Diffusion of one substance into a semi-infinite medium containing another with second-order reaction

Abstract

The system consists of one substance diffusing into a medium containing another substance with which it reacts according to a second-order equation. The latter substance can also diffuse within the medium. This system is represented by a pair of diffusion (heat-conduction) equations linked through a non-linear (negative) production term. When made dimensionless in the usual way, the pair of equations contain two significant dimensionless parameters: one is the ratio of the diffusion constants, the other the ratio of the initial uniform concentration of the substance present in the medium to the imposed boundary concentration of the diffusing substance. These parameters define a doubly infinite set of solutions. Analytic solutions are given for certain limiting cases, and where necessary, the structure of the asymptotic expansions valid near the limit is examined. Numerical solutions have been obtained for intermediate cases and are presented in brief. The aim has been to provide an exhaustive solution covering all possible cases, without specific reference to any particular physical circumstances.