  Confined suspensions of active particles show peculiar dynamics characterized
by wall accumulation, as well as upstream swimming, centerline 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 Peclet 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 which 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 ......
