Homogenisation of coupled reaction-diffusion systems in porous media with
Malte A. Peter (Centre for Industrial Mathematics, University of Bremen,
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.