Benard Layers with Heat or Mass Transfer

  • M. G. Velarde
Part of the International Centre for Mechanical Sciences book series (CISM, volume 428)


I have taken the Benard liquid-layer set-up to succintly illustrate the variety of phenomena and mathematical problems that the Marangoni stress can originate already in the simplest case of (l+l)D geometry. This surface (tangential) stress may be due to heat or mass transfer creating a surface tension gradient that ultimately leads to flow motion and/or instability and subsequent convection (Marangoni effect). For the (1+1)D geometry I provide, first, under drastically simplifying approximations the conditions for the onset and nonlinear evolution of steady convection (as a caricature of Benard cells). The evolutionary problem is presented for two different effective gravity levels, one of them corresponding to microgravity (the kind of effective gravity existing in the free fall of an Earth‘s orbital space lab. or station). Then I present the theory when instability leads to oscillatory motions and waves. I give discussion of physical mechanisms leading to either capillary-gravity (transverse) waves or dilational (sound-like, longitudinal) waves, and even internal waves, and their possible resonance. For the transverse mode I discuss its nonlin­ear evolution by deriving a (“long wave”, shallow layer) generalized (dissipation-modified) Boussinesq-Korteweg-de Vries equation. I discuss how solitary waves and solitons are expected to appear in the Benard layer. Finally, I comment on a few experimental results about dissipative solitons.


Solitary Wave Rayleigh Number Internal Wave Liquid Layer Marangoni Number 


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  1. Bar, D. and A. A. Nepomnyashchy (1995) Stability of periodic waves governed by the modified Kawahara equation. Physica D 86: 586–602.CrossRefMATHMathSciNetGoogle Scholar
  2. Bazin, H. (1865) Recherches hydrauliques experimentales sur la propagation des ondes entreposes par Darcy et continuees par M. Bazin. Mon. presentes par divers Savants Etrangers a l’Acad. Sci. Inst. France 19: 495–644Google Scholar
  3. Benard, H. (1900) Les tourbillons cellulaires dans une nappe liquide. Premiere partie: description generale des phenomenes. Rev Gen. Sci. Pures Appl. 11: 1261–1271.Google Scholar
  4. Benard, H. (1901) Les tourbillons cellulaires dans une nappe liquide transportant de le chaleur par convection en regime permanent. Ann. Chini. Phis. 23: 62–143.Google Scholar
  5. Benjamin, T. B. (1982) The solitary wave with surface tension. Quart. Appl. Maths 40: 231–234.MATHMathSciNetGoogle Scholar
  6. Berg, J.C., and Acrivos, A. (1965) The effect of surface active agents on convection cells induced by surface tension. Chew. Engng. Sci. 20: 737–745.CrossRefGoogle Scholar
  7. Berg, J.C., Acrivos, A., and Boudart, M. (1966) Evaporative convection. Ar. Chew. Eng. 6: 61–123. Bestehorn, M. (1996) Square patterns in Benard-Marangoni convection. Phys. Rer. Lett. 76: 46–49.Google Scholar
  8. Birikh, R. V., Briskman, V. A., Cherepanov, A. A., and Velarde, M. G. (2000) Faraday ripples, parametric resonance, and the Marangoni effect. J. Colloid Interface Sci. 238: 16–23.CrossRefGoogle Scholar
  9. Block, M.J. (1956) Surface tension as the cause of Benard cells and surface deformation in a liquid film. Nature (Lond.) 178: 650–651.Google Scholar
  10. Bouasse, H. (1924) Houle, Rides, Seiches et Marees. Delagrave (Paris). 291–292.Google Scholar
  11. Boussinesq, J.V. (1871) Theorie de l’intumescence liquide appelce onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Hehd. Seances Acad. Sci. (Paris) 72: 755–759.MATHGoogle Scholar
