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  • Cited by 80
Publisher:
Cambridge University Press
Online publication date:
September 2010
Print publication year:
2010
Online ISBN:
9780511730276

Book description

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.

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'There is no doubt that this volume would determine fruitful directions for future advanced study and research. So, this is an outstanding contribution of the author who spent a considerable amount of time and energy to write such a useful monograph.'

Source: Zentralblatt MATH

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Contents

References
References
[1] Acheson, D. J. 1990, Elementary Fluid Dynamics Google ScholarOxford University Press.
[2] Ackerberg, R. C. 1975, The effects of capillarity on free-streamline separation. J. Fluid Mech. 70 CrossRef | Google Scholar, 333–352.
[3] Akylas, T. R. 1993, Envelope solitons with stationary crests. Phys. Fluids A 5 CrossRef | Google Scholar, 789–791.
[4] Akylas, T. R. & Grimshaw, R. 1992, Solitary internal waves with oscillatory tails. J. Fluid Mech. 242 CrossRef | Google Scholar, 279–298.
[5] Amick, C. J., Fraenkel, L. E. & Toland, J. F. 1982, On the Stokes conjecture and the wave of extreme form. Acta Math. 148 CrossRef | Google Scholar, 193–214.
[6] Anderson, C. D. & Vanden-Broeck, J.-M. 1996, Bow flows with surface tension. Proc. Roy. Soc. Lond. A 452 CrossRef | Google Scholar, 1985–1997.
[7] Asavanant, J. & Vanden-Broeck, J.-M. 1994, Free-surface flows past a surface-piercing object of finite length. J. Fluid. Mech. 273 CrossRef | Google Scholar, 109–124.
[8] Batchelor, G. K.Fluid Dynamics Google Scholar. Cambridge University Press, 615 pp.
[9] Baker, G., Meiron, D. & Orszag, S. 1982, Generalised vortex methods for free surface flow problemsJ. Fluid Mech. 123 CrossRef | Google Scholar, 477–501.
[10] Beale, T. J. 1991, Solitary water waves with capillary ripples at infinity. Comm. Pure Appl. Maths 64 CrossRef | Google Scholar, 211–257.
[11] Benjamin, B. 1956, On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1 CrossRef | Google Scholar, 227–248.
[12] Benjamin, B. 1962, The solitary wave on a stream with arbitrary distribution of vorticity. J. Fluid Mech. 12 CrossRef | Google Scholar, 97–116.
[13] Billingham, J. & King, A. C. 2000, Wave Motion Google Scholar. Cambridge University Press.
[14] Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. 2005, Forced solitary waves and fronts past submerged obstacles. Chaos 15 CrossRef | Google Scholar | PubMed, 037106.
[15] Binder, B. J. & Vanden-Broeck, J.-M. 2005, Free surface flows past surfboards and sluice gates. Euro. J. Appl. Math. 16 CrossRef | Google Scholar, 601–619.
[16] Binder, B. J. & Vanden-Broeck, J.-M. 2007, The effect of disturbances on the flows under a sluice gate and past an inclined plate. J. Fluid Mech. 576 CrossRef | Google Scholar, 475–490.
[17] Binnie, A. M. 1952, The flow of water under a sluice gate. Q. J. Mech. Appl. Math. 5 CrossRef | Google Scholar, 395–407.
[18] Birkhoff, G. & Carter, D. 1957, Rising plane bubbles. J. Math. Phys. 6 Google Scholar 769–779.
[19] Birkhoff, G. & Zarantonello, E. 1957, Jets, Wakes and Cavities Google ScholarAcademic Press, 353 pp.
[20] Blyth, M. G. & Vanden-Broeck, J.-M. 2004, New solutions for capillary waves on fluid sheets. J. Fluid Mech. 507 CrossRef | Google Scholar, 255–264.
[21] Blyth, M. G. & Vanden-Broeck, J.-M. 2005, New solutions for capillary waves on curved sheets of fluids. IMA J. Appl. Math. 70 CrossRef | Google Scholar, 588–601.
[22] Brillouin, M. 1911, Les surfaces de glissement de Helmoltz at la résistance des fluides. Ann. de Chim. Phys. 23 Google Scholar, 145–230.
[23] Brodetsky, S. 1923, Discontinuous fluid motion past circular and elliptic cylinders. Proc. Roy. Soc. London A 102 CrossRef | Google Scholar, 1–14.
[24] Budden, P. & Norbury, J. 1982, Uniqueness of free boundary flows under gravity. Arch. Rat. Mech. Anal. 78 CrossRef | Google Scholar, 361–380.
