value of 9\ is chosen to be the smallest value of 9 > IT for which p = 0 and the cavity is then assumed to occupy the region 9\ < 9 < n + 9\. Documented evidence presented by Cameron [118] supports the view that this condition is a reasonable approximation to physical reality. A useful comparison of the various boundary conditions used to describe cavitation models for the journal bearing can be found in the paper of Gwynllyw et al. [259]. In the three-dimensional situation, however, it is no longer necessary to specify a priori the position of the cavitation region. Since ambient pressure conditions are imposed at the ends of the journal bearing the pressure is deter mined uniquely. Therefore, unlike the corresponding situation in two dimensions the pressure level does not need to be set by specifying the pressure at a single point in the domain. Davies and Li [163] proposed a viscosity cavitation model, so called since the definition of the cavitation region is related to the viscosity of the fluid. This idea 298
CHAPTER 9. BENCHMARK PROBLEMS II was presented by Davies and Li [163] for statically loaded journal bearings and implemented for dynamically loaded journal bearings by Gwynllyw et al. [260] in two dimensions. This model avoids the complex programming required to trace the moving boundary for the cavitation region, by incorporating a viscosity function for the lubricant that filters out the sub-ambient pressures within the cavitation region. In this model no attempt is made at modelling rupture, and one simply assumes that fluid remains in the cavitation region. In this region the viscosity function decreases quickly but smoothly to an asymptotic value, r)min. Ideally, r\min should be chosen to be approximately the viscosity of air but such a small value leads to difficulties in the numerical convergence of the algorithm. Gwynllyw et al. [259] used a value of 7jm;„ = 4.0 x 1 0 - 4 Pa.s in their numerical calculations. For the purpose of clarity we outline here the approach proposed by Davies and Li [163] for modelling cavitation: • The governing equations are solved over the whole region bounded by the journal and bearing. • The cavitation region is taken to be the region where the pressure in the fluid falls beneath a prescribed threshold p~. • The cavitation region is modelled by allocating a low viscosity to the fluid in this region using a viscosity 'cut-off' function. The effect of this is to quickly reduce the viscosity in the cavitation region to a level that is low enough to simulate the viscosity of air whilst high enough so as not to pose any numerical difficulties. Davies and Li [163] employed a tanh viscosity 'cut-off' function that possesses only a C° join with the viscosity function for the fluid. In the statically loaded case they reported no numerical difficulties with this approach when employed within a mono-domain spectral method. For the dynamically loaded problem Gwynllyw et al. [260] found that a C° join was too rough a join to give smooth journal trajectories. Therefore, they proposed the use of a C°° cavitation cut-off function that alleviates the detrimental effects of the C° function. The modified viscosity function introduced to model cavitation took the form f)(p) = rjmin[l - g(p)] + V9(p),
(9-36)
where g(p) is defined by
A form for g(p) that satisfies these conditions is g{jp)
-
(9 38)
-
where
*«-< JLrf-i/ri " l i t 299
<9-39>
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS Parameter Rj RB
eo P r)oo
m
a a m K0
Value 3.125 x lO - * (m) 3.129 x 10- 2 (m) 0.625 8.2 x 102 (kg m - 3 ) 4.5 x 10- 4 (Pa.s) 9.352 x 10- 4 (Pa.s) 1.119 x l O - ^ P a - 1 ) 2.39 x l O - ^ P a - 1 ) 0.545 5.45 x 10- 1 0 (s)
Table 9.7: Geometrical data and fluid parameters Here p~~ is the pressure cavitation threshold, and 5 is the length of the interval over which the join is applied. Gwynllyw et al. [260] chose p ~ = 0 Pa and <5=103 Pa. Davies and Li [163] also considered an approximation to the boundary lu brication model whereby an upper cut-off function was applied to the viscosity. However, in the situations modelled under dynamic loading conditions it has been found that the highest pressures do not reach typical boundary lubrica tion pressure thresholds and so a boundary lubrication model is not required. In the remainder of this section we examine the influence of viscoelasticity on the dynamics of the journal by presenting some of the results reported by Phillips et al. [459]. We assume that the journal is operating under isothermal conditions. We also recall a purely dynamical property of rigid journal bearings, namely, that as the rotational speed of the journal increases, so does the MOFT. In other words, at high speeds the journal path becomes more concentric and the eccentricity ratios drop in value. We look at two different rotation speeds (955 rpm and 4775 rpm), two different loading scenarios (a constant load and a variable load), and two values of A1/77, the ratio of relaxation time to viscosity. In this study fluid inertia has not been included and the fluid is assumed to be isothermal, i.e. /3, ft = 0. The fluid and geometric parameters are given in Table 9.7. Constant load Starting from rest, the variation of eccentricity ratio with time in response to a constant load given by F c = (0, — 105iV) and F p = 0, and a rotation speed of 955 rpm is shown in Fig. 9.7. Two cases (A1/77 = 2 x 1 0 - 4 P a - 1 , and \i/r) = 4 x 10~ 4 Pa _ 1 ) are shown in the same figure. In each case the journal quickly reaches its equilibrium position ( « 0.684 after about 0.2 seconds). The dotted line represents the path for the lubricant with the higher elastic content, for which the final eccentricity ratio is lower. This is a clear demonstration that increased elasticity increases the load-bearing capacity and hence the MOFT. In Fig. 9.8, the rotation speed is increased to 4775 rpm, everything else being kept the same. At this speed the journal takes longer to reach the equilibrium position (about 0.5 seconds), but again increased elasticity results in a larger MOFT. 300
CHAPTER 9. BENCHMARK PROBLEMS II
Figure 9.7: The variation of eccentricity ratio with time in response to a constant load given by F c = (0, — 105iV) and F p = 0, and a rotational speed of 955 rpm where Ai/»y = \1/T} = 2 X l O ^ P a " 1 and Ai/r? = A2/J? = 4 x l O ^ P a " 1 . The final eccentricity ratios are 0.68380 and 0.68378, respectively. From T. N. Phillips, R. E. Need, A. R. Davies, B. Williamson andL. E. Scales, The effect of viscoelasticity on the performance of dynamically loaded journal bearings, SAE paper 982639, 1998 ©Society of Automotive Engineers, Inc., with permission.
Figure 9.8: The variation of eccentricity ratio with time in response to a constant load given by F c = (0, —105iV) and F p = 0, and a rotational speed of 4775 rpm where AI/J? = A1/T? = 2 X l O ^ P a - 1 and Xi/n = A2/T? = 4 X 1 0 _ 4 P a _ 1 .
The
final eccentricity ratios are 0.19807 and 0.19806, respectively. From T. N. Phillips, R. E. Need, A. R. Davies, B. Williamson andL. E. Scales, The effect of viscoelasticity on the performance of dynamically loaded journal bearings, SAE paper 982639, 1998 ©Society of Automotive Engineers, Inc., with permission.
301
9.3. FLOW BETWEEN ECCENTRICALLY ROTATING CYLINDERS Variable load The effect of viscoelasticity under variable loads leads to much more complex behaviour than in the constant load case. An external load of the form (9.24) was applied by Phillips et al. [459] with Fc = (0,-Me5),
Fp = (0,FpSm(wt)),
(9.40)
where Me denotes the effective mass of the journal, u its angular speed and Fp = 5 x 105N. For the lower rotation speed of 935 rpm, the variation of eccentricity ratio with time in response to the variable load (9.40) is shown for two cases of the ratio A1/77 in Fig. 9.9. On the scale of Fig. 9.9 it is impossible to separate the two cases, but close-up views provided in the paper of Phillips et al. [459] show interesting effects. When the eccentricity ratio is high, increased elasticity comes to the rescue and reduces the eccentricity ratio, increasing the MOFT. On the other hand, when the eccentricity ratio is lower, increased elasticity has the reverse effect: it decreases the MOFT. The story is different at the higher rotation speed in Fig. 9.10. Here the eccentricity ratios are much lower because of the higher speed, and increased elasticity reduces the MOFT (increasing the eccentricity ratios). This high speed effect is difficult to explain. It is clearly a nonlinear effect, almost certainly dependent on the interaction of inertia and elasticity, and ap parently only arising when the eccentricity ratios are low. At high eccentricity ratios, viscoelasticity has a beneficial effect on the MOFT. Typically in all cases we have found that increased elasticity reduces torque and consequently friction.
Figure 9.9: The variation of eccentricity ratio with time in response to a variable load, Fp = 5 x 105 N and a rotational speed of 955 rpm where A1/77 = Al/77 = 2 x 1 0 _ 4 P a - 1 and A1/77 = A2/T? = 4 X 1 0 _ 4 P a _ 1 . From T. N. Phillips, R. E. Need, A. R. Davies, B. Williamson and L. E. Scales, The ef fect of viscoelasticity on the performance of dynamically loaded journal bearings, SAE paper 982639, 1998 ©Society of Automotive Engineers, Inc., with permis sion.
302
CHAPTER 9. BENCHMARK PROBLEMS II
Figure 9.10: The variation of eccentricity ratio with time in response to a variable load, Fp = 5 x 105 N and a rotational speed of 4775 rpm where A1/77 = Al/77 = 2 x l O ^ P a " 1 and Ai/»j = \2/r) = 4 x l O ^ P a " 1 . From T. N. Phillips, R. E. Need, A. R. Davies, B. Williamson and L. E. Scales, The ef fect of viscoelasticity on the performance of dynamically loaded journal bearings, SAE paper 982639, 1998 ©Society of Automotive Engineers, Inc., with permis sion.
303
Chapter 10
Error Estimation and Adaptive Strategies 10.1
Introduction
In the absence of analytic solutions for all but the very simplest problems in viscoelastic fluid mechanics many researchers in the field who have been concerned with demonstrating the accuracy of numerical solutions to complex problems have typically done so in one of three ways. The first is by comparing values of quantities of interest such as drag, pressure drop and flow resistance, with previously published results in the literature. However, it seems obvious that although uniformly accurate solutions to problems in viscoelastic fluid mechan ics should result in reliable predictions of the quantities mentioned above and others like them, the converse is not necessarily true. Consider, for example, the computation of the drag on a closed body having surface S and outward pointing unit normal n moving in a fluid in a fixed direction given by a unit vector e (say). Then since the drag is given by the expression drag =
crnedS,
(10.1)
where cr is the Cauchy stress tensor, generally not all the components of the elastic stress tensor or of the rate-of-strain tensor are involved explicitly in the computation. As a consequence, the drag value may not be sensitive to inaccuracies in the computation of the stress components not explicitly entering (10.1). Moreover, only an evaluation of
305
10.1. INTRODUCTION this approach, however, is the computational expense which might be incurred in such a fine mesh calculation. Given the drawbacks of the previous methods of demonstrating accuracy, a third way: the pursuit of affordable and reliable error indicators for viscoelastic flow calculations, has, since 1990, been attracting interest. This interest in error estimation has also been driven by the appreciation of the usefulness of error in dicators in guiding adaptive numerical methods. By channelling computational effort in an efficient way to those parts of the flow domain that require partic ular resolution of fine-scale structures it is possible to obtain solutions enjoying a global level of accuracy which could only be realized at much greater expense with uniform meshes. Beginning with the papers by Babuska and Rheinboldt in 1978 [31,32], the study of a posteriori error estimates for the finite element solution of ellip tic boundary value problems has generated intense interest and resulted in an enormous body of literature on the subject. A popular method of deriving a pos teriori error estimates is to use residual methods such as the element residual method where elemental residuals are used in the construction of local prob lems. In this approach Demkowicz et al. [175], Bank and Weiser [38] and Oden et al. [420] are pioneer workers. Element residual methods for the construction of a posteriori error estimates for the Stokes equations have been used by Bank and Welfert [39,40], Verfurth (see, for example, [595,596]), Bernardi et al. [59] and by Ainsworth and Oden (see, for example [8]) who adapted an a posteriori error estimator proposed by themselves in previous work [6] to the Stokes and Oseen equations. The a posteriori error estimate results of Ainsworth and Oden were then generalized by Oden et al. [422,423] for finite element solutions to the Navier-Stokes equations and by Legat and Oden [348] for free surface flows of a generalized Newtonian fluid with a power-law viscosity. A summary account of a posteriori error estimation for finite element solutions to problems in me chanics, including the Navier-Stokes equations and the Stokes problem, may be found in the monograph of Ainsworth and Oden [7]. Mention should also be made of the significant contribution over more than a decade of Zienkiewicz and co-workers to the field [94,148,649-656]. Starting with the introduction by Zienkiewicz and Zhu in 1987 [652] of procedures of error estimation using stress or gradient recovery, much progress has been made, especially with the improvement of recovery procedures, and, in particular, the development of so-called superconvergent patch recovery. However, evidence in the literature of similar progress in the development of numerical error estimates for viscoelastic calculations is more obscure. This is undoubtedly due to the high complexity of the governing equations found in the subject. Since 1990 there have been efforts to use both element residual methods and subdomain residual methods in order to obtain error indicators for guiding adaptive strategies. Evidence for the appropriateness of the residual methods used as a means of obtaining reliable error estimators has been empirical. Rao and Finlayson [488,489] used an adaptive finite element scheme in order to solve for the flow of a UCM fluid through an abrupt contraction. The error indicators they derived were obtained using element residual methods. In [490] the same authors considered Poiseuille flow of a UCM fluid (for which an exact solution is known) and performed a numerical study of the relationship between norms of the residual of one of the stress equations and the corresponding error norm. The norm of the error and the mean-square of the residual seemed to develop 306
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES an almost linear log-log relationship with mesh refinement. Since the residuals and errors decreased together with refinement, the authors claimed that the residual is an appropriate error indicator for use in mesh refinement. Jin and Tanner [309] derived an approximate error equation by substituting the errors into the weak form of the governing equations for an MUCM fluid, linearizing and then solving the linearized equations element by element. As with Rao and Finlayson [490] they considered Poiseuille flow in order to assess the accuracy of their error estimates. An exception to element residual methods for viscoelastic flow calculations is the paper by Jin [307] where empirical evidence for the suitability of the author's subdomain residual method was presented for finite element solutions to the flow of an MUCM fluid past a cylinder in a duct. Whereas the majority of papers dealing with error estimates for viscoelastic flow problems include estimates of the stress errors in their error estimates an exception is the work of Mutlu et al. [415]. Here the authors defined an a posteriori error estimate based, somewhat surprisingly, solely on the velocity gradients. The exact solution was approximated through a smoothing of the velocity gradients using a global recovery procedure and a Galerkin weighted residual approach. The adapted meshes were used for two test problems: flow past a rigid sphere in a tube and flow in an annular converging tube. In the case of the first problem, the adapted meshes gave cheaper solutions for a given level of accuracy than structured meshes and, at the same time, predicted drag coefficients that were closer to reference values. For the second problem it was observed that adaptive remeshing provided smoother results than for the non-adapted case and, moreover, that the predicted results for the adapted mesh were closer to analytical values at the tube exit than those predicted with the (more expensive) non-adapted mesh. Another exception is the estimate of the error used by Harlen et al. [269] in their Lagrangian-Eulerian method for simulating transient viscoelastic flows. Only squares of velocity differences across their linear finite elements were used although weighting terms involving the elastic stress were incorporated so as to magnify the effect of velocity errors in regions of high polymer stretch. The work of researchers in Louvain-la-Neuve has shown that, even in the absence of theoretical justification, the use of element residual methods for error estimation in adaptive procedures can lead to solutions of highly elastic liquids through complex geometries at costs that are less than those incurred with uniform meshes. Fan and Crochet [202] used a p-adaptive high-order SUPG finite element method for the simulation of the flow of a UCM fluid through a 4 : 1 axisymmetric smooth contraction. The maximum residual error at each of the nodes in a given element was taken as the elemental error indicator and solutions beyond a Weissenberg number We = 10 were obtained without difficulty. However, with the adaptive procedure used the rate at which the residual error was reduced decreased with mesh refinement at We = 10 leading the authors to conclude that a better adaptive procedure based on reliable error estimates was needed. An element residual method for viscoelastic flows proposed by Warichet and Legat [618,619] was an obvious example of an attempt to extend the element residual method used by Ainsworth and Oden [8] for the Stokes problem. No theoretical results were given by the authors, but in [618] good agreement was found between a Couette correction based error for flow of an MUCM fluid [13] through an abrupt contraction and an error index obtained from their element residuals. 307
10.2. PROBLEM DESCRIPTION Some of the most recent developments in error estimation and adaptivity for viscoelastic flow calculations have been made by Owens [435] and Chauviere and Owens [130]. Although in these papers spectral element methods were used for the discretization of the continuous problem, the error estimators presented are not restricted to this choice of discretization technique, and the results carry easily across to finite element methods, for example. The authors drew their original inspiration from the work of Warichet and Legat [618,619] and we will devote most of this chapter to an exposition of the main ideas involved.
10.2
Problem Description
Let Q. C R d be some bounded Lipschitz domain having boundary dQ,. We con sider the steady, inertialess and isothermal flow of an incompressible viscoelastic fluid for which conservation of linear momentum and mass require that the ve locity u and the Cauchy stress tensor a satisfy V - u = 0,
(10.2)
V • a + b = 0,
(10.3)
and
where b is a body force. The Cauchy stress tensor is given by the expression tr = -pl + 7?s7(u) + r ,
(10.4)
where rjs is the (Newtonian) solvent viscosity and r is the elastic stress tensor. If we choose the Oldroyd B model then r , the elastic contribution to the stress tensor, evolves according to T
+ Ai^"=»jp7(u),
(10.5)
where r)p is the polymeric viscosity, 7 is the usual rate-of-strain tensor and Ai is a characteristic relaxation time for the fluid. Given the velocity field u, the upper-convected derivative Vu m (10.5) is defined by T"=U-VT-(VU)
T
T-T(VU).
(10.6)
Breaking with convention, a subscript appears in the usual notation for the upper-convected derivative. This is done in order to make clear the choice of kinematic variable appearing in (10.6).
10.2.1
Weak formulation
Before writing down a weak statement of the governing equations we turn our attention to the question of an appropriate functional setting for the flow vari ables. Let us denote the function spaces for the velocities, pressure and stresses by V, Q and S, respectively. Then we take V Q
= =
(F 0 1 (O)) d , 2
L 0(n). 308
(10.7) (10.8)
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES Given the velocity field u the inflow boundary dfl way by dVT = {xedtt:
of 0 is defined in the usual
u(x) • n(x) < 0} ,
(10.9)
where 17 has outward pointing normal vector n. We then define E
=
( T : T G [ L 2 ( 0 ) ] f d , Ai T"e [ £ 2 ( « ) ] f d Vv € V and T = T inflow on <9fT } .
(10.10)
The choice of function space E will be explained shortly. Although, strictly speaking, the velocity space V chosen by Owens [435] and Chauviere and Owens [130] was some subspace of ( ^ ( f i ) ) larger than (if^(n)) , the analysis of this chapter is supposed to hold in either case by a suitable redefinition of the body force b in (10.3). Given V, Q and E as in (10.7), (10.8) and (10.10) we now define bilinear forms a : V x V —> R, b : Q x V —> R, c : E x V —> R thus: o(w,v)
=
i)sfvwT:Vvdx
Kq, v)
=
[ qV-vdx Jn
c(T, v)
=
f T : V v dx
Vw,ve7,
(10.11)
V « e Q , v £ V,
(10.12)
VT e E, v € V.
(10.13)
Note that for a solenoidal vector field w and symmetric tensor T the gradients of w and v appearing in (10.11) and (10-13) may be replaced with 2D(w) and D ( v ) , respectively, where D(v) = 7(v)/2 is a rate-of-strain tensor. In addition to a(-, ■), &(•, ■) and c(-, •) we introduce a trilinear form d : F x S x E —> R defined by d(w, T, S)
=
/ (T + Ai ( w - V T - ( V w ) r T - T(Vw))) : Sdx Jn in , / 7(w) : Sdx, Vw <E V, T, S € E. (10.14) Jn
With a ( v ) , b{-,-), c ( v ) and d(-,-,-) defined as in (10.11)-(10.14) the Galerkin weak form of the governing equations is: Find (u,p, r ) £ V x Q x E such that c(
=
(b,v), Vv € V,
(10.15)
6(g,u)
=
0, V g e Q ,
(10.16)
d(u,T,S)
=
0, V S G E ,
(10.17)
where (-, ■) denotes the usual L 2 inner product of two vectors. Norms on Q, V and E which will be used in this chapter may be denoted as follows:
IMIo = \\T\\l
=
/ 92dx
Jn
[T:Tdx Jn
V
9 e Q, lvl? = /
VvT :Vvdx
Jn
VT 6 E. 309
Vv e y
' ( 10 - 18 ) (10.19)
10.2. PROBLEM DESCRIPTION It will be noted that the H1 semi-norm | • |i is equivalent to the usual H1 norm by the Poincare-Priedrichs inequality. Note also that using the Cauchy-Schwarz inequality a few times it may be deduced that = <
||(u-V)r-(Vu)Tr-r(Vu)||^ 2(||(u.V)r!|^ + ||(Vu)Tr||J + ||r(Vu)||^).
