Equilibrium shapes of two and three dimensional two-phase rotating fluid drops with surface tension: effects of inner drop displacement

Abstract

The shapes of rotating fluid drops held together by surface tension is an important field of study in fluid mechanics. Recently, experiments with micron-scale droplets of liquid helium have been undertaken and it has proven useful to compare the shapes of the resultant superfluid droplets with classical analogs. If the helium is a mixture of He3 and He4, two phases are present. In a recent paper, the shapes of rotating two phase fluid droplets were calculated where the inner drop was constrained to stay at the drop center. The outer shapes and dimensionless rotation rate-angular momentum relationships were shown to be similar to single phase drops provided that the density and surface tension scales were chosen appropriately. In the current paper, I investigate models in which the inner drop can displace from the centre. In order to simplify the analyses, two dimensional drops are first investigated. I show that the inner drop is unstable in the centre position if its density is greater than the outer density and that the inner drop will move towards the outer boundary of the drop in these cases. When the inner drop has a higher density than the outer drop, the moment of inertia of displaced inner drops is increased relative to centered drops and hence the kinetic energy is decreased. Shapes of two and three dimensional drops, rotation rate-angular momentum and kinetic and surface energy relationships are investigated for off-axis inner drops with parameters relevant to recent liquid He experiments.