[25] Byatt-Smith, J. G. B. & Longuet-Higgins, M. S. 1976, On the speed and profile of steep solitary waves. Proc. Roy. Soc. London. A 350 CrossRef | Google Scholar, 175–189.
[26] Champneys, A. R., Vanden-Broeck, J.-M. & Lord, G. J. 2002, Do true elevation gravity–capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454 CrossRef | Google Scholar, 403–417.
[27] Chen, B. & Saffman, P. G. 1979, Steady gravity–capillary waves on deep water, Part I: Weakly nonlinear waves. Stud. Appl. Math. 60 CrossRef | Google Scholar, 183–210.
[28] Chen, B. & Saffman, P. G. 1980a, Numerical evidence for the existence of new types of gravity waves on deep water. Stud. Appl. Math. 62 CrossRef | Google Scholar, 1–21.
[29] Chen, B. & Saffman, P. G. 1980b, Steady gravity–capillary waves on deep water, Part II: Numerical results for finite amplitude. Stud. Appl. Math. 62 CrossRef | Google Scholar, 95–111.
[30] Chung, Y. K. 1972, Solution of flow under a sluice gates. ASCE J. Eng. Mech. Div. 98 Google Scholar, 121–140.
[31] Cokelet, E. D. 1977, Steep gravity waves in water of arbitrary uniform depth, Phil. Trans. Roy. Soc. London A 286 CrossRef | Google Scholar, 183–230.
[32] Collins, R. 1965, A simple model of a plane gas bubble in a finite liquid. J. Fluid Mech. 22 CrossRef | Google Scholar, 763–771.
[33] Concus, P. 1962, Standing capillary–gravity waves of finite amplitude. J. Fluid Mech. 14 CrossRef | Google Scholar, 568–576.
[34] Concus, P. 1964, Standing capillary–gravity waves of finite amplitude: Corrigendum. J. Fluid Mech. 19 CrossRef | Google Scholar, 264–266.
[35] Cooker, M. J., Weidman, P. D. & Bale, D. S. 1997, Reflection of a high-amplitude solitary wave at a vertical wall. J. Fluid Mech. 342 CrossRef | Google Scholar, 141–158.
[36] Couët, B. & Strumolo, G. S. 1987, The effects of surface tension and tube inclination on a two-dimensional rising bubble. J. Fluid Mech. 213 CrossRef | Google Scholar, 1–14.
[37] Crapper, G. D. 1957, An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 CrossRef | Google Scholar, 572–540.
[38] Crowdy, D. G. 1999, Exact solutions for steady capillary waves on a fluid annulus. J. Nonlinear Sci. 9 CrossRef | Google Scholar, 615–640.
[39] Cumberbatch, E. & Norbury, J. 1979, Capillarity modification of the singularity at a free-streamline separation point. Q. J. Mech. Appl. Math. 32 CrossRef | Google Scholar, 303–312.
[40] Dagan, G. & Tulin, M. P. 1972, Two-dimensional free surface gravity flows past blunt bodies. J. Fluid Mech. 51 CrossRef | Google Scholar, 529–543.
[41] Davies, T. V. 1951, Theory of symmetrical gravity waves of finite amplitude. Proc. Roy. Soc. London A 208 CrossRef | Google Scholar, 475–486.
[42] Dias, F. & Iooss, G. 1993, Capillary–gravity solitary waves with damped oscillations. Physica D 65 CrossRef | Google Scholar, 399–423.
[43] Dias, F. & Kharif, C. 1999, Nonlinear gravity and capillary–gravity waves. Ann. Rev. Fluid Mech. 31 CrossRef | Google Scholar, 301–346.
[44] Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996, Numerical study of capillary–gravity solitary waves. Eur. J. Mech. B – Fluids 15 Google Scholar, 17–36.
[45] Dias, F. & Vanden-Broeck, J.-M. 1989, Open channel flows with submerged obstructions. J. Fluid Mech. 206 CrossRef | Google Scholar, 155–170.
[46] Dias, F. & Vanden-Broeck, J.-M. 1992, Solitary waves in water of infinite depth and related free surface flows. J. Fluid Mech. 240 Google Scholar, 549–557.
[47] Dias, F. & Vanden-Broeck, J.-M. 1993, Nonlinear bow flows with splashes. J. Fluid Mech. 255 CrossRef | Google Scholar, 91–102.
[48] Dias, F. & Vanden-Broeck, J.-M. 2002, Generalized critical free-surface flows. J. Eng. Math. 42 CrossRef | Google Scholar, 291–301.
