Large amplitude motion of self-propelling slender filaments at low Reynolds numbers.

In the swimming of micro-organisms such as spermatozoa the Reynolds number involved in the flow is 0(10V3) which implies that propulsion is due predominantly to viscous forces. In such low Reynolds number problems. the equations governing the fluid are the linearized Navier-Stokes equations called the creeping motion or the Stokes equations. Taylor (1951) initiated the quantitative study of the motility of microscopic organisms as a low Reynolds number flow problem. He modelled the swimming spermatozoon as a two-dimensional. infinite. inextensible sheet of zero thickness with a sinusoidal wave travelling down the sheet. Although this geometry is unrealistic he did establish that propulsion solely by viscous forces was possible and that the fluid motion could be adequately described by the Stokes equations and did not require the more general Oseen equations. Hancock (1953) considered the more realistic geometric model of a self-propelling thin circular filament undergoing large amplitude motions. By placing a distribution of singularities such as doublets and stokelets on the centreline of the filament and formulating integral equations for the strengths of their distribution. he calculated propulsive velocities for several wave amplitudes and filament radii. Gray and Hancock (1955) later used Hancock’s analysis to develop a simplified theory of flagellar swimming. They assumed that the viscous forces acting on the filament could be described in terms of normal and tangential drag coefficients C, and C, Hancock’s earlier paper showed that for a vanishingly thin filament C, and C, are constants. independent of the form of the motion of the filament and that C,‘C, = 2. Although their expressions for propulsive velocity of a filament undergoing sinusoidal swimming motions are valid for motions of arbitrary amplitude. dependence on the filament thickness is not included. Subsequent papers. Brokaw (1965. 1966a.b. 1968. 1971). Carlson (1957). Gray (1958). Holwill and Burger (1963). Holwill (1966a), Holwill and Miles (1971). Machin (1958), and Rikmenspoel (1965). have considered more realistic planar swimming motions, while Chwang and Wu (1971). Coakley