Effects of surface tension and elasticity on critical points of the Kirchhoff-Plateau problem

TL;DR

This paper investigates how surface tension and elasticity influence the critical points of a modified Kirchhoff-Plateau problem, deriving equations and analyzing shape effects for different cross-sections.

Contribution

It introduces a new energy term to penalize shape changes, characterizes minimizers, and derives Euler-Lagrange equations for a planar version of the problem.

Findings

Existence of a unique critical point within physical parameters.

Derivation of Euler-Lagrange equations for different cross-sectional shapes.

Analysis of surface tension effects on equilibrium shapes.

Abstract

We introduce a modified Kirchhoff-Plateau problem adding an energy term to penalize shape modifications of the cross-sections appended to the elastic midline. In a specific setting, we characterize quantitatively some properties of minimizers. Indeed, choosing three different geometrical shapes for the cross-section, we derive Euler-Lagrange equations for a planar version of the Kirchhoff-Plateau problem. We show that in the physical range of the parameters, there exists a unique critical point satisfying the imposed constraints. Finally, we analyze the effects of the surface tension on the shape of the cross-sections at the equilibrium.