American Institute of Mathematical Sciences

Advanced
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.  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.   Google Scholar

[3]

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

[6]

D. Cioranescu and F. Murat, Un terme etrange venu d'ailleurs,, in Nonlinear partial differential equations and their applications, 70 (1982), 154.   Google Scholar

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Berlin-Heidelberg-New York-Tokyo, (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.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane,, J. Asymptotic Analysis, 86 (2012), 17.   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.  doi: 10.1080/00986449608936527.  Google Scholar

[14]

G. Nika and B. Vernescu, Dilute emulsions with surface tension,, Quart. Appl. Math., 74 (2016), 89.  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.   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.  doi: 10.1098/rspa.1932.0169.  Google Scholar

[17]

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

[18]

R. C. Tolman, The effect of droplet size on surface tension,, J. Chem. Phys., 17 (1949), 333.  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).   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.  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.   Google Scholar

[3]

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

[6]

D. Cioranescu and F. Murat, Un terme etrange venu d'ailleurs,, in Nonlinear partial differential equations and their applications, 70 (1982), 154.   Google Scholar

[7]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Berlin-Heidelberg-New York-Tokyo, (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.   Google Scholar

[9]

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

[10]

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

[11]

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

[12]

F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane,, J. Asymptotic Analysis, 86 (2012), 17.   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.  doi: 10.1080/00986449608936527.  Google Scholar

[14]

G. Nika and B. Vernescu, Dilute emulsions with surface tension,, Quart. Appl. Math., 74 (2016), 89.  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.   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.  doi: 10.1098/rspa.1932.0169.  Google Scholar

[17]

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

[18]

R. C. Tolman, The effect of droplet size on surface tension,, J. Chem. Phys., 17 (1949), 333.  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).   Google Scholar

[1]

Roman M. Taranets, Jeffrey T. Wong. Existence of weak solutions for particle-laden flow with surface tension. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4979-4996. doi: 10.3934/dcds.2018217
[2]

Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[3]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287
[4]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153
[5]

Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108
[6]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
[7]

Colette Calmelet, Diane Sepich. Surface tension and modeling of cellular intercalation during zebrafish gastrulation. Mathematical Biosciences & Engineering, 2010, 7 (2) : 259-275. doi: 10.3934/mbe.2010.7.259
[8]

Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023
[9]

Samuel Walsh. Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3241-3285. doi: 10.3934/dcds.2014.34.3241
[10]

Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[11]

Shengfu Deng. Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3419-3435. doi: 10.3934/dcds.2014.34.3419
[12]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[13]

Amina Mecherbet. Sedimentation of particles in Stokes flow. Kinetic & Related Models, 2019, 12 (5) : 995-1044. doi: 10.3934/krm.2019038
[14]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431
[15]

Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves. Networks & Heterogeneous Media, 2017, 12 (4) : 525-550. doi: 10.3934/nhm.2017022
[16]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[17]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
[18]

Nicolas Forcadel, Wilfredo Salazar, Mamdouh Zaydan. Homogenization of second order discrete model with local perturbation and application to traffic flow. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1437-1487. doi: 10.3934/dcds.2017060
[19]

Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339
[20]

Sergei A. Avdonin, Boris P. Belinskiy. Controllability of a string under tension. Conference Publications, 2003, 2003 (Special) : 57-67. doi: 10.3934/proc.2003.2003.57

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences