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Self-similar coalescence of clean foams

Published online by Cambridge University Press:  15 April 2013

Peter S. Stewart  and
Stephen H. Davis
Peter S. Stewart*
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, The University of Oxford, Oxford OX1 3LB, UK
Stephen H. Davis
Affiliation:
Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208, USA
*
†Email address for correspondence: peter.stewart@maths.ox.ac.uk
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Abstract

We consider the stability of a planar gas–liquid foam with low liquid fraction, in the absence of surfactants and stabilizing particles. We adopt a network modelling approach introduced by Stewart & Davis (J. Rheol., vol. 56, 2012, p. 543), treating the gas bubbles as polygons, the accumulation of liquid at the bubble vertices (Plateau borders) as dynamic nodes and the liquid bridges separating the bubbles as uniformly thinning free films; these films can rupture due to van der Waals intermolecular attractions. The system is initialized as a periodic array of equally pressurized bubbles, with the initial film thicknesses sampled from a normal distribution. After an initial transient, the first film rupture initiates a phase of dynamic rearrangement where the bubbles rapidly coalesce, evolving toward a new quasi-equilibrium. We present Monte Carlo simulations of this coalescence process, examining the time intervals over which large-scale rearrangement occurs as a function of the model parameters. In addition, we show that when this time interval is rescaled appropriately, the evolution of the normalized number of bubbles is approximately self-similar.

JFM classification

Complex Fluids: Complex Fluids Complex Fluids: Foams Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 645 - 664
Copyright
©2013 Cambridge University Press

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References

Anderson, A. M., Brush, L. N. & Davis, S. H. 2010 Foam mechanics: spontaneous rupture of thinning liquid films with plateau borders. J. Fluid Mech. 658, 63–88.CrossRefGoogle Scholar
Banhart, J. 2001 Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater. Sci. 46 (6), 559–632.Google Scholar
Banhart, J., Stanzick, H., Helfen, L. & Baumbach, T. 2001 Metal foam evolution studied by synchrotron radioscopy. Appl. Phys. Lett. 78, 1152–1154.Google Scholar
Bolton, F. & Weaire, D. 1992 The effects of plateau borders in the two-dimensional soap froth. II. General simulation and analysis of rigidity loss transition. Phil. Mag. B 65, 473–487.CrossRefGoogle Scholar
Breward, C. J. W. & Howell, P. D. 2002 The drainage of a foam lamella. J. Fluid Mech. 458, 379–406.Google Scholar
Brush, L. N. & Davis, S. H. 2005 A new law of thinning in foam dynamics. J. Fluid Mech. 534, 227–236.Google Scholar
Burnett, G. D., Chae, J. J., Tam, W. Y., de Almeida, R. M. C. & Tabor, M. 1995 Structure and dynamics of breaking foams. Phys. Rev. E 51 (6), 5788–5796.Google Scholar
Chae, J. J. & Tabor, M. 1997 Dynamics of foams with and without wall rupture. Phys. Rev. E 55 (1), 598–610.Google Scholar
Erneux, T. & Davis, S. H. 1993 Nonlinear rupture of free films. Phys. Fluids 5, 1117–1121.CrossRefGoogle Scholar
Garcia-Moreno, F., Rack, A., Helfen, L., Baumbach, T., Zabler, S., Babscan, N., Banhart, J., Martin, T., Ponchut, C. & Di Michiel, M. 2008 Fast processes in liquid metal foams investigated by high-speed synchrotron X-ray microradioscopy. Appl. Phys. Lett. 92, 134104.Google Scholar
Gratton, M. B. & Witelski, T. P. 2008 Coarsening of unstable thin films subject to gravity. Phys. Rev. E 77 (1), 16301.Google Scholar
Gratton, M. B. & Witelski, T. P. 2009 Transient and self-similar dynamics in thin film coarsening. Physica D 38, 2380–2394.Google Scholar
Hamsy, A., Paredes, R., Sonneville-Aubrun, O., Cabane, B. & Botet, R. 1999 Dynamical transition in a model for dry foams. Phys. Rev. Lett. 82 (16), 3368–3371.Google Scholar
Kern, N., Weaire, D., Martin, A., Hutzler, S. & Cox, S. J. 2004 The two-dimensional viscous froth model for foam dynamics. Phys. Rev. E 70, 041411.Google Scholar
van Lengerich, H. B., Vogel, M. J. & Steen, P. H. 2010 Coarsening of capillary drops coupled by conduit networks. Phys. Rev. E 82 (6), 066312.Google Scholar
Okuzono, T. & Kawasaki, K. 1995 Intermittent flow behavior of random foams: a computer experiment on foam rheology. Phys. Rev. E 51, 1246–1253.Google Scholar
Stewart, P. S. & Davis, S. H. 2012 Dynamics and stability of metallic foams: network modelling. J. Rheol. 56, 543–574.CrossRefGoogle Scholar
Vandewalle, N., Lentz, J. F., Dorbolo, S. & Brisbois, F. 2001 Avalanches of popping bubbles in collapsing foams. Phys. Rev. Lett. 86, 179–182.Google Scholar
Weaire, D., Bolton, F., Herdtle, T. & Aref, H. 1992 The effect of strain upon the topology of a soap froth. Phil. Mag. Lett. 66 (6), 293–299.Google Scholar
Weaire, D. & Hutzler, S. 1999 The Physics of Foams. Oxford University Press.Google Scholar

Stewart and Davis supplementary movie

Animation of bubble coalescence in a foam initially composed of 72 uniformly pressurized bubbles, corresponding to the snapshots shown in figure 4 and the time-traces shown in figures 5 and 6 in the paper. The movie illustrates how the breakage of the first film triggers a large-scale topological rearrangement of the foam, evolving toward a new quasi-equilibrium composed of only a few bubbles.

Download Stewart and Davis supplementary movie(Video)
Video 7.5 MB