A note on the reciprocal theorem for the swimming of simple bodies

TL;DR

This paper extends the reciprocal theorem to two-dimensional deforming bodies at low Reynolds number, providing concise formulas for their swimming velocities and linking these to Faxén's laws.

Contribution

It offers a straightforward method to apply the reciprocal theorem to 2D swimmers, overcoming previous difficulties with resistance problems.

Findings

Derived concise formulas for 2D swimmer velocities

Connected reciprocal theorem results with Faxén's laws

Addressed issues with resistance problems in lower-dimensional shapes

Abstract

The use of the reciprocal theorem has been shown to be a powerful tool to obtain the swimming velocity of bodies at low Reynolds number. The use of this method for lower-dimensional swimmers, such as cylinders and sheets, is more problematic because of the undefined or ill-posed resistance problems that arise in the rigid-body translation of these shapes. Here we show that this issue can be simply circumvented and give concise formulas obtained via the reciprocal theorem for the self-propelled motion of deforming two-dimensional bodies. We also discuss the connection between these formulae and Fax\'en's laws.