Stability of gravitactic micro-organisms in a fluid-saturated porous medium

A study is made of the stability of motile suspensions in a horizontal porous layer. The micro-organisms are assumed to have a gravitactic behaviour, swimming randomly, but on the average upward with a constant velocity V-c. The resulting equilibrium state is potentially unstable as a denser, layer of micro-organisms is formed on top of a lighter one. The basic mechanism is analogous to that of Benard convection in a fluid layer heated from below. The fluid flow is governed by the Darcy equation while the conservation of micro-organisms is described by a diffusion convection equation similar to the conservation of energy. The problem depends on two parameters, namely the Rayleigh number and the swimming velocity V-c. The present paper is focused on the stability of the equilibrium diffusive state. The stability diagram and the critical conditions for the onset of convection are obtained for a wide range of swimming velocity. It is found that if V-c is very small, the critical wavenumber is zero, corresponding to a very long cell (parallel flow), but as V-c is increased, the critical wavenumber also increases, corresponding to narrower flow patterns. (c) 2004 Elsevier Ltd. All rights reserved.