Thin viscous ferrofluid film in a magnetic field

Abstract

We consider a thin, ferrofluidic film flowing down an inclined substrate, under the action of a magnetic field, bounded above by an inviscid gas. Its dynamics are governed by a coupled system of the steady Maxwell’s, the Navier-Stokes, and the continuity equations. The magnetization of the film is a function of the magnetic field and may be prescribed by a Langevin function. We make use of a long-wave reduction in order to solve for the dynamics of the pressure and velocity fields inside the film. In addition, we investigate the problem in the limit of a large magnetic permeability. Imposition of appropriate interfacial conditions allows for the construction of an evolution equation for the interfacial shape via use of the kinematic condition. The resultant one-dimensional equations are solved numerically using spectral methods. The magnetic effects give rise to a non-local contribution. We conduct a parametric study of both the linear and nonlinear stabilities of the system in order to evaluate the effects of the magnetic field. Through a linear stability analysis, we verify that the Maxwell’s pressure generated from a normally applied magnetic field is destabilizing and can be used to control the size and shape of lobes and collars on the free surface. We also find that in the case of a falling drop, the magnetic field causes an increase in the velocity and capillary ridge of the drop.