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Published online by Cambridge University Press:  07 September 2010

Jean-Marc Vanden-Broeck
Jean-Marc Vanden-Broeck
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University College London
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Gravity–Capillary Free-Surface Flows , pp. 308 - 317
Publisher: Cambridge University Press
Print publication year: 2010

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References

[1] Acheson, D. J. 1990, Elementary Fluid DynamicsOxford University Press.Google Scholar
[2] Ackerberg, R. C. 1975, The effects of capillarity on free-streamline separation. J. Fluid Mech. 70, 333–352.CrossRefGoogle Scholar
[3] Akylas, T. R. 1993, Envelope solitons with stationary crests. Phys. Fluids A 5, 789–791.CrossRefGoogle Scholar
[4] Akylas, T. R. & Grimshaw, R. 1992, Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279–298.CrossRefGoogle Scholar
[5] Amick, C. J., Fraenkel, L. E. & Toland, J. F. 1982, On the Stokes conjecture and the wave of extreme form. Acta Math. 148, 193–214.CrossRefGoogle Scholar
[6] Anderson, C. D. & Vanden-Broeck, J.-M. 1996, Bow flows with surface tension. Proc. Roy. Soc. Lond. A 452, 1985–1997.CrossRefGoogle Scholar
[7] Asavanant, J. & Vanden-Broeck, J.-M. 1994, Free-surface flows past a surface-piercing object of finite length. J. Fluid. Mech. 273, 109–124.CrossRefGoogle Scholar
[8] Batchelor, G. K.Fluid Dynamics. Cambridge University Press, 615 pp.
[9] Baker, G., Meiron, D. & Orszag, S. 1982, Generalised vortex methods for free surface flow problemsJ. Fluid Mech. 123, 477–501.CrossRefGoogle Scholar
[10] Beale, T. J. 1991, Solitary water waves with capillary ripples at infinity. Comm. Pure Appl. Maths 64, 211–257.CrossRefGoogle Scholar
[11] Benjamin, B. 1956, On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1, 227–248.CrossRefGoogle Scholar
[12] Benjamin, B. 1962, The solitary wave on a stream with arbitrary distribution of vorticity. J. Fluid Mech. 12, 97–116.CrossRefGoogle Scholar
[13] Billingham, J. & King, A. C. 2000, Wave Motion. Cambridge University Press.Google Scholar
[14] Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. 2005, Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.CrossRefGoogle ScholarPubMed
[15] Binder, B. J. & Vanden-Broeck, J.-M. 2005, Free surface flows past surfboards and sluice gates. Euro. J. Appl. Math. 16, 601–619.CrossRefGoogle Scholar
[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, 475–490.CrossRefGoogle Scholar
[17] Binnie, A. M. 1952, The flow of water under a sluice gate. Q. J. Mech. Appl. Math. 5, 395–407.CrossRefGoogle Scholar
[18] Birkhoff, G. & Carter, D. 1957, Rising plane bubbles. J. Math. Phys. 6 769–779.Google Scholar
[19] Birkhoff, G. & Zarantonello, E. 1957, Jets, Wakes and CavitiesAcademic Press, 353 pp.Google Scholar
[20] Blyth, M. G. & Vanden-Broeck, J.-M. 2004, New solutions for capillary waves on fluid sheets. J. Fluid Mech. 507, 255–264.CrossRefGoogle Scholar
[21] Blyth, M. G. & Vanden-Broeck, J.-M. 2005, New solutions for capillary waves on curved sheets of fluids. IMA J. Appl. Math. 70, 588–601.CrossRefGoogle Scholar
[22] Brillouin, M. 1911, Les surfaces de glissement de Helmoltz at la résistance des fluides. Ann. de Chim. Phys. 23, 145–230.Google Scholar
[23] Brodetsky, S. 1923, Discontinuous fluid motion past circular and elliptic cylinders. Proc. Roy. Soc. London A 102, 1–14.CrossRefGoogle Scholar
[24] Budden, P. & Norbury, J. 1982, Uniqueness of free boundary flows under gravity. Arch. Rat. Mech. Anal. 78, 361–380.CrossRefGoogle Scholar
[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, 175–189.CrossRefGoogle Scholar
[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, 403–417.CrossRefGoogle Scholar
