Local drag of a slender rod parallel to a plane wall in a viscous fluid

TL;DR

This paper derives a comprehensive asymptotic expression for the viscous drag on a slender rod parallel to a wall across all separation distances, improving understanding of fluid-structure interactions in biological and industrial contexts.

Contribution

It introduces a unified asymptotic model for the local drag on a slender rod near a wall, valid for all separations, bridging previous regimes and enabling practical applications.

Findings

Derived a composite asymptotic representation valid for all separations.

Showed the model's accuracy and estimated errors as separation increases.

Demonstrated the model's application to a two-hinged swimmer near a wall.

Abstract

The viscous drag on a slender rod by a wall is important to many biological and industrial systems. This drag critically depends on the separation between the rod and the wall and can be approximated asymptotically in specific regimes, namely far from, or very close to, the wall, but is typically determined numerically for general separations. In this note we determine an asymptotic representation of the local drag for a slender rod parallel to a wall which is valid for all separations. This is possible through matching the behaviour of a rod close to the wall and a rod far from the wall. We show that the leading order drag in both these regimes has been known since 1981 and that they can used to produce a composite representation of the drag which is valid for all separations. This is in contrast to a sphere above a wall, where no simple uniformly valid representation exists. We estimate the error on this composite representation as the separation increases, discuss how the results could be used as resistive-force theory and demonstrate their use on a two-hinged swimmer above a wall.