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. C 2015 AIP Publishing LLC.

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

“…The governing equations describing ferrofluid films were presented previously in the work of Conroy and Matar [6]; thus, we only provide the dimensionless form, suitably extended to three dimensions. We scale the vertical dimension with the initial film height, z w , as (z, S) = z w (ž,Š), the horizontal dimensions as (x, y) = L(x,y), the fluid velocity as (u, v, w) = V (ǔ,v, δw), the pressure as p =pμV L/z 2 w , and time as t =ť L/V ; here, L is a characteristic lateral extent, V is a scale for the velocity to be defined later, μ is the fluid viscosity, and δ ≡ z w /L 1 is a small parameter, which will be used as the basis for the asymptotic reduction to be carried out below.…”

“…At the wall, z = −1, we fix the magnetic field as was done by [27], which is different from the model of [6]. The appropriate condition in this case for a flat surface is…”

“…In these models, magnetic effects are accounted for through the inclusion of Maxwell's stresses at the interface and in the bulk. The relation between the magnetic field and the magnetization in the case of ferrofluids is either linear [12] or, for sufficiently large magnetic fields, nonlinear [6]; in the former case, it is possible to rationalise ferrofluid flows by analogy with their electrified counterparts [17][18][19][20][21].…”

“…It is possible to examine the effect of a magnetic field on the fingering instability of a thin film or slender drop using the long-wave approximation [1,2] to couple the interfacial dynamics to the Maxwell stresses. Using this approach, the case of a highly conducting ferrofluid with a nonlinear magnetic susceptibility was investigated by Conroy and Matar [6] who found that the magnetic field has a destabilizing effect and led to a complex surface structure that increased in amplitude with increasing magnetic pressure; this work, however, was restricted to two-dimensional flows.…”

“…In this paper, we extend the model developed by Conroy and Matar [6], for the long-wave evolution of a ferrofluid film, to three dimensions. We solve the resultant equations numerically using pseudo-spectral methods, and investigate the spanwise instability of a falling drop down an inclined plane; the magnetic effects enter through a Maxwell pressure.…”