Commun. Comput. Phys., 12 (2012), pp. 479-493.


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

Yongsam Kim 1*, Yunchang Seol 1, Ming-Chih Lai 2, Charles S. Peskin 3

1 Department of Mathematics, Chung-Ang University, Dongjakgu Heukseokdong, Seoul 156-756, Korea.
2 Department of Applied Mathematics, Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, 1001, Ta Hsueh Road, Hsinchu 300, Taiwan.
3 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 USA.

Received 18 December 2010; Accepted (in revised version) 8 August 2011
Available online 20 February 2012
doi:10.4208/cicp.181210.080811s

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.

AMS subject classifications: 65-04, 65M06, 76D05, 76M20

Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164
Key words: Foam, permeability, capillary-driven motion, immersed boundary method, coarsening, topological changes, T1 and T2 processes.

*Corresponding author.
Email: kimy@cau.ac.kr (Y. Kim), ycseol@cau.ac.kr (Y. Seol), mclai@math.nctu.edu.tw (M.-C. Lai), peskin@cims.nyu.edu (C. S. Peskin)
 

The Global Science Journal