Non-Linear Thermal Convection

  • Enok Palm
Part of the NATO Advanced Study Institutes Series book series (NSSB, volume 77)


If a horizontal fluid layer is heated from below or cooled from above, the heat will be transported through the fluid by conduction alone, if the heating is very weak. If, however, the amount of heating is increased, the conduction state becomes unstable and a convective motion is set up.


Prandtl Number Rayleigh Number Thermal Convection Fluid Layer Stability Diagram 
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  1. Ahlers, G. and Behringer, R.P., 1978, Evolution of turbulence from the Rayleigh-Bénard instability, Phys. Rev. Lett. 40:712ADSCrossRefGoogle Scholar
  2. Bénard, H., 1900, Les tourbillons cellulaires dans une nappe liquide,Rev. Gén. Sci. Pures Appl. 11:1261–71, 1309–28Google Scholar
  3. Berge, P., 1976, J. Phys. (Paris) 37, Colloq. C1, 23ADSCrossRefGoogle Scholar
  4. Block, M.J., 1956, Nature 178:650ADSCrossRefGoogle Scholar
  5. Busse, F., 1962, Ph.D. thesis (Munich) unpublishedGoogle Scholar
  6. Busse, F., 1967, On the stability of two-dimensional convection in a layer heated from below, J. Math. and Phys. 46:140MATHGoogle Scholar
  7. Busse, F., 1967, The stability of finite amplitude cellular convection and its relation to an extremum principle, J. Fl. Mech. 30:625ADSMATHCrossRefGoogle Scholar
  8. Busse, F., 1971,spi Instability of continuous systems, ed. H. Leipholz, 41–47, Berlin, Heidelberg, New York:Springer, 422 pp.CrossRefGoogle Scholar
  9. Busse, F. and Whitehead, J.A., 1971, Instabilities of convection rolls in a high Prandtl number fluid, J. Fl. Mech. 47:305ADSCrossRefGoogle Scholar
  10. Busse, F., 1972, The oscillatory instability of convection rolls in a low Prandtl number fluid, J. Fl. Mech. 52:97ADSMATHCrossRefGoogle Scholar
  11. Busse, F., 1978, Instabilities of convection rolls in a fluid of moderate Prandtl number, J. Fl. Mech. 91:319ADSCrossRefGoogle Scholar
  12. Charlson, G.S. and Sani, R.L., 1970, Thermoconvective instability in a bounded cylindrical fluid layer, Int. J. Heat Mass Transfer 13:1479MATHCrossRefGoogle Scholar
  13. Charlson, G.S. and Sani, R.L., 1971, On thermoconvective instability in a bounded cylindrical fluid layer, Int. J. Heat Mass Transfer 14:2157CrossRefGoogle Scholar
  14. Charlson, G.S. and Sani, R.L., 1975, Thermoconvective flows in a cylindrical fluid layer, J. Fl. Mech. 71, part 1:1CrossRefGoogle Scholar
  15. Chen, M.M. and Whitehead, J.A., 1968, Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-number,J. Fl. Mech. 31:1ADSCrossRefGoogle Scholar
  16. Davies, S.H., 1968, Convection in a box: on the dependence of preferred wave number upon the Rayleigh number at finite amplitude, J. Fl. Mech. 32:619ADSCrossRefGoogle Scholar
  17. Dubois, M., 1976, J. Phys. (Paris) 37, Colloq. C1, 137Google Scholar
  18. Dubois, M. and Berge, P., 1978, Experimental study of the velocity field in a Rayleigh-Bénard convection, J. Fl. Mech. 85:Google Scholar
  19. Eckhaus, W., 1965/’Studies in non-linear stability theory”, Berlin, Heidelberg, New York, Springer, 177 ppMATHCrossRefGoogle Scholar
  20. Gorkov, L.P., 1957, Sov. Phys. JETP 6:311MathSciNetADSGoogle Scholar
