Temporal stability of small disturbances in MHD Jeffery-Hamel flows

In this paper, the temporal development of small disturbances in magnetohydrodynamic (MHD) Jeffery–Hamel flows is investigated, in order to understand the stability of hydromagnetic steady flows in convergent/divergent channels at very small magnetic Reynolds number Rm . A modified form of normal modes that satisfy the linearized governing equations for small disturbance development asymptotically far downstream is employed [A. McAlpine, P.G. Drazin, On the spatio-development of small perturbations of Jeffery–Hamel flows, Fluid Dyn. Res. 22 (1998) 123–138]. The resulting fourth-order eigenvalue problem which reduces to the well known Orr–Sommerfeld equation in some limiting cases is solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The results indicate that a small divergence of the walls is destabilizing for plane Poiseuille flow while a small convergence has a stabilizing effect. However, an increase in the magnetic field intensity has a strong stabilizing effect on both diverging and converging channel geometry. c © 2007 Elsevier Ltd. All rights reserved.