Draining of a Vertically-Oriented Liquid Free Film

Richard J. Braun, U of Delaware

Shailesh Naire, U of Delaware

Steven A Snow, Dow Corning

Udo C Pernisz, Dow Corning

The drainage of a lamella in a foam can be modelled by the drainage of a vertically oriented thin film; the film is "free" because both sides of the film are free surfaces. In some experiments carried out at Dow Corning by Snow, Pernisz and coworkers, detailed measurement of film shapes and correlation to drainage rate has resulted in a system where they can evaluate the relative merits of their products for some foam fabrication applications. They have been interested in the development of equally detailed mathematical theory as well; we have collaborated on the theory. A schematic of the situation of interest may be viewed.

We have used lubrication theory to develop nonlinear partial differential equation(s) that govern the free surface of the film (and other variables of interest). In the simplest case, the surface of the film is assumed to be tangentially immobile; then a single pde governs the shape of the free surface.

We have solved the pde numerically and in some cases have analytical approximation that work very well. For example, in the results shown, there is a capillary wave train that develops where the film meets the bath at the bottom. We can use matched asymptotics in a manner very similar to Jensen (JFM, 1997) to describe how quickly the bumps and dips decay relative to the long middle section of the film. For the biggest dip (at the bottom), analysis predicts power law decay with a -3/5 exponent; computations give an exponent (at long times) of -0.589. We can obtain similar results for the other bumps and dips. The comparison with experiment is also good. The tangentially-immobile case is a very tight bound on the slow-draining regime of Dow Corning's experiments.

This work has appeared in the Journal of Colloid and Interface Science (vol 219 (1999) 225-240) and has been published as a Dow Corning Corporation Research Report.

Shailesh Naire is currently investigating extensions to the theory where we relax the assumption of tangential immobility and allow a mobile surface with surfactant transport. The simplest model from relaxing the above assumptions gives a tight bound on the behavior of the fast draining regime of the experiments. A summary of his work is available on Shailesh's research page.

This work has been partially supported by Dow Corning and by the National Science Foundation under grant numbers 9623092, 9631287 and 9722854.