In an aqueous foam, gas can diffuse through the liquid phase, changing the sizes of the bubbles. On average, large bubbles grow while small bubbles shrink, and subsequently disappear. Thus the length scale of the foam increases, or coarsens. This is likely to reduce the efficacy of foams in applications such as soil remediation and firefighting.

Coarsening is well characterised in the dry (low) and wet (high) limits of liquid fraction. In the dry limit the gas flow is through the thin films, while in the wet limit there are no thin films and the gas flows through bulk liquid. Growth laws for individual bubbles are known, as is the exponent of time by which the average bubble size evolves in a scaling state. However, these properties have not yet been convincingly established for intermediate liquid fractions, which are found in most applications.

The growth rate of a bubble is determined by its pressure (and that of its neighbours),which is related to the curvature of its liquid/gas interfaces, and its surface area in contact with other bubbles. A first approximation [1] is to take these geometric properties as dependent only upon the bubble’s (equivalent spherical) radius. Guided by simulations, we derive improved mean-field approximations for a bubble’s pressure and contact area at arbitrary liquid fraction [2].

The approximations are combined to give a growth law for a bubble of a given size in a coarsening foam. From this growth law we are able to predict scaling-state bubble size distributions that are in agreement with the long-time evolution of the growth law calculated numerically. We present results for various liquid fractions [3], and compare them with recent experiments [4].

References

[1]   Lemlich, R. (1978). Prediction of changes in bubble size distribution due to interbubble gas diffusion in foam. Industrial & Engineering Chemistry Fundamentals, 17, 89-93.

[2]   J. Morgan and S.J. Cox (2024) Effects of liquid fraction and contact angle on structure and coarsening in two-dimensional foams. Journal of Fluid Mechanics 999: A10.

[3]   J. Morgan and S.J. Cox (2025) Mean-field model of bubble size distribution in coarsening wet foams. In preparation.

[4]    Galvani, N., Pasquet, M., Mukherjee, A., Requier, A., Cohen-Addad, S., Pitois, O., Höhler, R., Rio, E., Salonen, A., Durian, D. J. & Langevin, D. (2023). Hierarchical bubble size distributions in coarsening wet liquid foams. Proceedings of the National Academy of Sciences, 120, e2306551120.