Non-Newtonian Fluid Mechanics and Complex Flows: Levico Terme, Italy 2016 (Lecture Notes in Mathematics, 2212)

"This book presents a series of challenging mathematical problems which arise in the modeling of Non-Newtonian fluid dynamics. It focuses in particular on the mathematical and physical modeling of a variety of contemporary problems, and provides some results. The flow properties of Non-Newtonian fluids differ in many ways from those of Newtonian fluids. Many biological fluids (blood, for instance) exhibit a non-Newtonian behavior, as do many naturally occurring or technologically relevant fluids such as molten polymers, oil, mud, lava, salt solutions, paint, and so on. The term 'complex flows' usually refers to those fluids presenting an 'internal structure' (fluid mixtures, solutions, multiphase flows, and so on). Modern research on complex flows has increased considerably in recent years due to the many biological and industrial applications"--Page 4 of cover Preface 6 Contents 9 Hemorheology: Non-Newtonian Constitutive Models for Blood Flow Simulations 10 1 Introduction 10 2 Blood Rheology 13 2.1 Blood Components 13 2.2 Non-Newtonian Properties of Blood 14 2.2.1 Viscosity of Blood 15 2.2.2 Yield Stress of Blood 18 2.2.3 Viscoelasticity and Thixotropy of Blood 18 2.3 Constitutive Models for Blood 19 2.3.1 Constant Viscosity Models 20 2.3.2 Generalized Newtonian Models 22 2.3.3 Yield Stress Models 25 2.3.4 Viscoelastic Models 27 3 Numerical Simulations of Non-Newtonian Blood Flow Models 33 3.1 Numerical Simulations in Idealized Geometries 34 3.1.1 Stenosed Vessel 34 3.1.2 Curved Vessel 36 3.2 Numerical Simulations in a Realistic Geometry: Stenosed Carotid Bifurcation 41 References 48 Old Problems Revisited from New Perspectivesin Implicit Theories of Fluids 54 1 Introduction 54 2 Implicit Constitutive Models for the Cauchy Stress Tensor 58 3 Isochoric Motions of Fluids as Approximations Under Different Flow Regimes 61 3.1 Equations Governing the Flows in a Piezo-Viscous Fluid 63 3.2 Approximations 66 3.2.1 Generalized Oberbeck-Boussinesq Approximation 70 3.2.2 Generalized Navier-Stokes-Fourier Equations 71 4 Rayleigh-Bénard Problem for Fluids with Pressure- and Temperature Dependent Viscosities 71 4.1 Conduction Solution: Evolution Equations of Perturbations 72 4.2 Linear Stability Analysis 73 5 Parallel Shear Flows of Piezo-Viscous Fluids 78 5.1 Governing Equations 79 5.2 Couette Flows 80 5.3 Poiseuille Flows 81 6 Flow of Fluids with Pressure and Shear Dependent Viscosity Down an Inclined Plane 84 6.1 Basic Equations 86 6.2 Nearly Steady Uniform Regime 91 6.3 Viscous Regime 93 References 97 Lectures on Hyperbolic Equations and Their NumericalApproximation 100 1 Hyperbolic Equations 101 1.1 The Linear Advection Equation and Basic Concepts 101 1.2 Linear Systems 105 1.3 Non-linear Scalar Equations: Definitions and Examples 112 1.4 Numerical Approximation of Hyperbolic Equations 122 2 The Shallow Water Equations and the Riemann Problem 135 2.1 Equations, Properties and Wave Relations 135 2.2 The Riemann Problem 139 2.2.1 Wave Relations 139 2.2.2 Solution of Problem 1: The Star Problem 148 2.2.3 Solution of Problem 2: The Complete Solution 150 2.3 Concluding Remarks 152 3 Godunov's Method for the Shallow Water Equations 153 3.1 The Finite Volume Method 154 3.1.1 The Godunov Flux 155 3.1.2 Godunov Flux with the Exact Riemann Solver 156 3.2 A Simple Linearised Riemann Solver 157 3.3 A Two-Rarefaction Riemann Solver 158 3.4 The Harten-Lax-van Leer (HLL) Riemann Solver 159 3.5 The HLLC Riemann