PHYSICAL REVIEW FLUIDS 6, 014102 (2021) Editors’ Suggestion Integral-based spectral method for inextensible slender fibers in Stokes flow Ondrej Maxian , 1 , * Alex Mogilner, 1, 2 , † and Aleksandar Donev 1 , ‡ 1 Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA 2 Department of Biology, New York University, New York, New York 10012, USA (Received 22 July 2020; accepted 7 December 2020; published 14 January 2021) Every animal cell is filled with a cytoskeleton, a dynamic gel made of inextensible fibers, such as microtubules, actin fibers, and intermediate filaments, all suspended in a viscous fluid. Numerical simulation of this gel is challenging because the fiber aspect ratios can be as large as 10 4 . We describe a method for rapidly computing the dynamics of inextensible slender filaments in periodically sheared Stokes flow. The dynamics of the filaments is governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To enforce inextensibility, we parametrize the space of inextensible fiber motions and strictly confine the dynamics to the manifold of inextensible configurations. To do this, we introduce a set of Lagrange multipliers for the tensile force densities on the filaments and impose the constraint of no virtual work in an L 2 weak sense. We augment this approach with a spectral discretization of the local and nonlocal slender body theory operators which is linear in the number of unknowns and gives improved spatial accuracy over approaches based on solving a line-tension equation. For dynamics, we develop a second-order semi-implicit temporal integrator which requires at most a few evaluations of nonlocal hydrodynamics and a few block-diagonal linear solves per time step. After demonstrating the improved accuracy and robustness of our approach through numerical examples, we apply our formulation to a permanently cross-linked actin mesh in a background oscillatory shear flow. We observe a characteristic frequency at which the network transitions from quasistatic, primarily elastic behavior to dynamic, primarily viscous behavior. We find that nonlocal hydrodynamics increases the viscous modulus by as much as 25%. Most of this increase, in contrast to the smaller (about 10%) increase in the elastic modulus, is due to short-ranged intrafiber interactions. DOI: 10.1103/PhysRevFluids.6.014102 I. INTRODUCTION Interactions of long, thin, inextensible filaments with a viscous fluid abound in biology, engi- neering, physics, and medicine. In biology, the swimming mechanisms of flagellated organisms have been of interest for decades, with an initial cluster of studies on how force and torque balances lead to swimming [1–4], and a more recent focus on flagellar bundling and propulsion [5–7]. In physics and engineering, suspensions of high-aspect-ratio fibers have been observed to display non-Newtonian, viscoelastic behavior both experimentally [8] and computationally [9,10]. Our particular area of interest is the simulation of semiflexible filaments that make up the cell cy- toskeleton. These inextensible filaments, which include microtubules and actin filaments, maintain * [email protected] † [email protected] ‡ [email protected] 2469-990X/2021/6(1)/014102(55) 014102-1 ©2021 American Physical Society
Integral-based spectral method for inextensible slender fibers in Stokes flow
May 17, 2023
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