Theoretical Aspects of Interfacial Phenomena and Marangoni Effect

Modelling and Stability
  • R. Kh. Zeytounian
Part of the International Centre for Mechanical Sciences book series (CISM, volume 428)


In a liquid layer, wavy motion of the free surface open to ambiant passive air (at constant temperature Ta and constant pressure pa) represents one of the most important cases where capillary forces are displayed. The motion induced by tangential gradients of variable surface tension is customarily called the Marangoni effect (after one of the first scientists to give an explanation of this effect), and in the present Notes we consider mainly thermocapillarity effects and pose an equation of state: σ = σ(T), for the surface tension, as (only) function of the temperature T. Indeed, these Notes are devoted to a Theoretical Fluid Dynamics aspects and modelling of wave dynamics and thermocapillary instabilities processes in falling thin liquid films. The starting equations and the boundary conditions on the interface between two immiscible fluids, are the full Navier-StokesFourier (NS-F) equations for a viscous, thermally conducting, compressible (expansible) Newtonian fluid (liquid) and associated jump conditions for the stress tensor, the heat flux and the temperature across the interface (free surface). The simplified case we examine involve a simple geometry — the one-layer system — in which there is a liquid (weakly expansible) layer, whose lower boundary is a heated rigid plate and whose upper boundary is a deformable free surface with a passive gas (having negligible viscosity and density) — the so-called “Bénard thermal convection problem ”.


Free Surface Liquid Layer Marangoni Number Interfacial Phenomenon Marangoni Convection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alekseenko S.V. (1985). AIChE J., 31, 1446.CrossRefGoogle Scholar
  2. Alekseenko S.V. Nakoryakov V.E. and Pokusaev B.T. (1992). Wave Flow of Liquid Films. Nauka, Novosibirsk, original Russian ed.Google Scholar
  3. Bénard H. (1900). Rev. Gen. Sci. Pures Appl., 11, 1261.Google Scholar
  4. Bailly Ch. (1995). Modelisation asymptotique et mumérique de l’écoulement dû à des disques en rotation. Thése presentée à l’Université des Sciences et Technologies de Lille, Villeneuve d’Ascq Cedex. Soutenue le 18 avril 1995, N° d’ordre 1512, 160 pages.Google Scholar
  5. Benjamin T.B. (1957). Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefMathSciNetGoogle Scholar
  6. Benney D.J. (1966). Journal. Math. and Phys., 45, 150.MATHMathSciNetGoogle Scholar
  7. Bergeron A. et al. (1998). Marangoni convection in binary mixtures with Soret effect. J. Fluid Mech., 375, 143–177.CrossRefGoogle Scholar
  8. Betelù S.I. and Diez J.A. (1999). A two-dimensional similarity solution for capillary driven flows. Physica D, 126, 136–140.CrossRefMATHMathSciNetGoogle Scholar
  9. Binnie A.M. (1957). Experiments on the onset of wave formation on a film of water flowing down an inclined plane. J. Fluid Mech., 2, 551–553.CrossRefGoogle Scholar
  10. Block M.J. (1956). Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature, 178, 650–651.CrossRefGoogle Scholar
  11. Bobkov N.N. and Gupalo Yu.P. (1996). The flow pattern in a liquid layer and the spectrum of the boundary-value problem when the surface tension depends non-linearly of the temperature. Prikl. Mat. Mekh., 60, 6, 1021–1028, Russian original ed.Google Scholar
