numerical methods for non-newtonian fluids

Analy-More precisely, the Handbook will cover the basic methods of Numerical Analysis, gathered under the following general headings: – Solution of Equations in Rn, – Finite Difference M

Numerical Methods for Non-Newtonian Fluids Guest Editors: R Glowinski and Jinchao Xu

Handbook of

Numerical Analysis

General Editor:

P.G Ciarlet

Laboratoire Jacques-Louis Lions

Universit´e Pierre et Marie Curie

4 Place Jussieu

75005 PARIS, France

and Department of Mathematics

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11 12 10 9 8 7 6 5 4 3 2 1

General Preface

In the early eighties, when Jacques-Louis Lions and I considered the idea of a Handbook

of Numerical Analysis, we carefully laid out specific objectives, outlined in the followingexcerpts from the “General Preface” which has appeared at the beginning of each of thevolumes published so far:

During the past decades, giant needs for ever more sophisticated ical models and increasingly complex and extensive computer simulations

mathemat-have arisen In this fashion, two indissociable activities, mathematical eling and computer simulation, have gained a major status in all aspects of

mod-science, technology and industry

In order that these two sciences be established on the safest possiblegrounds, mathematical rigor is indispensable For this reason, two compan-

ion sciences, Numerical Analysis and Scientific Software, have emerged as

essential steps for validating the mathematical models and the computersimulations that are based on them

Numerical Analysis is here understood as the part of Mathematics that

describes and analyzes all the numerical schemes that are used on ers; its objective consists in obtaining a clear, precise, and faithful, represen-tation of all the “information” contained in a mathematical model; as such,

comput-it is the natural extension of more classical tools, such as analytic solutions,special transforms, functional analysis, as well as stability and asymptoticanalysis

The various volumes comprising the Handbook of Numerical siswill thoroughly cover all the major aspects of Numerical Analysis, bypresenting accessible and in-depth surveys, which include the most recenttrends

Analy-More precisely, the Handbook will cover the basic methods of Numerical Analysis, gathered under the following general headings:

– Solution of Equations in Rn,

– Finite Difference Methods,

– Finite Element Methods,

– Techniques of Scientific Computing

v

It will also cover the numerical solution of actual problems of rary interest in Applied Mathematics, gathered under the following generalheadings:

contempo-– Numerical Methods for Fluids,

– Numerical Methods for Solids

In retrospect, it can be safely asserted that Volumes I to IX, which wereedited by both of us, fulfilled most of these objectives, thanks to the emi-nence of the authors and the quality of their contributions

After Jacques-Louis Lions’ tragic loss in 2001, it became clear thatVolume IX would be the last one of the type published so far, i.e., edited

by both of us and devoted to some of the general headings defined above

It was then decided, in consultation with the publisher, that each future ume will instead be devoted to a single “specific application” and calledfor this reason a Special Volume “Specific applications” will include math-ematical finance, meteorology, celestial mechanics, computational chem-istry, living systems, electromagnetism, computational mathematics, etc

vol-It is worth noting that the inclusion of such “specific applications” in the

Handbook of Numerical Analysiswas part of our initial project

To ensure the continuity of this enterprise, I will continue to act as theEditor of each Special Volume, whose conception will be jointly coordi-nated and supervised by a Guest Editor

P.G CiarletJuly 2002

Contents of Volume XVI

Special Volume: Numerical Methods for Non-Newtonian Fluids

Numerical Methods for Grade-Two Fluid Models: Finite-Element

Discretizations and Algorithms, V Girault, F Hecht 1The Langevin and Fokker–Planck Equations in Polymer Rheology,

Viscoelastic Flows with Complex Free Surfaces: Numerical Analysis and

Simulation, Andrea Bonito, Philippe Cl´ement, Marco Picasso 305Stable Finite Element Discretizations for Viscoelastic Flow Models,

Positive Definiteness Preserving Approaches for Viscoelastic Flow of

Oldroyd-B Fluids: Applications to a Lid-Driven Cavity Flow and a

Particulate Flow, Tsorng-Whay Pan, Jian Hao, Roland Glowinski 433

On the Numerical Simulation of Viscoplastic Fluid Flow, Roland

Modeling, Simulation and Optimization of Electrorheological Fluids,

vii

Contents of the Handbook

Volume I

Finite Difference Methods (Part 1)

Finite Difference Methods for Linear Parabolic Equations, V Thom´ee 5

Splitting and Alternating Direction Methods, G.I Marchuk 197Solution of Equations in Rn(Part 1)

Volume II

Finite Element Methods (Part 1)

Basic Error Estimates for Elliptic Problems, P.G Ciarlet 17

Local Behavior in Finite Element Methods, L.B Wahlbin 353

Mixed and Hybrid Methods, J.E Roberts, J.-M Thomas 523

Volume III

Techniques of Scientific Computing (Part 1)

Historical Perspective on Interpolation, Approximation and

Approximation and Interpolation Theory, Bl Sendov, A Andreev 223

Numerical Methods for Solids (Part 1)

Numerical Methods for Nonlinear Three-Dimensional Elasticity,

ix

Solution of Equations in Rn(Part 2)

Numerical Solution of Polynomial Equations, Bl Sendov, A Andreev,

Volume IV

Finite Element Methods (Part 2)

Origins, Milestones and Directions of the Finite Element Method –

Automatic Mesh Generation and Finite Element Computation,

Numerical Methods for Solids (Part 2)

Limit Analysis of Collapse States, E Christiansen 193Numerical Methods for Unilateral Problems in Solid Mechanics,

Mathematical Modeling of Rods, L Trabucho, J.M ViaQno 487

Volume V

Techniques of Scientific Computing (Part 2)

