Transverse edge effects in a rectangular magnetohydrodynamic channel with a sectionalized lateral wall

A. I. Bertinov - L. K. Kovalev - V. K. Tyutin

Abstract
This paper considers a rectangular magnetohydrodynamic channel with solid electrodes having a width of 2L, a ratio of its sides A = L/H, and a lateral wall consisting of n equal conducting plates (modules), separated by insulating gaps with a height of 2∆. The electrodes are separated from the adjacent modules by insulators with a height ∆. The external magnetic field B and the conductivity are constant over the cross section. The velocity of the gas v is assumed to be an even function of the coordinate x, which corresponds approximately to the flow of a gas in a channel with small values of the parameters of the magnetohydrodynamic interaction and thin boundary layers at the electrodes. It is shown that the distribution of the current density in the channel is periodic with respect to y, with a period of 2H/n, and to determine the potential and the current it is sufficient to consider a cross section of the channel with a single module. A solution of the problem is obtained by conformal mapping of a cross section of the channel on a rectangle with insulated lateral walls. At Λ n ≥ 1, the edge effects at opposite lateral walls have only a slight effect on each other, which permits constructing an approximate solution to the problem for a semiinfinite band. The distribution of the potential and of the current in a cross section of the channel is obtained. The zone of nonhomogeneity in the distribution of the current in the channel is determined by measurement of a current eddy in a conducting module ( ∼ H/n). The distribution of the potential over the insulating gap near the edge of a module is nonlinear. The dependences of the Joule losses, of the current leaks, and of the coefficient of the no-load voltage drop k_x are determined as a function of the width of the channel Λ, the number of modules n, and of the fraction of insulating gaps on the lateral wall. With finite sectionalizing (∆n ≥ 4) and a fraction of insulating gaps ∆n/H ≥ 0.25, the effect of leakage currents and of eddy current zones on the overall characteristics of the channel is small.