Study of a three component Cahn-Hilliard flow model
ESAIM: Mathematical Modelling and Numerical Analysis - 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 : https://doi.org/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: Mathematical Modelling and Numerical Analysis - 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/}
}
Boyer, Franck; Lapuerta, Céline. Study of a three component Cahn-Hilliard flow model. ESAIM: Mathematical Modelling and Numerical Analysis - 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 0828.35105

[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 0936.65120

[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 0857.65137

[5] F. Boyer, Mathematical study of multiphase flow under shear through order parameter formulation. Asymptotic Anal. 20 (1999) 175-212. | Zbl 0937.35123

[6] F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (2002) 41-68. | Zbl 1057.76060

[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 1038064 | Zbl 0692.73003

[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 0853.73060

[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 0988.35019

[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 0936.82010

[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 0942.35095

[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 0966.76060

[16] D. Jacqmin, Contact-line dynamics of a diffuse fluid interface. J. Fluid Mechanics 402 (2000) 57-88. | Zbl 0984.76084

[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 2260612 | Zbl 1160.76366

[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 1329.76103

[20] J. Kim and J. Lowengrub, Phase field modeling and simulation of three-phase flows. Interfaces and Free Boundaries 7 (2005) 435-466. | Zbl 1100.35088

[21] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for ternary Cahn-Hilliard systems. Commu. Math. Sci. 2 (2004) 53-77. | Zbl 1085.65093

[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 1092.76069

[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 0927.76007

[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 1112.76437

[27] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68, Springer-Verlag, New York (1997). | MR 1441312 | Zbl 0871.35001

[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 1130.76437