# Vortex in liquid collects particles in center

At xmas, I had a cup of tea with some debris at the bottom from the leaves. With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin. At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.

What's the physical explanation of the accumulation of particles in the middle?

My guess is that it's something to do with a larger radius costing the particles more work through friction...

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The answer to this question was first given by Einstein in 1926, read this Wikipedia article: en.wikipedia.org/wiki/Tea_leaf_paradox – gigacyan Jan 20 '11 at 8:20
possible duplicate of Mixing a cup of tea and break of centrifugal force – FraSchelle Jun 30 '13 at 11:52
@Oaoa ahh, but that question is more recent, and was already closed as a duplicate of this one... – Chris White Jul 1 '13 at 4:44
This questions seems to be asked every few months or so. Most recently here, where there is a very good answer as well. – Chris White Jul 1 '13 at 4:45

The word viscosity hasn't been used above, and yet it's crucial to understand the problem. Since tea is viscous, it obeys a type of 'non-slip condition': it's completely at rest against the sides and bottom of the cup. The tea forms a small 'boundary layer' (of ~ 0.04 cm)on the bottom, where there is a large velocity gradient, since outside of this layer the tea behaves as if it were inviscid. This boundary layer is called an Ekman layer - this should allow you to look up good references on the subject. Georg's model explained above is quite correct. I just wanted to add that this is a classical but very subtle exercise (you need to use Navier-Stokes in cylindrical coordinates in a rotating frame in two different regimes...), but in the end you can actually show that inside the boundary layer $$v_r = -\frac{\Delta \Omega}{2} r e^{-\hat{z}}\sin{\hat{z}},$$ where $\Delta \Omega$ is the difference in rotational velocity between the bottom and the top of the cup, $\hat{z} = z \sqrt{\frac{\Omega}{\nu}}$ (thus giving a typical size for the boundary layer...), $\Omega$ is the typical rotational velocity and $\nu$ the tea's viscosity.

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A simple explanation:

Firstly, the vortex becomes stable after you stop shaking the cup because as you shake the cup in the conventional way, the forces acting on the whole fluid are uneven, when you stop shaking the cup the vortex is able to evenly apply force to the fluid. (Friction between the sides of the cup and the turning liquid helps to stabilise the vortex.)

The solids in the suspension are denser than the liquid solution and so more momentum is needed to increase the centripetal force and throw the solid out further. The solution however, tea, is much less dense than the solid so it is thrown out further.

Your guess is pretty much right, because the radius would be greater if the same force was applied to a less dense.

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Its the Centripetal force that causes the particles to come at the centre due to circular motion.

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Comment to the answer(v1): Schematically, the tealeaves travel radially to the center via a non-rotating Ekman layer near the bottom, so there is no circular motion involved. – Qmechanic Feb 8 '12 at 22:16

The water molecules move more quickly along the periphery than in the center of the cup. The resulting force on a macroscopic object is directed inward, as I try to show in diagram. Oops, as I have no points to render images I will substitute for text art:

The vertical lines are a representation of the velocity field

                                          |
|     |
|      |     |
Center  _____|______|______|_____| Periphery
0    1      2      3     4


Any macroscopic object that extends, for instance, between radius 2 and 3 will suffer a force directed to the center. Tea leaves will move to the center.

UPDATE: This answer is still incomplete/incorrect even after the clarifications. (Velez)

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The resulting force on a macroscopic object is directed inward Why does the velocity field imply that? – Malabarba Jan 18 '11 at 20:34
@Bruce: Momentum is proportional to mv² , each water atom has identical mass. Then the transfer of energy to the leaf is greater at a greater radius than in the inner part of the leaf, nearer to the configuration center. (I've considered a forced vortex where v=wr) see en.wikipedia.org/wiki/Vortex#Forced_.28rotational.29_vortex – Helder Velez Jan 18 '11 at 21:57
Momentum is proportional to $v^2$? And why is it that greater kinetic energy near the border implies a force directed inward? – Malabarba Jan 18 '11 at 22:02
@Bruce: (continuation) The sheet behaves like a solid bar and is hitted along its lengh by molecules with different momentum. – Helder Velez Jan 18 '11 at 22:09
@Helder: I see your point now. But that is not the case. If your logic was correct, then the leaves and particles would accumulate in the middle while the water spins at full speed. What happens is that they only accumulate after the water starts to decelerate. So your argument doesn't cause the leaves to move inward, it only causes them to spin. – Malabarba Jan 18 '11 at 22:17