Stability and Bifurcation of a Soap Film Spanning a Flexible Loop
- 387 Downloads
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.
KeywordsSurface tension Flexural rigidity Inextensibility Euler–Lagrange equations Second variation condition Plateau’s problem Thread problem Closed-curve problem
Mathematics Subject Classification (2010)49Q10 53A04 53A05 53A10 53A25 53C80 53Z05
- 7.Plateau, J.A.F.: Recherches expérimentales et théorique sur les figures d’équilibre d’une masse liquide sans pesanteur. Mém. Acad. R. Sci. Lett. Beaux-Arts Belg. 23, 1–151 (1849) Google Scholar
- 8.Singer, D.A.: Lectures on elastic curves and rods. In: Garay, O.J., García-Río, E., Vázquez-Lorenzo, R. (eds.) Curvature and Variational Modeling in Physics and Biophysics. Conference Proceedings of the American Institute of Physics, vol. 1002, pp. 3–32 (2008) Google Scholar
- 13.Chen, Y.-C.: Singularity theory and nonlinear bifurcation analysis. In: Fu, Y.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications. Cambridge University Press, Cambridge (2001) Google Scholar