Instability of a planar liquid layer in an alternating magnetic field

Abstract

Liquid metal interfaces, stressed by a high-frequency, alternating magnetic field are commonly observed to undulate. Even a planar interface stressed from above by a uniform magnetic field takes on an appearance that is very different from what is observed if the same layer is heated from below with about the same thermal input as associated with the eddy currents. This behaviour affects internal mixing and the transport of heat and material from interfaces. In applications where the interface is used to form glass or other materials, the undulations can be disasterous. A goal of this paper is to identify the circumstances under which this motion can be avoided. A theoretical model is developed for fluid motions, coupled to a magnetic flux density (having magnitude B0 and angular frequency ω) through a force density that is time averaged over one period of the alternating field. This theory, which does not include thermal effects, predicts a threshold for onset of instability determined by the ratio of layer thickness to skin depth and by the parameter M = B02/μηω where μ = 4π × 10−7 and and η is the viscosity. The instability has an internal nature in that it is predicted even when the liquid is bounded by rigid insulating materials. Threshold measurements are reported that agree with the predictions over more than an order-of-magnitude variation in frequency, including low frequencies, for which the finite depth of the liquid layer is important. However, observed growth times are far shorter than predicted. It is concluded that the observed motions are in fact thermally driven, but take on an appearance dictated by the hydromagnetics. A previously developed lumped parameter model, which includes thermally driven motion, does predict growth times on the order of those observed. In the lumped parameter model the critical field strength grossly affects the nonlinear saturation velocity. The critical M sets an upper limit on the extent to which a liquid metal can be levitated, depressed or transported magnetically at a given frequency without incurring interfacial undulations and an augmentation of mass and heat transfer.