Hartmann flow in a fluid layer with spatially inhomogeneous properties

In this study we consider the flow of a spatially-inhomogeneous electrically conductive fluid between parallel planes in a transverse magnetic field. The distributions of electrical conductivity and viscosity of the fluid are given by linear functions. The slopes of these distributions characterize the maximum deviation of the fluid properties from their mean values. We show that inhomogeneity of the fluid properties leads to distortion of the velocity profiles. The resulting profiles are asymmetric and have inflection points. We use a quantity equal to the ratio of flow rates in the upper and lower halves of the layer as a quantitative measure of asymmetry. We determine the relationship between this quantity, the average Hartmann number, and the parameters of the distributions of inhomogeneous properties. We show that starting from a relatively small mean Hartmann number, the inflection points in the velocity profiles appear for any values of the distribution parameters. We provide estimates of characteristic temperatures and concentrations of non-conducting impurity for liquid sodium, at which the described effects appear.