Solver 162 3.6 The Dumbser-Osher-Toro Riemann Solver: DOT 164 3.6.1 Definitions and Notation 164 3.6.2 The DOT Riemann Solver 165 3.6.3 Sample Numerical Results, Accuracy and Efficiency 166 3.6.4 Concluding Remarks 169 4 High Order Methods: The ADER Approach 169 4.1 Overview 169 4.2 ADER in the Finite Volume Framework 171 4.3 Ingredients of ADER 172 4.4 Generalized Riemann Problem 173 4.5 Numerical Examples 173 4.6 Concluding Remarks 176 References 177 An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media 179 1 Introduction to the Homogenization 179 2 Models for Quasi-Newtonian Fluids and a Derivation of the Filtration Laws by a Two-Scale Expansion 182 2.1 Continuum Physics Models for Quasi-Newtonian Fluids 182 2.2 The Geometry of a Periodic Porous Medium and a Priori Estimates 183 2.3 The Filtration Laws via Two-Scale Asymptotic Expansions: The Power-Law 186 2.4 The Filtration Laws via Two-Scale Asymptotic Expansions: Carreau Law 190 2.5 The Filtration Laws via Two-Scale Asymptotic Expansions: Bingham Fluid Case 193 3 An Introduction to the Two-Scale Convergence with Special Attention to the Two-Scale Lower Semi-Continuity 194 4 The a Priori Estimates for the Pressure and the Two-Scale Limits in the Case of the Power Law Viscosity 200 4.1 A Priori Estimates and the Two-Scale Convergence for the Case of the Law of Carreau 203 4.2 A Priori Estimates and the Two-Scale Convergence for the Case of the Bingham Flow 206 4.3 Concluding Remarks on Filtration Laws for Non-Newtonian Fluids 209 5 Homogenization of the Linearized Ionic Transport Equations in Rigid Periodic Porous Media 209 5.1 Equilibrium Solution 216 5.2 Linearization and the a Priori Estimates for the Perturbation 220 5.3 Homogenization via the Two-Scale Convergence 223 5.4 The Separation of the Fast and the Slow Scales and the Onsager Relations 227 References 232 Viscoplastic Fluids: Mathematical Modeling and Applications 236 1 Introduction 236 2 Constitutive Model 237 3 Flow in a Channel 242 4 Bingham Model with Deformable Core 246 4.1 Channel Flow of a Bingham-Like Fluid with Linear Elastic Core 247 4.2 Kinematics and Constitutive Equation 248 4.3 Flow in a Channel 249 4.3.1 Boundary Conditions 251 4.3.2 The Elastic Domain and the Viscous Domain 252 4.3.3 Asymptotic Expansion 253 4.3.4 First Case: Γ=O(1) 254 4.3.5 Second Case Γ=O() 259 4.3.6 Stationary Version of (54) 261 4.4 Numerical Simulations 263 5 Two Dimensional Channel Flow: A New Approach 265 5.1 The Physical Model 267 5.2 The Viscous Domain 268 5.3 The Rigid Domain 268 5.4 Scaling 269 5.5 The Leading Order Approximation 271 5.6 Flow Condition 274 5.7 Inner Core Appearance or Disappearance 276 5.8 Solution for an Almost Flat Channel 277 5.9 Numerical Simulations 280 5.10 Model with Pressure Dependent Viscosity 283 6 Planar Squeeze 286 6.1 Squeezing Between Parallel Plates 288 6.2 Problem at the Leading Order 290 6.3 Numerical Simulation 294 6.4 Squeezing Between Surfaces 295 References 304 Front Matter ....Pages i-ix Hemorheology: Non-Newtonian Constitutive Models for Blood Flow Simulations (Adélia Sequeira)....Pages 1-44 Old Problems Revisited from New Perspectives in Implicit Theories of Fluids (Giuseppe Saccomandi, Luigi Vergori)....Pages 45-90 Lectures on Hyperbolic Equations and Their Numerical Approximation (Eleuterio F. Toro)....Pages 91-169 An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media (Andro Mikelić)....Pages 171-227 Viscoplastic Fluids: Mathematical Modeling and Applications (Angiolo Farina, Lorenzo Fusi)....Pages 229-298 Back Matter ....Pages 299-300