Homogenisation of local colloid evolution induced by reaction and diffusion

Abstract

We consider the homogenisation of a coupled reaction-diffusion process in a porous medium with evolving microstructure. A concentrationdependent reaction rate at the interface of the pores with the solid matrix induces a concentration-dependent evolution of the domain. Hence, the evolution is fully coupled with the reaction-diffusion process. In order to pass to the homogenisation limit, we employ the two-scale-transformation method. Thus, we homogenise a highly non-linear problem in a periodic and in time cylindrical domain instead. The homogenisation result is a reaction-diffusion equation, which is coupled with an internal variable, representing the local evolution of the pore structure. Contents 1. Introduction 1 2. The mathematical model 3 2.1. Weak formulation 6 2.2. Transformation of the domain 6 2.3. Transformation of the weak form 10 3. Existence and uniform a priori estimates 10 4. Derivation of the limit problem for the periodic substitute problem 20 5. Back-transformation 30 6. Acknowledgements 32 References 32 2020 Mathematics Subject Classification. 35B27, 35K57, 35R35. Key words and phrases. Homogenization; evolving microstructure; free boundary problem; two-scale convergence; porous medium; reaction-diffusion process. D.W. was partially supported by a doctoral scholarship provided by the Studienstiftung des deutschen Volkes.