Now let us suppose that r G [H1(Q)]^xd. continuous in Q and therefore bounded,
(10.20)
Then, recalling that the velocity u is
||(U-V)T||2
jji(fi)> (10.21) for some constant c. In a similar fashion, exploiting the boundedness of each of the components of r , ||((Vu)Tr)g
(10.22)
where c' is a constant. It may thus be seen that
and is a proper subspace since the definition of £ includes, for example, stresses having discontinuous piecewise constant components. We note here that a discussion of errors is only meaningful when the weak problem statement (10.15)-(10.17) and its discrete counterpart (see §10.3) are well-posed. In fact, the function spaces required for establishing well-posedness of the problem (10.15)-(10.17) may well have to be smaller than V, Q and E as defined here (see [495], for example) since the choice of spaces V, Q and £ in (10.7), (10.8) and (10.10) is only guaranteed to ensure that the integrals in the Galerkin weak formulation (10.15)-(10.17) are bounded. There is only a small number of mathematical studies available in the literature on the well-posedness of flow problems involving viscoelastic models. Renardy has published existence proofs for slow steady flows of fluids of Maxwell type in smooth bounded domains [495] and demonstrated well-posedness of steady flows of Maxwell fluids [498] and fluids obeying the Jeffreys model [499] in a strip, subject to suitable inflow boundary conditions on the stresses. Existence of steady flows of Maxwell fluids subject to traction boundary conditions on the outflow of a region bounded by two parallel planes was considered by the same author [502]. More recently Renardy [504] has characterized inflow boundary conditions which lead to a wellposed initial boundary value problem for two-dimensional flow of a Maxwell fluid between parallel planes. In 2001, well-posedness results for the continuous three fields Stokes problem were established by Bonvin et al. [92]. The authors also proved convergence and stability of their Galerkin least-squares finite element method for this problem. In a yet more recent paper, Piccaso and Rappaz [465] proved existence in a convex polygon of a steady solution for a simplified Oldroyd B model where small relaxation times were assumed and the convective term was neglected. Existence, a priori and a posteriori error estimates for their Galerkin least-squares finite element method were proved. With the exception of these studies, another by Guillope and Saut [258] on existence results for differential 310
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES constitutive models of the Oldroyd type and a few other papers by Renardy (e.g. [496,497]) in the case of viscoelastic integral models, however, little is presently known about the well-posedness of boundary value problems for viscoelastic flows. In this chapter we will accordingly assume that the problem statements presented, both continuous and discrete, are well-posed.
10.3
Discretization and Error Analysis (Galerkin method)
Let V C V, Q C Q and E c E b e conforming discrete subspaces of V, Q and E, respectively. Then a discrete formulation of (10.15)-(10.17) is: Find (u,p, r ) 6 V x Q x E such that o(u,v)-ft(p,v)+c(f,v)
=
(b,v),
b(q,u)
=
0,
VqeQ,
(10.24)
=
0,
VSeE.
(10.25)
d(u,r,S)
WveV,
(10.23)
An important issue in mixed spectral/finite element discretizations of the gov erning equations for Newtonian flows and viscoelastic flows is that of compati bility of the approximation spaces so as to ensure a well-posed discrete problem (10.23)-(10.25). A full discussion on this topic in the context of spectral element methods and finite element methods may be found in §7.3.1. Assuming now that we have chosen our discrete approximation spaces to be compatible, let us define errors in the pressure, velocity and stress by E, e and e where E = p — p,
e = u - u
and
e = r — r,
(10.26)
and (u,p, T ) and (u,p, r ) are, respectively, the solutions to (10.15)-(10.17) and (10.23)-(10.25). To assist us in our description of the error equations we intro duce bilinear forms A:VxV —> R, B : Q x Q — > R a n d C A l : S x E — > R and define these as follows: i4(w,v)
=
T)s [ V w T : Vvdx,
Vw,v€V,
(10.27)
JQ
B(Z,Q)
=
[ &dx,
V£,q£Q,
(10.28)
JQ
C A l (T,S)
=
f
(T
+ Ai
T"]
: Sofoc,
VT,SGE.
(10.29)
In an attempt to follow the path beaten by Ainsworth and Oden we may, at this point, write down a global error problem: Find (0, ip, R) e V x Q x E such that A(4>,v)
=
a(e,v)-&(£,v)+c(e,v),
=
c(
=
-c(
Bfaq)
=
-6(9,e),
C 0 (R,S)
=
d(u,T,S)-d(u,r,S),
Vv € V,
V
311
(10.30) (10.31)
VS € E.
(10.32)
10.3. DISCRETIZATION AND ERROR ANALYSIS Equations (10.30)-(10.32) are the ones solved by Owens [435] who used a spectral element method with compatible velocity and pressure approximation spaces
v = vN = vn(vN(n))d, Q = QN = Qn7>jv_2(fi).
(10.33) (10.34)
The stress approximation space was chosen in [435] to be £ = T,N = {T € E : Tu G VN(P) n C°(0) V», j } .
(10.35)
In (10.33)-(10.35) we have used the notation VN(il) = {4>: \ak €TNk(nk)},
(10.36)
where, letting Nk for any 1 < k < K be a d—tuple (Nkti,... ,Nk,d), PNk(^k) denotes the space of all polynomials on Qk of degree less than or equal to Nkj in the j — th spatial direction. Owens [435] chose polynomial test functions q = q € QM C Q, v = v € F M C F and S = S 6 S M C S where the M superscript denotes an evaluation on a mesh finer than that used for the spectral element solution (u,p, f) of (10.23)-(10.25). This was necessary because the residuals are orthogonal to the original test spaces in a Galerkin method. The same choice of approximation spaces was adopted by Chauviere and Owens [130] in their development of the earlier work of Owens [435]. Owens [435] proved that, subject to restrictions on the relative magnitudes of the viscosities and relaxation time, a norm
^lelf + IIEHg + IHIg,
(10.37)
on the exact error could be bounded above and below by constant multiples of a suitably denned norm on the approximate errors <£, tp and R. A disadvantage of the method of Owens [435] was that it necessitated the solution of the problem (10.30)-(10.32) over all elements Qk simultaneously and was therefore not costeffective. Warichet and Legat [618] allowed the possible functional dependence of Ai on t r ( r ) , as in the so-called MUCM model [13], and then attempted to reduce the single global problem (10.30)-(10.32) to a sequence of independent local problems defined on each finite element flk C fi by adapting the approach of Ainsworth and Oden [8] and rewriting (10.30)-(10.32) in the form: Find (0fc,V>fc,R*) e Vk x Qk x Efc such that A
k{k,vk)
=
-Ck(tr(uk,Pk,Tk),v)
+
(nka(u,p,T))-vkds Jdfik
+(b*,v f c )*, Bk{i}>k,(ik) = Cojfc(Rfc,Sfc) =
Vvk£Vk,
h(qk,uk), Vgfc e Qk, -d(u fc> p fc ,S fc ), VSfcGEft,
(10.38) (10.39) (10.40)
where Vk, Qk and £& denote the restrictions of the spaces V, Q and E, re spectively, to Qk. Similarly, a k subscript attached to each of A(-,-), -B(-,-), C0(-, ■), b(-, •), c(-, ■), d(-, ■, ■), (•, ■), u, p, T and b in (10.38)-(10.40) above indi cate their restrictions to flk. It should be noted that in the work of Warichet 312
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES and Legat [618] the bilinear forms a(-, •), c(-, •) and A(-, ■) appearing in Eqns. (10.11), (10.13) and (10.27) were defined in terms of appropriate rate-of-strain tensors D, as explained previously. Warichet and Legat [618] actually took the negative of the right-hand side of (10.40) but this only changes the sign of the error estimate R of the elastic stress. The second term on the right-hand side of (10.38) is an averaged local flux across the boundary dttk of fifc. If TM is the interface dQk H dfli between two adjacent elements 0* and fy, then following Ainsworth and Oden [6,8] (n fc o-(u,p,r)>
=
nk
+
nkcr(ui,pi,Ti),
(10.41)
and ay : Tki -> H (i = 1,2) are smooth polynomial functions associated with Tki- Ainsworth and Oden provided details of how the polynomials ajj.y could be computed in the case of simple elliptic boundary value problems [4-6] and for the Stokes and Oseen equations [8] in order that a compatibility or equilibrium condition be satisfied. In a paper in 1997 Warichet and Legat [619] adopted a similar approach to the one described above for the localization of the error equations for the flow of a UCM fluid (TJS = 0). However, in addition to using a mixed Galerkin formu lation for the equations, stress splitting methods (EVSS, AVSS) and streamline upwinding (SUPG) were also implemented. Slightly different choices of finite element spaces were chosen by Warichet and Legat [619] depending on whether or not the mixed Galerkin method was used, and the reader is referred to equa tion sets (7) and (8) of [619] for further details. Using a p-adaptive strategy, similar to the rule for p enrichment in the original three-step strategy of Oden et al. [421], Warichet and Legat [619] were able to reach Deborah numbers of up to approximately 2.5 for the benchmark problem of flow past a sphere (see §9.2.2). In this section we attempt to show theoretically that the error tensor R computed by Owens [435] only controls part of the stress error, the so-called cell error, in any element. We shall draw on the work of Houston and Siili and co-workers [288,289] for linear hyperbolic problems in our efforts to show that one needs to take proper account of upstream residuals in order to compute the total stress error in a spectral/finite element, because of propagation effects. To this end, given any spectral or finite element 0,k m an elemental decomposition of fi, consider the solution r to f
(T + XI lr)
: Sdx
=
riP f
T
=
r,
-y(u) : Sdx,
VS <E S,
ondn^,
(10.42) (10.43)
where the exact velocity field u is used in the evaluation of the upper-convected derivative Vu- Prom the definition of the elemental stress residual r^: /
( r + Ai VTH : Sdx = TJP f
7(u) : Sdx - /
313
fk : Sdx,
VS € E. (10.44)
10.3. DISCRETIZATION AND ERROR ANALYSIS The stress cell error in the fcth element e£e" = Tk —Tk is the error incurred in this element due to the discretization of the constitutive equation (10.25) in the element. Subtracting (10.44) from (10.42) the cell error is thus seen to satisfy
/ (V
+ Ai4 e " n ) : S d x
efn = 0,
=
/rfc:Sdx,
VS £ E,
on dn~.
(10.45) (10.46)
In their papers on linear hyperbolic problems Houston, Siili and others [288,289] showed that the residual may only give a very poor estimate of the total local error when propagation effects are important. This is because the elemental residual does not give a proper measure of the so-called transmitted error; that is, the error which is created upwind of the element ilk and which is convected into it. This transmitted error ejj5an8 is defined by C ^ k - f * -
(10-47)
and consequently satisfies the equation etrans + A l £trans
/
. S d x = 0,
VS £ E,
(10.48)
Jnk 4 r a n 8 = Tb f c ~Tk,
ondil-k.
(10.49) eI1
rans
The total elemental error e* is then given by e* = e£ + e* . The basis of our criticism of the work of Owens [435] lies in the observation that the strong formulation of the discrete form of equation (10.32) satisfied by the error estimate R is R
=
- r - Ai (u ■ V f - ( V u ) T r - f ( V u ) ) + r)Pi (U) ,
«
- T - Ai (u ■ V f - ( V u ) T f - T(Vu)) + T)pj (u) = f. (10.50)
Therefore R|n fc behaves like the elemental residual rk and, for the reasons stated above, is an inadequate measure of e*. Similar criticism could be lev elled at other papers (e.g. [202,488-490,618,619]) where elemental residuals are calculated as indicators of the error. In an effort to obtain a better estimate for e^, Chauviere and Owens [130] formulated an equation satisfied by ek in element ilk- Adding (10.45)-(10.46) and (10.48)-(10.49) gives Ufc + Ai Ve"fc] : S d x = : f
/ JQk
\
)
rk : Sdx = - d ( u , T , S ) ,
VS e E,
(10.51)
JQk
,„_/ 4
on 50^ \ 50-,
where e~l denotes the value of e^ on the upstream side of the part of the inflow boundary of ilk having empty intersection with dil~. The hyperbolic problem (10.51)-(10.52) may then be approximated by solving f
( R * + Ai R k ) ■ S M d x = - d ( Q , f , S M ) ,
314
VS M € E M ,
(10.53)
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES
Rfc = { o
onSfi^n^fi-,
K
'
where T,M is a polynomial subspace of E of higher dimension than that used to approximate R. The upstream values RjJ" are determined by first solving in the domain of dependence of fi*. Thus the solution procedure proposed by Chauviere and Owens was to solve (10.53)-(10.54) on an element-by-element basis, starting with an element whose inflow boundary is a subset of dO,~. It is clear that a certain ordering of the elements is required in order to ensure that RjJ" is always available as inflow data for a particular elemental computation. Lesaint and Raviart [350] have shown that an ordering always exists for flows without recirculation. For flows with recirculation or where u • n changes sign along an element edge, it may be necessary to solve (10.53)-(10.54) over a block of elements. The proposed approach led to a considerable saving in storage and CPU over that employed by Owens [435] for solving (10.32), where a linear system arising from an assembly of the contributions from every element was inverted.
10.3.1
An error indicator
Let ( 0 , ^ , R ) be the solution to (10.30)-(10.31) and (10.53)-(10.54). Following Warichet and Legat [618], Chauviere and Owens [130] introduced an energy-like error norm ||(-, -,-)il* onVxQxH defined by \\(e,E,e)\\l^\(f>\\
+ \ml
+ \\K\\l,
(10.55)
where \-\A denotes the semi-norm induced on V by A(-, •). The following theorem was then proved: Theorem 10.1 Let (u,p,r) and (u,p,r) be solutions to the weak problem (10.15)-(10.17) and its discrete counterpart (10.23)-(10.25), respectively. Then assuming no quadrature errors in the evaluation of the linear forms in (10.23)(10.25) and provided that r)s, -^ and ^ are all sufficiently small there exist positive constants «i and «2 such that Kl\\(e,E,e)\\l
< {|e|? + ||E||g + ||e||g} <
K 2 ||(e,E ) € )||^.
(10.56)
Proof The proof follows an argument similar to that used in Theorem 5 of [435], except that now the properties of tensors in S and D M are exploited. (See [130] for the details). Remarks 1. The requirement that T}P/T]S is sufficiently small for the proof above to hold is admissible on physical grounds [425]. 2. Although the restrictions on the magnitudes of Ai, T)s and rjp/r)3 in the proof seem quite severe we shall see later in the chapter that even when these conditions are violated experimental evidence suggests that agree ment between the error indicator proposed in [130] and the exact error is good. 315
10.4. ADAPTIVE STRATEGIES
10.4
Adaptive Strategies
The aim of using an adaptive strategy is to achieve a prescribed level of accu racy at a given Deborah number at minimal cost. In that respect, it is desirable to devise a strategy which makes optimal or, at least, near optimal, use of the information supplied by a reliable error indicator. The adaptive strategies used in each of the papers by Oden et al. [422,423], Legat and Oden [348] and finally (in the case of viscoelastic flows) by Warichet and Legat [618,619] were inspired by the so-called "Texan 3-step" strategy originally devised by Oden et al. [421] for the finite element solution of Poisson problems in polygonal domains. In this strategy, the initial mesh details are specified and the problem is solved numerically. Then /i-refinements take place in order to obtain an intermediate global relative error whilst the global error is equidistributed amongst the ele ments. Finally, p-enrichments are carried out in order to obtain a final value for the global relative error with an equidistributed global error. Although this adaptive strategy is in widespread use, it rests on what the original authors ad mit to be shaky mathematical foundations and it has been the subject of some criticism in the literature [586]. As an engineering application of the derived error indicator ||(e,S,e)|| t (see Eqn. (10.55)), Chauviere and Owens [130] sought to use it to guide a p-adaptive spectral element method for the sphere in a tube problem (see §9.2). A spectral element mesh with 100 geometrically conforming elements, as shown in Fig. 10.1, was used.
Figure 10.1: The spectral element mesh with K = 100 used in [130] The enormous variation in solution magnitude exhibited by the stress vari ables over the flow domain prompted Chauviere and Owens to choose whether or not to p-enrich in a given spectral element flk according to the size of the elemental relative error 8k given by 6k
=
||(e,.E,e)U., t
yi<* + K,* + IMIS,*' «
"<e'g'g>"-»
(10.57)
y/\Kk+\mik + \\nu + \\{*,E,e)\\iik where the subscript k on a norm denotes its evaluation over £lk. A global relative
316
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES error 6 may also be calculated as
yJYLx
( N ? > + « , * + HrllS,* + IKe.-E.eJH^)
Once a maximum allowable relative error fl™3* (say) had been set any spectral element fie (say) where 6e > O™3* was jj-enriched. In recognition of the effect that residuals in the domain of dependence of Q,t can have on the elemental error the authors also p-enriched in the elements immediately upstream of $V In outline then, the procedure which Chauviere and Owens found to be the most efficient after experimenting with several different strategies was (i) Mark elements £lk where 0k > &£**. (ii) Mark elements in the domains of dependence of elements marked in (i). (iii) Select the component of the extra-stress which contributes the largest proportion of the stress error in those elements in (i) and (ii). (iv) Compute the average gradient in both spatial directions of that component and increase the polynomial order by one in the direction of the steeper gradient. (v) Sweep over all the spectral elements {fifc}fc=1 and increase polynomial or ders only so far as is necessary to ensure that the degree of approximation for all variables along common edges of contiguous elements is the same. Two points may be usefully made at this juncture by way of explanation of the procedure (i)-(v) outlined above. The first is that if elements immediately upstream of elements in (i) are not p—enriched then numerical experiments re vealed a very slow decrease, even an increase, in the elemental errors in marked elements. The second is that step (v) was only necessary for a standard con forming spectral element method [377,445]. Degrees of freedom could certainly have been saved by using a non-conforming mortar element method [375].
10.4.1
Numerical example: flow past a sphere in a t u b e
Although the theoretical development of ideas for error estimation and adaptivity from Ainsworth and Oden [4-6,8] and Oden et al. [348,421-423] through Warichet and Legat [618,619] to Owens [435] and Chauviere and Owens [130] is most easily accomplished in the Galerkin framework it has often been found nec essary to introduce stabilizing techniques in order to exceed the barrier of unity for the Deborah number in complex problems such as flow of a viscoelastic fluid past a sphere in a tube (see §9.2). As a consequence Chauviere and Owens [130] solved the Oldroyd B constitutive equations (10.5) by modifying the test tensor S in (10.25) to incorporate consistent streamline upwinding (SUPG) and solved the discrete set of equations on an element-by-element basis in the same fash ion as for the error tensor R. The improvement in solution smoothness over a standard Galerkin formulation using a Galerkin element-by-element method and an SUPG element-by-element (SUPG-EE) method, may be seen from Fig. 10.2. In this figure contours of the axial normal stress TZZ at a Deborah number 317
10.4. ADAPTIVE STRATEGIES
Figure 10.2: Contours of TZZ computed on a mesh with K = 100 and N = 5 at De = 0.9. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution? Error indicators for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science.
De = 0.9 computed on a 100 spectral element mesh (see Fig. 10.1) with poly nomial order N — 5 in each spatial direction are shown. The disastrous results arising from a Galerkin formulation when the discretization is not fine enough can be clearly seen. Using first a Galerkin and then an SUPG-EE approach Chauviere and Owens computed reference solutions to the problem on a fine mesh consisting of 100 uniform p spectral elements and compatible approxima tion spaces (see (10.33)-(10.35)) consisting of tensorized bases of polynomials of degree up to N = 8. By assuming that the reference solution was exact, a crude measure of the exact error (\e\l + \\E\\l + \\e\\l)1/2, 318
(10.59)
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES when coarser discretizations were used, was easily computable. The "exact" solution and error (10.59) were then used in two ways. The first was to assess what proportion of the error came from each of the velocity components, pres sure and stresses when a coarse mesh was employed. In Table 10.1 we show that by far the largest proportion of the error comes the elastic stress components, irrespective of whether a Galerkin or SUPG-EE formulation is used. For this reason it hardly seems worthwhile investing the computational effort calculating (j> and ip from (10.30)-(10.31) and certainly calls into question the usefulness of measures of the error published in the literature which rely solely on velocity data [269,415].
