Sigma Xi, The Scientific Research Society
A Note on the Bathtub Vortex and the Earth's Rotation: Do the same forces govern the
direction of rotation of a hurricane and that of water draining out of a bathtub?
Author(s): Merwin Sibulkin
Source: American Scientist, Vol. 71, No. 4 (July-August 1983), pp. 352-353
Published by: Sigma Xi, The Scientific Research Society
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A Note on the Bathtub
Vortex and the Earth's
Rotation
the same forces govern thedirection
of rotation of a hurricane and that of
water draining out of a bathtub?
Do
Merwin Sibulkin
in the
The draining of a liquid through an opening
bottom of its container is normally accompanied
by the
a phe
of vigorous
rotational motion,
development
nomenon often referred to as the "bathtub vortex." It is
a phenomenon
that has intrigued scientists for years. At
the age of seventeen,
in his first scientific publication
the
answered an in
(1865),
physicist Henry Rowland
on
cause
the
of
vortex
the
bathtub
quiry
by using the
analogy "that a ball and string will, if started,wind itself
up upon the hand." In 1908, Ottokar Tumlirz attempted
a proof of the earth's rotation by calculating
the direc
tions of streamlines in fluid draining from a tub. The
In
results, however, were inconclusive.
experimental
recent years, we have made
considerable
progress
toward explaining
the bathtub vortex and similar ex
of
fluid.
amples
rotating
Certain aspects of the vortex are easy to understand.
The increase of rotational velocity as the radius of the
motion decreases
is a familiar consequence
of conser
vation of angular momentum.
(A frequently used anal
ogy is the increase in the rate of spin of an ice skater who
reduces his radius of gyration by drawing his arms and
legs close to his body, the axis of rotation.) As the rota
tional speed of the liquid in a bathtub vortex increases,
a dimple appears on the surface that may deepen and
break through to the opening in the container, as shown
in Figure 1.A related but less well known phenomenon
occurs at the inlets of jet engines. Here the flow of the
fluid into the compressor
leads to the formation of a
core may bend and reach the ground.
vortex whose
Surface debris caught up in such a vortex has led to en
interesting technique for simulating such a flow in a
laboratory beaker.)
It is when
the direction of the bathtub vortex is
discussed that controversy arises. Do the same forces that
govern the rotation of hurricanes also control the di
rection of the bathtub vortex? The answer, as with many
questions,
can
be
yes
or no.
of con
Since the bathtub vortex is a manifestation
its direction is deter
servation of angular momentum,
mined by the initial direction of motion of the fluid in
an
inertial
coordinate
system?i.e.,
a coordinate
system
fixed in space. A circular tub fixed to the earth, however,
=
sin ,as
is itself rotating with an angular velocity
earth's
rotation
is
and
the
shown in Figure 2, where
are radians
and
is the local latitude. (The units of
radians.) An
per second, rad/s; one revolution equals 2
element of fluid that is at rest relative to the tub at radius
r therefore has an absolute velocity v0 = reo.The mag
is 7.3 X 10~5 rad/s (i.e., one revolution per
nitude of
^ 45?,
means
that at mid-latitude, where
which
day),
co^ 5 X 10~5 rad/s and if the radius is 1m, say, the ab
solute velocity is 5 X 10~5 m/s. The direction of v0 is
in the Northern Hemisphere
and
counterclockwise
clockwise in the Southern. Since the velocity in this case
is exceedingly small, it is generally correct to assume that
the direction of a bathtub vortex is an accidental conse
quence of the residual motion resulting from themethod
of filling the tub.
On the other hand, if sufficient care were taken to
reduce the residual velocities to a value less than v0, it
should be possible to deduce which hemisphere one is
located in from the direction of rotation of the bathtub
vortex. Experiments of this type were performed in the
Northern Hemisphere
by Shapiro (1962) at Cambridge,
and
then
Massachusetts,
by Trefethen and his colleagues
at Sydney, Australia.