  12. Boos W. and Thess A. (1999). Cascade of structures in long-wavelengt Marangoni instability. Phys. Fluids, 11 (6) 1484–1494.CrossRefMATHMathSciNetGoogle Scholar
  13. Bragard J. and Velarde M.G. (1997). Bénard Convection Flows. J. Non-Equilib. Therrrodyn., 22, 1–19.CrossRefMATHGoogle Scholar
  14. Chang H.-Ch., Demekhin E A and Kopelevich D.I. (1993). Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech., 250, 433–480.CrossRefMathSciNetGoogle Scholar
  15. Chang H.-Ch. (1994). Wave evolution on a falling film. Ann. Rev. Fluid Mech., 26, 103–136.CrossRefGoogle Scholar
  16. Chang H.-Ch. and Demekhin E.A. (1996). Solitary Wave Formation and Dynamics on Falling Films Advances in Appl. Mech. 32, 1–58.Google Scholar
  17. Charwat A.F., Kelly R.E. and Gazley C. (1972). The flow and stability of thin liquid films on a rotating disk. J. Fluid Mech., 53, part2, 227–255.Google Scholar
  18. Chhabra R.P. and de Kee D., Eds. (1992). Transport processes in bubles, drops and particles. New York, Hemisphere.Google Scholar
  19. Cowley S.J. and Davis S.H. (1983). Viscous thermocapillary convection at high Marangoni number. J. Fluid Mech., 135, 175–188.CrossRefMATHGoogle Scholar
  20. Cross M.C.,(1980). Phys. Fluids 23, 1727.Google Scholar
  21. Cross M.C. (1982). Phys. Rev. A 25, 1065.CrossRefGoogle Scholar
  22. Cross M.C. and Hohenberg P.C. (1993). Pattern formation outside of equilibrium. Rev. Mod. Phys., 65, 851.CrossRefGoogle Scholar
  23. Dauby P.C. and Lebon G. (1996). Bénard-Marangoni instability in rigid rectangular containers. Journal of Fluid Mechanics, 329, 25–64.CrossRefMATHGoogle Scholar
  24. Davis S.H. (1987). Thermocapillary instabilities. Ann. Rev. Fluid Mech., 19, 403–435.CrossRefMATHGoogle Scholar
  25. Durban D. and Pearson J.R.A., Eds. (1999). Non-linear Singularities in deformation and Flow. Kluwer Academic Publishers, Dordrecht- The Netherlands.Google Scholar
  26. Drazin P.G. and Reid W.H. (1981). Hydrodynamic Stability. Cambridge University Press, Cambridge.MATHGoogle Scholar
  27. Eggers J. (1997). Nonlinear dynamics and breakup of fre-surface flows. Reviews of Modern Physics, 69 (3), 835–929.CrossRefGoogle Scholar
  28. Finn R. (1986). Equilibrium Capillary Surfaces. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  29. Frenkel A.L. (1993). On evolution equations for thin films flowing down solid surfaces. Phys. Fluids, A 5, 2342–2347.CrossRefMATHMathSciNetGoogle Scholar
  30. Garazo A.N. and Velarde M.G. (1991). Dissipative Korteweg-de Vries description of solitary waves and their interaction in Marangoni-Bénard layers. Phys. Fluids, A 3, 2295–2300.CrossRefMATHGoogle Scholar
  31. Golovin A.A., Nepomnyaschy A.A. and Pismen L.M. (1994). Interaction between short-scale Marangoni convection and long-scale deformational instability. Phys. Fluids, 6 (1), 35–48.CrossRefGoogle Scholar
  32. Grigorian S.S. and Khairetdinov E.F. (1996). Solution of the equations of flow of a thin layer of heavy viscous liquid over a curvilinear surface. Vestnik MGU, Mat. Mekh., 6, 32–36.Google Scholar
  33. Grigorian S.S. and Khairetdinov E.F. (1998). Unsteady spreading of a thin layer of viscous liquid over a curved surface. Prikl. Mat. Mekh., 62, 1, 151–161, Russian original ed.Google Scholar
  34. Higgins B.G. (1986). Film flow on a rotating disk. Phys. Fluids, 29 (11), 3522–3529.CrossRefMATHGoogle Scholar
  35. Ida M.P. and Miksis M.J. (1995). Dynamics of a lamella in a capillary tube. SIAM J. Appl. Math., 55 (1), 23–57.CrossRefMATHMathSciNetGoogle Scholar
