BIMODAL DIFFUSION IN POWER-LAW SHEAR FLOWS

The motion of dynamically neutral Brownian particles that are influenced by a unidirectional velocity field of the form v(x,y) = upsilon-0\y\beta(sgn)(y)x, with beta greater-than-or-equal-to 0, is studied. Analytic expressions for the two-dimensional probability distribution are obtained for the special cases beta = 0 and 1. As a function of beta, the longitudinal probability distribution of displacements exhibits bimodality for beta < beta(c) and unimodality otherwise. A simple effective-velocity approximation is introduced that provides an integral form for the longitudinal probability distribution for general-beta and predicts the existence of this transition. A numerical exact enumeration of the probability distribution yields beta(c) = 3/4. The power-law model parallels the behavior found for tracer motion in a class of non-Newtonian fluids, where a unimodal-to-bimodal transition is also found to occur.