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Response of a two-dimensional liquid foam to air injection: swelling rate, fingering and fracture

Published online by Cambridge University Press:  02 January 2013

Imen Ben Salem ,
Isabelle Cantat  and
Benjamin Dollet
Imen Ben Salem
Affiliation:
Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes 1, Campus Beaulieu, Bâtiment 11A, 35042 Rennes CEDEX, France
Isabelle Cantat
Affiliation:
Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes 1, Campus Beaulieu, Bâtiment 11A, 35042 Rennes CEDEX, France
Benjamin Dollet*
Affiliation:
Institut de Physique de Rennes, UMR 6251 CNRS/Université de Rennes 1, Campus Beaulieu, Bâtiment 11A, 35042 Rennes CEDEX, France
*
†Email address for correspondence: benjamin.dollet@univ-rennes1.fr
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Abstract

The response of a two-dimensional liquid foam to a localized air injection is investigated experimentally and theoretically. The experiments show a rich phenomenology, with two essentially distinct behaviours, depending on the injection conditions. At low flux, the injected air forms a central bubble that grows inside the foam and induces plastic rearrangements, without film rupture. This ‘pure swelling’ regime is reminiscent of ductile fracture. In this regime, the central bubble shows fingering patterns beyond a certain velocity. The dependence among the swelling rate, the injection overpressure and the other control parameters, namely cell gap, bubble size and foam area, is captured by a simple balance between the pressure drop and bubble/wall friction under a radial assumption. Fingering is successfully modelled by the linear stability analysis of an azimuthal perturbation of the radial model; yield stress becomes an important parameter to determine the finger width. At high injection rate, films are broken and narrow cracks form rapidly through the foam, reminiscent of brittle fracture. Criteria for the transition between ductile and brittle behaviours are investigated, both at the local and global scales.

JFM classification

Complex Fluids: Complex Fluids Interfacial Flows (free surface): Fingering instability Complex Fluids: Foams
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 258 - 282
Copyright
©2013 Cambridge University Press

