Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Dilute emulsions with surface tension

Authors: Grigor Nika and Bogdan Vernescu
Journal: Quart. Appl. Math. 74 (2016), 89-111
MSC (2010): Primary 35J25, 35J20, 35K10; Secondary 76D07, 76T20
DOI: https://doi.org/10.1090/qam/1403
Published electronically: December 7, 2015
MathSciNet review: 3472521
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider an emulsion formed by two newtonian fluids in which one fluid is dispersed under the form of droplets of arbitrary shape in the presence of surface tension. We consider both cases of droplets with fixed centers of mass and of convected droplets. In the non-dilute case, for spherical droplets of radius $ a_\epsilon $ of the same order as the period length $ \epsilon $, the two models were studied by Lipton-Avellaneda (1990) and Lipton-Vernescu (1994). Here we are interested in the time-dependent, dilute case when the characteristic size of the droplets $ a_\epsilon $, of arbitrary shape, is much smaller than $ \epsilon $. We study the limit behavior when $ \epsilon \to 0$ in each of these two models. We establish a Brinkman type law for the critical size $ a_\epsilon = O(\epsilon ^3)$ in the first case, whereas in the second case there is no ``strange'' term, and in the limit the flow is unperturbed by the droplets.

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Additional Information

Grigor Nika
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: gnika@wpi.edu

Bogdan Vernescu
Affiliation: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts 01609
Email: vernescu@wpi.edu

DOI: https://doi.org/10.1090/qam/1403
Keywords: Stokes flow, Brinkman equations, surface tension, $\Gamma$-convergence, Mosco-convergence, emulsions
Received by editor(s): May 6, 2014
Published electronically: December 7, 2015
Article copyright: © Copyright 2015 Brown University

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