[49] Dias, F. & Vanden-Broeck, J.-M. 2004a, Trapped waves between submerged obstacles. J. Fluid Mech. 509 CrossRef | Google Scholar, 93–102.
[50] Dias, F. & Vanden-Broeck, J.-M. 2004b, Two-layer hydraulic falls over an obstacle. Eur. J. Mech. B – Fluids 23 CrossRef | Google Scholar, 879–898.
[51] Dingle, R. B. 1973, Asymptotic Expansions: Their Derivation and Interpretation Google Scholar. Academic Press.
[52] Eggers, J. 1995, Theory of drop formation. Phys. Fluids 7 CrossRef | Google Scholar, 941–953.
[53] Evans, W. A. B. & Ford, M. J. 1996, An exact integral equation for solitary waves (with new numerical results for some ‘internal’ properties). Proc. Roy. Soc. London A 452 CrossRef | Google Scholar, 373–390.
[54] Fangmeier, D. D. & Strelkoff, T. S. 1968, Solution for gravity flow under a sluice gate. ASCE J. Eng. Mech. Div. 94 Google Scholar, 153–176.
[55] Forbes, L.-K. 1981, On the resistance of a submerged semi-elliptical body. J. Eng. Math. 15 CrossRef | Google Scholar, 287–298.
[56] Forbes, L.-K. 1983, Free surface flow over a semicircular obstruction including the influence of gravity and surface tension. J. Fluid Mech. 127 CrossRef | Google Scholar, 283–297.
[57] Forbes, L.-K. 1988, Critical free-surface flow over a semi-circular obstruction. J. Eng. Math. 22 CrossRef | Google Scholar, 3–13.
[58] Forbes, L. K 1989, An algorithm for 3-dimensional free-surface problems in hydrodynamics. J. Comput. Phys. 82 CrossRef | Google Scholar, 330–347.
[59] Forbes, L. K. & Schwartz, L. W. 1982, Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114 CrossRef | Google Scholar, 299–314.
[60] Forbes, L. K. & Hocking, G. C. 1990, Flow caused by a point sink in a fluid having a free surface. J. Austral. Math. Soc. Ser. B 32 CrossRef | Google Scholar, 231–249.
[61] Friedrics, K. O. & Hyers, D. H. 1954, The existence of solitary waves. Comm. Pure Appl. Math. 7 CrossRef | Google Scholar, 517–550.
[62] Garabedian, P. R. 1957, On steady state bubbles generated by Taylor instability. Proc. Roy. Soc. London A 241 CrossRef | Google Scholar, 423–431.
[63] Garabedian, P. R. 1985, A remark about pointed bubbles. Comm. Pure Appl. Math. 38 CrossRef | Google Scholar, 609–612.
[64] Gleeson, H., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2007, A new application of the Korteweg–de-Vries Benjamin–Ono equation in interfacial electrohydrodynamics. Phys. Fluids 19 CrossRef | Google Scholar, 031703.
[65] Grandison, S. & Vanden-Broeck, J.-M. 2006, Truncation methods for gravity capillary free surface flows. J. Eng. Math. 54 CrossRef | Google Scholar, 89–97.
[66] Grilli, S. T., Guyenne, P. & Dias, F. 2001, A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom. Int. J. Numer. Meth. Fluids 35 CrossRef | Google Scholar, 829–867.
[67] Grimshaw, R. H. J. & Smyth, N. 1986, Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169 CrossRef | Google Scholar, 429–464.
[68] Groves, M. D. & Sun, M. S. 2008, Fully localised solitary-wave solutions of the three-dimensional gravity–capillary water-wave problem. Arch. Rat. Mech. Anal. 188 CrossRef | Google Scholar. 1–91.
[69] Gurevich, M. 1965, Theory of Jets and Ideal Fluids Google Scholar. Academic Press, 585 pp.
[70] Havelock, T. H. 1919, Periodic irrotational waves of finite amplitude. Proc. Roy. Soc. London Ser. A 95 CrossRef | Google Scholar, 38–51.
[71] Helmholtz, H. 1868, Über discontinuierliche Flüssigkeitsbewegungen. Monatsber, Berlin Akad. Google Scholar, 215–228, reprinted in Phil. Mag.36, 337–346.
[72] Hocking, G. C. & Vanden-Broeck, J.-M. 1997, Draining of a fluid of finite depth into a vertical slot. Applied Math. Modelling 21 CrossRef | Google Scholar, 643–649.
[73] Hocking, G. C., Vanden-Broeck, J.-M. & Forbes, L. K. 2002, A note on withdrawal from a fluid of finite depth through a point sink. ANZIAM J. 44 CrossRef | Google Scholar, 181–191.
