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

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 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. For regimes not accounted for with asymptotic analysis, i.e., large amplitude spiral and helical bending, our model results show a non-monotonic change in swimming speed with respect to the resistance parameter; a maximum swimming speed is observed when the resistance parameter is near one. The non-monotonic behavior is due to the emergent waveforms; as the resistance parameter increases, the swimmer becomes incapable of achieving the amplitude of its preferred configuration. We also show how simulation results of slower swimming speeds for larger resistance parameters are actually consistent with the asymptotic swimming speeds if work in the system is fixed.