A note on the helical movement of micro-organisms

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.