[27] Chen, B. & Saffman, P. G. 1979, Steady gravity–capillary waves on deep water, Part I: Weakly nonlinear waves. Stud. Appl. Math. 60, 183–210.CrossRefGoogle Scholar
[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, 1–21.CrossRefGoogle Scholar
[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, 95–111.CrossRefGoogle Scholar
[30] Chung, Y. K. 1972, Solution of flow under a sluice gates. ASCE J. Eng. Mech. Div. 98, 121–140.Google Scholar
[31] Cokelet, E. D. 1977, Steep gravity waves in water of arbitrary uniform depth, Phil. Trans. Roy. Soc. London A 286, 183–230.CrossRefGoogle Scholar
[32] Collins, R. 1965, A simple model of a plane gas bubble in a finite liquid. J. Fluid Mech. 22, 763–771.CrossRefGoogle Scholar
[33] Concus, P. 1962, Standing capillary–gravity waves of finite amplitude. J. Fluid Mech. 14, 568–576.CrossRefGoogle Scholar
[34] Concus, P. 1964, Standing capillary–gravity waves of finite amplitude: Corrigendum. J. Fluid Mech. 19, 264–266.CrossRefGoogle Scholar
[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, 141–158.CrossRefGoogle Scholar
[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, 1–14.CrossRefGoogle Scholar
[37] Crapper, G. D. 1957, An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 572–540.CrossRefGoogle Scholar
[38] Crowdy, D. G. 1999, Exact solutions for steady capillary waves on a fluid annulus. J. Nonlinear Sci. 9, 615–640.CrossRefGoogle Scholar
[39] Cumberbatch, E. & Norbury, J. 1979, Capillarity modification of the singularity at a free-streamline separation point. Q. J. Mech. Appl. Math. 32, 303–312.CrossRefGoogle Scholar
[40] Dagan, G. & Tulin, M. P. 1972, Two-dimensional free surface gravity flows past blunt bodies. J. Fluid Mech. 51, 529–543.CrossRefGoogle Scholar
[41] Davies, T. V. 1951, Theory of symmetrical gravity waves of finite amplitude. Proc. Roy. Soc. London A 208, 475–486.CrossRefGoogle Scholar
[42] Dias, F. & Iooss, G. 1993, Capillary–gravity solitary waves with damped oscillations. Physica D 65, 399–423.CrossRefGoogle Scholar
[43] Dias, F. & Kharif, C. 1999, Nonlinear gravity and capillary–gravity waves. Ann. Rev. Fluid Mech. 31, 301–346.CrossRefGoogle Scholar
[44] Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996, Numerical study of capillary–gravity solitary waves. Eur. J. Mech. B – Fluids 15, 17–36.Google Scholar
[45] Dias, F. & Vanden-Broeck, J.-M. 1989, Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.CrossRefGoogle Scholar
[46] Dias, F. & Vanden-Broeck, J.-M. 1992, Solitary waves in water of infinite depth and related free surface flows. J. Fluid Mech. 240, 549–557.Google Scholar
[47] Dias, F. & Vanden-Broeck, J.-M. 1993, Nonlinear bow flows with splashes. J. Fluid Mech. 255, 91–102.CrossRefGoogle Scholar
[48] Dias, F. & Vanden-Broeck, J.-M. 2002, Generalized critical free-surface flows. J. Eng. Math. 42, 291–301.CrossRefGoogle Scholar
[49] Dias, F. & Vanden-Broeck, J.-M. 2004a, Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.CrossRefGoogle Scholar
[50] Dias, F. & Vanden-Broeck, J.-M. 2004b, Two-layer hydraulic falls over an obstacle. Eur. J. Mech. B – Fluids 23, 879–898.CrossRefGoogle Scholar
[51] Dingle, R. B. 1973, Asymptotic Expansions: Their Derivation and Interpretation. Academic Press.Google Scholar
[52] Eggers, J. 1995, Theory of drop formation. Phys. Fluids 7, 941–953.CrossRefGoogle Scholar
[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, 373–390.CrossRefGoogle Scholar
[54] Fangmeier, D. D. & Strelkoff, T. S. 1968, Solution for gravity flow under a sluice gate. ASCE J. Eng. Mech. Div. 94, 153–176.Google Scholar
[55] Forbes, L.-K. 1981, On the resistance of a submerged semi-elliptical body. J. Eng. Math. 15, 287–298.CrossRefGoogle Scholar
