Rigorous homogenization of reactive transport and flow fully coupled to an evolving microstructure

Abstract

We consider the homogenization of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size epsilon. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer, which is assumed to be radially symmetric with a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection-reaction-diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for epsilon -> 0, based on epsilon-uniform a priori estimates. As a result of the passage to the limit, the processes on the macroscale are described by an advection-reaction-diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells that depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro-macro model. Homogenization results have been obtained recently for the case of reactive-diffusive transport coupled with microscopic domain evolution, but in the absence of advective transport. We extend these models by including the advective transport, which is driven by the Stokes equations in the a priori unknown evolving pore domain.