A diffuse interface approach to Hele-Shaw flow

A diffuse interface model for the one-phase Hele–Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Rat. Mech. Anal. 141 63). The resulting dynamical model yields a generalized form of Darcy's law, and reduces to a degenerate version of the well-known Cahn–Hilliard equation. Formal asymptotics illustrate the connection to the classical Hele–Shaw free boundary problem. Some example computations are carried out to demonstrate the flexibility of the modelling framework.

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