An 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

An integral model based on slender body theory, with applications to curved rigid fibers

“…While the fibers considered here are rigid, the model can also be used to simulate the dynamics of semiflexible filaments. The invertibility properties of the integral equation make it particularly well suited for handling simulations involving inextensible fibers, where an additional line tension equation must be solved at each time step [21,37]. We may also consider the effects of different choices of radius functions on the model properties, similar to what is done in [40], although we note the necessity of smooth decay in our radius function near the fiber endpoints.…”

“…Now, the velocity expression ( 9) is singular at x = X(s, t) and can be used only away from the fiber centerline; however, (9) presents a starting point for approximating the velocity of the slender body itself. Various methods can be used to obtain an expression for the relative velocity of the fiber centerline ∂X(s,t) ∂t which depends only on the arclength parameter s and time t [7,10,15,16,19,21,28,37]. Here we consider a different approach to deriving a limiting centerline expression from (9) which evidently results in a negative definite integral operator mapping f to u| ∂Σ .…”