Bath-Tub Vortex (논문)

아래 두 논문은 Massachusetts에서 연구한, 욕조에서 빠지는 물의 소용돌이 방향에 대한 논문이다. 내가 영어실력이 너무 약해서 정확한 의도를 파악할 수가 없는 것이 안타깝다. 대충 살펴본 바에 따르면 논문의 중요 요지가 있나 싶을 정도로밖에 파악되지 않기 때문에 하고자 하는 내용도 다 생략하기로 한다.

Nature 1962.12.15 vol.196 (p.1080-1081)
Bath-Tub Vortex

PERENNIALLY one hears controversy regarding the direction of the vortex in the drain of the bath-tub or the kitchen sink. Some claim that the direction of swirl is always the same in the northern hemisphere and that in the southern hemisphere it is always opposite to that for the northern hemisphere. Other claim that there is no unique direction of swirl in either hemisphere.
Both schools of thought are in some sense correct. For the everyday observations of the kitchen sink and bath-tub variety. the direction of the vortex seems to vary in an unpredictable manner with the date, the time of day, and the particular household of experimenter. But under well-controlled condition of experimentation, the observer looking downwarl at a drain in the northern hemisphere will always see a counter-clockwiae vortex, while one in the southern hemisphere will always see a clockwise vortex.
In a properly designed experiment, the vortex is produced by Coriolis forces. which are counter-clockwise in the northern hemisphere.
There are two reasons for the apparent haphazard results of the everday observations. The effective rim speed of a tank 2 ft. in diameter rotating with the earth at Boston, Massachusetts, is only about 0.04 in./min. This tiny Earth-induced motion may be much less than other motions too small to be perceived by ordinary observation. Unless the water in the vessel is allowed to settle for may hours, for example, the residual motions from the filling of the vessel may far exceed the foregoing figure. Moreover, the very act of pulling the plug from the bottom of the vessel may inadvertently create substantial velocities. Ever if many hours are allowed for settling, are currents in the room or thermal currents associated with nonuniform temperatures around the vessel may easily cause water velocities larger than that due to the Earth’s rotation. All these difficulties are quite real;this was amply proved by early experiments with the apparatus desribed later.
The Second source of confusion in the casual everyday observations is that the Coriolis forces due to the Earth’s rotation may in fact be much smaller than other forces normally prosent. For a speed of 0.1 in./sec toward the drain, the Coriolis force at latitude 42˚ north is about 3X10-7 times the force of gravity. If the vessel is not perfectly symmetrical about the axis of the drain, the asymmetrical viscous forces at the side walls and at the bottom of the vessel may produce torques larger than that due to the Coriolis force; the direction of the asymmetrical torque would, of course, vary from vessel to vessel. Even the extremely small forces of surface tension at the free surface may, owing to non-uniformities in temperature or chemical impurity, lead to torques.
A Meaningful experimental demonstration of ‘the bath-tub vortex’ must be designed to cope with the diffeculties mentioned here. Such an apparatus was made and operated near Boston for an educational motion picture entitled Verticity. The tank is circular, 6 ft. in diameter and 6 in. high, with a flat bottom. The drain hole, 3/8 in. in diameter, is centrally located and flush with the bottom. A length of hose some 20 ㄽ. long leads the water from the drain to a sump. A ‘vorticity float’ makes the vortex visible. This is a cross made of two slivers of wood, each about 1 in. long, pinned together with a fine wire through the centre of each. The cross floats in a horizontal plane just below the surface of the water, and a small length of wire extending above the surface of the water provides the necessary floating stability.
Before starting the experiment the end of the long hose is closed with a plug. The tank is then filled nearly to the brim with water swirling clockwise. This dirction is deliberately chosen to test whether sufficient time has elapsed for adequate decay of the original motion, so that one may be confident that the swirl seen in the experiment really arises from Coriolis forces rather than from initial swirl not yet sufficiently dissipated. After filling, the tank is covered with a plastic sheet to eliminate the effect of air currents, and the room is maintained at nearly uniform temperature. About 24 h are required for settling of the motion to a degree where the residual motions are unimportant compared with those dur to the Earth’s rotation. The plug is then pulled from the end of the long hose and the vorticity float is delicately inserted above the drain hole. About 20 min are required for the tank to drain completely. During the first 12-15 min no rotation of the float is perceptible to the eye. At about 15 min, however, the counter-clockwise rotation of the vorticity float becomes distinctly perceptible, and gradually increases. Near the end of the experiment, when the vessel is almost completely drained, the vorticity cross turns one revolution in about three or four seconds. This is about 30,000 times greater than the effective speed of rotation of the Earth at latitude 42˚ N. It agrees in order of magnitude with what one would calculate theoretically on the assumption that the rotation of the float is indeed due to the Coriolis forces of the Earth.
When all the precautions described were taken, the vortex was invariably in the counter-clockwise direction.
If the plug were pulled after only about 1-2 h, the vorticity cross would begin to spin clockwise within 1-2 min after the plug was pulled, and would reach a rate much greater than the rate of counter-clockwise spin for the correct experiment. This was because the angular velocity remaining from the clockwise filling action, although not readily perceptible, was nevertheless greater than that of the Earth.
In one experiment the original clockwise swirl was allowed to settle for four dor five hours before pulling the plug. An unusual result was observed. During the first two-thirds of the draining period, the vorticity float turned in the clockwise direction owing to the angular momentum of the initial swirl. After having reached a large clockwise angular velocity, the float began to slow down, ultimately came to rest, and then proceeded to accelerate in the counter-clockwise direction. This may be explained as follows. The water acquires the rotational velocity of the Earth through the diffusion of a viscous shear layer, mainly from the bottom of the tank. This viscous boundary layer on the bottom, in which the water has acquired the counter-clockwise angular velocity of the Earth, grows thicker with time. Above this boundary layer the water still retains the clockwise angular velocity associated with the initial swirl. when the vorticity float sinks to the appropriate level, it tends to reverse direction.
I am prepared to provide precise particulars of my apparatus to anyone in the southern hemisphere who wishes to perform an identical experiment for the sake of convincing any remaining doubters.
Incidentally, those who claim to have seen the direction of swirl change as a ship crosses the equator are surely pressing the case too far. At the equator the Coriolis forces vanish, and it would be virtually impossible to perform a valid experiment a shot distance from the equastor

