The tea leaves collect in the middle and the bottom, instead of along the rim.
The blue line is the secondary flow that pushes the tea leaves to the middle of the bottom.
Albert Einstein solved the paradox in 1926.
Visualization of secondary flow in river bend model (A.Ya.Milovich, 1913,[1] flow from right to left). Near-bottom streamlines are marked with dye injected by a pipette.

The tea leaf paradox is a phenomenon where tea leaves in a cup of tea migrate to the center and bottom of the cup after being stirred rather than being forced to the edges of the cup, as would be expected in a spiral centrifuge. The correct physical explanation of the paradox was for the first time given by James Thomson in 1857. He correctly connected the appearance of secondary flow (both Earth atmosphere and tea cup) with ″friction on the bottom″.[2] The formation of secondary flows in an annular channel was theoretically treated by Boussinesq as early as in 1868.[3] The migration of near-bottom particles in river-bend flows was experimentally investigated by A. Ya. Milovich in 1913.[1] The solution first came from Albert Einstein in a 1926 paper in which he explained the erosion of river banks, and repudiated Baer's law.[4][5]

## Explanation

Stirring the liquid causes a spiral flow schema by centrifugal action. As such, the anticipation is that tea leaves would, because of their mass, move to the edge of the cup. However, friction between moving water and the cup increases water pressure, resulting in a high pressure boundary layer. This high pressure boundary layer extends inwards and even overcomes the inertia of the tea leaf's centrifugally actuated mass. Therefore it is friction, between the cup and the water, that produces a centripetal force upon the mass of tea leaves.

The high pressure boundary layer also effects the flow schematic which produces the familiar spiral pattern. The high pressure boundary layer - caused by the stirring - forces water outward and up the edge of the cup, where pressure increases. Then the water moves downward, inward and then upward, about the centre (see diagram). In this way, the flow schematic exerts an inward force that exceeds the mass of the tea leaves, and effectively contains their outward (centrifugal) tendency, and causes the observable (centripetal) paradox.

Concurrently, the circular movement of water (in the x-axis) is slower at the bottom of the cup than at the top, because the friction surface at the bottom is greater. While there is sufficient water velocity, this difference can 'twist' the moving body of water into such a spiral.

## Applications

The phenomenon has been used to develop a new technique to separate red blood cells from blood plasma,[6][7] to understand atmospheric pressure systems,[8] and in the process of brewing beer to separate out coagulated trub in the whirlpool.[9]

• Baer–Babinet law, also known as Baer's law
• Ekman layer – Layer in a fluid where there is a force balance between pressure gradient force, Coriolis force and turbulent drag
• Secondary flow – Relatively minor flow superimposed on the primary flowby inviscid assumptions

## References

1. ^ a b His results are cited in: Joukovsky N.E. (1914). "On the motion of water at a turn of a river". Matematicheskii Sbornik. 28. Reprinted in: Collected works. Vol. 4. Moscow; Leningrad. 1937. pp. 193–216, 231–233 (abstract in English).
2. ^ James Thomson, On the grand currents of atmospheric circulation (1857). Collected Papers in Physics and Engineering, Cambridge Univ., 1912, 144-148 djvu file
3. ^ Boussinesq J. (1868). "Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides" (PDF). Journal de mathématiques pures et appliquées. 2e Série. 13: 377–424.
4. ^ Bowker, Kent A. (1988). "Albert Einstein and Meandering Rivers". Earth Science History. 1 (1). Retrieved 2008-12-28.
5. ^ Einstein, Albert (March 1926). "Die Ursache der Mäanderbildung der Flußläufe und des sogenannten Baerschen Gesetzes". Die Naturwissenschaften. Berlin / Heidelberg: Springer. 14 (11): 223–4. Bibcode:1926NW.....14..223E. doi:10.1007/BF01510300. S2CID 39899416. English translation: The Cause of the Formation of Meanders in the Courses of Rivers and of the So-Called Baer's Law, accessed 2017-12-12.
6. ^ Arifin, Dian R.; Leslie Y. Yeo; James R. Friend (20 December 2006). "Microfluidic blood plasma separation via bulk electrohydrodynamic flows". Biomicrofluidics. American Institute of Physics. 1 (1): 014103 (CID). doi:10.1063/1.2409629. PMC 2709949. PMID 19693352. Archived from the original on 9 December 2012. Retrieved 2008-12-28.
7. ^ Pincock, Stephen (17 January 2007). "Einstein's tea-leaves inspire new gadget". ABC Online. Retrieved 2008-12-28.
8. ^ Tandon, Amit; Marshall, John (2010). "Einstein's Tea Leaves and Pressure Systems in the Atmosphere". The Physics Teacher. 48 (5): 292–295. Bibcode:2010PhTea..48..292T. doi:10.1119/1.3393055. Retrieved 2019-09-25.
9. ^ Bamforth, Charles W. (2003). Beer: tap into the art and science of brewing (2nd ed.). Oxford University Press. p. 56. ISBN 978-0-19-515479-5.