[56] Forbes, L.-K. 1983, Free surface flow over a semicircular obstruction including the influence of gravity and surface tension. J. Fluid Mech. 127, 283–297.CrossRefGoogle Scholar
[57] Forbes, L.-K. 1988, Critical free-surface flow over a semi-circular obstruction. J. Eng. Math. 22, 3–13.CrossRefGoogle Scholar
[58] Forbes, L. K 1989, An algorithm for 3-dimensional free-surface problems in hydrodynamics. J. Comput. Phys. 82, 330–347.CrossRefGoogle Scholar
[59] Forbes, L. K. & Schwartz, L. W. 1982, Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.CrossRefGoogle Scholar
[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, 231–249.CrossRefGoogle Scholar
[61] Friedrics, K. O. & Hyers, D. H. 1954, The existence of solitary waves. Comm. Pure Appl. Math. 7, 517–550.CrossRefGoogle Scholar
[62] Garabedian, P. R. 1957, On steady state bubbles generated by Taylor instability. Proc. Roy. Soc. London A 241, 423–431.CrossRefGoogle Scholar
[63] Garabedian, P. R. 1985, A remark about pointed bubbles. Comm. Pure Appl. Math. 38, 609–612.CrossRefGoogle Scholar
[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, 031703.CrossRefGoogle Scholar
[65] Grandison, S. & Vanden-Broeck, J.-M. 2006, Truncation methods for gravity capillary free surface flows. J. Eng. Math. 54, 89–97.CrossRefGoogle Scholar
[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, 829–867.3.0.CO;2-2>CrossRefGoogle Scholar
[67] Grimshaw, R. H. J. & Smyth, N. 1986, Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.CrossRefGoogle Scholar
[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. 1–91.CrossRefGoogle Scholar
[69] Gurevich, M. 1965, Theory of Jets and Ideal Fluids. Academic Press, 585 pp.Google Scholar
[70] Havelock, T. H. 1919, Periodic irrotational waves of finite amplitude. Proc. Roy. Soc. London Ser. A 95, 38–51.CrossRefGoogle Scholar
[71] Helmholtz, H. 1868, Über discontinuierliche Flüssigkeitsbewegungen. Monatsber, Berlin Akad., 215–228, reprinted in Phil. Mag.36, 337–346.Google Scholar
[72] Hocking, G. C. & Vanden-Broeck, J.-M. 1997, Draining of a fluid of finite depth into a vertical slot. Applied Math. Modelling 21, 643–649.CrossRefGoogle Scholar
[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, 181–191.CrossRefGoogle Scholar
[74] Hogan, S. J. 1980, Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417–445.CrossRefGoogle Scholar
[75] Hunter, J. K. & Scherule, J. 1988, Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32, 253–268.CrossRefGoogle Scholar
[76] Hunter, J. K. & Vanden-Broeck, J.-M. 1983a, Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219.CrossRefGoogle Scholar
[77] Hunter, J. K. & Vanden-Broeck, J.-M. 1983b, Accurate computations for steep solitary waves. J. Fluid Mech. 136, 63–71.CrossRefGoogle Scholar
[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, 1–19.CrossRefGoogle Scholar
[79] Iooss, G. & Kirchgassner, K. 1990, Bifurcation d'ondes solitaires en présences d'une faible tension superficielle. C.R. Acad. Sci. Paris 311 I, 265–268.Google Scholar
[80] Iooss, G. & Kirchgassner, K. 1992, Water waves for small surface tension: an approach via normal form. Proc. Roy. Soc. Edinburgh 122A, 267–299.CrossRefGoogle Scholar
[81] Iooss, G., Plotnikov, P. & Toland, J. F. 2005, Standing waves on an infinitely deep perfect fluid under gravity. Arch. Rat. Mech. Anal. 177, 367–478.CrossRefGoogle Scholar
[82] Kang, Y. & Vanden-Broeck, J.-M. 2002, Stern waves with vorticityANZIAM J. 43, 321–332.Google Scholar
[83] Kawahara, T. 1972, Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33, 260–264.CrossRefGoogle Scholar
[84] Keller, H. B. 1977, Applications of Bifurcation Theory. Academic Press.Google Scholar
