Optimal swimming of flagellated micro-organisms

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Abstract
This paper studies flagellar undulations that propel a micro-organism at a given speed while minimizing its expenditure of hydrodynamical energy. The study is in two basic parts. The first part is a qualitative inquiry into the general nature of undulations that are hydrodynamically optimal in the instantaneous sense. The results indicate that an apparent sliding of the entire flagellum along its instantaneous axis is fundamental to such motions, although an additional deformation is necessary to compensate for the presence of the organism's head. Periodic or semi-periodic undulations are clearly suggested, and must consist of travelling waves propagated in the direction opposite to propulsion.The second part of the paper is a quantitative inquiry as to the values of parameters that optimize given periodic wave shapes in the time-average sense. The trade-off between wave amplitude and the number of wavelengths is of particular interest. Results are obtained for small amplitude sinusoidal waves and finite amplitude sawtooth waves. For the latter, a single wavelength with amplitude roughly one-sixth of the wavelength is optimal. The significance of the twitching movements of the head is investigated. The results are consistent with the qualitative study and emphasize the need to inhibit such motions. The implications of the dependence of resistive-force coefficients upon wave shape are considered, and the physical significance of rotational pitching motions is assessed.

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