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) = v sign (y)(X) over cap, 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(-1/4), compared to t(-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 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(-1/4) decay when the width of the strip diverges. Supporting numerical simulations of our analytical results are presented.