Confined suspensions of active particles show peculiar dynamics characterized by wall accumulation, as well as upstream swimming, centreline depletion and shear trapping when a pressure-driven flow is imposed. We use theory and numerical simulations to investigate the effects of confinement and non-uniform shear on the dynamics of a dilute suspension of Brownian active swimmers by incorporating a detailed treatment of boundary conditions within a simple kinetic model where the configuration of the suspension is described using a conservation equation for the probability distribution function of particle positions and orientations, and where particle–particle and particle–wall hydrodynamic interactions are neglected. Based on this model, we first investigate the effects of confinement in the absence of flow, in which case the dynamics is governed by a swimming Péclet number, or ratio of the persistence length of particle trajectories over the channel width, and a second swimmer-specific parameter whose inverse measures the strength of propulsion. In the limit of weak and strong propulsion, asymptotic expressions for the full distribution function are derived. For finite propulsion, analytical expressions for the concentration and polarization profiles are also obtained using a truncated moment expansion of the distribution function. In agreement with experimental observations, the existence of a concentration/polarization boundary layer in wide channels is reported and characterized, suggesting that wall accumulation in active suspensions is primarily a kinematic effect that does not require hydrodynamic interactions. Next, we show that application of a pressure-driven Poiseuille flow leads to net upstream swimming of the particles relative to the flow, and an analytical expression for the mean upstream velocity is derived in the weak-flow limit. In stronger imposed flows, we also predict the formation of a depletion layer near the channel centreline, due to cross-streamline migration of the swimming particles towards high-shear regions where they become trapped, and an asymptotic analysis in the strong-flow limit is used to obtain a scale for the depletion layer thickness and to rationalize the non-monotonic dependence of the intensity of depletion upon flow rate. Our theoretical predictions are all shown to be in excellent agreement with finite-volume numerical simulations of the kinetic model, and are also supported by recent experiments on bacterial suspensions in microfluidic devices.
Electrokinetic Transport of a Charged Dye in a Freely Suspended Liquid Film: Experiments and Numerical Simulations
