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

Abstract

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 new method for rapidly computing the dynamics of inextensible slender filaments in periodically-sheared Stokes flow. The dynamics of the filaments are governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To enforce inextensibility, we parameterize 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 quasi-static, primarily elastic behavior to dynamic, primarily viscous behavior. We find that nonlocal hydrodynamics increases the viscous modulus by as much as 25%, even for semi-dilute fiber suspensions.