Effective boundary conditions for magnetohydrodynamic flows with thin Hartmann layers.

In this study, we develop effective boundary conditions aimed at enhancing numerical calculations for magnetohydrodynamic (MHD) flows involving thin Hartmann layers adjacent to a locally plane wall. The primary objective is to eliminate the necessity for extremely fine meshing that is typically needed in such scenarios. We introduce wall models for both tangential and normal electric current densities and velocities, and derive a specific condition concerning the normal derivative of the tangential velocity. The research covers a broad range of problems, with the main requirement being that the magnetic Reynolds number must be sufficiently large at the scale of the Hartmann layer. We also investigate various scenarios including perfectly conducting, insulating walls, and thin conducting walls. One of the key findings is a condition for normal velocity that integrates inertial effects within the Hartmann layer, highlighting its critical role in accurately describing the flow dynamics. This approach offers a significant advancement in the computational modeling of MHD flows, providing robust methodologies for scenarios characterized by strong magnetic fields.