A 3-dimensional model of flagellar swimming in a Brinkman fluid

TL;DR

This paper develops a 3D model for flagellar swimming in a Brinkman fluid, incorporating resistive effects of fibers and out-of-plane motion, validated by comparison with asymptotic solutions.

Contribution

It extends the regularized Brinkmanlet method to include Kirchhoff rods, enabling simulation of 3D flagellar motion with resistive fibers and out-of-plane dynamics.

Findings

Swimming speed increases with fiber resistance for small amplitude waves.

The model captures both planar and helical waveforms.

Validation against asymptotic solutions confirms accuracy.

Abstract

We investigate 3-dimensional flagellar swimming in a fluid with a sparse network of stationary obstacles or fibers. The Brinkman equation is used to model the average fluid flow where a flow-dependent term, including a resistance parameter that is inversely proportional to the permeability, models the resistive effects of the fibers on the fluid. To solve for the local linear and angular velocities that are coupled to the flagellar motion, we extend the method of regularized Brinkmanlets to incorporate a Kirchhoff rod, discretized as point forces and torques along a centerline. Representing a flagellum as a Kirchhoff rod, we investigate emergent waveforms for different preferred strain and twist functions. Since the Kirchhoff rod formulation allows for out-of-plane motion, in addition to studying a preferred planar sine wave configuration, we also study the case with a preferred helical configuration. Our numerical method is validated by comparing results to asymptotic swimming speeds derived for an infinite-length cylinder propagating planar or helical waves. Similar to the asymptotic analysis for both planar and helical bending, we observe that with small amplitude bending, swimming speed is always enhanced relative to the case with no fibers in the fluid (Stokes) as the resistance parameter is increased....