Mathematical model of oxide film deformation on the surface of a metallic melt in an alternating magnetic field

We consider a thin oxide film on the surface of a molten metal during induction melting. The alternating electromagnetic field excites eddy currents in the metal volume, which heat it, and Lorentz force, which causes the forced convection of the melt, the heating and the flows in the metal are discussed briefly. The contribution to the mechanical stresses in the film that gives the alternating electromagnetic field, the thermal expansion of the film, and the melt motion are considered in detail. The equations system describing magnetic field diffusion, equations of motion and heat transfer in the melt are written in the axisymmetric formulation. The corresponding boundary conditions are described, the governing dimensionless criteria, which determine the structure and intensity of the melt flow, including those at the surface where the film is located, are given. The film elastic deformation equation is derived from Hooke's law and written in terms of displacement in dimensional and dimensionless forms. On the base of literature review, the values of physical characteristics of the film, which are not available for direct measurement, are proposed. The verification of the mathematical model is given. Possible flows in the melt are calculated, taking into account the action of dynamic and thermal action of the film on the surface. The unambiguous relation of the film stress state with these flows is shown. The influence of the magnetic field diffusion parameter and the Hartmann number, which determine, respectively, the structure and intensity of the forced flow, on the film deformations is demonstrated. The mode map of regimes is constructed that relates the integral deformation of the film to the parameters of the magnetic field and the initial size of the film. It is found that the situations are possible when the film in the stress-strain state does not change its total size and remains in stable equilibrium on the surface of the moving melt. Recommendations for the usage of the presented results are given.