Asymptotic Solutions for Low-Magnetic-Reynolds-Number Gas Flows Inside a Two-Dimensional Channel

This paper analyzes a near-continuum compressible magneto-gas-dynamic flow inside a two-dimensional microchannel or a two-dimensional channel of conventional dimensions, with a low-magnetic-Reynolds-number assumption. This work represents an extension from the classical Hartmann flow in a two-dimensional channel of infinite length to a microchannel of finite length. First, by comparing the magnitudes of different forces in the compressible gas flow, we obtain a nondimensional X momentum equation that relates the pressure ratio, Reynolds number, Mach number, magnetic Reynolds number, and magnetic force number. Second, for two cases of selected nondimensional parameters of comparable magnitude, we solve for asymptotic solutions of velocities, pressure, temperature, and mass flow rate of compressible gas flow based on the velocity-slip and temperature-jump wall boundary conditions while maintaining a consistent quasi-isothermal assumption. It is found that even with a small magnetic Reynolds number, the electric and magnetic field effects on the flow properties can be significant. Numerical solutions of the same formulation are obtained for validation of the present analytical solutions. The major work in this study is theoretical and the solutions are obtained in closed forms that can provide physical insights into flows inside a microchannel or a channel of conventional dimensions.