Introduction to Theoretical and Mathematical Fluid Dynamics

Bhimsen K. Shivamoggi, PhD, is Professor in the Departments of Mathematics and Physics at the University of Central Florida. He is a Senior Fellow of the Japan Society for the Promotion of Science. His research is focused on mathematical physics, fluid dynamics, stochastic processes, and nonlinear dynamics.

• ContentsPreface to the Third Edition xvAcknowledgments xviiPart I Basic Concepts and Equations of Fluid Dynamics 11 Introduction to the Fluid Model 31.1 The Fluid State 41.2 Description of the Flow-Field 51.3 Volume Forces and Surface Forces 71.4 Relative Motion Near a Point 101.5 Stress–Strain Relations 132 Equations of Fluid Flows 152.1 The Transport Theorem 162.2 The Material Derivative 182.3 The Law of Conservation of Mass 182.4 Equation of Motion 192.5 The Energy Equation 192.6 The Equation of Vorticity 222.7 The Incompressible Fluid 232.8 Boundary Conditions 242.9 A Program for Analysis of the Governing Equations 253 Hamiltonian Formulation of Fluid-Flow Problems 273.1 Hamiltonian Dynamics of Continuous Systems 283.2 Three-Dimensional Incompressible Flows 323.3 Two-Dimensional Incompressible Flows 354 Surface Tension Effects 394.1 Shape of the Interface between Two Fluids 394.2 Capillary Rises in Liquids 41Part II Dynamics of Incompressible Fluid Flows 455 Fluid Kinematics and Dynamics 475.1 Stream Function 475.2 Equations of Motion 505.3 Integrals of Motion 505.4 Capillary Waves on a Spherical Drop 515.5 Cavitation 545.6 Rates of Change of Material Integrals 555.7 The Kelvin Circulation Theorem 575.8 The Irrotational Flow 585.9 Simple-Flow Patterns 62(i) The Source Flow 62(ii) The Doublet Flow 63(iii) The Vortex Flow 66(iv) Doublet in a Uniform Stream 66(v) Uniform Flow Past a Circular Cylinder with Circulation 676 The Complex-Variable Method 716.1 The Complex Potential 716.2 Conformal Mapping of Flows 746.3 Hydrodynamic Images 826.4 Principles of Free-Streamline Flow 84(i) Schwarz-Christoffel Transformation 84(ii) Hodograph Method 937 Three-Dimensional Irrotational Flows 997.1 Special Singular Solutions 99(i) The Source Flow 99(ii) The Doublet Flow 1017.2 d’Alembert’s Paradox 1047.3 Image of a Source in a Sphere 1057.4 Flow Past an Arbitrary Body 1077.5 Unsteady Flows 1097.6 Renormalized (or Added) Mass of Bodies Moving through a Fluid 1118 Vortex Flows 1158.1 Vortex Tubes 1158.2 Induced Velocity Field 1178.3 Biot-Savart’s Law 1178.4 von Kármán Vortex Street 1218.5 Vortex Ring 1248.6 Hill’s Spherical Vortex 1298.7 Vortex Sheet 1318.8 Vortex Breakdown: Brooke Benjamin’s Theory 1359 Rotating Flows 1439.1 Governing Equations and Elementary Results 1439.2 Taylor-Proudman Theorem 1449.3 Propagation of Inertial Waves in a Rotating Fluid 1469.4 Plane Inertial Waves 1479.5 Forced Wavemotion in a Rotating Fluid 150(i) The Elliptic Case 153(ii) The Hyperbolic Case 1549.6 Slow Motion along the Axis of Rotation 1559.7 Rossby Waves 16010 Water Waves 16710.1 Governing Equations 16810.2 A Variational Principle for Surface Waves 16910.3 Water Waves in a Semi-Infinite Fluid 17110.4 Water Waves in a Fluid Layer of Finite Depth 17210.5 Shallow-Water Waves 174(i) Analogy with Gas Dynamics 175(ii) Breaking of Waves 17610.6 Water Waves Generated by an Initial Displacement over a Localized Region 17610.7 Waves on a Steady Stream 182(i) One-Dimensional Gravity Waves 183(ii) One-Dimensional Capillary-Gravity Waves 184(iii) Ship Waves 18510.8 Gravity Waves in a Rotating Fluid 18810.9 Theory of Tides 19310.10 Hydraulic Jump 195(i) Tidal Bores 195(ii) The Dam-Break Problem 19910.11 Nonlinear Shallow-Water Waves 202(i) Solitary Waves 206(ii) Periodic Cnoidal Waves 208(iii) Interacting Solitary Waves 214(iv) Stokes Waves 219(v) Modulational Instability and Envelope Solutions 22010.12 Nonlinear Capillary-Gravity Waves 230(i) Resonant Three-Wave Interactions 230(ii) Second-Harmonic Resonance 23511 Applications to Aerodynamics 24111.1 Airfoil Theory: Method of Complex Variables 242(i) Force and Moments on an Arbitrary Body 242(ii) Flow Past an Arbitrary Cylinder 245(iii) Flow Around a Flat Plate 248(iv) Flow Past an