De 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Velocity error
Pressure error
Extra-stress error
4.76 3.95 3.37 2.89 2.45 2.06 1.71 1.44
4.07 4.78 5.80 6.49 6.64 6.29 5.64 4.72
91.17 91.27 90.83 90.62 90.91 91.65 92.65 93.84
Table 10.1: Percentage contributions of velocity, pressure and stress errors of the "exact" error for solutions computed with N = 5 using the SUPG-EE method [130]. As such, Chauviere and Owens [130] proposed an error indicator based solely on the elastic stress errors, i.e. the authors used ||R||o instead of ||(e,.E,e)||,, where R was computed element-by-element from (10.53)-(10.54), as a measure of the error. The second way in which the "exact" error was used was to compare the effectivity indices (ratio of estimated error to exact error) for the extra-stress contribution to the error estimate calculated by Owens [435] from (10.32) and for the proposed error indicator ||R||o- In Table 10.2 we show results from [130] for the SUPG-EE method using degree N = 7 and N = 8 polynomials in each spatial variable for the stresses. Thanks to the SUPG-EE technique, the range of Deborah numbers over which the error indicator may be tested has been extended from De G [0.5,0.9] to De G [0.5,1.2]. The error indicator of Owens [435] grossly overpredicts the error by a factor of 6.57 for a Deborah number of 0.9. Furthermore, it becomes more and more unreliable as De increases. These results are not in contradiction with those in [435] where good ef fectivity indices were obtained. In [435] the Deborah numbers considered were very small (De G [0.05,0.1]). As Ai -» 0, the equation for the error indicator proposed in [130] becomes the same as that in [435], as can be seen from equa tions (10.32) and (10.53). The improved error indicator proposed in [130] yields effectivity indices that are close to unity, testifying to the fact that this error indicator may be used with confidence with SUPG-like methods. With a maximum elemental relative error fljj"1* set to 2% for all fc, Chauviere and Owens [130] were able to achieve a Deborah number of 1.2. Final polynomial 319
10.4. ADAPTIVE STRATEGIES
De 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Nref=
7
SUPG-EE [130] 1.01 1.01 1.00 0.98 0.94 0.89 0.84 0.84
Nref=
8
SUPG-EE [130] 1.01 1.02 1.01 1.00 0.96 0.91 0.86 0.82
NTef= 7
NTef= 7
Gal [130] 0.98 0.99 0.99 0.96 0.90 -
[435] 2.98 2.79 2.51 3.78 6.57 -
Table 10.2: Error effectivity indices for stress error when the reference solution is computed on a uniform p mesh N = 7 and TV = 8 for the SUPG-EE method [130], for N - 7 for the Galerkin method [130] and the indicator in [435]. orders in the 100 elements varied between JV = 5 and JV = 10. At De = 1.2, the axial normal elastic stress component TZZ was seen to exhibit a very steep gradi ent on the axis of symmetry in the wake of the sphere and the addition of 2568 degrees of freedom to the number required at De = 1.1 was needed to ensure that the relative error fell below 2% for all elements. In Fig. 10.3 we show the profiles of the non-dimensionalized stress component TZZ as functions of the nondimensionalized z—coordinate on and in the wake of the sphere for N = 8 using three adaptive calculations. In these calculations Chauviere and Owens [130] set the maximum relative error tolerances to 9™** = 1.5% (Adaptive 1), 0™ax = 2% (Adaptive 2) and fljj18* = 2.5% (Adaptive 3). The non-dimensionalizations were performed by dividing the original stress values by {i)p + T)a)U/a and the spa tial coordinates by a, so that the maximum stress values could be compared with those of Lunsmann et al. [365]. Convergence with mesh refinement on the sphere is a simple task in comparison with the wake region where more degrees of freedom are required. Lunsmann et al. [365] used up to 51,354 degrees of freedom for their EVSS calculations and reported a maximum value of 70.313 for the non-dimensional rzz on the sphere at De = 1.2. The value of 70.583 in [130] compared well and was obtained with 26,273 degrees of freedom. In Figs. 10.4 and 10.5 the development of the elemental errors using Adap tive 2 follows a similar pattern to the one observed for the Galerkin method; the maximum elemental relative errors gradually moving from the sphere sur face to the wake of the sphere. However, in the sequence of error diagrams for the SUPG-EE method the errors are typically smaller than for the Galerkin method when similar meshes are used at the same Deborah number. More over, for moderate Deborah numbers (0.5 — 0.9) the maximum relative errors no longer occur exclusively in the second layer of elements next to the sphere but are shared between the first two layers, testifying to the greater smoothness now realized in the approximation of the stress components. Meaningful comparisons in terms of accuracy, stability and cost with other results in the literature proved to be difficult for Chauviere and Owens [130] because, with the exception of Warichet and Legat [619], results of using reliable error indicators for flow of a viscoelastic fluid past a sphere in a tube have not 320
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES
Figure 10.3: Profiles of the non-dimensionalized TZZ on the sphere surface —1 < z < 1 and in the wake of the sphere z > 1 for different meshes for De = 1.2 using the SUPG-EE method. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution? Error indicators for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science. been made available. A relative error of about 10% at a Deborah number of 0.75 was reported by Warichet and Legat [619] for their adaptive EVSSGalerkin finite element method for flow of a UCM fluid past a sphere in a tube, using slightly fewer than 22,000 degrees of freedom. A relative error of between 0.1459% and 0.1551% was achievable with the adaptive method in [130] and fewer than 20,974 degrees of freedom were required. By using an adaptive AVSS-SUPG method Warichet and Legat were able to compute solutions for the sphere problem up to a Deborah number of approximately 2.5. However, the accuracy of the calculations as measurable by their error indicator was not reported and instead the authors chose to establish accuracy on the basis of the drag correction factor. As is now well established, this is not a reliable error indicator. At De = 1.3, the peak of TZZ in the wake of the sphere became so huge that trying to capture it by p-enrichment led the numerical method to break down if 0™ax was set at 2%. In Fig. 10.6 the most accurate (non-dimensionalized) stress solutions possible using the adaptive method with 0™** set equal to 2% (Adaptive 2) are shown. A maximum relative elemental error no better than 6.93% could be achieved with Adaptive 2, and oscillations are now evident in 321
10.4. ADAPTIVE STRATEGIES
Figure 10.4: Evolution of the elemental relative errors 9k with the adaptive strategy for 0.6 < De < 0.9. 6^ax = 2% for all fc. Only the first and last error plots for each De in the adaptive process are shown. SUPG-EE. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution? Error indica tors for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science.
322
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES
Figure 10.5: Evolution of the elemental relative errors 9 k with the adaptive strategy for 1.0 < De < 1.2. 9fax = 2% for all k. Only the first and last error plots for each De in the adaptive process are shown. SUPG-EE. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution1? Error indica tors for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science.
323
10.4. ADAPTIVE STRATEGIES
Figure 10.6: Contours of non-dimensionalized elastic stress components at De = 1.3. Adaptive 2. SUPG-EE method. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution? Error indicators for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science. 324
CHAPTER 10. ERROR ESTIMATION AND ADAPTIVE STRATEGIES
Figure 10.7: Profiles of the non-dimensionalized rzz on the sphere surface and in the wake of the sphere for different meshes at De = 1.3 using the SUPG-EE method. Reprinted from C. Chauviere and R. G. Owens, How accurate is your solution? Error indicators for viscoelastic flow calculations, J. Non-Newtonian Fluid Mech., 95:1-33, Copyright (2000), with permission from Elsevier Science.
the wake of the sphere for the zz component of the elastic stress. Notice in Fig. 10.7 that convergence on the sphere has occurred long before convergence in the wake of the sphere, and partly explains why the drag factor is such a poor indicator of the quality of the solution, particularly at higher Deborah numbers. The profiles shown for the non-dimensionalized stress values in this figure were produced with the uniform mesh N = 8 and the same three tolerancesfl™**as used in Fig. 10.3. This time, however, it was impossible to obtain the target relative errors using the adaptive strategy. The non-dimensionalized profiles T*Z of TZZ shown in Fig. 10.7 were obtained at the last adaptive step just before the procedure diverged at each of the three tolerance levels. The pressure and velocity components corresponding to all the stress calculations presented in this chapter remained well-behaved and smooth.
325
Chapter 11
C o n t e m p o r a r y Topics in Computational Rheology 11.1
Advances in Mathematical Modelling
The process of mathematical modelling involves a certain degree of pragmatism. For although the primary task of a mathematical modeller is to construct models that are able to predict phenomena which are observed in the laboratory or the physical world, pragmatism dictates that the level of sophistication of the model must be governed, to a certain extent, by the limited arsenal of analytical and numerical techniques (not to mention computational resources) at the modeller's disposal. The tension in the development and construction of a mathematical model is always between the level of sophistication, on the one hand, and its analytical or computational tractability, on the other. This tension exists in the development of mathematical models for describing the rheological properties of dilute and concentrated polymeric liquids. The quest is to derive a model that is as simple as possible, involving the minimum number of variables and parameters, and yet having the capability to predict the behaviour of polymeric liquids in complex flows. In this book we have concentrated on numerical methods for solving the equations resulting from a macroscopic description of a fluid. The macroscopic approach to the modelling of viscoelastic flows furnishes a closed form differen tial or integral constitutive equation that relates the stress to the deformation history. In many situations this approach is sufficient to provide a qualita tive description of the important flow phenomena. However, situations arise in which predictions using these models fail to give quantitative agreement with experimental measurements and observations. Therefore, it is necessary to look to more sophisticated models, based on a more detailed level of description of the fluid, to provide reliable predictions. Until relatively recently the alternative microscopic approach based on the kinetic theory of polymeric liquids was not computationally feasible. The task of computing with such models was thought to be beyond available computational power. For a dilute polymeric solution the microscopic approach focuses on the dynamics of polymer chains within a solvent. On the basis of the dynamics the stress can be computed. A closed form constitutive equation is not derived in 327
11.1. ADVANCES IN MATHEMATICAL MODELLING this approach and, in fact, it is impossible to write one down in the majority of cases. Traditionally, most of the effort expended in the field of viscoelastic nu merical simulation has been directed at pursuing a macroscopic approach to computer predictions. However, the increased computational resources avail able today mean that finer levels of description, such as those in the micromacro approach, can be contemplated. This chapter will describe some recent developments in the modelling of polymeric liquids and some of the numerical techniques that have been devised for performing simulations based on them. Keunings [327], in a recent review article on the modelling of polymeric liquids, described a number of levels of description of a fluid. Beginning with the finest level of description these are: • Quantum mechanics. Modelling based on this approach is not feasible at the start of the third millennium due to the vast number of microstructural degrees of freedom associated with polymers and the diversity of the time/length scales separating the atomistic and macroscopic processes. • Atomistic modelling. These models have been developed for the analysis of equilibrium structures and properties. This is the most detailed level of description of the rheological behaviour of polymers that can be contem plated at this moment in time. However, contemplation is the key word here because this description cannot be applied universally due to the computational resources involved in performing the calculations. The in tense nature of the computations means that the models used in numerical simulations are rather coarse. There is a strong belief that sophisticated versions of these models will be able to provide an explanation for phe nomena such as wall slip, in due course. Numerical techniques based on nonequilibrium molecular dynamics have been used to study the flow of polymers near walls and geometric singularities. Kinetic theory. This approach involves a coarse-grained model for poly mer conformations. Processes at the atomistic level are ignored. These models do not purport to provide a description of the fluid at a molecular level. Numerical methods for simulating models of this type are based on stochastic simulation or Brownian dynamics methods. An alternative ap proach, that also features a mesoscale description of the fluid microstructure, are lattice Boltzmann methods. These methods are highly parallelizable and therefore offer significant advantages for large-scale computations of viscoelastic fluids in complex geometries. • Continuum mechanics. In this approach the stress on a macroscopic fluid element is related to the deformation by a constitutive equation. Together with the conservation laws the constitutive equation yields a closed set of partial differential equations or integro-differential equations that can be discretized and solved using the techniques described in Chapters 5 and 6. In principle, macroscopic constitutive equations can be derived in closed form from kinetic theory. However, to realize this in the majority of situations requires a closure approximation. Several hybrid modelling approaches have been advocated, the most notable of which is the micro-macro approach. In this approach a coarse-grained micro328
CHAPTER 11. CONTEMPORARY TOPICS scopic description of the evolution of the stress is combined with a macroscopic description of the kinematics. Another hybrid approach worth exploring would be the combination of a description of the fluid near a wall based on atomistic modelling ideas with a more traditional macroscopic or microscopic approach away from the wall. This may circumvent some of the difficulties associated with the use of macroscopic models near walls and geometric singularities such as reentrant corners. The polymeric contribution to the extra-stress from kinetic theory consider ations may be determined using one of the two paths shown in Fig. 11.1. Models based on a kinetic theory approach provide a coarse-grained description of the polymer dynamics in terms of the microstructure. For example, for the dumb bell model for a polymer solution one can derive a diffusion or Fokker-Planck equation from the equations of motion of the beads. The solution of the FokkerPlanck equation provides the configuration probability density function (pdf), or configurational distribution function, as it is sometimes called. For the dumb bell model the configuration pdf yields information on the probability of finding a dumbbell with a given configuration at a particular material point. Unfortu nately, the Fokker-Planck equation is analytically intractable, even for simple flows. Furthermore, a numerical solution of this equation for a high-dimensional configuration space at all points in space and time using conventional numerical techniques would require extensive computational resources and cannot be con templated at this moment in time (see Ottinger [433]). Therefore, this approach is necessarily restricted to kinetic theory models with a low-dimensional config uration space. As alternatives to solving the Fokker-Planck equation one can either resort to closure approximations in which one seeks to eliminate the con figurational distribution function to obtain a closed form constitutive equation for the state variables or use an extremely important mathematical equivalence between the Fokker-Planck equation and a stochastic differential equation. In the first approach suitable state variables are introduced and finite dimen sional representations of the configurational distribution function are considered. This procedure leads to evolution equations for the state variables. The solu tion of these equations is used to construct the polymer stress. The drawback of this approach is that an approximation to the model is introduced through the restricted class of distribution functions that are considered. The second approach is based on the solution of a stochastic differential equation that is equivalent to the original Fokker-Planck equation. The numeri cal integration of stochastic differential equations is possible whereas, in general, the solution of the corresponding Fokker-Planck equations is not. However, it should be noted that the stochastic approach does not furnish the configura tional distribution function. Instead, it provides an ensemble of trajectories or configuration fields from which one can evaluate the polymer stress by taking averages over a large number of realizations. These two approaches are described in this chapter. In addition, we give an overview of the numerical methods that have been developed for solving the stochastic differential equations associated with kinetic theory models. Al though numerical methods in this field are in an embryonic stage of development, there have been a number of important contributions that have made these tech niques more competitive with macroscopic methods than their forerunners. It is our intention to describe some of these developments in this final chapter. In the past the choice of constitutive model was restricted in practice by 329
11.2. DYNAMICS OF DILUTE POLYMER SOLUTIONS Description of Polymer Dynamics using Kinetic Theory
4Fokker-Planck Equation for the Configurational Distribution Function
S
\
Closure Approximation
Differential Equation for an Equivalent Stochastic Process
I
I
Closed-Form Constitutive Equation for the State Variables
Evolution of Trajectories or Configuration Fields
i
4-
Stress Computed by Solving the Constitutive Equation using Conventional Numerical Methods
Stress Computed from an Ensemble Average over a Large Number of Realizations of the Stochastic Process
Figure 11.1: Determination of the polymer stress from kinetic theory consider ations what was tractable computationally. This restrictive choice may explain the many discrepancies that litter the field when comparisons with experimental observations are attempted. The computational power available today means that new classes of models may be investigated for the first time.
11.2
Dynamics of Dilute Polymer Solutions
The stress in any polymeric liquid depends on the conformations of the polymer molecules, viz., the orientation and degree of stretch of a molecule, since these contain information about the strain history. Kinetic theory provides a descrip tion of the polymer conformations based on coarse-grained molecular models of the polymers. Ensemble averages taken over many model polymers, each of which has its own conformation, are used to determine the stress in a fluid element. In this section we consider a description of the rheological behaviour of dilute polymer solutions based on kinetic theory. For a dilute polymer solution the
330
CHAPTER 11. CONTEMPORARY TOPICS interactions between different polymer chains are not considered. Kinetic theory models for describing concentrated polymer solutions and polymer melts take the interaction of polymer chains into account. These models are described in §11.5. There are a number of ways of describing dilute polymeric liquids within the framework of kinetic theory. These are based on mechanical models of polymers involving constraints and form a natural hierarchy in terms of level of sophistication. It is important to note that kinetic theory models have been developed with the goal of representing the polymer dynamics rather than the chemical structure of the polymer in any detail. One of the more sophisticated mechanical models is the Kramers chain which is a freely jointed bead-rod chain with N beads connected by N — 1 rigid rods of length L. In this model the beads do not represent the atoms along the backbone of the polymer chain but rather chain segments of around 10 or 20 monomers. The Kramers chain has a large number of internal degrees of freedom: it can be oriented, stretched and deformed, but it has a constant contour length. There fore, the model possesses properties that are important in formulating a kinetic theory model for dilute polymer solutions. For realistic simulations N is of the order of 100. Therefore, a large number of degrees of freedom is required to describe the conformation of model polymers based on the bead-rod model. A coarser level of description is the Rouse chain, which is a freely jointed bead-spring chain with N beads connected by N — 1 Hookean springs. In this model the spring represents several hundred atoms along the backbone of the polymer chain and the masses of these atoms are concentrated in the beads. This model will be described in more detail in §11.2.1. Finally, there is the dumbbell model (see §2.6.1), which is a special case of the Rouse chain with N = 2, i.e. two beads connected by a spring. When the beads are connected by Hookean springs the tension force in the elastic connector between the ith and (i + l)th beads is given by F( c ) = HQi,
(11.1)
where H is the spring constant and Q* is the vector connecting the ith and (i + l)th beads. This linear spring force law is only really valid in situations where the chains remain close to their equilibrium configuration. The problem with this law is that at high flow rates and in extensional flow the chains extend in length unboundedly whereas real polymers can only be extended, at most, to their fully stretched length provided they do not break. The property of finite extensibility of the chains can be incorporated in the model by introducing a finitely extensible nonlinear spring force:
F =
^Sl
ni2^
where Q0 is the maximum extension of each chain segment. This force law gives rise to the FENE model (see §2.6.1).
11.2.1
The Rouse model
In the Rouse model, also known as the freely jointed bead-spring model, the polymer is represented by linear chains of N identical spherical beads connected by N — 1 Hookean springs (see Fig. 11.2) 331
11.2. DYNAMICS OF DILUTE POLYMER SOLUTIONS
Arbitrary fixed point in space
Figure 11.2: The freely-jointed bead spring or Rouse model Each polymer molecule is a long chain composed of monomers. The beads in the Rouse model do not correspond to individual monomers but rather chain segments of a group of many monomers. The springs represent entropic effects due to the elimination of more local degrees of freedom rather than molecular interactions. The solvent is modelled as an incompressible Newtonian fluid with viscosity T)s. This model was developed for polymer solutions which are suffi ciently dilute that one can ignore the interactions between different chains. The model exhibits orientability and stretchability but the chain does not have a constant contour length. In fact, the polymer molecule modelled by the Rouse model can be stretched out to any length. The derivation of the Fokker-Planck equation for the Rouse model is a generalization of the derivation for the dumb bell model given in §2.6.1. The main steps in the derivation are presented here for completeness. The internal configuration of the chain is given by Qi = r i + i - rt, i = 1 , . . . , N - 1.
(11.3)
To write down the equation of motion for the beads it is necessary to identify the forces acting on each bead. In this model these are taken to be the following: • The forces on the beads exerted by the springs. The tension force in the elastic connector between the ith and (i + l)th bead given by (11.1). Thus the force F ; 6 ' on the ith bead exerted by the elastic connectors is given by
Fi c) , F (e)
F (c)
F (c)
t
-*N
»
332
= 1, = = N.
