THEORETICAL AND APPLIED RHEOLOGY
1 Tensor Notation 1 11 Cartesian Frame of Reference 1 111 Position Vector 3 12 FrameRotation 3 121 Orthogonal Matrix 3 122 RotationMatrix 4 13 Tensors 4 131 Zero-OrderTensors 4 132 First-OrderTensor 4 133 Outer Products 5 134 Second-Order Tensors 5 135 Third-OrderTensors 6 136 TransposeOperation 6 137 Decomposition 7 138 SomeCommonVectors 7 139 Gradient of a Scalar 7 1310 SomeCommonTensors 8 14 Tensor and Linear Vector Function 9 141 Claim 9 142 Dyadic Notation 10 15 TensorOperations 10 151 Substitution 10 152 Contraction 11 153 Transpose 11 154 Products of Two Second-Order Tensors 11 16 Invariants 12 161 Invariantof aVector 12 162 Invariantsof aTensor 12 17 Decompositions 13 171 Eigenvalue andEigenvector 13 172 Square Root Theorem 14 173 Polar Decomposition Theorem 14 174 Cayley–Hamilton Theorem 14 18 DerivativeOperations 15 181 Derivative of det(A) 16 182 Derivative of tr(A) 16 183 Derivative of tr(A2) 16 19 Gradientof aField 17 191 Field 17 192 CartesianFrame 17 193 Non-CartesianFrames 19 194 Spherical Coordinates 21 110 Integral Theorems 22 1101 Gauss Divergence Theorem 22 1102 Stokes Curl Theorem 23 1103 LeibnizFormula 23 111 Problems 24 2 Rheological Properties 29 21 Viscosity 29 211 Shear-Rate Dependent Viscosity 29 22 Normal Stress Differences 31 221 Weissenberg Rod-Climbing Effect 32 222 DieSwell 32 223 Flow Down an Inclined Channel 33 23 Transient Responses 34 231 Small Strain Oscillatory Flow 34 232 Stress Overshoot 35 233 StressRelaxation 36 234 Relaxation Modulus 36 235 Recoil 36 24 Elongational Flows 37 241 Elongational Viscosity 37 25 Viscoelastic Instabilities 38 3 Kinematics and Equations of Balance 41 31 Kinematics 41 311 Reference Configuration 41 312 Velocity and Acceleration Fields 42 313 MaterialDerivative 43 32 DeformationGradient andStrainTensors 43 321 DeformationGradient 43 322 Cauchy–Green Strain Tensor 44 323 RelativeStrainTensors 45 324 PathLines 46 325 Oscillatory Shear Flow 47 33 Rivlin–EricksenTensors 48 34 SmallStrain 50 35 Equations of Balance 51 351 Reynolds Transport Theorem 51 352 ConservationofMass 53 353 ConservationofMomentum 54 354 ConservationofEnergy 58 36 Problems 59 4 Constitutive Equation: General Principles 61 41 Some Well-Known Constitutive Equations 61 411 PerfectGas 61 412 InviscidFluid 61 413 Fourier’sLaw 62 414 HookeanSolid 63 415 NewtonianFluid 64 416 Non-NewtonianFluid 65 42 Weissenberg and Deborah Numbers 65 421 Deborah Number 66 422 Weissenberg Number 66 43 Some Guidelines in Constitutive Modeling 67 431 Oldroyd Approach 67 432 PrincipleofMaterialObjectivity 68 433 Objectivityof theStress 68 434 Frame Indifference 69 435 Principle of Local Action 71 436 PrincipleofDeterminism 71 44 IntegrityBases 72 441 Isotropic Scalar-Valued Functions 72 442 Isotropic Vector-Valued Functions 73 443 Isotropic Tensor-Valued Functions 73 45 SymmetryRestrictions 74 451 Unimodular Matrix 74 452 SymmetryGroup 75 453 IsotropicMaterials 75 46 IsotropicElasticMaterials 76 47 TheSimpleMaterial 77 471 SimpleFluid 78 472 Incompressible Simple Fluid 78 473 FadingMemory 79 48 OrderFluids 79 481 Unsteady Motion 79 482 Velocity Field in a Second-Order Fluid 81 49 Green–Rivlin Expansion 82 410 Problems 83 5 Inelastic Models and Linear Viscoelasticity 89 51 InelasticFluids 89 511 Carreau Model 90 512 Power-Law Model 90 52 Linear Viscoelasticity 92 521 Simple Shear Flow 94 522 StepStrain 96 523 Relaxation Spectrum 97 53 Correspondence Principle 98 531 Quasi-staticApproximation 98 532 CircularCouetteFlow 99 54 Mechanical Analogs 100 55 Problems 102 6 Steady Viscometric Flows 105 61 Kinematics 105 611 Steady Parallel Flow 106 612 Rectilinear Flow 107 613 Axial Fanned Flow 107 614 HelicalFlow 108 615 HelicoidalFlow 108 62 Stresses in Steady Viscometric Flows 109 621 Controllable andPartiallyControllableFlows 110 63 Problems 110 7 Polymer Solutions 113 71 Characteristicsof aPolymerChain 113 711 Random-Walk Model 113 72 ForcesonaChain 116 73 Fluctuation-Dissipation Theorem 119 731 Langevin Equation 119 732 Equi-partition of Energy 120 733 Fluctuation-Dissipation Theorem 121 734 DiffusivityStokes–EinsteinRelation 122 735 Fokker–Planck Equation 122 736 Smoothed-out Brownian Force 123 74 StressTensor 123 741 KramersForm 126 75 Elastic Dumbbell Model 127 751 Langevin Equations 127 752 AverageMotion 128 753 StrongandWeakFlows 129 754 Upper-Convected Maxwell Model 130 755 Oldroyd-B Model 131 76 Main Features of the Oldroyd-B Model 132 761 SimpleFlows 132 762 Multiple Relaxation Time UCM Model 134 77 Problems 135 8 Suspensions 137 81 Bulk Suspension Properties 138 82 Dilute Suspension of Spheroids 140 83 Problems 143 9 Dissipative Particle Dynamics (DPD) 147 91 1-D Model 148 92 DPDFluid 151 921 Langevin Equations 151 922 Phase-Space Description: Fokker–Planck Equation 156 923 Distribution Functions 157 924 Equation of Change 158 925 ConservationofMass 158 926 Conservation of Linear Momentum 159 927 EnergyEquation 164 93 SomeApproximateResults 166 931 HighDampingLimit 166 932 Standard DPD Parameters 168 933 EffectiveSizeof aDPDParticle 168 94 Modification of the Weighting Function 169 95 Numerical Implementation 171 951 Velocity Verlet Scheme 171 952 Exponential Time Differencing Scheme 172 953 Implementation of No-slip Boundary Conditions 175 954 Computationof InterparticleForces 177 955 CalculationofStressTensor 177 956 Complex-StructureFluid 177 96 FlowVerifications andSomeTypicalProblems 179 97 Epilogue 190 98 Problems 191 References 195