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 RmRm. 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.