On the Linear Stability Analysis of Plane Poiseuille Flow Due to Uniform Vertical Crossflow Under a Transverse Magnetic Field

Résumé

The temporal stability of magnetohydrodynamic (MHD) fluid flow in a rigid, horizontal porous channel under a vertical magnetic field is analysed, focusing on the influence of uniform crossflow via injection and suction. For most electrically conducting fluids of interest, the magnetic Prandtl number is extremely small, justifying the use of the low magnetic Prandtl number approximation. The linearized disturbance equations reduce to a fourth-order eigenvalue problem, which is solved numerically using the spectral collocation method applied to the modified Orr-Sommerfeld equation. Critical parameters such as Reynolds number, wave number, and wave speed are computed across a range of nondimensional parameters. Neutral stability curves in the ($Re$, $\beta$) - plane reveal that, for varying magnetic and crossflow strengths, the system exhibits instabilities analogous to Tollmien–Schlichting waves in Newtonian flows but modified by electromagnetic effects. Crossflow alters both the velocity and magnetic fields of the base state, resulting in either stabilizing or destabilizing effects depending on the parameter regimes. Growth rates of disturbances under combined magnetic and crossflow conditions are numerically assessed, confirming the dual effect of crossflow on the stability of the system.