[85] Keller, J. B. & Miksis, M. J. 1983, Surface tension driven flows. SIAM J. Appl. Math. 43, 268–277.CrossRefGoogle Scholar
[86] Keller, J. B., Milewski, P. & Vanden-Broeck, J.-M. 2000, Wetting and merging driven by surface tension. Euro. J. Mech. B – Fluids 19, 491–502.CrossRefGoogle Scholar
[87] Kim, B. & Akylas, T. R. 2005, On gravity–capillary lumps. J. Fluid Mech. 540, 337–351.CrossRefGoogle Scholar
[88] Kim, B. & Akylas, T. R. 2006, On gravity–capillary lumps, Part 2. Two dimensional Benjamin equation. J. Fluid Mech. 557, 237–256.CrossRefGoogle Scholar
[89] Kinnersley, W. 1976, Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229–241.CrossRefGoogle Scholar
[90] Kirchhoff, G. 1869, Zur Theorie freier Flüssigkeitsstrahlen. J. Reine Angew. Math. 70, 289–298.CrossRefGoogle Scholar
[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, 422–443.CrossRefGoogle Scholar
[92] Lamb, H. 1945, Hydrodynamics, 6th edn, Cambridge University Press.Google Scholar
[93] Larock, B. E. 1969, Gravity-affected flow from planar sluice gate. ASCE J. Engng Mech. Div. 96, 1211–1226.Google Scholar
[94] Lee, J. W. & Vanden-Broeck, J.-M. 1993, Two-dimensional jets falling from funnels and nozzles. Phys. Fluids A5, 2454–2460.CrossRefGoogle Scholar
[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, 332–349.CrossRefGoogle Scholar
[96] Lenau, C. W. 1966, The solitary wave of maximum amplitude. J. Fluid Mech. 26, 309–320.CrossRefGoogle Scholar
[97] Lombardi, E. 2000, Oscillatory Integrals on Phenomena Beyond All Orders: with Applications to Homoclinic Orbits in reversible systems. Lecture Notes in Mathematics 1741, Springer.CrossRefGoogle Scholar
[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, 344–357.CrossRefGoogle Scholar
[100] Lighthill, M. J. 1978, Waves in Fluids, Cambridge University Press, 504 pp.Google Scholar
[101] Longuet–Higgins, M. S. 1975, Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. London A 342, 157–174.CrossRefGoogle Scholar
[102] Longuet-Higgins, M. S. 1989, Capillary-gravity waves of solitary type on deep water. J. Fluid Mech. 200, 451–478.CrossRefGoogle Scholar
[103] Longuet-Higgins, M. S. 1993, Capillary–gravity waves of solitary type and envelope solitons on deep water. J. Fluid Mech. 252, 703–711.CrossRefGoogle Scholar
[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, 1–26.CrossRefGoogle Scholar
[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, 471–493.CrossRefGoogle Scholar
[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, 769–786.CrossRefGoogle Scholar
[107] Maneri, C. C. 1970, 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, 387–407.CrossRefGoogle Scholar
[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, 455–469.CrossRefGoogle Scholar
[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, 2652–2658.CrossRefGoogle Scholar
[111] Michallet, H. & Dias, F. 1999, Numerical study of generalized interfacial solitary waves. Phys. Fluids 11, 1502–1511.CrossRefGoogle Scholar
[112] Michell, J. H. 1883, The highest wave in water. Phil. Mag. 36, 430–437.CrossRefGoogle Scholar
[113] Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1981, Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–101.CrossRefGoogle Scholar
[114] Miksis, M., Vanden-Broeck, J.-M. & Keller, J. B. 1982, Rising bubbles. J. Fluid Mech. 123, 31–41.CrossRefGoogle Scholar
[115] Milewski, P. A. 2005, Three-dimensional localized solitary gravity–capillary waves. Comm. Math. Sc. 3, 89–99.CrossRefGoogle Scholar
[116] Nayfeh, A. H. 1970, Triple and quintuple-dimpled wave profiles in deep water. J. Fluid Mech. 13, 545–550.Google Scholar
[117] Ockendon, H. & Ockendon, J. R. 2004, Viscous Flow. Cambridge Texts in Applied Mathematics.Google Scholar
