A Hele-Shaw-Cahn-Hilliard Model for Incompressible Two-Phase Flows with Different Densities

Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn-Hilliard- Navier-Stokes model introduced by Abels et al. (Math Models Methods Appl Sci 22(3):1150013. 2012). which uses a volume-averaged velocity, we derive a diffuse interface model in a Hele-Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514-545, 2002). We recover the classical Hele-Shaw model as a sharp interface limit of the diffuse interface model. Furthermore. we show the existence of weak solutions and present several numerical computations including situations with rising bubbles and fingering instabilities.