Onsager Coefficients for Liquid Metal Flow in a Conduit under a Magnetic Field

Recently, interest has grown in applying liquid metals to miniature systems for energy and actuation applications. We analyze the flow of room and near room-temperature liquid metals in shallow, long rectangular conduits with two insulating and two perfectly conducting walls under a uniform magnetic field perpendicular to the flow direction and the insulating surfaces, focusing on moderate Hartmann numbers. A pressure gradient and Lorentz body forces may drive or oppose the flow. Analyzing such systems often requires repeated numerical solutions of the magnetohydrodynamic (MHD) equations during design, optimization, and control processes. We derive explicit expressions for the Onsager coefficients that relate the flow rate and electric current on the one hand to the potential difference across electrodes and the pressure gradient on the other hand. We further demonstrate that these coefficients satisfy Onsager-Casimir reciprocity. These simplified expressions provide a convenient framework for analyzing, optimizing, and controlling MHD machines operating with liquid metals in applications such as power conversion, energy harvesting, pumping, actuation, valving, breaking, and sensing without moving components.