Hostname: page-component-f554764f5-rvxtl Total loading time: 0 Render date: 2025-04-12T03:05:05.995Z Has data issue: false hasContentIssue false
  • English
  • Français

The amplitude equation near the convective threshold: application to time-dependent heating experiments

Published online by Cambridge University Press:  20 April 2006

Guenter Ahlers ,
M. C. Cross ,
P. C. Hohenberg  and
S. Safran
Guenter Ahlers
Affiliation:
Bell Laboratories, Murray Hill, New Jersey 07974 and Department of Physics, University of California, Santa Barbara, California 93106
M. C. Cross
Affiliation:
Bell Laboratories, Murray Hill, New Jersey 07974 and Department of Physics, University of California, Santa Barbara, California 93106
P. C. Hohenberg
Affiliation:
Bell Laboratories, Murray Hill, New Jersey 07974
S. Safran
Affiliation:
Bell Laboratories, Murray Hill, New Jersey 07974
Get access Rights & Permissions [Opens in a new window]

Abstract

High-resolution measurements have been performed of the convective heat current as a function of time when a Rayleigh-Bénard cell is swept through its threshold with a specified time-dependent heat input. The results are interpreted in terms of the amplitude equation which exactly describes the slow variations in space and time of hydrodynamic quantities near the threshold. A phenomenological forcing field is added to this equation, and its form and magnitude are fitted to the onset time of the convective heat current. A deterministic model in which the field is an adjustable constant yields a good fit to the data for both a step and a linear ramp in the heat input. An alternative stochastic model, in which the field is a Gaussian variable with zero mean and a white-noise spectrum, is adequate for the ramp experiments, but cannot fit the step data for any value of the mean-square field. The systematics of the field and onset time versus ramp rate are studied in both the deterministic and stochastic models, and attempts are made to interpret the field in terms of physical mechanisms. When the data for long times are analysed in terms of the amplitude equation, it is found that the state first excited at onset is not the roll pattern which is stable in steady state. Instead, the system goes first to an intermediate state, which we tentatively identify as a hexagonal configuration. The decay of this state is governed by a further adjustable field in the amplitude equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 110 , September 1981 , pp. 297 - 334
Copyright
© 1981 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. 1978 Bull. Am. Phys Soc. 23, 525.
Ahlers, G. 1980 J. Fluid Mech. 98, 137.
Ahlers, G. & Hohenberg, P. C. 1978 17the Int. Solvay Conf. on Physics, Brussels, Belgium.
Ahlers, G. & Walden, R. W. 1980 Phys. Rev. Lett. 44, 445.
Behringer, R. & Ahlers, G. 1977 Phys. Lett. A 62, 329.
Behringer, R. & Ahlers, G. 1982 (To be published).
Brown, S. N. & Stewartson, K. 1977 SIAM 57, 187.
Brown, S. N. & Stewartson, K. 1978 Proc. Roy. Soc. A 360, 455.
Brown, S. N. & Stewartson, K. 1979 SIAP (to be published).
Busse, F. 1967a J. Math. & Phys. 46, 140.
Busse, F. 1967b J. Fluid Mech. 30, 625.
Busse, F. 1978 Rep. Prog. Phys. 41, 1929.
Charlson, G. S. & Sani, R. L. 1975 J. Fluid Mech. 71, 209.
Cross, M. 1980 Phys. Fluids 23, 1727.
Cross, M., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1980 Phys. Rev. Lett. 45, 898.
Daniels, P. G. 1977 Proc. Roy. Soc. A 358, 173.
Davis, S. H. 1971 J. Fluid Mech. 45, 33.
Foster, T. D. 1965a Phys. Fluids 8, 1249.
Foster, T. D. 1965b Phys. Fluids 8, 1770.
Foster, T. D. 1969 Phys. Fluids 12, 2482.
Gollub, J. P. & Freilich, M. H. 1976 Phys. Fluids 19, 618.
Gollub, J. P. & Benson, S. V. 1980 J. Fluid Mech. 100, 449.
Graham, R. 1974 Phys. Rev. A 10, 1762.
Haake, F. 1978 Phys. Rev. Lett. 41, 1685.
Hall, P. & Walton, I. C. 1977 Proc. Roy. Soc. A 358, 199.
Homsy, G. M. 1973 J. Fluid Mech. 60, 129.
Jhaveri, B. & Homsy, G. M. 1980 J. Fluid Mech. 98, 329.
Joseph, D. D. 1976 Stability of Fluid Motions. Springer.
Kelly, R. E. & Pal, D. 1978 J. Fluid Mech. 86, 433.
Koschmieder, E. L. & Pallas, S. G. 1974 Int. J. Heat Mass Transfer 17, 991.
Krishnamurti, R. 1968a J. Fluid Mech. 33, 445.
Krishnamurti, R. 1968b J. Fluid Mech. 33, 457.
Libchaber, A. & Maurer, J. 1978 J. Physique Lett. 39, 369.
Lick, W. 1965 J. Fluid Mech. 21, 565.
Matkowski, B. J. & Reiss, E. L. 1977 SIAM J. Appl. Math. 33, 230.
Morton, R. R. 1957 Quart. J. Mech. Appl. Math. 10, 433.
Newell, A. C. & Whitehead, J. A. 1969 J. Fluid Mech. 38, 279.
Newell, A. C., Lange, C. G. & Aucoin, P. J. 1970 J. Fluid Mech. 40, 513.
Normand, C., Pomeau, Y. & Velarde, M. G. 1977 Rev. Mod. Phys. 49, 581.
Pasquale, F. De & Tombesi, P. 1979 Phys. Lett. A 72, 7.
Pellew, A. & Southwell, R. V. 1940 Proc. Roy. Soc. A 176, 312.
Robinson, J. L. 1967 J. Fluid Mech. 29, 461.
Sano, M. & Sawada, Y. 1978 Prog. Theor. Phys. (Kyoto) Suppl. 64, 202.
Schlüter, A., Lortz, D. & Busse, F. 1965 J. Fluid Mech. 23, 129.
Schryer, N. 1977 Bell Laboratories Computing Science Tech. Rep. 53.
Segel, L. A. 1969 J. Fluid Mech. 38, 203.
Shaumeyer, J. N., Behringer, R. P. & Baierlein, R. 1980 J. Fluid Mech. (to be published).
Spangenberg, W. G. & Rowland, W. R. 1961 Phys. Fluids 4, 743.
Stork, K. & Müller, U. 1975 J. Fluid Mech. 71, 231.
Suzuki, M. 1978 Phys. Lett. A 67, 339.
Swift, J. & Hohenberg, P. C. 1977 Phys. Rev. A 15, 319.
Walden, R. W. & Ahlers, G. 1981 J. Fluid Mech. (to be published).
Wankat, P. C. & Homsy, G. M. 1977 Phys. Fluids 20, 1200.
Wesfreid, J., Pomeau, Y., Dubois, M., Normand, C. & Bergé, P. 1978 J. Physique Lett. 39, 725.
Zaitsev, V. M. & Shliomis, M. I. 1970 Zh. Eksp. Teor. Fiz. 59, 1583 (also Sov. Phys., J. Exp. Theor. Phys. 32, 866, 1971).