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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: [email protected]
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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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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