Propulsion at low Reynolds numbers by multiple traveling waves

Microorganisms or micro-robotic swimmers employ traveling waves as a common swimming mechanism involving time-irreversible deformations of their outer surface. Normally, the deforming surfaces constitute of multiple spatial waves, some standing and others propagating forward or backward. A unique technique is developed here to experimentally decompose a waving surface into its spatial wavelengths in each time instance by processing a sequence of photographs. This information is curve fitted to yield the phase velocity, frequency, and amplitudes of the propagating and receding waves of each component. The significance of the harmonic decomposition is demonstrated using an experimental macro-scale swimmer that utilizes small amplitude circumferential waves. A numerical image processing and curve-fitting procedure is shown and a theoretical model is also developed to account for the hydrodynamic effects of multiple wavelengths. The theoretical results fit well with the experimental data at low speeds, although the contribution of higher harmonics was small in experiment, but the higher harmonics are clearly visible and successfully identified. Still, the importance of the multiharmonics analysis for swimmers, which utilize traveling waves mechanisms, found both in nature and in man-made machines, was formulated and partially verified.