Stokes force on a cylinder in the presence of fluid confinement

TL;DR

This paper revisits Stokes' calculation of the force on an oscillating cylinder in a viscous fluid, compares asymptotic and exact solutions, and extends the analysis to fluid confinement by an external cylinder.

Contribution

It provides a detailed review of Stokes' original asymptotic formulas and derives an exact solution for the force on a cylinder confined by an external boundary.

Findings

Stokes' asymptotic formulas match exact results in certain limits.

Exact force calculations depend on the ratio of cylinder radii.

Confinement significantly influences the fluid force on the cylinder.

Abstract

In this note, we present Stokes' calculation of the force exerted by the fluid on an oscillating cylinder. While the calculation of the similar problem in the case of the sphere is treated in several textbooks, the case of the cylinder is absent from these textbooks. Because modified Bessel functions were not defined in 1851 when Stokes made this calculation, Stokes was not able to express his results in closed forms but he gave asymptotic formulas valid in the two limits $a\ll \delta$ and $a \gg \delta$, where $a$ is the cylinder radius and $\delta$ is the viscous penetration depth. The closed form results were given by Stuart in 1963. We recall this calculation and we compare Stokes' asymptotic formulas to these exact results. Using modified Bessel functions, it is possible to calculate the force when the fluid is confined by an external cylinder of radius $b$ sharing the same axis: we review previous publications which have treated this problem and we present an exact calculation of this force which is also developed in powers of $\delta/a$, with the expansion coefficients being functions of the ratio $\gamma= a/b$ of the cylinder radii.