The Bénard (1900, 1901) Convection Problem, Heated from below

Chapter
Part of the Fluid Mechanics and its Applications book series (FMIA, volume 90)

In this chapter, we take into account the influence of a deformable free surface and, as a consequence, we revisit the mathematical formulation of the classical problem describing the Bénard instability of a horizontal layer of fluid, heated from below, and bounded by an upper deformable free surface. Because the deformation of the free surface, subject to a temperature-dependent surface tension, is taken into account in the full Bénard convection problem, heated from below, we have not specified this convection as being a ‘thermal convection’.

Indeed, in the full starting Bénard problem, heated from below, when we take into account the influence of a deformable free surface, subject to a temperature-dependent surface tension, the fluid being an expansible liquid, it is necessary to take into account, simultaneously, four main effects. Namely:
  1. (a)

    the conduction adverse temperature gradient (Bénard) effect in motionless steady-state conduction temperature

     
  2. (b)

    the temperature-dependent surface tension (Marangoni) effect

     
  3. (c)

    the heat flux across the upper, free surface (Biot) effect

     
  4. (d)

    the buoyancy (Archimedean—Boussinesq) effect arising from the volume (gravity) forces.

     

Keywords

Free Surface Liquid Layer Froude Number Biot Number Dimensionless Temperature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    H. Bénard, Les tourbillons cellulaires dans une nappe liquide. Revue Générale des Sciences Pures et Appliquées 11, 1261–1271 and 1309–1328, 1900. See also: Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent. Annales de Chimie et de Physique 23, 62–144, 1901.Google Scholar
  2. 2.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961. See also: Dover Publications, New York, 1981.MATHGoogle Scholar
  3. 3.
    E.L. Koschmieder, Bénard Cells and Taylor Vortices. Cambridge University Press, Cambridge, 1993.MATHGoogle Scholar
  4. 4.
    C. Marangoni, Sull'espansione delle gocce di un liquido gallegianti sulla superficie di altro liquido. Pavia: Tipografia dei fratelli Fusi, 1965 and Ann. Phys. Chem. (Poggendotff) 143, 337–354, 1871.Google Scholar
  5. 5.
    J. Plateau, Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moléculaires, Vol. 1. Gauthier-Villars, Paris, 1873.Google Scholar
  6. 6.
    R.Kh. Zeytounian, The Bénard–Marangoni thermocapillary-instability problem. Physics-Uspekhi 41(3), 241–267, March 1998 [English edition].CrossRefMathSciNetGoogle Scholar
  7. 7.
    D.A. Nield, Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19, 341–352, 1964.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M.J. Block, Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature 178, 650–651, 1956.CrossRefGoogle Scholar
  9. 9.
    R.Kh. Zeytounian, The Bénard—Marangoni thermocapillary instability problem: On the role of the buoyancy. Int. J. Engng. Sci. 35(5), 455–466, 1997.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M.G. Velarde and R.Kh. Zeytounian (Eds.), Interfacial Phenomena and the Marangoni Effect. CISM Courses and Lectures, No. 428, Udine. Springer-Verlag, Wien/New York, 2002.MATHGoogle Scholar
  11. 11.
    S.H. Davis, Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19, 403–435, 1987.MATHCrossRefGoogle Scholar
  12. 12.
    J.R.A. Pearson, On convection cells induced by surface tension. J. Fluid Mech. 4, 489– 500, 1958.MATHCrossRefGoogle Scholar
  13. 13.
    S.J. Vanhook et al., Long-wavelength instability in surface-tension-driven Bénard convection. Phys. Rev. Lett. 75, 4397, 1995.CrossRefGoogle Scholar
  14. 14.
    D.D. Joseph, Stability of Fluid Motions, Vol. II. Springer, Heidelberg, 1976.Google Scholar
  15. 15.
    P.M. Parmentier, V.C. Regnier and G. Lebond, Nonlinear analysis of coupled gravitational and capillary thermoconvection in thin fluid layers. Phys. Rev. E 54(1), 411–423, 1996.CrossRefGoogle Scholar
  16. 16.
    M. Takashima, J. Phys. Soc. Japan 50(8), 2745–2750 and 2751–2756, 1980.Google Scholar
  17. 17.
    R.Kh. Zeytounian, Int. J. Engng. Sci. 27(11), 1361, 1989.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    M.P. Ida and M.J. Miksis, The dynamics of thin films. I. General theory; II. Applications. SIAM J. Appl. Math. 58(2), 456–473 (I) and 474–500 (II), 1998.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    D.J. Needham and J.H. Merkin, J. Fluid Mech. 184, 357–379, 1987.MATHCrossRefGoogle Scholar
  20. 20.
    Ch. Bailly, Modélisation asymptotique et numérique de l'écoulement dû à des disques en rotation. Thesis, Université de Lille I, Villeneuve d'Ascq, No. 1512, 160 pp., 1995.Google Scholar
  21. 21.
    D.D. Joseph and Yu.R. Renardy, Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications. Springer-Verlag, 1993.Google Scholar
  22. 22.
    C. Ruyer-Quil, B. Scheid, S. Kalliadasis, M.G. Velarde and R.Kh. Zeytounian, Thermocapillary long-waves in a liquid film flow. Part 1: Low-dimensional formulation. J. Fluid Mech. 538, 199–222, 2005.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Oron, S.H. Davis and S.G. Bankoff, Long-scale evolution of thin liquid film. Rev. Modern Phys. 69(3), 931–980, July 1997.CrossRefGoogle Scholar
  24. 24.
    J.K. Platten and J.C. Legros, Convection in Liquids, 1st edn. Springer-Verlag, Berlin, 1984.MATHGoogle Scholar
  25. 25.
    B. Scheid, C. Ruyer-Quil, S. Kalliadasis, M.G. Velarde and R.Kh. Zeytounian, Thermocapillary long-waves in a liquid film flow. Part 2: Linear stability and nonlinear waves. J. Fluid Mech. 538, 223–244, 2005.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    D. Dowson, History of Tribology. Longmans, Green, London/New York, 1979.Google Scholar
  27. 27.
    H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York, 1968.Google Scholar
  28. 28.
    C.-S. Yih, Stability of parallel laminar flow down an inclined plane. Phys. Fluids 6, 321– 333, 1963.MATHCrossRefGoogle Scholar
  29. 29.
    T.B. Benjamin, Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574, 1957.CrossRefMathSciNetGoogle Scholar
  30. 30.
    S. Ndoumbe, F. Lusseyran, B. Izrar, Contribution to the modeling of a liquid film flowing down inside a vertical circular tube. C.R. Mecanique 331, 173–178, 2003.MATHCrossRefGoogle Scholar
  31. 31.
    C.M. Gramlich, S. Kalliadasis, G.M. Homsy and C. Messer, Optimal leveling of flow over one-dimensional topography by Marangoni stresses. Phys. Fluids 14(6), 1841– 1850, June 2002.CrossRefGoogle Scholar
  32. 32.
    L.E. Stillwagon and R.G. Larson, Leveling of thin film over uneven substrates during spin coating. Phys. Fluids A2, 1937, 1990.Google Scholar
  33. 33.
    S. Kalliadasis, C. Bielarz, and G.M. Homsy, Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889, 2000.CrossRefMathSciNetGoogle Scholar
  34. 34.
    L. Limat, Instability of a liquid hanging below a solid ceiling: influence of layer thickness. C.R. Acad. Sci. Paris, Ser. II 317, 563–568, 1993.MATHGoogle Scholar
  35. 35.
    D.H. Sharp, Physica D 12, 3–18, 1984.Google Scholar

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