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Tea leaf paradox

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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 describes 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 formation of secondary flows in annular channel was theoretically treated by Boussinesq as early as in 1868[2]. The migration of near-bottom particles in river bent flows was experimentally investigated by A.Ya.Milovich in 1913[1]. The solution first came from Albert Einstein in a 1926 paper where he used it to explain the erosion of river banks (Baer's law).[3][4]

Explanation

Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).

However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the centripetal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[5]

Because of inertia, the pressure is higher along the rim than in the middle. If all the liquid rotated as a solid body, the inward (centripetal) force would match the outward (inertial) force from the rotation and there would be no inward or outward movement.

In a teacup, where the rotation is slower at the bottom, the pressure gradient takes over and creates an inward flow along the bottom. Higher up, the liquid flows outward instead. This secondary flow travels inward along the bottom bringing the leaves to the center, then up, out and down near the rim. The leaves are too heavy to lift upwards, so they stay in the middle. Combined with the primary rotational flow, the leaves will spiral inward along the bottom.[4]

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]

See also

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).{{cite book}}: CS1 maint: location missing publisher (link)
  2. ^ Boussinesq J. (1868). "Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides" (PDF). Journal de mathématiques pures et appliquées 2 e série. 13: 377–424.
  3. ^ Bowker, Kent A. (1988). "Albert Einstein and Meandering Rivers". Earth Science History. 1 (1). Retrieved 2008-12-28.
  4. ^ 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. English translation: The Cause of the Formation of Meanders in the Courses of Rivers and of the So-Called Baer’s Law, accessed 2008-12-28.
  5. ^ "CEE 262A Hydrodynamics Lecture 18" (PPT). 2007. p. 35. Retrieved 2008-12-29.
  6. ^ Arifin, Dian R. (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. Retrieved 2008-12-28. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |laydate= ignored (help); Unknown parameter |laysource= ignored (help); Unknown parameter |laysummary= ignored (help)
  7. ^ Pincock, Stephen (17 January 2007). "Einstein's tea-leaves inspire new gadget". ABC Online. Retrieved 2008-12-28.
  8. ^ Tandon, Amit. "Einstein's Tea Leaves and Pressure Systems in the Atmosphere" (PDF). Retrieved 2008-12-29. {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  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.

Additional reading

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