Note: Intro -- Preface -- Acknowledgments -- Contents -- 1 The Basic Features of Viscoplasticity -- 1.1 Bingham Fluid at Rest in a Channel -- 1.2 Sign of the Shear Stress -- 1.3 Critical Pressure Drop and the Constitutive Relation -- 1.4 The Solution -- 1.5 Flow Rate -- 1.6 Inherent Nonlinearity -- 1.7 Non-dimensionalisation -- 1.8 The Buckingham Equation -- 1.9 Free Boundary Problems -- 1.10 The Minimiser and the Variational Inequality -- 1.11 Effects of Wall Slip -- 1.12 Experimental Support -- 1.13 Summary -- References -- 2 Kinematics of Fluid Flow -- 2.1 Kinematical Preliminaries -- 2.2 Relation Between the Velocity and Deformation Gradients -- 2.3 Rigid Motion -- 2.4 Polar Decomposition, Spin and Stretching -- 2.5 Steady Velocity Fields and Their Rivlin-Ericksen Tensors -- References -- 3 Fundamental Equations -- 3.1 Conservation of Mass -- 3.2 Cauchy's First Law -- 3.3 Cauchy's Second Law -- 3.4 Conservation of Energy -- 3.5 Control Volume and Control Surface -- Reference -- 4 Constitutive Equations -- 4.1 Pressure and Incompressibility -- 4.1.1 The Meaning of Pressure -- 4.2 Incompressible Viscoplastic Fluids -- 4.2.1 Equations of Motion for Incompressible Materials -- 4.3 Viscoplasticity Constraint Tensor -- 4.4 Regularisation -- 4.5 Compressible Viscoplastic Fluids -- 4.6 Analogues for Incompressible Viscoplastic Fluids -- 4.6.1 One Dimensional Models -- 4.6.2 Some Results from Tensor Analysis -- 4.6.3 Three Dimensional Models -- References -- 5 Analytic Solutions: Steady Flows -- 5.1 Simple Shearing Flow -- 5.2 Flow in a Channel -- 5.3 Flow Down an Inclined Plane -- 5.4 Flow in a Pipe of Circular Cross-Section -- 5.4.1 The Buckingham Equation -- 5.5 Axial Flow in a Concentric Annulus -- 5.6 Couette Flow -- 5.7 Helical Flow -- 5.8 Steady Flows of General Viscoplastic Fluids -- 5.9 Heat Transfer Problems. , 5.9.1 Heat Transfer Between Two Parallel Plates -- 5.9.2 More General Problems -- References -- 6 Analytic Solutions: Unsteady Shearing Flows -- 6.1 Unsteady Flow in a Channel -- 6.1.1 The Solution -- 6.1.2 Approximate Solution -- 6.1.3 Laplace Transform -- 6.1.4 Application of Maximum Principles -- 6.2 Unsteady Couette and Poiseuille Flows -- 6.3 Unsteady Flow in a Half-Space -- 6.3.1 An Initial Value Problem -- 6.3.2 Singular Surfaces in Motion -- 6.3.3 Hadamard Lemma and Unsteady Shearing Flows in Viscoplastic Fluids -- 6.3.4 Implications of the Continuity of σ/y at the Yield Surface -- 6.3.5 Extensions to Other Shearing Flows -- 6.3.6 Open Ended Problems -- References -- 7 Analytical Approximation Techniques -- 7.1 The Lubrication Paradox -- 7.2 Steady Flow in a Wavy Channel---The Periodic Case -- 7.2.1 Zeroth Order Solution -- 7.2.2 First Order Corrections -- 7.2.3 Breaking the Unyielded Plug -- 7.3 Hele-Shaw Flow Problems -- 7.3.1 The Viscometric Fluidity Function -- 7.3.2 Papanastasiou Model -- 7.3.3 The Symmetric Case -- 7.3.4 The Average Velocity Field in the Symmetric Case -- 7.3.5 Hele-Shaw Flow Equations -- 7.3.6 The Asymmetric Case -- 7.4 Linearised Stability Analysis -- 7.5 Summary -- References -- 8 Variational Principles and Variational Inequalities -- 8.1 Minimum and Maximum Principles for Incompressible Viscoplastic Fluids -- 8.1.1 Basic Definitions and