Differential-flow-induced instability in a cubic autocatalator system

Abstract

The formation of spatio-temporal stable patterns is considered for a reaction-diffusion-convection system based upon the cubic autocatalator, A + 2B → 3B, B → C, with the reactant A being replenished by the slow decay of some precursor P via the simple step P → A. The reaction is considered in a differential-flow reactor in the form of a ring. It is assumed that the reactant A is immobilised within the reactor and the autocatalyst B is allowed to flow through the reactor with a constant velocity as well as being able to diffuse. The linear stability of the spatially uniform steady state (a, b) = (µ⁻¹, µ), where a and b are the dimensionless concentrations of the reactant A and autocatalyst B, and µ is a parameter reflecting the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, flow-generated patterns is that the flow parameter φ > φ_c(µ, λ) where φ_c(µ,λ) is a (strictly positive) critical value of φ and λ is the dimensionless diffusion coefficient of the species B and also reflects the size of the system. Values of φ_c at which these bifurcations occur are derived in terms of µ and λ. Further information about the nature of the bifurcating branches (close to their bifurcation points) is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical Hopf bifurcations are possible. The bifurcating branches are then followed numerically by means of a path-following method, with the parameter φ as a bifurcation parameter, for representatives values of µ and λ. It is found that multiple stable patterns can exist and that it is also possible that any of these can lose stability through secondary Hopf bifurcations. This typically gives rise to spatio-temporal quasiperiodic transients through which the system is ultimately attracted to one of the remaining available stable patterns.