Hydrodynamics of a twisting, bending, inextensible fiber in Stokes flow

In swimming microorganisms and the cell cytoskeleton, inextensible fibers resist bending and twisting, and interact with the surrounding fluid to cause or resist large-scale fluid motion. In this paper, we develop a numerical method for the simulation of cylindrical fibers by extending our previous work on inextensible bending fibers [O. Maxian, Phys. Rev. Fluids 6, 014102 (2021)2469-990X10.1103/PhysRevFluids.6.014102] to fibers with twist elasticity. In our "Euler"model, twist is a scalar function that measures the deviation of the fiber cross section relative to a twist-free frame, the fiber exerts only torque parallel to the centerline on the fluid, and the perpendicular components of the rotational fluid velocity are discarded in favor of the translational velocity. In the first part of this paper, we justify this model by comparing it to another commonly used "Kirchhoff"formulation where the fiber exerts both perpendicular and parallel torque on the fluid, and the perpendicular angular fluid velocity is required to be consistent with the translational fluid velocity. Through asymptotics and numerics, we show that the perpendicular torques in the Kirchhoff model are small, thereby establishing the asymptotic equivalence of the two models for slender fibers. We then develop a spectral numerical method for the hydrodynamics of the Euler model. We define hydrodynamic mobility operators using integrals of the Rotne-Prager-Yamakawa tensor, and evaluate these integrals through a slender-body quadrature, which requires on the order of 10 points along a smooth fiber to obtain several digits of accuracy. We demonstrate that this choice of mobility removes the unphysical negative eigenvalues in the translation-translation mobility associated with asymptotic slender-body theories, and ensures strong convergence of the fiber velocity and weak convergence of the fiber constraint forces. We pair the spatial discretization with a semi-implicit temporal integrator to confirm the negligible contribution of twist elasticity to the relaxation dynamics of a bent fiber. We also study the instability of a twirling fiber, and demonstrate that rotation-translation and translation-rotation coupling change the critical twirling frequency at which a whirling instability onsets by ∼20%, which explains some (but not all) of the deviation from theory observed experimentally. When twirling is unstable, we show that the dynamics transition to a steady overwhirling state, and quantify the amplitude and frequency of the resulting steady crank-shafting motion.