2,...,N-1,
(11.4)
CHAPTER 11. CONTEMPORARY TOPICS The forces on the beads at the end of the chains are attached to only one connector which accounts for the special cases in the above formula for the elastic force. • The viscous drag force exerted on the beads by the solvent. For homoge neous flow fields the solvent velocity may be written as u(r) = u 0 + «r, where K = ( V u ) T is spatially constant. Thus, the drag force on the ith bead of the chain is given by F | d ) = -C(r< - no - «r,-), where the constant £ is the friction coefficient of a bead. For spherical beads we have £ = 6m]sa which follows from Stokes's law, where r)s is the viscosity of the solvent and a is the radius of a bead. • The force due to Brownian motion. This arises from the erratic bom bardment of the beads by the solvent molecules and is a correction to the continuum hypothesis. This force is only defined in a statistical sense and is taken to be random in direction and magnitude, so that the average force vanishes at all times, i.e. = o, where {•), defined by (2.96), represents an ensemble average. Since each bead experiences random impacts by the solvent molecules it is natural to assume that the stochastic forces have infinitesimally small correlation times compared to the timescale of the motion of the beads and that there is no correlation between the stochastic forces acting on different beads. Therefore, we have (i + T)> = CSWv,
(11.5)
where C is a constant tensor. If the velocity distribution is required to obey the Maxwell distribution at equilibrium then it can be shown that C = 2CATI, where T is absolute temperature and k is Boltzmann's constant. The equation of motion for the ith bead is then " • * | ( ^ t - u ( p * ) ) = F W + p W + p W , i = l,...1JV.
(11-6)
Neglecting the acceleration of the beads, the equations of motion for the beads reduce to force balance equations. Thus, we obtain the Langevin equations U = iio + « r i
+
i ( F i e ) + P | 6 ) ) , i = l,...,N.
(11.7)
The quantity V( r i>--- ,TN,t)dri ...drN, where ip(ri,... ,T^,t) is defined to be the configuration pdf, is the probability of finding a chain whose ith bead 333
11.2.
DYNAMICS OF DILUTE POLYMER SOLUTIONS
is located in the region {r< + fidrt : 0 < // < 1}, for i = 1 , . . . ,N, at time t. The diffusion or Fokker-Planck equation governs the evolution of ij). The derivation of the diffusion equation (2.95) for the elastic dumbbell model may be generalized to the Rouse model in which case we obtain the following diffusion equation for ip:
* = " £ * I y*+Kt'+<s
) *J+T £ w
(118)
'
Once the configuration pdf, ip, is known the polymer stress may be calculated using the Kramers expression, which for the Rouse model is N-l
T
= _ ( t f - i ) n m + n£
(11.9)
i=l
where n is the number of polymer molecules per unit volume. The Langevin equations (11.7) may be made dimensionless by introducing the non-dimensional variables =
XK,
(11.10) nkT'
v/fcT/F'
where A is a time constant. If we choose A = £/(4H) then the non-dimensional Langevin equations are rj = u$ + «*r* + i F ; ( e ) + F * ( 6 ) , i = l,...,N,
(11.11)
where the dimensionless Hookean spring force is given by
and F*^
is a stochastic process characterized by
< F * ( V ) > = 0,
(
V)F*
(
V+T*))
= \s(T*)SijI.
(11.12)
In Brownian dynamics computations only a limited number of chains are simu lated and the components of the polymer stress tensor are estimated by calculat ing the mean value of the components of the tensor Q F ^ once the configuration pdf is known. However, as we have already mentioned in the introduction to this chapter, for models with a large number of degrees of freedom it is not feasible, using conventional numerical techniques, to solve the Fokker-Planck equation for the configuration pdf. Fortunately, there exists an alternative ap proach for generating the distribution function based on an equivalent stochastic differential equation. This approach will be described in §11.4. The popularity of the Rouse, or bead-chain, model is due to its mathematical tractability, a consequence of the linear spring force law, which has enabled analytical solutions to a number of equilibrium and nonequilibrium problems in kinetic theory to be found. The use of finitely extensible springs, although more 334
CHAPTER 11. CONTEMPORARY TOPICS justified on physical grounds, precludes the determination of analytical solutions to many of these problems. However, real polymers are not infinitely extensible. Therefore, the linear spring force law is a poor approximation, particularly in extension-dominated flows. This approximation may be improved upon by means of a finitely extensible nonlinear spring force. Nonlinear spring force laws have already been introduced in §2.6.1 for the dumbbell models. Precisely the same laws may be incorporated into generalizations of the Rouse model.
11.2.2
Dumbbell models
There are many kinds of dumbbell models, each of which is characterized by the choice of spring force law. The dimensionless forms of the linear and Warner force laws introduced in §2.6.1 are F<e>(Q) = Q,
(11.13)
and
F(C)(Q)
= r^ Q '
(1L14)
respectively, where Q2 = tr Q Q and b = HQl/kT is a dimensionless finite extensibility parameter. The dimensionless form of the Fokker-Planck equation (2.95) for the Hookean and FENE dumbbell models is
*hm- <-> where F ^ is given by (11.13) and (11.14), respectively. The solution of (11.15) furnishes the probability ^>(Q(x, t))dQ of finding a dumbbell with configuration in the range Q to Q + dQ at (x, t). The dimensionless form of the Kramers expression for the polymer stress is T = - I + (QPW(Q)>,
(11.16)
where the ensemble average has been defined in (2.93).
11.3
Closure Approximations
In the last section the Fokker-Planck or diffusion equation for the configuration pdf was derived from a description of the dynamics of polymeric liquids using kinetic theory. In principle, this equation may be solved for the distribution function of the polymer configuration at each position within the flow domain. This, in turn, can be used to determine the stress using the Kramers expression. However, as was mentioned in §11.1, this procedure is not practical for kinetic theory models that possess a configuration space of large dimension. Therefore, its use is limited to kinetic theory models with a conformation space of low di mension. A more economic procedure must be established before kinetic theory models can be used as a basis for numerical simulations of polymeric liquids. There are two basic approaches that may be adopted. The first approach is based on the mathematical equivalence, under certain conditions, between the Fokker-Planck equation and a stochastic differential 335
11.3. CLOSURE APPROXIMATIONS equation. This is an extremely important and powerful result since stochastic differential equations are more tractable numerically than the diffusion equation. We will explore this approach more thoroughly in the next section. The second approach conforms to the more traditional approach to computa tional rheology in which a closed-form constitutive equation is sought. The idea is to derive a constitutive equation from a kinetic theory model. Unfortunately, this is decidedly more difficult than it sounds. This is because, apart from very simplistic models, closed-form constitutive equations cannot be forged from ki netic theory models without invoking a closure approximation. Thus, before incurring any discretization error the kinetic theory model is subject to approx imations that may be of a questionable nature. Indeed, some of the popular closure approximations are unable to reproduce the predicted behaviour of the original kinetic theory model in certain flow situations. More sophisticated clo sure approximations are required to broaden the range of flows that can be predicted accurately using these approximate kinetic theory models. We consider the closure problem for the kinetic theory of FENE dumbbells. Unlike the derivation of the constitutive equation for the Oldroyd B model from a molecular model consisting of Hookean dumbbells in a Newtonian solvent (see Chapter 2) it is not possible to derive an equivalent constitutive equation for FENE dumbbells, i.e. there is no direct closure. A suitable approximation must be made in order to obtain a closed-form constitutive equation. A/number of approximations, based on modifications of the spring force, have been proposed for the FENE model. The closure approximation due to Peterlin [449] is based on the modified law F
(Q) = r ^ Q >
("-IT)
in which the spring law is pre-averaged. This results in the FENE-P model and yields an evolution equation for the configuration tensor A = (QQ) given by f
+
u-VA-.A-A^
= I - ^ ^ A .
(11.18)
The polymeric contribution to the FENE-P stress is given by the Kramers ex pression
T
--I+r~ikmA-
(1L19)
In numerical computations using the FENE-P model the evolution equation (11.18) is solved for A. The polymer stress r is then computed using the ex pression on the right-hand side of (11.19) and this is inserted into the momentum equation. For steady state flows the FENE-P closure can provide reasonable qualitative agreement with FENE kinetic theory. However, major differences can occur for transient flows. For example, in the start-up of uniaxial extensional flow followed by relaxation the FENE-P closure is unable to predict the hysteretic behaviour exhibited in the plot of extensional viscosity against average molecular extension [536]. This behaviour is predicted by FENE kinetic theory [536]. The failure of the FENE-P model to reproduce the hysteretic behaviour of the FENE theory has provided the catalyst for the search for more realistic closure approximations for FENE kinetic theory. 336
CHAPTER 11. CONTEMPORARY TOPICS Lielens et al. [354,355] have responded to this mission by proposing a general framework for deriving closure approximations that was inspired by the work of Verleye and Dupret [598]. The idea is to describe the fluid by a finite number of state variables, Xiy i = 1 , . . . ,n. These variables are configuration space averages of a scalar or tensor function /,, say, of the configuration vector, Q: Xi = .
(11-20)
The state variables satisfy an evolution equation that is derived from the diffu sion equation (11.15) [73]. The task is to find evolution equations for the state variables that do not involve the configuration pdf, ■$. It is not possible to do this, in general, without committing an approximation in which only particular representations of the distribution function are considered. This is the closure problem in a nutshell. Lielens et al. [354,355] restricted the space of admissible distribution func tions to a so-called canonical subspace of finite dimension. This is the step at which the approximation is introduced. The exercise of deriving a particular closure approximation is simplified further by selecting the form of the canonical distribution, ipc, to be one that is decoupled with respect to the length (ipQ) and orientation (^ u ) of the dumbbells:
VC(Q) = V Q ( W » ,
(H-21)
where Q — |Q| and u = Q / Q are the length and unit orientation vector of the dumbbell, respectively. The notation u has already been used for the velocity vector. However, u is commonly used in the polymer dynamics literature to denote the unit orientation vector of a dumbbell or tube segment and it should be obvious from the context as to what is being referred to. The use of tjjc instead of ip in (11.20) results in approximations of the state variables. The next stage in the process is to define a radial distribution, pc(Q), related to i>Q(Q) by PC(Q) = Q2i>Q(Q), satisfying the normalization rs/b
/ Jo
P
(0) dQ = 1.
The use of the decoupled canonical distribution (11.21) results in the following evolution equations for the state variables Xi = A = (QQ) and X 2 = B = <(Q-Q) 2 >:
1*
=
1 0 M A )
+
4 ^ : A - 2 ^ ,
(11.23)
with
AC
= ^r^c(Q)dC?' 337
(1L24)
11.4. STOCHASTIC DIFFERENTIAL EQUATIONS The Kramers expression for the polymer stress is then given by A C
T
T =
-
I+
A
(11.26)
MA)A-
The FENE-P closure approximation results from a one-parameter canonical radial distribution, pca(Q), defined by Pea(Q) = Sa(Q)
=
S(Q-a),
with a single state variable A. To obtain a second-order closure approximation Lielens et al. [355] introduced a two-parameter canonical radial distribution, Pa,p(Q)- F ° r t n e FENE-L closure this canonical distribution assumes the form {
PaAQ) = a
^-^[l-Ha(Q)}+fida(Q),
(11.27)
where (a,/3) 6 [0, Vb] x [0,1] and Ha(Q) denotes the Heaviside function located at Q = a. The FENE-L closure expressions Ac and Bc are rather complicated in an algebraic sense and can be found in the paper of Lielens et al. [354]. A simplified closure model, known as the FENE-LS model, was derived [355] to overcome this drawback. The canonical radial distribution for this model is (11.28)
PaAQ) = V ~ P)&O/R(Q) + 0Sa(Q),
where R is a constant chosen to provide a good approximation of the FENE-L closure model. In deriving this model care has been taken to ensure that it is able to reproduce the predictions of the FENE-L model as much as possi ble. In the case of the FENE-LS closure approximation the closure expressions Ac(tr{A),B) and Bc{tr(A),B) have the explicit form Ac
_
BC =
2K
D\f-
D
D
d
Y
Ai^(l-?)\ 1-2JT V
6/
|
(W) 1 (/-/)
1,
4
i? (l-2X)J '
M129) (11.30) v
;
where
Computations with the FENE-L and FENE-LS models, with R2 = 5 and b = 50, show that they are superior to the FENE-P model in describing the transient flow behaviour predicted by FENE kinetic theory [355].
11.4
Stochastic Differential Equations
Many of the models that have been derived from kinetic theory considerations and used to describe the rheologieal behaviour of polymeric liquids can be for mulated in terms of a diffusion or Fokker-Planck equation for the configuration
338
CHAPTER 11. CONTEMPORARY TOPICS pdf, ift. Once this is known the macroscopic variables of relevance can be com puted as statistical averages of some function of the polymer conformations. The general Fokker-Planck equation for i)(x,t) is of the form j^(M)
= ~{A(x,^(x,t)} + ~|;:{D(x1tWx1t)})
(11.31)
where x is a ti-dimensional vector that defines the coarse-grained microstructure, A(x, t) is a d-dimensional vector representing the drift term and D(x,t) is a symmetric, positive definite dx d matrix representing the diffusion tensor. The quantities A and D define the deterministic and stochastic contributions to the model, respectively. Methods based on the numerical solution of the Fokker-Planck equation (11.31) are limited to kinetic theory models with a conformation space of small dimension. However, there is an important equivalence between Fokker-Planck equations and stochastic differential equations that has significant implications for numerical simulations of polymeric liquids since computations using the lat ter can be considerably simpler. The polymer dynamics described by the FokkerPlank equation (11.31) for ip(x,t) can be regarded as a Markovian stochastic process X(f) acting on the polymer molecules. This process satisfies a stochastic differential equation whose coefficients at any time depend only on the current configuration of the polymer molecules. The stochastic differential equation associated with the Fokker-Planck equation (11.31) is dX(t)
= A(X{t),t)dt
+ B(X(t),t)dW(t)
(11.32)
where the diffusion matrix D(x,i) = B ( x , i ) B T ( x , t ) ,
(11.33)
and the stochastic process W(i) is a multi-dimensional Wiener process. The components of W(i) are independent Wiener processes, i.e. Gaussian processes with zero mean and covariance (W(i)W(t')) = min(t,t')I. More precisely, we have the result [433]: Theorem 11.1 If the functions A and B on R d x T satisfy the Lipschitz con ditions |A(x,t)-A(y,t)|
<
c|x-y|,
|B(x,t)-B(y,i)|
<
c|x-y|,
.
. [n
^>
and the linear growth conditions |A(x,t)| < c(l + |x|), , , {il 6ii) |B(x,t)| < c(l + |x|), for all x £ JR,d, t £ T, for some constant c, then the unique solution of the stochastic differential equation (11.32) is a Markov process. If A and B are continuous functions, the infinitesimal generator of this Markov process is given by the second-order differential operator, Ct, defining the right-hand side of (11.31), i.e.
A = -|{A(x,*)} + i A | . : { D ( x > t ) } . 339
(n.36)
11.4. STOCHASTIC DIFFERENTIAL EQUATIONS In the case of the dimensionless form of the FENE dumbbell model, for example, we have in the notation of (11.31), X(t) = Q(i), A = #s(t)Q(t) - i p ( Q ) , D = I. The equivalent stochastic differential equation is dQ(t) = (K(t)Q(t)-^F(Q)\dt + dW(t).
(11.37)
Once the configurations are known macroscopic fields such as the stress tensor are obtained by taking an ensemble average over a large number of re alizations of the stochastic process X. Analytical solutions to the stochastic differential equation (11.32) can rarely be found. More importantly, there are no analytical solutions for the stochastic differential equations associated with the nonlinear models of polymer kinetic theory that are of interest. Therefore, one has to resort to numerical techniques for solving (11.32). The numerical integration of a stochastic differential equation generates an ensemble of trajec tories from which one can compute averages of the quantities of interest. Consider the numerical integration of the stochastic differential equation (11.32) with an initial condition X 0 at time t = 0. The time interval T = [0,T] is partitioned into subintervals [U, ij+i), i = 0 , . . . , n — 1, where 0 = t0 < h < ■ ■ ■ < tn = T. Let Xj denote the approximation to X(t) at time t — ti. To solve (11.32) an ensemble of trajectories is generated using a suitable integration scheme. For example, an algorithm based on the forward Euler scheme is X i + 1 = X i + A(X i ,i i )A
(11.38)
where Ati = i i + 1 — ti. This scheme is also known as the Euler-Maruyama scheme in recognition of the person who established the mean-square convergence of the method. Note that the time step used for the integration of the stochastic differential equation is typically smaller than the time step used for the solution of the conservation equations in a time-dependent computation. To obtain a quantitative description of the accuracy of an approximation scheme for solving the stochastic differential equation (11.32) the concept of the order of strong and weak convergence is introduced. Let At = maxi<j<„ Ati, then a given approximation scheme is said to converge strongly of order v > 0 at time T — tn if there exists a constant C independent of At such that (|X(T)-Xn|2)
< c(l + \x\)\t-t'\W, < c(l + \x\)\t-tf/2,
(il6y)
for all x G R and t, t' € T, where c is a constant. Then the Euler scheme (11.38) for solving (11.32) converges strongly with order v = 1/2. 340
CHAPTER 11. CONTEMPORARY TOPICS The order of convergence of the Euler scheme applied to a stochastic differential equation is one half lower than when the scheme is applied to a correspond ing deterministic equation. Ottinger [433] provided an heuristic argument that attributed this behaviour to the evaluation of the configuration-dependent dif fusion matrix B. When Xj+i is computed using the Euler scheme (11.38), it is assumed that the matrix B is evaluated at the initial configuration, X j , throughout the interval [ti,ti+i] instead of a time-dependent configuration. The time-dependent configurations differ from the initial ones by stochastic terms of i
0(At?+1) during a time interval of size Af j + 1 , and therefore so does B . The concept of strong convergence provides information on the accuracy of individual trajectories. If the main interest is in the accuracy of the averages of certain quantities then a more meaningful measure of accuracy is weak conver gence. An approximation scheme is said to converge weakly with order v > 0 at time T if, for all sufficiently smooth functions g : R d -> ]R with polyno mial growth, there exists a constant Cg > 0, independent of At, such that for sufficiently small At \{g(X(T)) - (g(Xn))\
<
Cg{Aty.
Under certain conditions on the coefficient functions A and B the Euler scheme (11.38) can be shown to be weakly convergent of order v — 1.
11.4.1
The CONNFFESSIT approach
In a viscoelastic flow calculation the conventional approach to determining the stress is to solve a closed form constitutive equation. The idea of using stochas tic simulations of the polymer dynamics as an alternative to solving constitu tive equations for the determination of the polymer stress is due to Laso and Ottinger [344]. This approach combines a finite element solution of the conserva tion equations with stochastic simulation techniques for computing the polymer stress. Laso and Ottinger [344] termed this hybrid method CONNFFESSIT (Calculation of Non-Newtonian Flow: Finite Elements and Stochastic Simula tion Technique). Since the approach combines a description of the microstructure of a polymeric liquid using kinetic theory with a macroscopic description of the flow this type of simulation method is known as a micro-macro approach. This approach allows for greater flexibility in the kinetic theory models that can be studied since it does not require the existence of an equivalent or approx imate closed-form constitutive equation. Therefore, models based on kinetic theory considerations such as the FENE model can be simulated without re sorting to closure approximations that are not universally accurate. Another advantage of the micro-macro approach is that effects such as polydispersity and liydrodynamic interactions can be easily incorporated into the numerical procedure since the motion of individual polymer molecules is simulated [207]. The function of the model polymer 'molecules' is to permit the computation of the polymer stress. They achieve this task through their configurations which contain information about the strain history. Numerical methods based on the micro-macro approach decouple the so lution of the conservation laws from the solution of the stochastic differential equation for the polymer conformations that serves to determine the polymer contribution to the extra-stress tensor. At each time step (for transient flows) or iteration (for steady flows) the micro-macro algorithm proceeds as follows: 341
11.4.
