# 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.

Figure 1 from [1]

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 $Q$ and observe the front of wetter foam penetrate through the column of dry foam at velocity $\nu_f$. The experimenters looked at three sizes of bubbles.

Old Theory: Equation 6 "rigid channel walls" $\nu_f=(V_0^{rigid}V_s)^{1/2}$, $V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}$

• the exponent of 1/2 is what the experimental data disagree with
Figure 3 from [1]

New Theory: "channel-slip theory" $\nu_f=((V_0^{slip})^2V_s)^{1/3}$, $V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}$

• I is dimensionless, viscous forces in nodes

## Soft Matter Details

Experimental Methods:

surface evolver