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International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 1, Pages 171-181

Finite bandwidth, long wavelength convection with boundary imperfectons: near-resonant wavelength excitation

Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana 61801, Illinois, USA

Received 12 June 1996

Copyright © 1998 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 with continuous finite bandwidth of long wavelength modes in a porous layer between two horizontal poorly conducting walls is studied when spatially nonuniform temperature is prescribed at the lower wall. The weakly nonlinear problem is solved by using multiple scales and perturbation techniques. The preferred long wavelength flow solutions are determined by a stability analysis. The case of near resonant wavelength excitation is considered to determine the non-modal type of solutions. It is found that, under certain conditions on the form of the boundary imperfections, the preferred horizontal structure of the solutions is of the same spatial form as that of the total or some subset of the imperfection shape function. It is composed of a multi-modal pattern with spatial variations over the fast variables and with non-modal amplitudes, which vary over the slow variables. The preferred solutions have unusual properties and, in particular, exhibit ‘kinks’ in certain vertical planes which are parallel to the wave vectors of the boundary imperfections. Along certain vertical axes, where some of these vertical planes can intersect, the solutions exhibit multiple ‘kinks’.