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Simulation of a Soap Film Möbius Strip Transformation

Published online by Cambridge University Press:  07 September 2017

Yongsam Kim*
Affiliation:
Department of Mathematics, Chung-Ang University, Dongjakgu Heukseokdong, Seoul 156-756, Korea
Ming-Chih Lai*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan
Yunchang Seol*
Affiliation:
National Center for Theoretical Sciences, No. 1, Sec. 4, Road. Roosevelt, National Taiwan University, Taipei 10617, Taiwan
*
*Corresponding author. Email addresses: kimy@cau.ac.kr (Y. Kim), mclai@math.nctu.edu.tw (M.-C. Lai), ycseol@ntu.edu.tw (Y. Seol)
*Corresponding author. Email addresses: kimy@cau.ac.kr (Y. Kim), mclai@math.nctu.edu.tw (M.-C. Lai), ycseol@ntu.edu.tw (Y. Seol)
*Corresponding author. Email addresses: kimy@cau.ac.kr (Y. Kim), mclai@math.nctu.edu.tw (M.-C. Lai), ycseol@ntu.edu.tw (Y. Seol)
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Abstract

If the closed wire frame of a soap film having the shape of a Möbius strip is pulled apart and gradually deformed into a planar circle, the soap film transforms into a two-sided orientable surface. In the presence of a finite-time twist singularity, which changes the linking number of the film's Plateau border and the centreline of the wire, the topological transformation involves the collapse of the film toward the wire. In contrast to experimental studies of this process reported elsewhere, we use a numerical approach based on the immersed boundary method, which treats the soap film as a massless membrane in a Navier-Stokes fluid. In addition to known effects, we discover vibrating motions of the film arising after the topological change is completed, similar to the vibration of a circular membrane.

