Dynamics of a thin ferrofluid film subjected to a magnetic field

We consider a thin film flowing down a rigid, impermeable inclined plane subjected to a magnetic field. The film corresponds to a ferrofluid and is bounded from above by a hydrodynamically-passive gas. The ferrofluid is considered to be weakly-conducting, and its dynamics are governed by the steady Maxwell's equations, coupled to the Navier-Stokes, and continuity equations. The magnetisation of the ferrofluid is a function of the magnetic field, which can be represented by a nonlinear Langevin function; in this work, however, we take the limit of small Langevin parameters, in which this function becomes linear. We use the long-wave limit to expand the governing equations associated with the film; no such approximation is applied in the gas phase in which the full Laplacian for the potential is solved. A one-dimensional partial differential equation is then derived that governs the nonlinear evolution of the interface. This equation is solved numerically for a wide range of system parameters. The results of this parametric study will be presented.