Instabilities of cubic autocatalytic waves on two and three dimensional domains

Abstract

Planar wave fronts in autocatalytic chemical systems propagate with a constant wave form and velocity provided that the reactant and autocatalytic species have similar diffusion coefficients. Such waves are also stable to spatial perturbations. Circular or spherical fronts show a constant wave form and a velocity that increases towards the planar wave velocity as the radius increases with time. These are again stable to spatial perturbation if the reactant and autocatalyst have similar diffusivities. However, if the ratio of the diffusion coefficients δ exceeds some critical value δ*≊2.3 a different situation arises. For cylindrical or spherical geometries, unperturbed waves decelerate as they expand if δ≳δ*. For all geometries, the smooth waves may become unstable to spatial perturbation if δ≳δ* although there are some additional requirements. In Cartesian systems, the width of the reaction zone transverse to the direction of propagation must exceed some minimum value W*≊6 (in dimensionless units) and the wave number of the imposed perturbation must be less than kcr,max≊0.15. For circular or spherical waves, the conditions for the growth of perturbations also involves the radius of the wave at the moment the perturbation is applied. A set of expansions based on small curvature and small departures of δ from unity have been derived allowing the instantaneous wave velocity to be written in the form v(θ,φ,τ)=c+v1κ+v2Δtrκ, where the coefficients c, v1, and v2 depend on δ and κ is the instantaneous curvature.