American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species
  • CPAA Home
  • This Issue
  • Next Article
    Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary
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 & 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,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 1175. doi: 10.1016/j.anihpc.2013.01.002. Google Scholar

[2]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Dyn. Partial Differ. Equ.}, 11 (2014), 1. doi: 10.4310/DPDE.2014.v11.n1.a1. Google Scholar

[3]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptot. Anal.}, 20 (1999), 175. Google Scholar

[4]

P.G. Ciarlet, The Finite Element Method for Elliptic Problems,, SIAM, (2002). doi: 10.1137/1.9780898719208. Google Scholar

[5]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls,, \emph{Numer. Math.}, 128 (2014), 517. doi: 10.1007/s00211-014-0618-0. Google Scholar

[6]

R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow,, \emph{Physical Review E}, 53 (1996). Google Scholar

[7]

A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system,, \emph{SIAM J. Numer. Anal.}, 53 (2015), 127. doi: 10.1137/130950628. Google Scholar

[8]

M. Doi, Dynamics of domains and textures,, \emph{Theoretical Challenges in the Dynamics of Complex Fluids}, (1996), 293. Google Scholar

[9]

S. Dong, On imposing dynamic contact-angle boundary conditions for wall-bounded liquid-gas flows,, \emph{Comput. Methods Appl. Mech. Engrg.}, (2012), 179. doi: 10.1016/j.cma.2012.07.023. Google Scholar

[10]

C.M. Elliott and D.A. French, A second order splitting method for the Cahn-Hilliard equation,, \emph{Numer. Math.}, 54 (1989), 575. doi: 10.1007/BF01396363. Google Scholar

[11]

X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 1049. doi: 10.1137/050638333. Google Scholar

[12]

X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids,, \emph{Math. Comp.}, 76 (2007), 539. doi: 10.1090/S0025-5718-06-01915-6. Google Scholar

[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,, \emph{SIAM J. Numer. Anal.}, 50 (2012), 1320. doi: 10.1137/110827119. Google Scholar

[14]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, (Encyclopedia of Mathematics and its Applications), (2008). doi: 10.1017/CBO9780511546754. Google Scholar

[15]

C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Ann. I. H. Poincar\'e-AN}, 27 (2010), 401. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar

[16]

C.G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines,, \emph{hal-01135747}, (2015). Google Scholar

[17]

M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 1363. doi: 10.1142/S0218202510004635. Google Scholar

[18]

V. Girault and A. Raviart, Finite Element Methods for Navier-Stokes equations: Theory and algorithms,, Springer-Verlag, (1981). doi: 10.1007/978-3-642-61623-5. Google Scholar

[19]

F. Hecht, New development in FreeFem++,, \emph{J. Numer. Math.}, 20 (2012), 251. Google Scholar

[20]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase field modeling,, \emph{J. Comput. Phys.}, 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar

[21]

D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system,, \emph{Interfaces Free Bound.}, 10 (2008), 15. doi: 10.4171/IFB/178. Google Scholar

[22]

D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in $2d$,, \emph{SIAM J. Sci. Comput.}, 29 (2007), 2241. doi: 10.1137/050648110. Google Scholar

[23]

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, DUNOD, (2002). Google Scholar

[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,, \emph{Phys. D}, 179 (2003), 211. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar

[25]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar

[26]

M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Asymptot. Anal.}, 90 (2014), 21. Google Scholar

[27]

Temam, Navier Stokes Equations. Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001). Google Scholar

[28]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

show all references

References:
[1]

H. Abels, D. Depner and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 30 (2013), 1175. doi: 10.1016/j.anihpc.2013.01.002. Google Scholar

[2]

S. Bosia and S. Gatti, Pullback exponential attractor for a Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Dyn. Partial Differ. Equ.}, 11 (2014), 1. doi: 10.4310/DPDE.2014.v11.n1.a1. Google Scholar

[3]

F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation,, \emph{Asymptot. Anal.}, 20 (1999), 175. Google Scholar

[4]

P.G. Ciarlet, The Finite Element Method for Elliptic Problems,, SIAM, (2002). doi: 10.1137/1.9780898719208. Google Scholar

