Solution of the Kirchhoff-Plateau problem

TL;DR

This paper proves the existence of equilibrium shapes for a liquid film spanning a flexible, thick filament modeled as a Kirchhoff rod, accounting for physical constraints like non-interpenetration and contact.

Contribution

It establishes a rigorous mathematical existence result for the equilibrium configurations of the Kirchhoff-Plateau problem with realistic physical constraints.

Findings

Existence of energy-minimizing equilibrium shapes proven.

Model accommodates contact points and finite thickness of the loop.

Results confirm physical plausibility of the model in experimental scenarios.

Abstract

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.