2024/04/02 by Markus Gahn, Gahn, Markus, Malte A. Peter +5
Chemical Engineering · Computer Science · Engineering · #35B27 #35K57 #35R35 #76M50 #76S05 #80A19 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Rheology and Fluid Dynamics Studies
paper · pdf · doi:10.48550/arxiv.2404.01983
openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the homogenisation of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size ε. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer having 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 ε → 0, based on ε-uniform a priori estimates. As a result of the limit passage, 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, which 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. For pure reactive-diffusive transport coupled with microscopic domain evolution but without advective transport, homogenisation results have recently been presented. We extend these models by advective transport which is driven by the Stokes equation in the a priori unknown evolving pore domain.