[74] Hogan, S. J. 1980, Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96 CrossRef | Google Scholar, 417–445.
[75] Hunter, J. K. & Scherule, J. 1988, Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32 CrossRef | Google Scholar, 253–268.
[76] Hunter, J. K. & Vanden-Broeck, J.-M. 1983a, Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134 CrossRef | Google Scholar, 205–219.
[77] Hunter, J. K. & Vanden-Broeck, J.-M. 1983b, Accurate computations for steep solitary waves. J. Fluid Mech. 136 CrossRef | Google Scholar, 63–71.
[78] Iooss, G. & Kirrmann, P. 1996, Capillary gravity waves on the free surface of an inviscid fluid of infinite depth – existence of solitary waves. Arch. Rat. Mech. Anal. 136 CrossRef | Google Scholar, 1–19.
[79] Iooss, G. & Kirchgassner, K. 1990, Bifurcation d'ondes solitaires en présences d'une faible tension superficielle. C.R. Acad. Sci. Paris 311 Google Scholar I, 265–268.
[80] Iooss, G. & Kirchgassner, K. 1992, Water waves for small surface tension: an approach via normal form. Proc. Roy. Soc. Edinburgh 122A CrossRef | Google Scholar, 267–299.
[81] Iooss, G., Plotnikov, P. & Toland, J. F. 2005, Standing waves on an infinitely deep perfect fluid under gravity. Arch. Rat. Mech. Anal. 177 CrossRef | Google Scholar, 367–478.
[82] Kang, Y. & Vanden-Broeck, J.-M. 2002, Stern waves with vorticityANZIAM J. 43 Google Scholar, 321–332.
[83] Kawahara, T. 1972, Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33 CrossRef | Google Scholar, 260–264.
[84] Keller, H. B. 1977, Applications of Bifurcation Theory Google Scholar. Academic Press.
[85] Keller, J. B. & Miksis, M. J. 1983, Surface tension driven flows. SIAM J. Appl. Math. 43 CrossRef | Google Scholar, 268–277.
[86] Keller, J. B., Milewski, P. & Vanden-Broeck, J.-M. 2000, Wetting and merging driven by surface tension. Euro. J. Mech. B – Fluids 19 CrossRef | Google Scholar, 491–502.
[87] Kim, B. & Akylas, T. R. 2005, On gravity–capillary lumps. J. Fluid Mech. 540 CrossRef | Google Scholar, 337–351.
[88] Kim, B. & Akylas, T. R. 2006, On gravity–capillary lumps, Part 2. Two dimensional Benjamin equation. J. Fluid Mech. 557 CrossRef | Google Scholar, 237–256.
[89] Kinnersley, W. 1976, Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77 CrossRef | Google Scholar, 229–241.
[90] Kirchhoff, G. 1869, Zur Theorie freier Flüssigkeitsstrahlen. J. Reine Angew. Math. 70 CrossRef | Google Scholar, 289–298.
[91] Korteweg, D. J. & G., de Vries 1895, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves. Phil. Mag. 39 CrossRef | Google Scholar, 422–443.
[92] Lamb, H. 1945, Hydrodynamics Google Scholar, 6th edn, Cambridge University Press.
[93] Larock, B. E. 1969, Gravity-affected flow from planar sluice gate. ASCE J. Engng Mech. Div. 96 Google Scholar, 1211–1226.
[94] Lee, J. W. & Vanden-Broeck, J.-M. 1993, Two-dimensional jets falling from funnels and nozzles. Phys. Fluids A5 CrossRef | Google Scholar, 2454–2460.
[95] Lee, J. W. & Vanden-Broeck, J.-M. 1998, Bubbles rising in an inclined two-dimensional tube and jets falling from along a wall. J. Austral. Math. Soc. B 39 CrossRef | Google Scholar, 332–349.
[96] Lenau, C. W. 1966, The solitary wave of maximum amplitude. J. Fluid Mech. 26 CrossRef | Google Scholar, 309–320.
[97] Lombardi, E. 2000, Oscillatory Integrals on Phenomena Beyond All Orders: with Applications to Homoclinic Orbits in reversible systems. Lecture Notes in Mathematics 1741 CrossRef | Google Scholar, Springer.
[98] Lighthill, M. J. 1946, A note on cusped cavities. Aero. Res. Councial Rep. and Mem. 2328 Google Scholar.
[99] Lighthill, M. J. 1953, On boundary layers and upstream influence, I. A comparison between subsonic and supersonic flows. Proc. Roy. Soc. London A 217 CrossRef | Google Scholar, 344–357.
