Dynamics of A + B --> C reaction fronts in the presence of buoyancy-driven convection

Chemically Enhanced Convective Dissolution of CO_{2}

Convective dissolution, one of the main mechanisms for geological storage of CO_{2}, occurs when supercritical or gas CO_{2} dissolves partially into an aqueous solution, thus triggering downward convection of the denser CO_{2}-enriched liquid. Chemical reaction in the liquid can greatly enhance the process. Here, experimental measurements of convective flow inside a cylinder filled with a sodium hydroxide (NaOH) solution show that the plume's velocity can be increased tenfold as compared to a situation with no NaOH. This tremendous effect is predicted by a model with no adjusting parameters.

Effect of variable solubility on reactive dissolution in partially miscible systems

When two partially miscible systems are put in contact, one phase, A, can dissolve into the other one with a given solubility. Chemical reactions in the host phase can impact this dissolution by consuming A and by generating products that impact the solubility of A. Here, we study theoretically the optimal conditions for transfer of a reactant A in a host phase containing a species B when a bimolecular A + B → C reaction generates a product C that linearly decreases the solubility of A. We have quantified numerically the influence of this variable solubility on the reaction-diffusion (RD) concentration profiles of all species in the host phase, on the temporal evolution of the position of the reaction front, and on the flux of A through the interface. We have also computed the analytical asymptotic concentration profiles, solutions at long times of the RD governing equations. For a fixed negative effect of C on the solubility of A, an increase in the initial concentration of reactant B or an increase in the diffusion rate of species B and C results in a larger flux of A and hence a larger amount of A dissolved in the host solution at a given time. However, when the influence of C on the solubility increases, the mass transfer decreases. Our results help understand to what extent a chemical reaction can optimize the reactive transfer of a solute to a host phase with application to, among other things, the geological sequestration of carbon dioxide in an aquifer.

Marangoni-driven nonlinear dynamics of bimolecular frontal systems: a general classification for equal diffusion coefficients

When bimolecular fronts form in solutions, their dynamics is likely to be affected by chemically driven convection such as buoyancy- and Marangoni-driven flows. It is known that front dynamics in the presence of buoyancy-driven convection can be predicted solely on the basis of the one-dimensional reaction-diffusion concentration profiles but that those predictions fail for Marangoni-driven convection. With a two-dimensional reaction-diffusion-Marangoni convection model, we analyze here convective effects on the time scalings of the front properties, together with the influence of reaction reversibility and of the ratio of initial reactants' concentrations on the front dynamics. The effect of buoyancy forces is here neglected by assuming the reactive system to be in zero-gravity condition and/or the solution density to be spatially homogenous. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Marangoni- vs. buoyancy-driven flows: competition for spatio-temporal oscillations in A + B → C systems

The emergence of self-organized behaviors such as spatio-temporal oscillations is well-known for complex reactions involving nonlinear chemical or thermal feedback. Recently, it was shown that local oscillations of the chemical species concentration can be induced under isothermal batch conditions for simple bimolecular A + B → C reactions, provided they are actively coupled with hydrodynamics. When two reactants A and B, initially separated in space, react upon diffusive contact, damped spatio-temporal oscillations could develop when the surface tension increases sufficiently in the reaction zone. Additionally, if the density decreases, the coupling of both surface tension- and buoyancy-driven contributions to the flow can further sustain this oscillatory instability. Here, we investigate the opposite case of a reaction inducing a localized decrease in surface tension and an increase in density in the reacting zones. In this case, the competition arising from the two antagonistic flows is needed to create oscillatory dynamics, i.e., no oscillations are observed for pure chemically driven Marangoni flows. We study numerically these scenarios in a 2-dimensional system and show how they are controlled by the following key parameters: (i) ΔM and ΔR governing the surface tension and density variation during the reaction, respectively, (ii) the layer thickness of the system, and (iii) its lateral length. This work is a further step toward inducing and controlling chemical oscillations in simple reactions.

