Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-29T16:39:29.802Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical experiments on two-dimensional foam

Published online by Cambridge University Press:  26 April 2006

Thomas Herdtle  and
Hassan Aref
Thomas Herdtle
Affiliation:
Department of Applied Mechanics and Engineering Science, University of California at San Diego, La Jolla, CA 92093-0411, USA
Hassan Aref
Affiliation:
Department of Applied Mechanics and Engineering Science, University of California at San Diego, La Jolla, CA 92093-0411, USA
Get access Rights & Permissions [Opens in a new window]

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’.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 233 - 260
Copyright
© 1992 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aboav D. A. 1970 The arrangement of grains in a polycrystal. Metallography 3, 383–390.Google Scholar
Aboav D. A. 1980 The arrangement of cells in a net. Metallography 13, 43–58.Google Scholar
Aref, H. & Herdtle T. 1990 Fluid networks. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober), pp. 745–764. Cambridge University Press.
Beenakker C. W. J. 1988 Numerical simulation of a coarsening two-dimensional network Phys. Rev. A 37, 1697–1702.Google Scholar
Berge B., Simon, A. J. & Libchaber A. 1990 Dynamics of gas bubbles in monolayers Phys. Rev. A 41, 6893–6900.Google Scholar
Bolton, F. & Weaire D. 1991 The effects of Plateau borders in the two-dimensional soap froth. I. Decoration lemma and diffusion theorem Phil. Mag. B 63, 795–809.Google Scholar
Bragg, L. & Nye J. F. 1947 A dynamical model of a crystal structure Proc. R. Soc. Lond. A 190, 474–481.Google Scholar
George, A. & Ng E. 1984 SPARSPAK: Waterloo Sparse Matrix Package. Dept. Computer Science, Univ. Waterloo, Res. Rep. CS-84–37, 47 pp.Google Scholar
Glazier J. A. 1989 Dynamics of cellular patterns. Ph.D. thesis, University of Chicago.
Glazier J. A., Gross, S. P. & Stavans J. 1987 Dynamics of two-dimensional soap froths Phys. Rev. A 36, 306–312.Google Scholar
Glazier, J. A. & Stavans J. 1989 Nonideal effects in the two-dimensional soap froth Phys. Rev. A 40, 7398–7401.Google Scholar
GruUnbaum, B. & Shephard G. C. 1987 Tilings and Patterns. W. H. Freeman.
Herdtle, T. & Aref H. 1989 Numerical simulation of two-dimensional foam. Bull. Am. Phys. Soc. 34, 2296 (abstract only).Google Scholar
Herdtle, T. & Aref H. 1991a Relaxation of fractal foam. Phil. Mag. Lett. 64, 335–340.Google Scholar
Herdtle, T. & Aref H. 1991b On the geometry of composite bubbles Proc. Roy. Soc. Lond. A 434, 441–447.Google Scholar
Isenberg C. 1978 The Science of Soap Films and Soap Bubbles. Somerset: Woodspring Press.
Kermode, J. P. & Weaire D. 1990 2D-FROTH: A program for the investigation of 2-dimensional froths. Comput. Phys. Commun. 60, 75–109 (referred to herein as KW).Google Scholar
Knobler C. M. 1990 Seeing phenomena in Flatland: Studies of monolayers by fluorescence microscopy. Sci. 249, 870–874.Google Scholar
Kraynik, A. M. & Hansen M. G. 1986 Foam and emulsion rheology: A quasistatic model for large deformations of spatially-periodic cells. J. Rheol. 30, 409–439.Google Scholar
Lewis F. T. 1928 The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis. Anat. Rec. 38, 341–376.Google Scholar
Lucassen J., Akamatsu, S. & Rondelez F. 1991 Formation, evolution and rheology of two-dimensional foams in spread monolayers at the air-water interface. J. Colloid Interface Sci. 144, 434–448.Google Scholar
Maddox J. 1989 Soap bubbles make serious physics. Nature 338, 293.Google Scholar
Neumann von J. 1952 Discussion remark concerning paper of C. S. Smith, ‘Grain shapes and other metallurgical applications of topology’. Metal Interfaces, pp. 108–110. Am. Soc. for Metals, Cleveland, Ohio.
Plateau J. A. F. 1873 Statique ExperimeAntale et TheAorique des Liquides Soumis aux Seules Forces MoleAculaires. Gauthier-Villars.
Princen, H. M. & Mason S. G. 1965 The permeability of soap films to gases. J. Colloid Sci. 20, 353–375.Google Scholar
Rivier N. 1990 Geometry of random packings and froths. In Physics of Granular Media (ed. D. Bideau & J. Dodds) Nova Science.
Stavans J. 1990 Temporal evolution of two-dimensional drained soap froths Phys. Rev. A 42, 5049–5051.Google Scholar
Stavans, J. & Glazier J. A. 1989 Soap froth revisited: dynamic scaling in the two-dimensional froth. Phys. Rev. Lett. 62, 1318–1321.Google Scholar
Sullivan J. M. 1991 Generating and rendering four-dimensional polytopes. Mathematica J. 1 (3), 76.Google Scholar
Weaire D. 1974 Some remarks on the arrangement of grains in a polycrystal. Metallography 7, 157–160.Google Scholar
Weaire, D. & Kermode J. P. 1983a The evolution of the structure of a two-dimensional soap froth Phil. Mag. B 47, L29–L31.Google Scholar
Weaire, D. & Kermode J. P. 1983b Computer simulation of two-dimensional soap froth I. Method and motivation Phil. Mag. B 48, 245–259.Google Scholar
Weaire, D. & Kermode J. P. 1984 Computer simulation of two-dimensional soap froth II. Analysis of results Phil. Mag. B 50, 379–395.Google Scholar
Weaire D., Kermode, J. P. & Wejchert J. 1986 On the distribution of cell areas in a Voronoi network Phil. Mag. B 53, L101–L105.Google Scholar
Weaire, D. & Lei H. 1990 A note on the statistics of the mature two-dimensional soap froth. Phil. Mag. Lett. 62, 427–430.Google Scholar
Weaire, D. & Rivier N. 1984 Soap, cells and statistics – Random patterns in two dimensions. Contemp. Phys. 25, 59–99.Google Scholar
Wejchert J., Weaire, D. & Kermode J. P. 1986 Monte Carlo simulation of the evolution of a two-dimensional soap froth Phil. Mag. B 53, 15–24.Google Scholar