Oscillating Cylinders and the Stokes' Paradox

Measurements are reported of the inertia coefficient $k$ and damping coefficient $\mathrm{k\prime }$ for a circular cylinder (radius $a$ ) oscillating at angular frequency $\omega$ in a fluid of kinematic viscosity $\nu$ . Good agreement with the theory of Stokes is obtained throughout the range ${\text{0.286<(2ν/ωa}}^2{)}^{\text{1/2}}\text{<4.13}$ even when the Reynolds number $R$ and the relative amplitude are not small compared with 1. Stokes' theory is expressed in terms of modified Bessel functions, and values of $k$ and $\mathrm{k\prime }$ are tabulated for ${\text{0.001<(a}}^2{\text{ω/4ν)}}^{\text{1/2}}\text{<0.5}$ . The relation to Stokes' paradox is considered by examining the damping force in the limit ${\mathrm{(\omega a}}^2{\mathrm{/\nu )}}^{\text{1/2}}\text{≪1}$ . The damping force for a cylinder, unlike that for a sphere, does not reduce to the low $R$ drag in this limit. However, comparison with Lamb's solution of the Oseen equations suggests an expression that agrees well with drag measurements up to $\text{R\hspace{0.17em}=\hspace{0.17em}2.5}$ .