Non-Newtonian fluid mechanics and complex flows

Preface......Page 6
Contents......Page 9
 1 Introduction......Page 10
  2.1 Blood Components......Page 13
  2.2 Non-Newtonian Properties of Blood......Page 14
   2.2.1 Viscosity of Blood......Page 15
   2.2.3 Viscoelasticity and Thixotropy of Blood......Page 18
  2.3 Constitutive Models for Blood......Page 19
   2.3.1 Constant Viscosity Models......Page 20
   2.3.2 Generalized Newtonian Models......Page 22
   2.3.3 Yield Stress Models......Page 25
   2.3.4 Viscoelastic Models......Page 27
 3 Numerical Simulations of Non-Newtonian Blood Flow Models......Page 33
   3.1.1 Stenosed Vessel......Page 34
   3.1.2 Curved Vessel......Page 36
  3.2 Numerical Simulations in a Realistic Geometry: Stenosed Carotid Bifurcation......Page 41
 References......Page 48
 1 Introduction......Page 54
 2 Implicit Constitutive Models for the Cauchy Stress Tensor......Page 58
 3 Isochoric Motions of Fluids as Approximations Under Different Flow Regimes......Page 61
  3.1 Equations Governing the Flows in a Piezo-Viscous Fluid......Page 63
  3.2 Approximations......Page 66
   3.2.1 Generalized Oberbeck-Boussinesq Approximation......Page 70
 4 Rayleigh-Bénard Problem for Fluids with Pressure- and Temperature Dependent Viscosities......Page 71
  4.1 Conduction Solution: Evolution Equations of Perturbations......Page 72
  4.2 Linear Stability Analysis......Page 73
 5 Parallel Shear Flows of Piezo-Viscous Fluids......Page 78
  5.1 Governing Equations......Page 79
  5.2 Couette Flows......Page 80
  5.3 Poiseuille Flows......Page 81
 6 Flow of Fluids with Pressure and Shear Dependent Viscosity Down an Inclined Plane......Page 84
  6.1 Basic Equations......Page 86
  6.2 Nearly Steady Uniform Regime......Page 91
  6.3 Viscous Regime......Page 93
 References......Page 97
Lectures on Hyperbolic Equations and Their NumericalApproximation......Page 100
  1.1 The Linear Advection Equation and Basic Concepts......Page 101
  1.2 Linear Systems......Page 105
  1.3 Non-linear Scalar Equations: Definitions and Examples......Page 112
  1.4 Numerical Approximation of Hyperbolic Equations......Page 122
  2.1 Equations, Properties and Wave Relations......Page 135
   2.2.1 Wave Relations......Page 139
   2.2.2 Solution of Problem 1: The Star Problem......Page 148
   2.2.3 Solution of Problem 2: The Complete Solution......Page 150
  2.3 Concluding Remarks......Page 152
 3 Godunov's Method for the Shallow Water Equations......Page 153
  3.1 The Finite Volume Method......Page 154
   3.1.1 The Godunov Flux......Page 155
   3.1.2 Godunov Flux with the Exact Riemann Solver......Page 156
  3.2 A Simple Linearised Riemann Solver......Page 157
  3.3 A Two-Rarefaction Riemann Solver......Page 158
  3.4 The Harten-Lax-van Leer (HLL) Riemann Solver......Page 159
  3.5 The HLLC Riemann Solver......Page 162
   3.6.1 Definitions and Notation......Page 164
   3.6.2 The DOT Riemann Solver......Page 165
   3.6.3 Sample Numerical Results, Accuracy and Efficiency......Page 166
  4.1 Overview......Page 169
  4.2 ADER in the Finite Volume Framework......Page 171
  4.3 Ingredients of ADER......Page 172
  4.5 Numerical Examples......Page 173
  4.6 Concluding Remarks......Page 176
 References......Page 177
 1 Introduction to the Homogenization......Page 179
  2.1 Continuum Physics Models for Quasi-Newtonian Fluids......Page 182
  2.2 The Geometry of a Periodic Porous Medium and a Priori Estimates......Page 183
  2.3 The Filtration Laws via Two-Scale Asymptotic Expansions: The Power-Law......Page 186
  2.4 The Filtration Laws via Two-Scale Asymptotic Expansions: Carreau Law......Page 190
  2.5 The Filtration Laws via Two-Scale Asymptotic Expansions: Bingham Fluid Case......Page 193
 3 An Introduction to the Two-Scale Convergence with Special Attention to the Two-Scale Lower Semi-Continuity......Page 194
 4 The a Priori Estimates for the Pressure and the Two-Scale Limits in the Case of the Power Law Viscosity......Page 200
  4.1 A Priori Estimates and the Two-Scale Convergence for the Case of the Law of Carreau......Page 203
  4.2  A Priori Estimates and the Two-Scale Convergence for the Case of the Bingham Flow......Page 206
 5 Homogenization of the Linearized Ionic Transport Equations in Rigid Periodic Porous Media......Page 209
  5.1 Equilibrium Solution......Page 216
  5.2 Linearization and the a Priori Estimates for the Perturbation......Page 220
  5.3 Homogenization via the Two-Scale Convergence......Page 223
  5.4 The Separation of the Fast and the Slow Scales and the Onsager Relations......Page 227
 References......Page 232
 1 Introduction......Page 236
 2 Constitutive Model......Page 237
 3 Flow in a Channel......Page 242
 4 Bingham Model with Deformable Core......Page 246
  4.1 Channel Flow of a Bingham-Like Fluid with Linear Elastic Core......Page 247
  4.2 Kinematics and Constitutive Equation......Page 248
  4.3 Flow in a Channel......Page 249
   4.3.1 Boundary Conditions......Page 251
   4.3.2 The Elastic Domain and the Viscous Domain......Page 252
   4.3.3 Asymptotic Expansion......Page 253
   4.3.4 First Case: Γ=O(1)......Page 254
   4.3.5 Second Case Γ=O()......Page 259
   4.3.6 Stationary Version of (54)......Page 261
  4.4 Numerical Simulations......Page 263
 5 Two Dimensional Channel Flow: A New Approach......Page 265
  5.1 The Physical Model......Page 267
  5.3 The Rigid Domain......Page 268
  5.4 Scaling......Page 269
  5.5 The Leading Order Approximation......Page 271
  5.6 Flow Condition......Page 274
  5.7 Inner Core Appearance or Disappearance......Page 276
  5.8 Solution for an Almost Flat Channel......Page 277
  5.9 Numerical Simulations......Page 280
  5.10 Model with Pressure Dependent Viscosity......Page 283
 6 Planar Squeeze......Page 286
  6.1 Squeezing Between Parallel Plates......Page 288
  6.2 Problem at the Leading Order......Page 290
  6.3 Numerical Simulation......Page 294
  6.4 Squeezing Between Surfaces......Page 295
 References......Page 304