# 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 new theory more closely matching the experimental results.

The network of channels in a dry foam (much more air than liquid) are 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 are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "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 older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.

Equation 6 from [1]: <math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <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, leads to plug flow, and viscous damping in the nodes.

Equation 9 from [1]: <math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math>

- I is dimensionless, viscous forces in nodes

The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.

## Soft Matter Details

**Experimental Methods:**

surface evolver

Question about constant foam generation

Testing that the theory is robust

Deciding which term dominates in equation 4

Dimensional analysis equation 7

Foam- further applications?