(1965) in the Southern Hemisphere
In both cases, circular, flat-bottomed tubs 6 ft. in diam
are
eter were filled with 6 in. of water (measurements
were
in
units
in
the
which
the
given
experiments
orig
inally conducted). The water inlets were arranged so that
the direction of rotation of thewater during filling was
in the Northern Hemisphere
and counter
clockwise
gine damage.
The bathtub vortex
is also similar tomore familiar
phenomena
involving the rotation of fluid. On an in
creasingly larger scale, these are encountered in the form
first of dust devils, then of tornadoes, and finally of
In all these movements
hurricanes.
of fluid the radius
of the flow decreases. However,
the fluid does not sink
in these larger rotations; rather, the direction of the in
ward flow turns vertically upward at the core of the
vortex. (Turner and Lilly (1963) have demonstrated
an
at New York University and the California Institute of
Sibulkin is at present Professor of Engineering at
Technology, Merwin
Brown University. His research interests are in the application of fluid
Educated
to such major problems as heat transfer to high-speed vehicles
and combustion of solid fuels, and to such minor problems as the bathtub
vortex and theHilsch tube. Address: Division of Engineering, Brown .
mechanics
University, Providence,
352
American
RI 02912.
Scientist,
a container
through a hole in the
Figure 1. As liquid drains out of
is
the hole, causing what
to rotate around
it begins
bottom,
as a bathtub vortex. The liquid rotates with
known
increasing
Education
Volume
into the hole
often extends
appears, which
here. (From Shapiro
1961; courtesy of
Center.)
Development
speed and a dimple
like the one shown
71
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clockwise in the Southern. Care was taken tominimize
thermal gradients to prevent convective motions in the
fluid, and the tubwas covered to prevent motions due
interval
to random air currents. After settling times?the
the order of 24 hours,
between filling and drainage?on
a drain plug % in. in diameter was removed, allowing
thewater to flow out over a period of some 20 minutes.
For the first 10 minutes no rotational motion was ap
parent. After this time, however, a small float at the
all the pre
center of the tub began to rotate, and?after
cautions described above had been taken?the direction
of rotation was consistently found to be counterclock
and clockwise in the
wise in theNorthern Hemisphere
Southern. (As Trefethen and his co-workers pointed out,
if a different result had been obtained, the experiments
would no doubt have been further refined.)
The rotational speed of the float was found to in
crease with the settling time up to a maximum of 1 rev
olution in 3 to 4 seconds. This result iswhat one would
for conservation of
expect if one applies the equation
1=
=
where H0 is
momentum,
constant,
H0
angular
the angular momentum
of the element of liquid and m
radius is re
is itsmass, to an element of liquid whose
the effects of vis
duced from 3 ft. to 3/16in. Neglecting
we
cosity does not seem tomake a difference here, but
will return to the subject of viscous effects in bathtub
the direction of rotation
experiments after considering
and
initially at rest relative to the coordinates
Figure 2. Liquid
the liquid,
are fixed with respect to the tub containing
y, which
if the tub were at rest in
not form a vortex during draining
would
on
the tub, represented
by the circle, is resting
space. However,
to the coordinates
x*
the earth and is therefore rotating relative
are fixed in space. The earth's rotation and the
and y*, which
,which,
its angular velocity,
latitude of the tub determine
the absolute
together with the radius of the tub, r, determine
velocity, v0.
of hurricanes.
are intense rotating systems of airflow
Hurricanes
vorticity was the
layer. I believe this boundary-layer
I
cause
direction
observed
reversal
of
of the
in the earth's atmosphere whose direction is always cy
(Sibulkin
the
in the Northern Hemi
counterclockwise
clonic?i.e.,
1962). Using a tub 1 ft. in diameter and allowing
a float rotating
I
found
that
are
to
10
for
water
settle
in
the
Southern.
