Markov Models of Coarsening in Two-Dimensional Foams with Edge Rupture

Abstract

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space $\ell_1(\mathbb N)$ which counts total numbers of cells with $n\ge 3$ sides ($n$-gons). Under a large cell limit, we show that number densities of $n$-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.