  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....
