Preface ........................................................ xi
1 Introduction ................................................. 1
2 Basic concepts ............................................... 7
2.1 The equations of fluid mechanics ........................ 7
2.2 Free-surface flows ...................................... 8
2.3 Two-dimensional flows .................................. 11
2.4 Linear waves ........................................... 15
2.4.1 The water-wave equations ........................ 15
2.4.2 Linear solutions for water waves ................ 17
2.4.3 Superposition of linear waves ................... 24
3 Free-surface flows that intersect walls ..................... 31
3.1 Free streamline solutions .............................. 33
3.1.1 Forced separation ............................... 33
3.1.2 Free separation ................................. 43
3.2 The effects of surface tension ......................... 58
3.2.1 Forced separation ............................... 58
3.2.2 Free separation ................................. 66
3.3 The effects of gravity ................................. 73
3.3.1 Solutions with β1=0 (funnels) ................... 83
3.3.2 Solutions with β1=0 (nozzles and bubbles) ....... 88
3.3.3 Solutions with β1= π/2 (flow under a gate
with gravity) ................................... 94
3.4 The combined effects of gravity and surface tension .... 98
3.4.1 Rising bubbles in a tube ........................ 99
3.4.2 Fingering in a Hele Shaw cell .................. 103
3.4.3 Further examples involving rising bubbles ...... 108
3.4.4 Exponential asymptotics ........................ 112
4 Linear free-surface flows generated by moving
disturbances ............................................... 114
4.1 The exact nonlinear equations ......................... 115
4.2 Linear theory ......................................... 116
4.2.1 Solutions in water of finite depth ............. 116
4.2.2 Solutions in water of infinite depth ........... 122
4.2.3 Discussion of the solutions .................... 124
5 Nonlinear waves - asymptotic solutions ..................... 129
5.1 Periodic waves ........................................ 129
5.1.1 Solutions when condition (5.55) is satisfied ... 135
5.1.2 Solutions when condition (5.55) is not
satisfied ...................................... 138
5.2 The Korteweg-de Vries equation ........................ 142
6 Numerical computations of nonlinear water waves ............ 148
6.1 Formulation ........................................... 148
6.2 Series truncation method .............................. 151
6.3 Boundary integral equation method ..................... 152
6.4 Numerical methods for solitary waves .................. 156
6.4.1 Boundary integral equation methods ............. 157
6.5 Numerical results for periodic waves .................. 160
6.5.1 Pure capillary waves (g = 0, T ≠ 0) ............ 160
6.5.2 Pure gravity waves (g ≠ 0, T = 0) .............. 164
6.5.3 Gravity-capillary waves (g ≠ 0, T ≠ 0) ......... 175
6.6 Numerical results for solitary waves .................. 181
6.6.1 Pure gravity solitary waves .................... 181
6.6.2 Gravity-capillary solitary waves ............... 186
7 Nonlinear free-surface flows generated by moving
disturbances ............................................... 191
7.1 Pure gravity free-surface flows in water of finite
depth ................................................. 192
7.1.1 Supercritical flows ............................ 192
7.1.2 Subcritical flows .............................. 195
7.2 Gravity-capillary free-surface flows .................. 201
7.2.1 Results in finite depth ........................ 201
7.2.2 Results in infinite depth (removal of the
nonuniformity) ................................. 203
7.3 Gravity-capillary free-surface flows with Wilton
ripples ............................................... 206
8 Free-surface flows with waves and intersections with
rigid walls ................................................ 210
8.1 Free-surface flow past a flat plate ................... 211
8.1.1 Numerical results .............................. 212
8.1.2 Analytical results ............................. 215
8.2 Free-surface flow past a surface-piercing object ...... 218
8.2.1 Numerical results .............................. 219
8.2.2 Analytical results ............................. 221
8.3 Flow under a sluice gate .............................. 226
8.3.1 Formulation .................................... 228
8.3.2 Numerical procedure ............................ 231
8.3.3 Discussion of the results ...................... 233
8.4 Pure capillary free-surface flows ..................... 236
8.4.1 Numerical results .............................. 236
8.4.2 Analytical results ............................. 239
9 Waves with constant vorticity .............................. 244
9.1 Solitary waves with constant vorticity ................ 245
9.1.1 Mathematical formulation ....................... 245
9.1.2 Numerical procedure ............................ 248
9.1.3 Discussion of the results ...................... 249
9.2 Periodic waves with constant vorticity ................ 267
9.2.1 Mathematical formulation ....................... 268
9.2.2 Numerical procedure ............................ 270
9.2.3 Numerical results .............................. 271
9.2.4 Discussion ..................................... 276
10 Three-dimensional free-surface flows ....................... 278
10.1 Green's function formulation for two-dimensional
problems .............................................. 278
10.1.1 Pressure distribution .......................... 278
10.1.2 Two-dimensional surface-piercing object ........ 283
10.2 Extension to three-dimensional free-surface flows ..... 286
10.2.1 Gravity flows generated by moving
disturbances in water of infinite depth ........ 286
10.2.2 Three-dimensional gravity-capillary free-
surface flows in water of infinite depth ....... 293
10.3 Further extensions .................................... 298
11 Time-dependent free-surface flows .......................... 301
11.1 Introduction .......................................... 301
11.2 Nonlinear gravity-capillary standing waves ............ 301
References .................................................... 308
Index ......................................................... 318
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