Study of a three component Cahn-Hilliard flow model
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 653-687.

In this paper, we propose a new diffuse interface model for the study of three immiscible component incompressible viscous flows. The model is based on the Cahn-Hilliard free energy approach. The originality of our study lies in particular in the choice of the bulk free energy. We show that one must take care of this choice in order for the model to give physically relevant results. More precisely, we give conditions for the model to be well-posed and to satisfy algebraically and dynamically consistency properties with the two-component models. Notice that our model is also able to cope with some total spreading situations. We propose to take into account the hydrodynamics of the mixture by coupling our ternary Cahn-Hilliard system and the Navier-Stokes equation supplemented by capillary force terms accounting for surface tension effects between the components. Finally, we present some numerical results which illustrate our analysis and which confirm that our model has a better behavior than other possible similar models.

DOI : 10.1051/m2an:2006028
Classification : 35B35, 35K55, 76T30
Mots-clés : multicomponent flows, Cahn-Hilliard equations, stability 
@article{M2AN_2006__40_4_653_0,
     author = {Boyer, Franck and Lapuerta, C\'eline},
     title = {Study of a three component {Cahn-Hilliard} flow model},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {653--687},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {4},
     year = {2006},
     doi = {10.1051/m2an:2006028},
     mrnumber = {2274773},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2006028/}
}
TY  - JOUR
AU  - Boyer, Franck
AU  - Lapuerta, Céline
TI  - Study of a three component Cahn-Hilliard flow model
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2006
SP  - 653
EP  - 687
VL  - 40
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2006028/
DO  - 10.1051/m2an:2006028
LA  - en
ID  - M2AN_2006__40_4_653_0
ER  - 
%0 Journal Article
%A Boyer, Franck
%A Lapuerta, Céline
%T Study of a three component Cahn-Hilliard flow model
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2006
%P 653-687
%V 40
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2006028/
%R 10.1051/m2an:2006028
%G en
%F M2AN_2006__40_4_653_0
Boyer, Franck; Lapuerta, Céline. Study of a three component Cahn-Hilliard flow model. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 653-687. doi : 10.1051/m2an:2006028. http://www.numdam.org/articles/10.1051/m2an:2006028/

[1] N.D. Alikakos, P.W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. An. 128 (1994) 165-205. | Zbl

[2] D.M. Anderson, G.B. Mcfadden and A.A. Wheeler, Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139-165.

[3] J.W. Barrett and J.F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy and a concentration dependent mobility matrix. Math. Mod. Meth. Appl. S. 9 (1999) 627-663. | Zbl

[4] J.F. Blowey, M.I.M. Copetti and C.M. Elliott, Numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111-139. | Zbl

[5] F. Boyer, Mathematical study of multiphase flow under shear through order parameter formulation. Asymptotic Anal. 20 (1999) 175-212. | Zbl

[6] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (2002) 41-68. | Zbl

[7] M.I.M. Copetti, Numerical experiments of phase separation in ternary mixtures. Math. Comput. Simulat. 52 (2000) 41-51.

[8] C.M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in Mathematical Models for Phase Change Problems, J.F. Rodrigues Ed., Birkhäuser Verlag Basel. Intern. Ser. Numer. Math. 88 (1989). | MR | Zbl

[9] C.M. Elliott and S. Luckhaus, A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. IMA Preprint Series 887 (1991).

[10] D.J. Eyre, Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686-1712. | Zbl

[11] H. Garcke and A. Novick-Cohen, A singular limit for a system of degenerate Cahn-Hilliard equations. Adv. Differ. Equ. 5 (2000) 401-434. | Zbl

[12] H. Garcke, B. Nestler and B. Stoth, On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Physica D 115 (1998) 87-108. | Zbl

[13] H. Garcke, B. Nestler and B. Stoth, A multi phase field: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60 (1999) 295-315. | Zbl

[14] G.A. Greene, J.C. Chen and M.T. Conlin, Onset of entrainment between immiscible liquid layers due to rising gas bubbles. Int. J. Heat Mass Tran. 31 (1988) 1309-1317.

[15] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1999) 96-127. | Zbl

[16] D. Jacqmin, Contact-line dynamics of a diffuse fluid interface. J. Fluid Mechanics 402 (2000) 57-88. | Zbl

[17] M. Jobelin, C. Lapuerta, J.-C. Latché, P. Angot and B. Piar, A finite element penalty-projection method for incompressible flows. J. Comput. Phys. (2006) (to appear). | MR | Zbl

[18] J. Kim, Modeling and simulation of multi-component, multi-phase fluid flows. Ph.D. thesis, Univeristy of California, Irvine (2002).

[19] J. Kim, A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204 (2005) 784-804. | Zbl

[20] J. Kim and J. Lowengrub, Phase field modeling and simulation of three-phase flows. Interfaces and Free Boundaries 7 (2005) 435-466. | Zbl

[21] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard systems. Commu. Math. Sci. 2 (2004) 53-77. | Zbl

[22] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a fourier-spectral method. Physica D 179 (2003) 211-228. | Zbl

[23] J.S. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Royal Soc. London, Ser. A 454 (1998) 2617-2654. | Zbl

[24] B. Piar, PELICANS: Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN (2004).

[25] J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity. Clarendon Press (1982).

[26] K.A. Smith, F.J. Solis and D.L. Chopp, A projection method for motion of triple junctions by level sets. Interfaces and Free Boundaries 4 (2002) 239-261. | Zbl

[27] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68, Springer-Verlag, New York (1997). | MR | Zbl

[28] P. Yue, J. Feng, C. Liu and J. Shen, A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mechanics 515 (2004) 293-317. | Zbl

Cité par Sources :