Stability analysis of the flow inside an electromagnetically levitated drop

Electromagnetic levitation is a well-known technique for containerless processing of metallic samples. If the sample is melted the electromagnetic forces cause a melt motion inside the droplet. This motion is of interest since it influences shape oscillations or heat transfer within the droplet, or may even effect the stability behaviour of the droplet at all. First, the axisymmetric case is considered allowing no variations of quantities around the vertical axis which is the pronounced axis due to the symmetry of the levitating or heating magnetic fields. Typical mean flow strucures are calculated depending on the skin-depth parameter and the magnetic field interaction parameter for the two cases of a uniform or linear magnetic field. The stability of this axisymmetric flow is analyzed showing that those flow fields become unstable only at high numbers of the magnetic field interaction parameter N. This is in contrast to a few experimental observations indicating that oscillatory motions take place already at much lower values of N.
Therefore, the three-dimensional stability of the flow is investigated. This stability analysis is performed using spectral methods. As expected the first instability is characterized by an azimuthal wave number of one. But it is found that the most dangerous three-dimensional instability occures at relatively small interaction parameters being by orders of magnitude lower than the critical values of the axisymmetric case.