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September  2019, 18(5): 2283-2298. doi: 10.3934/cpaa.2019103

Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  September 2017 Revised  September 2017 Published  April 2019

This paper is concerned with the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Thanks to the strong coupling at the boundary, it is very difficult to obtain the uniqueness of an energy solution for problem (1)-(3) even in two dimension. To overcome this difficulty, inspired by the idea of Sell's radical approach (see [49]) to the global attractor of the three dimensional Navier-Stokes equations, we prove the closedness of the set $ W $ of all global energy solutions for problem (1)-(3) equipped with some metric such that the $ \omega $-limit set of any bounded subset in $ W $ still stay in $ W, $ which is crucial to prove the existence of a global attractor for problem (1)-(3). In addition, we prove the existence of an absorbing set in $ W $ and the uniform compactness of the semigroup $ S_t $ for problem (1)-(3), which entails the existence of a global attractor in $ W $ for problem (1)-(3).

Citation: Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103
References:
[1]

H. Abels, Long-time behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Proceedings of the Conference Nonlocal and Abstract Parabolic Equations and their Applications, Bedlewo, Banach Center Publications, 86 (2009), 9-19.  doi: 10.4064/bc86-0-1.

[2]

H. Abels, On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.

[3]

H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.  doi: 10.1137/110829246.

[4]

H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.  doi: 10.1512/iumj.2008.57.3391.

[5]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.

[6]

V. E. BadalassiH. D. Ceniceros and S. Banerjee, Computation of multiphase systems with phase field models, J. Comput. Phys., 190 (2003), 371-397.  doi: 10.1016/S0021-9991(03)00280-8.

[7]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[8]

K. BaoY. ShiS. Sun and X. P. Wang, A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.  doi: 10.1016/j.jcp.2012.07.027.

[9]

A. BertiV. Berti and D. Grandi, Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids, Nonlinearity, 24 (2011), 3143-3164.  doi: 10.1088/0951-7715/24/11/008.

[10]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phy. D: Appl. Phys., 32 (1999), 1119-1123. 

[11]

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.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[12]

S. BosiaM. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[13]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. 

[14]

F. Boyer, Nonhomogenous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.

[15]

F. Boyer, Atheoretical and numerical model for the study of incompressiblemodel flows, Computers and Fluids, 31 (2002), 41-68. 

[16]

C. S. Cao and C. G. Gal, Global solutions for the 2D Navier-Stokes-Cahn-Hilliard model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.  doi: 10.1088/0951-7715/25/11/3211.

[17]

R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 53 (1996), 3832-3840. 

[18]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.

[19]

N. J. Cutland and H. J. Keisler, Attractors and neo-attractors for 3D stochastic Navier-Stokes equations, Stoch. Dyn., 5 (2005), 487-533.  doi: 10.1142/S0219493705001559.

[20]

E. B. DussanV, The moving contact line: the slip boundary condition, J. Fluid Mech., 77 (1976), 665-684. 

[21]

E. B. DussanV and S. H. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech., 65 (1974), 71-95. 

[22]

X. B. Feng, Fully discrete 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.

[23]

X. B. FengY. N. 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.

[24]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[25]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678.  doi: 10.1007/s11401-010-0603-6.

[26]

C. G. Gal and M. Grasselli, Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Phys. D, 240 (2011), 629-635.  doi: 10.1016/j.physd.2010.11.014.

[27]

C. G. GalM. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), 1-47.  doi: 10.1007/s00526-016-0992-9.

[28]

M. Gao and X. P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386.  doi: 10.1016/j.jcp.2011.10.015.

[29]

M. E. GurtinD. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.

[30]

M. Heida, On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system, Internat. J. Engrg. Sci., 62 (2013), 126-156.  doi: 10.1016/j.ijengsci.2012.09.005.

[31]

M. HeidaJ. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys., 63 (2012), 145-169.  doi: 10.1007/s00033-011-0139-y.

[32]

M. Hintermuller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772.  doi: 10.1137/120865628.

[33]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. 

[34]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modelling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.

[35]

D. Jasnow and J. Vinals, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669. 

[36]

D. KayV. 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.

[37]

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.

[38]

J. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12 (2012), 613-661.  doi: 10.4208/cicp.301110.040811a.

[39]

J. KimK. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511-543.  doi: 10.1016/j.jcp.2003.07.035.

[40]

A. G. LamorgeseD. Molin and R. Mauri, Phase field approach to multiphase flow modeling, Milan J. Math., 79 (2011), 597-642.  doi: 10.1007/s00032-011-0171-6.

[41]

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.

