The onset of instability in a hydromagnetic channel flow of Casson fluid: the accurate solutions

We study the temporal stability of linear two-dimensional disturbances of plane Poiseuille flow of a Casson fluid in a rigid parallel channel under the influence of a uniform magnetic field. When the disturbance is taken in normal mode it transforms the perturbed equa-tions to a fourth-order eigenvalue problem which is then solved by the spectral method. We first obtain sufficient conditions for the hydrodynamic stability in the channel by ana-lyzing the nature of the growth rate and the Reynolds number. For the chosen set of pa-rameters, the eigenspectra of the flow exhibit the familiar Y-shaped structure with three distinct modes depending on the phase speed approach. The eigenfunctions corresponding to the least stable and most unstable eigenmodes have the symmetric and antisymmetric modes. For a certain range of Casson parameter beta and the magnetic number M, instability always appears on the wall mode for which the eigenfunction is antisymmetric about the channel center-line. The neutral stability curves that correspond to various values of beta and M are found to be an extension of the continuation of the Tollmien-Schlichting instability that is noticed for a Newtonian channel flow. The regions of stability are enlarged for small beta and large M. (C) 2022 Published by Elsevier Inc.