Principle of Virtual Power -- 8.1.2 The Velocity and Stress Functionals -- 8.1.3 Proofs of the Theorems -- 8.1.4 Equality of Φ(u) and Ψ(T) -- 8.1.5 Shear Rate Dependent Yield Stress -- 8.1.6 Steady Flow in a Pipe of Uniform Cross-Section -- 8.2 Virtual Power and the Basic Inequality for Incompressible Viscoplastic Fluids -- 8.2.1 A Point-Wise Inequality: Isochoric Velocity Fields -- 8.2.2 The Integral Inequality -- 8.3 A General Energy Balance Equation for Viscoplastic Fluids. , 8.4 Fundamental Inequality: Non-isochoric Trial Velocity Fields -- 8.5 Variational Principles and Fundamental Inequality in the Presence of Wall Slip -- 8.6 Convex Analysis and Its Applications -- 8.6.1 The Direct Method -- 8.6.2 Convex Set and Convex Functionals -- 8.6.3 Existence and Uniqueness -- 8.6.4 Variational Inequality -- 8.6.5 Equivalence of the Minimiser and the Solution of the Variational Inequality -- 8.7 Equivalence of the Solution of the Variational Inequality and the Equations of Motion -- 8.8 Special Cases of the Variational Inequality -- 8.8.1 Flows with Zero Stress Power Difference -- 8.8.2 Flows with Non-zero Stress Power Difference -- 8.8.3 The Trilinear Functional Involving Acceleration Terms -- 8.9 Viscoplasticity Constraint Tensor: The Final Equivalence -- 8.10 The Basic Inequality for Compressible Viscoplastic Fluids -- References -- 9 Energy Methods in Action: Equality, Inequality and Stability -- 9.1 Axial Flow in a Pipe of Arbitrary Cross-Section -- 9.1.1 The Minimum Pressure Drop per Unit Length to Initiate a Steady Flow -- 9.1.2 Existence of Stagnant Zones -- 9.1.3 Bounds on the Magnitude of the Core and Its Maximum Velocity -- 9.2 Static Bubbles in Viscoplastic Fluids -- 9.2.1 Critical Value of the Bingham Number to Prevent Bubble Motion -- 9.2.2 Critical Value from Stress Maximisation -- 9.2.3 A Condition for a Bubble to Move: An Upper Bound for the Bingham Number -- 9.3 Motions of Rigid Bodies in Viscoplastic Fluids -- 9.4 Initiation and Cessation of Unsteady Shearing Flows -- 9.4.1 The Approach to the Steady State -- 9.4.2 The Proof of the Energy Inequality -- 9.4.3 Cessation of the Steady Flow in a Channel -- 9.4.4 Cessation of Steady Simple Shear Flow -- 9.4.5 Cessation of Steady Flow in a Pipe -- 9.4.6 Cessation of Steady Couette Flow -- 9.4.7 Effects of Wall Slip -- 9.5 Nonlinear Stability Analysis. , 9.5.1 Dissipation Terms -- 9.5.2 Global Stability Bounds -- 9.5.3 Conditional Stability -- References -- 10 Numerical Modelling -- 10.1 Augmented Lagrangian Methods: Finite Dimensional Case -- 10.2 Augmented Lagrangian Methods for Bingham Fluids -- 10.2.1 Optimality Conditions of the Augmented Lagrangian Functional -- 10.2.2 More General Problems -- 10.3 Operator-Splitting Method for Thermally Driven Flows -- 10.3.1 The Flow Problem and Mathematical Formulation -- 10.3.2 Non-dimensionalisation -- 10.3.3 Numerical Procedure -- 10.3.4 Discussion of the Results -- 10.4 Compressibility Effects: Numerical Experiments -- 10.4.1 Operator-Splitting Methods: Compressible Viscous Fluids -- 10.4.2 Compressible Viscoplastic Fluids: Isothermal Case -- 10.4.3 Operator-Splitting Method -- 10.5 Flow in a Cavity: Weakly Compressible Fluid -- 10.6 Regularised Models -- References -- Index.