[5]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls,, \emph{Numer. Math.}, 128 (2014), 517. doi: 10.1007/s00211-014-0618-0. Google Scholar

[6]

R. Chella and J. Vinals, Mixing of a two-phase fluid by a cavity flow,, \emph{Physical Review E}, 53 (1996). Google Scholar

[7]

A. Diegel, X. Feng and S. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system,, \emph{SIAM J. Numer. Anal.}, 53 (2015), 127. doi: 10.1137/130950628. Google Scholar

[8]

M. Doi, Dynamics of domains and textures,, \emph{Theoretical Challenges in the Dynamics of Complex Fluids}, (1996), 293. Google Scholar

[9]

S. Dong, On imposing dynamic contact-angle boundary conditions for wall-bounded liquid-gas flows,, \emph{Comput. Methods Appl. Mech. Engrg.}, (2012), 179. doi: 10.1016/j.cma.2012.07.023. Google Scholar

[10]

C.M. Elliott and D.A. French, A second order splitting method for the Cahn-Hilliard equation,, \emph{Numer. Math.}, 54 (1989), 575. doi: 10.1007/BF01396363. Google Scholar

[11]

X. Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows,, \emph{SIAM J. Numer. Anal.}, 44 (2006), 1049. doi: 10.1137/050638333. Google Scholar

[12]

X. Feng, Y. He and C. Liu, Analysis of finite element approximations of a phase field model for two-phase fluids,, \emph{Math. Comp.}, 76 (2007), 539. doi: 10.1090/S0025-5718-06-01915-6. Google Scholar

[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,, \emph{SIAM J. Numer. Anal.}, 50 (2012), 1320. doi: 10.1137/110827119. Google Scholar

[14]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, (Encyclopedia of Mathematics and its Applications), (2008). doi: 10.1017/CBO9780511546754. Google Scholar

[15]

C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Ann. I. H. Poincar\'e-AN}, 27 (2010), 401. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar

[16]

C.G. Gal, M. Grasselli and A. Miranville, Cahn-Hilliard-Navier-Stokes system with moving contact lines,, \emph{hal-01135747}, (2015). Google Scholar

[17]

M. Grasselli and M. Pierre, A splitting method for the Cahn-Hilliard equation with inertial term,, \emph{Math. Models Methods Appl. Sci.}, 20 (2010), 1363. doi: 10.1142/S0218202510004635. Google Scholar

[18]

V. Girault and A. Raviart, Finite Element Methods for Navier-Stokes equations: Theory and algorithms,, Springer-Verlag, (1981). doi: 10.1007/978-3-642-61623-5. Google Scholar

[19]

F. Hecht, New development in FreeFem++,, \emph{J. Numer. Math.}, 20 (2012), 251. Google Scholar

[20]

D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase field modeling,, \emph{J. Comput. Phys.}, 155 (1999), 96. doi: 10.1006/jcph.1999.6332. Google Scholar

[21]

D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system,, \emph{Interfaces Free Bound.}, 10 (2008), 15. doi: 10.4171/IFB/178. Google Scholar

[22]

D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in $2d$,, \emph{SIAM J. Sci. Comput.}, 29 (2007), 2241. doi: 10.1137/050648110. Google Scholar

[23]

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, DUNOD, (2002). Google Scholar

[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,, \emph{Phys. D}, 179 (2003), 211. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar

[25]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions,, \emph{Math. Methods Appl. Sci.}, 28 (2005), 709. doi: 10.1002/mma.590. Google Scholar

[26]

M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$,, \emph{Asymptot. Anal.}, 90 (2014), 21. Google Scholar

[27]

Temam, Navier Stokes Equations. Theory and Numerical Analysis,, AMS Chelsea Publishing, (2001). Google Scholar

[28]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Second edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[1]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[2]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[3]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[4]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[5]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[6]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[7]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[8]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[9]

Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829
[10]

Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517
[11]

Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405
[12]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[13]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[14]

Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[15]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[16]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
[17]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
[18]

Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
[19]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[20]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences