Theoretical and Computational Fluid Mechanics: Existence, Blow-up, and Discrete Exterior Calculus Algorithms

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Authors
Chapter 1 Introduction to Fluid Dynamics
	1.1 Introduction
		1.1.1 Steady Parallel Viscous Flow
		1.1.2 Properties of a Flow
		1.1.3 Exact Equation Leads to the Diffusion Equation
	1.2 Kinematics in Fluids
		1.2.1 Material Time Derivative
		1.2.2 Conservation of Mass
		1.2.3 Kinematic Boundary Conditions
		1.2.4 Streamfunction in the Incompressible Case
	1.3 Dynamics
		1.3.1 Navier-Stokes Equations
		1.3.2 Pressure
		1.3.3 Reynolds Number
	1.4 Scale Invariance of Navier-Stokes Equations
	1.5 Complex Potentials
		1.5.1 Stagnation Point Flow u = 0
	1.6 Inviscid Flow
		1.6.1 Momentum Equation for Inviscid ν = 0 Incompressible Fluid
		1.6.2 The Case of Linear Flows
		1.6.3 Vorticity Equation
	Exercises
Chapter 2 Geometric Algebra
	2.1 The Geometric Product
		2.1.1 Collinearity and Orthogonality
		2.1.2 The Dot and Wedge Products
		2.1.3 Geometric Algebra
	2.2 Rotations
		2.2.1 The Matrix Product
		2.2.2 The Geometric Product
	Exercises
Chapter 3 Compressible Navier–Stokes Equations
	3.1 Introduction
	3.2 A Solution Procedure For δ Arbitrarily Small in Quantity
	3.3 Characterization of the Sign of the Vorticity
	3.4 Non-Linear Further Reduction
	3.5 Stokes Theorem Applied to Dynamic Surfaces
	3.6 Analysis for Hunter Saxton Equation
	Exercises
Chapter 4 Hydrodynamic Stability and Maple
	4.1 Introduction
	4.2 Rayleigh–Taylor Instability
		4.2.1 Normal Mode Analysis
	4.3 The Rayleigh-Taylor Instability for Two Incompressible Fluids
	4.4 Rayleigh-Benard Convection
	4.5 Solution of Rayleigh Benard Convection with Maple
	4.6 Classical Kelvin-Helmholtz Instability
	Exercises
Chapter 5 Mathematics Preliminaries
	5.1 Introduction
	5.2 Mathematics Preliminaries
	5.3 The Lebesgue Integral
	Exercises
Chapter 6 Simplified Periodic Navier–Stokes (PNS) and Rayleigh-Plesset (RP) Equations
	6.1 Introduction
	6.2 Onset of Turbulence: Eddies and Vorticies in Incompressible Fluids
		6.2.1 3D Incompressible Navier–Stokes Equations
		6.2.2 Decomposition of NSE’s, Limit Cycles, and Vorticies
		6.2.3 Convergence & Singularities of Incompressible Eddies and Vortices along Edge of Cube Lattice
		6.2.4 Norm Analysis of Eq. (6.7)
	6.3 Novel Variational Formulation of Cavitation Dynamics
		6.3.1 Membrane Statistical Dynamics as a Variational Technique
		6.3.2 Identifying a Lagrangian Density for the Rayleigh-Plesset Equations
		6.3.3 Spherical Decomposition of Lagrangian Density
		6.3.4 Accounting for Energy Dissipation in a Rayleigh-Plesset Process
	6.4 Conclusion: Incompressible Eddies/Vortices & Cavitation Dynamics
	6.5 Appendix
Chapter 7 Introduction to Flows and Dynamical Systems
	7.1 Tangent Vectors
	7.2 Local Flows
	7.3 Applied Dynamical Systems and Bifurcation
		7.3.1 Flows on the Line
		7.3.2 Linear Stability Analysis
		7.3.3 Potential Functions
		7.3.4 Flows on the Circle
		7.3.5 Nonuniform Oscillator
	Exercises
Chapter 8 Numerical Analysis of 3D Periodic Navier–Stokes Equations and the Maple Environment
	8.1 Introduction to the Periodic Navier–Stokes Equations
	8.2 Equivalent Form of 3D Periodic Navier–Stokes Equations
		8.2.1 Decomposition of NSEs
		8.2.2 Liutex Vector and Respective Governing Equations
		8.2.3 Case 1
		8.2.4 JacobiSN Solution
		8.2.5 The Comparison of Blowup for Each of df**[sub(4)]/ds and Φ(s)
	8.3 Analysis of F**[sub(4)] (S) and Φ(S)
		8.3.1 Setting the Time Derivative of F[sub(4)] (s) Equal to – f[sub(0)] (s)
		8.3.2 Case 2
	8.4 General Case When Λ is Not Excluded
	8.5 No Finite Time Blowup When Pressure is Decreasing Cantor-Like Function
		8.5.1 The Cantor Set
	8.6 Observation of a Residual Set
		8.6.1 Base 3 Arithmetic
	8.7 The Cantor Function
	8.8 Lambert W Function
	8.9 Matching
