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Simple shearing flow of a dry Kelvin soap foam

Published online by Cambridge University Press:  26 April 2006

Douglas A. Reinelt  and
Andrew M. Kraynik
Douglas A. Reinelt
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, XX 75275-0156, USA
Andrew M. Kraynik
Affiliation:
Engineering Sciences Center, Sandia National Laboratories, Albuquerque, NM 87185-0834, USA
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Abstract

Simple shearing flow of a dry soap foam composed of identical Kelvin cells is analysed. An undeformed Kelvin cell has six planar quadrilateral faces with curved edges and eight non-planar hexagonal faces with zero mean curvature. The elastic-plastic response of the foam is modelled by determining the bubble shape that minimizes total surface area at each value of strain. Computer simulations were performed with the Surface Evolver program developed by Brakke. The foam structure and macroscopic stress are piecewise continuous functions of strain. Each discontinuity corresponds to a topological change (Tl) that occurs when the film network is unstable. These instabilities involve shrinking films, but the surface area and edge lengths of a shrinking film do not necessarily vanish smoothly with strain. Each Tl reduces surface energy, results in cell-neighbour switching, and provides a film-level mechanism for plastic yield behaviour during foam flow. The foam structure is determined for all strains by choosing initial foam orientations that lead to strain-periodic behaviour. The average shear stress varies by an order of magnitude for different orientations. A Kelvin foam has cubic symmetry and exhibits anisotropic linear elastic behaviour; the two shear moduli and their average over all orientations are Gmin = 0.5706, Gmax = 0.9646, and $\overline{G} = 0.8070$, where stress is scaled by T/V1/3, T is surface tension, and V is bubble volume. An approximate solution for the microrheology is also determined by minimizing the total surface area of a Kelvin foam with flat films.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 311 , 25 March 1996 , pp. 327 - 343
Copyright
© 1996 Cambridge University Press

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References

Almgren, F. & Taylor, J. 1976 The geometry of soap films and soap bubbles. Sci. Am. July, 82–93.Google Scholar
Bonnecaze, R.T. & Brady, J. F. 1992 Yield stresses in electrorheological fluids. J. Rheol. 36, 73–115.Google Scholar
Brakke, K. 1977 The Motion of a Surface by Its Mean Curvature. Princeton University Press.
Brakke, K. A. 1992 The surface evolver. Exp. Maths 1, 141–165.Google Scholar
Brakke, K. A. 1995 Surface Evolver Manual version 1.98, March.
Kelvin, Lord (Thompson, W.) 1887 On the division of space with minimum partitional area. Phil. Mag. 24, 503–514.
Khan, S. A. & Armstrong, R. C. 1986 Rheology of foams. I. Theory for dry foams. J. Non-Newtonian Fluid Mech. 22, 1–22.Google Scholar
Kraynik, A. M. 1988 Foam flows. Ann. Rev. Fluid Mech. 20, 325–357.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, 409–39.CrossRefGoogle Scholar
Kraynik, A.M. & Reinelt, D.A. 1996 The linear elastic behavior of dry soap foams. J. Colloid Interface Sci. Submitted.Google Scholar
Love, A. E. H. 1944 A Treatise on the Mathematical Theory of Elasticity. Dover.
Matzke, E. B. 1946 The three-dimensional shape of bubbles in foam-an analysis of the role of surface forces in three-dimensional cell shape determination. Am. J. Botany 33, 58–80.CrossRefGoogle Scholar
Nye, J. F. 1985 Physical Properties of Crystals. Clarendon Press.
Okuzono, T., Kawasaki, K. & Nagai, T. 1993 Rheology of random foams. J. Rheol. 37, 571–586.Google Scholar
Plateau, J. A. F. 1873 Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moleculaires. Gauthier-ViHiard.
Princen, H. M. 1983 Rheology of foams and highly concentrated emulsions. I. Elastic properties and yield stress of a cylindrical model system. J. Colloid Interface Sci. 91, 160–175.Google Scholar
Princen, H. M. 1985 Rheology of foams and highly concentrated emulsions. II. Experimental study of the yield stress and wall effects for concentrated oil-in-water emulsions. J. Colloid Interface Sci. 105, 150–171.Google Scholar
Princen, H. M. & Kiss, A. D. 1986 Rheology of foams and highly concentrated emulsions. III. Static shear modulus. J. Colloid Interface Sci. 112, 427–437.Google Scholar
Reinelt, D.A. 1993 Simple shearing flow of three-dimensional foams and highly concentrated emulsions with planar films. J. Rheol. 37, 1117–1139.Google Scholar
Reinelt, D.A. & Kraynik, A.M. 1993 Large elastic deformations of three-dimensional foams and highly concentrated emulsions. J. Colloid Interface Sci. 159, 460–470.Google Scholar
Rivier, N. 1994 Kelvin's conjecture on minimal froths and the counter-example of Weaire and Phelan. Phil. Mag. Lett. 69, 297–303.Google Scholar
Schwarz, H. W. 1965 Rearrangements in polyhedric foam. Recueil 84, 771–781.Google Scholar
Taylor, I.E. 1976 The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Maths 103, 489–539.Google Scholar
Weaire, D. & Kermode, J. P. 1983 Computer simulation of a two-dimensional soap froth.. Phil. Mag. B 48, 245–259.Google Scholar
Weaire, D. & Phelan, R. 1994a A counter-example to Kelvin's conjecture on minimal surfaces. Phil. Mag. Lett. 69, 107–110.Google Scholar
Weaire, D. & Phelan, R. 1994b The structure of monodisperse foam. Phil. Mag. Lett. 70, 345–350.Google Scholar