Numerical Methods for Non-Newtonian Fluids (eBook)

Front Cover 1
Handbook of Numerical Analysis 3
Copyright 5
General Preface 6
Table of Contents 8
Foreword 20
Numerical Methods for Grade-Two Fluid Models: Finite-Element Discretizations and Algorithms 24
Chapter 1. Theoretical Results 28
1.0. Foreword 28
1.1. Introduction and preliminaries 28
1.2. Constitutive and momentum equations 36
1.3. A brief survey of theoretical results 38
1.4. Splitting the two-dimensional problem 46
Chapter 2. Discretizing the Steady Split No-Slip Problem 54
2.1. General centered schemes 54
2.2. Centered schemes: Examples 69
2.3. Centered schemes: Successive approximations 82
2.4. Upwind schemes 89
Chapter 3. Discretizing the Time-Dependent No-Slip Problem 104
3.1. Introduction 104
3.2. Splitting the problem 107
3.3. Fully discrete centered schemes 126
3.4. Fully discrete upwind scheme with discontinuous Galerkin 137
Chapter 4. A Least-Squares Approach for the No-Slip Problem 148
4.1. Least-squares schemes for the steady no-slip problem 148
4.2. An approximate gradient algorithm 158
4.3. Application to the time-dependent problem 163
Chapter 5. The Steady Problem with Tangential Boundary Conditions 166
5.1. Some theoretical results 166
5.2. Centered schemes for the nonhomogeneous problem 172
5.3. Upwind schemes for the nonhomogeneous problem 185
Chapter 6. Numerical Experiments 196
6.1. The steady problem 196
6.2. The time-dependent case 213
References 224
List of Notation 230
The Langevin and Fokker–Planck Equations in Polymer Rheology 234
Chapter 1. Introduction 236
1.1. The Langevin and Fokker–Planck equations 237
1.2. Recent progress in the mathematical analysis and numerical simulation of flows of polymeric fluids 246
1.3. Article summary 250
Chapter 2. Stochastic Simulation Techniques 254
2.1. Introduction to stochastic differential equations 254
2.2. First-generation micro–macro techniques 258
2.3. Second-generation micro–macro techniques 264
2.4. Implicit micro–macro schemes 270
2.5. Stochastic methods for reptation models 272
Chapter 3. Fokker--Planck-Based Numerical Methods 276
3.1. Dilute solutions, locally homogeneous flows 276
3.2. Numerical methods for flows without the local homogeneity assumption 285
3.3. Numerical methods for concentrated solutions 291
3.4. Models with high-dimensional configuration spaces 292
Chapter 4. Numerical Results 306
4.1. Second-generation micro–macro techniques 307
4.2. Fokker–Planck-based numerical methods for locally homogeneous flows of dilute polymeric solutions 312
4.3. Fokker–Planck-based numerical methods for nonhomogeneous flows of dilute polymeric solutions: steady Poiseuille flow in a narrow channel 315
4.4. Fokker–Planck-based numerical methods for melts and concentrated polymeric solutions: Couette flow of a Doi–Edwards fluid 316
4.5. Fokker–Planck-based numerical methods for high-dimensional configuration spaces 317
Acknowledgments 320
Bibliography 322
Viscoelastic Flows with Complex Free Surfaces: Numerical Analysis and Simulation 328
Chapter 1. Modeling of Viscoelastic Flows with Complex Free Surfaces 330
1.1. Macroscopic models 333
1.2. Mesoscopic models 335
1.3. Initial and boundary conditions 340
1.4. Summary 342
Chapter 2. Numerical Analysis of Simplified Problems 344
2.1. Numerical models for viscoelastic flows: a chronological review 344
2.2. Time discretization: an operator splitting scheme 353
2.3. The three fields stokes problem 355
2.4. A simplified Oldroyd-B problem 361
2.5. A simplified Hookean dumbbells problem 364
Chapter 3. Numerical Simulation of Viscoelastic Flows with Complex Free Surfaces 370
