Use of successive approximations for calculating the boundary layer in MHD channel flow

Abstract

The general expressions are obtained for determining the boundary layer thickness and the wall friction stresses in the MHD duct for arbitrary outer flow velocity distribution along the duct length and arbitrary magnetic field strength. In [2] the Shvets method is used to solve the problem of flow of a conducting fluid past a slender profile in the absence of MHD forces in the outer flow, for the case of constant conductivity; in this case the author used the basic equation of motion from [8], in which a similar problem was solved for variable fluid conductivityinthe boundary layer. In contrast to [3], the author of [2] assumed the conductivity to be constant, although in so doing the equation of motion at the outer edge of the boundary layer is not satisfied. Nevertheless, it was shown in [2] that the approximate Shvets method can be applied to the problem of MHD flow in a boundary layer (in the formulation indicated above) and a solution in quadratures can be obtained. However, the question of the accuracy and range of applicability of the Shvets method in MHD problems was not examined. In the following, numerical calculations are made for special cases, and a comparison is made of the results with the exact solutions, i. e . , an attempt is made to evaluate the bounds of applicability of this technique for the solution of MHD problems, w We examine laminar flow of an incompressible conducting fluid with constant physical properties in a two-dimensional channel with crossed magnetic and electric fields. The channel flow scheme is shown in Fig. 1, The walI AA is insulated; BB are the electrodes. It is assumed that: the magnetic Reynolds number R m << 1; the Hall parameter w7 << 1; the applied electric field E is constant along the chaunel height; and the current density j in the channel is constant across the width. Constancy of the physical properties implies that the dynamic problem may be examined independently of the thermal problem, and the assumption that R m is small makes it possible to neglect the induced magnetic field in comparison with the applied field. The assumption that E is constant is essential, since otherwise the flow may have singularities which can not be accounted for within the framework of the approximate method [4]. However, the condition E = const is realized quite exactly for MHD generators and accelerators. The flow in the channel is broken down into two regions: the boundary layer and the outer flow (core flow). The core flow is examined using one-dimensionaI theory. The values of the core flow parameters are the boundary values for the boundary layer at its outer edge. The mutual effect of the boundary layers on the electrode walls and insulated walls is not considered. The problem is significantly different for the boundary layers on the channel wails which are oriented parallel to the magnetic field lines of force ('electrode" wails) and those perpendicular to these lines ('insulated" wails). In the first case the current flows perpendicular to the wall and the current density, and, therefore, the ponderomotive forces, are constant through the boundary layer thickness. In the second case, the electric field intensity remains constant through the boundary layer thickness, while the current density and, therefore, the ponderomotive forces will change across the boundary layer thickness. w Under these assumptions the equations of the dynamic boundary Iayer on the insulated walls have the form