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.