Cubic autocatalysis in a reaction–diffusion annulus: semi-analytical solutions

Abstract

Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reactiondiffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steadystate diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations. Dedication: This paper is dedicated to Professor James Hill on the occasion of his 70th birthday. Prof. Hill has been a source of inspiration to the authors, over many years, due to his significant contributions to Applied Mathematics and leadership of the field within Australia. He has also played a key role in the support and mentoring of a new generation of Applied Mathematicians that will influence the field for many years to come. Disciplines Engineering | Science and Technology StudiesAbstract. Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reactiondiffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.Mathematics Subject classifications. 35A35, 37C75, 80A30.