Optimal Transient Energy Growth of Two-Dimensional Perturbation in a Magnetohydrodynamic Plane Poiseuille Flow of Casson Fluid

The study investigates the influence of the Casson fluid parameter and the spanwise uniform magnetic field on the onset of instability against infinitesimal disturbances in an electrically conducting fluid flow between two parallel nonconducting rigid plates. The fourth-order linearized disturbance equation governing stability is solved using the spectral collocation method with Chebyshev-based polynomials. The aim is to analyze in detail the effect of the parameters involved in the problem using both modal and nonmodal linear stability analysis. The modal analysis provides accurate values of the critical Reynolds number, critical wave number, and critical wave speed, denoted as critical triplets (R_c, α_c, c_c). Additionally, it examines the eigen-spectrum, growth rate curves, and neutral stability curves. On the other hand, the nonmodal analysis investigates the transient energy growth G(t) of two-dimensional (2D) optimal perturbations, the pseudospectrum of the non-normal Orr–Sommerfeld (O–S) operator (⁠$ℒ$⁠), and the regions of stability, instability, and potential instability of the fluid flow system. The extensive examination of both long-term behavior through modal analysis and short-term behavior through nonmodal analysis reveals that the Hartmann number (Ha) acts as a stabilizing agent, delaying the onset of instability. Conversely, the Casson parameter $(η)$ acts as a destabilizing agent, advancing the onset of instability. The results obtained here are verified to be in good agreement with the existing literature in the absence of the Casson fluid flow parameter.