Breward, C. J. W. and Howell, P. D. (2002) The drainage of a foam lamella. Journal of Fluid Mechanics, 458 . pp. 379-406.
We present a mathematical model for the drainage of a surfactant-stabilised 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 centre-line 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.
|Uncontrolled Keywords:||surfactants, mathematical modelling, asymptotics|
|Subjects:||D - G > Fluid mechanics|
|Research Groups:||Oxford Centre for Industrial and Applied Mathematics|
|Deposited By:||Chris J.W. Breward|
|Deposited On:||26 Aug 2004|
|Last Modified:||20 Jul 2009 14:18|
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