  12. Boussinesq, J.V. (1872) Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horisontal en communiquant au liquid contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17: 55–108.MATHGoogle Scholar
  13. Boussinesq, J. V. (1877) Essai sur la theorie des eaux courantes. Mem. presentes par divers savants â 1’Acad. Sci. Inst. France 32: 1–680.MathSciNetGoogle Scholar
  14. Bragard, J., and Velarde, M. G. (1997) Benard convection flows. J. Nun Equilih. Thermodyn. 22: 1–19.MATHGoogle Scholar
  15. Bragard, J., and Velarde, M. G. (1998) Benard-Marangoni convection: Theoretical predictions about plan-forms and their relative stability. J. Fluid Mech. 368: 165–194.CrossRefMATHMathSciNetGoogle Scholar
  16. Busse, F. H. (1978) Non-linear properties of thermal convection. Rep. Prog. Phys. 41: 1929–1967.CrossRefGoogle Scholar
  17. Castillo, J. L., Garcia-Ybarra, P. L., and Velarde, M. G. (1988) in Synergetics and Dynamic Instabilities, edited by Caglioti, G., Haken, H., and Lugiato, L. A. ( North-Holland, Amsterdam ), 219–243.Google Scholar
  18. Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford.MATHGoogle Scholar
  19. Christov, C. I., and Velarde, M. G. (1993) On localized solutions in an equation in Benard convection. Appl. Math. Model. 17: 311–318.CrossRefMATHGoogle Scholar
  20. Christov, C. I., and Velarde, M. G. (1995) Dissipative solitons. Physica D 86: 323–347.CrossRefMATHMathSciNetGoogle Scholar
  21. Christov, C.I., Maugin, G. A., and Velarde, M. G. (1996) Well-posed Boussinesq Paradigm with purely spatial higher-order derivatives. Phys. Rev. E 54: 3621–3638CrossRefGoogle Scholar
  22. Chu, X.-L., and Velarde, M. G. (1988) Sustained transverse and longitudinal waves at the open surface of a liquid. Physicochem. Hydrodyn. 10: 727–737.Google Scholar
  23. Chu, X.-L., and Velarde, M. G. (1989) Transverse and longitudinal waves induced and sustained by surfactant gradients at liquid-liquid interfaces. J. Colloid Interface Sci. 131: 471–484.CrossRefGoogle Scholar
  24. Chu, X.-L., and Velarde, M. G. (1991) Korteweg-de Vries soliton excitation in Benard-Marangoni convection. Phys. Rev. A 43: 1094–1096.CrossRefGoogle Scholar
  25. Colinet, P., Legros, J. C., and Velarde, M. G. (2001) Nonlinear Dynamic of Surface Tension Driven Instabilities. Wiley-VCH, Weinheim.CrossRefGoogle Scholar
  26. Courant, R., and Friedrichs, K. O. (1948)Supersonic flow and shock waves. Interscience, N.Y. Dauzere C. (1908) Recherches sur la solidification cellulaire. J. Phys. (Paris) 7: 930–934.Google Scholar
  27. Dauzere C. (1912) Sur les changements qu’eprouvent les tourbillons cellulaires lorsque la temperature s’eleve. C. R. Hebd. Seances Acad. Sci. (Paris) 155: 394–398.Google Scholar
  28. Davis, S. H. (1987) Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19: 403–435.CrossRefMATHGoogle Scholar
  29. Davis S. H., and Segel, L. A. (1968) Effects of surface curvature and property variation on cellular convection. Phys. Fluids 11: 470–476.CrossRefMATHGoogle Scholar
  30. de Boer, P. C. T. (1984) Thermally driven motion of strongly heated fluids. hit. J. Heat Mass Transfer. 27: 2239–2251.CrossRefMATHGoogle Scholar
  31. de Boer, P. C. T. (1986) Thermally driven motion of highly viscous fluids. Int. J. Heat Mass Transfer. 29: 681–688.CrossRefMATHGoogle Scholar
  32. Drazin, P. G., and Reid, W. H. (1981)Hydrodynamic Stability. University Press, Cambridge. Drazin, P. G., and Johnson, R. S. (1989) Solitons. An Introduction. University Press, Cambridge.Google Scholar
