6 - Hydrodynamics of Slender Filaments

Summary

The viscous drag acting on a moving filament is the key to predicting correctly the direction of flagellar propulsive forces. In this sixth chapter we provide the mathematical basis for this result by evaluating asymptotically the hydrodynamic forces experienced by slender filaments. After revisiting the classical solution for Stokes flow due to the translation of a rigid sphere, we capture the flow along weakly bent, slender filaments using a centreline distribution of two hydrodynamic singularities. We show that in the slender limit the flow near any cross section of the filament is dominated by a local anisotropic force density, which arises from hydrodynamic singularities in the vicinity of the cross section, whose magnitudes depend logarithmically on the aspect ratio of the filament (resistive-force theory). The introduction of additional singularities far from the local cross section of the filament allows us to derive an improved nonlocal integral relationship relating the velocity of the filament to the distribution of hydrodynamic forces (slender-body theory). We close by comparing theoretical predictions with experiments on rotating helices.