A Cahn-Hilliard-Biot system and its generalized gradient flow structure

TL;DR

This paper introduces a new coupled model for flow in deformable porous media with two-phase materials, integrating Cahn-Hilliard and Biot's theories, and demonstrates its gradient flow structure with numerical insights.

Contribution

It develops a novel three-way coupled model combining Cahn-Hilliard, elasticity, and fluid flow, extending existing equations with a gradient flow framework.

Findings

The model captures complex interactions in porous media with two phases.

Numerical example shows flow influences solid phase evolution.

The system follows a generalized gradient flow structure.

Abstract

In this work, we propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties. The two phases of the porous material follow a Cahn-Hilliard type evolution, with additional impact from both elastic and fluid effects, and the coupling between flow and deformation is governed by Biot's theory. This results in a three-way coupled system which can be seen as an extension of the Cahn-Larch\'e equations with the inclusion of a fluid flowing through the medium. The model covers essential coupling terms for several relevant applications, including solid tumor growth, biogrout, and wood growth simulation. Moreover, we show that this coupled set of equations follow a generalized gradient flow framework. This opens a toolbox of analysis and solvers which can be used for further study of the model. Additionally, we provide a numerical example showing the impact of the flow on the solid phase evolution in comparison to the Cahn-Larch\'e system.