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We consider the interfacial dynamics of a thin, ferrofluid film flowing down an inclined substrate, under the action of a magnetic field, bounded above by an inviscid gas. The fluid is assumed to be weakly conducting, and its dynamics are governed by a coupled system of the steady Maxwell, Navier-Stokes, and continuity equations. The magnetization of the film is a function of the magnetic field, and is prescribed by a Langevin function. We make use of a long-wave reduction in order to solve for the dynamics of the pressure, velocity, and magnetic fields inside the film. The potential in the gas phase is solved by means of Fourier Transforms. Imposition of appropriate interfacial conditions allows for the construction of an evolution equation for the interfacial shape, via use of the kinematic condition, and the magnetic field. We study the three-dimensional evolution of the film to spanwise perturbations by solving the nonlinear equations numerically. The constant-volume configuration is considered, which corresponds to a slender drop flowing down an incline. A parametric study is then performed to understand the effect of the magnetic field on the stability and structure of the interface.

Dynamics and stability of three-dimensional ferrofluid films in a magnetic field