STOCHASTIC DIFFERENTIAL EQUATIONS
1. Using the current approximation to the polymer stress as a source term in the momentum equation the conservation equations are solved using standard finite element methods, for example, to obtain updated approx imations to the velocity and pressure fields. 2. The new velocity field is then used to convect a sufficiently large number of model polymer 'molecules' through the flow domain. This is achieved by integrating the stochastic differential equation associated with the kinetic theory model along particle trajectories. 3. The polymer stress within an element is determined from the configura tions of the polymer molecules in that element. These steps are repeated until convergence is obtained. Let us look at the stochastic part of the calculation for the FENE dumbbell model in more detail. The Euler-Maruyama scheme for solving the stochastic differential equation (11.37) corresponding to the FENE dumbbell model is
Q*+i = Qi+ UiQi - ^F(Q<)) &h + W i N /A^.
(11.40)
The scheme (11.40) may be used to generate a set of N independent trajectories, Q O ) , j = 1 , . . . , TV, by selecting realizations of the random vectors W,. These trajectories may be characterized in time by (QQ , . . . , Q„ ), j = 1 , . . . ,N, where n is the current value of the time counter. At each time t = ti the polymeric stress is computed by taking an ensemble average over individual realizations of the stochastic process Q, i.e. T{U)
1 N = -I+ ( Q < J ) ) .
(11.41)
Since Q ^ , j = 1 , . . . ,N, are independent, the strong law of large numbers guar antees that the arithmetic mean in (11.41) converges to the ensemble average as N -> oo. Higher-order integration schemes based on the predictor-corrector approach may also be used for stochastic differential equations. For example, a secondorder scheme based on the Euler-trapezoidal predictor-corrector pair is Qi+i
=
Qi + A ( Q i , f i ) A t i + B ( Q i , f i ) W i V / A t I ,
Qi+i
=
Q i + - [ A ( Q i + 1 , t i + 1 ) + A ( Q i , i i ) ] At»
+ i [B(Q i + 1 ,t i + 1 ) + B(Qi,ti)]
(n.42)
Wi^/KTi.
Laso and Ottinger [344] described the basic components of the CONNFFESSIT approach by simulating the time development of plane Couette flow using several model polymers such as Hookean and FENE dumbbells. In this onedimensional problem the gap between the two plates was divided into M finite elements and an ensemble of model polymers assigned to each element. Dur ing the transient development of the flow the dumbbells do not migrate across finite element boundaries. Instead they reside in their initial element for all 342
CHAPTER 11. CONTEMPORARY TOPICS time. In this problem, therefore, there is no need to keep track of the spatial locations of the dumbbells in order to allocate them to particular finite elements as a precursor to the evaluation of the polymer stress. In their computations Laso and Ottinger [344] chose M = 40 with O(106) dumbbells per finite ele ment. The time step for the stochastic part of the calculation was chosen to be ^tstoch = A/*/1000, where A# is the characteristic time of the dumbbell. The initial configuration vectors, Q0J , j = 1 , . . . ,N, were chosen from the known equilibrium distribution function [73], a three-dimensional Gaussian distribution with zero mean and unit covariance matrix. Comparisons of stochastic simulations using Hookean dumbbells were made with conventional finite element computations using the equivalent Oldroyd B model. Excellent agreement was found between the two approaches within statistical error. These initial studies using the CONNFFESSIT methodol ogy also revealed important differences in the dynamic behaviour of the FENE and FENE-P models. However, the early implementations of CONNFFESSIT proved to be considerably more expensive in terms of computer memory and CPU time than the corresponding macroscopic computations. The expense is attributable to the very large number of dumbbells that need to be simulated to obtain convergence and also the generation of MNd random numbers at each time step in the simulation of the stochastic differential equation in d dimen sions. A feature of stochastic simulations is the presence of temporal and spatial fluctuations in the computed stress and velocity fields. The temporal fluctua tions arise from the statistical error incurred in approximating ensemble averages such as (11.41), for example, using a limited number, N, of trajectories. These fluctuations may be controlled by increasing the number of trajectories that are simulated. However, as we have already remarked, this approach to statisti cal error reduction is undesirable because of the computational cost. Spatial fluctuations arise through the computation of the divergence of a nonsmooth stress field in the momentum equation. As we shall see these fluctuations may be reduced using correlated local ensembles of model polymers to approximate (11.41). Despite the advantages of the CONNFFESSIT approach in terms of the ki netic theory models that can be simulated there were a number of computational shortcomings in the original implementations of the idea. First, the trajectories of a large number of molecules have to be determined. Secondly, to evaluate the local polymer stress the model polymer molecules must be sorted according to elements. Thirdly, the computed stress may be nonsmooth and this may cause problems when it is differentiated to form the source term in the momentum equation. In subsequent developments of the technique these shortcomings have been overcome to a certain extent. Some of the new ideas will be described in the remainder of this section.
11.4.2
Variance reduction techniques
A means of reducing the statistical error in a stochastic simulation without increasing the number of trajectories that are simulated is to use variance re duction. To compute the polymeric stress in the Hookean dumbbell model at
343
11.4. STOCHASTIC DIFFERENTIAL EQUATIONS some time t we need to determine 1 N (Q(*)2> * ^ £ Q ( j ) ( i ) Q ( j ) ( ' ~ ) .
(11-43)
Let 9 = Var[Q(t)2] denote the variance of Q 2 . Then the statistical error incurred in approximating the ensemble (11.43) is \/Q/N. In a variance reduced simulation the idea is to construct a random variable Y such that {Y) — (Q 2 (£)) and Var(Y) = SY.< 6 . Melchior and Ottinger [397,398] proposed a number of variance reduction methods in the context of the CONNFFESSIT methodology based on impor tance sampling strategies and the idea of control variables. Large variances in stochastic simulations invariably occur when very few realizations make a sig nificant contribution to the mean value (11.43). Since the Gaussian distribution is peaked at the origin the majority of the configurations that are generated are centred there. These configurations do not make a significant contribution to the approximation of the ensemble average. The idea in importance sampling is to introduce a bias that gives greater weight to the realizations that make a substantial contribution to the average. The bias is constructed from an ap proximate solution of a stochastic differential equation for a modified stochastic process Q that gives greater weight to the important realizations [397]. In the second approach, based on control variables [398], the idea is to find a random variable that possesses the same fluctuations as the random variable of inter est, Q, but with a zero mean. When the control variable is subtracted from the original variable then the mean remains unchanged while the fluctuations are reduced. Melchior and Ottinger described two techniques for constructing control variables: direct control and parallel process simulations. Ottinger et al. [434] applied the control variable method to the problem of start-up of weak homogeneous shear flow in order to demonstrate the power of variance reduction techniques within the CONNFFESSIT framework for micromacro simulations. Two stochastic simulations were performed for an ensemble of FENE dumbbells, one assuming equilibrium (no flow) and the other under flowing conditions. Using the same initial ensemble and with the same sequence of random numbers in the simulations the fluctuations in the transient devel opment of the shear stress were virtually indistinguishable. In this application of the control variable technique the equilibrium simulation is suited to the role of parallel process simulation since it has virtually the same fluctuations as the original simulation at small shear-rates and the average of the shear stress vanishes at equilibrium due to symmetry considerations. Subtraction of the equilibrium control variable from the original variable results in substantial variance reduction and reliable results with far fewer dumbbells in the ensemble than would otherwise have been necessary. The construction of an appropriate control variable to be used in a parallel process simulation is not straightforward in general flow situations. An alterna tive approach is to use local ensembles of model polymers that are correlated. The idea is that corresponding model polymers in each material element feel the same Brownian force. More precisely, the same initial ensemble of model poly mers is defined in each material element and corresponding model polymers in each material element are allowed to evolve using the same sequence of random numbers. This approach leads to strong correlations in the stress fluctuations in 344
CHAPTER 11. CONTEMPORARY TOPICS neighbouring material elements. The evaluation of the divergence of the stress in the momentum equation involves the difference between stresses and leads to a cancellation of fluctuations and dramatic variance reduction. The Brownian configuration field method of Hulsen et al. [300,434] and the Lagrangian par ticle method of Halin et al. [264] are examples of variance reduced stochastic simulation methods based on the idea of correlated local ensembles of model polymers. Not only do these techniques reduce the spatial fluctuations in the computed velocity and stress fields but they also require the generation of fewer random numbers. This greatly reduces the computational cost associated with these stochastic simulation techniques. The cost of achieving variance reduction is that unphysical correlations in the random forces are introduced into the sim ulations. For problems in which physical fluctuations are important one must revert to calculations based on uncorrelated Brownian forces even though this is likely to be more expensive.
11.4.3
Lagrangian particle m e t h o d
In the version of the Lagrangian particle method [264] described in Chapter 6 the polymer contribution to the extra-stress tensor was computed by inte grating the constitutive equation along particle trajectories. Halin et al. [264] also described a version of the technique that is applicable to a description of the polymer dynamics using a kinetic theory model. Instead of integrating the constitutive equation along the computed particle trajectories, the stochastic differential equation is integrated by placing a large number of dumbbells at each particle location, i.e. a fixed number of dumbbells, iVj, is associated with each Lagrangian particle. Over each time interval [**,t»+i], the configuration of each dumbbell is determined by solving the stochastic differential equation (11.37) along the particle trajectory {r(t) :U
(11.44)
t=i
where QW is an individual realization of the stochastic process. Variance reduction is achieved through correlated ensembles of dumbbells. The implementation of this idea in the context of the Lagrangian particle method is accomplished by specifying that corresponding dumbbells in each Lagrangian particle have the same initial configuration and evolve using the same sequence of Brownian forces. Stability problems may be experienced when the Euler-Maruyama scheme is applied to the stochastic differential equation (11.37) for the FENE model. In particular, if the time step At used in the stochastic part of the calculation is too large then it is possible for the dimensionless length of a dumbbell to exceed Vb, the dimensionless finite extensibility parameter, which is clearly physically unrealistic. It is possible to circumvent this problem by either reducing the time step or ignoring all dumbbell realizations that lead to a value of Q2 larger than 345
11.4. STOCHASTIC DIFFERENTIAL EQUATIONS a prescribed value [433]. An alternative means of avoiding dumbbell configu rations that violate the maximum extensibility constraint is to use an implicit scheme such as the predictor-corrector scheme (11.42). However, note that the corrector stage of the scheme is a nonlinear equation for Q(r(£; + 1 )). In fact, the length of this vector satisfies a cubic equation. Ottinger [433] has shown that this equation possesses a unique solution between 0 and Vb for an arbitrary length of the vector on the right-hand side of (11.42).
11.4.4
Brownian configuration fields
To overcome the problem of having to track particle trajectories Hulsen et al. [300] developed a method based on the evolution of Brownian configuration fields. In addition, the method provides efficient variance reduction and may be interpreted as an Eulerian implementation of the idea of correlated local ensembles. This method departs from the standard micro-macro approach in that it is based on the evolution of a number of continuous configuration fields rather than the convection of discrete particles specified by their configuration vector. Dumbbell connectors with the same initial configuration and subject to the same random forces throughout the flow domain are combined to form a configuration field. The polymer dynamics is then described by the evolution of an ensemble of configuration fields instead of the evolution of local ensembles of model polymers. The method also provides a smooth spatial representation of the configuration field that can be differentiated to form the source term in the momentum equation. An ensemble of Nf configuration fields Q»(x,i), i = 1 , . . . ,Nf, is intro duced. Initially these fields are spatially uniform and their values are indepen dently sampled from the uniform distribution function of the dumbbell model. The evolution of a configuration field is governed by the stochastic differential equation dQ(x,i) = ( - u ( x , t ) - V Q ( x , t ) + i c ( x , t ) Q ( x , t ) - i F ( Q ) J < f t + «fW(t), (11.45) This equation is similar to (11.37) except that an additional term has been included to account for the convection of the configuration field by the flow. Note that the spatial gradients of the configuration fields are well-defined and smooth since dW(t) only depends on time. This procedure for determining the polymeric contribution to the extra-stress tensor is equivalent to the tracking of individual model polymer molecules. At each point (x, t) an ensemble of configuration vectors is generated that experienced the same history in terms of the kinematics but which underwent different stochastic processes. Hulsen et al. [300] used a discontinuous Galerkin method for solving the stochastic differential equation (11.45) for each configuration field. Let Q de note the appropriate function space for the configuration field. Then the weak formulation of (11.45) is: for each finite element e, find Qj 6 Q such that dQj + (u ■ VQj - KQJ + ^ F ( Q j ) ) At - dW(t),B.) +(n-u(Q^-Q>)dt,R)7«346
=
0,(11.46)
CHAPTER 11. CONTEMPORARY TOPICS V R e Q, for j = 1 , . . . ,Nf, where Q+ is the value of Qj in the neighbouring upstream element or the value imposed at inflow and 7 m is the part of the boundary of e for which u - n < 0 (n is the unit outward normal to the boundary of e). Discontinuous bilinear polynomials were used to approximate Qj in each element. The polymeric contribution to the extra-stress tensor is determined using r = - I + c, where the conformation tensor c = (QQ) is found by projecting the ensemble average onto Q, i.e. find c € Q such that 1 Nf \ c-w^2QiQi,-R\=0, VReQ.
(11-47)
Hulsen et al. [300] have used this technique to simulate the start-up of pla nar flow of an Oldroyd B fluid past a cylinder between two parallel plates. The Brownian configuration field method overcomes the weaknesses of the original methods based on Brownian simulation techniques in that it does not require individual particles to be tracked and sorted according to residency in a partic ular finite element at each time step. Another advantage is that the statistical error in the simulation is governed by the number, Nf, of configuration fields, which is independent of the mesh. Despite the very significant advances that have been made in the develop ment of accurate and efficient techniques based on Brownian dynamics simu lations the computational requirements remain considerably more demanding than macroscopic techniques. However, numerical techniques are now available for predicting the behaviour of kinetic theory models in complex flow situations and these may hold the key to achieving quantitative and qualitative agreement between numerical simulations and experimental observations of complex flows.
11.5
Dynamics of Polymer Melts
It is only fairly recently that sophisticated models have been developed for de scribing the dynamics of polymer melts. In this section we introduce two classes of models, the Doi-Edwards and pom-pom models, that have been developed for linear and branched polymers, respectively.
11.5.1
The Doi-Edwards model
The dynamics of each polymer molecule in a concentrated solution or melt is influenced by the surrounding polymer molecules. The rheological description of these materials is therefore quite distinct from the corresponding situation for dilute polymer solutions in which polymer molecules are sufficiently dispersed so as not to interact with each other. Doi and Edwards [181-183] developed a coarse-grained molecular theory for describing the dynamics of polymer melts in which the polymer configuration changes by reptation. The model is based on the idea that the motion of a molecule perpendicular to its backbone is strongly 347
11.5. DYNAMICS OF POLYMER MELTS reduced by the surrounding polymers, which may be assumed to form a tube. Effectively, the perpendicular motion of a polymer chain is constrained within a tube of given radius generated along its backbone while parallel to the tube a linear polymer chain is free to diffuse. The snake-like motion of the chain inspired de Gennes [167] to refer to this motion as reptation. For this reason the Doi-Edwards model is sometimes referred to as a reptation model. S=l
S=0 Figure 11.3: The Doi-Edwards model polymer. Although the description of the dynamics of polymer melts provided by the Doi-Edwards model is coarse-grained it is still too complicated to submit to analytical investigation. The complication is due to the coupling of different parts of the polymer as a result of retraction of the molecule along the length of the tube. Since a chain segment moves from one part of the tube to another by retraction the orientation of the segment is not determined by the part of the tube it originally occupied but by the part of the tube into which it moves. Therefore, the orientation of a given chain segment is difficult to predict in a general deformation history. This difficulty was circumvented by Doi and Edwards by making the simplification, known as the independent alignment assumption, that each segment deforms independently. With this assumption in place it is possible to study the behaviour of the Doi-Edwards model in simple flows. The Doi-Edwards model provides a description of the configuration pdf of the model polymer in terms of two variables: u and S (see Fig. 11.3). The variable u is a unit vector parallel to a tube segment and therefore provides information about the orientation of the polymer chain at the position S within the chain. The ends of the chain are labelled 5 = 0 and 5 = 1 . The FokkerPlanck equation derived by Doi and Edwards for describing the reptational motion of a polymer subject to the independent alignment assumption is
where K is the transpose of the velocity gradient and the quantity A is related to the so-called reptation or disengagement time r<j. This reptation time is the characteristic time for a polymer chain to escape from its original tube and is given by A = TT2Td. The boundary conditions for the Fokker-Planck equation (11.48) at 5 = 0 and S = 1 [433] are ^(u,0,t) = V(u,M) = ^ ( J ( H - l ) .
(11.49)
In equation (11.48) VKU> &: 0 1S t o be considered as the joint probability density of the two dynamic variables u and S rather than as the probability density of u that depends parametrically on S [433]. One can then use Theorem 11.1 to 348
CHAPTER 11. CONTEMPORARY TOPICS write down a stochastic differential equation that is equivalent to the FokkerPlanck equation (11.48). The evolution equations for the equivalent stochastic processes u(t) and S(t) are decoupled. The stochastic process u(f) satisfies a deterministic differential equation since there is only a first-order derivative with respect to u in (11.48). Therefore, we have ^ ^
= nu(t) - (K : u(t)u(t))u(t)
= (I - u(*)u(i))«u(i).
(11.50)
In this equation the term nu(t) communicates the fact that u(i) follows the flow field while the operator (I - u(t)u(t)) ensures that the property |u(i)| = 1 is preserved. The stochastic process S(t) is a pure diffusion process that describes the reptational motion of the chain. It specifies the part of the polymer chain that is currently resident in the tube segment with the orientation u. The evolution equation for S(t) describing the Brownian motion is given by the stochastic differential equation dS(t) = ^(2/X)dW(t).
(11.51)
The stochastic processes u(f) and S(t) are only coupled through the bound ary conditions (11.49) imposed when the chain escapes from its original tube. When the process reaches one of the boundaries, the tube orientation vector no longer follows the flow but is reset to be a randomly oriented unit vector. Ottinger [433] showed that the boundary conditions (11.49) imply that S = 0 and 5 = 1 constitute reflecting boundaries for the stochastic process S(t). This provides the mechanism for the reptation process. The stochastic differential equation (11.51) may be solved, for example, using either the Euler-Maruyama scheme Si+1
= Si + Wi^(2/X)Ati,
(11.52)
or the predictor-corrector scheme (11.42) described earlier in this chapter. If the value of Si+i obtained by this scheme lies outside the interval [0,1], it is replaced by the value obtained by reflection at the boundary it has just traversed, i.e. Si+i -> - S i + i for St+i < 0, and Sj+i -> 2 - S,+i for S i + 1 > 1. The Doi-Edwards model does not provide a constitutive equation of the differential type for the stress tensor. Instead it is calculated from the tube segment orientation tensor, i.e. r = G(uu),
(11.53)
where G is a constant and the ensemble average is taken over all tube segments present at time t. Micro-macro simulations of polymer melts can be performed in much the same way as for polymer solutions combining a macroscopic treatment of the conservation equations with stochastic techniques for determining the polymer stress. Very few numerical simulations have been performed using the DoiEdwards model in complex flows. We highlight the contribution of van Heel et al. [588] who compared two methods for simulating the Doi-Edwards model with the independent alignment assumption in the start-up of two-dimensional flow 349
11.5. DYNAMICS OF POLYMER MELTS past a cylinder confined between two parallel plates. The techniques, based on the dynamics of the tube segments described above, are known as the Brownian configuration field and deformation gradient field methods. The Brownian configuration field method has already been described for dumbbell models for dilute polymer solutions (see §11.4.4). The application of the technique to the Doi-Edwards model follows a similar procedure except that it is based on the evolution of a number of configuration fields that repre sent the orientation distribution of the tube segments rather than the dumbbell connector vector. Accordingly, consider a set of Nf configuration fields u^, k = 1 , . . . ,Nf. The configuration field approach avoids the need to follow the evolution of the orientation of tube segments associated with a number of dis crete particles, a process that involves the determination of the trajectories of these particles. Each of the Nf configuration fields is a global and continuous representation of the tube segment orientation. These fields evolve according to —— = KUjt - (K : ufcufc)ufc,
(11.54)
for k = 1 , . . . , Nf. Associated with each field is a stochastic process or random walker Sk that satisfies dSk = y/(2/X)dWkl
(11.55)
where Sk is a function of time but not of position. Whenever the random walker Sk is reflected the associated configuration field is removed and replaced by a new and spatially uniform random configuration. At the beginning of the simulation the configuration fields are initialized using a spatially uniform configuration, i.e. u fc (x,0) = u°k, where u° is drawn from the isotropic distribution on the surface of the unit sphere. The initial orientation of the configuration fields is uncorrelated. The associated random walkers Sk are set to independent random numbers from a uniform distribution in the interval [0,1]. The deformation gradient field method described in [588] is based on the in tegral form of the Doi-Edwards model with the independent alignment assump tion. The polymer stress at some time t is given by the integral representation r(x,t)
ft = G I
fi(x,t',t)Q(pc,t',t)dt',
(11.56)
J — oo
where /i(x, t',t) is the Doi-Edwards memory function given by . . .