Numerical Analysis for Nonlinear and Bifurcation Problems,

Volume VI

Numerical Methods for Solids (Part 3)

Iterative Finite Element Solutions in Nonlinear Solid Mechanics,

Numerical Analysis and Simulation of Plasticity, J.C Simo 183

Numerical Methods for Fluids (Part 1)

NavierStokes Equations: Theory and Approximation,

Volume VII

Solution of Equations in Rn(Part 3)

Gaussian Elimination for the Solution of Linear Systems of Equations,

Techniques of Scientific Computing (Part 3)

The Analysis of Multigrid Methods, J.H Bramble, X Zhang 173

Finite Volume Methods, R Eymard, T Gallou¨et, R Herbin 713

Volume VIII

Solution of Equations in Rn(Part 4)

Computational Methods for Large Eigenvalue Problems,

Techniques of Scientific Computing (Part 4)

Theoretical and Numerical Analysis of Differential-Algebraic Equations,

Numerical Methods for Fluids (Part 2)

Mathematical Modeling and Analysis of Viscoelastic Fluids of the

Oldroyd Kind, E Fern´andez-Cara, F Guill´en, R.R Ortega 543

Volume IX

Numerical Methods for Fluids (Part 3)

Finite Element Methods for Incompressible Viscous Flow, R Glowinski 3

Volume X

Special Volume: Computational Chemistry

Computational Quantum Chemistry: A Primer, E Canc`es,

The Modeling and Simulation of the Liquid Phase, J Tomasi,

An Introduction to First-Principles Simulations of Extended Systems,

Computational Approaches of Relativistic Models in Quantum Chemistry,

J.P Desclaux, J Dolbeault, M.J Esteban, P Indelicato, E S´er´e 453Quantum Monte Carlo Methods for the Solution of the Schr¨odinger

Equation for Molecular Systems, A Aspuru-Guzik, W.A Lester, Jr. 485Linear Scaling Methods for the Solution of Schr¨odinger’s Equation,

Finite Difference Methods for Ab Initio Electronic Structure and Quantum

Transport Calculations of Nanostructures, J.-L Fattebert,

Using Real Space Pseudopotentials for the Electronic Structure Problem,

J.R Chelikowsky, L Kronik, I Vasiliev, M Jain, Y Saad 613Scalable Multiresolution Algorithms for Classical and Quantum Molecular

Dynamics Simulations of Nanosystems, A Nakano, T.J Campbell,

R.K Kalia, S Kodiyalam, S Ogata, F Shimojo, X Su, P Vashishta 639

Simulating Chemical Reactions in Complex Systems, M.J Field 667Biomolecular Conformations Can Be Identified as Metastable Sets

Theory of Intense Laser-Induced Molecular Dissociation: From Simulation

to Control, O Atabek, R Lefebvre, T.T Nguyen-Dang 745Numerical Methods for Molecular Time-Dependent Schr¨odinger

Equations – Bridging the Perturbative to Nonperturbative Regime,

Control of Quantum Dynamics: Concepts, Procedures and Future

Volume XI

Special Volume: Foundations of Computational Mathematics

On the Foundations of Computational Mathematics, B.J.C Baxter,

Geometric Integration and its Applications, C.J Budd, M.D Piggott 35Linear Programming and Condition Numbers under the Real Number

Numerical Solution of Polynomial Systems by Homotopy Continuation

Volume XII

Special Volume: Computational Models for the Human Body

Mathematical Modeling and Numerical Simulation of the Cardiovascular

Computational Methods for Cardiac Electrophysiology, M.E Belik,

Mathematical Analysis, Controllability and Numerical Simulation

of a Simple Model of Avascular Tumor Growth, J.I D´ıaz,

Methods for Modeling and Predicting Mechanical Deformations of the

Breast under External Perturbations, F.S Azar, D.N Metaxas,

Volume XIII

Special Volume: Numerical Methods in Electromagnetics

Introduction to Electromagnetism, W Magnus, W Schoenmaker 3Discretization of Electromagnetic Problems: The “Generalized Finite

Finite-Difference Time-Domain Methods, S.C Hagness, A Taflove,

Discretization of Semiconductor Device Problems (I), F Brezzi,

Discretization of Semiconductor Device Problems (II), A.M Anile,

Modeling and Discretization of Circuit Problems, M G¨unther,

Simulation of EMC Behaviour, A.J.H Wachters, W.H.A Schilders 661

Solution of Linear Systems, O Schenk, H.A van der Vorst 755

Reduced-Order Modeling, Z Bai, P.M Dewilde, R.W Freund 825

Volume XIV

Special Volume: Computational Methods for the Atmosphere

and the Oceans

Finite-Volume Methods in Meteorology, Bennert Machenhauer,

Computational Kernel Algorithms for Fine-Scale, Multiprocess, Longtime

Oceanic Simulations, Alexander F Shchepetkin, James C McWilliams 121Bifurcation Analysis of Ocean, Atmosphere and Climate Models,

Time-Periodic Flows in Geophysical and Classical Fluid Dynamics,

Momentum Maps for Lattice EPDiff, Colin J Cotter, Darryl D Holm 247Numerical Generation of Stochastic Differential Equations in Climate

Large-eddy Simulations for Geophysical Fluid Dynamics, Marcel Lesieur,

Two Examples from Geophysical and Astrophysical Turbulence on

Modeling Disparate Scale Interactions, Pablo Mininni, Annick Pouquet,

Data Assimilation for Geophysical Fluids, Jacques Blum, Franc¸ois-Xavier

Energetic Consistency and Coupling of the Mean and Covariance

Boundary Value Problems for the Inviscid Primitive Equations

in Limited Domains, Antoine Rousseau, Roger M Temam,

Some Mathematical Problems in Geophysical Fluid Dynamics, Madalina

Volume XV

Special Volume: Mathematical Modeling and Numerical

Methods in Finance

Mathematical Models (Part I)