[100] Lighthill, M. J. 1978, Waves in Fluids Google Scholar, Cambridge University Press, 504 pp.
[101] Longuet–Higgins, M. S. 1975, Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. London A 342 CrossRef | Google Scholar, 157–174.
[102] Longuet-Higgins, M. S. 1989, Capillary-gravity waves of solitary type on deep water. J. Fluid Mech. 200 CrossRef | Google Scholar, 451–478.
[103] Longuet-Higgins, M. S. 1993, Capillary–gravity waves of solitary type and envelope solitons on deep water. J. Fluid Mech. 252 CrossRef | Google Scholar, 703–711.
[104] Longuet-Higgins, M. S. & Cokelet, E. 1976, The deformation of steep surface waves on water, I. A numerical method of computation. Proc. Roy. Soc. London A 350 CrossRef | Google Scholar, 1–26.
[105] Longuet-Higgins, M. S. & Fenton, J. D. 1974, On the mass, momentum, energy and circulation of a solitary wave, II. Proc. R. Soc. Lond. A 340 CrossRef | Google Scholar, 471–493.
[106] Longuet-Higgins, M. S. & Fox, M. J. H. 1978, Theory of the almost highest wave, Part 2. Matching and analytical extension. J. Fluid Mech. 85 CrossRef | Google Scholar, 769–786.
[107] Maneri, C. C. 1970 Google Scholar, The motion of plane bubbles in inclined ducts. Ph.D. thesis, Polytechnic Institute of Brooklyn, New York.
[108] McCue, S. W. & Forbes, L. K. 2002, Free surface flows emerging from beneath a semi-infinite plate with constant vorticity. J. Fluid Mech. 461 CrossRef | Google Scholar, 387–407.
[109] McLean, J. W. & Saffman, P. G. 1981, The effect of surface tension on the shape of fingers in a Hele Shaw cell. J. Fluid Mech. 102 CrossRef | Google Scholar, 455–469.
[110] Mekias, H. & Vanden-Broeck, J.-M. 1991, Subcritical flow with a stagnation point due to a source beneath a free surface. Phys. Fluids A 3 CrossRef | Google Scholar, 2652–2658.
[111] Michallet, H. & Dias, F. 1999, Numerical study of generalized interfacial solitary waves. Phys. Fluids 11 CrossRef | Google Scholar, 1502–1511.
[112] Michell, J. H. 1883, The highest wave in water. Phil. Mag. 36 CrossRef | Google Scholar, 430–437.
[113] Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1981, Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108 CrossRef | Google Scholar, 89–101.
[114] Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1982, Rising bubbles. J. Fluid Mech. 123 CrossRef | Google Scholar, 31–41.
[115] Milewski, P. A. 2005, Three-dimensional localized solitary gravity–capillary waves. Comm. Math. Sc. 3 CrossRef | Google Scholar, 89–99.
[116] Nayfeh, A. H. 1970, Triple and quintuple-dimpled wave profiles in deep water. J. Fluid Mech. 13 Google Scholar, 545–550.
[117] Ockendon, H. & Ockendon, J. R. 2004, Viscous Flow Google Scholar. Cambridge Texts in Applied Mathematics.
[118] Olfe, D. B. & Rottman, J. W. 1980, Some new highest-wave solutions for deep-water waves of permanent form. J. Fluid Mech. 100 CrossRef | Google Scholar, 801–810.
[119] Osher, S. & Fedkiw, R. 2003, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences 153 CrossRef | Google Scholar, Springer.
[120] Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2003, Large amplitude capillary waves in electrified fluid sheets. J. Fluid Mech. 508 CrossRef | Google Scholar, 71–88.
[121] Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2004, Antisymmetric capillary waves in electrified fluid sheets. Eur. J. Appl. Math. 15 CrossRef | Google Scholar, 609–623.
[122] Parau, E. & Vanden-Broeck, J.-M. 2002, Nonlinear two- and three-dimensional free surface flows due to moving disturbances. Eur. J. Mech. B – Fluids 21 CrossRef | Google Scholar, 643–656.
[123] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2005a, Nonlinear three dimensional gravity capillary solitary waves. J. Fluid Mech. 536 CrossRef | Google Scholar, 99–105.
[124] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2005b, Three-dimensional gravity–capillary solitary waves in water of finite depth and related problems. Phys. Fluids 17 CrossRef | Google Scholar, 122 101.
[125] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2007a, Three-dimensional capillary–gravity waves generated by a moving disturbance. Phys. Fluids 19 CrossRef | Google Scholar, 082 102.
