Numerical simulations of two-dimensional foam by the immersed boundary method

In this paper, we present an immersed boundary (IB) method to simulate a dry foam, i.e., a foam in which most of the volume is attributed to its gas phase. Dry foam dynamics involves the interaction between a gas and a collection of thin liquid-film internal boundaries that partition the gas into discrete cells or bubbles. The liquid-film boundaries are flexible, contract under the influence of surface tension, and are permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. Such problems are conventionally studied by assuming that the pressure is uniform within each bubble. Here, we introduce instead an IB method that takes into account the non-equilibrium fluid mechanics of the gas. To model gas diffusion across the internal liquid-film boundaries, we allow normal slip between the boundary and the gas at a velocity proportional to the (normal) force generated by the boundary surface tension. We implement this method in the two-dimensional case, and test it by verifying the von Neumann relation, which governs the coarsening of a two-dimensional dry foam. The method is further validated by a convergence study, which confirms its first-order accuracy.