# The Self-Propulsion of Microscopic Organisms through Liquids

G. J. Hancock

## Abstract

Since the Reynolds number of motion of microscopic organisms through liquids, defined as L$\rho$V/$\mu$, where L is the length of the organism, V the velocity with which it moves, $\rho$ the density of the liquid and $\mu$ the viscosity, is small, propulsion is due predominantly to the viscous forces, the effect of the inertial forces being negligible. The best-known problem that neglects all inertial forces is Stokes's solution for the slow steady fluid motion past a sphere, in which the velocity field can be described in terms of singularities situated at the centre of the sphere. The movement of microscopic organisms is determined by placing distributions of these singularities inside the surface of the organism and satisfying all boundary conditions. The motions that are considered are restricted to organisms which propagate some kind of disturbance along filaments of circular cross-section with small radius. The first problem to be considered is that of an infinite thin filament along which are propagated plane waves of lateral displacement. Formulae for the velocity of propulsion are obtained for (i) the limiting case of zero radius and (ii) the case when the amplitude of the displacement is small compared to the wave-length. Computations have been carried out to estimate the propulsion in the case of small non-zero filament radius when the amplitude is larger than that allowed for in case (ii) above. It is also shown that the propulsion of a finite filament which forms itself into a single wave is very near to that of an infinite filament with the same wave motion. The second problem is that of an infinite filament along which any general three-dimensional disturbance is propagated. The movement is then deduced for the propagation of a spiral wave along an infinite filament, and also for the propagation of longitudinal waves along a finite filament.