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Thermal rupture of a free liquid sheet

Published online by Cambridge University Press:  14 February 2018

G. Kitavtsev Open the ORCID record for G. Kitavtsev [Opens in a new window] ,
M. A. Fontelos  and
J. Eggers
G. Kitavtsev*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
M. A. Fontelos
Affiliation:
Instituto de Ciencias Matemáticas, (ICMAT, CSIC-UAM-UCM-UC3M), C/ Serrano 123, 28006 Madrid, Spain
J. Eggers
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider a free liquid sheet, taking into account the dependence of surface tension on the temperature or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi-one-dimensional thickness, velocity and temperature profiles in the pinch region in terms of similarity solutions, which possess a universal structure. Our analytical description agrees quantitatively with numerical simulations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Micro-/Nano-fluid dynamics: Microfluidics Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 840 , 10 April 2018 , pp. 555 - 578
Copyright
© 2018 Cambridge University Press 

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References

Bertozzi, A. L., Brenner, M. P., Dupont, T. F. & Kadanoff, L. P. 1994 Singularities and similarities in interface flows. In Applied Mathematics Series (ed. Sirovich, L.), vol. 100, p. 155. Springer.Google Scholar
Boulton-Stone, J. M. & Blake, J. R. 1993 Gas-bubbles bursting at a free surface. J. Fluid Mech. 254, 437–466.Google Scholar
Bowen, M. & Tilley, B. S. 2013 On self-similar thermal rupture of thin liquid sheets. Phys. Fluids 25, 102105.Google Scholar
Burton, J. C. & Taborek, P. 2007 2D inviscid pinch-off: an example of self-similarity of the second kind. Phys. Fluids 19, 102109.Google Scholar
Champougny, L., Rio, E., Restagno, F. & Scheid, B. 2017 The break-up of free films pulled out of a pure liquid bath. J. Fluid Mech. 811, 499–524.Google Scholar
Clasen, C., Eggers, J., Fontelos, M. A., Li, J. & McKinley, G. H. 2006 The beads-on-string structure of viscoelastic jets. J. Fluid Mech. 556, 283–308.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Duchemin, L., Popinet, S., Josserand, C. & Zaleski, S. 2002 Jet formation in bubbles bursting at a free surface. Phys. Fluids 14, 3000–3008.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.Google Scholar
Erneux, T. & Davis, S. H. 1993 Nonlinear rupture of free films. Phys. Fluids A 5 (5), 1117–1122.Google Scholar
Feng, J., Roché, M., Vigolo, D., Arnaudov, L. N., Stoyanov, S. D., Gurkov, T. D., Tsutsumanova, G. G. & Stone, H. A. 2014 Nanoemulsions obtained via bubble-bursting at a compound interface. Nat. Phys. 10, 606–612.Google Scholar
Fontelos, M., Kitavtsev, G. & Taranets, R.2017 Asymptotic decay and non-rupture of viscous sheets. arXiv:1711.10533.Google Scholar
Glasner, K., Otto, F., Rump, T. & Slepjev, D. 2008 Ostwald ripening of droplets: the role of migration. Eur. J. Appl. Maths 20 (1), 1–67.Google Scholar
Gordillo, J. M., Sevilla, A., Rodríguez-Rodríguez, J. & Martínez-Bazán, C. 2005 Axisymmetric bubble pinch-off at high Reynolds numbers. Phys. Rev. Lett. 95, 194501.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1993 The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A 5, 58–68.Google Scholar
Kitavtsev, G. 2014 Coarsening rates for the dynamics of slipping droplets. Eur. J. Appl. Maths 25 (1), 83–115.Google Scholar
Kitavtsev, G. & Wagner, B. 2010 Coarsening dynamics of slipping droplets. J. Engng Maths 66, 271–292.Google Scholar
Lamstaes, C. & Eggers, J. 2017 Arrested bubble rise in a narrow tube. J. Stat. Phys. 167, 656.CrossRefGoogle Scholar
Larson, R. G. 2014 Transport and deposition patterns in drying sessile droplets. AIChE J. 60, 1538–1571.Google Scholar
Lhuissier, H. & Villermaux, E. 2011 Bursting bubble aerosols. J. Fluid Mech. 696, 5–44.Google Scholar
Matar, O. K. 2002 Nonlinear evolution of thin free viscous films in the presence of soluble surfactant. Phys. Fluids 14, 4216.Google Scholar
Matsuuchi, K. 1976 Instability of thin liquid sheet and its breakup. J. Phys. Soc. Japan 41, 1410–1416.Google Scholar
Néel, B. & Villermaux, E. 2018 The spontaneous puncture of thick liquid films. J. Fluid Mech. 838, 192–221.Google Scholar
Peschka, D.2008 Self-similar rupture of thin liquid films with slippage. PhD thesis, Humboldt University of Berlin.Google Scholar
Peschka, D., Münch, A. & Niethammer, B. 2010 Thin-film rupture for large slip. J. Engng Maths 66, 33–51.Google Scholar
Pugh, M. C. & Shelley, M. J. 1998 Singularity formation in thin jets with surface tension. Commun. Pure Appl. Maths 51, 733–795.Google Scholar
Rowlinson, J. S. & Widom, B. 1982 Molecular Theory of Capillarity. Clarendon.Google Scholar
Seemann, R., Herminghaus, S. & Jacobs, K. 2001 Gaining control of pattern formation of dewetting liquid films. J. Phys.: Condens. Matter 13, 4925.Google Scholar
Spiel, D. E. 1998 On the births of film drops from bubbles bursting on seawater surfaces. J. Geophys. Res. 103, 907–924.Google Scholar
Stone, H. A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2, 111–112.Google Scholar
Tammisola, O., Sasaki, A., Lundell, F., Matsubara, M. & Söderberg, L. D. 2011 Stabilizing effect of surrounding gas flow on a plane liquid sheet. J. Fluid Mech. 672, 5–32.CrossRefGoogle Scholar
Thoroddsen, S. T., Thoraval, M.-J., Takehara, K. & Etoh, T. G. 2012 Micro-bubble morphologies following drop impacts onto a pool surface. J. Fluid Mech. 708, 469–479.Google Scholar
Tilley, B. S. & Bowen, M. 2005 Thermocapillary control of rupture in thin viscous fluid sheets. J. Fluid Mech. 541, 399–408.CrossRefGoogle Scholar
Vrij, A. 1966 Possible mechanism for the spontaneous rupture of thin, free liquid films. Discuss. Faraday Soc. 42, 23.Google Scholar
Wu, J. 1981 Evidence of sea spray produced by bursting bubbles. Science 212, 324–326.Google Scholar