Induced electric current-based formulation in computations of low magnetic Reynolds number magnetohydrodynamic flows

We use the induced electric current as the main electromagnetic variable to compute low magnetic Reynolds number magnetohydrodynamic (MHD) flows. The equation for the induced electric current is derived by taking the curl of the induction equation and using Ampère's law. Boundary conditions on the induced electric current are derived at the interface between the liquid and the thin conducting wall by considering the current loop closing in the wall and the adjacent liquid. These boundary conditions at the liquid-solid interface include the Robin boundary condition for the wall-normal component of the current and an additional equation for the wall potential to compute the tangential current component. The suggested formulation (denominated j-formulation) is applied to three common types of MHD wall-bounded flows by implementing the finite-difference technique: (i) high Hartmann number fully developed flows in a rectangular duct with conducting walls; (ii) quasi-two-dimensional duct flow in the entry into a magnet; and (iii) flow past a magnetic obstacle. Comparisons have been performed against the traditional formulation based on the induced magnetic field (B-formulation), demonstrating very good agreement.