We study theoretically and numerically the effects of the linear velocity field $\mathrm{v}=v_{0}y\hat{\mathrm{x}}$ 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 $t_c\sim v_0^{-1\mathrm{}}$. For $t\ll v_0^{-1\mathrm{}}$, the kinetics are unaffected by the shear and we retain both the effect of species segregation (for $d<\mathrm{}4\mathrm{}$) 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 $2<~d<\mathrm{}4\mathrm{}$, and give bounds in $d=4\mathrm{}$. However, for $t\gg v_0^{-1\mathrm{}}$, the critical dimension for anomalous kinetics is reduced to $d_c=2\mathrm{}$, with the density decay rate $B{t}^{-1\mathrm{}}$ holding for $d>~2$. Bounds are calculated for the amplitude $B$ in $d=2\mathrm{}$, which depend on the velocity gradient $v_0$ and the (equal) diffusion constants $D$. We also briefly consider the case of a nonlinear 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 $d=2\mathrm{}$ with results in agreement with theoretical predictions.

• Received 22 December 1995

DOI:https://doi.org/10.1103/PhysRevE.53.5949