3.1. Space discretization: structured cells and finite elements 370
3.2. Extension to mesoscopic models 378
3.3. Numerical results 378
Acknowledgment 385
Bibliography 386
Stable Finite Element Discretizations for Viscoelastic Flow Models 394
1. Introduction 395
2. Flow maps, generalized Lie derivatives, and Riccati equations 396
3. General macroscopic viscoelastic models 402
4. Basic mathematical and physical properties of the models 409
5. Existing numerical methods for viscoelastic fluid models 412
6. A family of Eulerian–Lagrangian finite element methods 417
7. Fast and robust solvers for Stokes-type systems 429
8. Stability analysis and existence of discrete solutions 433
9. Implementation details and numerical experiments 443
10. Concluding remarks 448
Acknowledgments 449
References 450
Positive Definiteness Preserving Approaches for Viscoelastic Flow of Oldroyd-B Fluids: Applications to a Lid-Driven Cavity Flow and a Particulate Flow 456
1. Introduction 456
2. Particulate flow 457
3. Cavity flow 484
Acknowledgments 501
Bibliography 502
On the Numerical Simulation of Viscoplastic Fluid Flow 506
Chapter 1. Viscoplastic Fluid Flow: A Review 510
1. Introduction 510
2. Applications 513
3. Constitutive laws 517
4. Numerical methods 521
5. A brief history of computational viscoplasticity 530
6. Conclusion 534
Chapter 2. Bingham Flow In Cylinders and Cavities 536
7. Introduction and Synopsis 536
8. On the modeling of Bingham viscoplastic flow 537
9. Bingham flow in cylinders: (I) Formulation 539
10. Bingham flow in cylinders: (II) the regularization approach 539
11. Bingham flow in cylinders: (III) variational inequality formulation. The multiplier approach 540
12. Bingham flow in cylinders: (IV) time-discretization of problem (11.1) 546
13. Bingham flow in cylinders: (V) steady flow 551
14. Bingham flow in cylinders: (VI) an augmented Lagrangian approach to the solution of problem (13.7) 563
15. Bingham flow in cylinders: (VII) finite-element approximation 566
16. Bingham flow in cylinders: (VIII) numerical experiments 568
17. Bingham flow in cavities 573
Chapter 3. Numerical Simulation of Nonisothermal, Compressible and Thixotropic Viscoplastic Flow: An Augmented Lagrangian Finite-Volume Approach 592
18. Generalities: synopsis 592
19. Governing equations 597
20. Augmented Lagrangian-based solution algorithms 600
21. A finite-volume scheme 610
22. Solution of the linear systems 624
23. Numerical experiments: wall-driven cavity creeping flow 627
24. Study of nonisothermal incompressible flow in pipelines 635
25. Transient isothermal compressible viscoplastic flow in a pipeline 650
26. Transient isothermal compressible and thixotropic flow in a pipeline: the isothermal restart of waxy crude oil flow 669
27. Additional comments on the augmented Lagrangian/finite-volume methodology: new challenges for waxy crude oil flow 680
Chapter 4. Application of Fictitious Domain Methods to the Numerical Simulation of Viscoplastic Flow 682
28. Introduction. Synopsis 682
29. Steady flow of a Bingham fluid through an eccentric annular cross-section 683
30. Dynamical simulation of particle sedimentation in a Bingham fluid 711
31. Further comments on distributed Lagrange multiplier/fictitious domain methods for Bingham fluid flow 731
Acknowledgments 733
References 734
Modeling, Simulation and Optimization of Electrorheological Fluids 742
1. Introduction 742
2. Mathematical models for electrorheological fluid flows 746
3. Numerical solution of electrorheological fluid flows 776
4. Numerical simulation and optimization of electrorheological devices 790
Acknowledgments 807
Bibliography 808
Index 818
Color plates 826