The bathtub vortex in a rotating container

Abstract

We study the time-independent free-surface flow which forms when a fluid drains out of a container, a so-called bathtub vortex. We focus on the bathtub vortex in a rotating container and describe the free-surface shape and the complex flow structure using photographs of the free surface, flow visualizations, and velocity measurements. We find that the velocity field in the bulk of the fluid agrees with predictions from linear Ekman theory for the boundary layer at the bottom, and we discuss the limitations of linear Ekman theory for the source–sink flow in the experiment. We introduce a radial expansion approximation of the central vortex core and reduce the model to a single first-order equation. We solve the equation numerically and find that the axial velocity depends linearly on height whereas the azimuthal velocity is almost independent of height. We discuss the model of the bathtub vortex introduced by Lundgren (J. Fluid Mech. vol. 155, 1985, p. 381) and compare it with our experiment. We find that the measured velocities and surface profiles are described well by the model when Ekman upflow and surface tension effects are included.