# Why there's a whirl when you drain the bathtub?

At first I thought it's because of Coriolis, but then someone told me that at the bathtub scale that's not the predominant force in this phenomenon.

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Related (perhaps duplicate even): physics.stackexchange.com/q/32 –  Marek Mar 29 '11 at 9:23
@Marek: partial duplicate. I was expecting a "definitive" answer as to what DOES cause the whirl. We can agree Coriolis is out... –  Dan Mar 29 '11 at 9:44

The whirl is due to the net angular momentum the water has before it starts draining, which is pretty much random.

If the circulation were due to Coriolis forces, the water would always drain in the same direction, but I did the experiment with my sink just now and observed the water to spin different directions on different trials.

The Coriolis force is proportional to the velocity of the water and the angular velocity of Earth. Earth's angular velocity is $2\pi/24\ {\rm hours}$, or about $10^{-4}\ s^{-1}$. If water's velocity as it drains is $v$ the Coriolis acceleration is about $10^{-4} v\ s^{-1}$.

The water moves about a meter while draining, which takes a time $1\ m/v$, so the total velocity imparted by Coriolis forces could be at most $10^{-4} v\ s^{-1} * 1\ m/v = 10^{-4} \ m/s$.

So the Coriolis effect is quite a small effect. But this first-order Coriolis effect does not cause the water to rotate.

The direction of Coriolis force depends on your direction of motion. All the water in your tub is moving the same direction, so the Coriolis force pushes it all the same direction. The effect is that if the bathtub starts out perfectly flat and begins draining (and it points north), all the water will get pushed east. The two edges of the tub will have very slightly different depths of water, because the Coriolis force is pushing sideways.

The Coriolis force could create "spinning" on uniformly-moving water, but only as a second-order effect. As you move away from the equator, the Coriolis force changes. This change in the Coriolis force is because the angle between "north" and the angular velocity vector of Earth changes as you move around; as you go further north (in the Northern Hemisphere) the "north" direction gets closer and closer to making a right angle with the angular velocity vector, so the Coriolis force increases in strength. The size of this effect would be proportional to the ratio of the size of your tub to the radius of Earth. That ratio is $10^{-7}$, so this effect is completely negligible.

The Coriolis force could also create some "spinning" if different parts of the water are moving different speeds. If the tub is draining to the north in the northern hemisphere, and water near the drain is moving faster than water far away, then the water near the drain would be pushed east more than water far away is. If you subtracted out the average effect of the Coriolis force, what remained would be an easterly push near the drain and a westerly push far away. This gives a clockwise spin as viewed from above.

We've already estimated the typical velocities as $\omega L$, so the angular momentum per unit mass induced this way would be on the order of $\omega L^2$ (but maybe smaller by a factor of 10). That's only $10^{-4}\ m^2/s$. To get an equivalent effect, in a tub of $100\ L$, you could give just one liter of water on the edge of the pool a velocity of a few cm/s, something you surely do many time over when removing your body from the tub.

This effect is too small to affect your bathtub, but it's still observable under the right conditions. According to Wikipedia, Otto Tumlirz conducted several experiments in the early 20th century that demonstrated the effects of the Coriolis forces on a draining tub of water. The tub was allowed to settle for 24 hours in a controlled environment before the experiment began. This was enough to damp out the residual angular momentum left over from filling the tub up to the point where Coriolis effects were dominant.

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Thanks! now I have to find a way to translate all this to my 4 year old daughter (who originally asked this question). –  Dan Mar 29 '11 at 14:44
If I keep the water still for a very long time, so that it has very little angular momentum to begin with, would the water swirl? –  Bernhard Heijstek Apr 26 '11 at 17:11
@phycker According to the reports of experiments, yes, eventually you would see a swirl. –  Mark Eichenlaub Apr 26 '11 at 18:47
Coriolis effect is not limited to that caused by the earth's rotation. It is just another consequence of conservation of angular momentum, regardless of the scale. –  Mike Dunlavey May 28 '12 at 3:25

Since you want to explain it to your daughter, take a plastic bottle, cut the bottom open, turn it upside town, hold the top closed and fill it with water. Give her that bottle and have her release the top (which is on the bottom now, sorry for the bad phrasing). The water will whirl in different orientations whenever you repeat this (if it whirls at all) and she can influence it by accelerating the bottle in a circular motion to understand that an initial disturbance is responsible for the whirl orientation.

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great idea! nothing better than hands-on science for a young inquisitive mind –  Dan Mar 31 '11 at 14:51
I am not sure if it is an honest answer. What is going to happen during the release is that the potential energy of the water will be moved to two componentes: the aceleration of the mass center, and the internal energy. The later will appear as a whirl around the mass center. –  arivero Mar 31 '11 at 20:54
@arivero: one could of course use a spoon to introduce a small turbulence instead, I was just focusing on using as little resources as possible... –  Tobias Kienzler Apr 1 '11 at 8:27
Tobias, my point is that even if you dont whirl by hand, it will appear. If it doesn't, it should mean that all the energy is going to energy of the mass center, which is unlikely. Of course I agree thatyou can influence it. –  arivero Apr 2 '11 at 19:15

The whirl happens in the draining tube, whose optimal solution to drain the bathtub is a laminar flow allowing for some rotation in the tube. What you see in the surface is the match between the solution of flow in the tube and the solution of flow in the surface.

Angular momentum of the flow gets modified a lot as the tube twists and twists, sometimes even siphoning up and down.

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You can think about it like this: It takes one day for the earth to perform a full rotation (about 86k seconds), on the other hand, it takes a few seconds for your sink to drain (lets say 10 seconds). So it takes 8600 times longer for the earth to do a full rotation than it takes the water to drain down the sink. It is not too hard to imagine that the earth's rotation can have no influence on the process of draining a sink.

