Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

Abstract

The shape of a weightless spinning liquid droplet is governed by the balance between the surface tension and centrifugal forces. The axisymmetric shape for slow rotation becomes unstable to a non-axisymmetric distortion above a critical angular velocity, beyond which the droplet progresses through a series of 2-lobed shapes. Theory predicts the existence of a family of 3-and 4-lobed equilibrium shapes at higher angular velocity. We investigate the formation of a triangular-shaped magnetically levitated water droplet, driven to rotate by the Lorentz force on an ionic current within the droplet. We also study equatorial traveling waves which give the droplet 3, 4 and 5-fold symmetry.In experiments published between 1863 and 1865, Plateau devised a way to study the behavior of liquids in the absence of gravity by suspending olive oil in a density-matched water/alcohol mixture [1]. In studies of a spinning oil droplet driven by a rotating shaft, he observed that the droplet shape, spherical at rest, became flattened at the poles as the droplet gained speed, whilst the equatorial diameter increased. Beyond a critical angular velocity, the droplet evolved into a spinning non-axisymmetric shape resembling a triaxial ellipsoid, which developed into a two-lobed shape with increasing speed. Plateau's experiments were inspired by the idea that the action of surface tension on the spinning droplet could model the influence of self-gravitation on the shape of spinning astronomical bodies [2]. It was later realized that the droplet model could provide insights into the behavior of atomic nuclei [2,3]. The analogies between the behavior of a liquid droplet and fundamental physics at both the astronomical and nuclear length scales has inspired some of the greatest mathematicians and physicists to study the instabilities of a rotating droplet (see [2]). Such analogies still attract great interest, as demonstrated by recent studies of the equilibrium shapes of relativistic rotating stellar masses [4,5] and of atomic nuclei [6].Although the simplicity of Plateau's technique for studying weightless fluids on Earth is attractive, comparison of the results with theory is complicated by shape deformations due to the viscous drag of the surrounding fluid. Here we avoid the problem of drag by using a strong magnetic field B to diamagnetically levitate centimetersized spinning water droplets in air. We have developed a 'liquid electric motor' technique to spin the droplet [7] and to realize a stable equilibrium droplet shape that has striking triangular symmetry in the plane of rotation.Brown and Scriven investigated the equilibrium shapes of a rigidly-rotating droplet theoretically using finite element analysis, grouping the shapes into 'families' according to their symmetry [8]. The 2-lobed shape family, which includes the ellipsoid-like shapes observed in Plateau's experiments and 2-lobed 'peanut' shapes observed in orbiting spacecraft [9,10] and in rolling 'liquid marbles' [11], branches from the axisymmetric family when the dimens...

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