Effect of radial geometry on autocatalytic reaction-diffusion-advection fronts

The understanding of the dynamics of reaction diffusion (RD) fronts is crucial for a wide variety of applications in chemistry, biology, physics and ecology, and it is especially important for hydrogeological problems involving chemical reactions. Reactive transport in geological media is generally controlled by the interplay of physical and chemical processes, which can give rise to complex dynamics of the reaction front. An important subset of RD fronts is represented by autocatalytic fronts, for which it is well known that the coupling of diffusion and chemical processes gives rise to self-organization phenomena and pattern forming instabilities [1]. When the initial interface between the reactant and the catalyst is a straight line, the autocatalytic front behaves as a solitary wave, which means that the shape of the front remains unchanged as it travels towards the nonreacted species [2]. The coupling with uniform advection does not change the picture, provided that the system is described in the proper comoving reference frame.

However, in this work we show that the geometrical properties of the injection source have a significant impact on the reaction front dynamics. Indeed, if the injection of one reactant into the other is performed radially at a constant flow rate, the pre-asymptotic dynamics of the front is strongly affected by the nonuniform velocity field. Moreover, although at long times the front still behaves as a solitary wave, the efficiency of the reaction is strongly increased in virtue of the increasing volume occupied by the radial front. We show how injecting a finite amount of reactant into the catalyst gives rise to collapsing fronts and we characterize their dynamics in terms of their position, width and the production rate. In contrast, when the reactant is injected into the catalyst at a constant flow rate, a stationary regime is reached where, unlike the case of solitary waves, the autocatalytic front does not move.      

 

References:

[1] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Dynamics: Oscillations, Waves, Patterns, and Chaos (Oxford University Press, Oxford, 1998)

[2] P. Gray, K. Showalter, and S. K. Scott, J. Chim. Phys. 84, 1329 (1987)