The boundary layer in crossed-fields m.h.d.

Abstract

This study of the boundary layer of steady, incompressible, plane, crossed-fields m.h.d. flow at large Reynolds numberReand magnetic Reynolds numberRmbegins with a review of Hartmann's case, where a boundary layer occurs whose thickness is proportional to (Re Rm)−½. Following this clue, it is shown that in general the boundary layer is a ‘local Hartmann boundary layer’. Its profiles are always exponential and it is determined completely by local quantities. The skin friction and the total electric current in the layer are proportional to the square root of the magnetic Prandtl number, i.e. to (Rm/Re)½. Thus the exterior-flow problem, the solution of which precedes a boundary-layer solution, generally involves a current sheet at the fluid-solid interface.This inviscid-flow problem becomes tractable if (Rm/Re)½is small enough to permit a linearized solution. The flow field about a flat plate at zero incidence is calculated in this approximation. It is pointed out that the thin-cylinder solutions of Sears & Resler (1959), which pertain toRm/Re= 0, can immediately be extended to small, non-zero values of this parameter by linear combination with this flat-plate solution.