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 Vc. 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 Bénard 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 Vc. 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 Vc is very small, the critical wavenumber is zero, corresponding to a very long cell (parallel flow), but as Vc is increased, the critical wavenumber also increases, corresponding to narrower flow patterns.