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Stability and Bifurcation of a Soap Film Spanning a Flexible Loop

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

  1. See, for instance, Proposition 2.1 of Gurtin and Murdoch [14].

References

  1. Dierkes, U., Hildebrandt, S., Tromba, A.J.: Regularity of Minimal Surfaces, 2nd edn. Springer, Berlin (2010)

    MATH  Google Scholar 

  2. Dierkes, U., Hildebrandt, S., Sauvigny, F.: Minimal Surfaces, 2nd edn. Springer, Berlin (2010)

    Book  Google Scholar 

  3. Bernatzki, F., Ye, R.: Minimal surfaces with an elastic boundary. Ann. Glob. Anal. Geom. 19, 1–9 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Giomi, L., Mahadevan, L.: Minimal surfaces bounded by elastic lines. Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468, 1851–1864 (2012)

    ADS  Article  MathSciNet  Google Scholar 

  5. Bernatzki, F.: Mass-minimizing currents with an elastic boundary. Manuscr. Math. 93, 1–20 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bernatzki, F.: On the existence and regularity of mass-minimizing currents with an elastic boundary. Ann. Glob. Anal. Geom. 15, 379–399 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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 

  9. do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976)

    MATH  Google Scholar 

  10. Efimov, N.V.: Some problems in the theory of space curves. Usp. Mat. Nauk 2, 193–194 (1947)

    MathSciNet  Google Scholar 

  11. Fenchel, W.: On the differential geometry of closed space curves. Bull. Am. Math. Soc. 57, 44–54 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, Berlin (1985)

    Book  MATH  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 

  14. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    MATH  MathSciNet  Google Scholar 

  15. Julicher, F.: Supercoiling transitions of closed DNA. Phys. Rev. E 49, 2429–2435 (1994)

    ADS  Article  Google Scholar 

  16. Dichmann, D.J., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mesirov, J.P., Schulten, K., Sumners, D.W. (eds.) Mathematical Approaches to Biomolecular Structure and Dynamics, pp. 71–113. Springer, Berlin (1996)

    Chapter  Google Scholar 

  17. Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 173–221 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ericksen, J.L.: The thermo-kinetic view of elastic stability theory. Int. J. Solids Struct. 2, 573–580 (1966)

    Article  Google Scholar 

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Correspondence to Eliot Fried.

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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

Mathematics Subject Classification (2010)

  • 49Q10
  • 53A04
  • 53A05
  • 53A10
  • 53A25
  • 53C80
  • 53Z05