International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 3, Pages 153-165
doi:10.1155/S0161171201004811
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 D. N. Riahi and Albert T. Hsui. 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.
Abstract
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.