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International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 3, Pages 153-165

Finite amplitude thermal convection with variable gravity

1Department of Theoretical and Applied Mechanics, 216 Talbot Laboratory, 104 S. Wright Street, University of Illinois at Urbana-Champaign, Urbana 61801, IL, USA
2Department of Geology, 245 Natural History Building, 1301 W. Green St., University of Illinois at Urbana-Champaign, Urbana 61801, IL, USA

Received 17 March 2000

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Finite amplitude thermal convection is studied in a horizontal layer of infinite Prandtl number fluid with a variable gravity. For the present study, gravity is restricted to vary quadratically with respect to the vertical variable. A perturbation technique based on a small parameter, which is a measure of the ratio of the vertical to horizontal dimensions of the convective cells, is employed to determine the finite amplitude steady solutions. These solutions are represented in terms of convective modes whose amplitudes can be either small or of order unity. Stability of these solutions is investigated with respect to three dimensional disturbances. A variable gravity function introduces two non-dimensional parameters. For certain range of values of these two parameters, double or triple cellular structure in the vertical direction can be realized. Hexagonal patterns are preferred for sufficiently small amplitude of convection, while square patterns can become dominant for larger values of the convective amplitude. Variable gravity can also affect significantly the wavelength of the cellular pattern and the onset condition of the convective motion.