Model Risk in Finance: Some Modeling and Numerical Analysis Issues,

Robust Preferences and Robust Portfolio Choice, Alexander Schied, Hans

Stochastic Portfolio Theory: an Overview, Ioannis Karatzas, Robert

Asymmetric Variance Reduction for Pricing American Options,

Downside and Drawdown Risk Characteristics of Optimal Portfolios

in Continuous Time, Dennis Yang, Minjie Yu, Qiang Zhang 189Investment Performance Measurement Under Asymptotically Linear

Malliavin Calculus for Pure Jump Processes and Applications to Finance,

Computational Methods (Part II)

On the Discrete Time Capital Asset Pricing Model, Alain Bensoussan 299Numerical Approximation by Quantization of Control Problems in

Finance Under Partial Observations, Huyˆen Pham, Marco Corsi,

Recombining Binomial Tree Approximations for Diffusions,

Partial Differential Equations for Option Pricing, Olivier Pironneau,

Advanced Monte Carlo Methods for Barrier and Related Exotic Options,

Applications (Part III)

Anticipative Stochastic Control for L´evy Processes With Application to

Insider Trading, Agn`es Sulem, Arturo Kohatsu-Higa, Bernt Øksendal,

Optimal Quantization for Finance: From Random Vectors to Stochastic

Stochastic Clock and Financial Markets, H´elyette Geman 649Analytical Approximate Solutions to American Barrier and Lookback

Asset Prices With Regime-Switching Variance Gamma Dynamics, Andrew

magnetic slurry: iron powder in a viscous liquid.”

Jack Vance

The Killing Machine

Book Two of

The Demon Princes, Volume One

Tom Doherty Associates Inc., New York, 1997

“Il est, il est, en lieu d’´ecumes et d’eaux vertes, comme aux clairi`eres en feu de la Math´ematique, des v´erit´es plus ombrageuses `a notre approche que l’encolure des bˆetes fabuleuses.”(*)

Few years ago, after the completion of Volume IX of the Handbook of Numerical Analysis,

one of the guest editors of the present volume wondered which topics deserve a dedicatedvolume Among the topics he considered, two in particular stood out: a methodology-

oriented topic, Operator-Splitting, and a thematic topic, Computational Non-Newtonian Fluid Mechanics As operator-splitting methods already had a strong presence in several

volumes of the Handbook of Numerical Analysis (starting with a 266-page article by G.I Marchuk in Volume 1), he focused on the second topic And, although the Handbook had

already covered some problems from non-Newtonian fluid mechanics, analytically and

com-putationally – problems from Viscoelasticity in Fern ´andez-Cara, Guill´en and Ortega [2002] and from Viscoelasticity and Viscoplasticity in Glowinski [2003] – more work

remained to be done Given that the first of these two articles is essentially analytical and thesecond is mostly dedicated to Newtonian flow, there is a strong rationale for a volume thatconcentrates on the numerical simulation of a variety of non-Newtonian fluid flows.There is no doubt that non-Newtonian flows and their numerical simulation have gener-

ated abundant literature, including the Journal of Non-Newtonian Fluid Mechanics (another

Elsevier publication) and books such as those by Bingham [1922], Lodge [1964], Duvautand Lions [1972a,b, 1976], Joseph [1990], Huilgol and Phantien [1997], and Owensand Phillips [2002], as well as additional publications, references to which can be found inthe articles of this volume This abundance of publications can be explained by the fact thatnon-Newtonian fluids occur in many real-life situations, such as the food industry, the oiland gas industry, chemical, civil and mechanical engineering, and the biosciences, to namejust a few Moreover, the mathematical and numerical analyses of non-Newtonian fluid flowmodels provide very challenging problems to partial differential equations specialists andapplied and computational mathematicians alike

Finite elements and finite volumes have been the methods of choice for the numerical

simulation of non-Newtonian fluid flows (see e.g., Marchal and Crochet [1986, 1987],Fortin and Fortin [1989], Fortin and Pierre [1989], El Hadj and Pa Tanguy [1990],Guenette and Fortin [1995], Fortin and Esselaoui [1987], Singh and Leal [1993],Baaijens [1994, 1998], Van Kemenade [1994a]; Van Kemenade and Deville [1994b],Fi´etier and Deville [2003], Xue et al [1998], Singh, Joseph, Hesla, Glowinski andPan [2000], Patankar et al [2000], Pillapakkam and Singh [2001], Chauvieres andOwens [2001], Behr, Arora, Coronado and Pasquali [2005], Coronado, Arora, Behrand Pasquali [2007], Dean, Glowinski and Guidoboni [2007]; see also the many refer-ences within these articles as well as in the articles in this volume)

xix

The purpose of this volume is twofold:

(1) Provide a review of well-known computational methods for the simulation of Newtonian fluid flows, particularly of the viscoelastic and viscoplastic types.(2) Discuss new numerical methods that have proven to be more efficient and more accu-rate than traditional methods

non-Even though the articles in this volume investigate a significant range of applications,

we strongly believe that the methods discussed herein will find applications in many moreareas

This volume is divided into three parts, each of which presents one or more articlesrelevant to a key problem inherent to non-Newtonian flows:

