MHD instability of the pressure-driven plane laminar flow in the presence of the uniform coplanar magnetic field: Linear stability analysis

The influence of the uniform longitudinal magnetic field on the stability against small disturbances of an electrically conducting Newtonian fluid flow between two parallel horizontal plates is investigated. The sixth-order system of disturbance equations is solved by the Chebyshev collocation method, and 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 the magnetic Reynolds number Rm and Alfven number A. Curves of wave number against Reynolds number for neutral stability are presented for different values of the parameters. The onset of instability is also discussed in detail using the growth rate curves for various parameters of the problem. It is observed that the effect of both conductivity of the fluid and the strength of the magnetic field is to decay the onset of instability. A comprehensive study is carried out at the critical state of the fluid using the graph of Re-c, alpha(c), and c(c) with respect to Rm for various values of A. The critical values at the onset of instability are also presented for both the Galerkin method and the Chebyshev collocation method.