# Why does bathwater make a vortex in the plughole?

When I google for this I just get stuff about whether or not the Coriolis effect makes it go clockwise or anticlockwise, but I don't care which direction it turns in. I want to know why it turns at all. Why doesn't the water just fall straight down the hole?

One wrong theory is that air wants to return up the pipe, but that only applies if there's a closed vessel like a wine bottle at one end of the pipe or the other. My bath drain empties into the back garden and the air can return via the bathroom window.

Another is that pre-existing rotational currents in the bath get amplified as the water is drawn to the plughole just like a ballerina pulling her arms in, but there are two problems with this. Firstly, under this theory you'd expect the speed of the vortex at the plughole to be proportional to the speed of the pre-existing currents, but experience suggests that every plughole has its own favourite speed that depends on its geometry, the water depth, etc, so that you get pretty much the same speed whether you stir the bath a little or a lot. Secondly, by leaving the tap running (taking care not to inject angular momentum) we can keep the vortex going forever, but if the only driver was the pre-existing currents, then surely they'd be depleted by viscous friction and falling down the plughole. Some other effect must be driving the vortex continuously. There's energy available from the loss of gravitational potential energy of the water, but why does it get turned sideways to make this vortex?

• Possible duplicates: physics.stackexchange.com/q/32/2451 , physics.stackexchange.com/q/7738/2451 and links therein. – Qmechanic Oct 26 '17 at 13:30
• If you think Coriolis theory is debunked see 20:00 mins into this video: youtube.com/watch?v=eImGuxgR8-A. Also if you used a highly viscous liquid such corn-syrup or honey, do you think it will rotate as it goes down the sink? – Deep Oct 27 '17 at 5:35
• It was a nice video, but I still don't believe that's the coriolis effect. Not only is the pool tiny, but the speed of the water movement is very slow because the hole is so small but the coriolis effect acts on things that are moving. If you do the maths I think you'll find that a pond skater in the water would be like a tsunami in comparison. – Adrian May Oct 27 '17 at 6:11
• About the video: see my comment on the other video below. I'm not asking about the direction, and the video does not claim that the coriolis effect explains the rapid rotation you get as time goes by. – Adrian May Oct 27 '17 at 6:20
• If you think about the water as layers stacked upon each other, then the bottom layer is the one that is most affected when the drain cork is removed, simply because it's closets. The bottom layer gets drained and as a result its shape will start to slightly resemble a funnel. Now think about the water above this funnel-shaped layer. If, for the sake of this thought, we allow this water to slide on top of the funnel, then we get the ballerina effect. However, our funnel isn't solid, so when the spinning gets faster close to the drain, it gets pushed upon harder and 'bent' to a vortex shape. – Nitsan Avni Feb 11 at 18:57

Do this at the equator, so you can forget about the angular momentum of things on the earth. Is the container circular, with no obstructions on the bottom? Is the drain in the center? Is the water in the container absolutely still? Like if you sprinkled some dust on it and came back 24 hours later it would not have moved? If so, when you start the drain, the water will flow straight into the drain, with no vortex.

Anything that gives it the slightest angular momentum, in other words, any motion that is not toward or away from the drain, will be magnified as the water approaches the drain. It's the same as a spinning figure skater pulling in his or her arms and legs. Any weight that's pulled toward the center obeys Kepler's equal-area rule for orbits, so reduced radius results in increased angular velocity.

• I already attacked the figure skater theory in the original post. – Adrian May Oct 26 '17 at 19:53
• @AdrianMay: It's called conservation of angular momentum. If you want to experiment with that, you need to control all the possible unknowns and collect numerical observations. You can't just say that that's not it because it doesn't look like it is. – Mike Dunlavey Oct 31 '17 at 4:02
• @dunlavey Suppose I put the bath out in the rain for a million years. I can keep the vortex going, but where does the angular momentum come from to replace that which falls out of the bottom or disperses against viscosity? – Adrian May Nov 1 '17 at 8:20

The velocity field of an incompressible fluid with no rotation obeys $\nabla \cdot v=0$ (incompressible) and $\nabla \times v=0$ (no rotational flow). As explained by Feynman, you can still have a fluid that moves in circles around a cylinder - it is just locally irrotational (zero vorticity; if you imagine putting in a tiny waterwheel in the fluid it will not turn). The velocity in this case scales as $v \propto 1/r$, which corresponds to a vortex.

