Cahn-Hilliard / Navier-Stokes model for the simulation of three-phase flows

International audienceIn this article, we describe some aspects of the diffuse interface modelling of incompressible flows, composed of three immiscible components, without phase change. In the diffuse interface methods, system evolution is driven by the minimisation of a free energy. The originality of our approach, derived from the Cahn-Hilliard model, comes from the par- ticular form of energy we proposed in Boyer and Lapuerta (M2AN Math Model Numer Anal, 40:653-987,2006), which, among other interesting properties, ensures consistency with the two-phase model. The modelling of three-phase flows is further completed by coupling the Cahn-Hilliard system and the Navier-Stokes equations where surface tensions are taken into account through volume capillary forces. These equations are discretized in time and space paying attention to the fact that most of the main properties of the original model (volume conservation and energy estimate) have to be maintained at the discrete level. An adaptive refinement method is finally used to obtain an accurate resolution of very thin moving internal layers, while limiting the total number of cells in the grids all along the simulation. Different numerical results are given, from the validation case of the lens spreading between two phases (contact angles and pressure jumps), to the study of mass transfer through a liquid/liquid inter- face crossed by a single rising gas bubble. The numerical applications are performed with large ratio between densities and viscosities and three different surface tensions