  33. Eckert, K., Besterhorn, M., and Thess, A. (1998) Square cells in surface-tension-driven Benard convection: experiment and theory. J. Fluid Mech. 356: 155–197.CrossRefMATHMathSciNetGoogle Scholar
  34. Garazo A. N., and Velarde, M. G. (1991) Dissipative Korteweg-de Vries description of Marangoni-Benard. convection. Phys. Fluids A 3: 2295–2300.CrossRefMATHGoogle Scholar
  35. Garcia-Ybarra, P. L., and Velarde, M. G. (1987) Oscillatory Marangoni-Benard interfacial instability and capillary-gravity waves in single-and two-component liquid layers with or without Soret thermal diffusion. Phys. Fluids 30: 1649–1655.CrossRefMATHGoogle Scholar
  36. Garcia-Ybarra, P.L., Castillo, J. L., and Velarde, M. G. (1987) Benard-Marangoni convection with a de-formable interface and poorly conducting boundaries. Phys. Fluids 30: 2655–2661.CrossRefMATHGoogle Scholar
  37. Gershuni, G. Z., and Zhukhovitsky, E. M. (1976)Convective Stability of Incompressible Fluids. Keter, Jerusalem.Google Scholar
  38. Golovin, A. A., Nepomnyashchy, A. A., and Pismen, L. M. (1994) Interaction between short-scale Marangoni convection and long-scale deformational instabilty. Phis. Fluids 6: 34–48.CrossRefMATHMathSciNetGoogle Scholar
  39. Hershey, A. V. (1939) Ridges in a liquid surface due to the temperature dependence of surface tension. Phys. Rev. 56: 204.CrossRefMATHGoogle Scholar
  40. Hornung, H. (1988) Regular and Mach reflection of shock waves. Ann. Rev. Fluid Mech. 18: 33–58.CrossRefGoogle Scholar
  41. Huang, G.-H., Velarde, M. G., and Kurdiumov, V. (1998) Cylindrical solitary waves and their interaction in Benard-Marangoni layers. Phys. Rev. E 57: 5473–5482.CrossRefGoogle Scholar
  42. Hyman, J. M., and Nicolaenko, B. (1987) Coherence and Chaos in the Kuramoto-Velarde equation, in Partial Differential Equations, edited by Crandall, M., Academic, New York.Google Scholar
  43. Kac, M., Uhlenbeck, G. E., and Hemmer, P. C (1963) On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys. 4: 216–228.CrossRefMATHMathSciNetGoogle Scholar
  44. Knobloch, E. (1990) Pattern selection in long-wavelength convection. Physica D 41: 450–479.CrossRefMATHMathSciNetGoogle Scholar
  45. Koschmieder, E.L. (1988) in Physicochemical Hydrodynamics: Interfacial Phenomena, edited by Velarde, M. G., Plenum, New York, 189–198.Google Scholar
  46. Koschmieder, E.L. (1993)Benard Cells and Taylor Vortices. University Press, Cambridge.Google Scholar
  47. Krehl, P., and M. van der Geest (1991) The discovery of the Mach reflection effect and its demonstration in an auditorium. Shock Waves 1: 3–15.CrossRefGoogle Scholar
  48. Linde, H., Chu, X.-L., and Velarde, M. G. (1993a) Oblique and head-on collisions of solitary waves in Marangoni-Benard convection. Phys. Fluids A 5. 1068–1070.CrossRefGoogle Scholar
  49. Linde, H., Chu, X.-L., Velarde, M. G., and Waldheim, W. (1993b) Wall reflection of solitary waves in Marangoni-Benard convection. Phys. Fluids A 5: 3162–3166.CrossRefGoogle Scholar
  50. Linde, H., Chu, X.-L., and Velarde, M. G. (1993c) Solitary waves driven by Marangoni stresses. Adv. Space Res. 13: 109–117.CrossRefGoogle Scholar
  51. Linde, H., Velarde, M. G., Wierschem, A., Waldheim, W., Loeschcke. K., and Rednikov, A. Ye. (1997) Interfacial wave motions due to Marangoni instability. I. Traveling periodic wave trains in square and annular containers. J. Colloid Interface Sci. 188: 16–26.Google Scholar
  52. Linde, H., Velarde, M. G., Waldheim, W., and Wierschem, A. (2001) Interfacial wave motions due to Marangoni instability. III. Solitary waves and (periodic) wavetrains and their collisions and reflections leading to dynamic network patterns. J. Colloid Interface Sci. 236: 214–224.CrossRefGoogle Scholar
