Homogenisation of coupled reaction-diffusion systems in porous media with evolving microstructure

Malte A. Peter (Centre for Industrial Mathematics, University of Bremen, Germany)

Chemical processes in porous media are modelled on the pore scale using reaction-diffusion equations. The resulting prototypical systems of coupled linear and nonlinear differential equations are homogenised in the context of periodic media.
The talk addresses two aspects: First, depending on the scaling of certain terms of the reaction-diffusion system with powers of the homogenisation parameter, different systems of equations arise in the homogenisation limit. The scaling arises from geometrical considerations or from the process itself. The resulting homogenised models are classified for the different choices of scaling powers using a unified approach based on two-scale convergence.
Second, chemical degradation mechanisms in porous materials often induce a change of the pore geometry. This effect cannot be captured by the standard periodic homogenisation method owing to the local evolution of the microscopic domain. A mathematically rigorous approach is suggested which makes use of a transformation of the evolving domain to a periodic reference domain. Two scenarios are considered: The evolution is given or it is induced by the reaction-diffusion process itself. In particular, we consider the prototypical situation where the reaction induces a change of pore-air volume. The well-posedness of the resulting systems of coupled partial and ordinary differential equations and their homogenisation are addressed.