INTRODUCTION TO THEORETICAL AND MATHEMATICAL FLUID DYNAMICS

A practical treatment of mathematical fluid dynamics

InIntroduction 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 effectsComprehensive explorations of the dynamics of incompressible fluid flows, fluid kinematics and dynamics, the complex-variable method, and three-dimensional irrotational flowsDetailed discussions of the dynamics of compressible fluid flows, including a review of thermodynamics, isentropic fluid flows, potential flows, and nonlinear theory of plane sound wavesSystematic discussions of the dynamics of viscous fluid flows, including shear-layer flow, jet flow and wake flow.

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.

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 Editionxv

Acknowledgmentsxvii

Part I Basic Concepts and Equations of Fluid Dynamics1

1 Introduction to the Fluid Model3

1.1 The Fluid State4

1.2 Description of the Flow-Field5

1.3 Volume Forces and Surface Forces7

1.4 Relative Motion Near a Point10

1.5 StressStrain Relations13

2 Equations of Fluid Flows15

2.1 The Transport Theorem16

2.2 The Material Derivative18

2.3 The Law of Conservation of Mass18

2.4 Equation of Motion19

2.5 The Energy Equation19

2.6 The Equation of Vorticity22

2.7 The Incompressible Fluid23

2.8 Boundary Conditions24

2.9 A Program for Analysis of the Governing Equations25

3 Hamiltonian Formulation of Fluid-Flow Problems27

3.1 Hamiltonian Dynamics of Continuous Systems28

3.2 Three-Dimensional Incompressible Flows32

3.3 Two-Dimensional Incompressible Flows35

4 Surface Tension Effects39

4.1 Shape of the Interface between Two Fluids39

4.2 Capillary Rises in Liquids41

Part II Dynamics of Incompressible Fluid Flows45

5 Fluid Kinematics and Dynamics47

5.1 Stream Function47

5.2 Equations of Motion50

5.3 Integrals of Motion50

5.4 Capillary Waves on a Spherical Drop51

5.5 Cavitation54

5.6 Rates of Change of Material Integrals55

5.7 The Kelvin Circulation Theorem57

5.8 The Irrotational Flow58

5.9 Simple-Flow Patterns62

(i) The Source Flow62

(ii) The Doublet Flow63

(iii) The Vortex Flow66

(iv) Doublet in a Uniform Stream66

(v) Uniform Flow Past a Circular Cylinder with Circulation67

6 The Complex-Variable Method71

6.1 The Complex Potential71

6.2 Conformal Mapping of Flows74

6.3 Hydrodynamic Images82

6.4 Principles of Free-Streamline Flow84

(i) Schwarz-Christoffel Transformation84

(ii) Hodograph Method93

7 Three-Dimensional Irrotational Flows99

7.1 Special Singular Solutions99

(i) The Source Flow99

(ii) The Doublet Flow101

7.2 dAlemberts Paradox104

7.3 Image of a Source in a Sphere105

7.4 Flow Past an Arbitrary Body107

7.5 Unsteady Flows109

7.6 Renormalized (or Added) Mass of Bodies Moving through a Fluid111

8 Vortex Flows115

8.1 Vortex Tubes115

8.2 Induced Velocity Field117

8.3 Biot-Savarts Law117

8.4 von Kármán Vortex Street121

8.5 Vortex Ring124

8.6 Hills Spherical Vortex129

8.7 Vortex Sheet131

8.8 Vortex Breakdown: Brooke Benjamins Theory135

9 Rotating Flows143

9.1 Governing Equations and Elementary Results143

9.2 Taylor-Proudman Theorem144

9.3 Propagation of Inertial Waves in a Rotating Fluid146

9.4 Plane Inertial Waves147

9.5 Forced Wavemotion in a Rotating Fluid150

(i) The Elliptic Case153

(ii) The Hyperbolic Case154

9.6 Slow Motion along the Axis of Rotation155

9.7 Rossby Waves160

10 Water Waves167

10.1 Governing Equations168

10.2 A Variational Principle for Surface Waves169

10.3 Water Waves in a Semi-Infinite Fluid171

10.4 Water Waves in a Fluid Layer of Finite Depth172

10.5 Shallow-Water Waves174

(i) Analogy with Gas Dynamics175

(ii) Breaking of Waves176

10.6 Water Waves Generated by an Initial Displacement over a Localized Region176

10.7 Waves on a Steady Stream182

(i) One-Dimensional Gravity Waves183

(ii) One-Dimensional Capillary-Gravity Waves184

(iii) Ship Waves185

10.8 Gravity Waves in a Rotating Fluid188

10.9 Theory of Tides193

10.10 Hydraulic Jump195

(i) Tidal Bores195

(ii) The Dam-Break Problem199

10.11 Nonlinear Shallow-Water Waves202

 (i) Solitary Waves206

(ii) Periodic Cnoidal Waves208

(iii) Interacting Solitary Waves214

(iv) Stokes Waves219

(v) Modulational Instability and Envelope Solutions220

10.12 Nonlinear Capillary-Gravity Waves230

(i) Resonant Three-Wave Interactions230

(ii) Second-Harmonic Resonance235

11 Applications to Aerodynamics241