  21. Herring, J.R., 1963, J. Atmos. Sci. 20:325ADSCrossRefGoogle Scholar
  22. Herring, J.R., 1964, J. Atmos. Sci. 21:277MathSciNetADSCrossRefGoogle Scholar
  23. Hoard, C.O., Robertson, C.R., Acrivos, A., 1970, Experiments on the cellular structure in Bénard convection, Int. J. Heat Mass Transfer 13:849CrossRefGoogle Scholar
  24. Malkus, W.V.R. and Veronis, G., 1958, Finite amplitude cellular convection, J. Fl. Mech. 4:225MathSciNetADSMATHCrossRefGoogle Scholar
  25. Mihaljan, J.M., 1962, Astrophys. J. 136:1126MathSciNetADSCrossRefGoogle Scholar
  26. Normand, C., Pomeau, J. and Velarde, M.G., 1977, Convective instability; A physicist’s approach, Rev. Mod. Phys. 49:581MathSciNetADSCrossRefGoogle Scholar
  27. Palm, E., 1960, On the tendency towards hexagonal cells in steady convection, J. Fl. Mech. 8:183ADSMATHCrossRefGoogle Scholar
  28. Palm, E. and Øiann, H., 1964, Contribution to the theory of cellular thermal convection, J. Fl. Mech. 19:353Google Scholar
  29. Palm, E., Ellingsen, T. and Gjevik, B., 1967, On the occurence of cellular motion in Bénard convection, J. Fl. Mech. 30:651Google Scholar
  30. Palm, E., 1972, A note on a minimum principle in Bénard convection,Int. J. Heat Mass Transfer 15:2409MATHCrossRefGoogle Scholar
  31. Palm, E. and Tveitereid, M., 1981, To be publishedGoogle Scholar
  32. Pearson, J.R.A., 1958, On convection cells induced by surface tension, J. Fl. Mech. 4:489ADSMATHCrossRefGoogle Scholar
  33. Rayleigh, Lord, 1916, Phil. Mag. 32:529Google Scholar
  34. Schlüter, A, Lortz, D. and Busse, F., 1965, On the stability of steady finite amplitude convection, J. Fl. Mech. 23:129Google Scholar
  35. Segel, L.A., 1965, The non-linear interaction of a finity number of disturbances to a layer of fluid heated from below, 21:359MathSciNetMATHGoogle Scholar
  36. Segel, L.A., 1969, Distant sidewalls cause slow amplitude modulation of cellular convection, J. Fl. Mech. 38:203ADSMATHCrossRefGoogle Scholar
  37. Segel, L.A. and Stuart, J.T., 1962, On the question of the preferred mode in cellular thermal convection, J. Fl. Mech. 13:289MathSciNetADSMATHCrossRefGoogle Scholar
  38. Spiegel, E.A., 1971, Ann. Rev. Astron. Astrophys. 9:323ADSCrossRefGoogle Scholar
  39. Spiegel, E.A., 1972, Ann. Rev. Astron. Astrophys. 10:260ADSCrossRefGoogle Scholar
  40. Spiegel, E.A. and Veronis, G., 1960, Astrophys. J. 131:442MathSciNetADSCrossRefGoogle Scholar
  41. Stork, K. and Millier, U., 1972, Convection in boxes: experiments, J. Fl. Mech., 54:599ADSCrossRefGoogle Scholar
  42. Stork, K. and Müller, U., 1975, Convection in boxes: an experimental investigation in vertical cylinders and annuli, J. Fl. Mech. 71:231ADSCrossRefGoogle Scholar
  43. Thomson, J., 1881, Proc. Glasgow Phil. Soc. 13:464Google Scholar
  44. Turcotte, D.L. and Oxburgh, E.R., 1972, Ann. Rev. Fl. Mech. 4:33ADSCrossRefGoogle Scholar
  45. Velarde, M.G. and Perez Gordon, R., 1976, J. Phys. (Paris) 37:177Google Scholar

Copyright information

© Plenum Press, New York 1982

Authors and Affiliations

  • Enok Palm
    • 1
  1. 1.Department of MathematicsUniversity of OsloBlindemNorway

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