Steady rotation of a tethered sphere at small, non-zero Reynolds and Taylor numbers: wake interference effects on drag

Abstract

Matched asymptotic expansion methods are used to solve the title problem. First-order Taylor number corrections to both the Stokes-law drag and Kirchhoff's-law couple on the sphere are obtained for Rossby numbers of order unity. This calculation fills a gap between the Proudman-Pearson (1957) rectilinear trajectory analysis, which includes Reynolds-number effects but does not address Taylor-number effects arising from the curvilinear trajectory, and the Herron, Davis & Bretherton (1975) curvilinear-trajectory analysis, which incorporates Taylor-number effects but ignores those arising from the Reynolds number. At the same Reynolds number, the drag on the sphere is found to be greater or less than the classical Oseen (1927)-Proudman & Pearson (1957) value, depending upon the magnitude of a certain dimensionless length parameter B measuring the tether radius to the sphere radius. This drag difference is attributed, in part, to the fact that the sphere runs into the disturbance created by its own wake.