Analysis of a two-scale system for gas-liquid reactions with non-linear Henry-type transfer

In this paper, we consider a coupled two-scale nonlinear reaction-diffusion system modelling gas-liquid reactions. The novel feature of the model is the nonlinear transmission condition between the microscopic and macroscopic concentrations, given by a nonlinear Henry-type transfer function. The solution is approximated by using a Galerkin method adapted to the multiscale form of the system. This approach leads to existence and uniqueness of the solution, and can also be used for numerical computations for a larger class of nonlinear multiscale problems

A multiscale Galerkin approach for a class of nonlinear coupled reaction–diffusion systems in complex media

AbstractA Galerkin approach for a class of multiscale reaction–diffusion systems with nonlinear coupling between the microscopic and macroscopic variables is presented. This type of models are obtained e.g. by upscaling of processes in chemical engineering (particularly in catalysis), biochemistry, or geochemistry. Exploiting the special structure of the models, the functions spaces used for the approximation of the solution are chosen as tensor products of spaces on the macroscopic domain and on the standard cell associated to the microstructure. Uniform estimates for the finite dimensional approximations are proven. Based on these estimates, the convergence of the approximating sequence is shown. This approach can be used as a basis for the numerical computation of the solution

Homogenization of a pore scale model for precipitation and dissolution in porous media

In this paper we employ homogenization techniques to provide a rigorous derivation of the Darcy scale model for precipitation and dissolution in porous media proposed in [19]. The starting point is the pore scale model in [12], which is a coupled system of evolution equations, involving a parabolic equation and an ordinary differential equation. The former models ion transport and is defined in a periodically perforated medium. It is further coupled through the boundary conditions to the latter, defined on the boundaries of the perforations and modelling the dissolution and precipitation of the precipitate. The main challenge is in dealing with the dissolution and precipitation rates, which involve a monotone but multi-valued mapping. Due to this, the micro-scale solution lacks regularity. With e being the scale parameter (the ratio between the micro scale and the macro scale length), we adopt the 2-scale framework to achieve the convergence of the homogenization procedure as e approaches zero

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