Flow of a thin liquid-metal film in a toroidal magnetic field

We investigate the gravity-driven flow of a thin film of liquid metal on a conducting conical substrate in the presence of a strong toroidal magnetic field (transverse to the flow and parallel to the substrate). We solve the leading-order governing equations in a physically relevant asymptotic limit to find the free-surface profile. We find that the leading-order fluid flow rate is a non-monotonic bounded function of the film height, and this can lead to singularities in the free-surface profile. We perform a detailed stability analysis and identify values of the relevant geometric, hydrodynamic and magnetic parameters such that the flow is stable.