In Introduction to Theoretical and Mathematical Fluid Dynamics, distinguished researcher Dr. Bhimsen K. Shivamoggi delivers a comprehensive and insightful exploration of fluid dynamics from a mathematical point of view. The book introduces readers to the mathematical study of fluid behavior and highlights areas of active research in fluid dynamics. With coverage of advances in the field over the last 15 years, this book provides in-depth examinations of theoretical and mathematical fluid dynamics with a particular focus on incompressible and compressible fluid flows.
Introduction to Theoretical and Mathematical Fluid Dynamics includes practical applications and exercises to illustrate the concepts discussed within, and real-world examples are explained throughout the text. Clear and explanatory material accompanies the rigorous mathematics, making the book perfect for students seeking to learn and retain this complex subject.
The book also offers:
- A thorough introduction to the basic concepts and equations of fluid dynamics, including an introduction to the fluid model, the equations of fluid flows, and surface tension effects
- Comprehensive explorations of the dynamics of incompressible fluid flows, fluid kinematics and dynamics, the complex-variable method, and three-dimensional irrotational flows
- Practical discussions of the dynamics of compressible fluid flows, including a review of thermodynamics, isentropic fluid flows, potential flows, and nonlinear theory of plane sound waves
Ideal for graduate-level students taking courses on mathematical fluid dynamics as part of a program in mathematics, engineering, or physics, Introduction to Theoretical and Mathematical Fluid Dynamics is also an indispensable resource for practicing applied mathematicians, engineers, and physicists.
Author(s): Bhimsen K. Shivamoggi
Edition: 1
Publisher: Wiley
Year: 2022
Language: English
Pages: 576
Tags: Fluid Dynamics, Incompressible Fluid Flow, Water Waves, Compressible Fluid Flow, Shock Waves, Aerodynamics, Viscous Fluid Flow, Reynolds Numbers
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 ?1 Is Real and Positive
(ii) ?1, ?2, and ?3 Are Real and Distinct
22.2 Flows at Low Reynolds Numbers
22.3 Flows at High Reynolds Numbers
References
Bibliography
Index
