We report on the numerical simulation of a two-fluid magnetohydrodynamics problem arising in the industrial production of aluminium. The motion of the two non miscible fluids is modeled through the incompressible Navier-St- okes equations coupled with the Maxwell equations. An Arbitrary Lagrangian Eulerian formulation is used to take into account the motion of the interface separating the two fuids. Details on the numerical analysis of the problem, with a special emphasis on conservation and stability properties, and on the surface tension discretization are provided. Various numerical simulations are eventually presented. Some of them reproduce complex physical phenomena.