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References

Arciniaga, M., Kuo, C.-C & Dennin, M. 2011 Size dependent brittle to ductile transition in bubble rafts. Colloids Surf. A 382, 36–41.Google Scholar
Arif, S., Tsai, J. C. & Hilgenfeldt, S. 2010 Speed of crack propagation in dry aqueous foam. Eur. Phys. Lett. 92, 38001.Google Scholar
Arif, S., Tsai, J. C. & Hilgenfeldt, S. 2012 Spontaneous brittle-to-ductile transition in aqueous foam. J. Rheol. 56, 485–499.Google Scholar
Baroud, C. N., Gallaire, F. & Dangla, R. 2010 Dynamics of microfluidic droplets. Lab on a Chip 10, 2032–2045.CrossRefGoogle ScholarPubMed
Baumberger, T., Caroli, C., Martina, D. & Ronsin, O. 2008 Magic angles and cross-hatching instability in hydrogel fracture. Phys. Rev. Lett. 100, 178303.Google Scholar
Ben Amar, M. 1995 Viscous fingering: a singularity in Laplacian growth models. Phys. Rev. E 51, R3819–R3822.Google Scholar
Ben Amar, M. & Corvera Poiré, E. 1999 Pushing a non-Newtonian fluid in a Hele–Shaw cell: from fingers to needles. Phys. Fluids 11, 1757–1767.CrossRefGoogle Scholar
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977–999.Google Scholar
Bonn, D., Kellay, H., Ben Amar, M. & Meunier, J. 1995 Viscous finger widening with surfactants and polymers. Phys. Rev. Lett. 75, 2132–2135.CrossRefGoogle Scholar
Bouchbinder, E., Fineberg, J. & Marder, M. 2010 Dynamics of simple cracks. Annu. Rev. Condens. Matter 1, 371–395.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., Höhler, R., Pitois, O., Rouyer, F. & Saint-Jalmes, A. 2010 Les Mousses. Structure et Dynamique. Belin.Google Scholar
Cantat, I., Kern, N. & Delannay, R. 2004 Dissipation in foam flowing through narrow channels. Europhys. Lett. 65, 726–732.Google Scholar
Cheddadi, I., Saramito, P., Dollet, B., Raufaste, C. & Graner, F. 2011 Understanding and predicting viscous, elastic, plastic flows. Eur. Phys. J. E 34, 1–15.Google Scholar
Chevalier, C., Lindner, A., Leroux, M. & Clément, E. 2009 Morphodynamics during air injection into a confined granular suspension. J. Non-Newtonian Fluid Mech. 158, 63–72.CrossRefGoogle Scholar
Chowdiah, P., Misra, B. R., Kilbane, J. J., Srivastava, V. J. & Hayes, T. D. 1998 Foam propagation through soils for enhanced in-situ remediation. J. Hazard. Mater. 62 (3), 265–280.Google Scholar
Cortet, P. P., Santucci, S., Vanel, L. & Ciliberto, S. 2005 Slow crack growth in polycarbonate films. Europhys. Lett. 71, 242–248.Google Scholar
Courty, S., Dollet, B., Elias, F., Heinig, P. & Graner, F. 2003 Two-dimensional shear modulus of a Langmuir foam. Europhys. Lett. 64, 709–715.Google Scholar
Coussot, P. 1999 Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.CrossRefGoogle Scholar
Denkov, N. D., Subramanian, V., Gurovich, D. & Lips, A. 2005 Wall slip and viscous dissipation in sheared foams: effect of surface mobility. Colloids Surf. A 263, 129–145.Google Scholar
Denkov, N. D., Tcholakova, S., Golemanov, K., Ananthapadmanabhan, K. P. & Lips, A. 2009a The role of surfactant type and bubble surface mobility in foam rheology. Soft Matt. 5, 3389–3408.Google Scholar
Denkov, N. D., Tcholakova, S., Golemanov, K. & Lips, A. 2009b Jamming in sheared foams and emulsions, explained by critical instability of the films between neighbouring bubbles and drops. Phys. Rev. Lett. 103, 118302.Google Scholar
Dennin, M. & Knobler, C. M. 1997 Experimental studies of bubble dynamics in a slowly driven monolayer foam. Phys. Rev. Lett. 78, 2485–2488.Google Scholar
Dollet, B. & Cantat, I. 2010 Deformation of soap films pushed through tubes at high velocity. J. Fluid Mech. 652, 529–539.CrossRefGoogle Scholar
Dollet, B. & Graner, F. 2007 Two-dimensional flow of foam around a circular obstacle: local measurements of elasticity, plasticity and flow. J. Fluid Mech. 585, 181–211.Google Scholar
Edwards, D. A., Brenner, H. & Wasan, D. T. 1991 Interfacial Transport Processes and Rheology. Butterworth–Heinemann.Google Scholar
Freund, L. B. 1990 Dynamic Fracture Mechanics. Cambridge University Press.CrossRefGoogle Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2002 Gouttes, Bulles, Perles et Ondes. Belin.Google Scholar
Gladden, J. R. & Belmonte, A. 2007 Motion of a viscoelastic micellar fluid around a cylinder: flow and fracture. Phys. Rev. Lett. 98, 224501.Google Scholar
Goerke, J. 1998 Pulmonary surfactant: functions and molecular composition. Biochem. Biophys. Acta 1408, 79–89.Google Scholar
