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.

Contents

Preface to the Third Edition xv

Acknowledgments xvii

Part I Basic Concepts and Equations of Fluid Dynamics 1

1 Introduction to the Fluid Model 3

1.1 The Fluid State 4

1.2 Description of the Flow-Field 5

1.3 Volume Forces and Surface Forces 7

1.4 Relative Motion Near a Point 10

1.5 Stress–Strain Relations 13

2 Equations of Fluid Flows 15

2.1 The Transport Theorem 16

2.2 The Material Derivative 18

2.3 The Law of Conservation of Mass 18

2.4 Equation of Motion 19

2.5 The Energy Equation 19

2.6 The Equation of Vorticity 22

2.7 The Incompressible Fluid 23

2.8 Boundary Conditions 24

2.9 A Program for Analysis of the Governing Equations 25

3 Hamiltonian Formulation of Fluid-Flow Problems 27

3.1 Hamiltonian Dynamics of Continuous Systems 28

3.2 Three-Dimensional Incompressible Flows 32

3.3 Two-Dimensional Incompressible Flows 35

4 Surface Tension Effects 39

4.1 Shape of the Interface between Two Fluids 39

4.2 Capillary Rises in Liquids 41

Part II Dynamics of Incompressible Fluid Flows 45

5 Fluid Kinematics and Dynamics 47

5.1 Stream Function 47

5.2 Equations of Motion 50

5.3 Integrals of Motion 50

5.4 Capillary Waves on a Spherical Drop 51

5.5 Cavitation 54

5.6 Rates of Change of Material Integrals 55

5.7 The Kelvin Circulation Theorem 57

5.8 The Irrotational Flow 58

5.9 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 67

6 The Complex-Variable Method 71

6.1 The Complex Potential 71

6.2 Conformal Mapping of Flows 74

6.3 Hydrodynamic Images 82

6.4 Principles of Free-Streamline Flow 84

(i) Schwarz-Christoffel Transformation 84

(ii) Hodograph Method 93

7 Three-Dimensional Irrotational Flows 99

7.1 Special Singular Solutions 99

(i) The Source Flow 99

(ii) The Doublet Flow 101

7.2 d’Alembert’s Paradox 104

7.3 Image of a Source in a Sphere 105

7.4 Flow Past an Arbitrary Body 107

7.5 Unsteady Flows 109

7.6 Renormalized (or Added) Mass of Bodies Moving through a Fluid 111

8 Vortex Flows 115

8.1 Vortex Tubes 115

8.2 Induced Velocity Field 117

8.3 Biot-Savart’s Law 117

8.4 von Kármán Vortex Street 121

8.5 Vortex Ring 124

8.6 Hill’s Spherical Vortex 129

8.7 Vortex Sheet 131

8.8 Vortex Breakdown: Brooke Benjamin’s Theory 135

9 Rotating Flows 143

9.1 Governing Equations and Elementary Results 143

9.2 Taylor-Proudman Theorem 144

9.3 Propagation of Inertial Waves in a Rotating Fluid 146

9.4 Plane Inertial Waves 147

9.5 Forced Wavemotion in a Rotating Fluid 150

(i) The Elliptic Case 153

(ii) The Hyperbolic Case 154

9.6 Slow Motion along the Axis of Rotation 155

9.7 Rossby Waves 160

10 Water Waves 167

10.1 Governing Equations 168

10.2 A Variational Principle for Surface Waves 169

10.3 Water Waves in a Semi-Infinite Fluid 171

10.4 Water Waves in a Fluid Layer of Finite Depth 172

10.5 Shallow-Water Waves 174

(i) Analogy with Gas Dynamics 175

(ii) Breaking of Waves 176

10.6 Water Waves Generated by an Initial Displacement over a Localized Region 176

10.7 Waves on a Steady Stream 182

(i) One-Dimensional Gravity Waves 183

(ii) One-Dimensional Capillary-Gravity Waves 184

(iii) Ship Waves 185

10.8 Gravity Waves in a Rotating Fluid 188

10.9 Theory of Tides 193

10.10 Hydraulic Jump 195

(i) Tidal Bores 195

(ii) The Dam-Break Problem 199

10.11 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 220

10.12 Nonlinear Capillary-Gravity Waves 230

(i) Resonant Three-Wave Interactions 230

(ii) Second-Harmonic Resonance 235

11 Applications to Aerodynamics 241

11.1 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 253

11.2 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  271

11.3 Slender-Body Theory 275

11.4 Prandtl’s Lifting-Line Theory for Wings 277

11.5 Oscillating Thin-Airfoil Problem: Theodorsen’s Theory 282

