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
Published electronically: December 7, 2015
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [1] Grégoire Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal. 113 (1990), no. 3, 209–259. MR 1079189, 10.1007/BF00375065
  • [2] Habib Ammari, Pierre Garapon, Hyeonbae Kang, and Hyundae Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, Asymptot. Anal. 80 (2012), no. 3-4, 189–211. MR 3025042
  • [3] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
  • [4] G. K. Batchelor, The stress system in a suspension of force-free particles, J. Fluid Mech. 41 (1970), 545-570.
  • [5] Eric Bonnetier, David Manceau, and Faouzi Triki, Asymptotic of the velocity of a dilute suspension of droplets with interfacial tension, Quart. Appl. Math. 71 (2013), no. 1, 89–117. MR 3075537, 10.1090/S0033-569X-2012-01275-7
  • [6] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
  • [7] Alain Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman’s law via epi-convergence methods, Ann. Fac. Sci. Toulouse Math. (5) 8 (1986/87), no. 2, 225–252 (English, with French summary). MR 928845
  • [8] S. J. Choi and W. R. Schowalter, Rheological properties of nondilute suspensions of deformable particles, Physics of Fluids 18 (1975), 420-427.
  • [9] D. Cioranescu and F. Murat, Un terme étrange venu d’ailleurs, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980) Res. Notes in Math., vol. 60, Pitman, Boston, Mass.-London, 1982, pp. 98–138, 389–390 (French, with English summary). MR 652509
    Doïna Cioranescu and François Murat, Un terme étrange venu d’ailleurs. II, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981) Res. Notes in Math., vol. 70, Pitman, Boston, Mass.-London, 1982, pp. 154–178, 425–426 (French, with English summary). MR 670272
  • [10] N. A. Frankel and A. Acrivos, The constitutive equation for a dilute emulsion, J. Fluid Mech. 28 (1970), 657-673.
  • [11] L. G. Leal, Laminar Flow and Convective Transport Processes, Butterworth-Heinemann, Stoneham, Massachusetts, 1992.
  • [12] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
  • [13] Robert Lipton and Marco Avellaneda, Darcy’s law for slow viscous flow past a stationary array of bubbles, Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), no. 1-2, 71–79. MR 1051608, 10.1017/S0308210500024276
  • [14] Robert Lipton and Bogdan Vernescu, Homogenisation of two-phase emulsions, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 6, 1119–1134. MR 1313192, 10.1017/S0308210500030146
  • [15] F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane, Asymptot. Anal. 86 (2014), no. 1, 17–48. MR 3177474
  • [16] W. R. Schowalter, C. C. Chaffey and H. Brenner, Rheological behavior of a dilute emulsion, J. Colloid Int. Sci., 26 (1968), 152-160.
  • [17] R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252
  • [18] Roger Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
  • [19] G. I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. Roy. Soc. London Ser. A 138 (1932), 41-48.
  • [20] B. Vernescu, Convergence results for the homogenization of flow in fractured porous media, IMA Preprint Series, 732, 1990, available at http://www.ima.umn.edu/preprints/Nov90Series/Nov90Series.html
  • [21] Z. Zapryanov and S. Tabakova, Dynamics of bubbles, drops and rigid particles, Fluid Mechanics and its Applications, vol. 50, Kluwer Academic Publishers Group, Dordrecht, 1999. MR 1673353

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35J25, 35J20, 35K10, 76D07, 76T20

Retrieve articles in all journals with MSC (2010): 35J25, 35J20, 35K10, 76D07, 76T20

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

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website