Diffusive Escape in a Nonlinear Shear Flow: Life and Death at the Edge of a Windy Cliff

The survival probability of a particle diffusing in the two-dimensional domain x > 0 near a "windy cliff" at x = 0 is investigated. The particle dies upon reaching the edge of the cliff: In addition to diffusion, the particle is influenced by a steady "wind shear" with velocity v(x,y)=vsign (y)~, i.e., no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time as t-~/4, compared to t-t/2 in the absence of wind. Scaling descriptions are developed to elucidate this behavior, as well as the survival probability within a semi-infinite strip of finite width lYl < w with particle absorption at x = 0. The behavior in the strip geometry can be described in terms of Taylor diffusion, an approach which accounts for the crossover to the t t / 4 decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.