8 ^
/
(2fc + l) 2 7r 2 (*-*')
(11.57)
and Q(x, t',t) denotes the actual orientation tensor of the tube segments, i.e. Q(x,i',i) = (u(x,t)u(x, *))«.,
(11.58)
where the average is only over those tube segments that were created in the time interval [t', f + dt']. The constant G is a modulus of rigidity and equals 6O770/A 350
CHAPTER 11. CONTEMPORARY TOPICS for the Doi-Edwards model, 770 being the zero shear-rate viscosity. The quantity /u(x, t',t)dt' is the number of tube segments that were created in the time interval [t',t' + dt'] and which have survived up to the present time t, relative to the total number of tube segments present. Thus, the memory function satisfies the property fj,(x,t',t) dt' = 1.
/' J
(11.59)
—c
Although the memory function (11.57) for the Doi-Edwards model is indepen dent of the flow history, the same is not true for more advanced reptation models. To evaluate the stress using (11.56) we need to know the orientation u(x, t) for each tube segment. Since the tube segments rotate affinely it can be shown, by integrating the evolution equation (11.50), that a tube segment that has orientation u(x,£') at a reference time t' has orientation U(X
F(x,f,t)u(x,f) '') = |F(*,*,t)u(x,*)|'
(11 60)
-
at a later time t where F is the deformation gradient tensor. Note, in fact, that putting t' = t and t" = t' in (6.108) we obtain I = F(x,i',i)F(x,t,i'),
(11.61)
F(x,t',«) = F - X ( x , t , t ' ) ,
(11.62)
so that
and is therefore the same as the displacement gradient tensor E(x, t, t1) of §2.5.1. Using a similar technique to that employed in §A.3 the tensor F(x,i',i) may be shown to evolve according to - ^ F ( x , i ' , i ) = #s(x,t)F(x,t',t).
(11.63)
Analogous with the method of deformation fields, described in §6.7.2 for integral constitutive equations of Maxwell type, the idea outlined in [588] was t o intro duce a number, ND, of deformation gradient fields F fc (x,i' fe ,i), k = 1 , . . . ,ND, which measure the deformation gradient at any point in the domain with re spect to the reference time t'k in the past. Note that the evolution of Ffc(x,t'fc,t) only requires knowledge of the velocity at the current time and is determined by solving (11.63) subject to the initial condition F,(x,i' f c ,4) = I. The method follows the same basic steps as for the method of deformation fields described in §6.7.2, with the polymer stress computed using the quadrature rule JVD-1
r(x,t) * G Y, t«*Q*(x,f*,t), fc=0
351
(H.64)
11.5. DYNAMICS OF POLYMER MELTS where Wk, k = 0 , . . . , No — 1, are weights that are calculated at the beginning of the calculation, and the orientation tensors in (11.64) are given by the following ensemble average Q fc (x,f fc> t) = t)u(x,t)> t i .
(11.65)
At each time step the oldest deformation gradient field is destroyed since its associated weight diminishes as the simulation proceeds. The number of fields is kept constant during a simulation by simultaneously creating a new field with a reference time that corresponds to the current time. In their simulations, van Heel et al. [588] replaced the integral over the actual tube segment distribution function by an average over a finite ensemble of tube segments. Each subensemble consists of NT tube segments and the actual tube orientation tensor of each sub-ensemble is approximated using 1 NT Qt(x,*'*.*) « j ^ 5 ] u j ( x , t ) u i ( x , t ) . T
(11.66)
3=1
For this part of the calculation to be efficient and accurate van Heel et al. [588] have shown that, at their time of creation t'k, the NT tube segments should be distributed evenly rather than randomly over the surface of the unit sphere since a random distribution of tube segment orientation vectors requires a con siderably larger sub-ensemble to produce results with a similar statistical error. In their simulations, van Heel et al. [588] chose Nf — 0(2000) in the method of Brownian configuration fields with ND = O(100) and NT = O(20) in the method of deformation gradient fields. Although the methods are theoretically equivalent the latter approach is more efficient computationally than the former. Furthermore, the large fluctuations that are present in the method of Brownian configuration fields are non-existent in the approach based on deformation fields. Therefore, the deformation field approach would seem to commend itself for the simulation of linear polymer melts. There are several modelling defects associated with the Doi-Edwards model. Perhaps the most significant of these is that the model predicts excessive shearthinning in steady shear flows leading to a maximum in the shear stress as a function of shear-rate. Marrucci and his co-workers [302,385,386] have proposed a number of modifications to the Doi-Edwards model. These include the ideas of tube stretch, convective constraint release (CCR) and force balance at entan glement nodes. One of the assumptions in the original Doi-Edwards theory for linear polymers is that after deformation, the polymer chains retract back to their equilibrium length. Marrucci and Grizzuti [386] removed this assumption and introduced the concept of tube stretch into the model by means of a finite retraction time. The relaxation mechanism of CCR, introduced into the DoiEdwards theory by lanniruberto and Marrucci [302], suppresses the tendency of polymers to align with the flow direction at high shear-rates through the con vection of entangled chains. The CCR mechanism is accounted for by means of the following expression - = -
+ £ « : T,
(11.67)
so that, in this model, the relaxation time, r , is a function of the flow. Ap propriate values of the coefficient /3, chosen to ensure that the shear stress is 352
CHAPTER 11. CONTEMPORARY TOPICS an increasing function of shear-rate, may be determined by examining the be haviour of the model in simple shear flow. For slow flows T & Td while for fast flows the CCR mechanism is activated, which has the effect of reducing the re laxation time of the entangled polymers. A complication with (11.67) is that it may provide negative relaxation times for viscoelastic flows since K : r may not be strictly positive. An ad hoc modification of (11.67) that ensures 0 < r < r^ is - = - + ^(K:T+\K:T\). r Td *u
(11.68)
A new model has been proposed by Marrucci et al. [385] that includes CCR and a force balance on the entanglement nodes. The latter results in a modified orientation tensor Q that depends on the Finger tensor, C _ 1 . In particular, we have Q(*,,,,) =
V
/
^ g g L trv/C-i(x,M')
(11.69)
With the orientation tensor thus defined it can be shown, by taking the trace of (11.56), that t r ( r ) = G. The main computational difficulty with models such as this one is that it is necessary to determine the memory function computa tionally since the relaxation time depends on the flow. The memory function is given by
*<*•<'<> = ^ ^ ( - / ' ; < S ) ) '
which, when inserted into (11.56), provides a double integral representation for the polymer stress. An alternative approach to computing the stress that avoids the evaluation of a double integral is to solve the equivalent evolution equation for the memory function: ■PM _ Dt ~
P(*,t,t) T
... {
'
7U
'
subject to the initial condition
«*>*>') = «hrr
(1L72)
The contribution of CCR to the memory function through the relaxation time may be decoupled using the decomposition fi = fj,DEnCCR [615], where \JPE is the memory function corresponding to a simplified version of the Doi-Edwards model in which only the longest relaxation time, Td, is retained, i.e. .DE Td
\
Td
)
and fj,CCR satisfies the evolution problem ^
-
= -HCC*£K
:r,
^^(x,i',t') = l + r d | 353
K
:r.
(11.73)
11.5. DYNAMICS OF POLYMER MELTS Note that pPE has been suitably normalized so that the property (11.59) holds. Marrucci et al. [385] derived a differential approximation to this integral model by considering a step strain deformation. The resulting differential equa tion is written in terms of the square of the polymer stress, i.e. ^ = K T
2
+
r2«T -
2T2(K
: T/G)
- \ [r2 - f r ) .
(11.74)
The solution of this differential equation can also be shown to satisfy t r ( r ) = G. The interest in deriving a differential model is due to computational expediency. Numerical simulations using advanced reptation models such as the model of Marrucci et al. [385] are in their infancy. However, recent numerical simulations using this model have been performed by Wapperom and Keunings [615]. In their method the integral model (11.56) is incorporated into the Lagrangian particle method using the deformation field approach. In the application of this technique the deformation history of the fluid is represented by a finite number of Finger tensor fields C^"1 (x, t, t'k), k = 0 , . . . , No — 1, associated with the reference times t'k. Each Finger tensor has an associated memory function fj,^CR(jc,t'k,t). The evolution, destruction and creation of these fields follows the procedure already outlined for the method of deformation fields. The Finger tensors are used to form the corresponding orientation tensors, using (11.69), that are needed in the evaluation of the polymer stress: ND-1
T(X,<)
= G£
rf;CRM,t)Qk(xJk,t),
wk
(11.75)
fc=0
where wk = [
tiDE(x,t',t)k(t')dt',
(11.76)
J — oo
where the functions (j>k,k = 0,... , N& — 1, are the standard linear finite element basis functions in time. Wapperom and Keunings [615] compared simulations based on the integral model (11.56) and its differential approximation (11.74) and found good agreement for the 4:1:4 contraction/expansion problem for low and high Weissenberg numbers. Finally, attention should be drawn to an alternative model that has been developed by Mead et al. [395] for linear polymers within the independent align ment assumption. This model, known as the Mead-Larson-Doi model, incorpo rates the CCR mechanism and tube stretch.
11.5.2
The pom-pom model
Polymer melts with long-chain branching possess rheological properties that are quite different from those of linear polymers. For example, polymer melts such as low density polyethylene (LDPE) exhibit a strain-hardening behaviour in uniaxial extensional flow that is not observed for linear polymers. McLeish and Larson [393] developed a coarse-grained molecular model, known as the pom pom model, for describing these materials. In order to model strain-hardening behaviour in extension and strain-softening behaviour in shear it was necessary to allow for multiple branching points on the same model polymer molecule. In 354
CHAPTER 11. CONTEMPORARY TOPICS this model the polymer molecules are represented by a backbone segment con necting two identical pom-poms each with q arms (see Fig. 11.4). The backbone segment is confined by a tube formed by other backbone segments. The pom pom model is one of the simplest model polymer molecules that possesses the facility for reproducing the rheological properties of highly-entangled branched polymer melts and represents a major advance in the mathematical description of these materials.
Figure 11.4: The 'pom-pom' model polymer molecule with q = 3. In their original model, McLeish and Larson [393] provided a description of the configuration distribution of the pom-pom molecule in terms of three variables: S, A and sc. The configuration tensor S = (uu), the second moment of the backbone tube orientation, describes the orientation distribution of the backbone segments. As for the Doi-Edwards model the vector u denotes a unit vector parallel to a tube segment. The variable A denotes the stretch of the backbone part of the pom-pom model with maximum relaxation time At,. The tension in the backbone is directly proportional to A. Since this tension cannot exceed the sum of the tensions in the arms, A cannot exceed q, and so the backbone segment is finitely extensible. Once the backbone segment of the molecule is fully extended, i.e. A = q, the arm segments are suddenly drawn into the tube; a process known as retraction. The variable sc denotes the length of the arms drawn into the tube. The sudden withdrawal of the branch points into the tube when A = q produces a discontinuity in the gradient of the extensional viscosity. Blackwell et al. [80] outlined a modified pom-pom model to circumvent this difficulty. In the improved model the sudden onset of retraction with an increasing degree of arm withdrawal as A -> q is smoothed out by a process known as local branch-point displacement. The improved model renders the variable s c redundant. McLeish and Larson [393] provided a rather formidable integral equation for
355
11.5. DYNAMICS OF POLYMER MELTS the evolution of S, viz.
(M)N
S/
f'
dt
'
f
dt
I*
" 1
= Lw^Hw)
(F(x,i',f)u)
F(x,t',t)uF(x,t',t)u^ F(x,t',t)u
(11.77)
Since this double integral representation for the configuration tensor S is too complicated to be used in numerical simulations of complex flows, McLeish and Larson [393] suggested using the following differential approximation to the model to determine an approximation S A to S:
x+
£( A -i , )-°' 8Afc ' ) -s$glr
<1L78)
The tensor A may be regarded as an auxiliary variable in the determination of the configuration tensor. The evolution of the backbone stretch is described by the equation ~
= A S A : Vu - j - (A - 1) e " ^ - 1 ) ,
A < q,
(11.79)
where the parameter v is taken to be 2/q [80] and A„ is the relaxation time for the stretch of the backbone. McLeish and Larson [393] showed that the polymeric contribution to the stress tensor is given by r =
^G
0
^A
2
SA,
(11.80)
where <> /& is the fraction of molecular weight contained in the backbone segment and Go is the plateau modulus obtained from the linear relaxation spectrum. However, Rubio and Wagner [520] showed, by considering the solution of the differential constitutive equation (11.78), given in terms of an integral, i.e. A(x,t)
= ~j_
exp(^)c-1(x,t,OA',
(11-81)
that when the differential approximation of the pom-pom model is used the coefficient in the expression for the polymer stress (11.80) should be 3 rather than ^ . Bishko et al. [79] have used the differential form (11.78) of the constitu tive equation for the original pom-pom model [393] in numerical simulations of branched polymer melts through a planar contraction. This benchmark prob lem was selected in order to predict the qualitative differences in the flow be haviour of branched and linear polyethylene melts that are observed in experi ments. Their algorithm is based on the Lagrangian-Eulerian method of Harlen et al. [269] (see §6.4.5) in which S A , A and sc, the variables that describe the conformation of the pom-pom molecule, are updated by integrating the evolu tion equations in time using a frame that deforms with the fluid. In this frame of reference the equations for A and sc are ordinary differential equations. Explicit schemes are used to determine A and A. However, the stiffness of the ordinary differential equation for sc means that an implicit scheme must be used for this part of the calculation. 356
CHAPTER 11. CONTEMPORARY TOPICS The main finding of their numerical simulations was that the size of the vor tex in the salient corner increases with the degree of branching caused by the increasing resistance to extension of the pom-pom molecule. Thus, their nu merical findings were in qualitative agreement with experiments on commercial branched polymers. Quantitative agreement is elusive because the pom-pom model, based on a mono-disperse description of a polymer melt, is used to model the behaviour of polydisperse polymer melts. The use of a polydisperse multi-mode pom-pom model [303], in which an assortment of different pom-pom molecules are simulated, is likely to provide quantitative as well as qualitative agreement with experiments using branched polymers such as LPDE in contrac tion flows. Confidence in this statement is based on recent work by Inkson et al. [303] in which a polydisperse multi-mode version of the pom-pom model was shown to provide a quantitative fit to transient shear and extensional data for LDPE. A modified pom-pom model has been proposed by Baaijens et al. [28] that remedies the prediction of a zero second normal stress coefficient furnished by the original model. In their 'enhanced' pom-pom model a Giesekus-like evolu tion equation for the configuration tensor is derived to invoke a non-zero second normal stress difference. Numerical calculations using this modified multi-mode pom-pom model have been performed by Baaijens et al. [28] within the frame work of the DEVSS/DG formulation in which a discontinuous approximation for the extra-stress is used. Direct comparisons between the numerical solu tion and the experimentally obtained birefringence data in the inhomogeneous cross-slot geometry showed that the multi-mode pom-pom model was able to predict the complex stress patterns including the fringe patterns near the stag nation point corresponding to steady planar extensional flow. These simulations showed that multi-mode pom-pom models can provide quantitative agreement with experimental observations in complex flows of polymer melts.
11.6
Lattice Boltzmann Methods
To conclude this book we describe another method suitable for large-scale (mas sively parallel) computations of viscoelastic fluids in complex geometries: the lattice Boltzmann (LB) method. This method features a natural mesoscale de scription of the fluid microstructure and offers computational advantages over other micro-macro approaches in terms of mesh generation for complex geome tries and parallelization. There has already been some progress [237,475] in the development of lattice Boltzmann methods for viscoelastic media, which augurs well for the application of the technique to the simulation of viscoelastic fluids through complex geometries. As with the earlier cellular automata (CA), LB methods (see Chopard and Droz [138], for example) are characterized by a lat tice (such as that shown in Fig. 11.5 where each node is labelled by its position r = (hj)) and some rule describing the manner in which particles move along lattice directions from one node to another. At each time step each particle jumps to a neighbouring lattice node and collides with other particles. During these collisions the particles scatter but mass and momentum are conserved. Suppose, for example, that n;(r, t) (= 0 or 1) represents the number of particles entering the node having label r at time t with velocity in the direction of the unit vector Cj. Denoting a time step by r and lattice spacing by A we 357
11.6. LATTICE BOLTZMANN METHODS
Figure 11.5: Hexagonal lattice showing direction vectors c, and node with label r. therefore have u^ = ^ c , and in a CA it is usual to write down some rule of the form n*(r + UjT,t + T) = n i ( r , i ) + n i ( n ) ,
(11.82)
where Qi(n) is a collision term whose precise definition depends upon the col lision rules defined for the particular CA being used. The disadvantage of a CA is that, since it deals with Boolean quantities, it tends to be very noisy and large-scale averages over several time steps may have to be taken in order to obtain macroscopic quantities of interest. In contrast, in LB methods the microdynamics are averaged before the simulation rather than after it. If we define an ensemble average iVj(r,t) = {n,j(r,£)} of the microscopic occupation variables, we see that Nt now represents a probability that a particle enters the node labelled r at time t with velocity u; = -Cj and the LB equation is written down as Ni(r + uiT,t + T) = Ni(r,t) + tl^N).
(11.83)
A popular simplification of (11.83) is to linearize the collision term, fij, about its local equilibrium solution and leads to so-called lattice BGK models: JV,(r + U|T, t + r)-
JV<(r, t) = Sl^N) = i (ivf } (r, t) - JV«(r, *)) .
(11.84)
where £ is a relaxation time. In LB fluids for nodes on a no-slip boundary, the bounce-back condition for an incoming JVj may be used. Macroscopic quantities such as the fluid density p and the momentum pu for a lattice having z different 358
CHAPTER 11. CONTEMPORARY TOPICS vectors c, associated with each node r are defined by z p = ^rriiNi,
z pu = ^TOjiViUj,
i=0
(11.85)
i=0
where m; is a mass (or weighting term) associated with each particle and i = 0 indicates the presence of a rest particle at the node r. The macroscopic governing equations may be derived from the mesoscopic description of the fluid using a multi-scale Chapman-Enskog expansion. LB methods were first used by McNamara and Zanetti [394] and have, since then, been widely used for simulating fluid flows (see the review by Qian et al. [476], for example). Only very recently, however, have attempts been made to simulate viscoelastic flows using LB methods. Qian and Deng [475] introduced an elasticity term into the equilibrium distribution N^"' of (11.84) in their twodimensional thirteen velocity (D2Q13) model for a Kelvin-Voigt-like material. Comparisons performed between the theoretical and computed values of speed and decay rate for transverse and longitudinal waves showed good agreement. Papers by Giraud and co-workers [236,237] have shown how the original LB method can be modified in order to model viscoelastic fluids. Giraud et al. [236] showed that the coupling of two new quasi-conserved non-propagating quanti ties with the viscous tensor in a fifteen velocity (D2Q15) model could create a viscoelastic behaviour. In their second paper [237], the same authors derived a model for a linear Jeffreys-like fluid using an eleven velocity two-dimensional LB method with several relaxation times. Giraud et al. [238] extended the results of their earlier papers to nonlinear viscoelastic fluids by introducing a double set of densities for each non-zero velocity, one of which was chosen to relax quickly, and the other to decay slowly, representing a viscoelastic stress. Wagner et al. [603] used the double density idea of Giraud et al. [238] in an eleven velocity two-dimensional LB simulation in the two-phase flow problem of a bubble rising in an Oldroyd B fluid. The experimentally observed cusp at the trailing end of the bubble was successfully reproduced by the simulations.
11.7
Closing Comments
Advances in the modelling of polymer solutions and melts using coarse-grained molecular models for the polymers, and the breathtaking rate at which the processing power of modern computing systems is increasing mean that compu tational rheology is entering an exciting phase in its development. The use of molecular-based models, it may be hoped, will enable the computational rheologist to move nearer the goal of obtaining numerical solutions that are in quantitative agreement with experiments. Future emphasis in the subject will almost certainly be focussed on the refinement of molecular-based models that incorporate more sophisticated 'physics' and the development of efficient nu merical methods for solving these models that are specifically tailored to the configuration of modern computing architectures.