However, if the sink was the size of lake Michigan and you were to drain it, Coriolis would play a role.

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The main effect is angular momentum (rotational inertia) in the water set up by various movements before you start observing, such as getting out of your bath.

This results in the water level being lower near the centre of rotation than further away, setting up centripital forces which maintain the rotation. When the difference in levels is significant relative to the average water level, you notice the typical whirl effect.

There are other things happening too, including the Coriolis force.

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A discussion by 'The Straight Dope' website

references experimental work carried out but Ascher Schapiro in 1962, which concluded something like it all depends on the shape of container and how its stirred before being left to empty.

Here is Schapiro's paper but I feel you will need academic access via a university or library to read the full PDF:

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=368912

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The "Straight Dope" article is a pretty muddy explanation, claiming "Coriolis forces in the Northern Hemisphere act in a counterclockwise direction." That's actually backwards. In the northern hemisphere, Coriolis forces are always making you turn to the right - clockwise. Hurricanes do turn counterclockwise, but that's because air is flowing in from all sides towards their low-pressure center. en.wikipedia.org/wiki/… –  Mark Eichenlaub Mar 29 '11 at 10:57
""In the northern hemisphere, Coriolis forces are always making you turn to the right - clockwise."" Yes, to the right, but not clockwise. The wind moving to the center of low pressure moves to the right, thus missing the center and that starts the counterclockwise movement of the anticyclone. Fig 13 in the page You liked is about that. –  Georg Mar 31 '11 at 21:32

Consider this picture of water running from a faucet:

The stream narrows as it falls. The shape of the stream has a definite mathematical shape, which follows from (1) conservation of energy, (2) conservation of mass, and (3) the approximation that the horizontal velocity of the water makes a negligible contribution to its kinetic energy. By conservation of energy, the water lower down is moving faster. If the stream didn't become narrower, then more water would be coming out at the bottom than went in at the top.

A draining bathtub is somewhat different because the water flows out at a rate that is approximately determined by the diameter of the drain, and there is also an open surface at the top, which can develop a conical indentation when there's rotation.

The open surface complicates things, so let's start by considering the case where there is no open surface. Then I think the same considerations apply, at least qualitatively, as in the case of the stream of water from the faucet. The bathtub doesn't have a shape that constricts strongly at the bottom. It has a nearly constant horizontal cross-sectional area. Presumably there is some very special shape for a bathtub that would allow the water to flow freely downward while satisfying constraints 1-3. For any other shape, the water has to violate #3 by rotating rapidly. This would suggest that the vortex forms spontaneously in all cases, and that it has a fixed speed of rotation regardless of the initial conditions. Feynman claims (Feynman 1964, section 40-4) that the flow is irrotational (in the sense that $\nabla\times\mathbf{v}=0$), so the speed of the rotation is proportional to $1/r$ (the same as the magnetic field around a wire). A possible loophole in all this is that according to the experiments and calculations in Andersen 2003, there is a layer of water near the drain hole that goes up and then comes back down. This allows us to have more kinetic energy for a fixed rate of flow.

In the case where there's an open surface, this gets even more complicated. Here we get an indentation, whose shape is shown by Feynman to be $z\propto 1/r^2$. The depth of this indentation is a variable that classifies different solutions. These different solutions have different rates of rotation, as predicted by Andersen. I don't know whether there is a solution for zero rotation. As discussed in the answers to another question, sufficiently careful experiments are able to see the tiny Coriolis effect, which suggests that there is a zero-rotation solution, but it's unstable.

In this experiment, they used two holes and found that only one whirlpool formed. This would seem consistent with my analysis in terms of conservation laws, since conservation laws are additive; only the total has to be conserved.

Andersen 2003 gives a complete mathematical analysis and comparison with experiment in the case where a cylindrical container with a hole at the bottom is intentionally rotated.

Andersen, Phys Rev Lett 91 (2003) 104502-1, http://sites.apam.columbia.edu/courses/apph4200x/prl-bathtub-vortex-2003.pdf

Feynman, The Feynman Lectures

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It's because of the Coriolis effect. In the Northern hemisphere, it goes around in one direction, in the Southern hemisphere around the opposite direction, and goes straight down bang on the equator.

Black natives demostrate the effect to white tourists in the video clip Water flow at the equator, Coriolis effect.

Edit:

The above is nonsense, but I'll leave it on to prevent others doing the same. Videos like the above fake the effect by pouring in the water close to the edge to give the required initial momentum.

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It's already been explained quite clearly that it is not due to the Coriolis effect. Further, if it were, you still wouldn't be able to see it at the equator because the Coriolis effect is zero there. –  Mark Eichenlaub Mar 29 '11 at 15:42
Pff. The video is a show for tourists, not a scientific experiment. The vessels are not equally shaped and are not filled the same way. Just look at the math done in one of the answers. –  Lagerbaer Mar 29 '11 at 15:49
@user2146 Sorry, for a moment there I forgot that all YouTube videos are completely credible and that guys running tourist traps in Kenya know more about physics than the entire scientific establishment. –  Mark Eichenlaub Mar 29 '11 at 15:53
I wish I had more karma so I could downvote this further –  Dan Mar 29 '11 at 16:00
@Larry: It is trivial to fake this. All you have to do is pour the water into the tub from slightly to one side or the other. Then it has angular momentum and coriolis force entirely due to the way you poured it. To Mark, I am very shy of argument by popularity ("entire scientific establishment"). There is an Einstein anecdote on that. –  Mike Dunlavey May 28 '12 at 14:01

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