Theoretical analysis and numerical simulations of the Darcy-Allen-Chan system

Michele Precuzzi (TU Delft) #

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 \(\Phi\) (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 \(\Phi \in [−1, 1]\) if the mass is conserved. However, by dropping the mass constraint, we can still prove that \(\Phi \in [−1, 1]\). 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.