# Dynamics of foam drainage

Original entry by Joerg Fritz, AP225 Fall 2009

## Contents

## Source

S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: *Physical Review E*, 1998, 58, pp 2097 to 2106

## Keywords

Liquid foam, Surface tension, Viscosity, Drainage, Wetting, Plateau Borders, Plateau's laws

## Summary

In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces, surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries.

## Foam Drainage Equation

Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area *A* of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient

<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math>

here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension.

A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.

## One Dimensional Drainage

The paper in detail examines three different cases of the foam drainage problem in one dimension:

- free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity
- wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading

front forms in the foam and again evolves toward a steady-state profile

- pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.

As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of *A* for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as *L* we can solve for the distribution

<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math>

In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions.

## Drainage in Higher Dimensions

In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius *R*. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes

<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2}} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2 \frac{\partial \epsilon}{\partial t} \right )</math>

## Conclusion

The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.