Free surface problems occur in many aspects of science and of everyday life such as the waves on a beach, bubbles rising in a glass of champagne, melting ice, pouring flows from a container and sails billowing in the wind. Consequently, the effect of surface tension on gravity-capillary flows continues to be a fertile field of research in applied mathematics and engineering. Concentrating on applications arising from fluid dynamics, Vanden-Broeck draws upon his years of experience in the field to address the many challenges involved in attempting to describe such flows mathematically. Whilst careful numerical techniques are implemented to solve the basic equations, an emphasis is placed upon the reader developing a deep understanding of the structure of the resulting solutions. The author also reviews relevant concepts in fluid mechanics to help readers from other scientific fields who are interested in free boundary problems.
Author(s): Vanden-Broeck J.-M.
Series: Cambridge Monographs on Mechanics
Publisher: CUP
Year: 2010
Language: English
Pages: 332
Tags: Механика;Механика жидкостей и газов;
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1 Introduction......Page 15
2.1 The equations of fluid mechanics......Page 21
2.2 Free-surface flows......Page 22
2.3 Two-dimensional flows......Page 25
2.4.1 The water-wave equations......Page 29
2.4.2 Linear solutions for water waves......Page 31
2.4.3 Superposition of linear waves......Page 38
3 Free-surface flows that intersect walls......Page 45
3.1.1 Forced separation......Page 47
3.1.2 Free separation......Page 57
3.2.1 Forced separation......Page 72
3.2.2 Free separation......Page 80
3.3 The effects of gravity......Page 87
3.3.1 Solutions with β1=0 (funnels)......Page 97
3.3.2 Solutions with β1=0 (nozzles and bubbles)......Page 102
3.3.3 Solutions with β1=π/2 (flow under a gate with gravity)......Page 108
3.4 The combined effects of gravity and surface tension......Page 112
3.4.1 Rising bubbles in a tube......Page 113
3.4.2 Fingering in a Hele Shaw cell......Page 117
3.4.3 Further examples involving rising bubbles......Page 122
3.4.4 Exponential asymptotics......Page 126
4 Linear free-surface flows generated by moving disturbances......Page 128
4.1 The exact nonlinear equations......Page 129
4.2.1 Solutions in water of finite depth......Page 130
4.2.2 Solutions in water of infinite depth......Page 136
4.2.3 Discussion of the solutions......Page 138
5.1 Periodic waves......Page 143
5.1.1 Solutions when condition (5.55) is satisfied......Page 149
5.1.2 Solutions when condition (5.55) is not satisfied......Page 152
5.2 The Korteweg–de Vries equation......Page 156
6.1 Formulation......Page 162
6.2 Series truncation method......Page 165
6.3 Boundary integral equation method......Page 166
6.4 Numerical methods for solitary waves......Page 170
6.4.1 Boundary integral equation methods......Page 171
6.5.1 Pure capillary waves (g=0, T≠0)......Page 174
6.5.2 Pure gravity waves (g≠0, T=0)......Page 178
6.5.2.1 Waves in water of infinite depth......Page 181
6.5.3 Gravity–capillary waves (g≠0, T≠0)......Page 189
6.5.3.1 Waves in water of infinite depth......Page 190
6.5.3.2 Waves in water of finite depth......Page 193
6.6.1 Pure gravity solitary waves......Page 195
6.6.1.1 A series truncation method for gravity solitary waves......Page 198
6.6.2 Gravity–capillary solitary waves......Page 200
7 Nonlinear free-surface flows generated by moving disturbances......Page 205
7.1.1 Supercritical flows......Page 206
7.1.2 Subcritical flows......Page 209
7.2.1 Results in finite depth......Page 215
7.2.2 Results in infinite depth (removal of the nonuniformity)......Page 217
7.3 Gravity–capillary free-surface flows with Wilton ripples......Page 220
8 Free-surface flows with waves and intersections with rigid walls......Page 224
8.1 Free-surface flow past a flat plate......Page 225
8.1.1 Numerical results......Page 226
8.1.2 Analytical results......Page 229
8.2 Free-surface flow past a surface-piercing object......Page 232
8.2.1 Numerical results......Page 233
8.2.2 Analytical results......Page 235
8.3 Flow under a sluice gate......Page 240
8.3.1 Formulation......Page 242
8.3.2 Numerical procedure......Page 245
8.3.3 Discussion of the results......Page 247
8.4.1 Numerical results......Page 250
8.4.2 Analytical results......Page 253
9 Waves with constant vorticity......Page 258
9.1.1 Mathematical formulation......Page 259
9.1.2 Numerical procedure......Page 262
9.1.3.1 Solitary wave branches......Page 263
9.1.3.2 More branches of solitary waves......Page 270
9.2 Periodic waves with constant vorticity......Page 281
9.2.1 Mathematical formulation......Page 282
9.2.2 Numerical procedure......Page 284
9.2.3 Numerical results......Page 285
9.2.4 Discussion......Page 290
10.1.1 Pressure distribution......Page 292
10.1.2 Two-dimensional surface-piercing object......Page 297
10.2.1.1 Formulation......Page 300
10.2.1.2 The numerical scheme......Page 303
10.2.1.3 Numerical results......Page 304
10.2.2 Three-dimensional gravity–capillary free-surface flows in water of infinite depth......Page 307
10.2.2.1 Formulation......Page 308
10.2.2.2 Results......Page 309
10.3 Further extensions......Page 312
11.2 Nonlinear gravity–capillary standing waves......Page 315
References......Page 322
Index......Page 332
