  The motion of microswimmers in complex flows is ruled by the interplay
between swimmer propulsion and the dynamics induced by the fluid velocity
field. Here we study the motion of a chiral microswimmer whose propulsion is
provided by the spinning of a helical tail with respect to its body in a simple
shear flow. Thanks to an efficient computational strategy that allowed us to
simulate thousands of different trajectories, we show that the tail shape
dramatically affects the swimmer's motion. In the shear dominated regime, the
swimmers carrying an elliptical helical tail show several different
Jeffery-like (tumbling) trajectories depending on their initial configuration.
As the propulsion torque increases, a progressive regularization of the motion
is observed until, in the propulsion dominated regime, the swimmers converge to
the same final trajectory independently on the initial configuration. Overall,
our results show that elliptical helix swimmer presents a much richer variety
of trajectories with respect to the usually studied circular helix tails.
