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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

 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.
Citation: Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure &amp; 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.  Google Scholar [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.  Google Scholar [3] F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20 (1999), 175-212.  Google Scholar [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, (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, Numer. Math., 128 (2014), 517-547. 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, 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, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628.  Google Scholar [8] M. Doi, Dynamics of domains and textures, Theoretical Challenges in the Dynamics of Complex Fluids, (1996), 293-314. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Numer. Anal., 50 (2012), 1320-1343. 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), Cambridge University Press, (2008). doi: 10.1017/CBO9780511546754.  Google Scholar [15] C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$, Ann. I. H. Poincaré-AN, 27 (2010), 401-436. 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, hal-01135747, 2015. Google Scholar [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.  Google Scholar [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.  Google Scholar [19] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, Phys. D, 179 (2003), 211-228. 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, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [26] M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$, Asymptot. Anal., 90 (2014), 21-51.  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, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1175-1190. 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 $2D$, Dyn. Partial Differ. Equ., 11 (2014), 1-38, 2014. 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, Asymptot. Anal., 20 (1999), 175-212.  Google Scholar [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, (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, Numer. Math., 128 (2014), 517-547. 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, 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, SIAM J. Numer. Anal., 53 (2015), 127-152. doi: 10.1137/130950628.  Google Scholar [8] M. Doi, Dynamics of domains and textures, Theoretical Challenges in the Dynamics of Complex Fluids, (1996), 293-314. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, SIAM J. Numer. Anal., 50 (2012), 1320-1343. 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), Cambridge University Press, (2008). doi: 10.1017/CBO9780511546754.  Google Scholar [15] C.G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in $2D$, Ann. I. H. Poincaré-AN, 27 (2010), 401-436. 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, hal-01135747, 2015. Google Scholar [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.  Google Scholar [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.  Google Scholar [19] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  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, Phys. D, 179 (2003), 211-228. 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, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590.  Google Scholar [26] M. Tachim, Pullback attractors for a non-autonomous Cahn-Hilliard-Navier-Stokes system in $2D$, Asymptot. Anal., 90 (2014), 21-51.  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, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar
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