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In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.
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A unified theory of free energy functionals and applications to diffusion
Free energy functionals of the Ginzburg–Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn–Hilliard and Swift–Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the free energy functionals frequently either are poorly justified or lead to physically opaque parameters. Here, we introduce a mathematically rigorous pathway for constructing free energy functionals that generalizes beyond the constraints of Ginzburg–Landau gradient expansions. We show that the formalism unifies existing free energetic descriptions under a single umbrella by establishing the criteria under which the generalized free energy reduces to gradient-based representations. Consequently, we derive a precise physical interpretation of the gradient energy parameter in the Cahn–Hilliard model as the product of an interaction length scale and the free energy curvature. The practical impact of our approach is demonstrated using both a model free energy function and the silicon–germanium alloy system.
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- PAR ID:
- 10393516
- Date Published:
- Journal Name:
- Proceedings of the National Academy of Sciences
- Volume:
- 119
- Issue:
- 23
- ISSN:
- 0027-8424
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1073/pnas.2203399119
