Theoretical fluid dynamics

Preface......Page 7
Acknowledgements......Page 8
 Useful Readings......Page 10
Contents......Page 12
1 Introduction......Page 16
 References......Page 19
 2.1 The Euler and Lagrange Picture......Page 20
 2.2 The Lagrange Derivative......Page 21
 2.3 Conservation Laws......Page 23
 2.4 Divergence-Free Vector Field......Page 25
 2.5 Fluid Boundaries......Page 27
 2.6 Phase Space Fluid......Page 29
 2.7 Moving Fluid Line......Page 30
 2.8 Internal Fluid Stress......Page 32
 2.9 Fluid Equations from Kinetic Theory......Page 43
 2.10 Streamlines and Pathlines......Page 46
 2.11 Vortex Line, Vortex Tube, and Line Vortex......Page 47
 2.12 Vortex Sheet......Page 50
 2.13 Vector Gradient in Cylindrical Coordinates......Page 52
 2.14 Vector Gradient in Orthogonal Coordinates......Page 54
 2.15 Vorticity Equation......Page 58
 2.16 Velocity from Vorticity......Page 61
 2.17 The Bernoulli Equation......Page 63
 2.18 The Euler–Lagrange Equations for Fluids......Page 65
 2.19 Water Waves from the Euler–Lagrange Equations......Page 71
 2.20 Stretching in an Isotropic Random Velocity Field......Page 76
 2.21 The Converse Poincaré Lemma......Page 78
 References......Page 91
 3.1 The Laplace Equation......Page 93
 3.2 Green's Theorems......Page 95
 3.3 The Dirichlet and Neumann Boundary Conditions......Page 96
 3.4 Mean Value and Maximum Property......Page 97
 3.5 Logarithmic Potential......Page 99
 3.6 Dirichlet's Principle......Page 102
 3.7 Streamfunction......Page 106
 3.8 Vorticity on a Sphere......Page 108
 3.9 Complex Speed and Potential......Page 110
 3.10 Analytic Functions. Conformal Transformation......Page 112
 3.11 The Schwarz–Christoffel Theorem......Page 114
 3.12 Mapping of Semi-infinite and Infinite Strips......Page 119
 3.13 The Riemann Surfaces......Page 122
 References......Page 127
 4.1 Straight Vortex......Page 129
 4.2 Corner Flow......Page 132
 4.3 Corner Flow with Viscosity......Page 135
 4.4 Flow Past a Flat Plate......Page 142
 4.5 The Blasius and Kutta–Joukowski Theorems......Page 144
 4.6 Plane Flow Past a Cylinder......Page 146
 4.7 The Kármán Vortex Street......Page 148
 4.8 Corner Eddy......Page 161
 4.9 Angular Momentum Transport......Page 166
 References......Page 173
 5.1 Free Streamlines......Page 175
 5.2 Flow Past a Step......Page 178
 5.3 Complex Potential and Speed Plane......Page 180
 5.4 Outflow from an Orifice......Page 181
 5.5 A Simple Wake Model......Page 186
 5.6 The Riabouchinsky Cavity......Page 192
 5.7 Levi-Civita's Method......Page 196
 5.8 Kolscher's Cusped Cavity......Page 198
 5.9 Re-Entrant Jet Cavity......Page 207
 5.10 Tilted Wedge......Page 208
 5.11 Weinstein's Theorem......Page 213
 References......Page 219
 6.1 The Kelvin–Helmholtz Circulation Theorem......Page 221
 6.2 The Bjerknes Circulation Theorem......Page 227
 6.3 The Kelvin–Helmholtz Instability......Page 230
 6.4 Vortex Chain Perturbation......Page 232
 6.5 Vortex Accumulation......Page 235
 6.6 Linear Stability Analysis......Page 239
 6.7 The Birkhoff–Rott Equation for Vortex Sheets......Page 244
 6.8 Curvature Singularity in Evolving Vortex Sheet......Page 248
 6.9 Subsequent Work on Moore's Singularity......Page 262
 6.10 Why Do Large Eddies Occur in Fast Flows?......Page 265
 6.11 Atmospheric Instability......Page 269
 6.12 The Rayleigh Inflexion Theorem......Page 271
 6.13 Kinematics of Vortex Rings......Page 272
 6.14 Curvature and Torsion......Page 275
 6.15 Helical Line Vortices......Page 277
 6.16 Knotted and Linked Vortex Rings......Page 280
 6.17 The Clebsch Coordinates and Knottedness......Page 284
 References......Page 285
7 Kinematics of Waves......Page 288
 7.1 Wave Basics......Page 289
 7.2 Group Speed......Page 290
 7.3 Kinematic Waves......Page 294
 7.4 The Wavefront......Page 295
 7.5 Waves and Instability from a Radiative Force......Page 297
 References......Page 306
8 Shallow Water Waves......Page 307
 8.1 Continuity Equation......Page 308
 8.2 The Euler Equations......Page 311
 8.3 Wave Equation for Linear Water Waves......Page 312
 8.4 Tides in Canals......Page 315
 8.5 Cotidal Lines and Amphidromic Points......Page 320
 8.6 Waves of Finite Amplitude......Page 324
 8.7 Nonlinear Tides in an Estuary......Page 327
 8.8 Similarity Solution: Dam Break......Page 335
 8.9 Non-breaking Waves......Page 340
 8.10 Bores......Page 346
 8.11 The Poincaré and Kelvin Waves......Page 353
 8.12 Wave Behind a Barrier......Page 358
 References......Page 376
9 Free Surface Waves......Page 379
 9.1 Dispersion Relation......Page 380
 9.2 Sudden Impulse......Page 383
 9.3 Refraction and Breaking at a Coast......Page 389
 9.4 Waves in a Nonuniform Stream......Page 397
 9.5 The Stokes Wave......Page 407
 9.6 The Stokes Singularity......Page 413
 9.7 The Crapper Wave......Page 420
 References......Page 428
 10.1 Introduction......Page 431
 10.2 Boundary Condition......Page 432
 10.3 Linear Integral Equations......Page 434
 10.4 Schmidt's Nonlinear Integral Equation......Page 445
 10.5 General Nonlinear Integral Equations......Page 451
 10.6 Integral Equations for Nonlinear Waves......Page 453
 References......Page 466
 11.1 Wave Equation......Page 468
 11.2 Acoustic Cutoff......Page 471
 11.3 The Schwarzschild Criterion......Page 475
 11.4 Gravo-Acoustic Waves......Page 478
 References......Page 482
 12.1 Shock Kinematics and Entropy......Page 484
 12.2 Jump Conditions at Shocks......Page 488
 12.3 Shock Speed......Page 495
 12.4 Shock Entropy and Supersonic Inflow......Page 496
 12.5 The Laval Nozzle and Solar Wind......Page 497
 12.6 Supersonic Spots......Page 503
 12.7 Solar Wind Exhibiting a Shock Pair......Page 513
 12.8 The Riemann Sheets for the Burgers Equation......Page 518
 12.9 Characteristics for First-Order Equations......Page 523
 12.10 Characteristics for Second-Order Equations......Page 530
 12.11 Derivatives on Characteristics......Page 532
 12.12 Simple Waves......Page 535
 References......Page 542
Appendix A Analytic and Meromorphic Functions......Page 545
 References......Page 554
Index......Page 555