Upscaling of Diffusion-Reaction Phenomena by Homogenisation Technique: Possible Appearance of Morphogenesis

The main objective of this work is to describe reaction-diffusion of two species in a porous medium. We aim at finding the macroscopic model equivalent to the description of the physics at the pore scale, with a peculiar attention to possible shape morphogenesis as introduced in Turing's seminal article in 1952 (Turing in Philos Trans R Soc Lond Ser B 237:37, 1952). The upscaling process makes use of the method of asymptotic expansions which takes advantage of the presence of the small parameter of separation of scales epsilon=l/L, where l and L are the characteristic sizes of the pores and the sample or the diffusion wavelength, respectively. Two different situations are investigated here: large reaction-diffusion and large diffusivity ratios. Among all possibilities, we investigate three orders of magnitude of the reaction terms and two orders of magnitude of the diffusivity ratio. In each of the situations, we demonstrate, if it exists, the macroscopic equivalent model at a scale L >> l, at the first order of approximation. Hence, we obtain a catalogue of macroscopic models for each considered case, their domain of validity, and the expression of the macroscopic properties. We underline the adequate conditions to account for the appearance of morphogenesis.