Curvature driven motion of soap cells in toroidal Hele-Shaw cells

TL;DR

This paper studies how the curvature of confining plates affects the equilibrium shape and energy of soap bubbles, revealing curvature-driven motion and energy variations in toroidal Hele-Shaw cells.

Contribution

It introduces a model linking plate curvature to bubble energy, validated by simulations, and demonstrates curvature-induced bubble motion in toroidal geometries.

Findings

Surface tension energy is lower between spherical plates than flat plates.

Energy varies with local Gaussian curvature of the plates.

Curvature acts as a geometric potential driving bubble motion.

Abstract

We investigate the equilibrium properties of a single area-minimising bubble trapped between two narrowly-separated parallel curved plates. We begin with the simple case of a a bubble trapped between concentric spherical plates. We develop a model that shows that the surface tension energy of the bubble is lower when confined between spherical plates as compared to a bubble trapped between flat plates. We confirm our findings by comparing against Surface Evolver simulations. Next, we derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the the model we consider a bubble trapped between concentric tori. In the toroidal case we find that the sensitivity of the bubble's energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature.