Abstract.
A liquid foam in contact with a solid surface forms a two-dimensional foam on the surface. We derive the equilibrium equations for this 2D foam when the solid surface is curved and smooth, generalising the standard case of flat Hele-Shaw cells. The equilibrium conditions at the vertices in 2D, at the edges in 3D, are invariant by conformal transformations. Regarding the films, conformal invariance only holds with restrictions, which we explicit for 3D and flat 2D foams. Considering foams confined in thin interstices between two non-parallel plates, normal incidence and Laplace’s law lead to an approximate equation relating the plate profile to the conformal map. Solutions are given for the logarithm and power laws in the case of constant pressure. The paper concludes on a comparison with available experimental data.
Similar content being viewed by others
References
D. Weaire, N. Rivier, Contemp. Phys. 25, 59 (1984).
D. Weaire, S. Hutzler, The Physics of Foams (Clarendon Press, Oxford, 1999).
S.J. Cox, Weaire D., M.F. Vaz, Eur. Phys. J. E 7, 311 (2002).
K. Brakke, F. Morgan, Eur. Phys. J. E 9, 453 (2002).
A.T. Fomenko, The Plateau Problem (Gordon and Breach, 1990).
J.M. DiMeglio, T. Senden, communication at Eufoam04, Marne la Vallée, France, July 2004, and in preparation.
W. Drenckhan, D. Weaire, S.J. Cox, Eur. J. Phys. 25, 429 (2004).
C. Moukarzel, Phys. Rev. E 55, 6866 (1997).
D. Weaire, Philos. Mag. Lett. 79, 491 (1999).
M. Mancini, C. Oguey, Philos. Mag. Lett. 83, 643 (2003).
F. Morgan, Riemannian Geometry: a Beginner’s Guide (AK Peters, Wellesley, MA, 1998).
T.J. Willmore, Riemannian Geometry (Clarendon Press, Oxford, 1996).
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern geometry -- Methods and Applications. Part 1. The Geometry of Surfaces, Transformation Groups, and Fields (Springer-Verlag, Berlin, 1992) 5, 8 and 15.
M. Mancini, C. Oguey, to be published in Colloids Surf. A (2005).
F. Rothen, P. Pieranski, N. Rivier, A. Joyet, Eur. J. Phys. 14, 227 (1993).
F. Rothen, P. Pieranski, Phys. Rev. E 53, 2828 (1996).
F. Elias, J.C. Bacri, H. de Mougins, T. Spengler, Philos. Mag. Lett. 79, 389 (1999).
N. Rivier, D. Reinelt, F. Elias, C. Vanden Driessche, in Proceedings of the International Workshop on Foams and Films, Leuven, Belgium, 5-6 March 1999, edited by D. Weaire and J. Banhart (Verlag MIT Publishers, 1999).
P. Concus, R. Finn, Phys. Fluids, 10, 39 (1998).
P.G. de Gennes, F. Brochard, D. Quéré, Gouttes, Bulles, Perles et Ondes (Belin, Paris, 2002).
M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Houston, 1979).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mancini, M., Oguey, C. Foams in contact with solid boundaries: Equilibrium conditions and conformal invariance. Eur. Phys. J. E 17, 119–128 (2005). https://doi.org/10.1140/epje/i2004-10133-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epje/i2004-10133-x