The onset of convection in porous layers with multiple horizontal partitions

Abstract

In this paper we investigate the onset of convection in a horizontally partitioned porous layer which is heated from below. Identical sublayers are separated by thin impermeable barriers. A linear stability analysis is performed, and dispersion relations are obtained directly and explicitly for two- and three-layer configurations. A systematic numerical procedure is devised to compute the dispersion relation for an arbitrary number of sublayers, but from this it is possible to guess the correct analytical form of the dispersion relation for general cases. Neutral stability curves are found to organise themselves into natural groups of N members when there are N sublayers. When the disturbance wavenumber, k, is large, each member of any group lies within an O( k −1) distance of all other members, but within an O(1) distance of other groups. When the number of sublayers is large, the system tends towards one with a critical Darcy–Rayleigh number of 12 and a critical wavenumber of zero; this is the well-known property of a single porous layer with constant heat flux boundary conditions. An asymptotic analysis is performed in order explore these two apparently disparate configurations. Finally, another asymptotic analysis is used to determine the critical Rayleigh number and its associated wavenumber when the number of sublayers is large.