Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Friday, January 19, 2007 - 04:00

Location:

Gilkey 113

Guest Speaker:

Institution:

Centre for Industrial Mathematics, University of Bremen

Abstract:

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, different scalings of certain terms of the reaction--diffusion system with powers of the homogenisation parameter are reasonable. The scaling arises from geometrical considerations or from the process itself. Depending on the particular choice of these scaling powers, different systems of equations arise in the homogenisation limit. The resulting models are classified using a unified approach based on two-scale convergence. Second, chemical degradation mechanisms of porous materials often induce a change of the pore geometry. This effect cannot be captured by the standard periodic homogenisation method due 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: First, for given evolution, it is shown that the transformed problem can be homogenised within the context of periodic media and that the homogenised problem can be transformed back to the evolving domain. Second, the additional terms in the transformed problem can also be interpreted physically and related to the process itself. For the prototypical situation where the reaction induces a change of pore-air volume, a model for the additional terms arising from the transformation is suggested, which relates the terms from the transformation to the reaction--diffusion process. The well-posedness of the resulting system of coupled partial and ordinary differential equations and its homogenisation are addressed.