Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains

Abstract

A reaction-diffusion system modeling concrete corrosion in sewer pipes is discussed. The system is coupled, semi-linear, and partially dissipative. It is defined on a locally periodic perforated domain with nonlinear Robin-type boundary conditions at water-air and solid-water interfaces. Asymptotic homogenization techniques are applied to obtain upscaled reaction-diffusion models together with explicit formulae for the effective transport and reaction coefficients. It is shown that the averaged system contains additional terms appearing due to the deviation of the assumed geometry from a purely periodic distribution of perforations for two relevant parameter regimes: (a) all diffusion coefficients are of order of O(1) and (b) all diffusion coefficients are of order of O(ε 2 ) except the one for H 2 S(g) which is of order of O(1). In case (a) a set of macroscopic equations is obtained, while in case (b) a two-scale reaction-diffusion system is derived that captures the interplay between microstructural reaction effects and the macroscopic transport.

Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

“…The striking thing is that in spite of the fact that the basic physical-chemistry of this relatively easy material is known [1], we have no control on how the microstructure changes (in time and space) and to which extent these spatio-temporal changes affect the observable macroscopic behavior of the material. The research reported here goes along the line open in [11], where a formal asymptotic expansion ansatz was used to derive macroscopic equations for a corrosion model, posed in a domain with locally-periodic microstructure (see [17] for a rigorous averaging approach of a reduced model defined in a domain with locallyperiodic microstructures). A two-scale convergence approach for periodic microstructures was studied in [10], while preliminary multiscale simulations are reported in [3].…”

Rigorous upscaling of rough boundaries for reactive flows

“…For g, we assume Lipschitz continuity and that it takes positive values. Such rates are considered for example in modeling the reactive flows in porous medium [24], in biological contexts in the diffusion of receptors in a cell [36], or in the description of sulphate attack for sewer pipes [19]. The extension to more number of species living on the boundary or inside the domain is analogous.…”

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

“…and we define the following 19) for j ∈ {1, 2}. To ensure the uniqueness of weak/strong solutions to this cell problem, suitable conditions on the spatial averages of cell functions need to be added.…”

“…The materials science scenario we have in view is motivated by a very practical problem: the sulphate corrosion of concrete. We refer the readers to [19] for a brief presentation of the physico-chemical problem and for the formal (two-scale) averaging of locally periodic distributions of unsaturated pores attacked by sulphuric acid.…”

“…the sulphatation reaction) in particular; see [19] for the description of the real-world problem we have in mind.…”