Pattern Selection in Surface Tension Driven Flows

  • H. A. Dijkstra
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 391)


When a motionless liquid layer is heated from below, spontaneous convection appears when the vertical temperature gradient exceeds a critical value. Under slightly supercritical conditions, the liquid organizes into steady regular polygonal patterns, for example rolls or hexagons. If the liquid has an upper free surface open to ambient air, both buoyancy gradients and surface tension gradients may be responsible for these flows. The latter effect is dominating in thin layers and in a micro-gravity environment and in that case usually hexagonal patterns are observed. In this chapter, an introduction is provided into the physics of these flows by giving an (incomplete) overview of theoretical, experimental and numerical results which have been obtained over the last decades. Focus is on the existence of the critical temperature gradient and the selection of steady patterns near critical conditions.


Surface Tension Bifurcation Diagram Pattern Selection Marangoni Number Amplitude Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Koschmieder, E.L. and D.W. Switzer: The wavenumbers of supercritical surfacetension-driven Bénard convection, J. Fluid Mech., 240 (1992), 533–548.ADSCrossRefGoogle Scholar
  2. [2]
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford 1961.MATHGoogle Scholar
  3. [3]
    Bénard, H.: Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur en régime permanent, Ann. Chem. Phys., 23 (1901), 62–144.Google Scholar
  4. [4]
    Koschmieder, E.L.: Bénard cells and Taylor Vortices, Cambridge University Press 1993.Google Scholar
  5. [5]
    Normand, C., Pomeau, Y., and M.G. Velarde: Convective instability: A physicist’s approach, Rev. Mod. Phys., 49 (1977), 581–624.ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    Rayleigh, Lord: On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., 32 (1916), 529–546.CrossRefGoogle Scholar
  7. [7]
    Jeffreys, H.: The surface elevation in cellular convection, Quart. J. Mech. Appl. Math., 4 (1951), 283–288.CrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    Block, M.J.: Surface tension as a cause of Bénard cells and surface deformation in a liquid film, Nature, 178 (1956), 650.ADSCrossRefGoogle Scholar
  9. [9]
    Pearson, J.R.A.: On convection cells induced by surface tension, J. Fluid Mech., 4 (1958), 489–500.ADSCrossRefMATHGoogle Scholar
  10. [10]
    Marangoni, C.: Ann. Phys. Lpz., 143 (1871), 337.ADSCrossRefGoogle Scholar
  11. [11]
    Thompson, J.J.: Phil. Mag., 10 (1855), 330.Google Scholar
  12. [12]
    Pantaloni, J., Bailleux, R., Salan, J. and M.G. Velarde: Rayleigh-BénardMarangoni instability: new experimental results, J. Non-Equlibr. Thermodyn., 4 (1979), 201–218.ADSGoogle Scholar
  13. [13]
    Cerisier, P., Perez-Garcia, C. and R. Occelli: Evolution of induced patterns in surface-tension-driven Bénard convection, Phys. Rev.E., 47 (1993), 3316–3325.Google Scholar
  14. [14]
    Koschmieder, E.L. and M.I. Biggerstaff: Onset of surface-tension-driven Bénard convection, J. Fluid Mech., 167 (1986), 49–64.ADSCrossRefGoogle Scholar
  15. [15]
    Levich, V.G. and V.S. Krylov: 1969, Surface tension driven phenomena, Ann. Rev. Fluid Mech., 1 (1969), 293–316.ADSCrossRefGoogle Scholar
  16. [16]
    Patberg, W.B., Koers, A., Steenge, W.D.E. and A.A.H. Drinkenburg: Effectiveness of mass transfer in a packed distillation column in relation to surface tension gradients, Chem. Engng. Sc., 38 (1983), 917–923.CrossRefGoogle Scholar
  17. [17]
    Davis, S.H.: 1987, Thermocapillary instabilities, Ann. Rev. Fluid Mech., 19 (1987), 403–435.ADSCrossRefMATHGoogle Scholar
  18. [18]
    Aris, R: Vectors, tensors, and the basic equations of fluid mechanics. Dover publications 1962.Google Scholar
  19. [19]
    Drazin, P.G. and W.H. Reid: Hydrodynamic Stability, Cambridge University Press 1981.Google Scholar
  20. [20]
    Joseph, D.D.: Stability of fluid motions, volumes I and II, Springer-Verlag 1976.Google Scholar
  21. [21]
    Vidal, A. and A. Acrivos, A.: Nature of the neutral state in surface tension driven convection, Phys. Fluids, 9 (1966), 615–616.ADSCrossRefGoogle Scholar
  22. [22]
    Davis, S.H.: On the principle of exchange of stabilities, Proc. Roy. Soc. A, 310 (1969), 341–358.ADSCrossRefMATHGoogle Scholar
  23. [23]