[118] Olfe, D. B. & Rottman, J. W. 1980, Some new highest-wave solutions for deep-water waves of permanent form. J. Fluid Mech. 100, 801–810.CrossRefGoogle Scholar
[119] Osher, S. & Fedkiw, R. 2003, Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences 153, Springer.CrossRefGoogle Scholar
[120] Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2003, Large amplitude capillary waves in electrified fluid sheets. J. Fluid Mech. 508, 71–88.CrossRefGoogle Scholar
[121] Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2004, Antisymmetric capillary waves in electrified fluid sheets. Eur. J. Appl. Math. 15, 609–623.CrossRefGoogle Scholar
[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, 643–656.CrossRefGoogle Scholar
[123] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2005a, Nonlinear three dimensional gravity capillary solitary waves. J. Fluid Mech. 536, 99–105.CrossRefGoogle Scholar
[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, 122 101.CrossRefGoogle Scholar
[125] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2007a, Three-dimensional capillary–gravity waves generated by a moving disturbance. Phys. Fluids 19, 082 102.CrossRefGoogle Scholar
[126] Parau, E., Vanden-Broeck, J.-M. & Cooker, M. 2007b, Nonlinear three dimensional interfacial flows with a free surface. J. Fluid Mech. 591, 481–494.CrossRefGoogle Scholar
[127] Pullin, D. I. & Grimshaw, R. H. J. 1988, Finite amplitude solitary waves at the interface between two homogeneous fluids. Phys. Fluids 31, 3550–3559.CrossRefGoogle Scholar
[128] Rayleigh, Lord 1883, The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. 15, 69–78.CrossRefGoogle Scholar
[129] Romero, L. 1982, 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, 567–581.CrossRefGoogle Scholar
[131] Saffman, P. G. 1986, Viscous fingering in Hele Shaw cells. J. Fluid Mech. 173, 73–94.CrossRefGoogle Scholar
[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, 312–329.CrossRefGoogle Scholar
[133] Schwartz, L. W. 1974, Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. 62, 553–578.CrossRefGoogle Scholar
[134] Schwartz, L. W. & Fenton, J. 1982, Strongly nonlinear waves. Ann. Rev. Fluid Mech. 14, 39–60.CrossRefGoogle Scholar
[135] Schwartz, L. W. & Vanden-Broeck, J.-M. 1979, Numerical solution of the exact equations for capillary–gravity waves. J. Fluid Mech. 95, 119–139.CrossRefGoogle Scholar
[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, 253–272.Google Scholar
[137] Sethian, J. A.Level Set Methods. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
[138] Sha, H. & Vanden-Broeck, J.-M. 1993, Two-layer flows past a semicircular obstaclePhys. Fluids A 5, 2661–2668.CrossRefGoogle Scholar
[139] Sha, H. & Vanden-Broeck, J.-M. 1997, Internal solitary waves with stratification in density. J. Austral. Math. Soc. B 38, 563–580.CrossRefGoogle Scholar
[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, 701–719.CrossRefGoogle Scholar
[141] Simmen, J. A. & Saffman, P. G. 1985, Steady deep water waves on a linear shear current. Stud. Appl. Maths 75, 35–57.CrossRefGoogle Scholar
[142] Southwell, R. V. & Vaisey, G. 1946, Fluid motions characterised by ‘free’ streamlines. Phil. Trans. Roy. Soc. A 240, 117–161.CrossRefGoogle Scholar
[143] Stokes, G. G. 1847, On the theory of oscillatory waves. Camb. Trans. Phil. Soc. 8, 441–473.Google Scholar
[144] Stokes, G. G. 1880, in Mathematical and Physical Papers, Vol. 1, p. 314, Cambridge University Press.Google Scholar
[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, 471–504.CrossRefGoogle Scholar
[146] Sun, S. M. 1999, Nonexistence of truly solitary waves in water with small surface tensionProc. Roy. Soc. London A 455, 2191–2228.CrossRefGoogle Scholar
[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, 533–566.CrossRefGoogle Scholar
