Energy stability analysis of MHD flow in a rectangular duct

Abstract

AbstractWe study the energy stability of pressure‐driven laminar magnetohydrodynamic flow in a rectangular duct in the presence of a transverse homogeneous magnetic field. The walls of the duct are electrically insulating. The quasistatic approximation of the induction equation is used. For sufficiently strong fields, the laminar velocity distribution has a uniform core and convex Hartmann and Shercliff boundary layers on the walls perpendicular and parallel to the magnetic field. The linear eigenvalue problem for the energy Reynolds number depends on the streamwise wavenumber, the Hartmann number, and the aspect ratio. We focus on duct geometries with small aspect ratios in order to compare with stability results from one‐dimensional channel flow. The lift‐up mechanism dominates in the limit of zero streamwise wavenumber and provides a linear dependence between the critical Reynolds and Hartmann number in the duct. For finite streamwise wavenumbers and decreasing aspect ratio, the duct results converge to Orr's original energy stability result for spanwise uniform perturbations to the plane Poiseuille base flow.