The long-time behavior of initially separated A+B→0 reaction-diffusion systems with arbitrary diffusion constants

Abstract

We examine the long time behaviour of A + B → 0 reaction diffusion systems with initially segregated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constants D A , D B , and initial concentrations a 0 and b 0 of A's and B's. We divide the domain of the partial differential equations describing the problem into several regions in which they can be reduced to simpler, solvable equations, and we merge the solutions. Thus we derive general formulae for the concentration profiles outside the reaction zone, the location of the reaction zone center, and the total reaction rate. An asymptotic condition for the reaction front to be stationary is also derived. The properties of the reaction layer are studied in the mean-field approximation, and we show that not only the scaling exponents, but also the scaling functions are independent of D A , D B , a 0 and b 0 .

Influence of buoyancy-driven convection on the dynamics of A+B→C reaction fronts in horizontal solution layers

“…The largest domain involved a computational grid of 1024 Â 64 points for L x ¼ 1024 and L z ¼ 20, and a typical time step dt ¼ 0:1ðdzÞ 2 $ 9:8 Â 10 À3 . For zero Rayleigh numbers, we also recovered the RD results previously obtained (see Galfi and R acz, 1988;Jiang and Ebner, 1990;Koza, 1996). We now recall the main characteristics of this RD system in the absence of flow.…”

“…For an infinite domain and in the asymptotic limit of large times, exact scaling laws characterizing the RD dynamics without convection were derived in the work of Galfi and Racz (1988) and in subsequent other theoretical studies including Jiang and Ebner (1990), Cornell et al (1995), Koza (1996) and Sinder and Pelleg (2000). They found that the position x f and the width w of the reaction front (defined as the position along x where the production rate is maximum and as the square root of the second moment of the production rate distribution, respectively) scale with time as x f $ t 1=2 and w $ t 1=6 .…”

“…They found that the position x f and the width w of the reaction front (defined as the position along x where the production rate is maximum and as the square root of the second moment of the production rate distribution, respectively) scale with time as x f $ t 1=2 and w $ t 1=6 . It can be shown that actually x f ¼ C f ffiffi t p with the constant C f ¼ C f ðb 0 =a 0 ; D a ; D b Þ, so that the motion of the reaction front depends only on the diffusion coefficients of the reactants and on the ratio of their initial concentrations, b 0 =a 0 , but not on the individual magnitudes of a 0 and b 0 , (Danckwerts, 1950;Koza, 1996). Further, Koza (1996) showed that the reaction front eventually propagates towards positive x (i.e.…”

“…It can be shown that actually x f ¼ C f ffiffi t p with the constant C f ¼ C f ðb 0 =a 0 ; D a ; D b Þ, so that the motion of the reaction front depends only on the diffusion coefficients of the reactants and on the ratio of their initial concentrations, b 0 =a 0 , but not on the individual magnitudes of a 0 and b 0 , (Danckwerts, 1950;Koza, 1996). Further, Koza (1996) showed that the reaction front eventually propagates towards positive x (i.e. to the right here) if and only if ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi D a =D b p 4 b 0 =a 0 .…”

“…Taitelbaum and coworkers have addressed the behavior of such systems in the non-diffusion limited or 'short time' regime, showing that such 'early time' behaviors can be characterized by completely different properties (see Taitelbaum et al, 1992;Koza and Taitelbaum, 1996). In the small time limit the position of the reaction front is also given by x f ¼ C f ffiffi t p , however, Koza (1996) showed that the constant C f ¼ C f ðD a ; D b Þ, i.e. C f is independent of the initial concentrations a 0 and b 0 .…”