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 \({\mathcal{O}(1)}\) and (b) all diffusion coefficients are of order of \({\mathcal{O}(\varepsilon^2)}\) except the one for H2S(g) which is of order of \({\mathcal{O}(1)}\). 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.
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Abbreviations
- PDE:
-
Partial differential equation
- ODE:
-
Ordinary differential equation
- RD:
-
Reaction diffusion
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Acknowledgments
We acknowledge fruitful multiscale-related discussions with Tycho van Noorden. Also, we would like to thank Prof. Luisa Mascarenhas for providing Ref. [21].
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Nasrin Arab—Visiting during spring 2009 CASA, Department of Mathematics and Computer Science, Technical University Eindhoven, The Netherlands.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Fatima, T., Arab, N., Zemskov, E.P. et al. Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains. J Eng Math 69, 261–276 (2011). https://doi.org/10.1007/s10665-010-9396-6
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DOI: https://doi.org/10.1007/s10665-010-9396-6