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 $k_c=F_0∕\left(\gamma l_0\right)$, which is determined by the linear density of the electromagnetic force $F_0$ acting on the edge, the surface tension $\gamma$, and the effective arclength of edge thickness $l_0$. Perturbations with wavelength shorter than critical are stabilized by the surface tension, whereas the growth rate of long-wave perturbations reduces as $\sim k$ for $k\to 0$. Thus, there is the fastest growing perturbation with the wave number $k_{\max }=2∕3k_c$. 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 $k_c=\sqrt{g\rho ∕\gamma }$, 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 $F_{0,c}=2k_c{l}_0\gamma$, which corresponds to the critical current amplitude $I_{0,c}=4\sqrt{\pi k_c{l}_{0}L\gamma ∕{\mu }_0}$ 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 $R_0$ placed in a transverse uniform high-frequency ac magnetic field with the induction amplitude $B_0$ becomes linearly unstable with respect to exponentially growing perturbation with the azimuthal wave number $m=2$ when the magnetic Bond number exceeds ${\mathrm{Bm}}_c=B_0^2{R}_0^2∕\left(2{\mu }_0{l}_0\gamma \right)=3\pi$. For $\mathrm{Bm}>{\mathrm{Bm}}_c$, the wave number of the fastest growing perturbation is $m_{\max }=[2\mathrm{Bm}∕(3\pi \left)\right]$. These theoretical results agree well with the experimental observations.

• Received 22 December 2005

DOI:https://doi.org/10.1103/PhysRevE.73.066303