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Tea leaf paradox

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In fluid dynamics, 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 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.
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 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 Joseph Valentin 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

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The stirring makes the water spin in the cup, causing a centrifugal force outwards. Near the bottom however, the water is slowed by friction. Thus the centrifugal force is weaker near the bottom than higher up, leading to a secondary circular (helical) flow that goes outwards at the top, down along the outer edge, inwards along the bottom, bringing the leaves to the center, and then up again.[5]

Applications

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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]

See also

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  • Baer–Babinet law, also known as Baer's law – Theory on the formation of rivers due to Earth's rotation
  • Ekman layer – Force equilibrium layer in a liquid
  • Secondary flow – Relatively minor flow superimposed on the primary flow by inviscid assumptions

References

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  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. 29. Reprinted in: Collected works. Vol. 4. Moscow; Leningrad. 1937. pp. 193–216, 231–233 (abstract in English).{{cite book}}: CS1 maint: location missing publisher (link)
  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. Archived from the original (PDF) on March 17, 2022.
  4. ^ Bowker, Kent A. (1988). "Albert Einstein and Meandering Rivers". Earth Science History. 1 (1): 45. Bibcode:1988ESHis...7...45B. doi:10.17704/eshi.7.1.yk72n55q84qxu5n6. Retrieved 2008-12-28.
  5. ^ a b Einstein, Albert (March 1926). "Die Ursache der Mäanderbildung der Flußläufe und des sogenannten Baerschen Gesetzes". Die Naturwissenschaften. 14 (11). Berlin / Heidelberg: Springer: 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. 1 (1). American Institute of Physics: 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. hdl:1721.1/118473.
  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.
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