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arXiv · 2101.00966

Markov models of coarsening in two-dimensional foams with edge rupture

Joseph Klobusicky
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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.

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Joseph Klobusicky. 2020-12-31. Markov models of coarsening in two-dimensional foams with edge rupture. https://doi.org/10.1007/s00332-021-09696-3

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