Shear induced migration of microswimmers in pressure-driven channel flow

Abstract

We study the shear induced migration of microswimmers (primarily, active Brownian particles or ABP's) in a plane Poiseuille flow. For wide channels characterized by $U_b/HD_r \ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of O($D^{-1}_r$)) and those that characterize migration across the channel (of O($H^{2}D_r/U^{2}_b$)), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_b$ is the swimming speed, $H$ is the channel half-width, and $D_r$ is the swimmer rotary diffusivity. The steady state concentration profile is a function of the P\'eclet number, $Pe = U_{f}/(D_r H)$ ($U_f$ being the channel centerline velocity), and the swimmer aspect ratio $\kappa$. Swimmers with $ \kappa \gg 1$ (with $ \kappa \sim$ O(1)), in the regime $1 \ll \textit{Pe} \ll \kappa^3$ ($Pe\sim$ O(1)), migrate towards the channel walls, corresponding to a high-shear trapping behavior. For $Pe \gg \kappa^3 $ ($Pe \gg $ 1 for $\kappa \sim$ O(1)), however, swimmers migrate towards the centerline, corresponding to a low-shear trapping behavior. Interestingly, within the low-shear trapping regime, swimmers with $\kappa < 2$ asymptote to a $Pe$-independent concentration profile for large $Pe$, while those with $\kappa \geq 2$ exhibit a `centerline-collapse' for $Pe \to \infty$. The prediction of low-shear-trapping, validated by Langevin simulations, is the first explanation of recent experimental observations [Barry $\textit{et al}$. (2015)]. We organize the high-shear and low-shear trapping regimes on a $Pe-\kappa$ plane, thereby highlighting the singular behavior of infinite-aspect-ratio swimmers.