The Helmholtz equation for convection in twodimensional porous cavities with conducting boundaries
 D. Andrew Rees,
 Peder A. Tyvand
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It is wellknown that every twodimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a secondorder problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finitedifference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains.
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 Title
 The Helmholtz equation for convection in twodimensional porous cavities with conducting boundaries
 Journal

Journal of Engineering Mathematics
Volume 49, Issue 2 , pp 181193
 Cover Date
 20040601
 DOI
 10.1023/B:ENGI.0000017494.18537.df
 Print ISSN
 00220833
 Online ISSN
 15732703
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 conducting boundaries
 degeneracy
 Helmholtz equation
 onset of convection
 porous media
 Industry Sectors
 Authors

 D. Andrew Rees ^{(1)}
 Peder A. Tyvand ^{(2)}
 Author Affiliations

 1. Department of Mechanical Engineering, University of Bath, Claverton Down, Bath, BA2 7AY, UK
 2. Department of Agricultural Engineering Agricultural, University of Norway, 1432, Ås, Norway