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)=${\mathit{v}}_0$‖y${\mathrm{\Vert }}^{\mathrm{\beta }}$sgn(y)x^, with β≥0, is studied. Analytic expressions for the two-dimensional probability distribution are obtained for the special cases β=0 and 1. As a function of β, the longitudinal probability distribution of displacements exhibits bimodality for β${\mathrm{\beta }}_{\mathit{c}}$ and unimodality otherwise. A simple effective-velocity approximation is introduced that provides an integral form for the longitudinal probability distribution for general β and predicts the existence of this transition. A numerical exact enumeration of the probability distribution yields ${\mathrm{\beta }}_{\mathit{c}}$=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.