Achim Feldmeier is an astrophysicist and obtained his PhD in astronomy from Ludwig-Maximilians-Universität in München in 1994. He was postdoc at the University of Kentucky in Lexington and at Imperial College in London.

Since 2000 he works at the Universität Potsdam, where he is apl professor since 2006. He gave numerous courses in hydrodynamics and his research work is on flow properties of stellar winds.

1 Description of fluids 5

1.1 Euler and Lagrange picture . . . . . . . . . . . . . . . . . . . . 5

1.2 Lagrange derivative . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Divergence-free vector field . . . . . . . . . . . . . . . . . . . . 10

1.5 Fluid boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Phase space fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.7 Moving fluid line . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Internal fluid stress . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.9 Fluid equations from kinetic theory . . . . . . . . . . . . . . . 29

1.10 Streamlines and Pathlines . . . . . . . . . . . . . . . . . . . . . 32

1.11 Vortex line, vortex tube and line vortex . . . . . . . . . . . . . 33

1.12 Vortex sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.13 Vector gradient in cylindrical coordinates . . . . . . . . . . . . 39

1.14 Vector gradient in orthogonal coordinates . . . . . . . . . . . . 41

1.15 Vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.16 Velocity from vorticity . . . . . . . . . . . . . . . . . . . . . . . . 47

1.17 Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.18 Euler-Lagrange equation for fluids . . . . . . . . . . . . . . . . 52

1.19 Water waves from Euler-Lagrange equations . . . . . . . . . . 58

1.20 Stretching in an isotropic random velocity field . . . . . . . . 63

1.21 Converse Poincaré lemma . . . . . . . . . . . . . . . . . . . . . 65

2 Flows in the complex plane 79

2.1 Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.2 Green's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.3 Dirichlet and Neumann boundary conditions . . . . . . . . . . 82

2.4 Mean value and maximum property . . . . . . . . . . . . . . . 83

2.5 Logarithmic potential . . . . . . . . . . . . . . . . . . . . . . . . 85

2.6 Dirichlet's principle . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.7 Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.8 Vorticity on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.9 Complex speed and potential . . . . . . . . . . . . . . . . . . . 96

2.10 Analytic functions, conformal transformation . . . . . . . . . 98

2.11 Schwarz-Christoffel theorem . . . . . . . . . . . . . . . . . . . . 100

2.12 Mapping of semi-infinite and infinite strips . . . . . . . . . . . 106

2.13 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3 Vortices, corner flow and flow past plates 117

3.1 Straight vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2 Corner flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3 Corner flow with viscosity . . . . . . . . . . . . . . . . . . . . . 122

3.4 Flow past a flat plate . . . . . . . . . . . . . . . . . . . . . . . . 129

3.5 Blasius and Kutta-Jukowski theorems . . . . . . . . . . . . . . 132

3.6 Plane flow past a cylinder . . . . . . . . . . . . . . . . . . . . . 135

3.7 Kármán vortex street . . . . . . . . . . . . . . . . . . . . . . . . 137

3.8 Corner eddy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.9 Angular momentum transport . . . . . . . . . . . . . . . . . . . 155

4 Jets, wakes and cavities 163

4.1 Free streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.2 Flow past a step . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

4.3 Complex potential and speed plane . . . . . . . . . . . . . . . . 169

4.4 Outflow from an orifice . . . . . . . . . . . . . . . . . . . . . . . 170

4.5 A simple wake model . . . . . . . . . . . . . . . . . . . . . . . . 175

4.6 Riabouchinsky cavity . . . . . . . . . . . . . . . . . . . . . . . . 181

4.7 Levi-Civita method . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.8 Kolscher's cusped cavity . . . . . . . . . . . . . . . . . . . . . . 188

