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Topology changes in multi-phase fluid flows are difficult to model within atraditional sharp interface theory. Diffuse interface models turn out to be anattractive alternative to model two-phase flows. Based on aCahn-Hilliard-Navier-Stokes model introduced by Abels, Garcke and Gr\"{u}n(Math. Models Methods Appl. Sci. 2012), which uses a volume averaged velocity,we derive a diffuse interface model in a Hele-Shaw geometry, which in the caseof non-matched densities, simplifies an earlier model of Lee, Lowengrub andGoodman (Phys. Fluids 2002). We recover the classical Hele-Shaw model as asharp interface limit of the diffuse interface model. Furthermore, we show theexistence of weak solutions and present several numerical computationsincluding situations with rising bubbles and fingering instabilities.

A Hele-Shaw-Cahn-Hilliard model for incompressible two-phase flows with different densities