Journal of Engineering Mathematics

, Volume 49, Issue 2, pp 181–193

The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

  • D. Andrew Rees
  • Peder A. Tyvand

DOI: 10.1023/B:ENGI.0000017494.18537.df

Cite this article as:
Rees, D.A. & Tyvand, P.A. Journal of Engineering Mathematics (2004) 49: 181. doi:10.1023/B:ENGI.0000017494.18537.df


It is well-known that every two-dimensional 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 second-order 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. Finite-difference 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.

conducting boundaries degeneracy Helmholtz equation onset of convection porous media 

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • D. Andrew Rees
    • 1
  • Peder A. Tyvand
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of Bath, Claverton DownBathUK
  2. 2.Department of Agricultural Engineering AgriculturalUniversity of NorwayÅsNorway