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, Deff , governing the long-time spread of the swimmer population along the flow (longitudinal) direction. This allows one to organize the different Deff -scaling regimes as a function of the rotary Péclet number ( Peᵣ) , where the latter parameter measures the relative importance of shear-induced rotation and relaxation of the swimmer orientation due to rotary diffusion. For large Peᵣ , Deff scales as O(Peᵣ⁴Dₜ) for 1 κ 2 , and as O(Peᵣ^ 10/3Dₜ) for κ = ∞ , with Dₜ being the intrinsic translational diffusivity of the swimmer arising from a combination of swimming and rotary diffusion, and κ being the swimmer aspect ratio; κ = 1 for spherical swimmers. For 2 κ < ∞ , the swimmers collapse onto the centreline with increasing Peᵣ , leading to an anomalously reduced longitudinal diffusivity of O(Peᵣ5-C(κ )Dₜ) . Here, C(κ )\!> \!1 characterizes the algebraic decay of swimmer concentration outside an O(Peᵣ⁻¹) central core, with the anomalous exponent $(5-C)$ governed by large velocity variations occasionally sampled by swimmers outside this core. Here, C(κ )> 5 for κ 10 , leading to Deff eventually decreasing with increasing Peᵣ , in turn implying a flow-independent maximum, at a finite Peᵣ , for the rate of slender swimmer dispersion.