Oblique axisymmetric stagnation flows in magnetohydrodynamics

Abstract

The present paper deals with the influence of the Lorentz force associated with an applied radial magnetic field on the axisymmetric stagnation flow impinging obliquely onto a uniformly rotating circular cylinder. It is found that the boundary layer flow is described by an exact solution of the Navier–Stokes equations when Hall effects are ignored. The stability of this basic solution is then considered in the framework of Görtler–Hammerlin assumption according to which linear disturbances inherit the underlying symmetry of the basic flow. The resulting eigenvalue problem is solved numerically by means of a pseudospectral method using Laguerre’s polynomials. The scheme is specifically designed to solve boundary layer equations. The numerical experiments indicate that the effect of cylinder rotation is to reduce the stability of the basic flow and most importantly the magnetic field acts to either increase or decrease it, depending on whether the rotation rate is smaller or greater than some critical value that changes with the Hartmann number. At criticality, the basic flow undergoes Hopf bifurcations leading to branching off solutions in the form of azimuthally travelling waves. In the case of axisymmetric disturbances the bifurcation remains of Hopf type provided that the Hartmann number is small enough, a saddle-node bifurcation is encountered in the opposite case.