  We present a quantitative analysis on the response of a dilute active
suspension of self-propelled rods (swimmers) in a planar channel subjected to
an imposed shear flow. To best capture the salient features of shear-induced
effects, we consider the case of an imposed Couette flow, providing a constant
shear rate across the channel. We argue that the steady-state behavior of
swimmers can be understood in the light of a population splitting phenomenon,
occurring as the shear rate exceeds a certain threshold, initiating the
reversal of swimming direction for a finite fraction of swimmers from down- to
upstream or vice versa, depending on swimmer position within the channel.
Swimmers thus split into two distinct, statistically significant and oppositely
swimming majority and minority populations. The onset of population splitting
translates into a transition from a self-propulsion-dominated regime to a
shear-dominated regime, corresponding to a unimodal-to-bimodal change in the
probability distribution function of the swimmer orientation. We present a
phase diagram in terms of the swim and flow Peclet numbers showing the
separation of these two regimes by a discontinuous transition line. Our results
shed further light on the behavior of swimmers in a shear flow and provide an
explanation for the previously reported non-monotonic behavior of the mean,
near-wall, parallel-to-flow orientation of swimmers with increasing shear
strength.
