Slender body theories for rotating filaments

Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness, 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 centreline 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.