The propulsion by large amplitude waves of uniflagellar micro-organisms of finite length

Abstract

The fluid mechanics of self-propelling, slender uniflagellar micro-organisms is examined theoretically. The mathematical analysis of these motions is based upon the Stokes equations, and the body is represented by a continuous distribution of stokeslets and doublets of undetermined strength. Since the body is self-propelling, additional constraints on the total force and moment upon it are applied. A system of singular integral and auxiliary equations, in which the propulsive velocity and viscous force per unit length are the unknowns, is derived. The vector integral equation is decomposed into near- and far-field contributions, and the solution is determined by a straightforward iterative procedure. The flagella considered are of constant radius and are restricted to planar undulations. The analysis is applied to a small amplitude wave form of infinite length, and a third-order analytic solution is obtained. By means of numerical computation, the method is extended to large amplitude wave forms of both infinite and finite length. The validity and accuracy of the solution method, the effect of local curvature, and an approximate model for an attached cell body-proper are evaluated in light of alternative theories. The solution method is systematically applied to a variety of wave-form shapes representative of actual flagella. For a sinusoidal wave form, the variations in propulsive velocity, power output and propulsive efficiency are examined as functions of the number of wavelengths on the flagellum, the amplitude and the flagellar radius. Wave forms of variable amplitude and variable wavelength are also considered. Among the significant results are the effect of the cell body on pitching, the significant differences between constant frequency and constant phase-speed undulations for variable wavelength wave forms, and comparisons with other pertinent theories.