Complex Fluids : Modeling and Algorithms

This book presents a comprehensive overview of the modeling of complex fluids, including many common substances, such as toothpaste, hair gel, mayonnaise, liquid foam, cement and blood, which cannot be described by Navier-Stokes equations. It also offers an up-to-date mathematical and numerical analysis of the corresponding equations, as well as several practical numerical algorithms and software solutions for the approximation of the solutions. It discusses industrial (molten plastics, forming process), geophysical (mud flows, volcanic lava, glaciers and snow avalanches), and biological (blood flows, tissues) modeling applications. This book is a valuable resource for undergraduate students and researchers in applied mathematics, mechanical engineering and physics. Preface 7 Contents 10 Notations 13 1 Navier--Stokes Equation 15 1.1 Conservation of Mass 15 1.2 Conservation of Momentum 18 1.3 Constitutive Equation 27 1.4 Incompressible Navier--Stokes Equations 29 1.5 Example: Poiseuille Flow 32 1.6 Example: Couette Flow 35 1.7 Time Discretization 40 1.8 Stokes Problem 44 1.9 Mixed Velocity--Pressure Discretization 50 1.10 Direct Resolution of the Stokes Problem 55 1.11 Space Discretization of Characteristics 58 1.12 Example: Driven Cavity Flow 63 1.13 Iterative Resolution of the Stokes Problem 70 1.14 Notes 74 2 Quasi-Newtonian Fluids 77 2.1 Viscosity Function 77 2.2 Problem Statement 80 2.3 Example: Poiseuille Flow 81 2.4 Example: Couette Flow 82 2.5 Time Discretization 85 2.6 Minimization Problem 86 2.7 Saddle Point and Variational Formulations 89 2.8 Fixed Point Algorithm 90 2.9 Example: Poiseuille Flow in a Square Section 94 2.10 Newton Algorithm 95 2.11 Notes 104 3 Viscoplastic Fluids 105 3.1 Energy of Dissipation 106 3.2 Subdifferential 107 3.3 Problem Statement 113 3.4 Rigid Regions 113 3.5 Example: Poiseuille Flow 115 3.6 Example: Couette Flow 118 3.7 Time Discretization 124 3.8 Variational Formulation 125 3.9 Regularization Method 126 3.10 Augmented Lagrangian Algorithm 129 3.11 Space Approximation 137 3.12 Example: Poiseuille Flow in a Square Section 142 3.13 Example: Flow Around an Obstacle 146 3.14 Example: Driven Cavity Flow 150 3.15 Notes 153 4 Viscoelastic Fluids 157 4.1 Constitutive Equations 157 4.2 Objective Tensor Derivatives 159 4.3 Problem Statement 167 4.4 Example: Poiseuille Flow 169 4.5 Example: Couette Flow 174 4.6 Time Discretization with the θ-Scheme Algorithm 181 4.7 Space Approximation 185 4.8 Tensorial Transport Subproblem 186 4.9 Variational Formulation 189 4.10 Discontinuous Galerkin Method 191 4.11 Example: Flow in an Abrupt Contraction 201 4.12 Log-Conformation Tensor Formulation 204 4.13 Energy Estimate 213 4.14 Example: Driven Cavity Flow 219 4.15 Notes 221 5 Elastoviscoplastic Fluids 228 5.1 Conservation of Energy 229 5.2 Second Principle of Thermodynamics 232 5.3 Framework for Constitutive Equations 235 5.3.1 Generalized Standard Materials 235 5.3.2 Example: Elastic Material 237 5.3.3 Example: Newtonian Fluid 238 5.3.4 Example: Herschel--Bulkley Model 240 5.3.5 Example: Oldroyd Model 243 5.4 Problem Statement 244 5.5 Example: Poiseuille Flow 248 5.6 Example: Couette Flow 255 5.7 Example: Flow Around an Obstacle 260 5.8 Notes 264 References 269 Index 282 Front Matter....Pages i-xvi Navier–Stokes Equation....Pages 1-62 Quasi-Newtonian Fluids....Pages 63-90 Viscoplastic Fluids....Pages 91-142 Viscoelastic Fluids....Pages 143-213 Elastoviscoplastic Fluids....Pages 215-255 Back Matter....Pages 257-276