Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-13T09:54:22.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenisation of two-phase emulsions

Published online by Cambridge University Press:  14 November 2011

Robert Lipton  and
Bogdan Vernescu
Robert Lipton
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester MA 01609, U.S.A.
Bogdan Vernescu
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester MA 01609, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1119 - 1134
Copyright
Copyright © Royal Society of Edinburgh 1994

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.)

References

1Allaire, G.. Homogenizataion and two-scale convergence. SIAM J. Math. An. 23 (1992), 1482.CrossRefGoogle Scholar
2Batchelor, G. K.. The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (1970), 545–570.CrossRefGoogle Scholar
3Cox, R. G.. The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37 (1969), 601–623.CrossRefGoogle Scholar
4Keller, J. B., Rubenfeld, L. A. and Molyneux, J. E.. Extremum principles for slow viscous flows with applications to suspensions. J. Fluid Mech. 30 (1967), 97–125.CrossRefGoogle Scholar
5Lipton, R. and Avellaneda, M.. Darcy's law for slow viscous flow past a stationary array of bubbles. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 71–79.CrossRefGoogle Scholar
6Lipton, R. and Vernescu, B.. Bulk properties of two phase emulsions with surface tension effects. Int. J. Engng. Sci. (submitted).Google Scholar
7Nguetseng, G.. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623.CrossRefGoogle Scholar
8Schowalter, W. R., Chaffey, C. E. and Brenner, H.. Rheological behavior of a dilute emulsion. J. Colloid Interface Sci. 26 (1968), 152–160.CrossRefGoogle ScholarPubMed
9Tartar, L.. Convergence of the homogenization process. In Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, 127 (Berlin: Springer, 1980).Google Scholar
10Taylor, G. I.. The viscosity of a fluid containing small drops of another fluid. Proc. Roy. Soc. London Ser.A 138 (1932), 41–48.Google Scholar
11Temam, R.. Navier-Stokes Equations (Amsterdam: North-Holland, 1984).Google Scholar
12Thurston, R. N.. Waves in solids. In Mechanics of Solids, vol. 4, ed. Truesdell, C. (Berlin: Springer, 1984).Google Scholar