  53. Linde, H., Velarde, M. G., Waldheim, W., Loeschcke, K., and Wierschem, A. (2002) Interfacial wave motion due to Marangoni instability. IV. Waves with anomalous dispersion or normal dispersion and dispesionfree waves and transitions among them with decreasing driving forcein instationary experiments. J. Colloid Interface Sci Google Scholar
  54. Lucassen, J. (1968) Longitudinal capillary waves, Part 1. Theory. Trans. Faraday Soc. 64: 2221–2229, Part 2. Experiments. ibidem 64: 2230–2235.Google Scholar
  55. Lucassen-Reynders, E. H. and Lucassen, J. (1969) Properties of capillary waves. Adv. Colloid Interface Sci. 2: 347–395.CrossRefGoogle Scholar
  56. Maxworthy, T. (1976) Experiments on collisions between solitary waves. J. Fluid Mech. 76: 177–185.CrossRefGoogle Scholar
  57. Melville, W.K. (1980) On the Mach reflexion of a solitary wave. J. Fluid Mech. 98: 285–297CrossRefGoogle Scholar
  58. Mihaljan, J. (1962) A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid. Astrophys. J 136: 1126–1133.CrossRefMathSciNetGoogle Scholar
  59. Miles, J. W. (1976) Korteweg-de Vries equation modified by viscosity. Phys. Fluids 19: 1063CrossRefMATHGoogle Scholar
  60. Miles, J. W. (1977) Obliquely interacting solitary waves. J. Fluid Mech. 79. 157–169.CrossRefMATHMathSciNetGoogle Scholar
  61. Miles, J. W. (1980) Solitary waves. Ann. Rev. Fluid Mech. 12: 11–43.CrossRefGoogle Scholar
  62. Nekorkin, V. I., and Velarde, M. G. (1994) Solitary waves of a dissipative Korteweg-de Vries equation describing Marangoni-Benard convection and other thermoconvective instabilities. Int. J. Bifurc. Chaos 4: 1135–1146.CrossRefMATHMathSciNetGoogle Scholar
  63. Nepomnyashchy, A. A., and Velarde, M. G. (1994) A three-dimensional description of solitary waves and their interaction in Marangoni-Benard layers. Phys. Fluids 6: 187–198.CrossRefMATHMathSciNetGoogle Scholar
  64. Nepomnyashchy, A. A., Velarde, M. G. and Colinet, P. (2001) Inter/acial Phenomena and Convection. Chapman and Hall/CRC, London.Google Scholar
  65. Nield, D. A. (1964) Surface tension and buoyancy effects in cellular convetion. J. Fluid Mech. 19: 341–352.CrossRefMATHMathSciNetGoogle Scholar
  66. Normand, C., Pomeau, Y., and Velarde, M. G. (1977) Convective instability: A Physicist’s Approach. Rev. Mod. Phys. 49: 581–624.CrossRefMathSciNetGoogle Scholar
  67. Ostrach, S. (1982) Low-gravity fluid flows. Ann. Rev. Fluid Mech. 14: 313–345.CrossRefGoogle Scholar
  68. Pearson, J. R. A. (1958) On convection cells induced by surface tension. J. Fluid Mech. 4: 489–500.Google Scholar
  69. Perez-Cordon, R. and Velarde, M. G. (1975) On the (non linear) foundations of Boussinesq approximation applicable to a thin layer of fluid. J. Phys. (Paris) 36: 591–601.CrossRefGoogle Scholar
  70. Pontes, J., Christov, C. I., and Velarde, M. G. (1996) Numerical study of patterns and their evolution in finite geometries. Int. J. Bifurc. Chaos 6: 1883–1890.CrossRefMATHGoogle Scholar
  71. Pontes, J., Christov, C. I., and Velarde, M. G. (2000) Numerical approach to pattern selection in a model problem for Benard convection in a finite fluid layer Prandtl number. Annu. Univ. Sofia 93: 157–175. Rayleigh, Lord (1876) On Waves. Phil. Mag. I: 257–279.Google Scholar
  72. Rayleigh, Lord (1916) On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Phil. Mag. 32: 529–536.CrossRefGoogle Scholar
  73. Rednikov, A. Ye., Colinet, P., Velarde, M. G., and Legros, J. C. (1998) Two-layer Benard-Marangoni instability and the limit of transverse and longitudinal waves. Phys. Rev. E 57: 2872–2884.CrossRefMathSciNetGoogle Scholar
