Slender fibers are ubiquitous in biology, physics, and engineering, with
prominent examples including bacterial flagella and cytoskeletal fibers. In
this setting, slender body theories (SBTs), which give the resistance on the
fiber asymptotically in its slenderness $\epsilon$, are useful tools for both
analysis and computations. However, a difficulty arises when accounting for
twist and cross-sectional rotation: because the angular velocity of a filament
can vary depending on the order of magnitude of the applied torque, asymptotic
theories must give accurate results for rotational dynamics over a range of
angular velocities. In this paper, we first survey the challenges in applying
existing SBTs, which are based on either singularity or full boundary integral
representations, to rotating filaments, showing in particular that they fail to
consistently treat rotation-translation coupling in curved filaments. We then
provide an alternative approach which approximates the three-dimensional
dynamics via a one-dimensional line integral of Rotne-Prager-Yamakawa
regularized singularities. While unable to accurately resolve the flow field
near the filament, this approach gives a grand mobility with symmetric
rotation-translation and translation-rotation coupling, making it applicable to
a broad range of angular velocities. To restore fidelity to the
three-dimensional filament geometry, we use our regularized singularity model
to inform a simple empirical equation which relates the mean force and torque
along the filament centerline to the translational and rotational velocity of
the cross section. The single unknown coefficient in the model is estimated
numerically from three-dimensional boundary integral calculations on a
rotating, curved filament.