Stability of a layered reactive channel flow

We analyse the linear stability of a reactive plane Poiseuille flow, where a reactant fluid A overlies another reactant B in a layered fashion within a two-dimensional channel. Both reactants are miscible and have the same viscosity, while upon reaction, they produce either a less or more-viscous product fluid C . The reaction kinetics is of simple A + B → C type, and the production of C occurs across the initial contact line of reactants A and B in a mixed zone of small and finite width. All three fluids have the same density. We demonstrate the effects of various controlling parameters such as the log-mobility ratio, Damköhler number, Schmidt number, Reynolds number, position and thicknesses of the reactive zone on the stability characteristics. We show that a tiny viscosity stratification by the reaction destabilizes the flow at a moderate (10–1000) and even at low Reynolds numbers (0.01–1). The maximum growth occurs for shorter waves than for the Tollmien–Schlichting eigenmode, and the ranges of unstable wavenumbers are wider than that known for non-reactive channel flow systems. In most cases, the instability occurs due to the overlap of the critical layer with the viscosity-stratified layer. Surprisingly for some parameters, it is observed that the reaction can make σ M decrease with increasing Reynolds number.

Chemo-hydrodynamic pulsations in simple batch A + B → C systems

Spatio-temporal oscillations can be induced under batch conditions with ubiquitous bimolecular reactions in the absence of any nonlinear chemical feedback, thanks to an active interplay between the chemical process and chemically driven hydrodynamic flows. When two reactants A and B, initially separated in space, react upon diffusive contact, they can power convective flows by inducing a localized variation of surface tension and density at the mixing interface. These flows feedback with the reaction-diffusion dynamics, bearing damped or sustained spatio-temporal oscillations of the concentrations and flow field. By means of numerical simulations, we detail the mechanism underlying these chemohydrodynamic oscillations and classify the main dynamical scenarios in the relevant space drawn by parameters ΔM and ΔR, which rule the surface tension- and buoyancy-driven contributions to convection, respectively. The reactor height is found to play a critical role in the control of the dynamics. The analysis reveals the intimate nature of these oscillatory phenomena and the hierarchy among the different phenomena at play: oscillations are essentially hydrodynamic and the chemical process features the localized trigger for Marangoni flows unstable toward oscillatory instabilities. The characteristic size of Marangoni convective rolls mainly determines the critical conditions and properties of the oscillations, which can be further tuned or suppressed by the buoyancy competition. We finally discuss the possible experimental implementation of such a class of chemo-hydrodynamic oscillator and its implications in fundamental and applied terms.

Spatial analog of the two-frequency torus breakup in a nonlinear system of reactive miscible fluids

We present a theoretical study on pattern formation occurring in miscible fluids reacting by a second-order reaction A+B→C in a vertical Hele-Shaw cell under constant gravity. We have recently reported that the concentration-dependent diffusion of species coupled with a frontal neutralization reaction can produce a multilayer system where low-density depleted zones could be embedded between the denser layers. This leads to the excitation of chemoconvective modes spatially separated from each other by a motionless fluid. In this Rapid Communication, we show that the layers can interact via a diffusion mechanism. Since diffusively coupled instabilities initially have different wavelengths, this causes a long-wave modulation of one pattern by another. We have developed a mathematical model which includes a system of reaction-diffusion-convection equations. The linear stability of a transient base state is studied by calculating the growth rate of the Lyapunov exponent for each unstable layer. Numerical simulations supported by phase portrait reconstruction and Fourier spectra calculation have revealed that nonlinear dynamics consistently passes through (i) a perfect spatially periodic system of chemoconvective cells, (ii) a quasiperiodic system of the same cells, and (iii) a disordered fingering structure. We show that in this system, the coordinate codirected to the reaction front paradoxically plays the role of time, time itself acts as a bifurcation parameter, and a complete spatial analog of the two-frequency torus breakup is observed.

Complex dynamics of interacting fronts in a simple A+B→C reaction-diffusion system

Pattern interaction has so far been restricted to systems with relatively complex reaction schemes, such as activator-inhibitor systems, that lead to rich spatio-temporal dynamics. Surprisingly, a simple second-order chemical reaction is capable of generating similar complex phenomena, such as attractive or repulsive interaction modes between the localized reaction zones (or fronts). We illustrate the latter statement both analytically and numerically with two initially separated A+B→C reaction-diffusion fronts when the solution of B is initially confined between two solutions of A. The nature of the front-front interaction changes from an attractive type to a repulsive one above a critical distance separating the two fronts initially. The complexity of the pattern dynamics emerges here due to finite-size effects. A scaling law relating the critical distance d_{c} above which the repulsion occurs and kinetic parameters gives insights into (i) extracting those parameters from experiments for bimolecular reactions and (ii) the control strategy of periodic patterns.

