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1 | (20) |
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2 | (6) |
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1.1.1 Physical Considerations |
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3 | (2) |
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1.1.2 Viscoelastic Fluids |
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5 | (1) |
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6 | (2) |
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8 | (4) |
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1.2.1 Physical Considerations: Circular Couette Flow |
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8 | (2) |
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10 | (1) |
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11 | (1) |
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12 | (1) |
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1.4 Examples of a Complex Fluid and Flow |
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12 | (5) |
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1.4.1 The Kaye Effect: Shear-Thinning Evidence |
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13 | (2) |
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1.4.2 Bouncing Newtonian Jet |
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15 | (1) |
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1.4.3 Turbulent Drag Reduction |
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15 | (2) |
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17 | (4) |
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2 Tensor Analysis, Invariants, and Representations |
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21 | (26) |
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2.1 Symmetries and Transformations |
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23 | (2) |
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2.2 Invariants and Traces of Matrix Polynomials |
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25 | (6) |
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2.2.1 Polynomial Invariants |
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26 | (2) |
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2.2.2 Traces of Matrix Polynomials |
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28 | (3) |
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2.3 Integrity Bases for Vectors and Tensors |
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31 | (7) |
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2.3.1 Integrity Basis for Vectors |
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32 | (1) |
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2.3.2 Integrity Bases for Symmetric Second-Order Tensors |
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32 | (2) |
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2.3.3 Integrity Bases for Vectors and Second-Order Tensors |
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34 | (4) |
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2.4 Polynomial Representations for Tensors and Vectors |
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38 | (9) |
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2.4.1 Proper Orthogonal Group |
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38 | (4) |
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2.4.2 Full Orthogonal Group |
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42 | (5) |
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3 Kinematics and Dynamics |
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47 | (22) |
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3.1 Material Elements and Deformation |
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47 | (8) |
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3.1.1 Decomposition of the Deformation |
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53 | (1) |
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3.1.2 Infinitesimal Strain and Rotation |
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53 | (2) |
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55 | (6) |
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3.2.1 Time Rate of Change |
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55 | (1) |
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3.2.2 Strain Rate and Rotation Rate Tensors |
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56 | (2) |
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58 | (1) |
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3.2.4 Rivlin-Ericksen Tensors |
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59 | (2) |
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3.3 Reynolds Transport Theorem |
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61 | (1) |
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3.4 Conservation Equations |
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62 | (7) |
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63 | (1) |
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3.4.2 Momentum Conservation |
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63 | (4) |
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3.4.3 Energy Conservation |
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67 | (2) |
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4 Constitutive Equations: General Principles |
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69 | (26) |
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69 | (1) |
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4.2 Methodological Principles |
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70 | (5) |
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4.2.1 Material Stress Field |
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71 | (2) |
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4.2.2 Turbulent Stress Field |
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73 | (2) |
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4.3 Frames, Transformations and Objectivity |
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75 | (9) |
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4.3.1 Transformations and Objectivity |
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75 | (4) |
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4.3.2 Objective Rates of the Stress Tensor |
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79 | (5) |
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4.4 Restrictions on Constitutive Relationships |
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84 | (5) |
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4.4.1 A Thermodynamic Constraint for Constitutive Relationships |
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84 | (1) |
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4.4.2 Objectivity Constraints on Material Constitutive Equations |
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85 | (4) |
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4.5 Deformation and Constant Stretch History Motion |
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89 | (6) |
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92 | (1) |
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93 | (1) |
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4.5.3 Viscometric Functions |
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94 | (1) |
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5 Non-Newtonian and Viscoelastic Fluids |
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95 | (54) |
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95 | (1) |
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5.2 Classical and Generalized Newtonian Models |
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96 | (2) |
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96 | (1) |
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5.2.2 Generalized Newtonian Fluids |
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97 | (1) |
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5.3 Linear Viscoelasticity |
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98 | (3) |
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99 | (1) |
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100 | (1) |
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100 | (1) |
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5.4 From a Simple Fluid to Viscoelasticity |
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101 | (2) |
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5.4.1 Reiner-Rivlin Fluid |
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101 | (1) |
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5.4.2 Elasticity as the Limit Case |
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102 | (1) |
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5.4.3 Design of a Viscoelastic Constitutive Equation |
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103 | (1) |
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5.5 Rivlin-Ericksen and Order Fluids |
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103 | (2) |
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5.5.1 Rivlin-Ericksen Fluids |
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104 | (1) |
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104 | (1) |
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5.5.3 Plane Shear Flow of a Second-Order Fluid |
