A note on the helical movement of micro-organisms

Abstract

This note seeks to evaluate the self-propulsion of a micro-organism, in a viscous fluid, by sending a helical wave down its flagellated tail. An explanation is provided to resolve the paradoxical phenomenon that a micro-organism can roll about its longitudinal axis without passing bending waves along its tail (Rothschild 1961, 1962; Bishop 1958; Gray 1962). The effort made by the organism in so doing is not torsion, but bending simultaneously in two mutually perpendicular planes. The mechanical model of the micro-organism adopted for the present study consists of a spherical head of radius a and a long cylindrical tail of cross-sectional radius b , along which a helical wave progresses distally. Under the equilibrium condition at a constant forward speed, both the net force and net torque acting on the organism are required to vanish, yielding two equations for the velocity of propulsion, U , and the induced angular velocity, Ω , of the organism. In order that this type of motion can be realized, it is necessary for the head of the organism to exceed a certain critical size, and some amount of body rotation is inevitable. In fact, there exists an optimum head-tail ratio a/b at which the propulsion velocity U reaches a maximum, holding the other physical parameters fixed. The power required for propulsion by means of helical waves is determined, based on which a hydromechanical efficiency η is defined. When the head-tail ratio a/b assumes its optimum value and when b is very small compared with the wavelength λ, η ≃ Ω/ω approximately ( Ω being the in­duced angular velocity of the head, ω the circular frequency of the helical wave). This η reaches a maximum at kh ≃ 0.9 ( k being the wavenumber 2π/ λ , and h the amplitude of the helical wave). In the neighbourhood of kh = 0.9, the optimum head-tail ratio varies in the range 15 < a/b < 40, the propulsion velocity in 0.08 < U/c < 0.2 ( c = ω/k being the wave phase velocity), and the efficiency in 0.14 < η < 0.24, as kb varies over 0.03 < kb < 0.2, a range of practical interest. Furthermore, a comparison between the advantageous features of planar and helical waves, relative to each other, is made in terms of their propulsive velocities and power consumptions.