Fluid dynamical aspects of the levitation-melting process

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Abstract
When a piece of metal is placed above a coil carrying a high frequency current, the induced surface currents in the metal can provide a Lorentz force which can support it against gravity; at the same time the heat produced by Joule dissipation can melt the metal. This is the process of ‘levitation melting’, which is a well-established technique in fundamental work in physical and chemical metallurgy. Most theoretical studies of magnetic levitation have dealt only with solid conductors and so have a voided the interesting questions of interaction between the free surface, the magnetic field and the internal flow. These fluid dynamical aspects of the process are studied in this paper. A particular configuration that is studied in detail is a cylinder levitated by two equal parallel currents in phase; this is conceived as part of a toroidal configuration which avoids a difficulty of conventional configurations, viz the leakage of fluid through the ‘magnetic hole’ at a point on the metal surface where the surface tangential magnetic field vanishes. The equilibrium and stability of the solid circular cylinder is first considered; then the dynamics of the surface film when melting begins; then the equilibrium shape of the fully melted body (analysed by means of a general variational principle proved in § 5); and finally the dynamics of the interior flow, which, as argued in § 2, is likely to be turbulent when the levitated mass is of the order of a few grams or greater.