The Immersed Boundary Method for Two-Dimensional Foam with Topological Changes

Abstract

We extend the immersed boundary (IB) method to simulate the dynamics of a 2D dry foam by including the topological changes of the bubble network. In the article [Y. Kim, M.-C. Lai, and C. S. Peskin, J. Comput. Phys. 229:5194-5207, 2010], we implemented an IB method for the foam problem in the two-dimensional case, and tested it by verifying the von Neumann relation which governs the coarsening of a two-dimensional dry foam. However, the method implemented in that article had an important limitation; we did not allow for the resolution of quadruple or higher order junctions into triple junctions. A total shrinkage of a bubble with more than four edges generates a quadruple or higher order junction. In reality, a higher order junction is unstable and resolves itself into triple junctions. We here extend the methodology previously introduced by allowing topological changes, and we illustrate the significance of such topological changes by comparing the behaviors of foams in which topological changes are allowed to those in which they are not.

Numerical Study on Statistical Behaviors of Two-Dimensional Dry Foam

“…Other topological changes are also possible through the combination of these two processes. In [24], the authors introduced the T1 and T2 processes to the IB computation for the foam dynamics in order to produce topological changes of a cell network.…”

Lubricated immersed boundary method in two dimensions

“…In the case of a flat lower boundary aligned with the x-axis, (N 0 · ∇) ≡ ∂/∂y, and (T 0 · ∇) ≡ ∂/∂x. Together, (5) and (6)- (7) define U h and U 0 ; to summarize, the standard immersed boundary formulation is used for the average velocity and a sharp formulation is used for the velocity difference. It is reasonable to compute the mean velocity using the standard immersed boundary method as in (5), since smoothing tends to yield the net motion as in the thought experiment above.…”

“…The immersed boundary method is a widely-used numerical method for fluidstructure interaction that has been applied to problems including blood clotting [1], osmotic swelling due to thermal fluctuations [2], sperm locomotion through viscoelastic fluids [3], and insect flight [4]. Several extensions of the immersed boundary method have been developed to incorporate realistic structural properties such as added mass [5], permeability [6,7], intrinsic twist [8], nonlinear constitutive laws [9], and variable viscosity and density [10,11]. The chief idea of the immersed boundary method is to use an Eulerian description of the fluid and a Lagrangian description of the structure, while coupling these descriptions through integral operators involving delta functions.…”

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

“…The gas diffusion through the liquid phase of the foam was modeled by allowing the internal boundaries to slip relative the fluid, at a velocity (speed and direction) proportional to the boundary force [20,21]. The paper [20] has shown that the IB method is a proper tool to simulate the 2D dry foam dynamics by verifying the vonNeumann relation and by simulating foams with an arbitrarily general shape, see also [22].…”