# 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. The main results for the three cases considered are

- In the free drainage problem there are asymptotically two regimes. The border between these two regimes is when the transient distribution of Plateau-area first shows a non-zero gradient at the top boundary (<math>\tau \approx 2</math> in figure 3). In the first regime the drainage profile <math>\partial A / \partial z</math> varies as <math>t^{-1}</math>. In the second regime the solution starts to forget about the initial conditions and a generic similarity solution shows very good agreement with the results from a numerical simulation
- For the wetting of a dry foam the profile <math>\partial A / \partial z</math> simply spreads as <math>t^{1/3}</math>
- For pulsed drainage the profile itself exhibits three distinct regions. An advancing nose which shows similar behavior as forced wetting (describe din earlier papers) that spreads as <math>t^{1/4}</math>, a middle region with maximum that spreads as <math>t^{1/2}</math> and decreases as <math>t^{-1/2}</math> and an advancing rear that shows similarities to the dry wetting case and thus scales with time as <math>t^{1/3}</math>

The results, especially of the numerical simulations solving exactly for the general foam drainage equation, agree very well with the few experimental results available.

## 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>

where <math>c_n</math> and <math>\delta</math> are constants describing the geometry.

This is essentially a non-linear diffusion equation with the diffusivity

<math>D = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta}</math>

We can thus see that our solution is not only valid for an initial spherical wet domain but for larger times asymptotically for any initial domain form.

Guided by this observation we can again find a (very complicated) similarity solution. Its main features are that the pulse spreads over time with <math>t^{2/7}</math> and its amplitude is reduced with time like <math>t^{-6/7}</math>.

Given that no experimental results in this parameter regime exist the authors state that they plan to perform experiments to test these predictions themselves.

## 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.