Rigorous justification of low-magnetic-Reynolds-number approximation for MHD duct flows

The low-magnetic-Reynolds-number approximation has been widely used in mathematical modeling of conducting fluids under the influence of external magnetic fields. The aim of this paper is to justify this approximation using a rigorous analysis in the appropriate functional setting. We derive absolute and relative error estimates for both three-dimensional magnetohydrodynamic duct flow and two-dimensional shallow-channel flow. The derived relative error estimate vanishes as the magnetic Reynolds number tends to zero, thereby validating the approximation and also providing its order of accuracy and the range of applicability.