The shrinking instability of toroidal liquid droplets in the Stokes flow regime

TL;DR

This paper investigates the unique shrinking instability of toroidal liquid droplets in Stokes flow, deriving analytical expressions for internal pressure, flow fields, and shrinking rates, revealing the dominant instability mode for fat tori.

Contribution

It provides the first analytical description of the pressure distribution and flow dynamics associated with the shrinking instability of toroidal droplets in the Stokes regime.

Findings

Identified a dominant shrinking instability specific to toroidal topology.

Derived an analytical expression for internal pressure distribution.

Calculated the shrinking rate of fat toroidal droplets using energy conservation.

Abstract

We analyze the stability and dynamics of toroidal liquid droplets. In addition to the Rayleigh instabilities akin to those of a cylindrical droplet there is a shrinking instability that is unique to the topology of the torus and dominates in the limit that the aspect ratio is near one (fat tori). We first find an analytic expression for the pressure distribution inside the droplet. We then determine the velocity field in the bulk fluid, in the Stokes flow regime, by solving the biharmonic equation for the stream function. The flow pattern in the external fluid is analyzed qualitatively by exploiting symmetries. This elucidates the detailed nature of the shrinking mode and the swelling of the cross-section following from incompressibility. Finally the shrinking rate of fat toroidal droplets is derived by energy conservation.