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Viscous sheet retraction

Published online by Cambridge University Press:  10 May 2009

NIKOS SAVVA  and
JOHN W. M. BUSH
NIKOS SAVVA
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
JOHN W. M. BUSH*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
‡E-mail address for correspondence: [email protected]
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Abstract

We present the results of a combined theoretical and numerical investigation of the rim-driven retraction of flat fluid sheets in both planar and circular geometries. Particular attention is given to the influence of the fluid viscosity on the evolution of the sheet and its bounding rim. In both geometries, after a transient that depends on the sheet viscosity and geometry, the film edge eventually attains the Taylor–Culick speed predicted on the basis of inviscid theory. The emergence of this result in the viscous limit is rationalized by consideration of both momentum and energy arguments. We first consider the planar geometry considered by Brenner & Gueyffier (Phys. Fluids, vol. 11, 1999, p. 737) and deduce new analytical expressions for the speed of the film edge at the onset of rupture and the evolution of the maximum film thickness for viscous films. In order to consider the expansion of a circular hole, we develop an appropriate lubrication model that predicts the form of the early stage dynamics of film rupture. Simulations of a broad range of flow parameters confirm the importance of geometry on the dynamics, verifying the exponential hole growth reported in early experimental studies. We demonstrate the sensitivity of the initial retraction speed on the film profile, and so suggest that the anomalous rate of retraction reported in these experiments may be attributed in part to geometric details of the puncture process.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 626 , 10 May 2009 , pp. 211 - 240
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
Bayvel, L. & Orzechowski, Z. 1993 Liquid Atomization. Taylor–Francis.Google Scholar
Bremond, N. & Villermaux, E. 2006 Atomization by jet impact. J. Fluid Mech. 549, 273–306.CrossRefGoogle Scholar
Brenner, M. P. & Gueyffier, D. 1999 On the bursting of viscous sheets. Phys. Fluids 11, 737–739.CrossRefGoogle Scholar
Buguin, A., Vovelle, L. & Brochard-Wyart, F. 1999 Shocks in inertial dewetting. Phys. Rev. Lett. 83, 1183–1186.CrossRefGoogle Scholar
Bush, J. W. M. & Hasha, A. E. 2004 On the collision of laminar jets: fluid chains and fishbones. J. Fluid Mech. 511, 285–310.CrossRefGoogle Scholar
Chepushtanova, S. V. & Kliakhandler, I. L. 2007 Slow rupture of viscous films between parallel needles. J. Fluid Mech. 573, 297–310.CrossRefGoogle Scholar
Culick, F. E. C. 1960 Comments on a ruptured soap film. J. Appl. Phys. 31, 1128–1129.CrossRefGoogle Scholar
Dalnoki-Veress, K., Nickel, B. G., Roth, C. & Dutcher, J. R. 1999 Hole formation and growth in freely standing polystyrene films. J. Fluid Mech. 59 (2), 2153–2156.Google Scholar
Debrégeas, G., de Gennes, G. P. & Brochard-Wyart, F. 1998 The life and death of viscous “bare” bubbles. Science 279, 1704–1707.CrossRefGoogle Scholar
Debrégeas, G., Martin, P. & Brochard-Wyart, F. 1995 Viscous bursting of suspended films. Phys. Rev. Lett. 75, 3886–3889.CrossRefGoogle ScholarPubMed
Deyrail, Y., Mesri, Z. El, Huneault, M., Zeghloul, A. & Bousmina, M. 2007 Analysis of morphology development in immiscible newtonian polymer mixtures during shear flow. J. Rheol. 51, 781–797.CrossRefGoogle Scholar
Dupré, M. A. 1867 Sixième memoire sur la theorie méchanique de la chaleur. Ann. Chim. Phys. 4 (11), 194–220.Google Scholar
Eggers, J. & Brenner, M. P. 2000 Spinning jets. In Proceedings of the IUTAM Symposium on Nonlinear Waves in Multi-Phase Flow (ed. Chang, H.-C.), pp. 185–194. Kluwer.CrossRefGoogle Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.CrossRefGoogle Scholar
Fullana, G. M. & Zaleski, S. 1999 Stability of a growing rim in a liquid sheet of uniform thickness. Phys. Fluids 11, 952–954.CrossRefGoogle Scholar
de Gennes, P. G., Brochart-Wyart, F. & Quéré, D. 2003 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.Google Scholar
Keller, J. B. 1983 Breaking of liquid films and threads. Phys. Fluids 26, 3451–3453.CrossRefGoogle Scholar
Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Math. 43, 268–277.CrossRefGoogle Scholar
Kelley, C. T. 1995 Iterative Methods for Linear and Nonlinear Equations. SIAM.CrossRefGoogle Scholar
Kelley, C. T. 2003 Solving Nonlinear Equations with Newton's Method. SIAM.CrossRefGoogle Scholar
Knoll, D. A. & Keyes, D. E. 2004 Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comp. Phys. 193, 357–397.CrossRefGoogle Scholar
Lefebvre, A. H. 1989 Liquid Atomization. Hemisphere.Google Scholar
McEntee, W. R. & Mysels, K. J. 1969 The bursting of soap films. I. An experimental study. J. Phys. Chem. 73, 3018–3028.CrossRefGoogle Scholar
Miyamoto, K. & Katagiri, Y. 1997 Curtain coating. In Liquid Film Coating (ed. Kistler, S. F. & Schweizer, P. M.), pp. 463–494. Chapman–Hall.CrossRefGoogle Scholar
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of contracting filaments. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
Pandit, A. B. & Davidson, J. F. 1990 Hydrodynamics of the rupture of thin liquid films. J. Fluid Mech. 212, 11–24.CrossRefGoogle Scholar
Pomeau, Y. & Villermaux, E. 2006 Two hundred years of capillarity research. Phys. Today 59, 39–44.CrossRefGoogle Scholar
Ranz, W. E. 1950 Some experiments on the dynamics of liquid films. J. Appl. Phys. 30, 1950–1955.CrossRefGoogle Scholar
Rayleigh, Lord 1891 Some applications of photography. Nature 44, 249–254.CrossRefGoogle Scholar
Roisman, I. V., Horvat, K. & Tropea, C. 2006 Spray impact: rim transverse instability initiating fingering and splash, and description of secondary spray. Phys. Fluids 18, 102104.CrossRefGoogle Scholar
Roth, C. B., Deh, B., Nickel, B. G. & Dutcher, J. R. 2005 Evidence of convective constraint release during hole growth in freely standing polystyrene films at low temperatures. Phys. Rev. E 72 (2), 021802.CrossRefGoogle ScholarPubMed
Savva, N. 2007 Viscous fluid sheets. PhD thesis, Massachusetts Institute of Technology.Google Scholar
Song, M. & Tryggvason, G. 1999 The formation of thick borders on an initially stationary fluid sheet. Phys. Fluids 11, 2487–2493.CrossRefGoogle Scholar
Sünderhauf, G., Raszillier, H. & Durst, F. 2002 The retraction of the edge of a planar liquid sheet. Phys. Fluids 14, 198–208.CrossRefGoogle Scholar
Taylor, G. I. 1959 The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc. R. Soc. Lond. Ser. A 253, 313–321.Google Scholar
Taylor, G. I. & Michael, D. I. 1973 On making holes in a sheet of fluid. J. Fluid Mech. 58, 625–639.CrossRefGoogle Scholar
Villermaux, E. 2007 Fragmentation. Annu. Rev. Fluid Mech. 39, 419–446.CrossRefGoogle Scholar