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1.

Vromans, Arthur

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Centre for Analysis, Computer Science and Applications (CASA), Department of Mathematics and Computers Science, Eindhoven University of Technology, Eindhoven, the Netherlands.

In this thesis, parabolic-pseudoparabolic equations are derived coupling chemical reactions, diffusion, flow and mechanics in a heterogeneous medium using the framework of mixture theory. The weak solvability in 1-D of the obtained models is studied. Furthermore, it is numerically illustrated that approximate solutions according to the Rothe method exhibit expected realistic behaviour. For a simpler model formulation, the periodic homogenization in higher space dimensions is performed.

In this dissertation, parabolic-pseudoparabolic equations are proposed to couple chemical reactions, diffusion, flow and mechanics in heterogeneous materials using the framework of mixture theory. The weak solvability is obtained in a one dimensional setting for the full system posed in a homogeneous domain - a formulation which we have obtained using the classical mixture theory. To give a glimpse of what each component of the system does, we illustrate numerically that approximate solutions according to the Rothe method exhibit realistic behaviour in suitable parameter regimes. The periodic homogenization in higher space dimensions is performed for a particular case of the initial system of partial differential equations posed in perforated domains. Besides obtaining upscaled model equations and formulas for computing effective transport coefficients, we also derive corrector/convergence estimates which delimitate the precision of the upscaling procedure. Finally, the periodic homogenization is performed for a thin vanishing multidomain. Corrector estimates are obtained for a comb-like domain placed on a thin plate in a monotone operator setting for pseudoparabolic equations.

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Eindhoven University of Technology, The Netherlands.

van de Ven, Fons

Eindhoven University of Technology, The Netherlands.

Muntean, Adrian

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).

We determine corrector estimates quantifying the convergence speed of the upscaling of a pseudo-parabolic system containing drift terms incorporating the separation of length scales with relative size 1. To achieve this goal, we exploit a natural spatial-temporal decomposition, which splits the pseudo-parabolic system into an elliptic partial differential equation and an ordinary differential equation coupled together. We obtain upscaled model equations, explicit formulas for effective transport coefficients, as well as corrector estimates delimitating the quality of the upscaling. Finally, for special cases we show convergence speeds for global times, i.e., t ∈ R+, by using time intervals expanding to the whole R+ simultaneously with passing to the homogenization limit ↓ 0.

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013). Tech Univ Eindhoven, Ctr Anal Sci Comp & Applicat CASA, Den Dolech 2, NL-5612 AZ Eindhoven, Netherlands.

Muntean, Adrian

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).

A 3-D continuum mixture model describing the corrosion of concrete with sulfuric acid is built. Essentially, the chemical reaction transforms slaked lime (calcium hydroxide) and sulfuric acid into gypsum releasing water. The model incorporates the evolution of chemical reaction, diffusion of species within the porous material and mechanical deformations. This model is applied to a 1-D problem of a plate-layer between concrete and sewer air. The influx of slaked lime from the concrete and sulfuric acid from the sewer air sustains a gypsum creating chemical reaction (sulfatation or sulfate attack). The combination of the influx of matter and the chemical reaction causes a net growth in the thickness of the gypsum layer on top of the concrete base. The model allows for the determination of the plate layer thickness h = h(t) as function of time, which indicates both the amount of gypsum being created due to concrete corrosion and the amount of slaked lime and sulfuric acid in the material. The existence of a parameter regime for which the model yields a non-decreasing plate layer thickness h(t) is identified numerically. The robustness of the model with respect to changes in the model parameters is also investigated.

The weak solvability of a nonlinearly coupled system of parabolic and pseudo-parabolic equations describing the interplay between mechanics, chemical reac- tions, diffusion and flow modelled within a mixture theory framework is studied via energy- like estimates and Gronwall inequalities. In analytically derived parameter regimes, these estimates ensure the convergence of discretized-in-time partial differential equa- tions. These regimes are tested and extended numerically. Especially, the dependence of the temporal existence domain of physical behaviour on selected parameters is shown.

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