Edge pinch instability of liquid metal sheet in a transverse high-frequency ac magnetic field

We analyze the linear stability of the edge of a thin liquid metal layer subject to a transverse high-frequency ac magnetic field. The layer is treated as a perfectly conducting liquid sheet that allows us to solve the problem analytically for both a semi-infinite geometry with a straight edge and a thin disk of finite radius. It is shown that the long-wave perturbations of a straight edge are monotonically unstable when the wave number exceeds the critical value k(c) = F0/(gamma l0), which is determined by the linear density of the electromagnetic force F0 acting on the edge, the surface tension gamma, and the effective arclength of edge thickness l0. Perturbations with wavelength shorter than critical are stabilized by the surface tension, whereas the growth rate of long-wave perturbations reduces as similar to k for k --> 0. Thus, there is the fastest growing perturbation with the wave number k max = 2/3 k(c). When the layer is arranged vertically, long-wave perturbations are stabilized by the gravity, and the critical perturbation is characterized by the capillary wave number k(c) = square root of (g rho/gamma), where g is the acceleration due to gravity and rho is the density of metal. In this case, the critical linear density of electromagnetic force is F(0,c) = 2k(c)l0 gamma, which corresponds to the critical current amplitude I(0,c) = 4 square root of (pi k(c) l0L gamma/mu 0) when the magnetic field is generated by a straight wire at the distance L directly above the edge. By applying the general approach developed for the semi-infinite sheet, we find that a circular disk of radius R0 placed in a transverse uniform high-frequency ac magnetic field with the induction amplitude B0 becomes linearly unstable with respect to exponentially growing perturbation with the azimuthal wave number m = 2 when the magnetic Bond number exceeds Bm(c) = B(0)2 R(0)2 / (2 mu 0 l0 gamma) = 3 pi. For Bm > Bm(c), the wave number of the fastest growing perturbation is m(max) = [2Bm/(3 pi)]. These theoretical results agree well with the experimental observations.