An effective two dimensional model for M

This paper presents a model for quasi-two-dimensional magnetohydrodynamic (MHD) flows between two planes with a small magnetic Reynolds number and a constant transverse magnetic field orthogonal to the planes. The objective is to develop a mathematical framework that incorporates three-dimensional effects into a two-dimensional equation of motion through a model for the transverse velocity profile. The methodology involves a double perturbation asymptotic development applied in both the core flow and in the Hartmann layers that form along the planes. By introducing a new model to capture inertial effects in these regions, two distinct classes of phenomena are identified: one linked to inertial effects in the Hartmann layer, proposing a model for recirculating flows, and the other enabling a transverse dependence of the velocity profile in the core flow. Analytical solutions for the proposed model are derived for two experimental configurations: isolated vortices created by a point electrode and axisymmetric parallel layers in the MATUR (MAgneticTURbulence) experiment. The theoretical predictions show satisfactory agreement with experimental observations, leading to the conclusion that recirculating flows significantly impact vortex core spreading and the dissipative behavior observed in axisymmetric sidewall layers.