Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-29T11:41:23.331Z Has data issue: false hasContentIssue false

Nonlinear convection in high vertical channels

Published online by Cambridge University Press:  20 April 2006

C. Normand
Affiliation:
CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France

Abstract

Application of Landau's ideas to the theory of weakly nonlinear instabilities shows that the amplitude of the unstable modes behaves as the square root of the reduced control parameter ε, its critical value being ε = 0. When applied to cellular structures the theory has been improved by taking into account the slow spatial variations of the amplitude and phase of the unstable modes. Until now the case of thermo-convective instabilities in high vertical channels has not been studied using this approach. In high vertical structures the nonlinear terms disappear in the limit of an infinite height, and the supercritical behaviour requires a specific treatment. It differs from the standard analysis valid for the case of fluid layers of infinite horizontal extent, where the nonlinearities and the finite-size effects are disconnected. In the limit of high aspect ratios (height [Gt ] horizontal extent) we have derived an amplitude equation for convective systems where the nonlinear terms contain derivatives at the lowest order. As a consequence the amplitude equation cannot be put into a variational form and the stability of the stationary solutions cannot be deduced from an ordering in decreasing values of a Lyapunov functional.

Type
Research Article
Copyright
© 1984 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.)

References

Azouni, T. 1977 J. Crystal Growth 42, 405.
Brown, S. N. & Stewartson, K. 1977 Stud. Appl. Maths 57, 187.
Catton, I. & Edwards, D. K. 1970 AIChE J. 16, 594.
Chapman, C. J. & Proctor, M. R. E. 1980 J. Fluid Mech. 101, 759.
Charlson, G. S. & Sani, R. L. 1975 J. Fluid Mech. 71, 209.
Gershuni, G. Z. & Zukhovitskii, E. M. 1972 Convective Stability of Incompressible Fluids. Israel Program For Scientific Translations.
Gorkov, L. P. 1958 Sov. Phys. JETP 6, 311.
Liang, S. F., Vidal, A. & Acrivos, A. 1969 J. Fluid Mech. 36, 239.
Lyubimov, D. V., Putin, G. F. & Chernatynskii, V. I. 1977 Sov. Phys. Dokl. 22, 360.
McHarg, E. A. 1947 J. Lond. Math. Soc. 22, 83.
Malkus, W. V. R. & Veronis, G. 1958 J. Fluid Mech. 4, 225.
Newell, A. C. & Whitehead, J. A. 1969 J. Fluid Mech. 38, 279.
Olson, J. M. & Rosenberger, F. 1979 J. Fluid Mech. 92, 609.
Ostrach, S. 1955 Unstable convection in vertical channels with heatings from below. NACA TN 3458.
Ostroumov, G. A. 1947 Natural convective heat transfer in closed vertical tubes. Izv. Estatstv. Nauch. Inot. Perm. Univ. 12, 113.Google Scholar
Palm, E. 1960 J. Fluid Mech. 8, 183.
Rosenblat, S. 1982 J. Fluid Mech. 122, 395.
Schlüter, A., Lortz, D. & Busse, F. 1965 J. Fluid Mech. 23, 129.
Segel, L. A. 1969 J. Fluid Mech. 38, 203.
Siggia, E. D. & Zippelius, A. 1981 Phys. Rev. Lett. 47, 835.
Sorokin, V. S. 1954 Prikl Mat. i Mekh 18, 197.
Wooding, R. A. 1960 J. Fluid Mech. 7, 501.
Yih, C. S. 1959 Q. Appl. Maths 17, 25.