  36. Ida M.P. and Miksis M.J. (1996). Thin film rupture. Appl. Math. Lett., 9 (3), 35–40.CrossRefMATHMathSciNetGoogle Scholar
  37. Ida M.P. and Miksis M.J. (1998a). The Dynamics of thin films I: General theory. SIAM J. Appl. Math., 58 (2), 456–473.CrossRefMATHMathSciNetGoogle Scholar
  38. Ida M.P. and Miksis M.J. (1998b). The Dynamics of thin films II: Applications. SIAM J. Appl. Math., 58 (2), 474–500.CrossRefMATHMathSciNetGoogle Scholar
  39. Isenberg C. (1992). The Science of Soap Films and Soap Bubles. Tieto, Clevendon, Avon, England (1978) - original edition; see also: Dower, New York (1992).Google Scholar
  40. Ivanov I.B., Ed., (1988). Thin Liquid Films. Marcel Dekker, Inc.. New York.Google Scholar
  41. Joo S.W. (1995)., Maraéngoni instabilities in liquid mixtures with Soret effects. J. Fluid Mech., 293, 127–145.Google Scholar
  42. Joseph D.D. (1976). Stability of Fluid Motions II. Springer-Verlag, Berlin.CrossRefMATHGoogle Scholar
  43. Joseph D.D. and Renardy Yu.R. (1993). Fundamentals of Two-Fluid Dynamics. Part (Mathematical Theory and Applications. Springer-Verlag, New York, Inc.Google Scholar
  44. Kapitza P.L. and Kapitza S.P. (1949). Wave flow of thin layers of a viscous fluid: II. Experimental study of undulatory flow conditions. J. Exp. Theor. Phys. 19, 105–120 (in Russian). See also:Collected Papers of P.L. Kapitza, ed.Google Scholar
  45. Ter Haar D. Pergamon, London, Vol. 2, 690–709, (1965).Google Scholar
  46. Kashdan D. et al. (1995). Nonlinear waves and turbulence in Marangoni convection. Phys. Fluids, 7 (11), 2679–2685.CrossRefMathSciNetGoogle Scholar
  47. Kopbosynov B.K. and Pukhnachev V.V. (1986). Thermocapillary flow in thin liquid films. Fluid Mech. Soy. Res. 15, 95.MATHGoogle Scholar
  48. Koschmieder E.L. (1993). Bénard Cells and Taylor Vortices. Cambridge Univ. Press, Cambridge, UK.Google Scholar
  49. Landau L.D. (1944). C.R. Akad. Sci. USSR, 44, 311.MATHGoogle Scholar
  50. Landau L.D. (1965). Collected Papers,387–391, Oxford.Google Scholar
  51. Limat L. (1993). Instabilité d’un liquide suspendu sous un surplomb solide: influence de l’ épaisseur de la couche. C.R. Acad. Sci. Paris, t. 217, Série II, 563–568.Google Scholar
  52. Lin S.P. (1969). Finite amplitude stability of a parallel flow with a free surface. J. Fluid Mech., 36, 113–126.CrossRefMATHGoogle Scholar
  53. Lin, S.P. (1970). J. Fluid Mech. 40, 307.CrossRefGoogle Scholar
  54. Lin S.P. (1974). Finite amplitude side-band stability of a viscous film. J. Fluid Mech., 63 (3), 417–429.CrossRefMATHGoogle Scholar
  55. Lin S P. and Chen J.N. (1998). The mechanism of surface wave suppression in film flow down a vertical plane. Phys. Fluids, 10 (8), 1787–1792.CrossRefMATHMathSciNetGoogle Scholar
  56. Liu J., Schneider J.B. and Gollub J.P. (1995). Three-dimensional instabilities of film flows. Phys. Fluids, 7 (1), 55–67.CrossRefMathSciNetGoogle Scholar
  57. Meyer, R.E., Ed. (1983). Waves on fluid interfaces. Academic Press, New York.MATHGoogle Scholar
  58. MyersT.G. (1998). Thin films with high surface tension. SIAM Rev., 40 (3), 441–462.CrossRefMathSciNetGoogle Scholar
  59. NeedhamD.J. and Merkin J.H. (1987). The development of nonlinear waves on the surface of a horizontal rotating thin liquid film. J. Fluid Mech., 184, 357–379.CrossRefGoogle Scholar
  60. Nepomnyaschy A.A. and Velarde M.G. (1994). A Three-dimensional description of solitary waves and their interaction in Managoni-Bénard layers. Phys. Fluids, 6 (1), 187–198.CrossRefMathSciNetGoogle Scholar
