Diffusive escape in a nonlinear shear flow: Life and death at the edge of a windy cliff

Abstract

The survival probability of a particle diffusing in the two-dimensional domainx>0 near a “windy cliff” atx=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 velocityv(x, y)=v sign(y)x, i.e., no average bias either toward or away from the cliff. For this semi-infinite system, the particle survival probability decays with time ast ^−1/4, compared tot ^−1/2 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 |y|<w with particle absorption atx=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^−1/4 decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.