[42]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[43]

J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos. Trans. R. Soc. London, 170 (1879), 704-712. 

[44]

T. Tachim. Medjo, Pullback attracots for a non-autonomous Cahn-Hilliard-Navier-Stokes system in 2D, Asymptot. Anal., 90 (2014), 21-51. 

[45]

H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18 (1964), 1-18. 

[46]

T. Z. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.

[47]

T. Z. QianX. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.

[48]

R. Ruiz and D. R. Nelson, Turbulence in binary fluid mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 23 (1981), 3224-3246. 

[49]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.

[50]

J. Shen and X. F. Yang, Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows, Chin. Ann. Math. Ser. B, 31 (2010), 743-758.  doi: 10.1007/s11401-010-0599-y.

[51]

J. ShenX. F. Yang and H. J. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), 617-630.  doi: 10.1016/j.jcp.2014.12.046.

[52]

Y. ShiK. Bao and X. P. Wang, 3D adaptive finite element method for a phase field model for the moving contact line problems, Inverse Probl. Imaging, 7 (2013), 947-959.  doi: 10.3934/ipi.2013.7.947.

[53]

E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 20 (1979), 595-605. 

[54]

V. N. Starovoitov, On the motion of a two-component fluid in the presence of capillary forces, Math. Notes, 62 (1997), 244-254.  doi: 10.1007/BF02355911.

[55]

Z. J. TanK. M. Lim and B. C. Khoo, An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model, J. Comput. Phys., 225 (2007), 1137-1158.  doi: 10.1016/j.jcp.2007.01.019.

[56]

X. P. Wang and Y. G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-294.  doi: 10.4310/MAA.2007.v14.n3.a6.

[57]

P. T. YueC. F. Zhou and J. J. Feng, Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., 645 (2010), 279-294.  doi: 10.1017/S0022112009992679.

[58]

L. Y. ZhaoH. Wu and H. Y. Huang, Convergence to equilibrium for a phase-field model for the mixture of two viscous incompressible fluids, Commun. Math. Sci., 7 (2009), 939-962. 

[59]

Y. Zhou and J. S. Fan, The vanishing viscosity limit for a 2D Cahn-Hilliard-Navier-Stokes system with a slip boundary condition, Nonlinear Anal. Real World Appl., 14 (2013), 1130-1134.  doi: 10.1016/j.nonrwa.2012.09.003.

show all references

References:
[1]

H. Abels, Long-time behavior of solutions of a Navier-Stokes/Cahn-Hilliard system, Proceedings of the Conference Nonlocal and Abstract Parabolic Equations and their Applications, Bedlewo, Banach Center Publications, 86 (2009), 9-19.  doi: 10.4064/bc86-0-1.

[2]

H. Abels, On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506.  doi: 10.1007/s00205-008-0160-2.

[3]

H. Abels, Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow, SIAM J. Math. Anal., 44 (2012), 316-340.  doi: 10.1137/110829246.

[4]

H. Abels and E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698.  doi: 10.1512/iumj.2008.57.3391.

[5]

D. M. AndersonG. B. McFadden and A. A. Wheeler, Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30 (1998), 139-165.  doi: 10.1146/annurev.fluid.30.1.139.

[6]

V. E. BadalassiH. D. Ceniceros and S. Banerjee, Computation of multiphase systems with phase field models, J. Comput. Phys., 190 (2003), 371-397.  doi: 10.1016/S0021-9991(03)00280-8.

[7]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502.  doi: 10.1007/s003329900037.

[8]

K. BaoY. ShiS. Sun and X. P. Wang, A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083-8099.  doi: 10.1016/j.jcp.2012.07.027.

[9]

A. BertiV. Berti and D. Grandi, Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids, Nonlinearity, 24 (2011), 3143-3164.  doi: 10.1088/0951-7715/24/11/008.

[10]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flows, J. Phy. D: Appl. Phys., 32 (1999), 1119-1123. 

[11]

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.  doi: 10.4310/DPDE.2014.v11.n1.a1.

[12]

S. BosiaM. Grasselli and A. Miranville, On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[13]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212. 

[14]

F. Boyer, Nonhomogenous Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 225-259.  doi: 10.1016/S0294-1449(00)00063-9.

[15]

F. Boyer, Atheoretical and numerical model for the study of incompressiblemodel flows, Computers and Fluids, 31 (2002), 41-68. 

[16]

C. S. Cao and C. G. Gal, Global solutions for the 2D Navier-Stokes-Cahn-Hilliard model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234.  doi: 10.1088/0951-7715/25/11/3211.

