Multiscale methods and modelling for chemical reactions on oscillating surfaces

Abstract

We study the dynamics of simple reactions where the chemical species reside on a non-uniform, oscillating surface, and are subject to externallyimposed stirring. We derive a model for this based on a reaction-advectiondiffusion equation. We first of all focus on the autocatalytic reaction, and determine whether stirring and surface oscillation can stabilise a particular homogeneous state. To do this, we use homogenisation methods: we show that the influence of the surface oscillation on the concentration field can be parametrised by an effective-diffusion operator. In practice, our criterion for extinction is difficult to interpret, and we therefore carry out numerical simulations. We focus on a chemical reaction occurring on the surface of a thin liquid film, and show that the surface oscillation can enhance the reaction yield. In this numerical context, we demonstrate that the extinction of the catalyst is unlikely. Qualitatively similar results arise when we consider purely homogeneous motion; here we consider both autocatalytic and bistable reaction kinetics. Fisher-KPP equation; Advection; Multiscale methods; Manifolds