Fluid Mechanics of Viscoplasticity

Preface
Acknowledgements
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 Evidence and Modelling
	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
	2.6 Appendix A: Divergence, Curl, Rivlin-Ericksen Tensor and Spin Tensor
	References
3 Fundamental Equations: Continuum Mechanics and Lattice Boltzmann Models
	3.1 Introduction
	3.2 Conservation of Mass
	3.3 Cauchy's First Law of Motion
	3.4 Cauchy's Second Law of Motion
	3.5 Conservation of Energy
	3.6 Control Volume and Control Surface
	3.7 Particle Based Models
	3.8 Evolution Equations for Particle Distribution Functions
	3.9 Fluid-Velocity and Particle-Velocity Lattice Boltzmann Methods
	3.10 Appendix A: Equations of Motion in Various Coordinates
	3.11 Appendix B: Equilibrium Particle Distribution Functions
	References
4 Constitutive Equations
	4.1 Pressure and Incompressibility
	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 Constitutive Models 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 Introduction
	5.2 Simple Shearing Flow
	5.3 Flow in a Channel
	5.4 Flow Down an Inclined Plane
	5.5 Poiseuille Flow
		5.5.1 The Velocity Field and the Flow Rate
		5.5.2 The Buckingham Equation
	5.6 Axial Flow in a Concentric Annulus
	5.7 Couette Flow
		5.7.1 Flow Due to Positive Shear Stress
		5.7.2 Lambert W Function
		5.7.3 Fully Sheared Flow
		5.7.4 Flow Due to Negative Shear Stress
	5.8 Axial Couette-Poiseuille Flow
		5.8.1 Axial Couette Flow
		5.8.2 Axial Couette-Poiseuille Flow
	5.9 Helical Flow
	5.10 Herschel-Bulkley and Casson Fluids: Shear Rate Formulation
		5.10.1 Herschel-Bukley Fluids
		5.10.2 Casson Fluids
	5.11 Herschel-Bukley Fluids: Partial Flow Rate Function Method
	5.12 Herschel-Bulkley Fluids: Antiplane Shear Flow
	5.13 Lambert W Function and the Papanastasiou Model
	5.14 Flows with Wall Slip
		5.14.1 Simple Shearing Flow
		5.14.2 Channel Flow
		5.14.3 Axisymmetric Poiseuille Flow
		5.14.4 Annular Poiseuille Flow
		5.14.5 Circular Couette Flow of a Bingham Fluid
		5.14.6 Torsional Parallel Flow
	5.15 Flows of Materials with Pressure Dependent Rheological Parameters
		5.15.1 Channel Flow of a Bingham Fluid
		5.15.2 Axisymmetric Poiseuille Flow of a Bingham Fluid
	5.16 Heat Transfer Problems
		5.16.1 Heat Transfer Between Parallel Plates
		5.16.2 More General Problems
	References
6 Unsteady Shearing Flows
	6.1 Unsteady Flow in a Channel
		6.1.1 The Solution
		6.1.2 Approximate Solution
	6.2 A Numerical Solution to the Velocity Field
		6.2.1 Approximate Evaluation
		6.2.2 Numerical Comparison
	6.3 Laplace Transform
	6.4 Application of Maximum Principles
	6.5 Unsteady Couette and Poiseuille Flows
	6.6 Unsteady Flow in a Half-Space
		6.6.1 An Initial Value Problem
		6.6.2 Singular Surfaces in Motion
		6.6.3 Hadamard Lemma and Unsteady Shearing Flows in Viscoplastic Fluids
		6.6.4 Implications of the Continuity of σ/y at the Yield Surface
		6.6.5 Extensions to Other Shearing Flows
		6.6.6 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 The Zeroth Order Solution
		7.2.2 First Order Corrections
		7.2.3 Breaking the Unyielded Plug
	7.3 Circumventing the Lubrication Paradox
		7.3.1 Flow of a Herschel–Bulkley Fluid in a Symmetric Channel
		7.3.2 The Zeroth Order Solution
		7.3.3 Flow in a Channel of Linearly Varying Width
		7.3.4 Viscoplastic Flows in Axisymmetric Tubes
	7.4 Slump Tests
		7.4.1 The Fifty Cent Rheometer
		7.4.2 Asymptotic Formulae for Cylinders of Large and Small Heights
		7.4.3 Height of the Incipient Failure of a Circular Cylinder
		7.4.4 The Dam Break and the Bostwick Consistometer
		7.4.5 The Twin-Fluid Model
	7.5 Hele–Shaw Flow Problems
		7.5.1 The Symmetric Case
		7.5.2 The Average Velocity Field in the Symmetric Case
		7.5.3 Hele–Shaw Flow Equations
		7.5.4 The Asymmetric Case
	7.6 Linearised Stability Analysis
	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 Sets 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 Solutions of the Variational Inequality …
	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
		9.3.1 Motion in an Unbounded Domain
		9.3.2 Settling in Bounded Domains and Cheeger Sets
	9.4 Initiation and Cessation of 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 Shooting Method for the Flow in an Annulus
		10.6.1 Helical Flows
	10.7 Flow in Pipes of Square and Circular Cross-Sections
		10.7.1 Theoretical Formulation
		10.7.2 The Numerical Method
		10.7.3 Boundary Conditions and Non-dimensional Variables
		10.7.4 The Algorithm
		10.7.5 The Lattice Speed σ
		10.7.6 Results and Discussion
		10.7.7 Flow in a Pipe of Circular Cross-Section
	10.8 Thermally Influenced Lid-Driven Flow in a Square Cavity
		10.8.1 Dimensional Equations
		10.8.2 Non-dimensional Equations
		10.8.3 The Continuity and Momentum Equations
		10.8.4 The Energy Equation
		10.8.5 Non-dimensional Variables
		10.8.6 The Algorithm
		10.8.7 Code Validation and Grid Independence
		10.8.8 Results and Discussion
	References
Index