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Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension
1. | Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, United States, United States |
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. |
[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. |
[3] |
H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. B. Keller, L. A. Rubenfeld and J. E. Molyneux, Extremum Principles for slow viscous flow, J. Fluid Mech., 30 (1967), 97-125. |
[9] |
L. Leal, Laminar flow and convective transport processes, Research article, (2007), 1137-1138.
doi: 10.1080/07373939308916891. |
[10] |
R. Lipton and B. Vernescu, Homogenization of two-phase emulsions, Proc. Roy. Soc. Edinburgh, 124 (1994), 1119-1134.
doi: 10.1017/S0308210500030146. |
[11] |
J.-L. Lions, Quelques Methodes de Resolution Des Problemes Aux Limites Non Lineaires, Dunod, Paris, 2002. |
[12] |
F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane, J. Asymptotic Analysis, 86 (2012), 17-48. |
[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. |
[14] |
G. Nika and B. Vernescu, Dilute emulsions with surface tension, Quart. Appl. Math., 74 (2016), 89-111.
doi: 10.1090/qam/1403. |
[15] |
G. Nika and B. Vernescu, Asymptotics for dilute emulsions with surface tension, J. Elliptic & Parabolic Equat., 1 (2015), 215-230. |
[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. |
[17] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. |
[18] |
R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys., 17 (1949), 333-337.
doi: 10.1063/1.1747247. |
[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. |
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. |
[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. |
[3] |
H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984. |
[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. |
[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. |
[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. |
[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. |
[8] |
J. B. Keller, L. A. Rubenfeld and J. E. Molyneux, Extremum Principles for slow viscous flow, J. Fluid Mech., 30 (1967), 97-125. |
[9] |
L. Leal, Laminar flow and convective transport processes, Research article, (2007), 1137-1138.
doi: 10.1080/07373939308916891. |
[10] |
R. Lipton and B. Vernescu, Homogenization of two-phase emulsions, Proc. Roy. Soc. Edinburgh, 124 (1994), 1119-1134.
doi: 10.1017/S0308210500030146. |
[11] |
J.-L. Lions, Quelques Methodes de Resolution Des Problemes Aux Limites Non Lineaires, Dunod, Paris, 2002. |
[12] |
F. Maris and B. Vernescu, Random homogenization for a fluid flow through a membrane, J. Asymptotic Analysis, 86 (2012), 17-48. |
[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. |
[14] |
G. Nika and B. Vernescu, Dilute emulsions with surface tension, Quart. Appl. Math., 74 (2016), 89-111.
doi: 10.1090/qam/1403. |
[15] |
G. Nika and B. Vernescu, Asymptotics for dilute emulsions with surface tension, J. Elliptic & Parabolic Equat., 1 (2015), 215-230. |
[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. |
[17] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, 1984. |
[18] |
R. C. Tolman, The effect of droplet size on surface tension, J. Chem. Phys., 17 (1949), 333-337.
doi: 10.1063/1.1747247. |
[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. |
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