Surface Instabilities and Patterning in Liquids: Exemplifications of the "Hairy Ball Theorem"

TL;DR

This paper applies the hairy ball theorem to analyze surface instabilities in liquids, demonstrating how zero velocity points relate to pattern formation during rapid evaporation, with implications for different topologies like tori.

Contribution

It introduces a novel application of the hairy ball theorem to liquid surface instabilities, linking topological constraints to observable patterning phenomena.

Findings

Zero velocity points are inherent in liquid surfaces with continuous tangential flow.

Surface patterning correlates with the theorem's predictions during evaporation.

Different topologies, such as tori, exhibit distinct instability patterns.

Abstract

Application of the "hairy ball theorem" to the analysis of the surface instabilities inherent for liquid/vapor interfaces is reported. When a continuous tangential velocity field exists on the surface of the liquid sample which is homeomorphic to a ball, zero velocity points will be necessarily present at the surface. The theorem is exemplified with the analysis of the instability occurring under the rapid evaporation of polymer solutions. Zero velocity points, accumulating pores, enable direct visualization of the instability. The patterning may be essentially different on the surface of a torus.