Tea leaf paradox
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 an annular channel was theoretically treated by Boussinesq as early as in 1868. The migration of near-bottom particles in river-bend flows was experimentally investigated by A.Ya.Milovich in 1913. 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).
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.
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.
The phenomenon has been used to develop a new technique to separate red blood cells from blood plasma, to understand atmospheric pressure systems, and in the process of brewing beer to separate out coagulated trub in the whirlpool.
- 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 4. Moscow; Leningrad. 1937. pp. 193–216; 231–233 (abstract in English).
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- Booker, John R. "Student Notes - Physics of Fluids - ESS 514/414" (PDF). Department of Earth and Space Sciences, University of Washington. ch. 5.8 p. 48. Retrieved 2008-12-29. See also figure 25 in figures.pdf
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