We report on the numerical simulation of the industrial production of aluminium. The motion of the two nonmiscible conducting fluids is modeled through the incompressible Navier-Stokes equations coupled with the (parabolic) Maxwell equations. Stabilized finite elements techniques and an arbitrary Lagrangian Eulerian formulation (for the motion of the interface separating the two-fluids) are used in the numerical simulation. The industrial problem and the modeling strategy are presented in details. An outline of the numerical analysis of the problem, with a special emphasis on conservation and stability properties is provided. Examples of numerical simulations of the industrial case are eventually presented.