  Biological and artificial microswimmers often self-propel in external flows
of vortical nature; relevant examples include algae in small-scale ocean
eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic
devices. A recent experiment has shown that swimming bacteria in model vortices
are expelled from the vortex all the way to a well-defined depletion zone
(Sokolov and Aranson (2016) "Rapid expulsion of microswimmers by a vortical
flow." Nature Comm. 7, 11114). In this paper, we propose a theoretical model to
investigate the dynamics of elongated microswimmers in elementary vortices,
namely active particles in two- and three-dimensional rotlets. A deterministic
model first reveals the existence of bounded orbits near the centre of the
vortex and unbounded orbits elsewhere. We further discover a conserved quantity
of motion that allows us to map the phase space according to the type of the
orbit (bounded vs unbounded). We next introduce translational and rotational
noise into the system. Using a Fokker--Planck formalism, we quantify the
quality of trapping near the centre of the vortex by examining the probability
of escape and the mean time of escape from the region of deterministically
bounded orbits. We finally show how to use these findings to formulate a
prediction for the radius of the depletion zone, which compares favourably with
the experiments of Sokolov and Aranson (2016).
