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

Abstract

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, Garcke and Grün (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 case of non-matched densities, simplifies an earlier model of Lee, Lowengrub and Goodman (Phys. Fluids 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.

41 pages, 75 figures

References in corpus (1)

• Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth

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