Analytical solution for the magnetohydrodynamic duct flow with slip condition on the perturbed boundary

We consider the flow of an electrically conducting fluid in a rectangular duct driven by the pressure gradient and under the action of a transverse magnetic field. A small boundary perturbation is applied on the cross-section of the duct and the hydrodynamic slip condition is imposed on this rough part of the boundary. A higher-order effective model is proposed via asymptotic expansion technique in the form of the explicit formulae for the velocity and the induced magnetic field. Based on the analytical results, numerical examples are provided indicating the nonlocal effects of the slip condition and the perturbed boundary.