[148] Tadjbakhsh, I. & Keller, J. B. 1960, Standing surface waves of finite amplitude. J. Fluid Mech. 8, 442–451.CrossRefGoogle Scholar
[149] Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987, Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.CrossRefGoogle Scholar
[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, 281–305.CrossRefGoogle Scholar
[151] Tooley, S. & Vanden-Broeck, J.-M. 2002, Waves and singularities in nonlinear capillary free-surface flows. J. Eng. Math. 43, 89–99.CrossRefGoogle Scholar
[152] Tsai, W. T. & Yue, D. K. 1996, Computation of nonlinear free surface flows. Ann. Rev. Fluid Mech. 28, 249–278.CrossRefGoogle Scholar
[153] Tseluiko, D., Blyth, M. & Papageorgiou, D. T. 2008a, Electrified viscous thin film over topography. J. Fluid Mech. 597, 449–475.CrossRefGoogle Scholar
[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, 042 103CrossRefGoogle Scholar
[155] Turner, R. E. L. & Vanden-Broeck, J.-M. 1986, The limiting configuration of interfacial gravity waves. Phys. Fluids 29, 372–375.CrossRefGoogle Scholar
[156] Turner, R. E. L. & Vanden-Broeck, J.-M. 1988, Broadening on interfacial solitary waves. Phys. Fluids 31, 2486–2490.CrossRefGoogle Scholar
[157] Turner, R. E. L. & Vanden-Broeck, J.-M. 1992, Long internal waves. Phys. Fluids A 4, 1929–1935.Google Scholar
[158] Vanden-Broeck, J.-M. 1980, Nonlinear stern waves. J. Fluid Mech. 96, 601–610.CrossRefGoogle Scholar
[159] Vanden-Broeck, J.-M. 1981, The influence of capillarity on cavitating flow past a flat plate. Quart. J. Mech. Appl. Math. 34, 465–473.CrossRefGoogle Scholar
[160] Vanden-Broeck, J.-M. 1983a, The influence of surface tension on cavitating flow past a curved obstacle. J. Fluid Mech. 133, 255–264.CrossRefGoogle Scholar
[161] Vanden-Broeck, J.-M. 1983b, Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–2034.CrossRefGoogle Scholar
[162] Vanden-Broeck, J.-M. 1983c, Some new gravity waves in water of finite depth. Phys. Fluids 26, 2385–2387.CrossRefGoogle Scholar
[163] Vanden-Broeck, J.-M. 1984a, The effect of surface tension on the shape of the Kirchhoff jet. Phys. Fluids 27, 1933–1936.CrossRefGoogle Scholar
[164] Vanden-Broeck, J.-M. 1984b, Numerical solutions for cavitating flow of a fluid with surface tension past a curved obstacle. Phys. Fluids 27, 2601–2603.CrossRefGoogle Scholar
[165] Vanden-Broeck, J.-M. 1984c, Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27, 1090–1093.CrossRefGoogle Scholar
[166] Vanden-Broeck, J.-M. 1984d, Rising bubbles in a two-dimensional tube with surface tension. Phys. Fluids 27, 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.CrossRefGoogle Scholar
[167] Vanden-Broeck, J.-M. 1984e, Nonlinear gravity–capillary standing waves in water of arbitrary uniform depth. J. Fluid Mech. 139, 97–104.CrossRefGoogle Scholar
[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, 1343–1344.CrossRefGoogle Scholar
[170] Vanden-Broeck, J.-M. 1986b, A free streamline model for a rising bubble. Phys. Fluids 29, 2798–2801.CrossRefGoogle Scholar
[171] Vanden-Broeck, J.-M. 1986c, Flow under a gate. Phys. Fluids 29, 3148–3151.CrossRefGoogle Scholar
[172] Vanden-Broeck, J.-M. 1986d, Steep gravity waves: Havelock's method revisited. Phys. Fluids 29, 3084–3085.CrossRefGoogle Scholar
[173] Vanden-Broeck, J.-M. 1987, Free-surface flow over an obstruction in a channel. Phys. Fluids 30, 2315–2317.CrossRefGoogle Scholar
[174] Vanden-Broeck, J.-M. 1988, Joukovskii's model for a rising bubble. Phys. Fluids 31, 974–977.CrossRefGoogle Scholar
[175] Vanden-Broeck, J.-M. 1989, Bow flows in water of finite depth. Phys. Fluids A1, 1328–1330.CrossRefGoogle Scholar
[176] Vanden-Broeck, J.-M. 1991a, Cavitating flow of a fluid with surface tension past a circular cylinder. Phys. Fluids A 3, 263–266.CrossRefGoogle Scholar
