  Using fully-resolved simulations, we investigate the torque experienced by a
finite-length circular cylinder rotating steadily perpendicularly to its
symmetry axis. The aspect ratio $\chi$, i.e. the ratio of the length of the
cylinder to its diameter, is varied from 1 to 15. In the creeping-flow regime,
we employ the slender-body theory to derive the expression of the torque up to
order 4 with respect to the small parameter $1/\ln(2\chi)$. Numerical results
agree well with the corresponding predictions for $\chi\gtrsim3$. We introduce
an \textit{ad hoc} modification in the theoretical prediction to fit the
numerical results obtained with shorter cylinders, and a second modification to
account for the increase of the torque resulting from finite inertial effects.
In strongly inertial regimes, a prominent wake pattern made of two pairs of
counter-rotating vortices takes place. Nevertheless the flow remains stationary
and exhibits two distinct symmetries, one of which implies that the
contributions to the torque arising from the two cylinder ends are identical.
We build separate empirical formulas for the contributions of pressure and
viscous stress to the torque provided by the lateral surface and the cylinder
ends. We show that, in each contribution, the dominant scaling law may be
inferred from simple physical arguments. This approach eventually results in an
empirical formula for the rotation-induced torque valid throughout the range of
inertial regimes and aspect ratios considered in the simulations.