Part I is dedicated to the numerical analysis and simulation of grade-two fluids

V Girault and F Hecht’s article addresses the mathematical and computational

dif-ficulties associated with the grade-two model, thereby providing a good introduction

to the analysis of flows with more complicated constitutive laws

Part II has four articles dedicated to the modeling and mathematical and

numer-ical analysis of viscoelastic flows The article by A Lozinski, R.G Owens and T.N Phillips follows the stochastic approach advocated by Laso and ¨Ottinger[1993] for deriving constitutive laws for polymeric flows The article takes these laws,which connect microscopic stochastic models with macroscopic ones, as the basisfor its approach because they are expected to be more accurate than the more phe-

nomenological ones encountered in the classical literature The article by A Bonito,

Ph Clement and M Picasso addresses the modeling, numerical analysis, and

simu-lation of viscoelastic flows, using models obtained via a two-scale analysis operating

at mesoscopic and macroscopic levels In addition, this article discusses the tion of viscoelastic flow with free surface, a highly nontrivial problem The article by

simula-Y.J Lee , J Xu, and C.S Zhang is mostly methodological and investigates the difficult

problem (at a large Weissenberg number) associated with the advection of the coelastic extra-stress tensor This article also shows that multilevel and parallelizationmethods can significantly speed up viscoelastic calculations Part II concludes with

vis-a vis-article by T.W Pvis-an, J Hvis-ao, vis-and R Glowinski, which investigvis-ates severvis-al methods

that can be used to guarantee the definite positiveness of the viscoelastic extra-stresstensor The article also discusses the numerical simulation of particulate flow for vis-coelastic fluids

Part III has two articles, both of which discuss the simulation of viscoplastic fluid flowswhere the viscoplastic properties are possibly coupled with additional physi-cal properties such as temperature dependence, compressibility, thixotropy, interac-

tion with solid particles, and an electric field The first article, by R Glowinski and

A Wachs, investigates a variety of viscoplastic flows encountered in the oil and gasindustry, such as waxy crude oil flow in pipelines at low temperatures The second

article, by R.H.W Hoppe and W.G Litvinov, is dedicated to the modeling and

simu-lation of electrorheological fluid flows and to the optimal design of devices that usethese fluids

This volume offers investigations, results, and conclusions that will no doubt be useful

to engineers and computational and applied mathematicians who are concerned with the

various aspects of non-Newtonian fluid mechanics Special thanks are due to Gavin Becker,Philippe G Ciarlet, Arjen Sevenster, Lauren Schultz, and Mageswaran Babusivakumar, all

of whom played major roles in bringing this volume into existence

Roland GlowinskiJinchao Xu

Bibliography

Baaijens, F.P.T (1994) Application of low-order Discontinuous Galerkin methods to the analysis of

vis-coelastic flows J Non-Newton Fluid Mech 52 (1), 37–57.

Baaijens, F.P.T (1998) Mixed finite element methods for viscoelastic flow analysis: a review J Newton Fluid Mech.79 (2–3), 361–385.

Non-Bingham, E.C (1922) Fluidity and Plasticity (McGraw-Hill, New York, NY).

Duvaut, G., Lions, J.L (1972) Les In´equations en M´ecanique et en Physique (Dunod, Paris).

Duvaut, G., Lions, J.L (1976) Inequalities in Mechanics and Physics (Springer, Berlin).

El Hadj, M., Tanguy, P.A (1990) A finite element procedure coupled with the method of characteristics

for simulation of viscoelastic fluid flow J Non-Newton Fluid Mech 36, 333–349.

Fi´etier, N., Deville, M.O (2003) Linear stability analysis of time-dependent algorithms with

spec-tral element methods for the simulation of viscoelastic flows J Non-Newton Fluid Mech 115 (2–3),

Non-Fortin, M., Pierre, R (1989) On the convergence of the mixed method of Crochet and Marchal for

viscoelastic flows Comput Methods Appl Mech Eng 73 (3), 341–350.

Guenette, R., Fortin, M (1995) A new mixed finite element method for computing viscoelastic flows.

J Non-Newton Fluid Mech.60, 27–52.

Huilgol, R.R., Phan-Thien, N (1997) Fluid Mechanics of Viscoelasticity (Elsevier, Amsterdam) Joseph, D.D (1990) Fluid Dynamics of Viscoelastic Liquids (Springer, Berlin).

Lodge, A.S (1964) Elastic Liquids (Academic Press, New York, NY).

Marchal, J.M., Crochet, M.J (1986) Hermitian finite elements for calculating viscoelastic flow.

J Non-Newton Fluid Mech.20, 187–207.

Marchal, J.M., Crochet, M.J (1987) A new mixed finite element for calculating viscoelastic flow.

J Non-Newton Fluid Mech.26 (1), 77–114.

Owens, R.G., Phillips, T.N (2002) Computational Rheology (Imperial College Press, London, UK).

Patankar, N.A., Singh, P., Joseph, D.D., Glowinski, R., Pan, T.W (2000) A new formulation of the

distributed Lagrange multiplier/fictitious domain method for particulate flows J Non-Newton Fluid Mech.26 (9), 1509–1524.

Singh, P., Joseph, D.D., Hesla, T.I., Glowinski, R., Pan, T.W (2000) A distributed Lagrange multiplier/

fictitious domain method for viscoelastic particulate flows J Non-Newton Fluid Mech 91 (2–3),

165–188.

Singh, P., Leal, L.G (1993) Finite-element simulation of the start-up problem for a viscoelastic fluid in

an eccentric rotating cylinder geometry using a third-order upwind scheme Theor Comput Fluid Dyn.

5 (2–3), 107–137.