  74. Rednikov, A. Ye., Colinet, P., Velarde, M. G., and Legros, J. C. (2000) Rayleigh-Marangoni oscillatory instability in a horizontal liquid layer heated from above: coupling between internal and surface waves. J. Fluid Mech. 405: 57–77.CrossRefMATHMathSciNetGoogle Scholar
  75. Rednikov, A. Ye., Colinet, P., Velarde, M. G., and Legros, J. C. (2000) Oscillatory thermocapillary instability in a liquid layer with deformable open surface: capillary-gravity waves, longitudinal waves and mode-mixing. J. Non-Equilib. Therm. 25: 381–405.CrossRefGoogle Scholar
  76. Rednikov, A. Ye., Velarde, M. G., Ryazantsev, Yu. S., Nepomnyashchy A. A., and Kurdyumov, V. (1995), Cnoidal wave trains and solitary waves in a dissipation-modified Korteweg-de Vries equation. Acta Appl. Math. 39: 457–475.CrossRefMATHMathSciNetGoogle Scholar
  77. Rodriguez- Bernal, A. (1992) Initial value problem and asymptotic low dimensional behavior in the Kuramoto-Velarde equation. Nonl. Anal. Theory Meth. Appl. 19: 643–685.CrossRefMATHMathSciNetGoogle Scholar
  78. Russell, J. S. (1844), Report on waves. Rep. 14th Meet. British Ass. Adv. Sci., York, 311–390, J. Murray, London; reprinted as Appendix in Russell (1885).Google Scholar
  79. Russell, J. S. (1885) The Wave of Translation in the Oceans of Water Air and Ether. Trubner, London.Google Scholar
  80. Sanfeld, A., Steinchen, A., Hennenberg, M., Bisch, P. M., D. Van Lamsweerde-Gallez, and Dalle-Vedove, W. in Dynamics and Instability of Fluid Interfaces edited by Sorensen, T. S., Springer-Verlag, Berlin 1979, 168–204.Google Scholar
  81. Henenberg, M., Bisch, P. M., Vignes-Adler, M. and Sanfeld, A., ibidem 229–259.Google Scholar
  82. Santiago-Rosanne, M., Vignes-Adler, M., and Velarde, M. G. (1997) Dissolution of a drop on a liquid surface leading to surface waves and interfacial turbulence. J. Colloid Interface Sci. 191: 65–80.CrossRefGoogle Scholar
  83. Schatz, M. F., Van Hook, S., Mc Comick, W., Swift. J. B., and Swinney, H. D. (1995) Onset of surfacetension-driven Benard convection. Phys. Rev. Lett. 75: 1938–1940.Google Scholar
  84. Scriven, L. E.,and Sternling C. V. (1960) The Marangoni effects. Nature 187: 186–188.CrossRefGoogle Scholar
  85. Simanovsky, I. B., and Nepomnyaschchy, A. A. (1993) Convective Instabilities in Systems with Interfaces. Gordon and Breach, N.Y.Google Scholar
  86. Smith, K. A. (1966) On convective instability induced by surface-tension gradients. J. Fluid Mech. 24: 401–414.CrossRefGoogle Scholar
  87. Sternling, C. V., and Scriven, L. E. (1959) Interfacial turbulence: Hydrodynamic instability and the Marangoni effect. A. I. Ch. E. Journal 5: 514–523.Google Scholar
  88. Tanford, Ch. (1989)Ben Franklin Stilled the Waves. An Informal History of Pouring Oil on Water With reflections on the ups and downs of scientific life in general. Duke Univ. Press, London.Google Scholar
  89. Ursell, F. (1953) The long-wave paradox in the theory of gravity waves. Proc. Cambridge Phil. Soc. 49: 685–694.CrossRefMATHMathSciNetGoogle Scholar
  90. Van Hook, S., Schatz, M. S., Mc Cornick, W., Swift, J. B., and Swinney, H. D. (1995) Long-wavelength instability in surface-tension-driven Benard convection. Phys. Rev. Lett. 75: 4397–4400.CrossRefGoogle Scholar
  91. Van Hook, S., Schatz, M. F., Mc Cornick, W., Swift, J. B., and Swinney,H. D. (1997) Long-wavelength surface-tension-driven Benard convection: experiment and theory. J. Fluid Mech. 345: 45–78.CrossRefMathSciNetGoogle Scholar
  92. Velarde M. G., and Normand, C. (1980) Convection. Sci. American 243 (1): 92–108.Google Scholar
  93. Velarde, M. G., and Chu, X.-L. (1988) The harmonic oscillator approximation to sustained gravity-capillary (Laplace) waves at liquid interfaces. Phys. Lett. A 131: 430–432.CrossRefMathSciNetGoogle Scholar
