Observations on the dynamics of electromagnetically levitated liquid metals and alloys at elevated temperatures
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I.
INTRODUCTION
ELECTROMAGNETIC fields are commonly used to levitate and melt metal specimens. Applications include metal shaping, welding, thermophysical property measurements, melt processing in a containerless environment avoiding crucible contamination, continuous casting of metals, and more recently, materials processing in the reduced gravity environment of space. The technique of electromagnetic levitation was patented by Muck [11 in 1923 and used experimentally by Okress et al. ~21for preparing high purity titanium. Since then, many attempts, both analytical and experimental, have been made to understand and quantitatively describe the phenomena involved in this process. Mestel I3] and Sneyd and MoffatI41 have investigated the fluid dynamical aspects of the levitation melting process. The stabilities of liquid metal-gas interfaces subjected to electromagnetic fields have been studied analytically and experimentally by McHale and MelchercSj and by Gamier and Moreau I61for a planar interface geometry. Busse ~71has studied analytically the effect of rotation on both the axisymmetric and non-axisymmetric modes of oscillation. He observed that the measured frequencies of the axisymmetric mode were different from the characteristic frequencies described by Lamb t81 due to the interaction of rotations with oscillations. In experiments involving levitation of spherical liquid metal droplets, sustained surface undulations were observed by the current authors. Shape oscillations of liquid droplets are caused by the restoring effect of surface tension, and the surface tension can be theoretically related to the characteristic oscillation frequency. However, in the case of electromagnetically levitated droplets, no such correlation seems to have been reported in the literature. The purpose of this paper is to report shape oscillations observed in electromagnetically levitated liquid metal droplets, and discuss available models in the light of the present experimental observations.
The free oscillations of a liquid droplet were first described by Lamb: t81 (0o.)2 = o'n(n -
1)(n + 2) (or3) ,
[1]
where o* is the characteristic oscillation frequency, or is the surface tension, p is the density, r is the spherical radius, and n is the mode of oscillation. This equation holds for a droplet oscillating in a medium of negligible viscosity and density. In terms of the droplet mass, we can rewrite Eq. [1] assuming a spherical drop shape: (0o.)2 = 4~rorn(n -
1)(n + 2) 3m '
[2]
where m is the mass of the droplet. On the other hand, Marstont91 has derived expressions for the frequency of acoustically driven axisymmetric shape oscillations about a spherical shape. Using an inviscid approximation, he obtained 1
,
1
0o~ = (0o*)~ - ~ t ~ V ~ * + ~ a ,
z
[3]
where 0ol is the driven frequency, co* is the Lamb frequency (Eq. [1]), and a is given by
(2n +
(IZ,lZoPiPo) |/2 1)z21'ZR[npo + (n + 1)p,][(lx,p,) v2 + (iZoPo)ln] "
[4] In Eq. [4] ~ and p refer to the viscosity and density, and the subscripts i and o refer to the i
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