TY - JOUR

T1 - Integral-based spectral method for inextensible slender fibers in Stokes flow

AU - Maxian, Ondrej

AU - Mogilner, Alex

AU - Donev, Aleksandar

N1 - Funding Information:
We thank those in the numerical analysis community who provided advice on some aspects of this work. F. B. Usabiaga provided results for the strong formulation that we use in Sec. . A.-K. Tornberg privately shared a preprint of Ref. with us, and L. af Klinteberg and A. Barnett supplied us with code on the special quadrature schemes for nearly singular integration. N. Trefethen and N. Hale pointed us to rectangular spectral collocation for the calculation of bending forces. Y. Mori and L. Ohm provided help with the theory of SBT. Thanks also to M. Shelley, L. Greengard, and W. Yan for helpful discussions on the numerical method and stress calculations. O. Maxian is supported by the National Science Foundation via Grant No. GRFP/DGE-1342536 and A. Mogilner is supported by U. S. Army Research Office Grant No. W911NF-17-1-0417. This work was also supported by the NSF through Research Training Group in Modeling and Simulation under Award No. RTG/DMS-1646339.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/1/14

Y1 - 2021/1/14

N2 - 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 104. 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 L2 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.

AB - 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 104. 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 L2 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.

UR - http://www.scopus.com/inward/record.url?scp=85100199197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85100199197&partnerID=8YFLogxK

U2 - 10.1103/PhysRevFluids.6.014102

DO - 10.1103/PhysRevFluids.6.014102

M3 - Article

AN - SCOPUS:85100199197

SN - 2469-990X

VL - 6

JO - Physical Review Fluids

JF - Physical Review Fluids

IS - 1

M1 - 014102

ER -