Breward, C. J. W. and Howell, P. D. (2002) The drainage of a foam lamella. Journal of Fluid Mechanics, 458 . pp. 379406.

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Abstract
We present a mathematical model for the drainage of a surfactantstabilised 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 thinfilm limit and perform an asymptotic decomposition of the liquid domain into a capillarystatic Plateau border, a timedependent thin film and a transition region between the two. By solving a quasisteady boundaryvalue 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 surfactantfree 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 fluidsurfactant 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.
Item Type:  Article 

Uncontrolled Keywords:  surfactants, mathematical modelling, asymptotics 
Subjects:  D  G > Fluid mechanics 
Research Groups:  Oxford Centre for Industrial and Applied Mathematics 
ID Code:  116 
Deposited By:  Chris J.W. Breward 
Deposited On:  26 Aug 2004 
Last Modified:  29 May 2015 18:16 
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