Theory analysis of steady magnetic field on rotational stability of electromagnetic levitation melting

The stability of an electrically conducting sphere levitated in a high frequency magnetic field with respect to rotational disturbances is analyzed theoretically with and without steady magnetic field respectively. The dimensionless angular frequency of inducer's current is (omega) over bar. Under the conditions of without steady magnetic field and the sphere is solid, there is exists a critical dimensionless angular frequency (omega) over bar (c). As (omega) over bar < (omega) over bar (c), the levitation of sphere sample is stable; as (omega) over bar > (omega) over bar (c), the levitation of sphere sample is unstable. Under the conditions of existing steady magnetic field and the sphere is solid, there are exist a lower limit dimensionless angular frequency (omega) over bar (c,1) and an upper limit dimensionless angular frequency (omega) over bar (c,2). As ($) over bar < (omega) over bar (c,1) or (omega) over bar > (omega) over bar (c,2), the levitation of sphere sample is stable; as (omega) over bar (c,1) < (omega) over bar < (omega) over bar (c,2) the levitation of sphere sample is unstable. The increasing of steady magnetic field can make (omega) over bar (c,1) increase and (omega) over bar (c,2) decrease. As the strength of steady magnetic field B-d reaches the critical value B-d,B-e, the levitation of sphere sample is always stable. Relative to that of solid sphere, the levitation stability of liquid sphere is decreasing. Steady magnetic field can reduce the hydrodynamic destabilization and melt splashing and improve the levitation stability of liquid sphere sample.