A transient flow problem in magnetohydrodynamics

Abstract

A flat plate x = 0, |y| < L is initially at rest in an electrically conducting, inviscid, incompressible fluid permeated by a uniform magnetic field (B0, 0, 0). The plate is impulsively accelerated to a small velocity (— U, 0, 0) which is then kept constant. It is assumed that LV/λ 1 and U/V [Lt ] 1, where V is the Alfven velocity, and λ is the magnetic diffusivity.Four stages in the development of the flow are distinguished, the last three being: (ii) L [Lt ] Vt [Lt ] (λ)½. During this stage the initial potential flow is being disturbed by propagation of electric current and vorticity from the plate. The initial discontinuity on the plate has only propagated a small distance away compared to L, but a large distance compared to the length scale of diffusion (λt)½. Exact solutions to the flow are found in the neighbourhood of y = — L and y = 0 (x = 0).(iii) Vt [Lt ] L [Lt ] (λt)½. The asymptotic behaviour of the electric current and vorticity on the plate are determined showing that a column of fluid of length Vt moves with the plate, aligned to the magnetic field. The transverse diffusion of the current sheets bounding the column accelerates the fluid in layers of thickness O(λt)½.(iv) (λt)½ [Lt ] L. Unlike stages (ii) and (iii), where the motion is dominated by the propagation of vorticity and electric current as Alfven waves from the plate, diffusive mechanisms completely dominate the motion. ‘Slug’ flow is maintained. The nature of the flow, including the structure of the layers bounding the column are determined for |x| [Lt ] (λt)½.