Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers

TL;DR

This paper investigates the shapes, stability, and bifurcation behavior of axisymmetric fluid membranes with resistance to bending, connecting two rings, revealing multiple solution branches, critical conditions, and differences from soap film stability.

Contribution

The study introduces a comprehensive analysis of axisymmetric membrane shapes, including multiple solution branches and stability properties, extending understanding beyond classical soap film models.

Findings

Multiple solution branches for membrane shapes as functions of ring separation.

Catenoid shape is the maximal separation at critical area.

Additional tether solutions exist for larger separations.

Abstract

We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.