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Asmar Nakhle, H., Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall (2005).Google Scholar
[2] Biance, A.-L., Cohen-Addad, S. and Höhler, R., Topological transition dynamics in a strained bubble cluster, Soft Matter 5, 46724679 (2009).CrossRefGoogle Scholar
[3] Boudaoud, A., Partíco, P. and Amar Ben, M., The helicoid versus catenoid: Geometrically induced bifurcations, Phys. Rev. Lett. 83, 38363839 (1999).CrossRefGoogle Scholar
[4] Brakke, K., The surface Evolver, Exp. Math. 1, 141165 (1992).CrossRefGoogle Scholar
[5] Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., Höhler, R., Pitois, O., Rouyer, F. and Saint-Jalmes, A., Foams: Structure and Dynamics, Oxford University Press (2013).CrossRefGoogle Scholar
[6] Cecil, T., A numerical method for computing minimal surfaces in arbitrary dimension, J. Comput. Phys. 206, 650660 (2005).CrossRefGoogle Scholar
[7] Chen, Y.-J. and Steen, P.H., Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge, J. Fluid. Mech. 341, 245267 (1997).CrossRefGoogle Scholar
[8] Chopp, D.L., Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106, 7791 (1993).CrossRefGoogle Scholar
[9] Colding, T.H. and Minicozzi, W.P. II, Shapes of embedded minimal surfaces, Proc. Nat. Acad. Sci. USA 103, 38363839 (2006).CrossRefGoogle ScholarPubMed
[10] Concus, P., Numerical solution of the minimal surface equation., Math. Comp. 21, 340350 (1967).CrossRefGoogle Scholar
[11] Courant, R., Soap film experiments with minimal surfaces, Am. Math. Mon. 47, 167174 (1940).Google Scholar
[12] Douglas, J., A method of the numerical solution of the Plateau problem, Ann. Math. 29, 180188 (1928).CrossRefGoogle Scholar
[13] Douglas, J., The mapping theorem of Koebe and the problem of Plateau, J. Math. Phys. 10 (1931) 106130.CrossRefGoogle Scholar
[14] Douglas, J., Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33, 263321 (1931).CrossRefGoogle Scholar
[15] Durand, M. and Stone, H.A., Relaxation time of the topological T1 process in a two-dimensional foam, Phys. Rev. Lett. 97, 226101 (2006).CrossRefGoogle Scholar
[16] Dziuk, G. and Hutchinson, J.E., The discrete Plateau problem: Algorithm and numerics, Math. Comp. 68, 123 (1999).CrossRefGoogle Scholar
[17] Eggers, J., Nonlinear dynamics and the breakup of free-surface flows, Rev. Mod. Phys. 69, 865929 (1997).CrossRefGoogle Scholar
[18] Eri, A. and Okumura, K., Bursting of a thin film in a confined geometry: Rimless and constant-velocity dewetting, Phys. Rev. E 82, 030601(R) (2010).Google Scholar
[19] Goldstein, R.E., Moffatt, H.K., Pesci, A.I. and Ricca, R.L., Soap-film Möbius strip changes topology with a twist singularity, Proc. Nat. Acad. Sci. USA 107, 2197921984 (2010).CrossRefGoogle Scholar
[20] Harbrecht, H., On the numerical solution of Plateau's problem, Appl. Numer. Math. 59, 27852800 (2009).CrossRefGoogle Scholar
[21] Harrison, J., Soap film solutions to Plateau's problem, J. Geom. Anal. 24, 271297 (2014).CrossRefGoogle Scholar
[22] Hinata, M., Shimasaki, M. and Kiyono, T., Numerical solution of Plateau's problem by a finite element method, Math. Comp. 28, 4560 (1974).Google Scholar
[23] Hu, W.-F. and Lai, M.-C., An unconditionally energy stable immersed boundary method with application to vesicle dynamics, East Asian J. Appl. Math. 3, 247262 (2013).CrossRefGoogle Scholar
[24] Hutzler, S., Saadatfar, M., van der Net, A., Weaire, D., Cox, S.J., The dynamics of a topological change in a system of soap films, Colloid. Surface A 323, 123131 (2008).CrossRefGoogle Scholar
[25] Jones, S.A. and Cox, S.J., The transition from three-dimensional to two-dimensional foam structures, Eur. Phys. J. E 34:82, (2011).CrossRefGoogle ScholarPubMed
[26] Kim, Y. and Peskin, C.S., 2-D parachute simulation by the immersed boundary method, SIAM J. Sci. Comput. 28, 22942312 (2006).CrossRefGoogle Scholar
[27] Kim, Y. and Lai, M.-C., Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys. 229, 48404853 (2010).CrossRefGoogle Scholar
[28] Kim, Y., Lai, M.-C. and Peskin, C.S., Numerical simulations of two-dimensional foam by the immersed boundary method, J. Comput. Phys. 229, 51945207 (2010).CrossRefGoogle Scholar
[29] Kim, Y., Lai, M.-C., Peskin, C.S. and Seol, Y., Numerical simulations of three-dimensional foam by the immersed boundary method, J. Comput. Phys. 269, 121 (2014).CrossRefGoogle Scholar
[30] Maggioni, F. and Ricca, R.L., Writhing and coiling of closed filaments, Proc. R. Soc. A 462, 31513166 (2006).CrossRefGoogle Scholar
[31] Le Merrer, M., Cohen-Addad, S. and Höhler, R., Bubble rearrangement duration in foams near the jamming point, Phys. Rev. Lett. 108, 188301 (2012).CrossRefGoogle ScholarPubMed
[32] Meeks, W.W. III and Yau, S.-T., The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179, 151168 (1982).CrossRefGoogle Scholar
[33] Nitsche, J.C.C., A new uniqueness theorem for minimal surfaces, Arch. Ration. Mech. Anal. 52, 319329 (1973).CrossRefGoogle Scholar
[34] Nitsche, M. and Steen, P.H., Numerical simulations of inviscid capillary pinchoff, J. Comput. Phys. 200, 299324 (2004).CrossRefGoogle Scholar
[35] Osher, S.J. and Sethian, J.A., Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulation, J. Comput. Phys. 79, 1249 (1988).CrossRefGoogle Scholar
[36] Osserman, R., A Survey of Minimal Surfaces, Dover Publications (2014).Google Scholar
[37] Peskin, C.S., Flow patterns around heart valves: A numerical method, J. Comput. Phys. 10, 252271 (1972).CrossRefGoogle Scholar
[38] Peskin, C.S. and McQueen, D.M., Three dimensional computational method for flow in the heart: Immersed elastic fibers in a viscous incompressible fluid, J. Comput. Phys. 81, 372405 (1989).CrossRefGoogle Scholar
[39] Peskin, C.S., The immersed boundary method, Acta Numerica, 11, 479517 (2002).CrossRefGoogle Scholar
[40] Pickover, C.A., The Möbius strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology, Basic Books (2007).Google Scholar
[41] Plateau, J., Statique Expérimentale et Théoretique des Liquides Soumis aux Seules Forces Moléculaires, Gauthier-Villars (1873).Google Scholar
[42] Rado, T., The problem of the least area and the problem of Plateau, Math. Z. 32, 763796 (1930).CrossRefGoogle Scholar
[43] Rado, T., On Plateau's problem, Ann. Math. 31, 457469 (1930).CrossRefGoogle Scholar
[44] Robinson, N.D. and Steen, P.H., Observations of singularity formation during the capillary collapse and bubble pinch-off a soap film bridge, J. Colloid. Interf. Sci. 241, 448458 (2001).CrossRefGoogle Scholar
[45] Saye, R.I. and Sethian, J.A., Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams, Science 340, 720724 (2013).CrossRefGoogle ScholarPubMed
[46] Taylor, J.E., Selected Works of Frederick J. Almgren, Jr., American Mathematical Society (1999).Google Scholar
[47] Tråsdahl, Ø. and Rønquist, E.M., High order numerical approximation of minimal surfaces, J. Comput. Phys. 230, 47954810 (2011).CrossRefGoogle Scholar
[48] Weaire, D. and Hutzler, S., The Physics of Foams, Oxford University Press (1999).Google ScholarPubMed
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