Mean-field model for the bubble size distribution in coarsening wet foams

Aqueous foams are subject to coarsening, whereby gas from the bubbles diffuses through the liquid phase. Gas is preferentially transported from small to large bubbles, resulting in a gradual decrease of the number of bubbles and an increase in the average bubble size. Coarsening foams are expected to approach a scaling state at late times in which their statistical properties are invariant. However, a model predicting the experimentally observed bubble-size distribution in the scaling state of foams with moderate liquid content, as a function of the liquid fraction $\phi$, has not yet been developed. To this end, we propose a three-dimensional mean-field bubble growth law for foams without inter-bubble adhesion, validated against bubble-scale simulations, and use it to derive a prediction of the scaling-state bubble-size distribution for any $\phi$ from zero up to the unjamming transition ${\phi }_{\text{c}}\approx 36\mathrm{\%}$. We verify that the derived scaling state is approached from a variety of initial conditions using mean-field simulations implementing the proposed growth law. Comparing our predicted bubble-size distribution with previous simulations and experimental results, we likewise find a large population of small bubbles when $\phi >0$, but there are qualitative differences from prior results which we attribute to the absence of rattlers, i.e. bubbles not pressed into contact with their neighbours, in our model.