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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