Effect of AC magnetic field on the damping of shape oscillationsof liquid metal droplets

Shape oscillations of electromagnetically-levitated metal droplets can be used to measure the surface tension and viscosity of liquid metals. The first determines the frequency while the second accounts for the attenuation rate of the oscillations. Until now the effect of the magnetic field on the frequency spectrum has been considered only in the perfect-conductor approximation where the action of the magnetic field reduces to the magnetic pressure on the liquid surface. In this approximation, magnetic field influences droplet oscillations in two ways. First, it causes a static deformation of the droplet. So the oscillations in the magnetic field occur about an aspheric basic shape. Second, the coupling between the shape variation and the magnetic field gives rise to a new oscillation mode called hydromagnetic surface wave. For small droplets magnetic field causes a slight perturbation of oscillations with both aforementioned effects resulting in the same order small corrections to the frequency spectrum of an inviscid spherical droplet. Magnetic field yields the same order relative correction also for the damping rate of oscillations of a viscous droplet. But there is no direct influence of the magnetic field on the damping rate of oscillations that is a serious drawback of the perfect-conductor approximation. Since for low-viscosity metal droplets the damping rate itself is small, even a weak direct effect of the magnetic field might be significant.
This work is dealt with the effect a high-frequency alternating magnetic field on the damping of the shape oscillations of viscous metal droplet. Conversely to the viscosity, high-frequency magnetic field is found to reduce the damping of the oscillations. We find the leading order solution of the damping rate by assuming the viscosity to be small but the frequency of the magnetic field to be high. The magnetic pressure is considered to be small with respect to the capillary pressure so that the static deformation of the droplet may be neglected. Besides, we assume the Reynolds number of the fluid flow driven by the magnetic field at finite skin depth in the droplet to be small so that the coupling between the base flow and shape oscillations may be neglected. Under these assumptions we formulate a theory for the case when both the magnetic field and oscillations are axisymmetric. The theory is applied to calculate the effect of a uniform magnetic field on the damping of the axisymmetric fundamental mode.