Study of a three component Cahn-Hilliard flow model

¹ Laboratoire d'Analyse, Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. fboyer@cmi.univ-mrs.fr
² Institut de Radioprotection et de Sûreté Nucléaire, DPAM/SEMIC/LMPC, BP 3, 13115 Saint-Paul-lez-Durance, France.

Received: 18 October 2005
Revised: 31 March 2006

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.