Isogeometric Analysis of a Phase Field Model for Darcy Flows with Discontinuous Data

Abstract

The authors consider a phase field model for Darcy flows with discontinuous data in porous media; specifically, they adopt the Hele-Shaw-Cahn-Hillard equations of [Lee, Lowengrub, Goodman, Physics of Fluids, 2002] to model flows in the Hele-Shaw cell through a phase field formulation which incorporates discontinuities of physical data, namely density and viscosity, across interfaces. For the spatial approximation of the problem, the authors use NURBS—based isogeometric analysis in the framework of the Galerkin method, a computational framework which is particularly advantageous for the solution of high order partial differential equations and phase field problems which exhibit sharp but smooth interfaces. In this paper, the authors verify through numerical tests the sharp interface limit of the phase field model which in fact leads to an internal discontinuity interface problem; finally, they show the efficiency of isogeometric analysis for the numerical approximation of the model by solving a benchmark problem, the so-called “rising bubble” problem.

Keywords

Darcy flows / Phase field / Hele-Shaw cell / Cahn-Hilliard equation / Sharp interface limit / Isogeometric analysis

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