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 evolve according to a generalized Ginzburg–Landau energy functional, 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–Larché 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–Larché system.
Highlights
In this letter, we develop a general model with the ability to capture situations with flow through a deformable porous medium that changes character in terms of stiffness, permeability, compressibility, and poroelastic coupling strength due to Cahn-Hilliard-type phase changes in the solid matrix
We propose a new model for flow through deformable porous media, where the solid material has two phases with distinct material properties
We develop a general model with the ability to capture situations with flow through a deformable porous medium that changes character in terms of stiffness, permeability, compressibility, and poroelastic coupling strength due to Cahn-Hilliard-type phase changes in the solid matrix
Summary
We develop a general model with the ability to capture situations with flow through a deformable porous medium that changes character in terms of stiffness, permeability, compressibility, and poroelastic coupling strength due to Cahn-Hilliard-type phase changes in the solid matrix. Coupling the Cahn-Hilliard model with elasticity, is often called the Cahn-Larche model due to its origination [19], and several applications have been considered with this model in mind, including li-ion batteries [20], and tumor evolution [9, 8]. Conservation laws for each of the three coupled processes; phase-field evolution, elasticity, and fluid flow are introduced, the free energy of the system is proposed together with constitutive relations to close the system.
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