Oscillations of weakly viscous conducting liquid drops in a strong magnetic field

Abstract

We analyse small-amplitude oscillations of a weakly viscous electrically conducting liquid drop in a strong uniform DC magnetic field. An asymptotic solution is obtained showing that the magnetic field does not affect the shape eigenmodes, which remain the spherical harmonics as in the non-magnetic case. Strong magnetic field, however, constrains the liquid flow associated with the oscillations and, thus, reduces the oscillation frequencies by increasing effective inertia of the liquid. In such a field, liquid oscillates in a twodimensional (2D) way as solid columns aligned with the field. Two types of oscillations are possible: longitudinal and transversal to the field. Such oscillations are weakly damped by a strong magnetic field -the stronger the field, the weaker the damping, except for the axisymmetric transversal and inherently 2D modes. The former are overdamped because of being incompatible with the incompressibility constraint, whereas the latter are not affected at all because of being naturally invariant along the field. Since the magnetic damping for all other modes decreases inversely with the square of the field strength, viscous damping may become important in a sufficiently strong magnetic field. The viscous damping is found analytically by a simple energy dissipation approach which is shown for the longitudinal modes to be equivalent to a much more complicated eigenvalue perturbation technique. This study provides a theoretical basis for the development of new measurement methods of surface tension, viscosity and the electrical conductivity of liquid metals using the oscillating drop technique in a strong superimposed DC magnetic field.

Levitated Liquid Dynamics in Reduced Gravity and Gravity-Compensating Magnetic Fields

“…The frequency of the L = 3 mode is reduced about Ö3 times relative to the nonmagnetic case and closely matches the theoretical value. 7 The flow is purely laminar as can be seen from the lack of mixing that gives the radiationcooling-dominated temperature distribution in Fig. 7.…”

“…Some modes of surface motion are modified in the presence of strong uniform field. 6,7 The damping of this oscillation is relatively low and persists even at even higher magnetic fields.…”

“…The motions of an electrically conducting levitated droplet are expected to be damped in a strong DC magnetic field; however, some types of movement are not affected in particular field configurations. In the microgravity case (g = 0) and a uniform vertical magnetic field, the asymptotic solution 7 shows that odd axisymmetric oscillation modes are very moderately damped and the frequencies are reduced significantly. It is not immediately clear if this behavior will stay in the presence of terrestrial gravity and the gradient magnetic field required for the magnetic levitation.…”

“…6 The flow in typical melting conditions is approaching the conditions with laminar viscosity and heat transfer when a uniform DC magnetic field exceeds about 4-5 T. In addition to the modified damping, the electrically conducting liquid droplet in a high DC magnetic field behaves quite differently to the nonconducting one: some oscillation modes are damped completely, and other modes are persisting for long times or damped moderately. 7 On a larger scale, the magnetic suspension of liquid reactive materials, for example titanium alloys, can be used for high-quality castings. However, it is often difficult to achieve the required superheat in the melt with traditional ''cold'' crucible-type furnaces due to a partial contact with the water-cooled copper walls.…”

Shape oscillations of an electrically charged diamagnetically levitated droplet

“…Although a strong magnetic field is expected to damp the motion of a conducting liquid through the field via the Lorentz force 30 , the conductivity of the distilled water used here is too small for any effect on the oscillation frequencies, or damping of the oscillations to be observed.…”