Introduction to Fluid Dynamics in Physics and Astrophysics

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Chapter 1: Preliminaries
	1.1. Vectors and Their Bases
		1.1.1. Matrix Notation
		1.1.2. Limitations of Matrix Notation
		1.1.3. The Cross-Product and Coordinate Transformations
		1.1.4. Gradient, Divergence and Curl
		1.1.5. Gradient, Divergence, Curl and Integration
		1.1.6. Divergence, Curl and Curvilinear Coordinates
	1.2. Index Notation
		1.2.1. The Cross Product and Index Notation
		1.2.2. Four-vectors, Indices and (Square) Brackets
	1.3. Covariant Components and the Dual Basis
	1.4. Partial and Full Differentiation
		1.4.1. Exact Differentials
		1.4.2. Inexact Differentials
Chapter 2: The Conservation Laws of Fluid Dynamics
	2.1. A Derivation of the Conservation Laws
		2.1.1. Mass Conservation: The Continuity Equation
		2.1.2. Sneak Peek at Full Set of Euler’s Equations
		2.1.3. Derivation of the Momentum Equation
		2.1.4. Derivation of the Energy Equation
	2.2. Fluid Dynamics and the Continuum Approximation
		2.2.1. The Continuum Approximation and the Mean Free Path
	2.3. Energy Transport in Atmospheres and Other Fluids
	2.4. Boltzmann’s Equation and Fluid Dynamics
		2.4.1. The Continuity Equation
		2.4.2. The Momentum Equation
		2.4.3. Equations of State
Chapter 3: Lagrangian Fluid Dynamics
	3.1. Fluid Parcels
	3.2. Derivation of Conservation Laws in a Lagrangian Approach
		3.2.1. Example: Stellar Wind
	3.3. Mass Coordinates
	3.4. Polytropic Processes and Thermodynamics
		3.4.1. The Polytropic Exponent of a Gas Dominated by Radiation Pressure
Chapter 4: Hydrostatics, Atmospheres and Stellar Structure
	4.1. Hydrostatics of a Plane-Parallel Atmosphere
	4.2. An Isothermal Slab Model
		4.2.1. Some Quick Notes on Gravity
		4.2.2. General Isothermal Slab Density Profiles
	4.3. Idealized Stellar Structure Models
		4.3.1. Isothermal Spheres
		4.3.2. Polytropic Stellar Structure Models
		4.3.3. The Physics of Polytropes
	4.4. Stellar Structure Modelling Using Mass Coordinates
Chapter 5: Sound Waves and Sub-/supersonic Flow
	5.1. The Wave Nature of Sound
	5.2. Acoustic Waves
		5.2.1. Wave Packets and Fourier Analysis
		5.2.2. The Energy of an Acoustic Sound Wave
		5.2.3. Spherical Acoustic Waves
	5.3. Sound Waves in a Gravitational Field
Chapter 6: Properties and Kinematics of Fluid Flow
	6.1. An Overview of Terminology
	6.2. Streamlines, Streaklines and Path Lines
	6.3. Flow Lines of an Incompressible Fluid
	6.4. Bernoulli’s Equation
	6.5. The de Laval Nozzle
	6.6. Vorticity
	6.7. Potential Flow, Irrotational Flow and Incompressible Flow
Chapter 7: Shock Waves
	7.1. The Shock-Jump Conditions
	7.2. Compression Shocks, Rarefaction Waves and Contact Discontinuities
	7.3. The Entropy Change Across a Shock
	7.4. Blast Waves
	7.5. Self-Similar Explosions
	7.6. First-Order Fermi Acceleration Across Strong Shocks
Chapter 8: Fluid Dynamics in Special Relativity
	8.1. Core Concepts in Special Relativity
		8.1.1. Four-Dimensional Spacetime
		8.1.2. Point Particles
	8.2. Special Relativistic Fluid Dynamical Equations
		8.2.1. The Continuity Equation
