The propulsive effect generated by fast travelling-wave vibrations

Abstract

A prototype problem for the propulsion arising from a spatially-dependent source is analysed asymptotically. Such driving is a phenomenon that, while fairly common in biological settings, is less well understood in mechanical and engineering contexts. If two parallel plates bound a fluid and the plates are free to translate relative to each other, recent calculations have shown that vibrations applied to one plate in the form of travelling waves may induce motion of the other. The degree of the propulsion obtained is a function of both the wavelength and the phase speed of the imposed vibrations and there seems to an optimal wavelength for which propulsion is greatest. In order to shed light on these observations, here we undertake a formal asymptotic analysis of the process in the circumstance when the vibrations move quickly. It is shown how the flow structure takes on a form consisting of a bulk core layer supplemented by suitable edge layers attached to the two plates. The efficiency of the mechanism relies on the generation of a mean flow component whose properties are considered. It is found that there is excellent agreement between our analytical predictions and the previous simulations; in particular we are able to find the critical wavelength for maximum propulsion and explain the modifications that arise in the short and long wavelength limits.