Fluid Mechanics of Viscoelasticity (eBook)

Front Cover 1
Fluid Mechanics of Viscoelasticity 4
Copyright Page 5
Contents 14
Part 1: General Principles 22
Chapter I. Kinematics of Fluid Flow 22
1 Kinematical Preliminaries 24
2 Path Lines 32
3 The Relative Deformation Gradient and Strain Tensors 44
4 Rivlin-Ericksen Tensors 46
5 Approximations to the Relative Strain Tensor 50
6 Flows such that An-0 for any Odd Integer and Higher 56
7 Viscometric Flows 61
8 Deformation Gradients Equivalent to Exponential Functions 68
9 Motions with Constant Stretch History 72
10 Effects on Local Kinematics from Translation and Rotation 81
11 Zorawski Velocity Fields - Global Effects of Translation and Rotation 85
12 Local Change of Reference Configuration 91
Appendix to Chapter 1: Basic Results from Tensor and Dyadic Analysis 95
Chapter 2. Balance Equations for Smooth and Non-Smooth Regions 106
13 Reynolds' Transport Theorem . 106
14 Conservation of Mass 109
15 Balance of Linear Momentum 110
16 Balance of Angular Momentum 116
17 Balance of Energy 118
18 Other Forms of Equations of Motion 119
19 The Length of a Curve 120
20 Fractal Dimension 122
21 The Box or Minkowski Dimension 128
22 Unit Tangent and Unit External Normal 130
23 Flux across a Fractal Curve: The Divergence Theorem 134
24 Stokes' Theorem 136
25 A Coordinate Free Proof of the Existence of the Stress Tensor 138
26 Stress Singularities and Shape of a Body 141
Part 2: Constitutive Modelling 146
Chapter 3. Formulation of Constitutive Equations--The Simple Fluid 146
27 Constitutive Relations and General Principles of Formulation 147
28 Symmetry Restrictions on Constitutive Equations of Elastic Materials 151
29 Objectivity Restrictions on Constitutive Equations of Elastic Materials 155
30 Integrity Basis for an Elastic Fluid 158
31 Restrictions due to Symmetry: Simple Fluids 159
32 Objectivity Restrictions on Constitutive Equations of Simple Materials 161
33 The Incompressible Simple Fluid 163
Appendix A to Chapter 3: Exploiting Integrity Bases 165
Appendix B to Chapter 3: Constitutive Approximations 169
Chapter 4. Constitutive Equations Derived From Microstructures 176
34 Dilute Polymer Solutions 178
35 Network Theories 240
36 Reptation Models 251
37 Suspension Models 260
Part 3: Analytical and Numerical Techniques 292
Chapter 5. The Shape and Nature o f General Solutions 292
38 Some Consequences of the Isotropy of the Constitutive Functional 293
39 Equations of Motion in Curvilinear Coordinates 301
40 Viscometric Flows 304
41 Rectilinear Motions 323
42 Non-Viscometric Flows 330
43 Dynamically Compatible Unsteady Flows 336
Chapter 6. Simple Models and Complex Phenomena 344
44 Conditions for Identical Velocity Fields in Newtonian and Some Non- Newtonain Fluids 345
45 The Role of the Second Normal Stress Difference in Rectilinear Flows 350
46 Plane Creeping Flows and the Relevance of the First Normal Stress Difference 359
47 Experiments and Theoretical Results to Delineate a Simple Fluid 367
48 Cessation of One History and Continuation with Another History 375
49 Linearised Stability and Bifurcation 380
50 Qualitative Dynamics 406
Chapter 7.Computational Viscoelastic Fluid Dynamics 418
51 Overview 419
52 Finite Difference Method 434
53 Finite Volume Method 443
54 Finite Element Method 468
55 Other Methods 477
56 Epilogue 482
Appendix to Chapter 7 484
Index 494