Van Kemenade, V., Deville, M.O (1994a) Application of spectral elements to viscoelastic creeping

flows J Non-Newton Fluid Mech 51 (3), 277–308.

Van Kemenade, V., Deville, M.O (1994b) Spectral elements for viscoelastic flows with change of type.

J Rheol.38 (2), 291–307.

Xue, S.C., Phan-Thien, N., Tanner, R.I (1998) Three dimensional numerical simulations of viscoelastic

flows through planar contractions J Non-Newton Fluid Mech 74 (1–3), 195–245.

Numerical Methods for Grade-Two

Fluid Models: Finite-Element

Discretizations and Algorithms

Vivette Girault

UPMC, Univ Paris 06, UMR 7598, F-75005 Paris, France and

Department of Mathematics, TAMU, College Station TX 77843, USA

1

3.4 Fully discrete upwind scheme with discontinuous Galerkin 114

2

Chapter 5 The Steady Problem with Tangential Boundary Conditions 143

Theoretical Results

1.0 Foreword

The numerical analysis of schemes and algorithms used in discretizing non-Newtonian fluidmodels is a challenging task To this date, there are only very few models for which a com-plete numerical analysis, namely stability, error estimates, and convergence of algorithms, isknown The two-dimensional grade-two fluid model with tangential Dirichlet boundary con-ditions studied in this work is one of them This is made possible by the fact that, owing to thedimension, this model has a formulation that yields good discrete a priori estimates In threedimensions, discrete a priori estimates for the same formulation are not yet known Tangen-tial boundary conditions alone, i.e., with no inflow or outflow, are studied here because theproblem may be ill-posed if complete Dirichlet boundary conditions are prescribed.The material in this work is fairly well self-contained and all prerequisite notions arerecalled It is accessible to advanced graduate students and part of this work was taught

by the first author in an advanced graduate course at the Mathematics Department of theUniversity of Pittsburgh

This work is divided into six chapters In order to present clearly the main ideas, withoutobscuring the discussion by too many technical details, the first four chapters are devoted

to the problem with homogeneous Dirichlet boundary conditions The first chapter presents

a short survey of theoretical results with particular emphasis on the two-dimensional lem Chapter 2 is devoted to the discretization of the steady-state problem, and Chapter 3 isdevoted to the discretization of the time-dependent problem Chapter 4 presents an interest-ing heuristic least-squares scheme and gradient algorithm for the steady and unsteady prob-lems The steady model with tangential Dirichlet boundary conditions is treated in Chapter

prob-5 Numerical experiments are presented in Chapter 6

1.1 Introduction and preliminaries

A grade-two fluid belongs to the class of non-Newtonian fluids of differential type Newtonian fluid models are used to describe the behavior of liquids frequently encountered

Non-in nature and Non-industry, such as many polymeric liquids, biological fluids, foams, and slurries.Unlike water, these liquids are characterized by the fact that they exhibit at least one behav-ior such as shear-thinning or shear-thickenning, stress-relaxation, nonlinear creep, normalstress differences or yielding Grade-two fluids cannot exhibit stress-relaxation, but they candevelop normal stress differences and they can experience creep

5

In a fluid of differential type, be it Newtonian or non-Newtonian, the Cauchy stress sor is determined explicitly by the symmetric part of the velocity gradient and possibly itsvarious higher time derivatives But in contrast to Newtonian fluid models where the consti-tutive relation for the Cauchy stress tensor is a linear function of the symmetric part of thevelocity gradient, in a non-Newtonian fluid model, this constitutive relation is nonlinear.

ten-A grade-two fluid is considered an appropriate model for the motion of a water solution

of polymers, cf Dunn and Rajagopal [1995] Interestingly, its equations can also be preted as a model of turbulence; we refer to the work of Holm, Marsden and Ratiu (cf forinstance [1998a, 1998b]) In the simplest case, its equations of motion have the form

force, such as an L2 force, to mention just these two “simple” questions At least for thesteady two-dimensional problem, we can handle tangential Dirichlet boundary conditions,i.e., with no ingoing or outgoing flow But if there is an ingoing or outgoing flow, the problem

is ill-posed and we still do not know what additional boundary condition must be added tomake the problem well-posed

In contrast, numerical results obtained so far are very scanty We now know how to dothe numerical analysis of some carefully chosen schemes for the steady and time-dependent

problems in dimension d D 2 But up to now, the numerical analysis of schemes that imate this problem in dimension d D 3, be it steady or unsteady, is not resolved The expla-

approx-nation is simple: we lack some discrete a priori estimates, estimates that appear plausible,but for which we have yet no proof, except perhaps for very crude schemes These estimatesare a crucial ingredient in the numerical analysis of several models of non-Newtonian fluids,and this analysis will remain an open question as long as such estimates are not established.For this reason, the present work is dedicated only to numerical methods for the model

in two dimensions

1.1.1 Notation

The following notation will be used in the sequel We state them in dimension d D 3 because

the theoretical problem is, of course, three-dimensional, but the numerical study will be done

mainly in dimension d D 2 Unless otherwise specified, the domains of interest will all be

bounded, connected, and have a boundary@ that is at least C0;1, i.e., Lipschitz-continuous

(cf Grisvard [1985]), and we shall call them Lipschitz-continuous domains We denote by

D / the subspace of functions of C1 / with compact support in Let k D k1; k2; k3/ be

a triple of non-negative integers and set jkj D k C k C k ; we define the partial derivative

Recall the standard Sobolev spaces, for a non-negative integer m and a number r 1(cf Adams [1975] or Neˇcas [1967])

jkjDm

Zj@k vjr dx

35

1=r

;and the norm (for which it is a Banach space)

kvk W m ;r /D

24X

0 k m jvj r W k ;r /

35

1=r

;

with the usual modification when r D 1; we refer to Grisvard [1985], Lions and Magenes [1968] or Adams [1975] for extending this definition to fractional Sobolev spaces When r D

