Shear Flows and Segregation in the Reaction $A+B\to \varnothing$

Abstract

We study theoretically and numerically the effects of the linear velocity field $\mathbf{v}=v_{0}y\hat{\mathbf{x}}$ on the irreversible reaction $A+B\to \varnothing$. Assuming homogeneous initial conditions for the two species, with equal initial densities, we demonstrate the presence of a crossover time $t_c\sim v_0^{-1}$. For $t\ll v_0^{-1}$, the kinetics are unaffected by the shear and we retain both the effect of species segregation (for $d<4$) and the density decay rate $A{t}^{-\alpha }$, where $\alpha =\min (\frac{d}{4},1)$. We calculate the amplitude $A$ to leading order in a small density expansion for $2\le d<4$, and give bounds in $d=4$. However, for $t\gg v_0^{-1}$, the critical dimension for anomalous kinetics is reduced to $d_c=2$, with the density decay rate $B{t}^{-1}$ holding for $d\ge 2$. Bounds are calculated for the amplitude $B$ in $d=2$, which depend on the velocity gradient $v_0$ and the (equal) diffusion constants $D$. We also briefly consider the case of a non-linear shear flow, where we give a more general form for the crossover time $t_c$. Finally, we perform numerical simulations for a linear shear flow in $d=2$ with results in agreement with theoretical predictions.

26 pages latex document, 13 figures, uses epsf.sty

References in corpus (7)

• Renormalization Group Calculation for the Reaction $kA\to \varnothing$
• Renormalization Group Study of the A+B->0 Diffusion-Limited Reaction
• The Reaction-Diffusion Front for $A+B\to \varnothing$ in One Dimension
• FLUCTUATION EFFECTS AND MULTISCALING OF THE REACTION-DIFFUSION FRONT FOR A+B->0
• Kinetics of A+B--->0 with Driven Diffusive Motion
• Asymptotic behavior of A + B --> inert for particles with a drift
• Spatial Organization in the Reaction A + B --> inert for Particles with a Drift

Cited by in corpus (3)

• Applications of Field-Theoretic Renormalization Group Methods to Reaction-Diffusion Problems
• Master equations and the theory of stochastic path integrals
• The A+B -> 0 annihilation reaction in a quenched random velocity field