Abstract
The Euler–Plateau problem, proposed by Giomi and Mahadevan in Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012), 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 a spanning surface with energy density σ and a bounding loop with length 2πR and flexural rigidity a, the first bifurcation, during which the spanning surface remains flat but the bounding loop becomes noncircular, occurs at R 3 σ/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.
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Notes
See, for instance, Proposition 2.1 of Gurtin and Murdoch [14].
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Dedicated to Roger Fosdick.
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Chen, Yc., Fried, E. Stability and Bifurcation of a Soap Film Spanning a Flexible Loop. J Elast 116, 75–100 (2014). https://doi.org/10.1007/s10659-013-9458-x
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DOI: https://doi.org/10.1007/s10659-013-9458-x
Keywords
- Surface tension
- Flexural rigidity
- Inextensibility
- Euler–Lagrange equations
- Second variation condition
- Plateau’s problem
- Thread problem
- Closed-curve problem