Making a Simple A+B→C Reaction Oscillate by Coupling to Hydrodynamic Effect

We present a new mechanism through which chemical oscillations and waves can be induced in batch conditions with a simple A+B→C reaction in the absence of any nonlinear chemical feedback or external trigger. Two reactants A and B, initially separated in space, react upon diffusive contact and the product actively fuels in situ convective Marangoni flows by locally increasing the surface tension at the mixing interface. These flows combine in turn with the reaction-diffusion dynamics, inducing damped spatiotemporal oscillations of the chemical concentrations and the velocity field. By means of numerical simulations, we single out the detailed mechanism and minimal conditions for the onset of this periodic behavior. We show how the antagonistic coupling with buoyancy convection, due to concurrent chemically induced density changes, can control the oscillation properties, sustaining or suppressing this phenomenon depending on the relative strength of buoyancy- and surface-tension-driven forces. The oscillatory instability is characterized in the relevant parametric space spanned by the reactor height, the Marangoni (Ma_{i}) and the Rayleigh (Ra_{i}) numbers of the ith chemical species, the latter ruling the surface tension and buoyancy contributions to convection, respectively.

Surface tension- and buoyancy-driven flows across horizontally propagating chemical fronts

Chemical reactions can interplay with hydrodynamic flows to generate various complex phenomena. Because of their relevance in many research areas, chemically-induced hydrodynamic flows have attracted increasing attention in the last decades. In this context, we propose to give a review of the past and recent theoretical and experimental works which have considered the interaction of such flows with chemical fronts, i.e. reactive interfaces, formed between miscible solutions. We focus in particular on the influence of surface tension- (Marangoni) and buoyancy-driven flows on the dynamics of chemical fronts propagating horizontally in the gravity field.

Shock-wave-like structures induced by an exothermic neutralization reaction in miscible fluids

We report shock-wave-like structures that are strikingly different from previously observed fingering instabilities, which occur in a two-layer system of miscible fluids reacting by a second-order reaction A+B→S in a vertical Hele-Shaw cell. While the traditional analysis expects the occurrence of a diffusion-controlled convection, we show both experimentally and theoretically that the exothermic neutralization reaction can also trigger a wave with a perfectly planar front and nearly discontinuous change in density across the front. This wave propagates fast compared with the characteristic diffusion times and separates the motionless fluid and the area with anomalously intense convective mixing. We explain its mechanism and introduce a new dimensionless parameter, which allows to predict the appearance of such a pattern in other systems. Moreover, we show that our governing equations, taken in the inviscid limit, are formally analogous to well-known shallow-water equations and adiabatic gas flow equations. Based on this analogy, we define the critical velocity for the onset of the shock wave which is found to be in the perfect agreement with the experiments.

Enhanced steady-state dissolution flux in reactive convective dissolution

Chemical reactions can accelerate, slow down or even be at the very origin of the development of dissolution-driven convection in partially miscible stratifications when they impact the density profile in the host fluid phase. We numerically analyze the dynamics of this reactive convective dissolution in the fully developed non-linear regime for a phase A dissolving into a host layer containing a dissolved reactant B. We show for a general A + B → C reaction in solution, that the dynamics vary with the Rayleigh numbers of the chemical species, i.e. with the nature of the chemicals in the host phase. Depending on whether the reaction slows down, accelerates or is at the origin of the development of convection, the spatial distributions of species A, B or C, the dissolution flux and the reaction rate are different. We show that chemical reactions can enhance the steady-state flux as they consume A and can induce more intense convection than in the non-reactive case. This result is important in the context of CO₂ geological sequestration where quantifying the storage rate of CO₂ dissolving into the host oil or aqueous phase is crucial to assess the efficiency and the safety of the project.

