Does the Coriolis Force Affect Tea Leaf Movement in a Stirred Cup?

[daniel_i_l]

When you stir a cup of tea with leaves at the bottom then when you stop stirring the leaves go to the center. I always thought that the cause was the centrifugal force and since the leaves where lighter than the water the water was pushed out and the leaves sucked in. But I just read in SA that the cause was the coriolis force. Can someone explain how the coriolis force is responsible and what's wrong with my explanation?
Thanks.

 

[Shooting Star]

Tea leaves are heavier than water. I just went up to the kitchen to verify. Some may float initially because of the air trapped in them, but I’m sure that after soaking in water, they will sink, for any brand of tea.

In the rotating frame of water, the effective gravity is in the outward direction and also slightly pointing downward because of earth’s g. By your logic, if the tea leaves are lighter than water, they should all “float” up to the centre. But since they are heavier than water, we would expect them to collect on the walls, which they don’t. Of course, when they are moving wrt the water, there is Coriolis force changing their direction of motion. The net result have to be calculated.

The sinking is explained because they are heavier than water, but why they collect at the centre is a mystery to me. Things are sucked into a vortex which the rotating water creates. So, that may be an explanation.

After writing all this, I found these two sites. You can check them out.

http://www.abc.net.au/catalyst/stories/s1982571.htm
http://www.madsci.org/posts/archives/1997-12/875655758.Ph.r.html

 

[Meir Achuz]

But I just read in SA that the cause was the coriolis force.

Can you give a reference to the SA article?

 

[daniel_i_l]

Here's the source:
http://www.sciam.com/askexpert_question.cfm?articleID=ABE57453-E7F2-99DF-32538FF7C7B37F20

 

[Shooting Star]

Well, at least they say that the leaves should have been pushed to the walls, due to centrifugal force. But the rest explanation is not so clear, but may be correct.

The three links gives different explanations. What about the one with Einstein’s theory (the first link of the two I had mentioned)?

We really have to sit down and do some hard calculations.

 

[daniel_i_l]

I liked the explanation here the best:
http://www.ucalgary.ca/~kmuldrew/river.html

 

[Meir Achuz]

That does seem simple and correct. Daniel: Why don't you email Politzer to ask him about his Coriolis force. The Coriolis force acts in the theta direction and wouldn't cause this.

 

[cesiumfrog]

When you have a rotating fluid in the cup then, throughout the body of the fluid, the pressure is higher (at a given height) towards the circumference due to (and in balance with) centripetal force. But in the boundary layers (since the fluid closest to the cup itself must be co-stationary), and for the bottom of the cup in particular, this less-rotating layer experiences less centrifugal force despite the same radial pressure gradient, and therefore flows from the outside to the centre (collecting the tea leaves there).

I've above avoided mention of Coriolis force by choosing an inertial frame of reference and a fairly idealised case, but it's really a little more complicated (it's a major topic in ocean and atmosphere dynamics: Ekman layers, http://soconnell.web.wesleyan.edu/ees106/lecture_notes/lecture13_106ocean_cir1/sld010.htm, etc).

 

[daniel_i_l]

I emailed Politzer and here's what he said:
In the water frame, the water is stationary except for a thin layer next to the sides and the bottom that rotates with the spinning cup because of friction. Since the thin layer on the bottom is spinning in the opposite direction of the spinning of the water (in the cup frame) then the Coriolis force pulls it towards the center.
He agreed the in the cup frame the link that I posted was accurate.

 

[pixel01]

May I explain in kinetic energy aspect.
Every system should try to settle at the lowest potential if possible. When the water swirls in the cup, the linear velocity is getting smaller at the center, so anything more dense than water will try to migrate to the center to get the lowest kinetic energy.

 

[Meir Achuz]

May I explain in kinetic energy aspect.
Every system should try to settle at the lowest potential if possible. When the water swirls in the cup, the linear velocity is getting smaller at the center, so anything more dense than water will try to migrate to the center to get the lowest kinetic energy.

But they migrate to the edge at the top of the cup.
Lowest energy works if you are not putting energy into the system, but not if you are stirring.

