A numerical method for inextensible elastic filaments in viscous fluids

Abstract

The deformations of flagella are important in the motility of single-and multi-flagellated bacteria. Existing numerical methods have treated flagella as extensible filaments with a large extensional modulus, resulting in a stiff numerical problem and long simulation times. However, flagella are nearly inextensible, so to avoid large extensional stiffness, we introduce inextensible elastic rod models with hydrodynamics treated by a surface distribution of regularized Stokeslets.We benchmark this new model against previously described models of extensible elastic rods with hydrodynamics treated by a centerline distribution of regularized Stokeslets and rotlets, as well as a surface distribution of regularized Stokeslets. We compare the accuracy of the inextensible model with the extensible models and illustrate for which ratios of stretching/bending stiffness (which depend on the diameter of filament) the inextensible model is accurate and more efficient. We show that our inextensible approach can be markedly more efficient than the extensible models for many biological filaments. We also compare the accuracy of the centerline distribution of the Stokeslets and rotlets to the more accurate surface distribution of Stokeslets for modeling fluid-structure interactions of filaments.

An integral-based spectral method for inextensible slender fibers in Stokes flow

“…An extension of this work is to account for twist by using the Kirchhoff rod model instead of the Euler beam. The Kirchhoff rod model has previously been used in combination with the immersed boundary method [45], RPY tensor [71], and method of regularized Stokeslets [62,32] to model a bent twisted fiber interacting with a fluid, with the methods of [32,71] even preserving discrete inextensibility. As we've discussed at length, these regularization-based models become impractical as the fiber becomes slender since 1/ regularized points are required to properly resolve the fiber thickness.…”

“…Since the principle of virtual work (32) must hold for any inextensible motion, it must hold for all U and all sufficiently smooth g 1 and g 2 . Therefore, we must have, for all s,…”

“…All of these regularization methods have been used to model immersed rods, but generally with penalty terms to enforce inextensibility [62,45]. To our knowledge, only the recent approaches of Schoeller et al [71] and Jabbarzadeh and Fu [32] enforce inextensibility rigorously with Lagrange multipliers. In the case of [32], the fiber is broken up into segments of regularized point forces, and each segment is updated via a rotation that preserves inextensibility exactly.…”

“…Yet both [71] and [32] suffer from the same pitfall as all regularization methods: when modeling slender fibers, the width/radius of the regularization function must be on the order of the fiber radius [13]. Since the regularization width is also tied to the fiber discretization spacing (and, in the IB method, the fluid grid spacing), semi-flexible slender fibers must be discretized with many more points than would be necessary in a continuum, SBT-based approach.…”

“…In our approach, the fibers are evolved via a rotation of the tangent vector on the unit sphere, and the fiber positions are then obtained by integration. This approach is similar to that of [32], but unique because we consider the fiber as a continuum, rather than a collection of discrete line segments. In this way, we maintain strict inextensibility of the fibers without introducing a penalty parameter.…”