Bimodal Diffusion in Power-Law Shear Flows,
E. Ben-Naim, S. Redner, and D. ben-Avraham
The motion of dynamically-neutral Brownian particles which are
influenced by a unidirectional velocity field of the form $\vec
v(x,y)=v_0 \vert y\vert{}^{\beta }\, {\rm sgn}(y)\hat x$, with $\beta
\ge 0$, is studied. Analytic expressions for the two-dimensional
probability distribution are obtained for the special cases $\beta =0$\
and $\beta =1$. 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, which provides an integral form for the
longitudinal probability distribution for general $\beta$, and which
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.
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