Part III Dynamics of Compressible Fluid Flows 297

12 Review of Thermodynamics 299

12.1 Thermodynamic System and Variables of State 299

12.2 The First Law of Thermodynamics and Reversible and Irreversible Processes 300

12.3 The Second Law of Thermodynamics 303

12.4 Entropy 304

12.5 Liquid and Gaseous Phases 307

13 Isentropic Fluid Flows  309

13.1 Applications of Thermodynamics to Fluid Flows 309

13.2 Linear Sound Wave Propagation 310

13.3 The Energy Equation 310

13.4 Stream-Tube Area and Flow Velocity Relations 312

14 Potential Flows 317

14.1 Governing Equations 317

14.2 Streamline Coordinates 319

14.3 Conical Flows: Prandtl-Meyer Flow 320

14.4 Small Perturbation Theory 324

14.5 Characteristics 326

(i) Compatibility Conditions in Streamline Coordinates 328

(ii) A Singular-Perturbation Problem for Hyperbolic Systems 331

15 Nonlinear Theory of Plane Sound Waves 343

15.1 Riemann Invariants 343

15.2 Simple Wave Solutions 344

15.3 Nonlinear Propagation of a Sound Wave 352

15.4 Nonlinear Resonant Three-Wave Interactions of Sound Waves 355

15.5 Burgers Equation 361

16 Shock Waves 371

16.1 The Normal Shock Wave 371

16.2 The Oblique Shock Wave 384

16.3 Blast Waves: Taylor’s Self-similarity and Sedov’s Exact Solution 387

17 The Hodograph Method 393

17.1 The Hodograph Transformation of Potential Flow Equations 393

17.2 The Chaplygin Equation 394

17.3 The Tangent-Gas Approximation 396

17.4 The Lost Solution 401

17.5 The Limit Line 402

18 Applications to Aerodynamics 411

18.1 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 417

18.2 Slender Bodies of Revolution 420

18.3 Oscillating Thin Airfoil in Subsonic Flows: Possio’s Theory 427

18.4 Oscillating Thin Airfoils in Supersonic Flows: Stewartson’s Theory 435

Part IV Dynamics of Viscous Fluid Flows 439

19 Exact Solutions to Equations of Viscous Fluid Flows 441

19.1 Channel Flows  442

19.2 Decay of a Line Vortex: The Lamb-Oseen Vortex  443

19.3 Line Vortex in a Uniform Stream 446

19.4 Diffusion of a Localized Vorticity Distribution 446

19.5 Burgers Vortex  451

19.6 Flow Due to a Suddenly Accelerated Plane  453

19.7 The Round Laminar Jet: Landau-Squire Solution 456

19.8 Ekman Layer at a Free Surface in a Rotating Fluid 459

19.9 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution 462

19.10 Shock Structure: Becker’s Solution 464

19.11 Couette Flow of a Gas 467

20 Flows at Low Reynolds Numbers 469

20.1 Dimensional Analysis 469

20.2 Stokes’ Flow Past a Rigid Sphere: Stokes’ Formula 470

20.3 Stokes’ Flow Past a Spherical Drop 474

20.4 Stokes’ Flow Past a Rigid Circular Cylinder: Stokes’ Paradox 478

20.5 Oseen’s Flow Past a Rigid Sphere 479

20.6 Oseen’s Approximation for Periodically Oscillating Wakes 483

21 Flows at High Reynolds Numbers 489

21.1 Prandtl’s Boundary-Layer Concept  489

21.2 The Method of Matched Asymptotic Expansions 490

21.3 Location and Nature of the Boundary Layers 497

21.4 Incompressible Flow Past a Flat Plate  500

(i) The Outer Expansion  501

(ii) The Inner Expansion 502

(iii) Flow Due to Displacement Thickness 507

21.5 Separation of Flow in a Boundary Layer: Landau’s Theory 509

21.6 Boundary Layers in Compressible Flows 512

(i) Crocco’s Integral 514

(ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation 516

21.7 Flow in a Mixing Layer between Two Parallel Streams 517

(i) Geometrical Characteristics of the Mixing Flow 520

21.8 Narrow Jet: Bickley’s Solution  521

21.9 Wakes  524

21.10 Periodic Boundary Layer Flows 524

22 Jeffrey-Hamel Flow  529

22.1 The Exact Solution 529

(i) Only 𝑒1 Is Real and Positive 531

(ii) 𝑒1, 𝑒2, and 𝑒3 Are Real and Distinct  532

22.2 Flows at Low Reynolds Numbers 535

22.3 Flows at High Reynolds Numbers 541

References 545

Bibliography 549

Index 551