Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T00:04:45.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Simulation of a Soap Film Möbius Strip Transformation

Part of: Thin bodies, structures Incompressible viscous fluids

Published online by Cambridge University Press:  07 September 2017

Yongsam Kim ,
Ming-Chih Lai  and
Yunchang Seol
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)
Get access

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.

Keywords

Möbius stripminimal surfacetwist singularitytopological changeNavier-Stokes equationimmersed boundary method

MSC classification

Secondary: 68U20: Simulation 74K35: Thin films 76D05: Navier-Stokes equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 615 - 628
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 4672–4679 (2009).Google Scholar
[3] Boudaoud, A., Partíco, P. and Amar Ben, M., The helicoid versus catenoid: Geometrically induced bifurcations, Phys. Rev. Lett. 83, 3836–3839 (1999).CrossRefGoogle Scholar
[4] Brakke, K., The surface Evolver, Exp. Math. 1, 141–165 (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).Google Scholar
[6] Cecil, T., A numerical method for computing minimal surfaces in arbitrary dimension, J. Comput. Phys. 206, 650–660 (2005).Google 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, 245–267 (1997).Google Scholar
[8] Chopp, D.L., Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106, 77–91 (1993).Google Scholar
[9] Colding, T.H. and Minicozzi, W.P. II, Shapes of embedded minimal surfaces, Proc. Nat. Acad. Sci. USA 103, 3836–3839 (2006).Google Scholar
[10] Concus, P., Numerical solution of the minimal surface equation., Math. Comp. 21, 340–350 (1967).CrossRefGoogle Scholar
[11] Courant, R., Soap film experiments with minimal surfaces, Am. Math. Mon. 47, 167–174 (1940).Google Scholar
[12] Douglas, J., A method of the numerical solution of the Plateau problem, Ann. Math. 29, 180–188 (1928).CrossRefGoogle Scholar
[13] Douglas, J., The mapping theorem of Koebe and the problem of Plateau, J. Math. Phys. 10 (1931) 106–130.Google Scholar
[14] Douglas, J., Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33, 263–321 (1931).Google 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, 1–23 (1999).Google Scholar
[17] Eggers, J., Nonlinear dynamics and the breakup of free-surface flows, Rev. Mod. Phys. 69, 865–929 (1997).Google 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, 21979–21984 (2010).CrossRefGoogle Scholar
[20] Harbrecht, H., On the numerical solution of Plateau's problem, Appl. Numer. Math. 59, 2785–2800 (2009).CrossRefGoogle Scholar
[21] Harrison, J., Soap film solutions to Plateau's problem, J. Geom. Anal. 24, 271–297 (2014).Google Scholar
[22] Hinata, M., Shimasaki, M. and Kiyono, T., Numerical solution of Plateau's problem by a finite element method, Math. Comp. 28, 45–60 (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, 247–262 (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, 123–131 (2008).Google 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).Google Scholar
[26] Kim, Y. and Peskin, C.S., 2-D parachute simulation by the immersed boundary method, SIAM J. Sci. Comput. 28, 2294–2312 (2006).Google Scholar
[27] Kim, Y. and Lai, M.-C., Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method, J. Comput. Phys. 229, 4840–4853 (2010).Google 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, 5194–5207 (2010).Google 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, 1–21 (2014).Google Scholar
[30] Maggioni, F. and Ricca, R.L., Writhing and coiling of closed filaments, Proc. R. Soc. A 462, 3151–3166 (2006).Google 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).Google Scholar
[32] Meeks, W.W. III and Yau, S.-T., The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179, 151–168 (1982).CrossRefGoogle Scholar
[33] Nitsche, J.C.C., A new uniqueness theorem for minimal surfaces, Arch. Ration. Mech. Anal. 52, 319–329 (1973).Google Scholar
[34] Nitsche, M. and Steen, P.H., Numerical simulations of inviscid capillary pinchoff, J. Comput. Phys. 200, 299–324 (2004).Google 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, 12–49 (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, 252–271 (1972).Google 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, 372–405 (1989).Google Scholar
[39] Peskin, C.S., The immersed boundary method, Acta Numerica, 11, 479–517 (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, 763–796 (1930).CrossRefGoogle Scholar
[43] Rado, T., On Plateau's problem, Ann. Math. 31, 457–469 (1930).Google 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, 448–458 (2001).Google Scholar
[45] Saye, R.I. and Sethian, J.A., Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams, Science 340, 720–724 (2013).Google Scholar
[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, 4795–4810 (2011).Google Scholar
[48] Weaire, D. and Hutzler, S., The Physics of Foams, Oxford University Press (1999).Google Scholar