11.1 Airfoil Theory: Method of Complex Variables242

(i) Force and Moments on an Arbitrary Body242

(ii) Flow Past an Arbitrary Cylinder245

(iii) Flow Around a Flat Plate248

(iv) Flow Past an Airfoil250

(v) The Joukowski Transformation253

11.2 Thin Airfoil Theory259

(i) Thickness Problem262

(ii) Camber Problem264

(iii) Flat Plate at an Angle of Attack269

(iv) Combined Aerodynamic Characteristics271

(v) The Leading-Edge Problem of a Thin Airfoil271

11.3 Slender-Body Theory275

11.4 Prandtls Lifting-Line Theory for Wings277

11.5 Oscillating Thin-Airfoil Problem: Theodorsens Theory282

Part III Dynamics of Compressible Fluid Flows297

12 Review of Thermodynamics299

12.1 Thermodynamic System and Variables of State299

12.2 The First Law of Thermodynamics and Reversible and Irreversible Processes300

12.3 The Second Law of Thermodynamics303

12.4 Entropy304

12.5 Liquid and Gaseous Phases307

13 Isentropic Fluid Flows309

13.1 Applications of Thermodynamics to Fluid Flows309

13.2 Linear Sound Wave Propagation310

13.3 The Energy Equation310

13.4 Stream-Tube Area and Flow Velocity Relations312

14 Potential Flows317

14.1 Governing Equations317

14.2 Streamline Coordinates319

14.3 Conical Flows: Prandtl-Meyer Flow320

14.4 Small Perturbation Theory324

14.5 Characteristics326

(i) Compatibility Conditions in Streamline Coordinates328

(ii) A Singular-Perturbation Problem for Hyperbolic Systems331

15 Nonlinear Theory of Plane Sound Waves343

15.1 Riemann Invariants343

15.2 Simple Wave Solutions344

15.3 Nonlinear Propagation of a Sound Wave352

15.4 Nonlinear Resonant Three-Wave Interactions of Sound Waves355

15.5 Burgers Equation361

16 Shock Waves371

16.1 The Normal Shock Wave371

16.2 The Oblique Shock Wave384

16.3 Blast Waves: Taylors Self-similarity and Sedovs Exact Solution387

17 The Hodograph Method393

17.1 The Hodograph Transformation of Potential Flow Equations393

17.2 The Chaplygin Equation394

17.3 The Tangent-Gas Approximation396

17.4 The Lost Solution401

17.5 The Limit Line402

18 Applications to Aerodynamics411

18.1 Thin Airfoil Theory411

(i) Thin Airfoil in Linearized Supersonic Flows411

(ii) Far-Field Behavior of Supersonic Flow Past a Thin Airfoil414

(iii) Thin Airfoil in Transonic Flows417

18.2 Slender Bodies of Revolution420

18.3 Oscillating Thin Airfoil in Subsonic Flows: Possios Theory427

18.4 Oscillating Thin Airfoils in Supersonic Flows: Stewartsons Theory435

Part IV Dynamics of Viscous Fluid Flows439

19 Exact Solutions to Equations of Viscous Fluid Flows441

19.1 Channel Flows442

19.2 Decay of a Line Vortex: The Lamb-Oseen Vortex443

19.3 Line Vortex in a Uniform Stream446

19.4 Diffusion of a Localized Vorticity Distribution446

19.5 Burgers Vortex451

19.6 Flow Due to a Suddenly Accelerated Plane453

19.7 The Round Laminar Jet: Landau-Squire Solution456

19.8 Ekman Layer at a Free Surface in a Rotating Fluid459

19.9 Centrifugal Flow Due to a Rotating Disk: von Kármán Solution462

19.10 Shock Structure: Beckers Solution464

19.11 Couette Flow of a Gas467

20 Flows at Low Reynolds Numbers469

20.1 Dimensional Analysis469

20.2 Stokes Flow Past a Rigid Sphere: Stokes Formula470

20.3 Stokes Flow Past a Spherical Drop474

20.4 Stokes Flow Past a Rigid Circular Cylinder: Stokes Paradox478

20.5 Oseens Flow Past a Rigid Sphere479

20.6 Oseens Approximation for Periodically Oscillating Wakes483

21 Flows at High Reynolds Numbers489

21.1 Prandtls Boundary-Layer Concept489

21.2 The Method of Matched Asymptotic Expansions490

21.3 Location and Nature of the Boundary Layers497

21.4 Incompressible Flow Past a Flat Plate500

(i) The Outer Expansion501

(ii) The Inner Expansion502

(iii) Flow Due to Displacement Thickness507

21.5 Separation of Flow in a Boundary Layer: Landaus Theory509

21.6 Boundary Layers in Compressible Flows512

(i) Croccos Integral514

(ii) Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation516

21.7 Flow in a Mixing Layer between Two Parallel Streams517

(i) Geometrical Characteristics of the Mixing Flow520

21.8 Narrow Jet: Bickleys Solution521

21.9 Wakes524

21.10 Periodic Boundary Layer Flows524

22 Jeffrey-Hamel Flow529

22.1 The Exact Solution529

(i) Only𝑒1 Is Real and Positive531

(ii)𝑒1,𝑒2, and𝑒3 Are Real and Distinct532

22.2 Flows at Low Reynolds Numbers535

22.3 Flows at High Reynolds Numbers541

References545

Bibliography549

Index551