4.9 Re-entrant jet cavity . . . . . . . . . . . . . . . . . . . . . . . . . 197

4.10 Tilted wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

4.11 Weinstein theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5 Kelvin-Helmholtz instability 211

5.1 Kelvin-Helmholtz circulation theorem . . . . . . . . . . . . . . 211

5.2 Bjerknes circulation theorem . . . . . . . . . . . . . . . . . . . 217

5.3 Kelvin-Helmholtz instability . . . . . . . . . . . . . . . . . . . . 220

5.4 Vortex chain perturbation . . . . . . . . . . . . . . . . . . . . . 222

5.5 Vortex accumulation . . . . . . . . . . . . . . . . . . . . . . . . . 226

5.6 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . 230

5.7 Birkhoff-Rott equation for vortex sheets . . . . . . . . . . . . . 235

5.8 Curvature singularity in evolving vortex sheet . . . . . . . . . 239

5.9 Subsequent work on Moore's singularity . . . . . . . . . . . . 254

5.10 Nonlinear stages of K-H instability . . . . . . . . . . . . . . . . 257

5.11 Why do large eddies occur in fast flows? . . . . . . . . . . . . . 259

5.12 Atmospheric instability . . . . . . . . . . . . . . . . . . . . . . . 262

5.13 Rayleigh inflexion theorem . . . . . . . . . . . . . . . . . . . . . 264

5.14 Kinematics of vortex rings . . . . . . . . . . . . . . . . . . . . . 266

5.15 Curvature and torsion . . . . . . . . . . . . . . . . . . . . . . . . 269

5.16 Helical line vortices . . . . . . . . . . . . . . . . . . . . . . . . . 271

5.17 Knotted and linked vortex rings . . . . . . . . . . . . . . . . . . 274

5.18 Clebsch coordinates and knottedness . . . . . . . . . . . . . . . 278

6 Kinematics of waves 279

6.1 Wave basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

6.2 Group speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

6.3 Kinematic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

6.4 The wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

6.5 Waves and instability from a radiative force . . . . . . . . . . 289

7 Shallow water waves 299

7.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . 300

7.2 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

7.3 Wave equation for linear water waves . . . . . . . . . . . . . . 304

7.4 Tides in canals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7.5 Cotidal lines and amphidromic points . . . . . . . . . . . . . . 312

7.6 Waves of finite amplitude . . . . . . . . . . . . . . . . . . . . . . 317

7.7 Nonlinear tides in an estuary . . . . . . . . . . . . . . . . . . . 321

7.8 Similarity solution: dam break . . . . . . . . . . . . . . . . . . 329

7.9 Non-breaking waves . . . . . . . . . . . . . . . . . . . . . . . . . 334

7.10 Bores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

7.11 Poincaré and Kelvin waves . . . . . . . . . . . . . . . . . . . . . 348

7.12 Wave behind a barrier . . . . . . . . . . . . . . . . . . . . . . . . 353

8 Free surface waves 373

8.1 Dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . 374

8.2 Sudden impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

8.3 Refraction and breaking at a coast . . . . . . . . . . . . . . . . 383

8.4 Waves in a non-uniform stream . . . . . . . . . . . . . . . . . . 392

8.5 Stokes wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

8.6 Stokes singularity . . . . . . . . . . . . . . . . . . . . . . . . . . 409

8.7 Crapper wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

9 Existence proof for weakly nonlinear water waves 427

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

9.2 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . 429

9.3 Linear integral equations . . . . . . . . . . . . . . . . . . . . . . 430

9.4 Schmidt's nonlinear integral equation . . . . . . . . . . . . . . 441

9.5 General nonlinear integral equations . . . . . . . . . . . . . . 447

9.6 Integral equations for nonlinear waves . . . . . . . . . . . . . 449

10 Sound and internal gravity waves 463

10.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

10.2 Acoustic cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

10.3 Schwarzschild criterion . . . . . . . . . . . . . . . . . . . . . . . 470

10.4 Gravo-acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . 473

11 Supersonic flow and shocks 479

11.1 Shock kinematics and entropy . . . . . . . . . . . . . . . . . . . 479

11.2 Jump conditions at shocks . . . . . . . . . . . . . . . . . . . . . 484

11.3 Shock speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

11.4 Shock entropy and supersonic inflow . . . . . . . . . . . . . . . 492

11.5 Laval nozzle and solar wind . . . . . . . . . . . . . . . . . . . . 493

11.6 Supersonic spots . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

11.7 Solar wind exhibiting a shock pair . . . . . . . . . . . . . . . . 509

11.8 Riemann sheets for the Burgers equation . . . . . . . . . . . . 514

11.9 Characteristics for first-order equations . . . . . . . . . . . . . 520

11.10Characteristics for second-order equations . . . . . . . . . . . 527

11.11Derivatives on characteristics . . . . . . . . . . . . . . . . . . . 529

11.12Simple waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

A Analytic and meromorphic functions 541