 

[Meir Achuz]

I emailed Politzer and here's what he said:
In the water frame, the water is stationary except for a thin layer next to the sides and the bottom that rotates with the spinning cup because of friction. Since the thin layer on the bottom is spinning in the opposite direction of the spinning of the water (in the cup frame) then the Coriolis force pulls it towards the center.
He agreed the in the cup frame the link that I posted was accurate.

The Nobel prize is like a PhD. They can't take it back.
I am sure if Politzer thought about it he would agree that the Coriolis force has no radial component.

 

[daniel_i_l]

Why no radial component? Fc = -2m * angular_velocity X velocity
where X is the cross product. So if "angular_velocity" points up and "velocity" is tangent to the edge of the cup in the direction opposite to the spinning then Fc points towards the center.

 

[pixel01]

But they migrate to the edge at the top of the cup.
Lowest energy works if you are not putting energy into the system, but not if you are stirring.

When you stir it, it receive a bit of energy in the form of kinetics. If you stop doing so, the heavier tea will migrate to the center, the floating ones will be close to the edge, because it density is lower. If you sirt it too rapidly, other effects appear,say, tubulences..

 

[marcusl]

I emailed Politzer and here's what he said:
In the water frame, the water is stationary except for a thin layer next to the sides and the bottom that rotates with the spinning cup because of friction. Since the thin layer on the bottom is spinning in the opposite direction of the spinning of the water (in the cup frame) then the Coriolis force pulls it towards the center.
He agreed the in the cup frame the link that I posted was accurate.

Spinning the cup is not the same as stirring the water+tea as your original post! Furthermore, the quote above makes no sense to me.

I throw my vote in with Albert E.

Edit: He also explains why light floating leaves go towards the rim.

 

[Meir Achuz]

Why no radial component? Fc = -2m * angular_velocity X velocity
where X is the cross product. So if "angular_velocity" points up and "velocity" is tangent to the edge of the cup in the direction opposite to the spinning then Fc points towards the center.

You are right about radial, but they don't take back PhDs.
I was oversimplifying from the hurricane example.
However if you go into the frame that is rotating at the rate of the bottom part of the fluid
(and not the cup frame), there is no Coriolis force at all since v=0.
There will be a smaller centrifugal force than at the top, so Einstein is right.
I'm sure he will welcome my approval.

 

[daniel_i_l]

You are right about radial, but they don't take back PhDs.
I was oversimplifying from the hurricane example.
However if you go into the frame that is rotating at the rate of the bottom part of the fluid
(and not the cup frame), there is no Coriolis force at all since v=0.
There will be a smaller centrifugal force than at the top, so Einstein is right.
I'm sure he will welcome my approval.

Yes, basically what I was saying is that the explanation depends on your frame of reference.

 

[cesiumfrog]

The Nobel prize is like a PhD. They can't take it back.
I am sure if Politzer thought about it[..]

Such dazzling confidence, to conclude that if either you or the Nobel prize committee has misunderstood then it has to be to be the Nobel prize committee.

You are right [that Coriolis force can be radial,] I was oversimplifying

...

However if you go into the frame that is rotating at the rate of the bottom part of the fluid (and not the cup frame), there is no Coriolis force at all since v=0. There will be a smaller centrifugal force than at the top, so Einstein is right.

So any explanations involving Coriolis are inferior because from one perspective that force disappears? (Except we need to ignore that the frame co-rotating with the bottom part of the fluid is the cup frame, and that your centrifugal force concept is equally "fictitious".)

But to demonstrate your worthiness of the welcome by Einstein (whose legacy incidentally is yet another fictitious force), how about you clarify something his paper omitted: Why exactly does having the smaller centrifugal force at the bottom give rise to the inward motion?

Every system should try to settle at the lowest potential [so] the heavier tea will migrate to the center, the floating ones will be close to the edge

Pixel, that explanation is less satisfactory because it explains neither why a dye injected at the side will circulate down (then across the bottom and up again through the centre) nor why the heavier tea leaves do not end up stationary at the circumference.

 

[Meir Achuz]

"Why exactly does having the smaller centrifugal force at the bottom give rise to the inward motion?"

"this less-rotating layer experiences less centrifugal force despite the same radial pressure gradient, and therefore flows from the outside to the centre (collecting the tea leaves there)."