The magnetohydrodynamic boundary layer in the presence of a pressure gradient

Abstract

Abstract The present paper is concerned with the study of the two-dimensional, magnetohydrodynamic boundary layer, in steady, incompressible flow, under the influence of an external magnetodynamic pressure gradient (characterized by B) proportional to some power of distance along the boundary. The problem, which has also been considered by Davies (1963), may be regarded as that of extending to conducting fluids the solutions of Falkner and Skan well known in the theory of non-conducting fluids, and here again the ‘similarity’ assumption enables the differential equations to be written with but one independent variable. Solutions of these equations are sought for the cases when e, the ratio of diffusivities (viscous to magnetic), has small or large values. Hence the layers adjacent to the surface in which the diffusion of vorticity and electric current is important have thicknesses which, although both small compared with a typical streamwise length, are of different orders of magnitude. Physically the two cases correspond to fluids whose electric conductivities are sufficiently small or large. Series expansions in e, in each layer, are assumed and the form of the series, as well as boundary conditions satisfied by their coefficients, are determined from ‘matching’ the series in the two layers. The method is equivalent to that used by Glauert (1961, 1962) who, in the former paper, treated the special case to which the present problem reduces when β = 0. For this case, when e < 1, terms of 0 (e In e) must be included in the series but the present analysis shows that that is the only case where the ‘matching' requires this logarithmic term; the presence of a pressure gradient enables the ‘matching' to be achieved with purely algebraic dependence on e at least up to terms of O (e). For large e logarithmic terms are present (of O(e-1 In e)) and the results include those of Glauert (1961) showing no distinctive features. Throughout, the magnetic field parameter S (ratio of magnetic and dynamic pressures in the main stream) must be restricted so that 1 — S is not small. To obtain ‘physically acceptable' solutions when the pressure gradient is adverse, further criteria, discussed by Stewartson (1954), may be invoked. Otherwise, the solutions are non-unique and some remarks are made on the effect of this non-uniqueness on the ‘matching’. Results, in the form of series for the skin friction and field intensity at the wall, indicate a reduction of skin friction with increase of magnetic field.