		8.2.2. Conservation of Energy-Momentum
	8.3. The Microphysics of Relativistic Gases
	8.4. Relativistic Shocks
		8.4.1. Strong Shocks
	8.5. Relativistic Blast Waves
	8.6. Self-Similar Relativistic Explosions
	8.7. First-Order Fermi Shock-Acceleration Revisited
Chapter 9: Viscous Flow
	9.1. The Navier-Stokes Equation
		9.1.1. Angular Momentum and the Stress Tensor
		9.1.2. Bulk Viscosity
		9.1.3. Shear Viscosity
	9.2. Viscosity and Dissipation
	9.3. Physical Interpretation of Shear Viscosity
	9.4. Flow Through a Pipe
	9.5. Two Example Similarity Parameters in Viscous Flows
		9.5.1. The Reynold’s Number
		9.5.2. The Prandtl Number
Chapter 10: Fluid Instabilities
	10.1. Convection and Stability
	10.2. The Rayleigh-Taylor Instability
	10.3. The Kelvin-Helmholtz Instability
	10.4. Gravitational Instability
	10.5. Thermal Instability
	10.6. Homogeneous and Isotropic Turbulence
Chapter 11: Accretion Flow
	11.1. Accretion as a Source of Energy in Astrophysics
	11.2. Bondi Accretion
		11.2.1. The Sonic Point
		11.2.2. The Bondi Accretion Rate
	11.3. The Eddington Luminosity
	11.4. Accretion Discs
		11.4.1. Thin Discs
		11.4.2. Accretion Discs and Viscosity
Chapter 12: Concepts in Plasma Physics
	12.1. Introduction
	12.2. Incorporating Maxwell’s Equations
		12.2.1. The Magnetic Field Equation
		12.2.2. The Equations of Magnetohydrodynamics
	12.3. The Nature of Plasmas
	12.4. Field Freezing
	12.5. Magnetohydrodynamic Waves
	12.6. Two Example Similarity Parameters in Plasma Physics
Chapter 13: Computational Fluid Dynamics
	13.1. Euler’s Equations in Terms of a State Vector Equation
	13.2. Rudimentary Finite Difference Schemes
	13.3. Stability, Accuracy and Diffusion
		13.3.1. Stability
		13.3.2. Accuracy
		13.3.3. Diffusion and Dispersion
		13.3.4. Lax-Friedrichs & Lax-Wendroff
	13.4. Boundary Conditions
	13.5. Finite Volume Methods
		13.5.1. FDM vs FVM
		13.5.2. Basic Finite Volume Methods
		13.5.3. Working Code
		13.5.4. The HLL Method and Godunov Approach
		13.5.5. Wave Speed Estimates
		13.5.6. The HLLC Method
	13.6. Higher Order in Time and Space
		13.6.1. Higher Order in Space
		13.6.2. Higher Order in Time
	13.7. Alternatives and Extensions to the Finite Volume Approach
		13.7.1. Adapting the Fixed Mesh
		13.7.2. Arbitrary Lagrangian-Eulerian Methods
		13.7.3. Smoothed Particle Hydrodynamics
		13.7.4. Finite Element Methods
	13.8. Computational Hydrodynamics and Special Relativity
Appendix A: Concepts from Thermodynamics
	A.1. The First and Second Law of Thermodynamics
	A.2. Legendre Transforms
	A.3. Heat Capacities
	A.4. The Ideal Gas Law and Perfect Gases
	A.5. The Energy Equation in Stellar Structure Modelling
Appendix B: Vector Identities and Derivatives
	B.1. Vector Product Rules
	B.2. Higher-Order Derivatives
	B.3. Dot and Cross-Product Combinations
	B.4. Spherical Coordinates
	B.5. Cylindrical Coordinates
Appendix C: Euler’s Equations in Non-Cartesian Coordinate Systems
	C.1. Cylindrical Coordinates
	C.2. Spherical Coordinates
Appendix D: List of Symbols
Appendix E: Abbreviated Answers to Selected Problems
References
Index