Anomalous dispersion of microswimmer populations

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 $P{e}_r$, $P{e}_r$ being the rotary Peclet number, this diffusivity scales as $O(P{e}_r^4{D}_t)$ for $1\le \kappa \lesssim 2$, and as $O(P{e}_r^{\frac{10}{3}}{D}_t)$ for $\kappa =\mathrm{\infty }$, $D_t$ being the (bare),swimmer translational diffusivity and $\kappa$ the swimmer aspect ratio. For $2\lesssim \kappa <\mathrm{\infty }$, swimmers collapse onto the centerline with increasing $P{e}_r$, leading to an anomalously reduced diffusivity of $O(P{e}_r^{5+C(\kappa )}{D}_t)$. Here, $C(\kappa )\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}-1$ characterizes the algebraic decay of swimmer concentration outside an $O(P{e}_r^{-1})$ 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.