  94. Velarde, M. G., and Chu, X.-L. (1989a) Waves and turbulence at interfaces. Phys. Scr. T25: 231–237.CrossRefGoogle Scholar
  95. Velarde, M. G., and Chu, X.-L. (1989b) Dissipative hydrodynamic oscillators. I. Marangoni effect and sustained longitudinal waves at the interface of two liquids. Il Nuovo Cimento DI I: 707–716.Google Scholar
  96. Velarde, M. G., and Chu, X.-L. (1992) Dissipative thermohydrodynamic oscillators. Adv. Thermodyn. 6: 110–145.Google Scholar
  97. Velarde, M. G., and Perez-Cordon, R. (1976) On the (non-linear) foundations of Boussinesq approximation applicable to a thin layer of fluid. II. Viscous dissipation and large cell gap effects. J. Phys. (Paris) 37: 177.CrossRefGoogle Scholar
  98. Velarde, M. G., Garcia-Ybarra, P. L., and Castillo, J. L. (1987) Interfacial oscillations in Benard-Marangoni layers. Physicochem. Hydrodyn. 9: 387–392.Google Scholar
  99. Velarde, M. G., Nekorkin, V. I., and Maksimov, A. G. (1995) Further results on the evolution of solitary waves and their bound states of a dissipative Korteweg-de Vries equation. Int. J. Bifurc. Chaos 5: 83 1839.Google Scholar
  100. Velarde, M. G., Linde, H., Nepomnyashchy, A. A. and Waldhelm, W. (1995) Further evidence of solitonic behavior in Benard-Marangoni convection: periodic wave trains, in Fluid Physics, edited by Velarde, M. G. and Christov, C. I., World Scientific, Singapore, 433–441.Google Scholar
  101. Velarde, M. G., and Rednikov, A. Ye. (1998) Time-dependent Benard-Marangoni instability and waves, in Time-Dependent Nonlinear Convection, edited by P.A. Tyvand, Computational Mechanics Publications, Southampton, 177–218.Google Scholar
  102. Velarde, M. G., Rednikov, A. Ye., and Linde, H. (1999) Waves generated by surface tension gradients and Instability, in Fluid Dynamics at Interfaces, edited by W. Shyy, University Press, Cambridge, 43–56.Google Scholar
  103. Velarde, M. G., Nepomnyashchy, A. A., and Hennenberg, M. (2000) Onset of oscillatory interfacial instability and wave motions in Benard layers. Adv. Appl. Mech. 37: 167–237.CrossRefGoogle Scholar
  104. Vidal, A. and Acrivos, A. (1966) Nature of the neutral state in surface-tension driven convection. Phys. Fluids 9: 615–616.CrossRefGoogle Scholar
  105. Weidman, P. and Maxworthy, T. (1978) Experiments on strong interactions between solitary waves. J. Fluid Mech. 85: 417–431.CrossRefGoogle Scholar
  106. Weidman, P., Linde, H., and Velarde, M. G. (1992) Evidence for solitary waves in Marangoni-driven unstable liquid layers. Phys. Fluids A 4: 921–926.CrossRefGoogle Scholar
  107. Wierschem, A., Velarde, m. G., Linde, H., and Waldhelm, W. (1999) Interfacial wave motions due to Marangoni instability. Il. Three dimensional characteristics of surface waves. J. Colloid Interface Sci. 212: 365–383.CrossRefGoogle Scholar
  108. Wierschem, A., Linde, H., and Velarde, M. G. (2000) Internal waves excited by the Marangoni effect. Phys. Rev. E 62: 6522–6530.CrossRefGoogle Scholar
  109. Wierschem, A., Linde, H., and Velarde, M. G. (2001) Properties of surface wave trains excited by mass transfer through aliquid surface. Phys. Rev. E 64: 22601–1–4.Google Scholar
  110. Zabusky, N., and Kruskal, M. D. (1965) Interaction of “solitons” in collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15: 57–62.CrossRefGoogle Scholar

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© Springer-Verlag Wien 2002

Authors and Affiliations

  • M. G. Velarde
    • 1
  1. 1.Instituto PluridisciplinarUCMMadridSpain

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