Interaction between buoyancy and diffusion-driven instabilities of propagating autocatalytic reaction fronts. I. Linear stability analysis.

The interaction between buoyancy-driven and diffusion-driven instabilities that can develop along a propagating reaction front is discussed for a system based on an autocatalytic reaction. Twelve different cases are possible depending on whether the front is ascending or descending in the gravity field, whether the reactant is heavier or lighter than the products, and whether the reactant diffuses faster, slower, or at the same rate as the product. A linear stability analysis (LSA) is undertaken, in which dispersion curves (plots of the growth rate sigma against wave number k) are derived for representative cases as well as an asymptotic analysis for small wave numbers. The results from the LSA indicate that, when the initial reactant is denser than the reaction products, upward propagating fronts remain unstable with the diffusion-driven instability enhancing this instability. Buoyantly stable downward propagating fronts become unstable when the system is also diffusionally unstable. When the initial reactant is lighter than the reaction products, any diffusionally unstable upward propagating front is stabilized by small buoyancy effects. A diffusional instability enhances the buoyant instability of a downward propagating front with there being a very strong interaction between these effects in this case.

Interaction between buoyancy and diffusion-driven instabilities of propagating autocatalytic reaction fronts. II. Nonlinear simulations.

The nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for fronts ascending or descending in the gravity field and for various values of the relevant parameters, the Rayleigh numbers R(a) and R(b) of the reactant A and autocatalytic product B, respectively, and the ratio D=D(B)/D(A) of the diffusion coefficients of the two key chemical species. The interaction between the coarsening dynamics characteristic of the RT instability and the fixed short wavelength dynamics of the diffusive instability leads in some parameter regimes to complex dynamics dominated by the irregular succession of birth and death of fingers. Large single convective fingers with a tip deformed by the short wavelength diffusive instability are also observed. If D is sufficiently small and the RT instability is active, the concentration of the slower diffusing species B can be convected to values above its fully reacted concentration. Experimental conditions that would allow the observation of the dynamics predicted here are described.

Front fingering and complex dynamics driven by the interaction of buoyancy and diffusive instabilities.

Traveling fronts can become transversally unstable either because of a diffusive instability arising when the key variables diffuse at sufficiently different rates or because of a buoyancy-driven Rayleigh-Taylor mechanism when the density jump across the front is statically unfavorable. The interaction between such diffusive and buoyancy instabilities of fronts is analyzed theoretically for a simple model system. Linear stability analysis and nonlinear simulations show that their interplay changes considerably the stability properties with regard to the pure Rayleigh-Taylor or diffusive instabilities of fronts. In particular, an instability scenario can arise which triggers convection around statically stable fronts as a result of differential diffusion. Moreover, spatiotemporal chaos can be observed when both buoyancy and diffusive effects cooperate to destabilize the front. Experimental conditions to test our predictions are suggested.

Flow-field development during finger splitting at an exothermic chemical reaction front.

Fingertip splitting may be observed at chemical reaction fronts subject to buoyancy-induced Rayleigh-Taylor fingering, as investigated in ascending fronts of the iodate-arsenous acid reaction in vertical Hele-Shaw cells. We study the properties of the flow-field evolution during a tip-splitting event both experimentally and theoretically. Experimental particle-image velocimetry techniques show that the flow field associated to a finger displays a quadrupole of vortices. The evolution of the flow field and the reorganization of the vortices after a tip-splitting event are followed experimentally in detail. Numerical integration of a model reaction-diffusion-convection system for an exothermic reaction taking into account possible heat losses through the walls of the reactor shows that the nonlinear properties of the flow field are different whether the walls are insulating or conducting. In insulating systems, the flow field inside one finger features only one pair of vortices. A quadrupole of flow vortices arranged around a saddle-node structure similar to the one observed experimentally is obtained in the presence of heat losses suggesting that heat effects, even if of very small amplitude, are important in understanding the nonlinear properties of fingering of exothermic chemical fronts.

Effects of constant electric fields on the buoyant stability of reaction fronts.