clockwise
and
minutes,
intense,
Why
sphere
in the same direction inwhich the tubwas filled slowed
the earth's
large-scale storms always cyclonic? Above
to the
and then reversed its direction shortly before the
to
down
be
tends
airflow
the
parallel
layer
boundary
traces of dye
was
tub
This
isobars, the lines of constant barometric pressure.
completely drained. Subsurface
confirmed that the fluid had also reversed its direction
movement of air, which is called the geostrophic wind,
of rotation, eliminating the fear that the float had been
the pressure gradient
results from a balance between
tension. An
,
and the Coriolis pseudoforce ?2
force (1/p) V
unduly influenced by the effects of surface
to
and
his
and
are
the density, pressure,
,pr and
where
reproduce
attempt by Kelley
colleagues (1964)
velocity
this reversal phenomenon was only partially successful,
of the air, respectively. In order for these forces to be in
an alternate explanation based on
and they proposed
the direction of the flow must be cyclonic
balance,
surface waves
around a low-pressure center and anticyclonic around
resulting from laboratory vibrations. A
related study of steady viscous, swirling flow has pre
center (e.g., Dobbins
a high-pressure
1979). Frictional
cause
air
the possibility of a reversal of flow (Ackerberg
dicted
the
in
the atmospheric boundary layer
forces
flow to be directed inward toward an area of low pres
1973), but a definitive mathematical
analysis of the vis
cous bathtub vortex remains to be done.
sure and outward from a region of high pressure. As the
a low-pressure center,
air approaches
inward-flowing
References
as
to
rises
it
and
it turns upward,
higher altitudes it tends
at a free streamline.
R. C. 1973. Boundary-layer
separation
Ackerberg,
the rising air ismoist, the en
to cool. However, when
of separation.
flow and flow downstream
Part 3. Axisymmetric
/.
its temperature.
Fluid Mech.
59:645-63,
ergy released by condensation wilfraise
The result is a convective updraft with an associated
and Air Pollution. Wiley.
R. A. 1979. Atmospheric Motion
Dobbins,
increase in the rate of flow into the system. The moving
1964. A further note on
E.
S.
W.
and
B.
D.
Martin,
L.,
Taylor.
Kelley,
air now resembles an inverted bathtub vortex, with the
19:539-42.
the bathtub vortex. /. Fluid Mech.
Sei. Am. 13:308.
rotational velocity increasing as the low-pressure center
H. A. 1865. The vortex problem.
Rowland,
is approached. Under suitable atmospheric conditions,
16 mm film. Distributed
1961.
A.
H.
by En
Vorticity.
(prod.).
Shapiro,
Britannica Educational
this system of air
which are as yet poorly understood,
Corp. ? Education Development
cyclopedia
Center, Newton, MA.
flow intensifies into a hurricane.
_
196:1080-81.
1962. Bath-tub vortex. Nature
we have learned a great deal about the
Although
14:
on the bathtub vortex. /. Fluid Mech.
note
A
1962.
M.
Sibulkin,
bathtub vortex and other examples of rotating fluid,
21-24.
new questions continue to arise. For instance, does the
Trefethen, L. M., R. W. Bilger, R T. Fink, R. E. Luxton, and R. I. Tanner.
water flows out
draining fluid reverse direction? As the
Nature
vortex
in the southern
1965. The bath-tub
hemisphere.
of a tub, viscosity will cause a boundary layer to form on
207:1084-85.
the bottom of the tub in which the fluid has a vertical
Beweis
f?r die Achsen
Tumlirz, O, 1908. Ein neuer physikalischer
a radial component of vorticity.
velocity gradient and
drehung der Erde. Akad. Wiss. Wien, Abt. IIa, 117:819-41.
Near the end of the drainage process, all the fluid being
tornado
Turner,
J. S., and D. K. Lilly. 1963. The carbonated-water
vortex. /. Atmos. Sci. 20:468-71.
come from the region of the boundary
discharged will
1983 July-August
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353