Analytical modelling of a thin liquid metal layer submitted to an ac magnetic field

A cylindrical thin liquid metal layer is submitted to a uniform ac magnetic field. When the intensity of the electromagnetic field exceeds a critical value, an opening in the liquid is shaped from outside to inside. At a given intensity of the electromagnetic field, this opening is in a frozen state, that is, the liquid metal layer reaches a new equilibrium shape. In this paper, we show that this equilibrium corresponds to a minimum of the total energy of the system. This total energy is equal to the sum of the magnetic energy and the mechanical energy. The magnetic energy is computed by assuming that the induced eddy current flowing through the liquid metal layer is concentrated in the cross-section Sc equal to the product of the skin depth and the thickness of the layer. This assumption leads us to study an equivalent electrical circuit.