The effects that applying constant electric fields have on the buoyant instability of reaction fronts propagating vertically in a Hele-Shaw cell are investigated for a range of electric field strengths and fluid parameters. The reaction produces a decrease in density across the front such that upwards propagating fronts are buoyantly unstable in the field-free situation. The reaction kinetics are modeled by cubic autocatalysis. A linear stability analysis reveals that a positive electric field increases the stability of a reaction front and can stabilize an otherwise unstable front. A negative field has the opposite effect, making the reaction front more unstable. Numerical simulations of the full nonlinear problem confirm these predictions and show the development of cellular fingers on unstable fronts. These simulations show that the electric field effects on the reaction within the front can alter the fluid density so as to give the possibility of destabilizing an otherwise stable downward propagating front.

Effects of a constant electric field on the diffusional instability of cubic autocatalytic reaction fronts.

An electric field applied in the direction of propagation of a chemical reaction-diffusion front can affect the stability of this front with regard to diffusive instabilities. The influence of an applied constant electric field is investigated by a linear stability analysis and by nonlinear simulations of a simple chemical system based on the cubic autocatalytic reaction A-+2B--->3B-. The diffusional stability of the front is seen to depend on the intensity E and sign of the applied field, and D, the ratio diffusion coefficients of the reactant species. Depending on E, the front can become more or less diffusively unstable for a given value of D. Above a critical value of E, which depends on D, electrophoretic separation of the two fronts is observed.

On the classification of buoyancy-driven chemo-hydrodynamic instabilities of chemical fronts.

Exothermic autocatalytic fronts traveling in the gravity field can be deformed by buoyancy-driven convection due to solutal and thermal contributions to changes in the density of the product versus the reactant solutions. We classify the possible instability mechanisms, such as Rayleigh-Benard, Rayleigh-Taylor, and double-diffusive mechanisms known to operate in such conditions in a parameter space spanned by the corresponding solutal and thermal Rayleigh numbers. We also discuss a counterintuitive instability leading to buoyancy-driven deformation of statically stable fronts across which a solute-light and hot solution lies on top of a solute-heavy and colder one. The mechanism of this chemically driven instability lies in the coupling of a localized reaction zone and of differential diffusion of heat and mass. Dispersion curves of the various cases are analyzed. A discussion of the possible candidates of autocatalytic reactions and experimental conditions necessary to observe the various instability scenarios is presented.

Reaction driven convection around a stably stratified chemical front.

A vertical stratification of a light and hot fluid over a heavy and cold one is expected to be stable with regard to buoyancy-driven convection. Here we show that chemical reactions can trigger convection around chemical fronts even in cases where concentration and heat both contribute to a stable density stratification. The balance between intrinsic thermal and solutal density gradients initiated by a spatially localized reaction zone and double diffusive mechanisms are at the origin of a new convective instability, the mechanism of which is explained by a displaced particle argument. Linear stability analysis of a reaction-diffusion-convection model confirmed by nonlinear simulations delimits the instability region in the parameter space spanned by the thermal and solutal Rayleigh numbers. Experimental systems in which to test our theoretical predictions are proposed.

Fingering of exothermic reaction-diffusion fronts in Hele-Shaw cells with conducting walls.

We consider the influence of heat losses through the walls of a Hele-Shaw cell on the linear stability and nonlinear dynamics of exothermic chemical fronts whose solutal and thermal contributions to density changes have the same signs. Our analysis is based on the reaction-diffusion-convection equations obtained from the Darcy-Boussinesq approximation. The parameters governing the equations are the Damkohler number, a kinetic parameter d, the Lewis number Le, the thermal-expansion coefficient gammaT, and a heat-transfer coefficient alpha which measures heat losses through the walls. We show that for thermally insulating walls, the temperature profile is a front that follows the concentration profile, while in the presence of heat losses, the temperature profile becomes a pulse that leads to a nonmonotonic density profile which in turn may lead to a destabilization of an otherwise stable front.