Anomalous scaling of the dispersion of microswimmer populations

We examine the dispersion of prolate spheroidal microswimmers in pressure-driven channel flow, with the emphasis on a novel anomalous scaling regime. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are all well separated, a multiple scales analysis leads to a closed form expression for the shear-enhanced diffusivity, $D_{\it{eff}}$ , governing the long-time spread of the swimmer population along the flow (longitudinal) direction. This allows one to organize the different $D_{\it{eff}}$ -scaling regimes as a function of the rotary Péclet number ( ${\it{{\it{Pe}}}}_r)$ , where the latter parameter measures the relative importance of shear-induced rotation and relaxation of the swimmer orientation due to rotary diffusion. For large ${\it{{\it{Pe}}}}_r$ , $D_{\it{eff}}$ scales as $O({\it{{\it{Pe}}}}_r^4D_t)$ for $1 \leqslant \kappa \lesssim 2$ , and as $O({\it{{\it{Pe}}}}_r^{ {10}/{3}}D_t)$ for $\kappa = \infty$ , with $D_t$ being the intrinsic translational diffusivity of the swimmer arising from a combination of swimming and rotary diffusion, and $\kappa$ being the swimmer aspect ratio; $\kappa = 1$ for spherical swimmers. For $2 \lesssim \kappa \lt \infty$ , the swimmers collapse onto the centreline with increasing ${\it{{\it{Pe}}}}_r$ , leading to an anomalously reduced longitudinal diffusivity of $O({\it{{\it{Pe}}}}_r^{5-C(\kappa )}D_t)$ . Here, $C(\kappa )\!\gt \!1$ characterizes the algebraic decay of swimmer concentration outside an $O({\it{{\it{Pe}}}}_r^{-1})$ central core, with the anomalous exponent $(5-C)$ governed by large velocity variations occasionally sampled by swimmers outside this core. Here, $C(\kappa )\gt 5$ for $\kappa \gtrsim 10$ , leading to $D_{\it{eff}}$ eventually decreasing with increasing ${\it{{\it{Pe}}}}_r$ , in turn implying a flow-independent maximum, at a finite ${\it{{\it{Pe}}}}_r$ , for the rate of slender swimmer dispersion.