Universal stability of a hydromagnetic convective flow in a porous medium

Abstract

The linear and nonlinear stability of a conducting convective fiow of a fluid in a porous medium is investigated. The analysis is restricted to Darey's law, small magnetic Reynolds number and Boussinesq approximation. In the case of linear theory the condition for marginal stability is obtained in terms of the Rayleigh number Re, wave nunber a and the Hantmann number M; the criterion for the convective flow is given. A marginal stability curve is drawn. It is shown that the magnetic field inhibits the onset of convection. In the case of nonlinear theory a universal stability estimate, namely a stability limit for motions subject to arbitrary nonlinear disturbsances, is obtained in terms of Rayleigh number R/sub a/, Reynolds number R/sub e/ and Hartmann number M for the flow. The existence of an open region of certain stability near the origin of the (R/sub a/, E/sub e/) Cartesian plane for a fixed M is drawn. The universal stability limit can then be improved by suitably defining a maximum problem using variational techniques. It is found that, as in the case of linear theory, the magnetic field inhibits the onset of convection. It is also shown that the effect of nonlinear disturbances is to reduce the critical Rayleigh number by an amount of R/sub e/ compared to that of the linear theory.