Instability of a planar liquid layer in a magnetic field with alternating longitudinal and uniform transverse components

Abstract

In many cases of levitation, melting of metals and shaping of the liquid metal interface to form certain materials like glass, the interface is found to undulate in an alternating magnetic field. As undulations have undesirable effects, it is of considerable importance to study the stability of the liquid under similar circumstances. Earlier, investigations had been made to determine the stability criterion of a planar conducting liquid layer in an alternating longitudinal field and the critical value of the parameter M = B 0 2 μηω ( B 0, ω, μ and η are the amplitude of the alternating field, the angular frequency of the field, the magnetic permeability and the viscosity, respectively) was determined. It was shown that if M c was the critical value of M then the layer was stable or unstable according as M < or > M c . Since there is some possibility of additional magnetic fields such as the geomagnetic field or any other stray field to act along with the alternating field and affect the stability, we have investigated in this note the stability of a planar liquid layer in the presence of a uniform transverse field acting along with the alternating longitudinal field. The layer has been considered as thick with a perfectly conducting rigid boundary. After comparing with the earlier results we find that the critical value M c minimized with respect to the wave number decreases from the approximate value of 425 to an approximate value 80 even when B 1 is as small as 10 −4, where B 1 is the ratio of the strength of the uniform transverse field to the amplitude of the alternating field. It is also found that M c decreases further as B 1 increases and for B 1 = 10 there is no stability.