# Shear Flows and Segregation in the Reaction $A+B\\to\\emptyset$ Source: [Shear Flows and Segregation in the Reaction $A+B\\to\\emptyset$]() Collection index: [llms.txt]() Data updated: 2025-10-24T09:00:16.554Z Authors: M.J. Howard, G.T. Barkema Published: 1996-01-22T23:50:27.000Z ## Abstract 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()