The many-body reciprocal theorem and swimmer hydrodynamics

TL;DR

This paper extends the reciprocal theorem for swimmer hydrodynamics to complex interactions and geometries, providing new analytical tools and exact results for two-dimensional flows, enhancing understanding of swimmer behavior near surfaces.

Contribution

It generalizes the reciprocal theorem for swimmers, enabling analysis of interactions and flows in complex environments, including surfaces and two-dimensional cases.

Findings

Recovered standard results for a squirmer near a surface

Extended the theorem to general squirming motions and ciliated surfaces

Provided exact solutions for two-dimensional hydrodynamics

Abstract

We present a reinterpretation and extension of the reciprocal theorem for swimmers, extending its application from the motion of a single swimmer in an unbounded domain to the general setting, giving results for both swimmer interactions and general hydrodynamics. We illustrate the method for a squirmer near a planar surface, recovering standard literature results and extending them to a general squirming set, to motion in the presence of a ciliated surface, and expressions for the flow field throughout the domain. Finally, we present exact results for the hydrodynamics in two dimensions which shed light on the near-field behaviour.