Airfoil 250(v) The Joukowski Transformation 25311.2 Thin Airfoil Theory 259(i) Thickness Problem 262(ii) Camber Problem  264(iii) Flat Plate at an Angle of Attack 269(iv) Combined Aerodynamic Characteristics  271(v) The Leading-Edge Problem of a Thin Airfoil  27111.3 Slender-Body Theory 27511.4 Prandtl’s Lifting-Line Theory for Wings 27711.5 Oscillating Thin-Airfoil Problem: Theodorsen’s Theory 282Part III Dynamics of Compressible Fluid Flows 29712 Review of Thermodynamics 29912.1 Thermodynamic System and Variables of State 29912.2 The First Law of Thermodynamics and Reversible and Irreversible Processes 30012.3 The Second Law of Thermodynamics 30312.4 Entropy 30412.5 Liquid and Gaseous Phases 30713 Isentropic Fluid Flows  30913.1 Applications of Thermodynamics to Fluid Flows 30913.2 Linear Sound Wave Propagation 31013.3 The Energy Equation 31013.4 Stream-Tube Area and Flow Velocity Relations 31214 Potential Flows 31714.1 Governing Equations 31714.2 Streamline Coordinates 31914.3 Conical Flows: Prandtl-Meyer Flow 32014.4 Small Perturbation Theory 32414.5 Characteristics 326(i) Compatibility Conditions in Streamline Coordinates 328(ii) A Singular-Perturbation Problem for Hyperbolic Systems 33115 Nonlinear Theory of Plane Sound Waves 34315.1 Riemann Invariants 34315.2 Simple Wave Solutions 34415.3 Nonlinear Propagation of a Sound Wave 35215.4 Nonlinear Resonant Three-Wave Interactions of Sound Waves 35515.5 Burgers Equation 36116 Shock Waves 37116.1 The Normal Shock Wave 37116.2 The Oblique Shock Wave 38416.3 Blast Waves: Taylor’s Self-similarity and Sedov’s Exact Solution 38717 The Hodograph Method 39317.1 The Hodograph Transformation of Potential Flow Equations 39317.2 The Chaplygin Equation 39417.3 The Tangent-Gas Approximation 39617.4 The Lost Solution 40117.5 The Limit Line 40218 Applications to Aerodynamics 41118.1 Thin Airfoil Theory 411(i) Thin Airfoil in Linearized Supersonic Flows 411(ii) Far-Field Behavior of Supersonic Flow Past a Thin Airfoil 414(iii) Thin Airfoil in Transonic Flows 41718.2 Slender Bodies of Revolution 42018.3 Oscillating Thin Airfoil in Subsonic Flows: Possio’s Theory 42718.4 Oscillating Thin Airfoils in Supersonic Flows: Stewartson’s Theory 435Part IV Dynamics of Viscous Fluid Flows 43919 Exact Solutions to Equations of Viscous Fluid Flows 44119.1 Channel Flows  44219.2 Decay of a Line Vortex: The Lamb-Oseen Vortex  44319.3 Line Vortex in a Uniform Stream 44619.4 Diffusion of a Localized Vorticity Distribution 44619.5 Burgers Vortex  45119.6 Flow Due to a Suddenly Accelerated Plane  45319.7 The Round Laminar Jet: Landau-Squire Solution 45619.8 Ekman Layer at a Free Surface in a Rotating Fluid 45919.9 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution 46219.10 Shock Structure: Becker’s Solution 46419.11 Couette Flow of a Gas 46720 Flows at Low Reynolds Numbers 46920.1 Dimensional Analysis 46920.2 Stokes’ Flow Past a Rigid Sphere: Stokes’ Formula 47020.3 Stokes’ Flow Past a Spherical Drop 47420.4 Stokes’ Flow Past a Rigid Circular Cylinder: Stokes’ Paradox 47820.5 Oseen’s Flow Past a Rigid Sphere 47920.6 Oseen’s Approximation for Periodically Oscillating Wakes 48321 Flows at High Reynolds Numbers 48921.1 Prandtl’s Boundary-Layer Concept  48921.2 The Method of Matched Asymptotic Expansions 49021.3 Location and Nature of the Boundary Layers 49721.4 Incompressible Flow Past a Flat Plate  500(i) The Outer Expansion  501(ii) The Inner Expansion 502(iii) Flow Due to Displacement Thickness 50721.5 Separation of Flow in a Boundary Layer: Landau’s Theory 50921.6 Boundary Layers in Compressible Flows 512(i) Crocco’s Integral 514(ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation 51621.7 Flow in a Mixing Layer between Two Parallel Streams 517(i) Geometrical Characteristics of the Mixing Flow 52021.8 Narrow Jet: Bickley’s Solution  52121.9 Wakes  52421.10 Periodic Boundary Layer Flows 52422 Jeffrey-Hamel Flow  52922.1 The Exact Solution 529(i) Only 𝑒1 Is Real and Positive 531(ii) 𝑒1, 𝑒2, and 𝑒3 Are Real and Distinct  53222.2 Flows at Low Reynolds Numbers 53522.3 Flows at High Reynolds Numbers 541References 545Bibliography 549Index 551