The effect of the order of autocatalysis for reaction fronts in vertical slabs

Abstract

The coupling of molecular diffusion and chemical autocatalysis leads to propagating chemical fronts. The specific type of autocatalysis determines the thickness and speed of the front. Chemical fronts in liquids separate reacted from unreacted fluids with a corresponding mass density gradient. These differences may lead to convective flow which enhances the speed and determines the curvature of the front. The transition to convection is determined not only by the spatial geometry and density differences, but also by the type of chemical reaction. We determine the transition to convection for chemical fronts with different autocatalytic reactions of different order. We study fronts propagating in porous media, in viscous fluids, and fluids confined in a vertical slab. We compare these results with the results based on a thin front approximation using an eikonal relation.

Rayleigh–Taylor instability of pulled versus pushed fronts

“…Some pulled fronts become indeed very sensitive to noise or slight changes in the equations [20], but in other cases it appears that an empirical moving boundary (or thin front) approximation still leads to sensible results for the stability of the fronts, provided one takes into account that the convergence of the pulled fronts to their asymptotic speed and shape is quite slow. The results of Coroian and Vazquez [18] and similar results for combustion fronts [22] indicate that this is the case with chemical and combustion fronts. Indeed, as we shall see in this paper, for the "quadratic" autocatalytic fronts with a Rayleigh-Taylor instability, the main implication of the pulled nature of these fronts appears to be that the Rayleigh-Taylor instability emerges only very slowly in time, much slower than in the pushed regime.…”

“…The nonlinear pushed fingers are robust with regard to fluctuations, which allows their fingering dynamics to be mapped into that of a sharp interface. This explains why the dispersion curves for the cubic kinetics of the iodate-arsenous acid reaction [9,[11][12][13], the fourth-order kinetics of the chlorite-tetrathionate reaction [10,14,15] and more generally other typical pushed kinetics of order larger than two [18,19] all have similar properties. In the same spirit, their nonlinear dynamics [1][2][3]16] show similar features that could be mapped into a sharp interface description independently of the details of the pushed kinetics.…”

“…This might be a reason for the difference in stability properties between an order two pulled kinetics and higher-order pushed kinetics noted by Coroian and Vasquez [18]. Unfortunately, insufficient information about the simulations in [18] is available to make quantitative estimates of the importance of such effects.…”

“…(7) below) [17][18][19]. For example, in a careful study of Coroian and Vasquez [18], it was found that going from cubic to higher-order autocatalytic reactions only induces very small shifts in the critical Rayleigh number at which the front becomes unstable in wide cells, while in their widest cells the critical Rayleigh number was found to be about three times as big when they switched from cubic to quadratic kinetics. In other words, fronts with order two kinetics were found to be much more stable with regard to fingering than higher-order ones which converged to the results for an eikonal relation in wide cells.…”

Density fingering in spatially bimodulated Hele-Shaw cells

“…Several experimental [5][6][7][8] and numerical studies [9][10][11][12][13] have been carried out in order to quantitatively describe the effect of convection in wide and narrow reaction vessels, so called Hele-Shaw cells [14]. Dispersion curves were introduced to characterize the initial pattern evolution where the growth rate is given as a function of its wave number.…”