When the liquid drains, if it has some angular momentum then it must obviously start to circulate and the inward current will concentrate the vorticity. But even without angular momentum it is tricky to simultaneously maintain energy, momentum and mass conservation with a purely radial flow. So if the fluid starts circulating it gets an extra degree of freedom that allows it to satisfy all conditions, and it does not have to increase its vorticity to do so. Apparently the amount needed to make a swirling appear is fairly small. However, the swirl is temporary as angular momentum is gradually depleted by outflow.

(This leaves out a fair bit of nonlinearity, see Andersen et al. for more stuff).

• You're right that angular momentum leaks out of the bottom, in fact, I think there's an angular recoil on the bucket, but it's not true that the swirl is only temporary, and that's why your answer doesn't suffice. Also, the extra degree of freedom was always there because nobody tried to constrain the water into moving radially. The question still remains of why it can better satisfy one conservation law or another by going sideways. I could only read the abstract of that article. – Adrian May Oct 26 '17 at 19:52
• Your answer can't be wrong because it doesn't put many of its cards on the table. Can you tell us which of those various conservation laws can't be satisfied with purely radial flow, or what solutions might appear when circumferential flows are allowed? In some circumstances radial solutions do exist, like with a slow dribble into a large trumpet-shaped hole, so what are the conditions for a vortex to form? – Adrian May Oct 26 '17 at 20:38

Let me try to explain it. When water flows towards the hole, it chooses the shortest path (and we'll have next to no error if we assume that shortest path is a straight line). However, the Earth is rotating, and if you observe the straight-line motion of object from the rotating frame of reference, a motion relative to you will look curved. So, in fact the water is not rotating, it's the Earth "under" it that's rotating. Coriolis force you mentioned does not exist. It's just the effect of non-inertiality of FOR.

For example, when the uniformly moving bus brakes, you feel like you are dragged forward, but in reality, you are just continuing the uniform motion (or, at least, trying to continue it), while the bus is moving with acceleration (deceleration in that case), making it's FOR non-inertial. And you feel that non-inertiality as fictional "force" dragging you forward (relative to the bus).

• The other related links suggest the Coriolis effect is not sufficient to explain the behaviour we see in drains. – JMac Oct 26 '17 at 14:56
• Yes, I decided to make an estimate after I submitted an answer, 'cuz the situation looked odd to me. Will edit the answer as soon as I figure out an intuitive way to explain turbulence where no obstructions exist. – Tajimura Oct 26 '17 at 15:06
• I said in my post that I don't care about the direction. That's discussed all over the web and we can agree to differ about the strength of the coriolis effect. The 50s video and its remake both claim that the coriolis effect can determine whether the water goes clockwise or anticlockwise, but not that it can explain the vigorous rotation you get when the pool is nearly empty. The latter is what I'm asking about. – Adrian May Oct 27 '17 at 6:18
• That new video is a fake. The food colouring shows laminar flow, not a vortex, and it's orders of magnitude stronger than in the old video. – Adrian May Oct 27 '17 at 7:30
• Correction: The food colouring shows laminar flow AS WELL AS a vortex. Look near the edges of the pool. The water was already stirred. – Adrian May Oct 27 '17 at 7:37

If the slightest asymmetry of the tub, air movement on top, convection current from uneven temperature etc., or disturbance causes a slight sideways movement of the water any where around the hole, it will deflect the incoming water slightly, as in the diagram. That will cause the incoming water to push the water around causing it to rotate. This will deflect the incoming water more, increasing the rotation, & so on. It's like standing on a wheel. If you are right in the middle, where your weight is straight down towards the shaft, it will not rotate. But move slightly to the side, & it will start to turn, moving your feet to the side, which will make it accelerate more, moving your feet more to the side & so on.