Introduction to Theoretical and Mathematical Fluid Dynamics

Introduction to Theoretical and Mathematical Fluid Dynamics
Contents
Preface to the Third Edition
Acknowledgments
Part I Basic Concepts and Equations of Fluid Dynamics
	1 Introduction to the Fluid Model
		1.1 The Fluid State
		1.2 Description of the Flow-Field
		1.3 Volume Forces and Surface Forces
		1.4 Relative Motion Near a Point
		1.5 Stress–Strain Relations
	2 Equations of Fluid Flows
		2.1 The Transport Theorem
		2.2 The Material Derivative
		2.3 The Law of Conservation of Mass
		2.4 Equation of Motion
		2.5 The Energy Equation
		2.6 The Equation of Vorticity
		2.7 The Incompressible Fluid
		2.8 Boundary Conditions
		2.9 A Program for Analysis of the Governing Equations
	3 Hamiltonian Formulation of Fluid-Flow Problems
		3.1 Hamiltonian Dynamics of Continuous Systems
		3.2 Three-Dimensional Incompressible Flows
		3.3 Two-Dimensional Incompressible Flows
	4 Surface Tension Effects
		4.1 Shape of the Interface between Two Fluids
		4.2 Capillary Rises in Liquids
Part II Dynamics of Incompressible Fluid Flows
	5 Fluid Kinematics and Dynamics
		5.1 Stream Function
		5.2 Equations of Motion
		5.3 Integrals of Motion
		5.4 CapillaryWaves on a Spherical Drop
		5.5 Cavitation
		5.6 Rates of Change of Material Integrals
		5.7 The Kelvin Circulation Theorem
		5.8 The Irrotational Flow
		5.9 Simple-Flow Patterns
			(i) The Source Flow
			(ii) The Doublet Flow
			(iii) The Vortex Flow
			(iv) Doublet in a Uniform Stream
			(v) Uniform Flow Past a Circular Cylinder with Circulation
	6 The Complex-Variable Method
		6.1 The Complex Potential
		6.2 Conformal Mapping of Flows
		6.3 Hydrodynamic Images
		6.4 Principles of Free-Streamline Flow
			(i) Schwarz-Christoffel Transformation
			(ii) Hodograph Method
	7 Three-Dimensional Irrotational Flows
		7.1 Special Singular Solutions
			(i) The Source Flow
			(ii) The Doublet Flow
		7.2 d'Alembert's Paradox
		7.3 Image of a Source in a Sphere
		7.4 Flow Past an Arbitrary Body
		7.5 Unsteady Flows
		7.6 Renormalized (or Added) Mass of Bodies Moving through a Fluid
	8 Vortex Flows
		8.1 Vortex Tubes
		8.2 Induced Velocity Field
		8.3 Biot-Savart’s Law
		8.4 von Kármán Vortex Street
		8.5 Vortex Ring
		8.6 Hill's Spherical Vortex
		8.7 Vortex Sheet
		8.8 Vortex Breakdown: Brooke Benjamin's Theory
	9 Rotating Flows
		9.1 Governing Equations and Elementary Results
		9.2 Taylor-Proudman Theorem
		9.3 Propagation of InertialWaves in a Rotating Fluid
		9.4 Plane InertialWaves
		9.5 ForcedWavemotion in a Rotating Fluid
			(i) The Elliptic Case
			(ii) The Hyperbolic Case
		9.6 Slow Motion along the Axis of Rotation
		9.7 Rossby Waves
	10 Water Waves
		10.1 Governing Equations
		10.2 A Variational Principle for SurfaceWaves
		10.3 WaterWaves in a Semi-Infinite Fluid
		10.4 WaterWaves in a Fluid Layer of Finite Depth
		10.5 Shallow-Water Waves
			(i) Analogy with Gas Dynamics
			(ii) Breaking ofWaves
		10.6 Water Waves Generated by an Initial Displacementover a Localized Region
		10.7 Waves on a Steady Stream
			(i) One-Dimensional GravityWaves
			(ii) One-Dimensional Capillary-GravityWaves
			(iii) ShipWaves
		10.8 GravityWaves in a Rotating Fluid
		10.9 Theory of Tides
		10.10 Hydraulic Jump
			(i) Tidal Bores
			(ii) The Dam-Break Problem
		10.11 Nonlinear Shallow-WaterWaves
			(i) SolitaryWaves
			(ii) Periodic CnoidalWaves
			(iii) Interacting SolitaryWaves
			(iv) StokesWaves
			(v) Modulational Instability and Envelope Solutions
		10.12 Nonlinear Capillary-GravityWaves
			(i) Resonant Three-Wave Interactions
			(ii) Second-Harmonic Resonance
	11 Applications to Aerodynamics
		11.1 Airfoil Theory: Method of Complex Variables
			(i) Force and Moments on an Arbitrary Body
			(ii) Flow Past an Arbitrary Cylinder
			(iii) Flow Around a Flat Plate
			(iv) Flow Past an Airfoil