Nature 1965.09.04 vol.207 (p.1084-1085)
The Bath-Tub Vortex in the Southern Hemisphere

It has long been thought that water draining from a tank would rotate counter-clockwise in the northern hemisphere and clockwise in the southern hemisphere, provided other influences were kept small compared with the influence of the rotation of the Earth. This idea has only recently been tested, by Shapiro in Watertown, Massachusetts, as part of a film on vorticity[footnote]Shapiro, Ascher, H., film, Vorticity (Educational Services Inc., Watertown, Mass, 1961)[/footnote], and later by Binnie in Cambridge, England[footnote]Binnie, A.M., J. Mech, Eng. Sci., 6.256(1964).[/footnote]. Shapiro and Binnie both acquired confidence, after surmounting difficulties in their early experiments, that the counter-clockwise rotations observed in their later experiments were due to the rotation of the Earth.
The experiment has now been performed in Sydney, Australia. For this we should like to express appreciation for assisance from the University of Sydney, Tufts University, and the U.S. National Science Foundation.
The apparatus was modelled on Shapiro’s. It was a tank, 6 ft. in diameter and 9 in. high, with a central drain pipe 0.375 in. in diameter conneted to a draining hose. The tank differed from Shapiro’s in ways suggested by early difficulties Shapiro and Binnie experienced. It was made of ply-wood instead of metal, to reduce thermal convection. The drain pipe projected up 1 in. from the bottom, and was tapered to a sharp-edged opening; and the tank was located in a small, cement-walled basement room which both rom temperature and inlet-water temperature remained within a degree of 20℃ during the tests. As in Shapiro’s experiments, the tank was filled by hose to a depth of about 6 in. above the orifice. The hose was directed so as leave water swirling counter-clockwise in the tank.
Initially, the apparatus did not work as expected. In the first test, after a 60 h settling period, dust particles on the water surface showed no discernible rotation at any time during an 80 min draining period. The apparntly meant that the water was moving to the centre so slowly that the viscous damping effect of the bottom was cancelling the water’s angular momentum to the point where no angular momentum was perceptible in the draining fluid. To speed draining, the tank was raised to 30 in. above the floor. Form then on, with the increased drop of the draining hose, drainage took about 22 min, and some form of rotation was always observed above the orifice.
Clockwise rotation was observed in all five of the later tests that had settling times of 18 h or more. During the first 10 ro 12 min of drainage, no rotation was apparent. Rotation then developed as drainage progressed. In three of these tests a float 0.625 in. in diameter, a slice from a wine cork, was used to indicate rotation. It reached speeds of about one revolution in 8 sec for the runs with settling times of 18 and 20 h, and one revolution in 3 sec in the one run with a settling time of 70 h. One revolution in 3 sec is what one would expect of a ring of particles rotating with the surface of the Earth at the latitude of Sydney, and then brought in from a diameter of 6 ft. to a diameter of 0.375 in., provided the ring conserves its angular momentum.
To provide a comparison with Shapiro’s work, a floating cross made of two  matchstick segments 1 in. long was alternated in one test with the cork float. Both cross and float rotated at approximately the same speed. Shapiro used a comparable corss. He reported[footnote]Shapiro, Ascher H., Nture, 196, 1080 (1962)[/footnote] and filmed[footnote]Shapiro, Ascher H., four-min film loop No. FM-15. The Bathlub Vortex (Educational Services Inc., Watertown, Mass., 1963).[/footnote] rotation speeds of one revolution in about 3-4 sec. His drainago time was also comparable, that is, 20-24 min, and he also reports rotation only after about 10-12 min of drainage.