[126] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2007b, Nonlinear three dimensional interfacial flows with a free surface. J. Fluid Mech. 591 CrossRef | Google Scholar, 481–494.
[127] Pullin, D. I. & Grimshaw, R. H. J. 1988, Finite amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31 CrossRef | Google Scholar, 3550–3559.
[128] Rayleigh, Lord 1883, The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. 15 CrossRef | Google Scholar, 69–78.
[129] Romero, L. 1982 Google Scholar, Ph.D. thesis, California Institute of Technology.
[130] Saffman, P. G. 1980, Long wavelength bifurcation of gravity waves on deep water. J. Fluid Mech. 101 CrossRef | Google Scholar, 567–581.
[131] Saffman, P. G. 1986, Viscous fingering in Hele Shaw cells. J. Fluid Mech. 173 CrossRef | Google Scholar, 73–94.
[132] Saffman, P. G. & Taylor, G. I. 1958, The penetration of a fluid into a porous medium or Hele Shaw cell containing a more viscous fluid. Proc. Roy. Soc. London A 245 CrossRef | Google Scholar, 312–329.
[133] Schwartz, L. W. 1974, Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62 CrossRef | Google Scholar, 553–578.
[134] Schwartz, L. W. & Fenton, J. 1982, Strongly nonlinear waves. Ann. Rev. Fluid Mech. 14 CrossRef | Google Scholar, 39–60.
[135] Schwartz, L. W. & Vanden-Broeck, J.-M. 1979, Numerical solution of the exact equations for capillary–gravity waves. J. Fluid Mech. 95 CrossRef | Google Scholar, 119–139.
[136] Schultz, W. W., Vanden-Broeck, J.-M., Jiang, L. & Perlin, M. 1998, Highly nonlinear water waves with small capillary effect. J. Fluid Mech. 369 Google Scholar, 253–272.
[137] Sethian, J. A.Level Set Methods. Cambridge Monographs on Applied and Computational Mathematics Google Scholar, Cambridge University Press.
[138] Sha, H. & Vanden-Broeck, J.-M. 1993, Two-layer flows past a semicircular obstaclePhys. Fluids A 5 CrossRef | Google Scholar, 2661–2668.
[139] Sha, H. & Vanden-Broeck, J.-M. 1997, Internal solitary waves with stratification in density. J. Austral. Math. Soc. B 38 CrossRef | Google Scholar, 563–580.
[140] Shen, S.-P. 1995, On the accuracy of the stationary forced Korteweg-de-Vries equation as a model equation for flows over a bump. Quart. J. Appl. Math. 53 CrossRef | Google Scholar, 701–719.
[141] Simmen, J. A. & Saffman, P. G. 1985, Steady deep water waves on a linear shear current. Stud. Appl. Maths 75 CrossRef | Google Scholar, 35–57.
[142] Southwell, R. V. & Vaisey, G. 1946, Fluid motions characterised by ‘free’ streamlines. Phil. Trans. Roy. Soc. A 240 CrossRef | Google Scholar, 117–161.
[143] Stokes, G. G. 1847, On the theory of oscillatory waves. Camb. Trans. Phil. Soc. 8 Google Scholar, 441–473.
[144] Stokes, G. G. 1880, in Mathematical and Physical Papers, Vol. 1 Google Scholar, p. 314, Cambridge University Press.
[145] Sun, S. M. 1991, Existence of generalized solitary wave solution for water with positive Bond number less than ⅓. J. Math. Anal. Appl. 156 CrossRef | Google Scholar, 471–504.
[146] Sun, S. M. 1999, Nonexistence of truly solitary waves in water with small surface tensionProc. Roy. Soc. London A 455 CrossRef | Google Scholar, 2191–2228.
[147] Sun, S. M. & Shen, M. C. 1993, Exponentially small estimate for the amplitude of capillary ripples of generalised solitary waves. J. Math. Anal. Appl. 172 CrossRef | Google Scholar, 533–566.
[148] Tadjbakhsh, I. & Keller, J. B. 1960, Standing surface waves of finite amplitude. J. Fluid Mech. 8 CrossRef | Google Scholar, 442–451.
[149] Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987, Instability and breaking of a solitary wave. J. Fluid Mech. 185 CrossRef | Google Scholar, 235–248.
[150] Teles da Silva, A. F. & Peregrine, D. H. 1988, Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 195 CrossRef | Google Scholar, 281–305.
[151] Tooley, S. & Vanden-Broeck, J.-M. 2002, Waves and singularities in nonlinear capillary free-surface flows. J. Eng. Math. 43 CrossRef | Google Scholar, 89–99.