Flow Control of A+B→C Fronts by Radial Injection

The dynamics of A+B→C fronts is analyzed theoretically in the presence of passive advection when A is injected radially into B at a constant inlet flow rate Q. We compute the long-time evolution of the front position, r_{f}, of its width, w, and of the local production rate R of the product C at r_{f}. We show that, while advection does not change the well-known scaling exponents of the evolution of corresponding reaction-diffusion fronts, their dynamics is however significantly influenced by the injection. In particular, the total amount of product varies as Q^{-1/2} for a given volume of injected reactant and the front position as Q^{1/2} for a given time, paving the way to a flow control of the amount and spatial distribution of the reaction front product. This control strategy compares well with calcium carbonate precipitation experiments for which the amount of solid product generated in flow conditions at fixed concentrations of reactants and the front position can be tuned by varying the flow rate.

Chemical control of dissolution-driven convection in partially miscible systems: nonlinear simulations and experiments

Numerical simulations combined with experimental results from two laboratory-scale model systems show how to control convective dissolution by chemical reactions.

Chemo-hydrodynamic patterns in porous media

Chemical reactions can interplay with hydrodynamic flows to generate chemo-hydrodynamic instabilities affecting the spatio-temporal evolution of the concentration of the chemicals. We review here such instabilities for porous media flows. We describe the influence of chemical reactions on viscous fingering, buoyancy-driven fingering in miscible systems, convective dissolution as well as precipitation patterns. Implications for environmental systems are discussed.This article is part of the themed issue 'Energy and the subsurface'.

Density profiles around A+B→C reaction-diffusion fronts in partially miscible systems: A general classification

Various spatial density profiles can develop in partially miscible stratifications when a phase A dissolves with a finite solubility into a host phase containing a dissolved reactant B. We investigate theoretically the impact of an A+B→C reaction on such density profiles in the host phase and classify them in a parameter space spanned by the ratios of relative contributions to density and diffusion coefficients of the chemical species. While the density profile is either monotonically increasing or decreasing in the nonreactive case, reactions combined with differential diffusivity can create eight different types of density profiles featuring up to two extrema in density, at the reaction front or below it. We use this framework to predict various possible hydrodynamic instability scenarios inducing buoyancy-driven convection around such reaction fronts when they propagate parallel to the gravity field.

Cross-diffusion-driven hydrodynamic instabilities in a double-layer system: General classification and nonlinear simulations

Cross diffusion, whereby a flux of a given species entrains the diffusive transport of another species, can trigger buoyancy-driven hydrodynamic instabilities at the interface of initially stable stratifications. Starting from a simple three-component case, we introduce a theoretical framework to classify cross-diffusion-induced hydrodynamic phenomena in two-layer stratifications under the action of the gravitational field. A cross-diffusion-convection (CDC) model is derived by coupling the fickian diffusion formalism to Stokes equations. In order to isolate the effect of cross-diffusion in the convective destabilization of a double-layer system, we impose a starting concentration jump of one species in the bottom layer while the other one is homogeneously distributed over the spatial domain. This initial configuration avoids the concurrence of classic Rayleigh-Taylor or differential-diffusion convective instabilities, and it also allows us to activate selectively the cross-diffusion feedback by which the heterogeneously distributed species influences the diffusive transport of the other species. We identify two types of hydrodynamic modes [the negative cross-diffusion-driven convection (NCC) and the positive cross-diffusion-driven convection (PCC)], corresponding to the sign of this operational cross-diffusion term. By studying the space-time density profiles along the gravitational axis we obtain analytical conditions for the onset of convection in terms of two important parameters only: the operational cross-diffusivity and the buoyancy ratio, giving the relative contribution of the two species to the global density. The general classification of the NCC and PCC scenarios in such parameter space is supported by numerical simulations of the fully nonlinear CDC problem. The resulting convective patterns compare favorably with recent experimental results found in microemulsion systems.

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.

Buoyancy-driven instabilities around miscible A+B→C reaction fronts: a general classification

Upon contact between miscible solutions of reactants A and B along a horizontal interface in the gravity field, various buoyancy-driven instabilities can develop when an A+B→C reaction takes place and the density varies with the concentrations of the various chemicals. To classify the possible convective instability scenarios, we analyze the spatial dependence of the large time asymptotic density profiles as a function of the key parameters of the problem, which are the ratios of diffusion coefficients and of solutal expansion coefficients of species A, B, and C. We find that 62 different density profiles can develop in the reactive problem, whereas only 6 of them can be obtained in the nonreactive one.

