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July  2016, 15(4): 1419-1449. doi: 10.3934/cpaa.2016.15.1419

On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions

Laurence Cherfils 1, and  Madalina Petcu 2,

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

Received  December 2015 Revised  January 2016 Published  April 2016

In the present article we study the viscous Cahn-Hilliard-Navier-Stokes model, endowed with dynamic boundary conditions, from the theoretical and numerical point of view. We start by deducing results on the existence, uniqueness and regularity of the solutions for the continuous problem. Then we propose a space semi-discrete finite element approximation of the model and we study the convergence of the approximate scheme. We also prove the stability and convergence of a fully discretized scheme, obtained using the semi-implicit Euler scheme applied to the space semi-discretization proposed previously. Numerical simulations are also presented to illustrate the theoretical results.
Keywords: viscous Cahn-Hilliard equations, finite element method, well-posedness, error estimates, backward Euler scheme., Navier-Stokes equations.
Mathematics Subject Classification: 65M60, 65M1.
Citation: Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419
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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