Hostname: page-component-76c49bb84f-887v8 Total loading time: 0 Render date: 2025-07-02T07:15:49.444Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase-winding solutions in a finite container above the convective threshold

Published online by Cambridge University Press:  20 April 2006

M. C. Cross ,
P. G. Daniels ,
P. C. Hohenberg  and
E. D. Siggia
M. C. Cross
Affiliation:
Bell Laboratories, Murray Hill, NJ 07974
P. G. Daniels
Affiliation:
Department of Mathematics, City University, London
P. C. Hohenberg
Affiliation:
Bell Laboratories, Murray Hill, NJ 07974
E. D. Siggia
Affiliation:
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853
Get access Rights & Permissions [Opens in a new window]

Abstract

An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L [Gt] 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band |q| ∼ [(R − R0)/R0]½ in the laterally infinite system to a band |q| ∼ (R − R0)/R0 in a system with sidewalls (R is the Rayleigh number and R0 its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R − R0)/R0]½, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newell & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q0. The applicability of the results to convection in a rectangular container is briefly discussed.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 127 , February 1983 , pp. 155 - 183
Copyright
© 1983 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ahlers, G. & Behringer, R. P. 1978 Phys. Rev. Lett. 40, 712.
Brown, S. N. & Stewartson, K. 1977 SIAM J. Appl. Math. 57, 187.
Brown, S. N. & Stewartson, K. 1978 Proc. R. Soc. Lond. A, 360, 455.
Busse, F. H. 1978 Rep. Prog. Phys. 41, 1929.
Busse, F. H. & Clever, R. M. 1982 Phys. Fluids 25, 931.
Cross, M. C. 1980 Phys. Fluids 23, 1727.
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1980 Phys. Rev. Lett. 45, 898.
Daniels, P. G. 1977 Proc. R. Soc. Lond. A, 358, 173.
Daniels, P. G. 1978 Mathematika 25, 216.
Daniels, P. G. 1981 To be published.
Davis, S. H. 1967 J. Fluid Mech. 30, 465.
Drazin, P. G. 1975 Z. angew. Math. Phys. 26, 239.
Dubois, M. & BERGÉ, P. 1978 J. Fluid Mech. 85, 641.
Fauve, S. D. & Libchaber, A. 1981 In Chaos and Order in Nature (ed. H. Haken). Springer.
Hall, P. & Walton, J. C. 1977 Proc. R. Soc. Lond. A, 358, 199.
Joseph, D. D. 1976 Stability of Fluid Motions, chap. 11. Springer.
Koschmieder, E. L. 1974 Adv. Chem. Phys. 26, 177.
Luijkx, J. M. & Platten, J. K. 1981 J. Non-Equilib. Thermodyn. 6, 141.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Newell, A. C. & Whitehead, J. A. 1969 J. Fluid Mech. 38, 279.
Pomeau, Y. & Manneville, P. 1980 Phys. Lett. 75 A, 296.
Pomeau, Y. & Manneville, P. 1981 J. Phys. (Paris) 42, 1067.
Pomeau, Y. & Zaleski, S. 1980 C. R. Acad. Sci., Paris 290B, 505.
Schlüter, A., Lortz, D. & Busse, F. H. 1965 J. Fluid Mech. 23, 129.
Segel, L. A. 1969 J. Fluid Mech. 38, 203.