Hilgenfeldt, S., Arif, S. & Tsai, J. C. 2008 Foam: a multiphase system with many facets. Phil. Trans. R. Soc. Lond. A 366, 2145–2159.Google Scholar
Hirasaki, G. J. & Lawson, J. B. 1985 Mechanisms of foam flow in porous media: apparent viscosity in smooth capillaries. Soc. Petrol. Engng J. 176–190.Google Scholar
Höhler, R. & Cohen-Addad, S. 2005 Rheology of liquid foam. J. Phys.: Condens. Matter 17, R1041–R1069.Google Scholar
Holtzman, R. & Juanes, R. 2010 Crossover from fingering to fracturing in deformable disordered media. Phys. Rev. E 82, 046305–046309.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.Google Scholar
Kondic, L., Palffy-Muhoray, P. & Shelley, M. J. 1996 Models of non-Newtonian Hele–Shaw flow. Phys. Rev. E 54, R4536–R4539.Google Scholar
Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 1998 Non-Newtonian Hele–Shaw flow and the Saffman–Taylor instability. Phys. Rev. Lett. 80, 1433–1436.CrossRefGoogle Scholar
Kraynik, A. M. & Hansen, M. G. 1987 Foam rheology: a model of viscous phenomena. J. Rheol. 31, 175–205.Google Scholar
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.Google Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Lemaire, E., Levitz, P., Daccord, G. & van Damme, H. 1991 From viscous fingering to viscoelastic fracturing in colloidal fluids. Phys. Rev. Lett. 67, 2009–2012.Google Scholar
Lindner, A., Bonn, D., Corvera Poiré, E., Ben Amar, M. & Meunier, J. 2002 Viscous fingering in non-Newtonian fluids. J. Fluid Mech. 469, 237–256.Google Scholar
Lindner, A., Coussot, P. & Bonn, D. 2000 Viscous fingering in a yield-stress fluid. Phys. Rev. Lett. 85, 314–317.Google Scholar
Livne, A., Bouchbinder, E. & Fineberg, J. 2008 Breakdown of linear elastic fracture mechanics near the tip of a rapid crack. Phys. Rev. Lett. 101, 264301.Google Scholar
Mann, E. K. & Primak, S. V. 1999 Stability of two-dimensional foams in Langmuir monolayers. Phys. Rev. Lett. 83, 5397–5400.Google Scholar
Marmottant, P. & Raven, J. P. 2009 Microfluidics with foams. Soft Matt. 5, 3385–3388.Google Scholar
Park, S. S. & Durian, D. J. 1994 Viscous and elastic fingering instabilities in foam. Phys. Rev. Lett. 72, 3347–3350.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele–Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Princen, H. M. 1983 Rheology of foams and highly concentrated emulsions. I. Elastic properties and yield stress of a cylindrical model system. J. Colloid Interface Sci. 91, 160–175.Google Scholar
Ratulowski, J. & Chang, H.-C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1, 1642–1655.Google Scholar
Raufaste, C., Foulon, A. & Dollet, B. 2009 Dissipation in quasi-two-dimensional flowing foams. Phys. Fluids 21, 053102–053110.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Sandnes, B., Flekkøy, E. G., Knudsen, H. A., Måløy, K. J. & See, H. 2011 Patterns and flow in frictional fluid dynamics. Nat. Commun. 2, 288.Google Scholar
Tabuteau, H., Mora, S., Porte, G., Abkarian, M. & Ligoure, C. 2009 Microscopic mechanisms of the brittleness of viscoelastic fluids. Phys. Rev. Lett. 102, 155501.Google Scholar
Terriac, E., Etrillard, J. & Cantat, I. 2006 Viscous force exerted on a foam at a solid boundary: influence of the liquid fraction and of the bubble size. Europhys. Lett. 74, 909–915.Google Scholar
Vaz, M. F. & Cox, S. J. 2005 Two-bubble instabilities in quasi-two-dimensional foams. Phil. Mag. Lett. 85, 415–425.Google Scholar
Wilson, S. D. R. 1990 The Taylor–Saffman problem for a non-Newtonian liquid. J. Fluid Mech. 220, 413–425.Google Scholar
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries. J. Fluid Mech. 292, 71–94.Google Scholar

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 14.2 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 6.5 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 8.7 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 37.5 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 603.9 KB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 3.8 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 9.2 MB

Ben Salem et al. supplementary movie

Movies a to d illustrate the snapshots shown on figure 1a to d: (a) quasistatic response of foam to air injection (accelerated 420 times); (b) intermediate regime, showing the development of ductile fingers (slowed down 35 times); (c) and (d) high-speed regime, showing the development of (c) single or (d) multiple, branched fragile cracks (slowed down 100 times).

Download Ben Salem et al. supplementary movie(Video)
Video 4.5 MB