Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-18T19:07:07.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Study of a three component Cahn-Hilliard flow model

Published online by Cambridge University Press:  15 November 2006

Franck Boyer  and
Céline Lapuerta
Franck Boyer
Affiliation:
Laboratoire d'Analyse, Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected]
Céline Lapuerta
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire, DPAM/SEMIC/LMPC, BP 3, 13115 Saint-Paul-lez-Durance, France.
Get access

Abstract

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.

Keywords

Multicomponent flowsCahn-Hilliard equationsstability.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 4 , July 2006 , pp. 653 - 687
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alikakos, N.D., Bates, P.W. and Chen, X., Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. An. 128 (1994) 165–205. CrossRef
Anderson, D.M., McFadden, G.B. and Wheeler, A.A., Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139–165. CrossRef
Barrett, J.W. and Blowey, J.F., 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. CrossRef
Blowey, J.F., Copetti, M.I.M. and Elliott, C.M., Numerical analysis of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16 (1996) 111–139. CrossRef
Boyer, F., Mathematical study of multiphase flow under shear through order parameter formulation. Asymptotic Anal. 20 (1999) 175–212.
Boyer, F., A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31 (2002) 41–68. CrossRef
Copetti, M.I.M., Numerical experiments of phase separation in ternary mixtures. Math. Comput. Simulat. 52 (2000) 41–51. CrossRef
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).
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).
Eyre, D.J., Systems of Cahn-Hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686–1712. CrossRef
Garcke, H. and Novick-Cohen, A., A singular limit for a system of degenerate Cahn-Hilliard equations. Adv. Differ. Equ. 5 (2000) 401–434.
Garcke, H., Nestler, B. and Stoth, B., On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Physica D 115 (1998) 87–108. CrossRef
Garcke, H., Nestler, B. and Stoth, B., A multi phase field: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60 (1999) 295–315. CrossRef
Greene, G.A., Chen, J.C. and Conlin, M.T., Onset of entrainment between immiscible liquid layers due to rising gas bubbles. Int. J. Heat Mass Tran. 31 (1988) 1309–1317. CrossRef
Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1999) 96–127. CrossRef
Jacqmin, D., Contact-line dynamics of a diffuse fluid interface. J. Fluid Mechanics 402 (2000) 57–88. CrossRef
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).
J. Kim, Modeling and simulation of multi-component, multi-phase fluid flows. Ph.D. thesis, Univeristy of California, Irvine (2002).
Kim, J., A continuous surface tension force formulation for diffuse-interface models. J. Comput. Phys. 204 (2005) 784–804. CrossRef
Kim, J. and Lowengrub, J., Phase field modeling and simulation of three-phase flows. Interfaces and Free Boundaries 7 (2005) 435–466. CrossRef
Kim, J., Kang, K. and Lowengrub, J., Conservative multigrid methods for ternary Cahn-Hilliard systems. Commu. Math. Sci. 2 (2004) 53–77.
Liu, C. and Shen, J., 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. CrossRef
J.S. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. Royal Soc. London, Ser. A 454 (1998) 2617–2654.
B. Piar, PELICANS: Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN (2004).
J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity. Clarendon Press (1982).
Smith, K.A., Solis, F.J. and Chopp, D.L., A projection method for motion of triple junctions by level sets. Interfaces and Free Boundaries 4 (2002) 239–261.
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68, Springer-Verlag, New York (1997).
Yue, P., Feng, J., Liu, C. and Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mechanics 515 (2004) 293–317. CrossRef