Probability distribution of a trapped brownian particle in plane shear flows.

We investigate the statistical properties of an overdamped brownian particle that is trapped by a harmonic potential and simultaneously exposed to a linear shear flow or to a plane Poiseuille flow. Its probability distribution is determined via the corresponding Smoluchowski equation, which is solved analytically for a linear shear flow. In the case of a plane Poiseuille flow, analytical approximations for the distribution are obtained by a perturbation analysis and are substantiated by numerical results. There is good agreement between the two approaches for a wide range of parameters.