In this work, linear stability of an electrically conductive fluid experiencing Poiseuille flow for minimum Reynolds value under a normal magnetic field is analysed using the Chebyshev collocation method. The neutral curves of linear instability are derived by utilising Qualitat and Zuverlassigkeit (QZ) method. Instability of the magnetohydrodynamics for plane Poiseuille flowis introduced by solving the generalised Orr–Sommerfeld equation to determine the growth rates, wave number and spatial shapes of the eigenmodes. To solve linear problems, we use numerical methods which help us at each time step of the simulation, uncoupled by physical processes, which can improve the computational performance. This article provides the stability and error analysis, presents a concise study of the Poiseuille flow, and produces computational tests to support the given theory.

Magnetohydrodynamics (MHD) is a very challenging problem which affects the stability of Poiseuille flow. Therefore, in this work we investigate the instability of an electrically conductive fluid between two parallel plates under the influence of a transverse magnetic field. We apply the Chebyshev collocation method to solve the generalised Orr–Summerfield equations to determine wave number, growth rates and spatial modes of the eigenmodes. To get the neutral curves of MHD instability, the QZ method is used. It is observed that the magnetic field has a stabilising effect on the flow and the stability increases as we increase the Hartmann number and for various wave numbers, magnetic field put down the growth of perturbation. It is concluded that effect of perturbations is little in span-wise direction for different Hartmann numbers that increase the critical values of Reynolds numbers.