Shear flows and segregation in the reaction A+B→∅

We study theoretically and numerically the effects of the linear velocity field v=v0yx^ on the irreversible reaction A+B→∅. Assuming homogeneous initial conditions for the two species, with equal initial densities, we demonstrate the presence of a crossover time tc∼v−10. For t≪v−10, 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−α, where α=min(d4, 1). We calculate the amplitude A to leading order in a small density expansion for 2<~d<4, and give bounds in d=4. However, for t≫v−10, the critical dimension for anomalous kinetics is reduced to dc=2, with the density decay rate Bt−1 holding for d>~2. Bounds are calculated for the amplitude B in d=2, which depend on the velocity gradient v0 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 tc. Finally, we perform numerical simulations for a linear shear flow in d=2 with results in agreement with theoretical predictions.