The drainage of a foam lamella
We present a mathematical model for the drainage of a surfactant-stabilized foam lamella, including capillary, Marangoni and viscous effects and allowing for diffusion, advection and adsorption of the surfactant molecules. We use the slender geometry of a lamella to formulate the model in the thin-f...
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Published in | Journal of fluid mechanics Vol. 458; pp. 379 - 406 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cambridge, UK
Cambridge University Press
10.05.2002
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Subjects | |
Online Access | Get full text |
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Summary: | We present a mathematical model for the drainage of a surfactant-stabilized foam
lamella, including capillary, Marangoni and viscous effects and allowing for diffusion,
advection and adsorption of the surfactant molecules. We use the slender geometry
of a lamella to formulate the model in the thin-film limit and perform an asymptotic
decomposition of the liquid domain into a capillary-static Plateau border, a time-dependent
thin film and a transition region between the two. By solving a quasi-steady
boundary-value problem in the transition region, we obtain the flux of liquid from
the lamella into the Plateau border and thus are able to determine the rate at which
the lamella drains. Our method is illustrated initially in the surfactant-free case. Numerical results
are presented for three particular parameter regimes of interest when surfactant is
present. Both monotonic profiles and those exhibiting a dimple near the Plateau
border are found, the latter having been previously observed in experiments. The
velocity field may be uniform across the lamella or of parabolic Poiseuille type, with
fluid either driven out along the centreline and back along the free surfaces or vice
versa. We find that diffusion may be negligible for a typical real surfactant, although
this does not lead to a reduction in order because of the inherently diffusive nature
of the fluid–surfactant interaction. Finally, we obtain the surprising result that the
flux of liquid from the lamella into the Plateau border increases as the lamella thins,
approaching infinity at a finite lamella thickness. |
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Bibliography: | PII:S0022112002007930 istex:53405289B9389FF79F0C9BD00A28922BC48552E3 ark:/67375/6GQ-Z0VBL5CF-V ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0022-1120 1469-7645 |
DOI: | 10.1017/S0022112002007930 |