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Gravity-Capillary Free-Surface Flows
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Preface page xi 1 Introduction 1 2 Basicconcepts 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 vii viii Contents 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 Contents ix 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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