Further studies of liquid-metal flows and power losses in ducts with a moving conducting wall and a skewed magnetic field

Abstract

In a previous paper the authors initiated studies of fully developed laminar liquid-metal flows, currents, and power losses in a rectangular channel with a moving perfectly conducting wall and with a skewed homogeneous external magnetic field for high Hartmann numbers, high interaction parameters, low magnetic Reynolds numbers, and different aspect ratios. The channel had insulating side walls that were skewed to the external magnetic field, while the perfectly conducting moving top wall with an external potential and the stationary perfectly conducting bottom wall at zero potential acted as electrodes. These electrodes were also skewed to the external magnetic field. A mathematical solution was obtained for high Hartmann numbers by dividing the flow into three core regions, two free shear layers, and six Hartmann layers along the channel walls. The free shear layers were treated rigorously and in detail with fundamental magnetohydrodynamic theory. The previous work, however, left the solution for the velocity profiles in terms of a complex integral equation which was not solved. In the present work the integral equation is solved numerically by the method of quadratures to give the velocity profiles, viscous dissipation and Joulean losses in the free shear layers. In addition, expressions for the viscous dissipation in the six Hartmann layers are presented. The best approximation to the viscous dissipation in the channel is the sum of the O(M3/2) contributions from the two free shear layers, the O(M3/2) contributions from the two Hartmann layers separating the free shear layers from the insulators, and the O(M) contributions from three of the Hartmann layers separating core regions from the walls. The best approximation to the Joulean power losses in the channel is the sum of the O(M2) contribution from the central core region which carries an O(1) current between the electrodes and the O(M3/2) contributions from the free shear layers. The expressions for the viscous dissipation and Joulean losses in each region involve the products of universal constants, electrical potentials and geometric factors. The theoretical magnetohydrodynamic model presented here was developed to provide data to help in the design of liquid-metal current collectors.