One early test did not fit the pattern of settling time influence that emerged later from the tests. the tank, after a settling time of only 4 h 40 min, performed as if it had had 20 h of settling time. Near the end of the drainage period, a 0.5 in. patch of dust particles that had accumulated at the centre was rotating one revolution in 8 sec, clockwise. This was the only test in which the tank had not been mostly or wholly covered during the settling period. It was an unusually windy day outside, and just before drainage dust specks on different areas of the water surface were moving in several directions at large speeds of about 1 em/min. The test may therefore have been significantly influenced by air currents. For subsequent tests, the louvre in the ceiling, whith had been blocked off, though not completel, by ply-wood sheets, was additionally covered outside with ‘Pliofilm’. The door was kept closed, except briefly to allow entry, and the tank was kept mostly covered by two ply-wood sheets resting on two angle irons placed diametrically across the tank. The angle irons were usually spaced about 3 in. apart, so that surface motions could be observed between them, and the direction of these supporting beams in relation to the room was varied from test to test in an attempt to detect any remaining influence of air currents. Air currents did not appear to be a significant influence in the later tests, but it is our opinion that they are likely to have been the largest of the disturbing influences.
In one early test, after a settling period of only 13 h, the no-rotation period was followed by a period of counter-clockwise rotation, which changed to clockwise near the end of the draining period. (In this particular test it is not known how the tank had been filled.) Shapiro reports one similar test, experienced after a settling time of 4-5 h, attributed to undamped initial angular velocity residing in the upper water, while the water nearer the bottom was rotating in the direction of the Earth’s rotation. In another of our tests, with a deliberately short settling time of 3 h, water drained out counter-clockwise during all but an initial 2 min of no rotation. Shapiro reports a similar result. In fact, it would seem that the results of these experiments at Sydney are quite similar to those obtained by Shapiro in the northern hemisphere, with one exception. After suitable settling periods, Shapiro observed counter-clockwise rotation. We, in Australia, observed the opposite.
These tests posed for us an unusual problem in experimental work. Normally, one does experiments in which there is some uncertainty in the expected outcom. In these experiments, however, our confidence in the idea that the Earth rotates, and in the applicability of conservation of angular momentum to masses of fluid, was probably so strong that experimental denial would have been almost inadmissible. We should have gone to unusual lengths to get the apparatus to work as expected. Realizing this, we found ourselves reluctant to accept as conclusive the results we were getting, results which apparently confirmed our ideas. One can never prove, for example, that it was not some small air current which persistently maintained a circulation that gave the results we observed, and that a quantitatively comparable, but oppositely directed, air current caused Shapiro’s results. There is, in principle, an infinite number of hyposheses that can explain any set of observations. This difficulty in validation of scientific theories is not aj new one and, in this instance, as in all instances, it cannot be proved that any one hypothesis is correct. Nevertheless, we have acquired confidence in the hypothesis that carefully performed experiments on liquid drainage from a tank will show clockwise rotation, if done in the southern hemisphere.