2, this space is the Hilbert space H m / In particular, the scalar product of L2 / is denoted

by ; / These definitions are extended straightforwardly to vector-valued functions, with

the same notation, except for non-Hilbert norms In the case of a vector or tensor u, we set

kuk L r /D

24Z

ju.x/j r d x

35

We shall frequently use Sobolev imbeddings: for a real number p 2 IR, p 1 in dimension

dD 2 or 1 p 6 in dimension d D 3, the space H1 / is imbedded into L p / In

partic-ular, there exists a constant S p (that depends only on p, the dimension and the domain) such

that

When p D 2, this is Poincar´e’s inequality and S2 is Poincar´e’s constant In the case of themaximum norm, the following imbedding holds:

where n D n1; n2; n3/ is the unit normal vector to @ , directed outside , and v D

.v1; v2; v3/ An easy application of Peetre–Tartar’s Theorem (cf Peetre [1966], and tar [1978], or Girault and Raviart [1986]) proves the analog of Sobolev’s imbeddings in

Tar-H1 / for any real number p 1 if d D 2 or 1 p 6 if d D 3:

In particular, for p D 2, the mapping v 7! jvj H1 /is a norm on H1 /, equivalent to the H1

norm and QS2is the analog of Poincar´e’s constant Moreover, the analog of (1.1.5) holds: for

each r > d, there exists a constanteS 1;r, such that

These definitions carry over to d D 2 with one exception: when d D 2, the curl operator is

considered a scalar because it has only one component:

As usual, for handling time-dependent problems, it is convenient to consider functions

defined on a time interval ]a ; b[ with values in a functional space, say X (cf Lions and

Magenes [1968]) More precisely, let k kX denote the norm of X; then for any number r,

1=r

;

with the usual modification if r D 1 It is a Banach space if X is a Banach space and, when

r D 2, it is a Hilbert space if X is a Hilbert space For example, L2.a; bI H m // is a Hilbert

space and, in particular, L2.a; bI L2 // coincides with L2 ]a ; b[/ In addition, we shall

also use spaces with derivatives in time, such as

H1.a; bI X/ D f f 2 L2.]a; b[I X/I @f

@t 2 L2.]a; b[I X/g;

equipped with the graph norm

1.1.2 Properties of the Laplace and Stokes operators

We close this introduction by recalling useful properties of the Laplace and Stokes

equa-tions in dimension d D 2 or d D 3 The presentation is restricted to homogeneous Dirichlet

boundary conditions

Let us start with the Laplace equation with a homogeneous Dirichlet boundary condition

in a bounded Lipschitz domain: For f given in H 1 /, find u in H1

0 / such that

It can be set into the following equivalent variational formulation: Find u in H1 / solving

8v 2 H01 /; r u; r v/ D h f ; vi:

By Lax–Milgram’s Lemma (cf Lax and Milgram [1954]), this problem has one and only

one solution that depends continuously on f Furthermore, increasing the regularity of f , increases up to a certain extent, the regularity of u This is stated in the following theorems;

the first one is proved by Grisvard [1985] and the second one by Dauge [1992]

Theorem 1.1.1 Let be a polygon in IR2 If f belongs to L r / for some r with 1 < r

4=3, then the solution u of (1.1.17) belongs to W2;r / with continuous dependence on f

Theorem 1.1.2 Let be a polyhedron in IR3 with a Lipschitz-continuous boundary If f belongs to H s 1 / for some s with 0 s < 1=2, then the solution u of (1.1.17) belongs

to H sC1 / with continuous dependence on f If f belongs to L3=2 /, then u belongs to

H3=2 / with continuous dependence on f

When f is smoother than in the above statements, the solution is also smoother provided

the inner angles of@ are suitably restricted For instance, it is well known that the nextregularity holds in a convex domain (cf Grisvard [1985])

Theorem 1.1.3 If f belongs to L2 / and the domain is a convex polygon or polyhedron, then the solution u of (1.1.17) belongs to H2 /, with continuous dependence on f

None of the results listed above address the major question: When is the solution in

W1;1? This property has no clear-cut answer (cf Dauge [1992], Kozlov, Maz’ya andRossmann [2000]), but a sufficient condition can be given in view of the Sobolev imbedding

(1.1.4) applied to gradients: for each r > d, there exists a constant C 1;rsuch that

8v 2 W2;r /; kr vk L1 / C 1;r kvk W2;r /: (1.1.18)

Thus, the question can be reformulated as follows : When does a right-hand side f in L r /

for some real number r > d imply that u belongs to W2;r /? The answer is given by

Grisvard [1985] when d D 2 and by Dauge [1992] when d D 3.

Theorem 1.1.4 (1) Let be a convex polygon in IR2 Then there exists a real number

r > 2 depending on the largest inner angle of @ such that for all r with 2 r r , f in

L r / implies that the solution u of (1.1.17) belongs to W2;r / with continuous dependence

on f

(2) In IR3, let be a polyhedron with its largest inner dihedral angle strictly smaller than

2 =3 Then there exists a real number r > 3 depending on the largest inner dihedral angle

of @ such that for all r with 2 r r , f in L r / implies that the solution u of (1.1.17) belongs to W2;r / with continuous dependence on f

Now, we turn to the Stokes problem with homogeneous Dirichlet boundary conditions

in a bounded, connected Lipschitz domain It reads: For f given in H 1 /d and constant

This is equivalent to the inf-sup condition (cf Babuˇska [1973], Brenner and Scott [1994],Brezzi [1974], Brezzi and Fortin [1991], Dur ´an and Muschietti [2001], and Giraultand Raviart [1986], or Ern and Guermond [2004]):

8q 2 L20 /; sup

v2H1 /d

1

jvj H1 /Z

q div v dx kqk L2 /: (1.1.26)

The regularity properties of the solution of the Stokes problem are fairly similar to those

of the Laplace equation The following result is now well known (cf Kellog and Osborn

[1976], or Grisvard [1985], if d D 2, and Dauge [1989], if d D 3).

