Minimum Dissipation Theorem for Microswimmers

TL;DR

This paper establishes a fundamental theorem providing a lower bound on energy dissipation for microswimmers at low Reynolds number, linking it to passive body resistance tensors, and introduces a new efficiency measure.

Contribution

It derives a minimum dissipation theorem for microswimmers of arbitrary shape and proposes a novel efficiency metric that is bounded by unity.

Findings

Minimum dissipation can be expressed via resistance tensors of passive bodies.

Optimal surface velocity matches flow around a perfect-slip body.

New efficiency measure cannot exceed unity.

Abstract

We derive a theorem for the lower bound on the energy dissipation rate by a rigid surface-driven active microswimmer of arbitrary shape in a fluid at low Reynolds number. We show that, for any swimmer, the minimum dissipation at a given velocity can be expressed in terms of the resistance tensors of two passive bodies of the same shape with a no-slip and perfect-slip boundary. To achieve the absolute minimum dissipation, the optimal swimmer needs a surface velocity profile that corresponds to the flow around the perfect-slip body, and a propulsive force density that corresponds to the no-slip body. Using this theorem, we propose an alternative definition of the energetic efficiency of microswimmers that, unlike the commonly-used Lighthill efficiency, can never exceed unity. We validate the theory by calculating the efficiency limits of spheroidal swimmers.