  61. Newell A.C. (1974). Envelope equations, in Lectures in Applied Math., 15, ed. A.C. Newell, 157.Google Scholar
  62. Normand Ch., Pomeau Y. and Velarde M.G. (1977). Convective instability: A Physicist’s approach. Rev. Mod. Phys., 49 (3), 581–624.CrossRefMathSciNetGoogle Scholar
  63. Or A.C., Kelly R.E., Cortelezzi L. and Speyer J.L. (1999). Control of long-wavelength Marangoni- Bénard convection. J. Fluid Mech., 387, 321–341.CrossRefMATHGoogle Scholar
  64. Oron A. and Rosenau Ph. (1992). Journal Physique II France, 2, 131.Google Scholar
  65. Oron A. and Rosenau Ph. (1994). On a nonlinear thermocapillary effect in thin liquid layers. J. Fluid Mech., 273, 361–374.CrossRefMATHMathSciNetGoogle Scholar
  66. Oron A., Davis S.H. and Bankoff, S.G. (1997). Long-scale evolution of thin liquid films. Reviews of Modern Physics, 69 (3), 931–960.CrossRefGoogle Scholar
  67. Parmentier P.M., Regnier V.C. and Lebon G. (1996). Nonlinear analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Physical Review E, 54 (1), 411–423.CrossRefGoogle Scholar
  68. Pavithran S.and Redeekopp L.G. (1994). Studies in Appl. Maths., 93, 209.Google Scholar
  69. Pearson J.R.A. (1958). On convection cells induced by surface tension. J. Fluid Mech., 4, 489–500.CrossRefMATHGoogle Scholar
  70. Pérez- Garcia C. and Carneiro G. (1991). Linear stability analysis of BénardMarangoni convection in fluids with a deformable free surface. Phys. Fluids, A3, 292.CrossRefGoogle Scholar
  71. Platten J.K. and Legros J.C. (1984). Convection in Liquids. 1st ed., Springer-Verlag, Berlin.Google Scholar
  72. Probstein R.F. (1994)., Physicochemical Hydrodynamics: An Introduction. Wiley - Interscience Publ., second ed., New York.CrossRefGoogle Scholar
  73. Rath H.J., Ed. (1992). Microgravity Fluid Mechanics. Springer -Verlag.Google Scholar
  74. Regnier V.C. and Lebon G. (1995). Time-growth and correlation length of fluctuations in thermocapillary convection with surface deformation. Q. Journal. Mech. Appl. Math. 48, Pt. 1, 57–75.CrossRefGoogle Scholar
  75. Shkadov V.Ya. (1967). Izvestiya Akad. SSSR, Mech. Zhidkosti i Gaza, 1, 43–50.Google Scholar
  76. Shkadov V.Ya. (1973). Some Methods and Problems of the Theory of Hydrodynamic Stability. Moscow, Izd. MGU (in Russian).Google Scholar
  77. Scriven L.E. and Sterling C.V. (1964). J. Fluid Mech., 19, 321.CrossRefMATHMathSciNetGoogle Scholar
  78. Shugai G.A. and Yakubenko P.A. (1998). Spatio-temporal instability in free ultra-thin films. Eur. J. Mech. B/Fluids, 17, n° 3, 371–384.CrossRefMathSciNetGoogle Scholar
  79. Simanovskii J.B. and Nepomnyaschy A.A. (1993). Convective Instabilities in Systems with Interface. Gordon and Breach.Google Scholar
  80. Sisoev G.M. and Shkadov V.Ya. (1987). Flow stability of a film of viscous liquid on a rotating disk../. Eng. Physics (English ed., translated from Russian), 52, 671–674.Google Scholar
  81. Sisoev G.M. and Shkadov V.Ya. (1988). J. Eng. Physics,55, N°3, 419–423.Google Scholar
  82. Sisoev G.M. and Shkadov V.Ya. (1990). J. Eng. Physics,58, N°4, 573–577.Google Scholar
  83. Sisoev G.M. and Shkadov V.Ya. (1997). Dominant waves in a viscous liquid flowing in a thin sheet. Physics-Doklady, 42 (12), 683–686.MATHGoogle Scholar
  84. Sisoev G.M. and Shkadov V.Ya. (1997). Development of dominating waves from small disturbances in falling viscous-liquid films. Fluid Dynamics, 32, (6), 784–792.MATHMathSciNetGoogle Scholar
  85. Sorensen T.S. (1978). Dynamics and Instability of Fluid Interfaces. SpringerVerlag, Berlin.Google Scholar