359
Appendix A
Some Results about Tensors A.l
Existence and Symmetry of the Stress Tensor
Theorem A . l Let T> be some bounded region in R 3 and let s n (x, t) be the stress vector introduced in §2.1.2 and defined throughout D. Then there exists a second-order stress tensor er(x,t) such that throughout T>, (i) sn = n - a, (ii) P~pr- = V ■ er + ph, (Cauchy's equation of motion), (Hi) a is symmetric. Proof. (i) We fix a time t and any point P in the fluid. We now take a fixed frame of reference with origin at P and rectangular Cartesian coordinate axes in the direction of orthonormal base vectors ex, e 2 and e 3 , as shown in Fig. A.l. Then we may write n = n^e; and we assume that nin 2 n3 7^ 0. Let the infinite family of tetrahedra {Te}i>0 be such that for any e > 0 Tt is the region in R 3 bounded by the three coordinate planes through P and the plane whose vector equation is x n = e.
(A.l)
The plane described in (A.l) is just the plane a distance e away from P in the direction of n. We now label the faces of the tetrahedron having outward unit normal — e* by Vi, (i = 1,2,3), and let the corresponding areas of these faces be At. The face defined by (A.l) is labelled Vn and is defined to have area An. Then it is easy to show that e2 e2 €2 e2 M = 7, » A2 = , As = and An = , 2n2ri3 2nin 3 2n\n,2 2nin 2 ri3
361
(A.2)
A.l. EXISTENCE AND SYMMETRY OF THE STRESS TENSOR
Figure A.l: The tetrahedron %■ so that Ai=mAn,
(i = 1,2,3).
(A.3)
We now let V(t) = Tt in (2.11) to obtain
k=1
T,
Vk
Vn
T.
and let c be a positive constant chosen so that, for some finite E > 0, and for all 0 < e < E, the uniform bound (A.5)
£
f s_ e , (x, t)dS + f s n (x, t)dS
k=1
Vk
Qnin2ii3
3
cAne.
(A .6)
Vn
Dividing throughout by An and taking note of the identities in (A.2) and (A.3) we therefore have 3
E fc=1
^
1 / S-efe (X, t)dS + -£- | S „(X, t)dS < 3 « , n n v«.
(A.7)
so that, allowing e —> 0,
- E n*s-«* (°>*) = s»(°' *)• A=l
362
(A.8)
APPENDIX A. SOME RESULTS ABOUT TENSORS By defining the stress tensor a to have (j, «)th component — (s-Cj )± we see that at the point P (s n ). = nj<jji,
(A.9)
as required. (ii) Prom (2.11) we have, on replacing s n by n • er, that / P ^ V(t)
d V
=
f nadS+ S(t)
f phdV, V(t)
■
(A.10)
and, after using the divergence theorem on the first integral on the right-hand side, we get
/ {pl£ ~V-
(A- 11 )
V(t) Since this result holds for arbitrary V we have
P
^ = V'°" + /9b'
(A 12)
"
as required, assuming that the integrand is continuous.
■
(iii) We substitute s„ = n • a into the fcth component of (2.13) to see that / ekjiXjp-~-dV
=
V(t)
/ ekjiXjUiaudS + / ekjiXjpbidV, S(t)
(A.13)
V(t)
where e is the third-order alternating tensor defined by
{
0 if two or more of the suffixes i, j , k are equal, 1 if i, j , k is an even permutation of the numbers 1, 2, 3, - 1 if i, j , k is an odd permutation of the numbers 1 , 2 , 3 . (A.14)
Applying the divergence theorem to the first term on the right-hand side of (A.13) now gives us / ejfcji f xiP-lj7 V(t)
/ ««i*i (rj£
~ ■g-(x3au)
- ^
~ xiPbi jdV = 0,
- /*,) dV - j ekji^aHdV
= 0,
V(t) V(t) f I tkjiPjidV = 0, (A.15) V{t) since the integrand in the first integral in (A.15) is identically zero from (A.12) dXj and -ggdxi = Sji, the Kronecker delta. By the usual arguments, (A.15) implies that f-kiiOji = 0,
(A.16)
and from the definition of c we conclude that u^ = Cji, \fi,j G {1,2,3}, i.e. a is symmetric. ■ 363
A.2 SMALL DISPLACEMENT GRADIENT LIMIT OF yW(x, t, t')
A.2
Small Displacement Gradient Limit of 7^(x, t, f)
With the relative displacement gradient z = x' — x as before we see that 7« ( x , t , f ) = - 7 i i ( x , < ? , t ) = g i + | | ,
(A.17)
and zi — 3J» 3J i IS the ith component of the position vector of a particle at time t' relative to that of the same particle at time t. Then 1 7g (x,M')
=
difattf-Sij, dx
=
_
dx
'k
'k
5i
;
dxi dxj *3' d(zk +xk)d(zk + xk) "»j j dxi dxj dzk dzk dzk dxk dxk dzk dxi dxj dxi dxj dxi dxj dxi dxj
dxi
3
l
dxk dxk dxi dxj
l3
'
dxj'
In the small displacement gradient limit we neglect the first (quadratic) term on the right-hand side of (A. 18), so that J 7i° (x,M')
-> dxi i f + £dxjr = 7«(x,M'),
(A-19)
i.e.
7[01(x,i,f')^7(x,t,*'), as required.
A.3
(A.20) ■
Partial Time Derivative of the Deformation Gradient Tensor F(x,t,/ / )
To demonstrate (2.57) we take the partial time derivative with respect to t' of the (i,j)th component of the deformation gradient tensor F as follows: dFtj dt'
_
d (dx'i\_du'i dt' \dxjj dxj'
9K dx'k dx'k
dxj'
(A.21)
and hence fiF(x,t,t') dt'
= (Vu)/TF(X)fji/)
(A22)
In an entirely analogous way we may also show that dFT(x,t,t') m,
=FT(x,M')(Vu)'.
364
(A.23)
Appendix B
Governing Equations in Orthogonal Curvilinear Coordinates B.l
Differential Relations and Identities
In this section <j> is a scalar function, u, v and w denote vectors and r is a symmetric second-order tensor. (u x v) x w
= V x V<j> = V • (V x v) = V • (
V x (<j>u) = V x (u x v) = V • (u x v) = V ( u • v)
= V ■ (uv) =
(u • w)v — (v • w)u,
o, o,
(B.l) (B.2)
4>V ■ u + u • V<j>,
(B.3) (B.4)
0 V x u + V 0 x u, (v • V ) u - (u • V ) v + u( V • v) --v ( V - u ) , v ■ (V -u) - u - ( V x v),
(B.6) (B.7)
(B.5)
(Vu) • v + (Vv) ■ u, u x (V x v) + v x (V x u) + ( u • V ) v + (v • V ) u , u • V v + v ( V • u),
(B.8) (B.9)
u-Vu
=
iv(|u|2)-ux(Vxu),
(B.10)
V2u
=
V(V •
- V x ( V x u),
(B.ll)
r : Vu
=
V • (TU) -
• (V • T )
(B.12)
VL)
U
365
B.2. DIFFERENTIAL OPERATORS IN ORTHOGONAL COORDINATES
B.2
Differential Operators in Orthogonal Curvi linear Coordinates
Consider the change from rectangular Cartesian coordinates (xi,x2,x3) to gen eral orthogonal curvilinear coordinates (xi,x2,x3), denned by the equations Xi = X i ( x i , X 2 , X 3 ) , X2 = X2(x~i,X2,X3),
X3 =
X3(x1,X2,X3).
(B.13)
Then if r = Yli=i xiei denotes the position vector of a point with coordinates (xi,x2,x3), orthonormal base vectors e 1 ; e 2 and e 3 for the new coordinate system may be defined as 1 dr hi dxi'
1 dr h2 3x2 '
1 dr h3 dx3 '
(B.14)
where h\, h2 and h3 are scale factors, given by
dr dr dr , h2 = , h3 = dx dx dx\ 2 3
hi
(B.15)
Note that _ dr dxi
_
y—^ dr ^-i dxa a=l 3
dxa dxi'
dxa^
= Y,ha OXi
e a
~~Q
^
3
'
JL_ dxi ^
(B.16)
and d _ ^-» dxp dx
i
f^[
dx
i
d
(B.17)
dx
P
From (B.16) and (B.17) we deduce that the del operator V is o
Y^ T~! i=l
B.2.1
r\
O
O
O
-
" _ V* V^ V* % dx i i^ia^ia^iha
r\
r\
r\""-
* d a
^
P "dXi
a dx
P
o
Rectangular coordinates (£i,X2,£3) = hx = 1, hy = 1, hz = 1. _
d
d
366
d
<-*■
r\
V e<* ^a = l ha dx°
(B.18)
(x,y,z) (B.19)
(B.20)
APPENDIX B. GOVERNING EQUATIONS Velocity gradient t e n s o r dux
(Vu),
(Vu)» y =_
UUy
(B.22)
dx ' 9uz dx ' dux dy ' dv,y dy '
(Vu) I Z = (vu)„ = (Vu)w = fX7
(B.21)
(B.23) (B.24) (B.25)
9U
\
(B.26)
*
(Vu)zx
(B.27)
=
92 '
(Vu)ZJ/ = (Vu) 2
-£,
(B.28)
duz dz '
(B.29)
Laplacian of t h e velocity
=
d2ux dx2
_
92Uy
>
~
2
(V2u),
=
(V2u)x ,r,2v KW
v
dx d2uz dx2
+
d2ux dy2
+
d2Uy +
+
d2u, dz2 '
(B.30)
d2Uy
2
+
dy d2uz dy2
+
(B.31)
dz2 ' d2u dz2
(B.32)
Convective derivative of a t e n s o r 9TXX drxx u*xx—dx ^Yu"" v—dy drxy drXy U *-dx-+Uv-dy-
u ■ Vr)xy u-vr)„ u
• Vr)yx
U
WT)m
U
S/T)yZ
— , u-Vr),x
drxx \-uz- dz
(B.33)
9TX + U
> dz ' 9TXZ
(B.34)
ux— 1-«„— httj dx ^"y dy ^ z dz ' dr, dT, dr, = u. yx + Uy yx + uz yx dz ' dx dy drv drv dr, z ~ Ux dx + v dy dz'
(B.35)
-
UX ^
(B.38)
=
drzx Ux—
=
+Uy
+
367
^
drzx Uy—
+UZ
+ Uz
^
drzx — ,
,
(B.36) (B.37)
(B.39)
B.2.
B.2.2
DIFFERENTIAL OPERATORS IN ORTHOGONAL COORDINATES d
(u-Vr)Z!/
=
U x
-^-
(u-Vr)„
=
„ . _
^.,
(B.40)
+« , , _ + « , _ .
(B.41)
+ U y
^
+ U z
Cylindrical polar coordinates (^1,^2, ^3) =
(f,0,z)
hr = 1, he = r, hz = 1.
(B.42)
ee 9 _ ~ 8 ~ 8 V=er—• + r d9+*zdz' or
(B.43)
: r and z but 5er
^
dez
9eg
~89 ~* ' ~8J = — e r , 9
w = 0-
(B.44)
Velocity gradient t e nsor nsor dur dr due dr duz dr 1 duT
(Vu)rr
=
(Vu)r9
=
(Vu)„
=
(Vu)fr
=
rW
(Vu)M
=
(Vu)9z
=
1 due r"d9 1 9u z
(Vu)„
=
(Vu)rf
=
(Vu)«
=
(B.45) (B.46) (B.47) ue r
(B.48)
, Ur
(B.49)
r
(B.50)
)
dur ~8z~' due ~dz~' duz ~dz'
(B.51) (B.52) (B.53)
Laplacian of t h e velocity Q
/I
P
\
1 «2„,
(vu)
p2
5
' = a ^ r ^ H ^ ^ * " * " "■;**"' (B-54) (Vu) + + p. + (B 55)
» =4 ^ H
2
1 S / 5wz\
^
2
1 d uz
368
822i
^
d uz
^
-
APPENDIX B. GOVERNING EQUATIONS Convective derivative of a tensor _ . u-Vr)rr TT \ u-Vr)r9
= =
8TTZ
u • VT)„ u ■
^
u$ 8TTT 8TTT ue. . y-^-+^-^--y(rr»+rer), ue drre drre ue , , — — + uz — + — {T„-T99), U$ 8TTZ
Ur~^— H 8r r
VT)9T
U-Vr)g9
8rrr r-^r + ®Tre , ur— +
u
8T0T
8r + Ur-^r=
ue 8T0T
*
8T
U98T0Z
8T0Z
ue
r 89 us 8TZT r 86
oz 8rzr oz
r ue r
U98TZ9
8TZ0
U0
oz
r
8TZB
N
r
8TZZ
N
+ yiTrS+Ter),
or 8TZT
89
U98TZZ
8r
8TZZ
r 89
(B.59)
U0
8T9Z
8r
B.2.3
T0Z, r
9T r 89 dz — + Uz~^+ — {Trr-m),
or
_.
ue
^ ~ + Uz— 89 oz
Ur-Q-T + y-Qf+Uz-^
. _.
8TTZ
. . (B.57) ,_ _ 0 . (B.58)
dz
(B.60) (B.61)
(B.62) (B.63) (B.64) (B.65)
Spherical polar coordinates (x\, £27 £3) = {r,0,(j)) hr = 1 , he =r, V =
erl
dr
+
h^ = rsin6.
hd+
r 89
*+
(B.66)
d
r sin 6 8<j)'
(B.67)
with ea independent of r but *-ys
8er 89 der
_ ~ =
y^ O^* ~ r8e, ^ 8e^ _ _ _ 8e^ _ e *' 89 ~ 6 r ' 89 ' 8e ^9 8QJ. sinfle^,, —— ^ - = — s i n # e r — cosfiep. d = c o s ^ e ^ , , -8(f>
(B.68)
Velocity gradient tensor (Vu)rr err ^
(Vu)r9
,_
(B.69)
»
(B.70)
~dV' OUs
18U0
=
(B.71)
r¥"7'
=
v
(Vu)^.
—, du
=
N
(VuW ,_
=
Ur
1 8u$ r 89 ' 1 8ur u<j, rsva.9 8 r 369
(B.72) (B.73) (B.74) (B.75)
B.2. DIFFERENTIAL OPERATORS IN ORTHOGONAL COORDINATES
(Vu)„
- ^ ? T - - c o ^ . rsmd ocp r 1 du,j, ur ug cot 9
=
,_ .
Vuw
■a J + — + —
=
rsmff
ocp
r
o
Q
(B.76) (B.77)
• r
Laplacian of the velocity
»2„.
1 2
2
r sin 9 ocp 1 ^ / 2^M
/«2 N
r sin 0 o9 \ d ( \
1 fl2u9 2 r sin 0 d<£2
2 dtt r r 2 30
2
d2u
1
o
2
2
r 2 sin 9 dcp2
2
d .
r sin 0 d . „ '
2cotfl d
dur
r2 sin 9 dcj>
CR7Q^ '
l
2cot6> dug r2 sin 8 dcp'
Convective derivative of a tensor
(u-Vr)rr
=
drrr uedrrr « r —— + 5Z~+ or
r o9
u$ drrr • a~5I rsmd
ocp
ug (T r e+r« r ) r
-^(r^+r0r), / _ v (u-Vr)rg
/
„
=
v
drre u9 drrg u$ drrg ug ur — 1 £3~H r^-^rH far or r 08 rsmd ocp r -^[^e+Tre cot6], r dTrd, , ue dTT
(u-Vr)er
=
(u-VT)gg
=
Y
(B.81)
ur— or - — Ur — or -^(re* r or +^[T9r r
- Tee) (B.82)
(B.83)
^ - +—r^-5-r"" 1 far-Tee) r o9 rsm8 ocp r fo*+i>Pcot0), (B.84) 1 53"+ • Q-5T +—fa<> + T»r) r o9 rsm9 ocp r +T^cobO, (B.85)
1
r
08
rsm.8 dcp
+ fag - TU) cot 9},
370
r (B.86)
APPENDIX B. GOVERNING EQUATIONS /
_
dTd,r
N
, Ug dTSr
or
,
«
r off
0
dTAr
rsmff
Ug
oq>
r
+ — [(Trr ~ TH) + TgT COt 0], /
—
(B.87)
dr^g ue dr^e , uj, dr^g ug or r off rsmff dq> r +—[Tre + (reo - r 0 0 ) cot0],
N
(n [U X7r-\,. VTjW
--
U 9T „ dT W I+ 9r M m U r ^
(B.88)
dT ■+ rus* . n ^ Q(f)
+ — [ ( r r 0 + r 0 r ) + (r 9 0 + r 0 e ) cot $].
B.3
(B.89)
Conservation Equations fdu
DM
_ \
V u
B.3.1
_ = V-T-Vp + ph, = 0.
(B.90) (B.91)
Rectangular coordinates (x,y,z) (dux {-oT
P
dux dux dux\ -b^+U^+U*-oj) =
dTxx 8Tyx dTz. ~dx~ + ~dy~ + ^z
+ Ux
- ~ + Ph, P
+Ux
[dt
duz
.
dx
+Uy
dy
duz
+Uz
duz
dz) ~
duz\
u p(^fdt + ^-^dx + v^7 dy + ^^f dz J
d*-xy
. V-Lyy .
dx
dy
B.3.2
duy t du dy dz
Cylindrical polar coordinates (r, 8, z) dur dt
dur dr Id r dr
ug dur r dff
u\ r
1 dT9r , dTzr r dff dz 371
dlzy
dz
~~ + pby, (B.93) dTLxz dT„. J xz . '~' -yz . dT, IS-LZZ = dx - ^ +dy ~^r +dz dp -^+pbz, (B.94)
Tz
dux dx
(B.92)
duT dz Tgg dp r dr
B.3. CONSERVATION EQUATIONS
(
dug
dug
Ug dug
1 d r2 dr
UrUg
1 dTgg r d6
2
dug \
dTzg dz
Tgr - Trg r
1 dp r dQ (B.97)
fduz
p
{-bT
+ Ur
duz
ugduz
-fr+T-d6-
duz\
+u
*-dz-)
i a , m , I a r ^ ar^ ap i a . . I ai*e B.3.3
C P
( du£dt
duz „
N
.„
Spherical polar coordinates (r, 9,4>) dur
dur
ug dur
U0 dur
uB + u^
~dT+Ur~dV + V^e+ rsinO d<j> r i2 9i r,( 2r mT r>r ) .+ —r-ZJZiiTsrSin0) i d , „ . „, +. i ar* = "a r 9r r sin 0 90 r sin 0 a^> (rgg+T^) dp dr+/*r,
( dug "ST
\
/n
+ W
OT
dug Ug dug r^- + ^~+
ar
r
duj, dr
Us dug UrUg ■ a aj. +
W? ^COt0
r dv rsm.0 d(p r r 1 9 , 3, _ , . 1 a / m . „v . 1 dTA . T —(r- Tr(,) + ^ - r ^ ( T ^ s i n 0 ) + H dr rsm6d6 rsm8 d + - {{TgT -Tr9) -T^cote) - -%+pbg, r r uu ug du^ r 89
u,/, duj, uru^> „„ „ v r sin 6 d<j> r r
1 d , 3_ , . 1 d ,„ r r6 dr r sin 6 dd
. „x .
(B.101)
)
1 S^ r sin 6 d(j)
+1 ({T^r - Tr4>) + T+e cat$) - ^ 4 ^ ^ + /**,
^ | " ( r 2 u r ) + - ^ ( s i i i f t , , ) + - J - ^ = 0. H ar r sin 0 dd r sin S 3 ^ 372
(B.100)
(B.102)
(B.103)
APPENDIX B. GOVERNING EQUATIONS
B.4
Some Important Theorems in Vector and Tensor Calculus
B.4.1
The Reynolds transport theorem
Let V(t) be a material volume and G(x,t) any scalar or vector function. Then 4- f GdV = f (^dt Jv(t) Jv(t) \Dt
B.4.2
+ GV-v\
dV.
(B.104)
)
The divergence theorem
Let the region V be bounded by a simple closed surface S with outward pointing unit normal n. Then if F (vector or tensor) and its divergence are denned throughout V f n ■ FdS = / V • FdV. Js Jv
B.4.3
(B.105)
Stokes's theorem
Let the vector or tensor field F, together with V x F, be defined everywhere on a simple open surface S having a unit normal vector n and correspondingly oriented boundary C traversed in the direction of its unit tangent vector t. Then it-¥ds=
[ n • V x YdS.
