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Fully developed, viscous liquid‐metal velocity profiles and induced magnetic field contours were studied for Hartmann numbers of M=2 and 10 and for different load currents for a particular rectangular channel configuration (two‐dimensional Couette flow). The rectangular channel was assumed to have a homogeneous external (axial) magnetic field parallel to the moving, perfectly conducting top wall and the stationary, perfectly conducting bottom wall. The two stationary side walls were also perfect conductors. The small gap between the moving wall and each side wall was an insulating, free surface. The method of weighted residuals was used to obtain truncated series solutions for the variables of interest. The heavy load currents across the channel were obtained by simulating an external potential to the conducting moving wall. The load currents in each case were opposed by the induced electric field. Since there is no pressure gradient, the flow along the channel is driven by the viscous effects of the moving wall and the Lorentz body force and is retarded by the stationary walls. In the case where no load current is applied across the channel, the current circulates in the channel. The circulation is driven by the generator that is due to the axial variation of velocity in an axial magnetic field. The numerical results presented show that the radial gap and the free surface region represent electrical resistances in parallel between the perfectly conducting stationary wall and the perfectly conducting moving wall. The numerical results also show that the resistance of the radial gap increases as M2 while that of the free surface increases by M or M1/2. Thus, as M increases, the division of current shifts to the free surface region and the current density in the radial gap decreases as M-1. The theoretical magnetohydrodynamic model presented here was developed to provide numerical parameters to help in the design of liq- uid‐metal current collectors. Numerical results were computed for one‐dimensional Couette flow with no pressure gradient in an external, homogeneous axial magnetic field. One‐dimensional Couette flow has no end effects, and thus the numerical results were compared with corresponding numerical results for the two‐dimensional Couette flow case to determine end effects.