On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer

Using slender-body hydrodynamics in the inertialess limit, we examine the motion of Purcell's swimmer, a planar, fore-aft-symmetric three-link flagellum or propulsive mechanism that translates by alternately moving its front and rear segments. Purcell (1976) concluded via symmetry arguments that the net displacement of such a swimmer must follow a straight line, but the direction and other details of the motion have never been investigated. Numerical results indicate that the direction of net translation and the speed of Purcell's swimmer depend on the angular amplitude of the swimming strokes as well as on the relative length of the links. Analytical results are presented for small rotations about the straightened configuration, and physical arguments are given to qualitatively explain the propulsive dynamics. The optimal swimmer configurations under the conditions of constant forcing and of minimum mechanical work are determined. We use a definition of efficiency based on the straightened configuration as a reference state to compare Purcell's swimmer with the previously treated swimming motions of an undulating rod and a rotating helix. Finally, we demonstrate the importance of the anisotropy in the local hydrodynamic slender-body drag to swimming motions at low Reynolds number by showing that, in general, any inextensible swimmer in an otherwise quiescent fluid cannot alter its average position under conditions of locally isotropic drag.