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Coupled reaction-diffusion processes inducing an evolution of the microstructure: Analysis and homogenization. (English) Zbl 1151.35308

Summary: Chemical processes in porous media often cause a change of the microstructure of the porous material due to interaction with the solid matrix, by reaction or adsorption, e.g. We consider a reaction-diffusion problem where a solid matrix constituent is converted into another one of different density. Thus, the solid matrix locally grows or shrinks in volume, which in turn changes the pore-air volume. This affects the transport of reactants in the pore air. The homogenization of this problem with evolving microstructure is performed using the method of transformation to a periodic reference domain, which has recently been put forward by the author. The final system to be homogenized consists of three coupled partial differential equations for the concentrations coupled to one ordinary differential equation for a quantity describing the evolution of the pore-air volume.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K57 Reaction-diffusion equations
74F25 Chemical and reactive effects in solid mechanics
92E20 Classical flows, reactions, etc. in chemistry
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[1] J. Kropp, Relations between transport characteristics and durability, in: J. Kropp, H.K. Hilsdorf (Eds.), Performance Criteria for Concrete Durability, RILEM Report 12, E & FN SPON, 1995, pp. 97-137
[2] T.A. Bier, Karbonatisierung und Realkalisierung von Zementstein und Beton, Ph.D. Dissertation, University of Karlsruhe, 1988
[3] Meier, S.A.; Peter, M.A.; Böhm, M., A two-scale modelling approach to reaction-diffusion processes in porous materials, Comput. mat. sci., 39, 1, 29-34, (2007)
[4] S.A. Meier, M.A. Peter, A. Muntean, M. Böhm, J. Kropp, A two-scale approach to concrete carbonation, in: R.M. Ferreirad, J. Gulikers, C. Andrade (Eds.), RILEM Proc. PRO 56: Proc. Int. RILEM Workshop on Integral Service Life Modelling of Concrete Structures, 2007, pp. 3-10
[5] Peter, M.A., Homogenisation in domains with evolving microstructure, C. R. Mécanique, 335, 7, 357-362, (2007) · Zbl 1132.74036
[6] Conca, C.; Díaz, J.I.; Timofte, C., Effective chemical processes in porous media, Math. models methods appl. sci., 13, 10, 1437-1462, (2003) · Zbl 1058.76071
[7] M.A. Peter, M. Böhm, Scalings in homogenisation of reaction, diffusion and interfacial exchange in a two-phase medium, in: M. Fila, A. Handlovicova, K. Mikula, M. Medved, P. Quittner, D. Sevcovic (Eds.), Proc. Equadiff-11, 2005, pp. 369-376
[8] M.A. Peter, Coupled reaction-diffusion processes and evolving microstructure: mathematical modelling and homongenisation, Ph.D. Dissertation, University of Bremen, 2006. Also: Logos Verlag Berlin, 2007
[9] Cioranescu, D.; Damlamian, A.; Griso, G., Periodic unfolding and homogenization, C. R. acad. sci. Paris, ser. I, 335, 99-104, (2002) · Zbl 1001.49016
[10] Logan, J.D., Transport modeling in hydrogeochemical systems, (2001), Springer · Zbl 0982.86001
[11] A. Muntean, A moving-boundary problem: Modeling, analysis and simulation of concrete carbonation, Ph.D. Dissertation, University of Bremen, 2006. Also: Cuvillier, 2006
[12] Wilmański, K., Thermomechanics of continua, (1998), Springer · Zbl 0917.73001
[13] Bear, J., Dynamics of fluids in porous media, (1972), Elsevier · Zbl 1191.76001
[14] Bear, J.; Bachmat, Y., Introduction to modeling of transport phenomena in porous media, (1990), Kluwer · Zbl 0743.76003
[15] Peter, M.A., Homogenisation of a chemical degradation mechanism inducing an evolving microstructure, C. R. Mécanique, 335, 11, 679-684, (2007) · Zbl 1136.35310
[16] Showalter, R.E., Monotone operators in Banach space and nonlinear partial differential equations, (1997), American Mathematical Society · Zbl 0870.35004
[17] Clark, G.W.; Showalter, R.E., Two-scale convergence of a model for flow in a partially fissured medium, Electron. J. differential equations, 1999, 2, 1-20, (1999) · Zbl 0914.35013
[18] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. math. anal., 20, 3, 608-629, (1989) · Zbl 0688.35007
[19] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. anal., 23, 6, 1482-1518, (1992) · Zbl 0770.35005
[20] Allaire, G.; Damlamian, A.; Hornung, U., Two-scale convergence on periodic surfaces and applications, (), 15-25
[21] Neuss-Radu, M., Some extensions of two-scale convergence, C. R. acad. sci. Paris, ser. I, 322, 899-904, (1996) · Zbl 0852.76087
[22] Allaire, G.; Briane, M., Multiscale convergence and reiterated homogenisation, Proc. roy. soc. edinb., 126A, 297-342, (1996) · Zbl 0866.35017
[23] Lukkassen, D.; Nguetseng, G.; Wall, P., Two-scale convergence, Int. J. pure appl. math., 2, 1, 35-86, (2002) · Zbl 1061.35015
[24] Holmbom, A., Homogenization of parabolic equations an alternative approach and some corrector-type results, Appl. math., 42, 5, 321-343, (1997) · Zbl 0898.35008
[25] M.A. Peter, M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium, Math. Methods Appl. Sci., in press (doi:10.1002/mma.966) · Zbl 1154.35008
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