Optimal propulsive flapping in Stokes flows

TL;DR

This paper analytically determines the optimal flapping motion for a rigid spheroid in Stokes flow, revealing that optimal kinematics resemble insect wing movements and could inform propulsion across various Reynolds numbers.

Contribution

It provides an exact hydrodynamic model and computes the optimal flapping kinematics analytically and numerically for low Reynolds number flows.

Findings

Optimal flapping kinematics resemble insect wing figure-eight motion.

Flapping efficiency depends weakly on flapper shape.

Results suggest flapping as a versatile propulsion mechanism.

Abstract

Swimming fish and flying insects use the flapping of fins and wings to generate thrust. In contrast, microscopic organisms typically deform their appendages in a wavelike fashion. Since a flapping motion with two degrees of freedom is able, in theory, to produce net forces from a time-periodic actuation at all Reynolds number, we compute in this paper the optimal flapping kinematics of a rigid spheroid in a Stokes flow. The hydrodynamics for the force generation and energetics of the flapping motion is solved exactly. We then compute analytically the gradient of a flapping efficiency in the space of all flapping gaits and employ it to derive numerically the optimal flapping kinematics as a function of the shape of the flapper and the amplitude of the motion. The kinematics of optimal flapping are observed to depend weakly on the flapper shape and are very similar to the figure-eight motion observed in the motion of insect wings. Our results suggest that flapping could be a exploited experimentally as a propulsion mechanism valid across the whole range of Reynolds numbers.