Theorem 1.1.5 If f belongs to L2 /d and the domain is a convex polygon or polyhedron, then the solution .u; p/ of (1.1.19)–(1.1.20) belongs to H2 /d H1 /, with continuous dependence on f

Of course when is convex, we obtain by interpolation for 0 s 1, that.u; p/ belongs

to H sC1 /d H s /, with continuous dependence on f, whenever f belongs to H s 1 /d

But for small s, the restrictions on the angles of the domain can be substantially relaxed.

Indeed, without restriction on the angles of@ , the following theorems hold, analogous toTheorems 1.1.1 and 1.1.2; the first one can be found in Grisvard [1985] and the second one

in Dauge [1989]

Theorem 1.1.6 Let be a polygon in IR2 If f belongs to L r /2 for some r with1<

r 4=3, then the solution u; p/ of (1.1.19)–(1.1.20) belongs to W2;r /2 W1;r / with continuous dependence on f

Theorem 1.1.7 Let be a polyhedron in IR3 with a Lipschitz-continuous boundary If

f belongs to H s 1 /3for some s with0 s < 1=2, then the solution u; p/ of (1.1.19)–

(1.1.20) belongs to H sC1 /3 H s / with continuous dependence on f.

The result for the borderline case s D 1=2, which extends a result of Fabes, Kenig and

Verchotta [1988], is due to Dauge and Costabel [2000] and can be found in Giraultand Lions [2001a]:

Theorem 1.1.8 Let be a polyhedron in IR3with a Lipschitz-continuous boundary If f

belongs to L3=2 /3, then the solution .u; p/ of (1.1.19)–(1.1.20) belongs to H3=2 /3

H1=2 / with continuous dependence on f.

The case when the velocity is in W1;1 will play an important part in the sequel.

Again, we formulate it as follows: When does a right-hand side f in L r /d for some

real number r > d imply that u belongs to W2;r /d

? The answer is given by Grisvard

[1985] when d D 2 and by Dauge [1989], Kozlov, Maz’ya and Rossmann [2000] when

dD 3

Theorem 1.1.9 (1) Let be a convex polygon in IR2 Then there exists a real number

r > 2 depending on the largest inner angle of @ such that for all r with 2 r r , f in

L r /2implies that the solution .u; p/ of (1.1.19)–(1.1.20) belongs to W2;r /2 W1;r /

with continuous dependence on f

(2) In IR3, let be a polyhedron with its largest inner dihedral angle strictly smaller than

2 =3 Then there exists a real number r > 3 depending on the largest inner dihedral angle

of @ such that for all r with 2 r r , f in L r /3 implies that the solution .u; p/ of

(1.1.19)–(1.1.20) belongs to W2;r /3 W1;r / with continuous dependence on f.

1.2 Constitutive and momentum equations

There are several references on the mechanics of grade-two fluid models; for example,the reader can refer to Truesdell and Rajagopal [2000], Dunn and Fosdick [1974], orTruesdell and Noll [1975] Before writing the constitutive equation of a grade-two fluid,let us recall the Rivlin–Ericksen tensors They are defined recursively by (cf Rivlin andEricksen [1955]):

the equation of a differential fluid because T is defined explicitly in terms of A1 and A2

Furthermore, the presence of A21and of the products in the definition of A2makes this tion nonlinear To compare, the constitutive relation for the Navier–Stokes fluid model is thelinear relation

rela-T D rela-T.u; / D I C A1: (1.2.5)

We observe that when the normal stress moduli vanish, (1.2.4) and (1.2.5) coincide

When substituting (1.2.4) into the balance of linear momentum:

Remark 1.2.1 The condition 0 has been (and is still) a source of rough controversy;

we refer to Dunn and Rajagopal [1995] for an interesting discussion on this subject Apartfrom mechanical considerations, mathematically speaking, the term @

@t 1 u in the

left-hand side of the momentum equation makes the model unstable when is negative (seeRemark 1.3.3), and therefore, we shall not study this case here

1.3 A brief survey of theoretical results

The results presented here are for homogeneous boundary conditions The theory ofthe steady two-dimensional problem with tangential boundary conditions is discussed inChapter 5

1.3.1 The no-slip three-dimensional problem

Let [0; T] be an interval of time, with T > 0, and let be a bounded, connected domain of

IR3, with a Lipschitz-continuous boundary@ Consider the problem: Find a velocity vector

u and a scalar pressure p, solution of

u 0/ D u0in with div u0D 0 in and u0D 0 on @ : (1.3.4)Remark 1.3.1 Considering that (1.3.1) involves a third derivative, we can ask the ques-tion: does (1.3.3) impose enough boundary conditions to determine the solution of (1.3.1)–(1.3.4)? We shall see further on that the answer is “yes.” More generally, Girault and Scott

