The hydrodynamics of a twisting, bending, inextensible fiber in Stokes flow

Abstract

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 novel numerical method for the simulation of cylindrical fibers by extending our previous work on inextensible bending fibers [Maxian et al., Phys. Rev. Fluids 6 (1), 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 novel slender-body quadrature, which requires on the order of 10 points along the 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 1

Slender body theories for rotating filaments

“…In a companion paper [25], we develop efficient quadrature schemes to evaluate the integrals (23) to numerical, rather than asymptotic, precision. The RPY kernels change their form when the distance between the spheres is less than 2â, i.e., when the spheres overlap, thus regularizing the singularity that occurs in the interactions between point forces.…”

“…If the rod is inextensible and unshearable, the perpendicular components of rotational velocity are not independent and must correspond to the translational velocity U . Thus we only deal with parallel rotational velocity here[25].…”

Mapping flagellated swimmers to surface-slip driven swimmers

“…Therefore, at least a 100 steps are required for a swimmer to move one body length (∼ 10 µm) [14]. When the flexibility of the appendages is modeled additional care and time steps restrictions are necessary [15,16,17,18].…”

Well-posedness and applications of classical elastohydrodynamics for a swimming filament

“…A weaker notion of inextensibility-perhaps related to these reformulations-may overcome this difficulty. Moreover, a broader aim of this paper is to serve as a starting point for a full numerical analysis of the various formulations of fibre inextensibility used in numerical simulations, ranging from penalization methods [55] to variants of the more recent approaches which avoid solving for the fibre tension altogether [17,32,33,35,58,59]. Our own numerical method is validated against a direct implementation of the classical formulation of filament elastohydrodynamics.…”