Numerical experiments on two-dimensional foam

Abstract

The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.

Bubble statistics and coarsening dynamics for quasi-two-dimensional foams with increasing liquid content

“…This includes direct measurements on dry soap froths [5][6][7][8][9][10], soap froths with different boundary conditions [11][12][13], and measurements on lipid monolayers [14,15]. Simulations have also been performed [16][17][18][19][20][21]. This body of work shows good general agreement with von Neumann's law.…”

Something old, something new

“…In the case of the relaxing foam, the distribution function for the number of faces of the constituent bubbles assumes a universal form. Numerical experiments (Herdtle & Aref 1992) suggest that this distribution is lognormal. 3 …”

Topological correlations and asymptotic freedom in cellular aggregates

“…This expression follows from inserting the solutions α j , β j into (7,8) and either summing the layer charges (6) from 0 to j or integrating the difference equation (13). It is consistent with (21).…”

“…The present analysis is mostly devoted to bi-dimensional foams, subject to extensive theoretical and experimental investigations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and where the progresses are ahead of those on 3D foams.…”