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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Eliot Fried
McGill University
Title: Stability and bifurcation of a soap film spanning an elastic loop

Abstract: The Euler--Plateau problem, proposed by Giomi $\&$ Mahadevan in 2013, 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 work described in this talk, which is joint with Yi-chao Chen, 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 a 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 flat but the bounding loop becomes noncircular, occurs at $\sigma R^3/a=3$, confirming a result obtained previously via an energy comparison. All other bifurcation solution branches emanating from the flat circular solution branch, including those to nonplanar solution branches, are found to be unstable. Finally, relevance to the modeling of discoidal high-density lipoproteins is discussed.

Date: Thursday, September 26, 2013
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer