Stability and Bifurcation of a Soap Film Spanning an Elastic Loop

IAM Seminar
January 21, 2013 11:00 pm

Speaker:  Prof. Eliot Fried, Mechanical Engineering, McGill University, Montreal, Quebec

URL for Speaker:

Location:  LSK 460

Intended Audience:  Public

We reformulate the Euler-Plateau problem of Giomi & Mahadevan to obtain a boundary-value problem for a vector field that parametrizes both the spanning surface and the bounding loop.  Using the first and second variations of the relevant free-energy functional, we perform detailed bifurcation and stability analyses.  For spanning surface with energy density $\sigma$ and a bounding loop with length $2\pi R$ and bending rigidity $a$, the first bifurcation, during which the spanning surface remains planar but the bounding loop becomes noncircular, occurs at $\sigma R^3/a=3$, confirming a result obtained previously via an energy comparison.  To provide a firm basis for the analysis of the subsequent bifurcation to out-of-plane configurations, we provide a complete characterization of all nontrivial planar solutions.