Optimal transport of surface-actuated microswimmers

Abstract

We analyze the transport behavior of surface-actuated spheroidal microswimmers that locomote steadily with or without a spatiotemporally uniform external forcing. The surface actuation is in the form of either a tangential surface motion or a zero-net-mass-flux wall-normal transpiration. Starting from a general modal expansion in terms of an appropriate basis set, we link the surface actuation, the force exerted on the spheroid, and its forward speed through a Stokesian representation of the microhydrodynamics. Our analysis is generic and enables a systematic investigation over the complete range of aspect ratios from zero (streamlined needlelike spheroid) to infinity (disc-shaped spheroid). We identify a critical aspect ratio of 1.82 below and above which tangential and wall-normal surface actuations enable transport at minimal energetic cost, irrespective of whether the spheroidal microswimmer is free or forced. Crucially, we find the propulsive performance of a forced spheroidal swimmer to be appreciably higher than the one of an analogous self-propelled swimmer. Most importantly, the optimal energy expenditure minimizing tangential or wall-normal surface actuation for forced transport is passive overall so that the power requirement arises solely from the rate at which work is done by the external forcing. We highlight the complementing roles of external forcing and surface actuation over moderate and extreme aspect ratios and also exemplify the crucial disparities between optimal transport in free and forced environments. Our results indicate that a combination of external forcing and an optimal surface actuation could substantially enhance the transport of generic streamlined and bluff microswimmers.