The action of waving cylindrical rings in a viscous fluid

Abstract

Dinoflagellates (Pfisteria piscicida) are unicellular micro-organisms that swim due to the action of two eucaryotic flagella: a trailing, longitudinal flagellum that propagates planar waves and a transverse flagellum that propagates helical waves. Motivated by the wish to understand the role of the transverse flagellum in dinoflagellate motility, we study the fundamental fluid dynamics of a waving cylindrical tube wrapped into a closed helix. Given an imposed travelling wave on the structure, we determine that the helical ring propels itself in the direction normal to the plane of the circular axis of the helix. The magnitude of this translational velocity is proportional to the square of the helix amplitude. Additionally, the helical ring exhibits rotational motion tangential to its axis. These calculated swimming velocities are consistent when using the method of regularized Stokeslets with prescribed wave kinematics, regularized Stokeslets with dynamic forcing and Lighthill's slender-body theory, except in cases where the slenderness parameter is not small. The translational velocity results are nearly indistinguishable using the three approaches, leading to the conjecture that the main contribution to this velocity at a cross-section is the far-field flow generated by the portion on the opposite side of the ring. The largest contribution to the rotational velocity at a cross-section comes from the cross-section itself and others nearby, thus the geometric details of the slender body have a larger effect on the results.