An oscillating-boundary-layer theory for ciliary propulsion

Abstract

This paper analyses the locomotion of a finite body propelling itself through a viscous fluid by means of travelling harmonic motions of its surface. The methods are developed with application to the propulsion of ciliated micro-organisms in mind. Provided that the metachronal wavelength (of the surface motions) is much smaller than the overall dimensions of the body, the flow can be divided into an oscillating-boundary-layer flow to which is matched an external complementary Stokes flow. The present paper employs the envelope model of fluid/cilia interaction to construct equations of motion for the oscillating boundary layer. The final solution for the propulsive velocity is obtained by application of the condition of zero total force on the self-propelling body; alternatively, if the organism is held at rest, the thrust it generates can be computed. Various optimum propulsive velocities for self-propelling bodies and optimum thrusts for restrained bodies are analysed in some simple examples. The results are compared with the relatively sparse observations for a number of micro-organisms.