Slender body theories for rotating filaments

Abstract

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 ǫ, 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 the cross section can range from O ǫ −2 to O(1), asymptotic theories must give accurate results for rotational dynamics and rotation-translation coupling 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. We then provide an alternative approach which uses regularized Rotne-Prager-Yamakawa (RPY) singularities to represent the fiber centerline.We show that, unlike existing SBTs, this approach gives a grand mobility with symmetric rotation-translation and translation-rotation coupling, making it fit to handle rotational velocity of arbitrary order. Our asymptotics reveal that the regularized singularity radius â can be chosen to match either the slender body expression for translation from force or rotation from torque, but not both.