  We theoretically study the conformational and dynamical properties of
semiflexible active polar ring polymers under linear shear flow. A ring is
described as a continuous Gaussian polymer with a tangential active force of a
constant density along its contour. The linear but non-Hermitian equation of
motion is solved using an eigenfunction expansion, which yields
activity-independent, but shear-rate-dependent, relaxation times and
activity-dependent frequencies. As a consequence, the ring's stationary-state
properties are independent of activity, and its conformations as well as
rheological properties are equal to those of a passive ring under shear. The
presence of characteristic time scales by the relaxation and the frequency
gives rise to a particular dynamical behavior. A tank-treading-like motion
emerges for large relaxation times and high frequencies, specifically for
stiffer rings, governed by the activity-dependent frequencies. In the case of
very flexible polymers, the relaxation behavior dominates over tank-treading.
Shear strongly affects the crossover from a tank-treading to a relaxation-time
dominated dynamics and suppresses tank-treading. This is reflected in the
tumbling frequency, which exhibits two shear-rate dependent regimes, with an
activity-dependent plateau at low shear rates followed by a power-law regime
with increasing tumbling frequency for large shear rates.
