Anomalous dispersion of microswimmer populations

Abstract

We examine the longitudinal dispersion of spheroidal microswimmers in pressure-driven channel flow. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are well separated, a multiple scales analysis leads to the shear-enhanced diffusivity governing the long-time spread of the swimmer population along the flow\,(longitudinal) direction. For large $Pe_r$, $Pe_r$ being the rotary Peclet number, this diffusivity scales as $O(Pe_r^4D_t)$ for $1 \leq \kappa \lesssim 2$, and as $O(Pe_r^{\frac{10}{3}}D_t)$ for $\kappa = \infty$, $D_t$ being the (bare)\,swimmer translational diffusivity and $\kappa$ the swimmer aspect ratio. For $2 \lesssim \kappa < \infty$, swimmers collapse onto the centerline with increasing $Pe_r$, leading to an anomalously reduced diffusivity of $O(Pe_r^{5+C(\kappa)}D_t)$. Here, $C(\kappa)\!<\!-1$ characterizes the algebraic decay of swimmer concentration outside an $O(Pe_r^{-1})$ central core, with the anomalous exponent governed by large velocity variations sampled by the few swimmers outside this core. $C(\kappa)$ dips below $-5$ for $\kappa \gtrsim 10$, leading to a flow-independent bound of $O(\kappa^{10}D_t)$ for the dispersion of sufficiently slender swimmers.