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

In this paper, we present an immersed boundary (IB) method to simulate a 2 dimensional wet foam. Whereas most of the volume in a dry foam is attributed to its gas phase, a wet foam contains a considerable amount of liquid volume which comprises the Plateau borders as well as thin liquid boundaries separating cells. Wet foam dynamics involves the interaction between a surrounding gas, a liquid inside the Plateau borders, and a collection of thin liquid-film boundaries that partition a domain filled with gas and liquid into discrete cells and Plateau borders. The liquid-film boundaries are flexible, contract under the influence of surface tension. The boundary between cells is permeable to the gas, which moves across them by diffusion at a rate proportional to the local pressure difference across the boundary. The boundary between a cell and a Plateau border is impermeable to the gas, however. We here introduce an IB method that takes into account the non-equilibrium fluid mechanics of the fluid. To model gas diffusion across the internal boundaries between cells, we allow normal slip between the boundary and the gas at 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 modified von-Neumann relation, which governs the coarsening of a two-dimensional wet foam. The method is further validated by a convergence study, which confirms its first-order accuracy.