Chemical convection in the methylene-blue-glucose system: Optimal perturbations and three-dimensional simulations

A case of convection driven by chemical reactions is studied by linear stability theory and direct numerical simulations. In a plane aqueous layer of glucose, the methylene-blue-enabled catalytic oxidation of glucose produces heavier gluconic acid. As the oxygen is supplied through the top surface, the production of gluconic acid leads to an overturning instability. Our results complement earlier experimental and numerical work by Pons et al. First, we extend the model by including the top air layer with diffusive transport and Henry's law for the oxygen concentration at the interface to provide a more realistic oxygen boundary condition. Second, a linear stability analysis of the diffusive basic state in the layers is performed using an optimal perturbation approach. This method is appropriate for the unsteady basic state and determines the onset time of convection and the associated wavelength. Third, the nonlinear evolution is studied by the use of three-dimensional numerical simulations. Three typical parameters sets are explored in detail showing significant differences in pattern formation. One parameter set for which the flow is dominated by viscous forces, displays persistently growing convection cells. The other set with increased reaction rate displays a different flow regime marked by local chaotic plume emission. The simulated patterns are then compared to experimental observations.

Control of convective dissolution by chemical reactions: general classification and application to CO(2) dissolution in reactive aqueous solutions

In partially miscible two-layer systems within a gravity field, buoyancy-driven convective motions can appear when one phase dissolves with a finite solubility into the other one. We investigate the influence of chemical reactions on such convective dissolution by a linear stability analysis of a reaction-diffusion-convection model. We show theoretically that a chemical reaction can either enhance or decrease the onset time of the convection, depending on the type of density profile building up in time in the reactive solution. We classify the stabilizing and destabilizing scenarios in a parameter space spanned by the solutal Rayleigh numbers. As an example, we experimentally demonstrate the possibility to enhance the convective dissolution of gaseous CO_{2} in aqueous solutions by a classical acid-base reaction.

Thermal effects on the diffusive layer convection instability of an exothermic acid-base reaction front

A buoyancy-driven hydrodynamic instability appearing when an aqueous acid solution of HCl overlies a denser alkaline aqueous solution of NaOH in a vertically oriented Hele-Shaw cell is studied both experimentally and theoretically. The peculiarity of this reactive convection pattern is its asymmetry with regard to the initial contact line between the two solutions as convective plumes develop in the acidic solution only. We investigate here by a linear stability analysis (LSA) of a reaction-diffusion-convection model of a simple A+B→C reaction the relative role of solutal versus thermal effects in the origin and location of this instability. We show that heat effects are much weaker than concentration-related ones such that the heat of reaction only plays a minor role on the dynamics. Computation of density profiles and of the stability analysis eigenfunctions confirm that the convective motions result from a diffusive layer convection mechanism whereby a locally unstable density stratification develops in the upper acidic layer because of the difference in the diffusion coefficients of the chemical species. The growth rate and wavelength of the pattern are determined experimentally as a function of the Brinkman parameter of the problem and compare favorably with the theoretical predictions of both LSA and nonlinear simulations.

Flow-induced transitions in bistable systems

We studied transitions between spatiotemporal patterns that can be induced in a spatially extended nonlinear chemical system by a unidirectional flow in combination with constant inflow concentrations. Three different scenarios were investigated. (i) Under conditions where the system exhibited two stable fixed points, the propagation direction of trigger fronts could be reversed, so that domains of the less stable fixed point invaded the system. (ii) For bistability between a stable fixed point and a limit cycle we observed that above a critical flow velocity, the unstable focus at the center of the limit cycle could be stabilized. Increasing the flow speed further, a regime of damped flow-distributed oscillations was found and, depending on the boundary values at the inflow, finally the stable fixed point dominated. Similarly, also in the case of spatiotemporal chaos (iii), the unstable steady state could be stabilized and was replaced by the stable fixed point with increasing flow velocity. We finally outline a linear stability analysis that can explain part of our findings.

