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On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions
1. | Université de La Rochelle, Laboratoire de Mathématiques Images et Applications EA 3165, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1 |
2. | Laboratoire de Mathématiques et Applications UMR CNRS 6086, Université de Poitiers, Téléport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil |
References:
[1] |
H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1175-1190.
doi: 10.1016/j.anihpc.2013.01.002. |
[2] |
S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in $2D$, Dyn. Partial Differ. Equ., 11 (2014), 1-38, 2014.
doi: 10.4310/DPDE.2014.v11.n1.a1. |
[3] |
F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. |
[4] |
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, (2002).
doi: 10.1137/1.9780898719208. |
[5] |
L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-547.
doi: 10.1007/s00211-014-0618-0. |
[6] |
R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow, Physical Review E, 53 (1996). |
[7] |
A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152.
doi: 10.1137/130950628. |
[8] |
M. Doi, Dynamics of domains and textures, Theoretical Challenges in the Dynamics of Complex Fluids, (1996), 293-314. |
[9] |
S. Dong, On imposing dynamic contact-angle boundary conditions for wall-bounded liquid-gas flows, Comput. Methods Appl. Mech. Engrg., (2012), 179-200.
doi: 10.1016/j.cma.2012.07.023. |
[10] |
C.M. Elliott and D.A. French, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590.
doi: 10.1007/BF01396363. |
[11] |
X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072.
doi: 10.1137/050638333. |
[12] |
X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571.
doi: 10.1090/S0025-5718-06-01915-6. |
[13] |
X. Feng and S. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50 (2012), 1320-1343.
doi: 10.1137/110827119. |
[14] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, (Encyclopedia of Mathematics and its Applications), Cambridge University Press, (2008).
doi: 10.1017/CBO9780511546754. |
[15] |
C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$, Ann. I. H. Poincaré-AN, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[16] |
C.G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, hal-01135747, 2015. |
[17] |
M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term, Math. Models Methods Appl. Sci., 20 (2010), 1363-1390.
doi: 10.1142/S0218202510004635. |
[18] |
V. Girault and A. Raviart, Finite Element Methods for Navier-Stokes equations: Theory and algorithms, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
doi: 10.1007/978-3-642-61623-5. |
[19] |
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. |
[20] |
D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase field modeling, J. Comput. Phys., 155 (1999), 96-127.
doi: 10.1006/jcph.1999.6332. |
[21] |
D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces Free Bound., 10 (2008), 15-43.
doi: 10.4171/IFB/178. |
[22] |
D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in $2d$, SIAM J. Sci. Comput., 29 (2007), 2241-2257.
doi: 10.1137/050648110. |
[23] |
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, DUNOD, 2002. |
[24] |
C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211-228.
doi: 10.1016/S0167-2789(03)00030-7. |
[25] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[26] |
M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$, Asymptot. Anal., 90 (2014), 21-51. |
[27] |
Temam, Navier Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, (2001). |
[28] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
show all references
References:
[1] |
H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1175-1190.
doi: 10.1016/j.anihpc.2013.01.002. |
[2] |
S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in $2D$, Dyn. Partial Differ. Equ., 11 (2014), 1-38, 2014.
doi: 10.4310/DPDE.2014.v11.n1.a1. |
[3] |
F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. |
[4] |
P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, (2002).
doi: 10.1137/1.9780898719208. |
[5] |
L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-547.
doi: 10.1007/s00211-014-0618-0. |
[6] |
R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow, Physical Review E, 53 (1996). |
[7] |
A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system, SIAM J. Numer. Anal., 53 (2015), 127-152.
doi: 10.1137/130950628. |
[8] |
M. Doi, Dynamics of domains and textures, Theoretical Challenges in the Dynamics of Complex Fluids, (1996), 293-314. |
[9] |
S. Dong, On imposing dynamic contact-angle boundary conditions for wall-bounded liquid-gas flows, Comput. Methods Appl. Mech. Engrg., (2012), 179-200.
doi: 10.1016/j.cma.2012.07.023. |
[10] |
C.M. Elliott and D.A. French, A second order splitting method for the Cahn-Hilliard equation, Numer. Math., 54 (1989), 575-590.
doi: 10.1007/BF01396363. |
[11] |
X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072.
doi: 10.1137/050638333. |
[12] |
X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids, Math. Comp., 76 (2007), 539-571.
doi: 10.1090/S0025-5718-06-01915-6. |
[13] |
X. Feng and S. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM J. Numer. Anal., 50 (2012), 1320-1343.
doi: 10.1137/110827119. |
[14] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, (Encyclopedia of Mathematics and its Applications), Cambridge University Press, (2008).
doi: 10.1017/CBO9780511546754. |
[15] |
C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$, Ann. I. H. Poincaré-AN, 27 (2010), 401-436.
doi: 10.1016/j.anihpc.2009.11.013. |
[16] |
C.G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, hal-01135747, 2015. |
[17] |
M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term, Math. Models Methods Appl. Sci., 20 (2010), 1363-1390.
doi: 10.1142/S0218202510004635. |
[18] |
V. Girault and A. Raviart, Finite Element Methods for Navier-Stokes equations: Theory and algorithms, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
doi: 10.1007/978-3-642-61623-5. |
[19] |
F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. |
[20] |
D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase field modeling, J. Comput. Phys., 155 (1999), 96-127.
doi: 10.1006/jcph.1999.6332. |
[21] |
D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system, Interfaces Free Bound., 10 (2008), 15-43.
doi: 10.4171/IFB/178. |
[22] |
D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in $2d$, SIAM J. Sci. Comput., 29 (2007), 2241-2257.
doi: 10.1137/050648110. |
[23] |
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, DUNOD, 2002. |
[24] |
C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Phys. D, 179 (2003), 211-228.
doi: 10.1016/S0167-2789(03)00030-7. |
[25] |
A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[26] |
M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$, Asymptot. Anal., 90 (2014), 21-51. |
[27] |
Temam, Navier Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, (2001). |
[28] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
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