The Rayleigh Problem for the Interior of a Torus

Abstract

This paper contains a description of the low Reynolds number flow inside a torus which has been set impulsively in motion about its central axis. The moving wall drags with it a primary flow confined to a sheath that grows steadily toward the center of the torus cross section. The primary flow in turn produces centrifugal and Coriolis accelerations which lead to secondary flows in the cross-sectional plane. At very long time the secondary flows subside, and the primary flow approaches a condition of solid-body rotation. The present analysis treats this problem in the thin-torus limit, where the cross-sectional radius is small compared to the toroidal radius, and is restricted to wall velocities small enough to support a low Reynolds-number assumption. At this level of approximation, the flow is characterized by a single dimensionless parameter, analogous to the Dean number.