Anatomy of a BathtubVortex
J. Juul Rasmussen,
and B. Lautrup
The Technical University of Denmark, Department of Physics, DK-2800 Kgs. Lyngby, Denmark
Risø National Laboratory, Optics and Fluid Dynamics Department, DK-4000 Roskilde, Denmark
The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
(Received 11 March 2003; published 5 September 2003)
We present experiments and theory for the ‘‘bathtub vortex,’’ which forms when a Fuid drains out of
a rotating cylindrical container through a small drain hole. The fast down-Fow is found to be con±ned
to a narrow and rapidly rotating ‘‘drainpipe’’ from the free surface down to the drain hole. Surrounding
this drainpipe is a region with slow upward Fow generated by the Ekman layer at the bottom of the
container. This Fow structure leads us to a theoretical model similar to one obtained earlier by
Lundgren [J. ²luid Mech.
, 381 (1985)], but here including surface tension and Ekman upwelling,
comparing favorably with our measurements. At the tip of the needlelike surface depression, we
observe a bubble-forming instability at high rotation rates.
PACS numbers: 47.32.–y, 47.45.Gx
The generation of strongly localized vorticity is a fas-
cinating and complicating ingredient of a broad variety of
Fuid Fows ranging from vortex shedding at solid surfaces
(such as paddles, sand ripples, or insect wings) over Fows
through turbines to large-scale tornadoes . In general,
these Fows are poorly understood, since the interplay
between fast axial motion and intense, localized vorticity
leads to dif±cult mathematical problems outside the com-
fortable realm of classical subjects such as potential Fow
or standard boundary layer theory.
One of the most well-known examples of such Fows is
the so-called ‘‘bathtub vortex,’’ which forms when water
drains out of a container. The strong, localized deforma-
tion of the free surface makes the vortex beautifully
visible, and has made the bathtub vortex the prototype
‘‘vortex.’’ This popularity is in stark contrast to the
attention which the phenomenon receives in the literature.
The few classic papers about it either neglect the axial
Fow  or consider the problem without a free surface .
Similarly, textbooks very seldom mention the bathtub
vortex, and if they do [4,5] the Fow is modeled within
potential theory (with the inclusion of an
core ) without incorporating the axial motion. In the
vortex core the axial velocity can be high, an essential
ingredient of the strong ‘‘swirl’’  which makes the Fow
so fascinating. Our aim in this Letter is to provide basic
understanding of the stationary bathtub vortex: the Fow
structure, the shape of the free surface, and the interde-
pendence of important characteristics such as the size of
the central surface depression, the rate of the out-Fow,
and the rotation rate in the vortex core.