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

We study theoretically and numerically the effects of the linear velocity field ${\bf v}=v_0y{\bf\hat x}$ on the irreversible reaction $A+B\rightarrow\emptyset$. 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 $At^{-\alpha}$, where $\alpha=\min({d\over 4},1)$. We calculate the amplitude $A$ to leading order in a small density expansion for $2\leq d<4$, 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 $Bt^{-1}$ holding for $d\geq 2$. 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.