Applied Analysis SeminarTheoretical Analysis and Numerical Simulations of the Darcy-Allen-Chan

We consider a diffuse interface model of incompressible binary fluid in a bounded domain. This model consists of the Allen-Cahn equation coupled with a Darcy's law through a Korteweg force and equipped with homogeneous Neumann boundary condition. The present system can be considered as a variant of the well-known Cahn-Hilliard-Darcy system when the total mass is conserved. The latter property holds when the Allen-Cahn equation is, for instance, enriched with a non-local term. We first analyze this case assuming that the potential density appearing in the free energy is the so-called Flory-Huggins potential (i.e. the mixing entropy is the Boltzmann entropy). Therefore, the order parameter Φ (i.e. the difference of the relative concentrations) necessarily takes its values in the physical range [−1,+1]. We prove the existence of a finite energy solution as a limit of finite energy solutions to an extended system characterized by a time relaxation of the Darcy's law as the relaxation parameter goes to 0. Then we consider the case in which the potential density is approximated by a double well polynomial function. In this case we cannot ensure that Φ∈[−1,1] if the mass is conserved. However, by dropping the mass constraint, we can still prove that Φ∈[−1,1]. This fact allows us to use a semi-Galerkin scheme to prove the existence of a finite energy solution to the extended system and then recover a similar solution to the original problem for the relaxation parameter tending to 0. We provide and discuss numerical simulations to sustain our results.