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October  2016, 9(5): 1553-1564. doi: 10.3934/dcdss.2016062

Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension

Grigor Nika 1, and  Bogdan Vernescu 1,

1. 

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, United States, United States

Received  June 2015 Revised  November 2015 Published  October 2016

In this paper we are interested in a problem of dilute emulsions of two immiscible viscous fluids, in which one is distributed in the other in the form of droplets of arbitrary shape, with non-uniform surface tension due to surfactants. The problem includes an essential kinematic condition on the droplets. In the periodic homogenization framework, it can be shown using Mosco-convergence that, as the size of the droplets converges to zero faster than the distance between the droplets, the emulsion behaves in the limit like the continuous phase. Here we determine the rate of convergence of the velocity field for the emulsion to that of the velocity for the one fluid problem and in addition, we determine the corrector in terms of the bulk and surface polarization tensors.
Keywords: surface tension, homogenization., Stokes flow, Emulsions.
Mathematics Subject Classification: Primary: 35J25, 35J20; Secondary: 76D07, 76T2.
Citation: Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062
References:
[1]

G. Allaire, Homogenization of Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rat. Mech. Anal., 113 (1991), 209-259. doi: 10.1007/BF00375065.  Google Scholar

[2]

H. Ammari, P. Garapon, H. Kang and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, J. Asymptotic Analysis, 80 (2012), 189-211.  Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.  Google Scholar

[4]

E. Bonnetier, D. Manceau and F. Triki, Asymptotic of the velocity of a dilute suspension of droplets with interfacial tension, Quart. Appl. Math., 71 (2013), 89-117. doi: 10.1090/S0033-569X-2012-01275-7.  Google Scholar

[5]

A. Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman's law via epi-convergence methods, Ann. Fac. Sci. Toulouse, 8 (1986), 225-252.  Google Scholar

[6]

D. Cioranescu and F. Murat, Un terme etrange venu d'ailleurs, in Nonlinear partial differential equations and their applications, College de France Seminar, II & III (eds. H.Brezis and J.L.Lions), Pitman, 70 (1982), 154-178.  Google Scholar

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[8]

J. B. Keller, L. A. Rubenfeld and J. E. Molyneux, Extremum Principles for slow viscous flow, J. Fluid Mech., 30 (1967), 97-125. Google Scholar

[9]

L. Leal, Laminar flow and convective transport processes, Research article, (2007), 1137-1138. doi: 10.1080/07373939308916891.  Google Scholar

[10]

R. Lipton and B. Vernescu, Homogenization of two-phase emulsions, Proc. Roy. Soc. Edinburgh, 124 (1994), 1119-1134. doi: 10.1017/S0308210500030146.  Google Scholar

[11]

J.-L. Lions, Quelques Methodes de Resolution Des Problemes Aux Limites Non Lineaires, Dunod, Paris, 2002. Google Scholar

[12]

F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane, J. Asymptotic Analysis, 86 (2012), 17-48.  Google Scholar

[13]

A. Nadim, A concise introduction to surface rheology with application to dilute emulsions of viscous drops, Chem. Engin. Comm., 148/150 (1996), 391-407. doi: 10.1080/00986449608936527.  Google Scholar

[14]

G. Nika and B. Vernescu, Dilute emulsions with surface tension, Quart. Appl. Math., 74 (2016), 89-111. doi: 10.1090/qam/1403.  Google Scholar

[15]

G. Nika and B. Vernescu, Asymptotics for dilute emulsions with surface tension, J. Elliptic & Parabolic Equat., 1 (2015), 215-230.  Google Scholar

[16]

G. I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. Roy. Soc. London Ser. A, 138 (1932), 41-48. doi: 10.1098/rspa.1932.0169.  Google Scholar

[17]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar

[18]

R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys., 17 (1949), 333-337. doi: 10.1063/1.1747247.  Google Scholar

[19]

B. Vernescu, Convergence results for the homogenization of flow in fractured porous media, IMA Preprint Series, 732 (1990). Available from: http://www.ima.umn.edu/preprints/Nov90Series/Nov90Series.html. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization of Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rat. Mech. Anal., 113 (1991), 209-259. doi: 10.1007/BF00375065.  Google Scholar

[2]

H. Ammari, P. Garapon, H. Kang and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, J. Asymptotic Analysis, 80 (2012), 189-211.  Google Scholar

[3]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.  Google Scholar

[4]

E. Bonnetier, D. Manceau and F. Triki, Asymptotic of the velocity of a dilute suspension of droplets with interfacial tension, Quart. Appl. Math., 71 (2013), 89-117. doi: 10.1090/S0033-569X-2012-01275-7.  Google Scholar

[5]

A. Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous media. Brinkman's law via epi-convergence methods, Ann. Fac. Sci. Toulouse, 8 (1986), 225-252.  Google Scholar

[6]

D. Cioranescu and F. Murat, Un terme etrange venu d'ailleurs, in Nonlinear partial differential equations and their applications, College de France Seminar, II & III (eds. H.Brezis and J.L.Lions), Pitman, 70 (1982), 154-178.  Google Scholar

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[8]

J. B. Keller, L. A. Rubenfeld and J. E. Molyneux, Extremum Principles for slow viscous flow, J. Fluid Mech., 30 (1967), 97-125. Google Scholar

[9]

L. Leal, Laminar flow and convective transport processes, Research article, (2007), 1137-1138. doi: 10.1080/07373939308916891.  Google Scholar

[10]

R. Lipton and B. Vernescu, Homogenization of two-phase emulsions, Proc. Roy. Soc. Edinburgh, 124 (1994), 1119-1134. doi: 10.1017/S0308210500030146.  Google Scholar

[11]

J.-L. Lions, Quelques Methodes de Resolution Des Problemes Aux Limites Non Lineaires, Dunod, Paris, 2002. Google Scholar

[12]

F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane, J. Asymptotic Analysis, 86 (2012), 17-48.  Google Scholar

[13]

A. Nadim, A concise introduction to surface rheology with application to dilute emulsions of viscous drops, Chem. Engin. Comm., 148/150 (1996), 391-407. doi: 10.1080/00986449608936527.  Google Scholar

[14]

G. Nika and B. Vernescu, Dilute emulsions with surface tension, Quart. Appl. Math., 74 (2016), 89-111. doi: 10.1090/qam/1403.  Google Scholar

[15]

G. Nika and B. Vernescu, Asymptotics for dilute emulsions with surface tension, J. Elliptic & Parabolic Equat., 1 (2015), 215-230.  Google Scholar

[16]

G. I. Taylor, The viscosity of a fluid containing small drops of another fluid, Proc. Roy. Soc. London Ser. A, 138 (1932), 41-48. doi: 10.1098/rspa.1932.0169.  Google Scholar

[17]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar

[18]

R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys., 17 (1949), 333-337. doi: 10.1063/1.1747247.  Google Scholar

[19]

B. Vernescu, Convergence results for the homogenization of flow in fractured porous media, IMA Preprint Series, 732 (1990). Available from: http://www.ima.umn.edu/preprints/Nov90Series/Nov90Series.html. Google Scholar

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