Analytical solutions to slender-ribbon theory

TL;DR

This paper derives analytical solutions for the hydrodynamics of slender ribbons at low Reynolds numbers, simplifying the force distribution calculation and extending resistive-force theory to ribbon-like structures.

Contribution

It provides the first analytical solution for force distribution across slender ribbons, simplifying the equations and extending resistive-force theory to ribbons.

Findings

Analytical force distribution across ribbon width derived

Reduced surface integrals to line integrals in theory

Extended resistive-force theory to ribbon geometries

Abstract

The low-Reynolds number hydrodynamics of slender ribbons is accurately captured by slender-ribbon theory, an asymptotic solution to the Stokes equation which assumes that the three length scales characterising the ribbons are well separated. We show in this paper that the force distribution across the width of an isolated ribbon located in a infinite fluid can be determined analytically, irrespective of the ribbon's shape. This, in turn, reduces the surface integrals in the slender-ribbon theory equations to a line integral analogous to the one arising in slender-body theory to determine the dynamics of filaments. This result is then used to derive analytical solutions to the motion of a rigid plate ellipsoid and a ribbon torus and to propose a ribbon resistive-force theory, thereby extending the resistive-force theory for slender filaments.