Stability and bifurcation of a soap film spanning an elastic loop

Abstract

The Euler--Plateau problem, proposed by \cite{gm}, concerns a soap film spanning a flexible loop. The shapes of the film and the loop are determined by the interactions between the two components. In the present work, the Euler--Plateau problem is reformulated to yield a boundary-value problem for a vector field that parameterizes both the spanning surface and the bounding loop. Using the first and second variations of the relevant free-energy functional, detailed bifurcation and stability analyses are performed. For spanning surface with energy density $σ$ and a bounding loop with length $2πR$ and bending rigidity $a$, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at $σR^3/a=3$, confirming a result obtained previously via an energy comparison. Other bifurcation solution branches, including those emanating from the flat circular solution branch to nonplanar solution branches, are also shown to be unstable.