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105 | (1) |
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5.6 Constant Stretch History Flows |
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105 | (2) |
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5.7 Constitutive Equations of the Rate Type |
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107 | (8) |
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107 | (6) |
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5.7.2 Improved Rate Type Models |
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113 | (1) |
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5.7.3 Relation Between Rate Type and Integral Models |
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114 | (1) |
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115 | (6) |
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115 | (1) |
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5.8.2 The Hookean Dumbbell |
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116 | (1) |
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117 | (1) |
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118 | (1) |
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119 | (2) |
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5.8.6 The Giesekus Model Revisited |
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121 | (1) |
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5.9 Dumbbells and Stochastic Differential Equations |
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121 | (7) |
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5.9.1 The Fokker-Planck Equation |
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121 | (2) |
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123 | (1) |
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5.9.3 Nonlinear Dumbbells |
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124 | (3) |
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5.9.4 Dumbbells with Hydrodynamic Interactions |
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127 | (1) |
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5.10 The Micro-Macro Description |
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128 | (2) |
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5.10.1 Solving the Fokker-Planck Equation |
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129 | (1) |
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5.10.2 Brownian Configuration Fields |
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130 | (1) |
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5.11 Consequences of Non-affine Motion |
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130 | (3) |
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5.11.1 Dumbbells with Non-affine Motion and the Gordon-Schowalter Model |
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130 | (2) |
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5.11.2 Modeling Polymeric Networks |
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132 | (1) |
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5.12 Modeling of Polymer Melts |
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133 | (16) |
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134 | (3) |
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5.12.2 Differential Form of the Doi-Edwards Model |
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137 | (2) |
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139 | (4) |
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5.12.4 The Extended Pom-Pom Model |
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143 | (2) |
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5.12.5 Linear Entangled Polymer Chains and the Rolie-Poly Equation |
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145 | (4) |
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149 | (66) |
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6.1 Homogeneity and the Spectral Cascade |
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149 | (3) |
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6.2 Numerical Solution Methodologies |
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152 | (21) |
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6.2.1 Direct Numerical Simulation (DNS) |
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154 | (4) |
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6.2.2 Scale Resolving Simulations |
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158 | (10) |
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6.2.3 Mean Equation Methods |
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168 | (5) |
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6.3 Mean Equation Closure |
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173 | (10) |
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6.3.1 Reynolds Stress Tensor |
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175 | (4) |
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6.3.2 Dissipation Rate Tensor |
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179 | (4) |
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6.4 Reynolds Stress Transport Equation Closure |
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183 | (11) |
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6.4.1 Pressure-Strain Rate Correlation |
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184 | (4) |
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6.4.2 Turbulent Transport |
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188 | (6) |
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6.5 Polynomial Representations of the Turbulent Stress Tensor |
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194 | (18) |
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6.5.1 Turbulent Stress of a Simple Fluid |
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195 | (2) |
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6.5.2 Turbulent Stress from Invariant Bases |
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197 | (13) |
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6.5.3 Constraints Imposed by Solid Boundaries |
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210 | (2) |
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212 | (3) |
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215 | (28) |
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216 | (4) |
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216 | (2) |
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7.1.2 Continuous Boltzmann Equation |
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218 | (1) |
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7.1.3 Boltzmann-BGK Based Continuous Equations |
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219 | (1) |
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7.2 Hermite Function Approximation |
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220 | (2) |
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222 | (1) |
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7.4 Chapman-Enskog Expansion |
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223 | (5) |
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7.4.1 Zero Order Approximation |
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224 | (1) |
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7.4.2 First Order Approximation |
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224 | (4) |
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7.5 Lattice Boltzmann Method |
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228 | (5) |
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7.6 Multiple Relaxation Time Boltzmann Equation |
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233 | (2) |
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7.6.1 Linearized Boltzmann Equation |
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233 | (1) |
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7.6.2 MRT Lattice Boltzmann Method |
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234 | (1) |
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7.7 LBM for Viscoelastic Fluids |
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235 | (5) |
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7.7.1 Advection-Diffusion Equation with a Source Term |
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236 | (2) |
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7.7.2 Computation of the Constitutive Equation |
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238 | (1) |
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7.7.3 Description of the Algorithm |
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239 | (1) |
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7.8 LBM for Turbulent Flows |
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240 | (3) |
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Appendix Properties of the Hermite Polynomials |
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243 | (6) |
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243 | (2) |
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Gauss-Hermite Quadrature Rule |
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245 | (1) |
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246 | (3) |
References |
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249 | (12) |
Index |
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261 | |