Higher-order large-time asymptotics for a reaction of the form nA+mB-->C

Chemical control of dissolution-driven convection in partially miscible systems: theoretical classification

Dissolution-driven convection occurs in the host phase of a partially miscible system when a buoyantly unstable density stratification develops upon dissolution. Reactions can impact such convection by changing the composition and thus the density of the host phase. Here we study the influence of A + B → C reactions on such convective dissolution when A is the dissolving species and B a reactant initially in solution in the host phase. We perform a linear stability analysis of related reaction-diffusion density profiles to compare the growth rate of the instability in the reactive case to its non reactive counterpart when all species diffuse at the same rate. We classify the stabilizing or destabilizing influence of reactions on the buoyancy-driven convection in a parameter space spanned by the solutal Rayleigh numbers RA,B,C of chemical species A, B, C and by the ratio β of initial concentrations of the reactants. For RA > 0, the non reactive dissolution of A in the host phase is buoyantly unstable. In that case, we show that the reaction is enhancing convection provided C is sufficiently denser than B. Increasing the ratio β of initial reactant concentrations increases the effect of chemistry but does not significantly impact the stabilizing/destabilizing classification. When the non reactive case is buoyantly stable (RA≤ 0), reactions can create in time an unstable density stratification and trigger convection if RC > RB. Our theoretical approach allows classifying previous results in a unifying picture and developing strategies for the chemical control of convective dissolution.

Scaling and crossover dynamics in the hyperbolic reaction-diffusion equations of initially separated components

In this paper we investigate the dynamics of front propagation in the family of reactions (nA + mB (k)→ C) with initially segregated reactants in one dimension using hyperbolic reaction-diffusion equations with the mean-field approximation for the reaction rate. This leads to different dynamics than those predicted by their parabolic counterpart. Using perturbation techniques, we focus on the initial and intermediate temporal behavior of the center and width of the front and derive the different time scaling exponents. While the solution of the parabolic system yields a short time scaling as t(1/2) for the front center, width, and global reaction rate, the hyperbolic system exhibits linear scaling for those quantities. Moreover, those scaling laws are shown to be independent of the stoichiometric coefficients n and m. The perturbation results are compared with the full numerical solutions of the hyperbolic equations. The crossover time at which the hyperbolic regime crosses over to the parabolic regime is also studied. Conditions for static and moving fronts are also derived and numerically validated.

Analytical small-time asymptotic properties of A+B-->C fronts

The small-time asymptotic properties of the reaction front formed by a reaction A+B-->C coupled to diffusion are considered. Reactants A and B are initially separately dissolved in two identical solvents. The solvents are brought into contact and the reactants meet through diffusion. The small-time asymptotic position of the center of mass of the reaction rate is obtained analytically. When one of the reactants diffuses much faster than the other reactant then the position of the local maximum in the reaction rate travels on a length scale related to the diffusion coefficient of the slowest diffusing reactant while the first moment of the reaction rate and the width of the reaction front are on a length scale related to the diffusion coefficient of the fastest diffusing reactant. If the sum of the initial reactant concentrations is fixed, then the fastest reaction rate is obtained when equal concentrations are used. The first-order solutions are analytically obtained, however, each solution involves an integral which requires numerical evaluation. Various small-time asymptotic analytical reaction front properties are obtained. In particular, one finds that the position of the center of mass of the product concentration distribution is initially located at three quarters of the position of the center of mass of the reaction rate.