Modal and Nonmodal Instabilities of a Two-Dimensional Channel Flow Subject to Wall Slip and Transverse Magnetic Field

Abstract

This study investigates the linear stability of two-dimensional channel flows subjected to a transverse magnetic field, with walls coated by super-hydrophobic materials that introduce slip boundary conditions. Both symmetric and asymmetric slip conditions are considered, with the stability analysis performed using both modal and nonmodal approaches. A numerical solution is obtained using a Chebyshev collocation method (CCM) via an Orr–Sommerfeld normal-mode approach. The Hartmann number (Ha) quantifies the magnetic field, and its influence on flow stability is examined. The results from both modal and nonmodal analysis indicate that the Hartmann number exerts a stabilizing effect in both symmetric and asymmetric slip configurations, enhancing the critical Reynolds number for the onset of instability. In symmetric slip flows, slip consistently enhances stability, and for sufficiently large slip lengths, the upper and lower branches of the neutral stability curve coalesce, rendering the flow linearly stable for all Reynolds numbers and wavelengths. However, the role of slip in asymmetric flows is more complex. While slip generally acts as a stabilizing agent, it can also induce destabilization under certain conditions, with this destabilizing effect becoming more pronounced as the Hartmann number increases. This dual role of slip highlights its critical influence in modulating the stability of magnetohydrodynamic (MHD) channel flows, especially in the presence of asymmetric wall conditions. The results from the nonmodal analysis appear to align with those from the modal analysis and previous literature.