Pattern formation in a reaction-diffusion-advection system with wave instability

In this paper, we show by means of numerical simulations how new patterns can emerge in a system with wave instability when a unidirectional advective flow (plug flow) is added to the system. First, we introduce a three variable model with one activator and two inhibitors with similar kinetics to those of the Oregonator model of the Belousov-Zhabotinsky reaction. For this model, we explore the type of patterns that can be obtained without advection, and then explore the effect of different velocities of the advective flow for different patterns. We observe standing waves, and with flow there is a transition from out of phase oscillations between neighboring units to in-phase oscillations with a doubling in frequency. Also mixed and clustered states are generated at higher velocities of the advective flow. There is also a regime of "waving Turing patterns" (quasi-stationary structures that come close and separate periodically), where low advective flow is able to stabilize the stationary Turing pattern. At higher velocities, superposition and interaction of patterns are observed. For both types of patterns, at high velocities of the advective field, the known flow distributed oscillations are observed.

Convective mixing induced by acid-base reactions

When two miscible solutions, each containing a reactive species, are put in contact in the gravity field, local variations in the density due to the reaction can induce convective motion and mixing. We characterize here both experimentally and theoretically such buoyancy-driven instabilities induced by the neutralization of a strong acid by a strong base in aqueous solutions. The diverse patterns obtained are shown to depend on the type of reactants used and on their relative concentrations. They have their origin in a combination of classical hydrodynamic instabilities including differential diffusion of the solutes involved while temperature effects only play a marginal role.

Influence of double diffusive effects on miscible viscous fingering

Miscible viscous fingering classically occurs when a less viscous fluid displaces a miscible more viscous one in a porous medium. We analyze here how double diffusive effects between a slow diffusing S and a fast diffusing F component, both influencing the viscosity of the fluids at hand, affect such fingering, and, most importantly, can destabilize the classically stable situation of a more viscous fluid displacing a less viscous one. Various instability scenarios are classified in a parameter space spanned by the log-mobility ratios R(s) and R(f) of the slow and fast component, respectively, and parametrized by the ratio of diffusion coefficients δ. Numerical simulations of the full nonlinear problem confirm the existence of the predicted instability scenarios and highlight the influence of differential diffusion effects on the nonlinear fingering dynamics.

Chemically driven hydrodynamic instabilities

In the gravity field, density changes triggered by a kinetic scheme as simple as A+B-->C can induce or affect buoyancy-driven instabilities at a horizontal interface between two solutions containing initially the scalars A and B. On the basis of a general reaction-diffusion-convection model, we analyze to what extent the reaction can destabilize otherwise buoyantly stable density stratifications. We furthermore show that, even if the underlying nonreactive system is buoyantly unstable, the reaction breaks the symmetry of the developing patterns. This is demonstrated both numerically and experimentally on the specific example of a simple acid-base neutralization reaction.

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.

Analytical asymptotic solutions of nA+mB-->C reaction-diffusion equations in two-layer systems: a general study

Large time evolution of concentration profiles is studied analytically for reaction-diffusion systems where the reactants A and B are each initially separately contained in two immiscible solutions and react upon contact and transfer across the interface according to a general nA+mB-->C reaction scheme. This study generalizes to immiscible two-layer systems the large time analytical asymptotic limits of concentrations derived by Koza [J. Stat. Phys. 85, 179 (1996)] for miscible fluids and for reaction rates of the form A;{n}B;{m} with arbitrary diffusion coefficients and homogeneous initial concentrations. In addition to a dependence on the parameters already characterizing the miscible case, the asymptotic concentration profiles in immiscible systems depend now also on the partition coefficients of the chemical species between the two solution layers and on the ratio of diffusion coefficients of a given species in the two fluids. The miscible time scalings are found to remain valid for the immiscible fluids case. However, for immiscible systems, the reaction front speed is enhanced by increasing the stoichiometry of the invading species over that of the species being invaded. The direction of the front propagation is found to depend on the diffusion coefficient of the invading species in its initial fluid but not on its value in the invading fluid. Hence, a reaction front in immiscible fluids can travel in the opposite direction to the reaction front formed in miscible fluids for a range of parameter values. The value of the invading species partition coefficient affects the magnitude of the front speed but it cannot alter the direction of the front. For sufficiently large times, the total amount of product produced in time is independent of the rate of the reaction. The centre of mass of the product can move in the opposite direction to the center of mass of the reaction rate.