			(v) The Joukowski Transformation
		11.2 Thin Airfoil Theory
			(i) Thickness Problem
			(ii) Camber Problem
			(iii) Flat Plate at an Angle of Attack
			(iv) Combined Aerodynamic Characteristics
			(v) The Leading-Edge Problem of a Thin Airfoil
		11.3 Slender-Body Theory
		11.4 Prandtl’s Lifting-Line Theory for Wings
		11.5 Oscillating Thin-Airfoil Problem: Theodorsen’s Theory
Part III Dynamics of Compressible Fluid Flows
	12 Review of Thermodynamics
		12.1 Thermodynamic System and Variables of State
		12.2 The First Law of Thermodynamics and Reversible and Irreversible Processes
		12.3 The Second Law of Thermodynamics
		12.4 Entropy
		12.5 Liquid and Gaseous Phases
	13 Isentropic Fluid Flows
		13.1 Applications of Thermodynamics to Fluid Flows
		13.2 Linear SoundWave Propagation
		13.3 The Energy Equation
		13.4 Stream-Tube Area and Flow Velocity Relations
	14 Potential Flows
		14.1 Governing Equations
		14.2 Streamline Coordinates
		14.3 Conical Flows: Prandtl-Meyer Flow
		14.4 Small Perturbation Theory
		14.5 Characteristics
			(i) Compatibility Conditions in Streamline Coordinates
			(ii) A Singular-Perturbation Problem for Hyperbolic Systems
	15 Nonlinear Theory of Plane Sound Waves
		15.1 Riemann Invariants
		15.2 Simple Wave Solutions
		15.3 Nonlinear Propagation of a Sound Wave
		15.4 Nonlinear Resonant Three-Wave Interactions of Sound Waves
		15.5 Burgers Equation
	16 Shock Waves
		16.1 The Normal Shock Wave
		16.2 The Oblique Shock Wave
		16.3 Blast Waves: Taylor's Self-similarity and Sedov's Exact Solution
	17 The Hodograph Method
		17.1 The Hodograph Transformation of Potential Flow Equations
		17.2 The Chaplygin Equation
		17.3 The Tangent-Gas Approximation
		17.4 The Lost Solution
		17.5 The Limit Line
	18 Applications to Aerodynamics
		18.1 Thin Airfoil Theory
			(i) Thin Airfoil in Linearized Supersonic Flows
			(ii) Far-Field Behavior of Supersonic Flow Past a Thin Airfoil
			(iii) Thin Airfoil in Transonic Flows
		18.2 Slender Bodies of Revolution
		18.3 Oscillating Thin Airfoil in Subsonic Flows: Possio’s Theory
		18.4 Oscillating Thin Airfoils in Supersonic Flows: Stewartson’s Theory
Part IV Dynamics of Viscous Fluid Flows
	19 Exact Solutions to Equations of Viscous Fluid Flows
		19.1 Channel Flows
		19.2 Decay of a Line Vortex: The Lamb-Oseen Vortex
		19.3 Line Vortex in a Uniform Stream
		19.4 Diffusion of a Localized Vorticity Distribution
		19.5 Burgers Vortex
		19.6 Flow Due to a Suddenly Accelerated Plane
		19.7 The Round Laminar Jet: Landau-Squire Solution
		19.8 Ekman Layer at a Free Surface in a Rotating Fluid
		19.9 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution
		19.10 Shock Structure: Becker’s Solution
		19.11 Couette Flow of a Gas
	20 Flows at Low Reynolds Numbers
		20.1 Dimensional Analysis
		20.2 Stokes’ Flow Past a Rigid Sphere: Stokes’ Formula
		20.3 Stokes’ Flow Past a Spherical Drop
		20.4 Stokes’ Flow Past a Rigid Circular Cylinder: Stokes’ Paradox
		20.5 Oseen’s Flow Past a Rigid Sphere
		20.6 Oseen’s Approximation for Periodically Oscillating Wakes
	21 Flows at High Reynolds Numbers
		21.1 Prandtl’s Boundary-Layer Concept
		21.2 The Method of Matched Asymptotic Expansions
		21.3 Location and Nature of the Boundary Layers
		21.4 Incompressible Flow Past a Flat Plate
			(i) The Outer Expansion
			(ii) The Inner Expansion
			(iii) Flow Due to Displacement Thickness
		21.5 Separation of Flow in a Boundary Layer: Landau’s Theory
		21.6 Boundary Layers in Compressible Flows
			(i) Crocco’s Integral
			(ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation
		21.7 Flow in a Mixing Layer between Two Parallel Streams
			(i) Geometrical Characteristics of the Mixing Flow
		21.8 Narrow Jet: Bickley’s Solution
		21.9 Wakes
		21.10 Periodic Boundary Layer Flows
	22 Jeffrey-Hamel Flow
		22.1 The Exact Solution
			(i) Only