[177] Vanden-Broeck, J.-M. 1991b, Elevation solitary waves with surface tensionPhys. Fluids A 3, 2659–2663.CrossRefGoogle Scholar
[178] Vanden-Broeck, J.-M. 1994, Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 274, 339–348.CrossRefGoogle Scholar
[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, 761–774.Google Scholar
[180] Vanden-Broeck, J.-M. 1996a, Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Math. 56, 207–217.CrossRefGoogle Scholar
[181] Vanden-Broeck, J.-M. 1996b, Numerical calculations of the free-surface flow under a sluice gate. J. Fluid Mech. 330, 339–347.CrossRefGoogle Scholar
[182] Vanden-Broeck, J.-M. 2002, Wilton ripples generated by a moving pressure distribution. J. Fluid Mech. 451, 193–201.CrossRefGoogle Scholar
[183] Vanden-Broeck, J.-M. 2004, Nonlinear capillary free-surface flows. J. Eng. Math. 50, 415–426.CrossRefGoogle Scholar
[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, 549–557.CrossRefGoogle Scholar
[185] Vanden-Broeck, J.-M. & Keller, J. B. 1980, A new family of capillary waves. J. Fluid Mech. 98, 161–169.CrossRefGoogle Scholar
[186] Vanden-Broeck, J.-M. & Keller, J. B. 1989, Surfing on solitary waves. J. Fluid Mech. 198, 115–125.CrossRefGoogle Scholar
[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, 403–409.CrossRefGoogle Scholar
[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, 892–903.CrossRefGoogle Scholar
[189] Vanden-Broeck, J.-M. & Schwartz, L. W. 1979, Numerical computation of steep gravity waves in shallow water. Phys. Fluids 22, 1868–1871.CrossRefGoogle Scholar
[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, 207–224.CrossRefGoogle Scholar
[191] Vanden-Broeck, J.-M. & Shen, M. C. 1983, A note on solitary and periodic waves with surface tension. Z. Angew. Math. Phys. 34, 112–117.CrossRefGoogle Scholar
[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, 371–381.Google Scholar
[193] Vanden-Broeck, J.-M. & Tuck, E. O. 1994, Steady inviscid rotational flows with free surfacesJ. Fluid Mech. 258, 105–113.CrossRefGoogle Scholar
[194] Villat, H. 1914, Sur la validité des solutions de certains problèmes d'hydrodynamique. J. de Math. 10, 231–290.Google Scholar
[195] Whitham, G. B. 1974, Linear and nonlinear waves. Wiley Interscience, John Wiley & Sons.Google Scholar
[196] Wehausen, J. V. & Laitone, E. V. 1960, Surface waves. In Handbuch der Physik, C., Truesdell, ed., Vol. IX, pp. 446–778, Springer.Google Scholar
[197] Williams, J. M. 1981, Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139–188.CrossRefGoogle Scholar
[198] Wilton, J. R., 1915, On ripples. Phil. Mag. 29, 688–700.CrossRefGoogle Scholar
[199] Zufuria, J. A. 1987, Symmetry breaking in periodic and solitary gravity–capillary waves on water of finite depth. J. Fluid Mech. 184, 183–206.CrossRefGoogle Scholar

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  • References
  • Jean-Marc Vanden-Broeck, University College London
  • Book: Gravity–Capillary Free-Surface Flows
  • Online publication: 07 September 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730276.013
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  • References
  • Jean-Marc Vanden-Broeck, University College London
  • Book: Gravity–Capillary Free-Surface Flows
  • Online publication: 07 September 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730276.013
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  • References
  • Jean-Marc Vanden-Broeck, University College London
  • Book: Gravity–Capillary Free-Surface Flows
  • Online publication: 07 September 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511730276.013
Available formats
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