The combined effect of porosity of porous media and longitudinal magnetic field on stability of the modified plane Poiseuille flow

The combined influence of the porosity of a porous medium and the longitudinal magnetic field on the stability against small disturbances of the pressure-driven laminar fluid flow between two parallel planes is studied. It is assumed that the channel walls have zero conductivity and the region outside the channel is electrically non-conducting. A sixth-order system of disturbance equations is derived and it is solved by the Galerkin method using the Legendre polynomials as its basis. The critical Reynolds number Re-c, the critical wave number alpha(c) and the critical wave speed c(c) are computed for a wide range of porosity parameter sigma(p), magnetic Reynolds number Rm and Alfven number A. Curves of wave number against Reynolds number for neutral stability are presented for a wide range of values of certain parameters. The onset of instability is discussed in detail using the growth rate curves for various parameters present in the problem. The combined influence of the parameters sigma(p) and A to delay the onset of instability in comparison with their influence in isolation is observed and the corresponding results are presented in detail.