[1999] prove that in dimension d D 2, the answer is also “yes” for the steady-state problem

in the case when (1.3.3) is replaced by a tangential Dirichlet condition:

u D g on @ ]0; T[ with g n D 0; (1.3.5)

see Section 5.1.2 It is likely that, with adequate conditions on g, this result extends to

the evolution equation (1.3.1)–(1.3.4) But when the boundary values are not tangential,there are examples where the problem is ill-posed, cf Rajagopal [1995], Rajagopal andKaloni [1989], and Remarks 1.3.4, 6.2.1, parts (2) and (3)

Problem (1.3.1)–(1.3.4) is difficult because its nonlinear term involves a third-orderderivative, whereas its elliptic part only comes from a Laplace operator; for this reason,

it behaves mostly as a hyperbolic problem From 1993 onward, many publications havebeen devoted to this problem, but by far the best proof of existence, due to Cioranescu andOuazar, goes back to more than 25 years ago (1981) and is found in the thesis of Ouazar[1981]; it was published later by Cioranescu and Ouazar [1984a, 1984b] The reader canalso refer to Cioranescu, Girault, Glowinski and Scott [1999] and to Cioranescu andGirault [1997]

Here is a brief description of the construction of solutions by Cioranescu and Ouazar.Some of its ideas will be very helpful for discretizing the problem First, we make preciseassumptions on the data and the domain: simply-connected with boundary of classC3 ;1, f

in L2.0; TI H1 /3/ and u0in H3 /3 Formally, observe first that (1.3.1) yields the energyequality:

dt ju.t/j2H1 /C ju.t/j2H1 /D f t/; u.t//: (1.3.6)

This equality shows in particular that, if a solution u exists, then it is unconditionally

bounded in L1.0; TI H1 /3/ by the data f Now, set

This choice is crucial because Cioranescu and Ouazar prove that if a function u 2 V

satis-fies curl.u 1 u/ 2 L2 /3and is simply connected, then u 2 H3 /3\ V and there exists a constant C such that

Next, take formally the curl of (1.3.1); this gives a transport equation, (that we multiply here

By applying the Sobolev bound (1.1.18) to kr u.t/k L1 /and by using (1.3.8) and (1.3.7),

we obtain with another constant C

kr u.t/k L1 / C kz.t/k L2 /:

Then by substituting this bound into the left-hand side of (1.3.11), and by substituting the

estimate deduced from (1.3.6) to bound kcurl u.t/k L2 / in its right-hand side, we find that

kz.t/k2L2 /is bounded by the solution of a Riccati differential equation on the time interval[0; T ], for some T > 0, T T This shows that, if a solution u exists, then it is bounded

in L1.0; T I H3 /3/, see Coddington and Levinson [1955] Finally, on multiplying mally (1.3.1) by@u=@t and using the previous bound for u, we infer that @u=@t is also

for-bounded in L2.0; T I H1 /3/

These bounds only hold provided a solution exists, but constructing a solution by makinguse of (1.3.1), (1.3.7), and (1.3.9) is very difficult because these three equations are redun-dant and no fixed-point can use all three at the same time The originality and power ofconstruction by Cioranescu and Ouazar lie in that they did use all three equations Theiridea consists in discretizing (1.3.1) by a Galerkin method with the basis of eigenfunctions

of the operator curl curl.u 1 u/ This special basis has the effect that, on multiplying

the ith equation that discretizes (1.3.1) by the eigenvalue i and on summing over i, we

derive a discrete version of the transport equation (1.3.9) This allows to recover (1.3.11)

in the discrete case Thus, we construct a discrete solution u mthat is bounded uniformly in

L1.0; T I H3 /3/, with @u m =@t also bounded uniformly in L2.0; T I H1 /3/ Note thatall the above steps (which were hitherto formal), and in particular the delicate Green’s for-mula (1.3.10), are justified because the basis functions are sufficiently smooth Furthermore,passing to the limit is standard because this limit is only taken in the discrete version of(1.3.1) The above bounds allow us to pass to the limit in the discrete equations and provelocal existence in time of a solution Global existence in time for suitably restricted datacan also be established, by taking better advantage of the small damping effect of the vis-cous term 1 u The precise conditions are somewhat technical, and we refer the reader to

Cioranescu and Girault [1997] The next theorem summarizes the local existence resultthat was obtained by Cioranescu and Ouazar [1984a, 1984b]

Theorem 1.3.2 Let be simply connected with boundary of class C3 ;1 Then, for any

force f in L2.0; TI H1 /3/, any initial velocity u0in H3 /3and any parameters > 0 and

> 0, there exists a time T > 0, such that problem (1.3.1)–(1.3.4) has a unique solution

.u; p/ in L1.0; T I H3 /3/ L2.0; T I L2

0 // with @u=@t in L2.0; T I H1 /3/.

Regarding the regularity hypotheses on the data, it follows from (1.3.11) that curl f 2

L2 /3is sufficient (instead of f in H1 /3) Furthermore, finding u in H3 /3is not essary; if we accept solutions that are less smooth, we can lower the regularity of@ Indeed,

nec-(1.3.11) only requires u in W1;1 /3 Thus applying Sobolev’s imbedding (1.1.18), it

suf-fices that u 2 W2;r /3for some r > 3 This is also sufficient for estimating k@u=@tk L2 /.

As (1.3.8) is based on the regularity of a Stokes problem with data in H1 /3, it can

be replaced by a weaker statement with data in L r /3, and Theorem 1.1.9 in the case

dD 3 implies that it suffices that the largest inner dihedral angle of @ be strictly smallerthan 2 =3 Finally, Bernard [1998] and Bernard [1999] prove that can be multiply-connected, if@ is of class C2;1 This makes use of the material in Amrouche, Bernardi,Dauge and Girault [1998]

Remark 1.3.3 The importance of the positivity of is made clear by the energyequality (1.3.6)