    Nield, D.A.: Surface tension and buoyancy effects in cellular convection, J. Fluid Mech., 19 (1964), 341–352.ADSCrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    Scriven, L.E. and C.V. Sternling: On cellular convection driven by surface tension gradients: effect of mean surface tension and surface viscosity, J. Fluid Mech., 19 (1964), 321–340.ADSCrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    Smith, K.A.: On convective instability induced by surface tension gradients, J. Fluid Mech., 24 (1966), 410–414.ADSGoogle Scholar
  26. [26]
    Takashima, M.: Surface tension driven instability in a horizontal liquid layer with a deformable surface. I. Stationary convection, J. Phys. Soc. Japan, 50 (1981), 2745–2750.ADSCrossRefGoogle Scholar
  27. [27]
    Davis, S.H.: Buoyancy-surface tension instability by the method of energy, J. Fluid Mech., 39 (1969), 347–359.ADSCrossRefMATHGoogle Scholar
  28. [28]
    Davis, S.H. and G.M. Homsy: 1980, Energy stability theory for free surface problems: buoyancy-thermocapillary layers, J. Fluid Mech., 98 (1980), 527–553.ADSCrossRefMATHMathSciNetGoogle Scholar
  29. [29]
    Castillo, J.L. and M.G. Velarde: Buoyancy-thermocapillary instability: the role of interfacial deformation in one-and two-component fluid layers heated from below or above, J. Fluid Mech., 125 (1982), 463–474.ADSCrossRefMATHGoogle Scholar
  30. [30]
    Cloot, A. and G. Lebon: A nonlinear stability analysis of the Bénard-Marangoni problem, J. Fluid Mech., 145 (1984), 447–469.ADSCrossRefMATHGoogle Scholar
  31. [31]
    Bestehorn, M.: Phase and amplitude instabilities for Bénard-Marangoni convection in fluid layers with large aspect ratio, Phys. Rev. E., 48 (1993), 3622–3634.MathSciNetGoogle Scholar
  32. [32]
    Schlüter, A., Lortz, D. and F.H. Busse: On the stability of steady finite amplitude convection, J. Fluid Mech., 23 (1965), 129–144.ADSCrossRefMATHMathSciNetGoogle Scholar
  33. [33]
    Scanlon, J.W. and L.A. Segel: Finite amplitude cellular convection induced by surface tension, J. Fluid Mech., 30 (1967), 149–162.ADSCrossRefMATHGoogle Scholar
  34. [34]
    Thess, A. and S.A. Orszag: Surface tension driven Bénard convection at infinite Prandtl number, J. Fluid Mech., 283 (1995), 201–230.ADSCrossRefMATHMathSciNetGoogle Scholar
  35. [35]
    Koschmieder, E.L. and S.A. Prahl: Surface tension driven convection Bénard convection in small containers, J. Fluid Mech., 215 (1990), 571–583.ADSCrossRefGoogle Scholar
  36. [36]
    Rosenblat, S., Homsy, G.M. and S.H. Davis: Nonlinear Marangoni convection in bounded layers, Part 2. Rectangular cylindrical containers, J. Fluid Mech., 120 (1982), 123–138.ADSCrossRefMATHMathSciNetGoogle Scholar
  37. [37]
    Rosenblat, S., Homsy, G.M. and S.H. Davis: Eigenvalues of the Rayleigh-Bénard and Marangoni problems, Phys. Fluids, 24 (1991), 2115–2117.ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    Kuznetsov, Y.A.: Elements of applied bifurcation theory, Springer Verlag 1995.Google Scholar
  39. [39]
    Nayfeh, A.H. and B. Balachandran: Applied nonlinear dynamics, John Wiley 1995.Google Scholar
  40. [40]
    Dijkstra, H.A.: On the structure of cellular patterns in Rayleigh-BénardMarangoni flows in small-aspect-ratio containers, J. Fluid Mech., 243 (1992), 73–102.ADSCrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    Dauby, P.C., Lebon, G., Colinet, P. and J.C. Legros: Hexagonal Marangoni convection in a rectangular box with slippery walls, Q. J. Mech. Appl. Math., 46 (1993), 683–707.CrossRefMATHGoogle Scholar
  42. [42]
    Van de Vooren, A.I. and H.A. Dijkstra, A finite-element stability analysis of the Marangoni problem in a two-dimensional container with rigid sidewalls, Computers and Fluids, 17 (1989), 467–485.ADSCrossRefMATHGoogle Scholar
  43. [43]
    Dijkstra, H.A.: Surface tension driven cellular patterns in three-dimensional boxes Linear Stability, Microgravity Science and Technology, VII /4 (1995), 307–312.Google Scholar
  44. [44]
    Dijkstra, H.A.: Surface tension driven cellular patterns in three-dimensional boxes A bifurcation study, Microgravity Science and Technology, VIII /2 (1995), 7077.MathSciNetGoogle Scholar
  45. [45]
    Dijkstra, H.A.: Surface tension driven cellular patterns in three-dimensional boxes The formation of hexagonal patterns, Microgravity Science and Technology, VIII/3, 155–162.Google Scholar
  46. [46]
    Dauby, P.C. and G. Lebon: Bénard-Marangoni instability in rigid rectangular containers, J. Fluid Mech., 329 (1996), 25–64.ADSCrossRefMATHGoogle Scholar
  47. [47]
    Dijkstra, H.A.: Surface tension driven cellular patterns in three-dimensional boxes On the preference of hexagonal patterns, in preparation, (1998).Google Scholar

Copyright information

© Springer-Verlag Wien 1998

Authors and Affiliations

  • H. A. Dijkstra
    • 1
  1. 1.Utrecht UniversityUtrechtThe Netherlands

Personalised recommendations