Instability of the plane parallel flow through a saturated porous medium in the presence of a longitudinal magnetic field using the Chebyshev collocation method

Abstract

The onset of instability against small disturbances of the pressure driven laminar fluid flow through a saturated porous medium in the presence of horizontal magnetic field between the two parallel impermeable plates is studied numerically. Assuming the channel walls have zero conductivity. The Chebyshev collocation method is used to solve the 6th order generalized Orr–Sommerfeld equation. The critical values of the Reynolds number Rec, the wave number αc and the wave speed cc are computed for various sets of the values of the porous parameter σp, magnetic Reynolds number Rm and Alfven number A. The marginal stability curves are drawn for various values of the physical parameters present in the problem. The onset of instability is also discussed in detail using the curves of growth rate of the disturbances and the critical values for various parameters present in the problem. It is observed that the role of σp, Rm and A are to delay the onset of instability of the system. The degree of stabilization is moderate up to certain value of A but it is so rapid when the Alfven number exceeds that value is observed. It is also seen that, the role of the combined influence of porosity of the porous medium and the strength of the magnetic field together suppress the growth of the disturbances very significantly. In this study the results computed are also compared with the existing literature as a particular case for ordinary Newtonian fluids.