[17]

R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, 53 (1996), 3832-3840. 

[18]

A. Cheskidov and C. Foias, On global attractors of the 3D Navier-Stokes equations, J. Differential Equations, 231 (2006), 714-754.  doi: 10.1016/j.jde.2006.08.021.

[19]

N. J. Cutland and H. J. Keisler, Attractors and neo-attractors for 3D stochastic Navier-Stokes equations, Stoch. Dyn., 5 (2005), 487-533.  doi: 10.1142/S0219493705001559.

[20]

E. B. DussanV, The moving contact line: the slip boundary condition, J. Fluid Mech., 77 (1976), 665-684. 

[21]

E. B. DussanV and S. H. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech., 65 (1974), 71-95. 

[22]

X. B. Feng, Fully discrete 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.

[23]

X. B. FengY. N. 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.

[24]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436.  doi: 10.1016/j.anihpc.2009.11.013.

[25]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678.  doi: 10.1007/s11401-010-0603-6.

[26]

C. G. Gal and M. Grasselli, Instability of two-phase flows: a lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Phys. D, 240 (2011), 629-635.  doi: 10.1016/j.physd.2010.11.014.

[27]

C. G. GalM. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines, Calc. Var. Partial Differential Equations, 55 (2016), 1-47.  doi: 10.1007/s00526-016-0992-9.

[28]

M. Gao and X. P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386.  doi: 10.1016/j.jcp.2011.10.015.

[29]

M. E. GurtinD. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831.  doi: 10.1142/S0218202596000341.

[30]

M. Heida, On the derivation of thermodynamically consistent boundary conditions for the Cahn-Hilliard-Navier-Stokes system, Internat. J. Engrg. Sci., 62 (2013), 126-156.  doi: 10.1016/j.ijengsci.2012.09.005.

[31]

M. HeidaJ. Málek and K. R. Rajagopal, On the development and generalizations of Cahn-Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys., 63 (2012), 145-169.  doi: 10.1007/s00033-011-0139-y.

[32]

M. Hintermuller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system, SIAM J. Control Optim., 52 (2014), 747-772.  doi: 10.1137/120865628.

[33]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. 

[34]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modelling, J. Comput. Phys., 155 (1999), 96-127.  doi: 10.1006/jcph.1999.6332.

[35]

D. Jasnow and J. Vinals, Coarse-grained description of thermo-capillary flow, Phys. Fluids, 8 (1996), 660-669. 

[36]

D. KayV. 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.

[37]

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.

[38]

J. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys., 12 (2012), 613-661.  doi: 10.4208/cicp.301110.040811a.

[39]

J. KimK. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), 511-543.  doi: 10.1016/j.jcp.2003.07.035.

[40]

A. G. LamorgeseD. Molin and R. Mauri, Phase field approach to multiphase flow modeling, Milan J. Math., 79 (2011), 597-642.  doi: 10.1007/s00032-011-0171-6.

[41]

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.

[42]

J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[43]

J. C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos. Trans. R. Soc. London, 170 (1879), 704-712. 

[44]

T. Tachim. Medjo, Pullback attracots for a non-autonomous Cahn-Hilliard-Navier-Stokes system in 2D, Asymptot. Anal., 90 (2014), 21-51. 

[45]

H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech., 18 (1964), 1-18. 

[46]

T. Z. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.

[47]

T. Z. QianX. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360.  doi: 10.1017/S0022112006001935.

[48]

R. Ruiz and D. R. Nelson, Turbulence in binary fluid mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 23 (1981), 3224-3246. 

[49]

G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33.  doi: 10.1007/BF02218613.

[50]

J. Shen and X. F. Yang, Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows, Chin. Ann. Math. Ser. B, 31 (2010), 743-758.  doi: 10.1007/s11401-010-0599-y.

[51]

J. ShenX. F. Yang and H. J. Yu, Efficient energy stable numerical schemes for a phase field moving contact line model, J. Comput. Phys., 284 (2015), 617-630.  doi: 10.1016/j.jcp.2014.12.046.

[52]

Y. ShiK. Bao and X. P. Wang, 3D adaptive finite element method for a phase field model for the moving contact line problems, Inverse Probl. Imaging, 7 (2013), 947-959.  doi: 10.3934/ipi.2013.7.947.

[53]

E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Phys. Rev. A: At, Mol. Opt. Phys., 20 (1979), 595-605. 

[54]

V. N. Starovoitov, On the motion of a two-component fluid in the presence of capillary forces, Math. Notes, 62 (1997), 244-254.  doi: 10.1007/BF02355911.

[55]

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