[152] Tsai, W. T. & Yue, D. K. 1996, Computation of nonlinear free surface flows. Ann. Rev. Fluid Mech. 28 CrossRef | Google Scholar, 249–278.
[153] Tseluiko, D., Blyth, M. & Papageorgiou, D. T. 2008a, Electrified viscous thin film over topography. J. Fluid Mech. 597 CrossRef | Google Scholar, 449–475.
[154] Tseluiko, D., Blyth, M. & Papageorgiou, D. T. 2008b, Effect of an electric field on film flow down a corrugated wall at zero Reynolds number. Phys. Fluids 20 CrossRef | Google Scholar, 042 103
[155] Turner, R. E. L. & Vanden-Broeck, J.-M. 1986, The limiting configuration of interfacial gravity waves. Phys. Fluids 29 CrossRef | Google Scholar, 372–375.
[156] Turner, R. E. L. & Vanden-Broeck, J.-M. 1988, Broadening on interfacial solitary waves. Phys. Fluids 31 CrossRef | Google Scholar, 2486–2490.
[157] Turner, R. E. L. & Vanden-Broeck, J.-M. 1992, Long internal waves. Phys. Fluids A 4 Google Scholar, 1929–1935.
[158] Vanden-Broeck, J.-M. 1980, Nonlinear stern waves. J. Fluid Mech. 96 CrossRef | Google Scholar, 601–610.
[159] Vanden-Broeck, J.-M. 1981, The influence of capillarity on cavitating flow past a flat plate. Quart. J. Mech. Appl. Math. 34 CrossRef | Google Scholar, 465–473.
[160] Vanden-Broeck, J.-M. 1983a, The influence of surface tension on cavitating flow past a curved obstacle. J. Fluid Mech. 133 CrossRef | Google Scholar, 255–264.
[161] Vanden-Broeck, J.-M. 1983b, Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26 CrossRef | Google Scholar, 2033–2034.
[162] Vanden-Broeck, J.-M. 1983c, Some new gravity waves in water of finite depth. Phys. Fluids 26 CrossRef | Google Scholar, 2385–2387.
[163] Vanden-Broeck, J.-M. 1984a, The effect of surface tension on the shape of the Kirchhoff jet. Phys. Fluids 27 CrossRef | Google Scholar, 1933–1936.
[164] Vanden-Broeck, J.-M. 1984b, Numerical solutions for cavitating flow of a fluid with surface tension past a curved obstacle. Phys. Fluids 27 CrossRef | Google Scholar, 2601–2603.
[165] Vanden-Broeck, J.-M. 1984c, Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27 CrossRef | Google Scholar, 1090–1093.
[166] Vanden-Broeck, J.-M. 1984d, Rising bubbles in a two-dimensional tube with surface tension. Phys. Fluids 27 CrossRef | Google Scholar, 2604–2607 and 1992, Rising bubbles in a two-dimensional tube: asymptotic behavior for small values of the surface tension, Phys. Fluids A4, 2332–2334.
[167] Vanden-Broeck, J.-M. 1984e, Nonlinear gravity–capillary standing waves in water of arbitrary uniform depth. J. Fluid Mech. 139 CrossRef | Google Scholar, 97–104.
[168] Vanden-Broeck, J.-M. 1985, Nonlinear free-surface flows past two-dimensional bodies. In Advances in Nonlinear Waves, Vol. II, L., Debnath, ed., Boston, Pitman Google Scholar.
[169] Vanden-Broeck, J.-M. 1986a, Pointed bubbles rising in a two dimensional tube. Phys. Fluids 29 CrossRef | Google Scholar, 1343–1344.
[170] Vanden-Broeck, J.-M. 1986b, A free streamline model for a rising bubble. Phys. Fluids 29 CrossRef | Google Scholar, 2798–2801.
[171] Vanden-Broeck, J.-M. 1986c, Flow under a gate. Phys. Fluids 29 CrossRef | Google Scholar, 3148–3151.
[172] Vanden-Broeck, J.-M. 1986d, Steep gravity waves: Havelock's method revisited. Phys. Fluids 29 CrossRef | Google Scholar, 3084–3085.
[173] Vanden-Broeck, J.-M. 1987, Free-surface flow over an obstruction in a channel. Phys. Fluids 30 CrossRef | Google Scholar, 2315–2317.
[174] Vanden-Broeck, J.-M. 1988, Joukovskii's model for a rising bubble. Phys. Fluids 31 CrossRef | Google Scholar, 974–977.