373
(B.106)
This page is intentionally left blank
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Index acceleration, 19, 166 Adams-Bashforth method, 152, 153 adaptive strategy, 316-317 area-weighting technique, 121 arm withdrawal, 355 Arnoldi method, 145 Arrhenius law, 81 artificial diffusion, 119, 178 artificial viscosity, 100 asymptotic rate of convergence, 137 atomistic modelling, 328 AVSS formulation, 181, 193-194, 260, 321 AVSS-SI, 277 azimuthal shearing flow, 288
branch points, 355 branched polymer, 354, 357 Brownian configuration fields, 263, 266, 286, 345-347, 352 Brownian dynamics methods, 328 Brownian forces, 345 Brownian motion, 349 bubble stabilization, 129
backbone stretch, 355, 356 Barus law, 81, 296 benchmark problems contraction flows, 15, 186-187, 201246, 356-357 contraction/expansion problem, 221-222, 354 eccentrically rotating cylinder prob lem, 127, 153, 287-303 flow in an undulating tube, 127128 flow past a confined cylinder, 247266, 347 asymmetrically placed, 251-253 flow past a confined sphere, 15, 75, 171, 181, 186-187, 267286 BiCGStab method, 149, 243, 244, 262 biorthogonality, 67 Biot boundary condition, 296 Biot number, 296 birefringent strand, 253, 271, 278, 281 Boger fluids, 38, 74, 290 changes in contraction geometry, 206-209 flow transitions in contractions, 202206 boundedness criterion, 115, 119
411
canonical distribution, 337, 338 capillary rheometer, 202 Carreau model, 78 Cauchy-Green strain tensor, 28, 167 cavitation, 297-300 half-Sommerfeld model, 298 oscillating 7r-film model, 298 viscosity model, 298 CEF equation, 32-33 cell error, 314 cellular automata, 357 CG method, 141-142, 144 change of type, 195-199 characteristic equation, 26 characteristic function, 168 characteristic variables, 61-64 Chebyshev polynomials, 122, 294 closure approximations, 335-338, 341 FENE-L, 44 FENE-LS, 44 FENE-P, 40-43, 336, 338 co-deforming frame, 162 coercivity, 105 collocation method, 123, 153 compatible approximation spaces, 187190 finite elements, 110, 112-113 spectral elements, 130-133 spectral methods, 132 complex shear modulus, 84 compressible fluid dynamics, 155 concentrated polymer solutions changes in contraction geometry, 213-216 flow transitions in contractions, 209212
INDEX
condition number, 142 configuration tensor, 355-357 configurational distribution function, 3 5 36, 329, 348 conformation tensor, 154, 160, 161 conforming approximation space, 104 CONNFFESSIT, 263, 341-344 conservation equations, 341, 371-372 mass, 168 conservation laws angular momentum, 21 energy, 23, 296 linear momentum, 21 mass, 21 continuity equation, 154, 161, 165, 168 control variable method, 344 control variables, 344 convection-diffusion problem, 96, 113, 178 convective constraint release, 171, 352 Couette correction, 221, 239, 245, 307 Couette flow, 16, 159, 289 Crank-Nicolson method, 152, 156 Cross model, 77, 81, 296 cut-off error, 171 cylindrical bipolar coordinates, 293, 294 cylindrical polar coordinates, 368, 371
dilute polymer solutions, 327, 330-335 direct methods, 138-140 Dirichlet boundary condition, 296 displacement gradient tensor, 28, 164, 169, 351 dissipative stresses, 230, 285, 286 divergence theorem, 363, 373 divergent ("bulb") flow, 205, 207, 210, 225 Doi-Edwards model, 47, 347-354 drag, 247, 305 increase in, 271, 284 influence of aspect ratio, 273-274 influence of Theological properties, 270-273 reduction in, 272-274 steady, 274 drag coefficient, 249, 268-274 drag enhancement, 249, 273, 274 drag factor, 274, 277 drift term, 339 dumbbell model, 335 dumbbell models, 33-37, 161, 329 dynamic experiments, 84 eccentricity ratio, 288, 294, 297, 300 EEME formulation, 52, 72, 96, 181, 197-199, 227, 241, 245, 275, 320 effective pressure, 153 effectivity index, 319 elastic instability, 205, 206, 208, 209, 254-257, 275, 278, 279, 289291 elasticity number, 203, 216-218, 225, 234, 278, 282, 290 element residual methods, 306, 397 elementary sampling function, 93 energy equation, 23 enhanced constitutive models, 75 ensemble average, 36, 349, 352, 358 entanglement nodes, 352 equilibrium modulus, 84 error estimates finite element, 105, 107 spectral, 125 viscoelastic flows, 182, 310, 315 error projection property, 105 Euler method, 153, 160, 169, 171, 340 Euler-Maruyama scheme, 340, 342, 345, 349 Eulerian formulation, 160, 164 Eulerian techniques, 170-171
DAVSS formulation, 260 DAVSS-w, 262, 263 DAVSS-G, 260 DAVSS-G/DG, 260, 262 Deborah number, 7, 290, 294 deformation gradient tensor, 27, 167, 351, 364 deformation history, 135, 167, 170, 171, 354 deformation tensor, 164-166 Delaunay triangulation, 163 deviatoric stress tensor, 24 DEVSS formulation, 186, 194-195, 244, 259, 260, 263 DEVSS-w, 262 DEVSS-G, 257, 258, 260, 263 DEVSS-G/DG, 262 DEVSS/DG, 72, 148, 186, 2 4 1 243, 266, 275, 357 DG formulation, 171,182-187, 192, 195, 245, 252, 275, 346 diagonal dominance, 99, 102, 115 die-swell, 167 differential relations and identities, 365 diffusion equation, 329, 337 diffusion tensor, 339
412
INDEX
EVSS formulation, 181, 182, 185-187, 190-194, 198, 241, 242, 244, 245, 251, 252, 275, 283, 321 EVSS/DG, 187 EVSS-G, 159, 193, 258 EVSS/DG, 186, 187, 245 explicit method, 136, 150, 156 exponential sampling, 94 extensional stress growth coefficient, 212 extra-stress tensor, 24
friction coefficient, 34 frontal solver, 138 full-film condition, 298 Galerkin approximation, 105, 106, 123 Galerkin method, 105, 113, 155-157, 180 Gauss-Lobatto points, 123 Gauss-Seidel method, 140 Gaussian elimination, 139 Gaussian process, 339 generalized coordinates, 126 generalized eigenvalue problem, 257, 289 generalized Newtonian fluid, 25-27 generalized Stokes problem, 52, 144, 148, 154, 158 Gibbs phenomenon, 125 Giesekeus expression for the stress ten sor, 37, 41 Giesekus model, 45, 258, 285 global error, 98 GMRES method, 145-149, 182, 186, 243, 275 governing equations boundary conditions, 64 change of type, 61 classification, 59-61 existence and uniqueness of solu tions, 51-54, 310 inflow data, 53 Oldroyd B fluid, 54 type, 55 Green's function technique, 153 growth rate, 57, 289 Gortler number, 255
falling-ball viscometer, 267, 268 FENE model, 40-45, 336, 341-343, 345 FENE-CR model, 43-44, 73, 154, 161, 258 FENE-L model, 338 FENE-LS model, 338 FENE-P model, 258, 266, 336, 343 Finger strain tensor, 28, 164, 170, 353, 354 finite difference methods, 97-104 central differences, 97 upwind differences, 99, 100, 102 finite element basis functions bilinear, 108, 109, 168 biquadratic, 108 constant, 168 cubic Hermite, 294 linear, 105, 354 quadratic, 294 finite element methods, 104-113, 156, 161, 165, 168 h — p version, 108 p version, 129 finite elements Hermitian, 110, 133, 190, 232 quadrilateral, 107, 109 stress subelements, 113, 179, 232 triangular, 107, 162 finite extensibility parameter, 335 finite volume methods, 113-121 cell centre, 114 cell vertex, 114 fluctuation distribution, 119 flux limiter, 119 flux-corrected transport (FCT) scheme, 119 Fokker-Planck equation, 35-36, 329, 335, 338, 348, 349 Fourier modes, 289 Fourier series, 121, 294 free surface flows, 11-13 frequency, 289 friction, 302
Hadamard instability, 56-59 half-speed whirl, 298 Helmholtz problem, 154 Hencky strain, 205 Hessenberg form, 145 hierarchical basis, 108 high Weissenberg number problem, 1718, 49, 173-199, 274 Hookean dumbbell model, 343 Hookean elastic solid, 2 Hopf bifurcation, 205, 257, 289, 290 hybrid schemes finite element/finite volume meth ods, 119, 238 hydrodynamic interactions, 341 implicit method, 135, 151 importance sampling strategies, 344
413
INDEX
incompressibility constraint, 153, 156, 158 incompressible fluid, 21 independent alignment assumption, 348, 350, 354 influence matrix, 154 influence matrix method, 153-155 initial value problem, 160, 165 instability elastic, 205, 206, 208, 209, 255257, 275, 278, 279, 289-291 inertial, 289, 290 numerical, 275 integral models, 45-47 isoparametric mapping, 127, 166 isotropic material, 24 iterative methods, 140-149 nonstationary, 141-149 stationary, 140
linear regression, 86-88 linear stability analysis, 275, 289, 290 linear viscoelastic fluid, 83 Maxwell model, 4-7 lip vortices, 201, 203, 205-207, 209, 212, 213, 215-218, 221, 222, 225, 228, 229, 232, 233, 235, 237, 238, 244 spurious numerical, 237, 238 load, 288, 291 constant, 300 variable, 302 local branch-point displacement, 355 local truncation error, 98 Lodge network model, 47 loss modulus, 84 loss of evolution, 56-59, 195-197 LU decomposition, 138 lubrication, 287 lubrication theory, 291-293 long bearing, 292 short bearing, 292 LUST, 130, 262, 263
Jacobi method, 140 Jacobi polynomials, 122 Jacobian, 108, 109, 126, 137, 147, 168, 170 journal bearing simulator, 297 journal bearings, 287 dynamically loaded, 288, 297-303 statically loaded, 288, 291, 293297, 299
MAC method, 101, 103, 104, 231 mass matrix, 106 master curve, 80 material derivative, 20, 160 material particles, 160, 162, 164, 165, 167 mathematical modelling, 327 maximum principle, 99 Maxwell model, 4-7 MCR model, 180 Mead-Larson-Doi model, 171, 354 memory function, 6, 45, 164, 169, 170, 350, 351, 354 memory integral, 164, 165 mesh distortion, 163, 170 mesh Peclet number, 97, 107 method of deformation fields, 170, 275, 350, 351 method of false transients, 295 method of frozen coefficients, 58 micro-macro approach, 230, 266, 328, 341, 344, 349 minimum oil-film thickness, 288, 300, 302 minimum residual property, 145 MIX1 formulation (Fan et a l ) , 259 mixed formulation, 110 mortar element method, 129 moving-average formulae, 93 MUCM model, 180
K-BKZ model, 47, 167, 273 Kelvin model, 7-8 kinetic theory, 327, 328, 341 Kramers expression for the stress ten sor, 36, 41, 334, 338 Krylov subspace, 141, 145, 149 Lagrange interpolant, 122 Lagrangian formulation, 160, 164, 168 Lagrangian particle method, 163, 345346, 354 Lagrangian techniques, 160-163, 167170 Laguerre polynomials, 165 lattice Boltzmann methods, 328, 357359 Lax-Wen droff methods, 155 LBB condition, 112, 174, 188-189 LDV, 278 least squares problem, 145, 163 Legendre polynomials, 122, 124 limit point, 174-176, 233, 239, 241, 275, 277 linear polymer, 348, 352, 354
414
INDEX
multi-domain techniques, 129 multigrade oils, 287, 296, 297
polydispersity, 341, 357 polymer melts, 347-357 changes in contraction geometry, 213-216 flow transitions in contractions, 209212 polymer stress, 163, 338, 341, 343, 350, 354, 356 dissipative contribution to, 230, 285, 286 pom-pom model, 354-357 enhanced, 357 modified, 355 multi-mode, 357 power law model, 79 preconditioners, 142-144 predictor-corrector methods, 156, 157, 166, 167 pressure correction method, 152, 156 pressure drop, 27, 174, 201, 205, 207209, 218-222, 230, 234, 305 pressure mass matrix, 144 pressure Poisson problem, 103, 151 primary flow, 289 primitive variable formulation, 101, 153 principal invariants, 26 probability density function, 35-36, 329, 348 projection methods, 150-153, 156 protean coordinates, 127 pseudospectral differentiation matrix, 124 pseudospectral methods, 128, 290 P T T model, 45, 159
Navier-Stokes equations, 150, 156 negative wakes, 251 steady flow, 277-281 Neumann boundary condition, 151 neutral stability curve, 289 Newton's method, 136, 137, 147, 170 Newtonian flows, 157 Newtonian fluids, 2, 24-25 non-Newtonian fluids, 8-16 definition of, 8, 77 nonlinear regression, 88-91 nonuniform mesh, 100 normal stress coefficients, 11, 32, 291 normal stress differences, 10, 38, 291, 297, 357 normal stresses, 10-11 numerical diffusion, 100 numerical methods for differential mod els steady flows, 137-149 transient flows, 149-163 numerical methods for integral models background, 163-164 steady flows, 164-167 transient flows, 167-171 Oldroyd 8 constant model, 291 Oldroyd B model, 33, 37-40, 57-58, 137, 154, 157, 159, 161, 251, 290, 343 open syphon effect, 15 order fluids, 16, 28-32 orientation tensor, 352-354 orientation vector, 337 orthogonal curvilinear coordinates, 366371 orthogonal transformations, 145 OS method, 192, 277 overstability, 289
QR factorization, 146 quadrature rules, 109, 165, 170, 351 quantum mechanics, 328 quasilinear system of partial differen tial equations, 54 QUICK scheme, 115-117, 231 random variable, 344 rate-of-deformation tensor, 24 rate-of-strain tensor, 24 recovery, 157 rectangular coordinates, 366, 371 reduced variables, 80 reference time, 170 regularization techniques, 87 relative deformation gradient, 3 relative finite strain tensors, 31, 46 relaxation modulus, 6, 83 relaxation parameter, 141
parameter estimation, 73 partial viscosity, 84 particle tracking, 162 particle trajectory, 160, 161, 164-166 PCG method, 143 penalty term, 161 Petrov-Galerkin method, 157 Picard iteration, 136 pivoting, 139 Poiseuille flow, 16 Poisson equation, 157
415
INDEX
relaxation spectrum continuous, 83 determination of, 83-94 discrete, 84, 89 relaxation time, 5, 352, 355 reptation, 349 reptation model, 348 reptation time, 348 retardation time, 7 retarded motion expansions, 32 retraction, 348, 355 retraction time, 352 Reynolds transport theorem, 21-23, 373 Reynolds's equation, 291-292 rheology, 2 rheometry, 77 rheopectic fluids, 9 rigidity modulus, 3 Rivlin-Ericksen tensors, 28, 29 Rivlin-Sawyers equation, 46^47 rod-climbing effect, 13 Rouse model, 331-335 Runge-Kutta methods, 160, 167
spherical polar coordinates, 369, 372 spring force law Hookean, 37 Peterlin, 40 Warner, 40 spurious modes, 117 stabilization, 96, 161, 179 stabilization techniques, 157 high-order methods, 119, 130 staggered grid, 103, 117 stagnation point, 279 statistical error, 343 stick-slip problem, 113, 180 stiffness matrix, 106 stochastic differential equations, 329, 338-347, 349 stochastic process, 340, 349 stochastic simulation methods, 328 Stokes problem, 131, 147, 157, 294 Stokes's law, 34, 333 Stokes's theorem, 373 storage modulus, 84 strain, 3 strain history, 95, 164-166, 341 strain-hardening, 354 stream function, 61, 101, 127 stream function/vorticity formulation, 101 streamline diffusion method. 111 streamline patterns, 247 streamlines, 165-167 downstream shift of, 248, 249, 251, 255, 258, 270, 278, 280 upstream shift of, 248-250, 252, 278-280 streamtube method, 240 stress boundary layers, 130 stress diffusion, 129, 154 stress history, 3 stress intensity factor, 65 stress tensor, 19, 22, 26, 361 stress vector, 19, 20 stress wake, 259 stress-conformation hysteresis, 208, 230, 285, 336 strong convergence, 340 Sturm-Liouville problem, 121 SU method, 112, 177-182 subdomain residual methods, 306, 307 superconvergence, 142 superconvergent patch recovery, 306 SUPG method, 111, 112,127,130, 177182, 186-187 SUPG-EE, 317-320
sampling localization, 91-94 secondary flow, 290 semi-Lagrangian method, 120 shear stress, 3 shear-rate, 9 shear-thickening fluids, 9, 68, 77 shear-thinning fluids, 9, 14, 31, 33, 41, 45, 68, 77, 78, 159, 185, 202, 203, 211, 212, 219, 220, 223, 226, 228-230, 239, 249-252, 255, 262, 269-274, 277-280, 283, 296, 352 simple shear flow, 9, 10, 25, 38, 41, 352, 353 SIMPLER algorithm, 239, 262 SIMPLEST algorithm, 231 singular value decomposition, 87 singularities, 294 elliptic problems, 65-68 Stokes, 66-68 viscoelastic, 68-71 SMART scheme, 117, 263 Smoluchowski equation, 35 SOR method, 140 sparse matrix, 139 spectral accuracy, 122 spectral element methods, 129-133, 148 spectral methods, 121-129, 294, 299 spectral radius, 140 sphere bouncing, 281, 284
416
INDEX
swirling flows, 13 symbol of a differential operator, 59 Taylor number, 288, 290 Taylor series expansions, 155, 169 Taylor-Couette problem, 287-291 Taylor-Galerkin methods, 155-157, 235, 238, 244, 251 tension-thickening fluids, 82, 206, 216, 219, 224 tension-thinning fluids, 82, 212, 219, 220 test fluids M l , 207, 271 SI, 90-92, 212, 220, 221, 230, 272 theta method, 149, 157-159, 224, 235, 240, 257 thixotropic fluids, 9 time lapse, 165, 171 time splitting schemes, 149 torque, 292, 293, 302 total variation diminishing (TVD) scheme, 119 transfinite mapping, 126, 129 transient extensional flow, 44 transmitted error, 314 Trouton ratio, 14,15, 82, 212, 219, 220, 223, 272, 280 transient, 212 tube orientation vector, 349 tube segment, 348, 349, 352 tube stretch, 171, 352, 354 UCM model, 37, 58, 167,169, 289, 290, 294 integral form, 56, 163 uniaxial extensional flow, 14, 38, 336, 354 upper-convected derivative, 36, 161 Uzawa algorithm, 152
extensional, 14, 39, 41, 82-83, 355 pressure coefficient of, 296 pressure dependence of, 81-82, 296 shear, 9, 32, 76-79, 296 temperature dependence of, 7981, 296 Vogel law, 296 vortex enhancement, 15, 202, 203, 206, 207, 213, 216, 218-220, 222, 224, 226, 228, 230, 232, 240, 241 mechanisms for enhancement, 216225 spurious numerical, 226, 244 stress-relief mechanism, 208, 209, 219, 224 vorticity, 101, 102 vorticity equation, 61 wall correction factor Newtonian, 268, 269 non-Newtonian, 269 viscoelastic, 269, 270, 272, 273 wave number, 57, 58, 289 weak convergence, 341 weak formulation, 169 weights, 171 Weissenberg number, 96, 294 well-posedness continuous problem, 51-54, 56, 59, 64, 89, 103, 150, 174, 310, 311 discrete problem, 110, 112, 113, 132, 173, 188, 190, 191, 310, 311 whirl instability, 298 White-Metzner model, 295, 296 Wiener process, 339 zero-shear-rate viscosity, 77
variance, 344 variance reduction, 343-345 velocity gradient, 157, 159, 167 velocity overshoot, 250, 251, 280-284 velocity vector, 19 Verhoef model, 285 viscoelastic Mach number, 217, 218, 234 viscometric flows, 16 viscometric functions, 10 viscosity, 299 apparent, 9 asymptotic high shear-rate, 296 constant, 4, 38, 295
417