		8.9.1 Analysis for the Non-Blowup on Turbulent Cantor-Dust
	8.10 Cantor Function Replaced by Linear Form
		8.10.1 Quantitative Bounds for Critically Bounded Solution to Navier–Stokes Equations
	8.11 Figures Confirming No Blowup for Third Component of Velocity of PNS Solution for Sums of Cantor Functions
	8.12 General Solution With No Restrictions on Forcing and Spatial Velocities
	8.13 Discussion and Conclusion
	8.14 Appendix 1
	8.15 Appendix 2
	Exercises
Chapter 9 Introduction to Fractional Calculus
	9.1 Introduction
		9.1.1 The Gamma Function
		9.1.2 Properties of Gamma Function
		9.1.3 An Important Representation of the Gamma Function
	9.2 Beta Function
	9.3 The Mittag-Leffler Function
	9.4 Riemann-Liouville Fractional Derivatives
	9.5 Integer Order Integration-Differentiation
	9.6 Integrals of Arbitrary Order
	9.7 Derivatives of Arbitrary Order
	9.8 Fractional Derivative Example
	9.9 Composition with Integer-Order Derivatives
	9.10 Composition with Fractional Derivatives
	9.11 Caputo’s Fractional Derivative
	9.12 Caputo Fractional Differential Operator
	9.13 Main Properties
		9.13.1 Representation of Caputo Fractional Derivative
		9.13.2 Interpolation Theory
		9.13.3 Linearity of Operator
		9.13.4 Non-Commutation Property
	9.14 The Laplace Transform
	9.15 Caputo Versus Riemann-Liouville Operator
	9.16 The Constant Function
	9.17 Connection with the Riemann-Liouville Operator
	9.18 Examples of Fractional Derivatives
	9.19 The Constant Function
	9.20 The Power Function
	9.21 The Exponential Function
	9.22 Some Other Functions
	9.23 Fractional Time and Multi-Fractional Space Incompressible and Compressible Navier–Stokes Equations
	9.24 Continuity Equation of Unsteady Fluid Flow in Fractional Time and Multi-Fractional Space
	9.25 Momentum Equations of Unsteady Flow in Fractional Time and Multi-Fractional Space
	Exercises
Chapter 10 Introduction to Simplicial Complexes, and Discrete Exterior Calculus (DEC)
	10.1 Introduction
		10.1.1 Previous Numerical Methods
		10.1.2 Discrete Exterior Calculus as Alternative to FD/FE-Methods
	10.2 Exterior Calculus Preliminaries
		10.2.1 Exterior Derivative on k-Forms, (dΣ)
		10.2.2 Hodge Star Operator, (*Σ)
		10.2.3 Co-Differential Operator and Analogues to the Divergence of Vector Fields on Manifolds
		10.2.4 Laplace-deRham Operator and Analogues to the Laplace-Beltrami Operator
		10.2.5 Integration, Stokes Theorem, and Ostrogradsky’s Theorem
	10.3 Exterior Calculus Discretization
		10.3.1 Simplicial Complexes
		10.3.2 Discretized Exterior Forms
		10.3.3 Exterior Derivative on Simplicial Complexes
		10.3.4 Hodge Star Operator on Simplicial Complexes
		10.3.5 Hodge-deRham Co-Homology
		10.3.6 Specialized Differential Operators
		10.3.7 Graphical Hodge-deRham Commutative Diagrams
	10.4 Conclusion
Chapter 11 Applications of Discrete Exterior Calculus (DEC) to Fluid Mechanics and Fluid-Structure Interactions
	11.1 Introduction
		11.1.1 Partial Differential Equation (Scalar): Poisson’s and Laplace’s Equation
		11.1.2 Partial Differential Equation (Vector): Poiseuille Flow and Stokes Flow
	11.2 Methods
		11.2.1 DEC Poisson Equation on Plane
		11.2.2 DEC Laplace Equation on Annulus
		11.2.3 DEC Poiseuille Flow with Steady-State Boundary and Initial Conditions
		11.2.4 DEC Stokes Flow around Hole with Steady-State Boundary and Initial Conditions
		11.2.5 Ptačkova Reconstruction
	11.3 Results
		11.3.1 Poisson Equation Solution on Plane
		11.3.2 Laplace Equation Solution on Annulus
		11.3.3 Poiseuille Flow with Steady-State Boundary and Initial Conditions
		11.3.4 Stokes Flow around Hole with Steady-State Boundary and Initial Conditions
	11.4 Discussion
		11.4.1 Solving Scalar PDEs
		11.4.2 Validation of Poiseuille Flow
		11.4.3 Observations on Stokes Flow
	11.5 Conclusion
Appendix A Computer Programs
Bibliography
Index