Keywords: oscillating drop technique, high magnetic field, liquid metal, frequency spectrum, magnetic damping, asymptotic analysis.

Shape oscillations of electromagnetically levitated metal droplets can be used to measure the surface tension and viscosity of liquid metals. Theoretically, the former determines the frequency while the latter accounts for the damping rate of the oscillations. Unfortunately, the measurements may strongly be disturbed by the electromagnetically-driven flow in the droplet. In order to reduce the strength of the AC magnetic field necessary for levitation and, thus, to minimise the internal flow, it is not unusual to conduct such experiments under the microgravity conditions during parabolic flights or on the board of space stations. This, however, makes such experiments rather expensive. A much cheaper alternative is conduct experiments under the terrestrial conditions in a sufficiently strong DC magnetic field which can suppress the undesirable internal flows driven by the AC magnetic field. In order to determine the surface tension and viscosity or the electrical conductivity one needs to relate the observed surface oscillations to the relevant thermophysical properties of the liquid. Such simple theoretical models exist for the oscillations of nearly spherical droplets without significant internal flow in the absence of magnetic field. Oscillations of a conducting droplet in a homogeneous magnetic field have been considered by Gailitis more than 40 years ago [1]. His analysis, which is carried out for arbitrary field strength but restricted to inviscid droplets, is formidably complex and incomplete that makes its practical application difficult.

In the present work, we analyse theoretically three-dimensional free oscillations of a viscous conducting droplet in a homogeneous magnetic field. Viscosity is assumed to be small but the magnetic field strong. This approach allows us to obtain a relatively simple asymptotic solution of the problem. First, we show that a strong magnetic field does not affect the shape of surface eigenmodes which remains the same as without the magnetic field, i.e. the spherical harmonics. The magnetic field, however, changes the internal flow associated with surface oscillations and, thus, the frequency of the eigenmodes. As the droplet oscillates in a strong magnetic field, the liquid moves in solid columns aligned with the field. Two types of such oscillations are possible: longitudinal and transversal to the magnetic field. Both of these oscillation modes, which are essentially three-dimensional, are weakly damped by the magnetic field. The magnetic damping decreases with the field strength. As a result, viscous damping becomes dominant in a sufficiently strong magnetic field. This does not apply to axisymmetric modes which are not compatible with the columnar liquid oscillations and, thus, are strongly over-damped by the magnetic field. The relaxation time of these axisymmetric transversal modes is related to the electrical conductivity.

This theoretical model provides a basis for the development of new measurement methods of surface tension, viscosity and electrical conductivity of liquid metals using the oscillating drop technique in a strong superimposed DC magnetic field.

1

A. Gailitis, "Oscillations of a conducting drop in a magnetic field", Magnetohydrodynamics, 2, 47-53, 1966.

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