Catenaries in viscous fluid

TL;DR

This paper develops analytical models for flexible, slender structures in viscous fluids, extending classical catenaries to include flow effects, and explores their shapes, tensions, and boundary conditions.

Contribution

It introduces a five-parameter family of catenary-like solutions for structures in viscous flow, including new tension behaviors and boundary value problem solutions.

Findings

Derived analytical solutions for translating equilibria in viscous flow.

Identified parameter regimes for different cable configurations.

Discovered counterintuitive tension behaviors in infinite-length cables.

Abstract

This work explores a simple model of a slender, flexible structure in a uniform flow, providing analytical solutions for the translating, axially flowing equilibria of strings subjected to a uniform body force and drag forces linear in the velocities. The classical catenaries are extended to a five-parameter family of curves. A sixth parameter affects the tension in the curves. Generic configurations are planar, represented by a single first order equation for the tangential angle. The effects of varying parameters on representative shapes, orbits in angle-curvature space, and stress distributions are shown. As limiting cases, the solutions include configurations corresponding to "lariat chains" and the towing, reeling, and sedimentation of flexible cables in a highly viscous fluid. Regions of parameter space corresponding to infinitely long, semi-infinite, and finite length curves are delineated. Almost all curves subtend an angle less than $\pi$ radians, but curious special cases with doubled or infinite range occur on the borders between regions. Separate transitions in the tension behavior, and counterintuitive results regarding finite towing tensions for infinitely long cables, are presented. Several physically inspired boundary value problems are solved and discussed.