A thermo-diffusion system with Smoluchowski interactions: well-posedness and homogenization

We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients

Derivation of a macroscopic model for nutrient uptake by a single branch of hairy-roots

In this article the process of nutrient uptake by a single branch of a root is studied. We consider diffusion and active transport of nutrients dissolved in water. The uptake of nutrients happens on the surface of thin root hairs distributed periodically and orthogonal to the root surface. Water velocity is defined by the Stokes equations. Macroscopic model for nutrient uptake by a hairy root is derived. The macroscopic model consists of a reaction-diffusion equation in the domain with hairs, and diffusion-convection equation in the domain without hairs. The macroscopic water velocity is described by the Stokes system in the domain without hairs, with no-slip condition on the boundary between domains with hairs and of free fluid

The cardiac bidomain model and homogenization

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on Mathematical Methods for Systems Biolog

Rigorous upscaling of rough boundaries for reactive flows

We consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude e. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non-linear and multi-valued rates and is posed in a fixed geometry with rough boundaries. We provide a rigorous justification for the upscaling process in which we define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process. Keywords: Reactive flows; rough boundaries; homogenization

Upscaling of reactive flows

The thesis deals with the upscaling of reactive flows in complex geometry. The reactions which may include deposition or dissolution take place at a part of the boundary and depending on the size of the reaction domain, the changes in the pore structure that are due to the deposition process may or may not be neglected. In mathematical terms, the models are defined in a fixed, respectively variable geometry, when the deposition layer generates a free boundary at the pore scale. Specifically, for the chemical vapor deposition (CVD) process on a trenched geometry, we have developed mathematical models for both situations. For the multi-scale computations, numerical methods inspired from domain decomposition ideas have been proposed and the convergence of the scheme has been proved. Computing the full solution in a domain with oscillating boundary requires a lot of computational effort, as one has to achieve an accuracy that agrees with the scale of oscillations. To approximate these solutions, one defines equations in a simpler domain, where flat boundaries but modified boundary conditions approximate the rough one. The two situations mentioned before were considered: the fixed geometry case, and the time dependent geometry at the microscale (free boundaries). We have derived an approximating (effective) model where a flat boundary is replacing the oscillatory boundary, but defining an effective boundary condition. In the fixed geometry case, we provide rigorous mathematical proofs for the upscaling procedure. The second case, when we take into account the geometry changes at the microscale, is more involved, and we use formal asymptotic methods to derive these boundary conditions. Our contributions in this respect are in dealing with non-Lipschitz reactive terms on the boundary in the fixed geometry case and the formal asymptotic approach for the moving boundary. Both add to the present literature. Next, to understand the flow in a domain with variable geometry, we have considered a thin strip with reactions taking place at the lateral boundaries of the strip under dominant transport conditions. Reactions take place at the lateral boundaries of the strip (the walls), where the reaction product can deposit in a layer with a non-negligible thickness compared to the width of the strip. This leads to a free boundary problem, in which the moving interface between the fluid and the deposited (solid) layer is explicitly taken into account. Using asymptotic expansion methods, we derive an upscaled, one-dimensional model by averaging in the transversal direction. The upscaled equations are similar to the Taylor dispersion and we have performed numerical simulations to compare the upscaled equations with other simpler upscaled equations and the transversally averaged, two-dimensional solution. The derivation introduces new terms originating from the changing geometry. The numerical computations also provide an insight into the regimes where such an upscaling is useful. We have further studied the rigorous homogenization process for the reactive flows for a periodic array of cells and proved the validity of upscaled equations. These reactive flows model the precipitation and dissolution processes in a porous medium. We define a sequence of microscopic solutions u" and obtain the upscaled equations as the limit of e \ 0. We adopt the 2-scale framework to achieve this. The challenges are in dealing with the low regularity of microscopic solutions and particular non-linearities in the reaction term. This rigorous derivation closes the gap of the rigorous transition from a given pore scale model to the heuristically proposed macroscopic model. In addition, numerical methods to compute the solution for an upscaled model have been proposed. The upscaled model describes the reactive flow in a porous medium. The reaction term, especially, the dissolution term has a particular, multi-valued character, which leads to stiff dissolution fronts. We have considered both the conformal and mixed schemes for the analysis including both the semi-discrete (time-discretization) and the fully discrete (both in space and time) cases. The fully discrete schemes correspond to the finite element method and the mixed finite element method for conformal, respectively mixed schemes. The numerical schemes have been analyzed and the convergence to the continuous formulation has been proved. Apart from the proof for the convergence, this also yields an existence proof for the solution of the upscaled model. Numerical experiments are performed to study the convergence behavior. The challenges are in dealing with the specific non-linearities of the reaction term. We deal with them by using the translation estimates which are adapted to the specific numerical scheme. The applications are in the development of all-solid state rechargeable batteries having a high dtorage capacity. Such devices have a complex 3D geometry for the electrodes to enhance the surface area. The challenges are in the development of the appropriate technologies for the formation of these electrodes. In particular we focus on chemical vapor deposition processes (CVD), with the aim of getting a deeper understanding of the reactions taking place in a complex geometry. Other applications include flows in porous media, bio-film growth etc

Reaction-diffusion-ODE systems: de-novo formation of irregular patterns and model reduction

Classical models of pattern formation in systems of reaction-diffusion equations are based on diffusion-driven instability (DDI) of constant stationary solutions. The destabilisation may lead to emergence of stable, regular Turing patterns formed around the destabilised equilibrium. In this thesis it is shown that coupling reaction-diffusion equations with ordinary differential equations may lead to de-novo formation of far from equilibrium steady states. In particular, conditions for so called (ε0 , A)-stability (resp. stability in epi-graph-topology) are given, yielding from bistability and hysteresis effects in the null sets of nonlinearities. A model exhibiting coexistence of Turing-type destabilisation and stable far from equilibrium steady states, is proposed. It is shown, under suitable conditions, that DDI and (in)stability can be derived from so called quasi-stationary model reduction. Moreover, similar to a result for ordinary differential equations, proved by Tikhonov, the dynamical behaviour of the reduced and the unreduced model are similar. It is shown that the spectral properties of the operators resulting from linearisation of the unreduced system, determining the long-term behaviour around a steady state, are reflected in the spectral properties of the operators resulting from linearisation of the reduced system. The given conditions are satisfied by a larger range of classical models, as illustrated by application to a degenerate version of the Lengyel-Epstein model. The dynamical behaviour of reaction-diffusion equations for large diffusion and on finite time intervals is essentially reflected by their so called shadow systems. In this hesis, existence and stability of steady states with jump-type discontinuity is investigated and compared for this reduction. The results show that, in case of static patterns, not only the short-term behaviour, but also the long-term behaviour of the reduced system is reflected in the unreduced system. Moreover, a result showing Turing-type destabilisation for such shadow systems, given in a joint-paper, is generalised. Finally, such shadow systems are reduced by application of a quasi-stationary model reduction leading to a scalar integro-differential equation. It is shown that the quasi-stationary model reduction is regular in the sense of Turing-type destabilisation and dynamical behaviour on finite time intervals. Hence, reaction-diffusion-ODE models may be reduced to scalar integro-differential equations in order to investigate the qualitative behaviour around homogeneous steady states and the qualitative behaviour on finite time intervals. A hypothesis is that the long-term behaviour is similar, but a proof is missing. The result shows that a link between reaction-diffusion-ODE systems and scalar integro-differential equations exists and that the mechanisms of pattern formation may be investigated based on the reduction