[175] Vanden-Broeck, J.-M. 1989, Bow flows in water of finite depth. Phys. Fluids A1 CrossRef | Google Scholar, 1328–1330.
[176] Vanden-Broeck, J.-M. 1991a, Cavitating flow of a fluid with surface tension past a circular cylinder. Phys. Fluids A 3 CrossRef | Google Scholar, 263–266.
[177] Vanden-Broeck, J.-M. 1991b, Elevation solitary waves with surface tensionPhys. Fluids A 3 CrossRef | Google Scholar, 2659–2663.
[178] Vanden-Broeck, J.-M. 1994, Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 274 CrossRef | Google Scholar, 339–348.
[179] Vanden-Broeck, J.-M. 1995, New families of steep solitary waves in water of finite depth with constant vorticity. Eur. J. Mech. B – fluids 14 Google Scholar, 761–774.
[180] Vanden-Broeck, J.-M. 1996a, Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Math. 56 CrossRef | Google Scholar, 207–217.
[181] Vanden-Broeck, J.-M. 1996b, Numerical calculations of the free-surface flow under a sluice gate. J. Fluid Mech. 330 CrossRef | Google Scholar, 339–347.
[182] Vanden-Broeck, J.-M. 2002, Wilton ripples generated by a moving pressure distribution. J. Fluid Mech. 451 CrossRef | Google Scholar, 193–201.
[183] Vanden-Broeck, J.-M. 2004, Nonlinear capillary free-surface flows. J. Eng. Math. 50 CrossRef | Google Scholar, 415–426.
[184] Vanden-Broeck, J.-M. & Dias, F. 1992, Gravity–capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240 CrossRef | Google Scholar, 549–557.
[185] Vanden-Broeck, J.-M. & Keller, J. B. 1980, A new family of capillary waves. J. Fluid Mech. 98 CrossRef | Google Scholar, 161–169.
[186] Vanden-Broeck, J.-M. & Keller, J. B. 1989, Surfing on solitary waves. J. Fluid Mech. 198 CrossRef | Google Scholar, 115–125.
[187] Vanden-Broeck, J.-M. & Keller, J. B. 1997, An axisymmetric free surface with a 120 degree angle along a circle. J. Fluid Mech. 342 CrossRef | Google Scholar, 403–409.
[188] Vanden-Broeck, J.-M. & Miloh, T. 1995, Computations of steep gravity waves by a refinement of the Davies–Tulin approximation. Siam J. Appl. Math. 55 CrossRef | Google Scholar, 892–903.
[189] Vanden-Broeck, J.-M. & Schwartz, L. W. 1979, Numerical computation of steep gravity waves in shallow water. Phys. Fluids 22 CrossRef | Google Scholar, 1868–1871.
[190] Vanden-Broeck, J.-M., Schwartz, L. W. & Tuck, E. O. 1978, Divergent low-Froude-number series expansion in nonlinear free-surface flow problems. Proc. Roy. Soc. London A 361 CrossRef | Google Scholar, 207–224.
[191] Vanden-Broeck, J.-M. & Shen, M. C. 1983, A note on solitary and periodic waves with surface tension. Z. Angew. Math. Phys. 34 CrossRef | Google Scholar, 112–117.
[192] Vanden-Broeck, J.-M. & Tuck, E. O. 1977. Computation of near-bow or stern flows, using series expansion in the Froude number. In Proc. 2nd Int. Conf. on Num. Ship Hydrodynamics, Berkeley, California Google Scholar, 371–381.
[193] Vanden-Broeck, J.-M. & Tuck, E. O. 1994, Steady inviscid rotational flows with free surfacesJ. Fluid Mech. 258 CrossRef | Google Scholar, 105–113.
[194] Villat, H. 1914, Sur la validité des solutions de certains problèmes d'hydrodynamique. J. de Math. 10 Google Scholar, 231–290.
[195] Whitham, G. B. 1974, Linear and nonlinear waves. Wiley Interscience Google Scholar, John Wiley & Sons.
[196] Wehausen, J. V. & Laitone, E. V. 1960, Surface waves. In Handbuch der Physik, C., Truesdell, ed., Vol. IX Google Scholar, pp. 446–778, Springer.
[197] Williams, J. M. 1981, Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302 CrossRef | Google Scholar, 139–188.
[198] Wilton, J. R., 1915, On ripples. Phil. Mag. 29 CrossRef | Google Scholar, 688–700.
[199] Zufuria, J. A. 1987, Symmetry breaking in periodic and solitary gravity–capillary waves on water of finite depth. J. Fluid Mech. 184 CrossRef | Google Scholar, 183–206.

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