# Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation

## Overview

- [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters.
**82**, 21. 4232-4235 (1999). -
**Keywords:**Foam, Drainage, Plateau Border, Tetrakaidecahedron (Kelvin Cell)

## Summary

Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.

The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.

The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.

The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?

The researchers consider two models of flow through a monodisperse foam with tetrakaidecahedral cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>: <math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>

with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math>

The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is: <math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>

with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math>

Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes

The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.

## Soft Matter Details

**Experimental Methods:**

The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescence, so are able to see and record the front of wet foam progress.

To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the critical micelle concentration), and determine that these changes had no effect.

To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.

It would be great to verify the flow profile, Poiseuille or plug, in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.

**Theoretical Methods:**

To create Figure 1, Koehler *et. al.* used Surface Evolver from the University of Minnesota Geometry Center.

The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!

The authors use dimensional analysis in developing their new theory focused on damping in the nodes.