  86. Stewartson K. and Stuart J.T. (1971). J. Fluid Mech., 48, 529.CrossRefMATHMathSciNetGoogle Scholar
  87. Straughan B. (1992). The Energy Method, Stability, and Nonlinear Convection. Springer - Verlag, New York, Inc.Google Scholar
  88. Stuart J.T. (1960). J. Fluid Mech., 9, 353.CrossRefMATHMathSciNetGoogle Scholar
  89. Takashima M. (1981a). J. of the Physical Soc.of Japan,Vol. 50, n° 8, 2745–2750.Google Scholar
  90. Takashima M. (198lb). idem,2751–2756.Google Scholar
  91. Thess A. and Orszag S.A. (1995). Surface-tension-driven Bénard concetion at infinite Prandtl number. J. Fluid Mech., 283, 201–230.CrossRefMATHMathSciNetGoogle Scholar
  92. Trifonov Yu.Ya. and Tsvelodub O.Yu. (1991). J. Fluid Mech., 229, 531.CrossRefMATHMathSciNetGoogle Scholar
  93. VanHook S.J. et al. (1995). Long-wavelength instability in surface-tension -driven Bénard convection. Phys. Rev. Lett., 75, 4397.CrossRefGoogle Scholar
  94. Van Vaerenberg S., Colinet P. and Legros J.C. (1990). The Role of the Soret Effect on Marangoni-Bénard Stability. Springer.Google Scholar
  95. Velarde M.G., Ed. (1987). Physicochemical Hydrodynamics; Interfacial Phenomena. Plenum Press, N. Y.Google Scholar
  96. Velarde M.G., Ed. (1988). Physicochemical Hydrodynamics. NATO ASI Series, Vol. 174, PLenum Press.Google Scholar
  97. Velarde M.G. and Rednikov A. Ye. (1998). Time-dependent Bénard-Marangoni instability. In: Time-Dependent Nonlinear Convection, Chapter 6, 177–218, Tyvand P.A. Editor. Adv. in Fluid Mech., 19, Computational Mech. Publ., Southampton, UK.Google Scholar
  98. Velarde M.G. (1998). Drops, liquid layers and the Marangoni effect. Phil. Trans. R. Soc., London A 356, 829–844.CrossRefMATHGoogle Scholar
  99. Vince Jean-Marc. (1994). Ondes propagatives dans des systèmes convectifs soumis à des effets de tension superficielle. Doctoral thesis, University Paris 7, N° 345, 185 pages.Google Scholar
  100. Wilson S.K. (1994). The onset of steady Marangoni convection in a spherical geometry. J. of Engineering Math., 28, 427–445.CrossRefMATHGoogle Scholar
  101. Wilson S.K. and Thess A. (1997). On the linear growth rates of the long-wave modes in Bénard-Marangoni convection. Phys. Fluids 9 (8), 2455–2457.CrossRefGoogle Scholar
  102. Yih C.-S. (1963). Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–330.CrossRefMATHGoogle Scholar
  103. Zeytounian R.Kh. (1989). Intern. Journal Engng. Sciences, 27 (11), 1361.CrossRefMATHMathSciNetGoogle Scholar
  104. Zeytounian R.Kh. (1994). Modelisation asymptotique en mécanique des fluides newtoniens. SMAI - Mathématiques et Applications, Vol. 15. Springer-Verlag, Berlin.Google Scholar
  105. Zeytounian R.Kh. (1995). Long-Waves on Thin Viscous Liquid Film: Derivation of Model Equations.In: Asymptotic Modelling in Fluid Mechanics, Lecture Notes in Physics (LNP 442)„ Springer.Google Scholar
  106. Zeytounian R.Kh. (1997). The Bénard–Marangoni thermocapillary instability problem: on the rôle of the buoyancy. Int. J. of Enginering Sciences. 35 (5), 455–466.CrossRefMATHMathSciNetGoogle Scholar
  107. Zeytounian R.Kh. (1998). The Bénard-Marangoni thermocapillary instability problem. Uspekhi Fizicheskikh Nauk, 168 (3), 259–286, Russian original ed.Google Scholar
  108. Zierep J. and Oertel H., Eds. (1982). Convective transport and instability phenomena. Braun — Verlag, Karlsruhe.Google Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • R. Kh. Zeytounian
    • 1
    • 2
  1. 1.ParisFrance
  2. 2.France

Personalised recommendations