Einstein, tea leaves, meandering rivers, and beer

If you make your tea the old-fashioned way, ending up with a few tea leaves at the bottom of the teacup, and you start stirring the tea, you would expect the leaves to move outward, due to the push of the centrifugal force. Instead the leaves follow a spiral trajectory toward the center the cup. The physical processes that result in this 'tea leaf paradox' are essentially the same as the ones responsible for building point bars in meandering rivers. It turns out that the first scientist to make this connection and analogy was none other than Albert Einstein.

In a paper published in 1926 (English translation here), Einstein first explains how the velocity of the fluid tea flow is smaller at the bottom of the cup than higher up, due to friction at the wall. [The velocity has to decrease to zero at the wall, a constraint called 'no-slip condition' in fluid mechanics.] To Einstein it is obvious that "the result of this will be a circular movement of the liquid" in the vertical plane, with the liquid moving toward the center at the bottom of the cup and outward at the surface (see the figure below). For us, it is probably useful to think things out in a bit more detail.

Einstein's illustration of secondary flow in a teacup

A smaller velocity at the bottom means a reduced centrifugal force as well. But overall, the tea is being pushed toward the sidewalls of the cup, and this results in the water surface being higher at the sidewalls than at the center. The pressure gradient that is created this way is constant throughout the whole water tea column, and overall it balances the centrifugal force (unless you stir so hard that the tea spills over the lips). This means that the centrifugal force wins at the top, creating a velocity component that points outward, but loses at the bottom, creating a so-called secondary flow that is pointing inward. The overall movement of the liquid has a helical pattern; in fact, those components of the velocity that act in a direction perpendicular to the main rotational direction are usually an order of magnitude smaller than the primary flow.

Einstein goes on to suggest that the "same sort of thing happens with a curving stream". He also points out that, even if the river is straight, the strength of the Coriolis force resulting from the rotation of the Earth will be different at the bottom and at the surface, and this induces a helical flow pattern similar to that observed in meandering rivers. [This force and its effects on sedimentation and erosion are much smaller than the 'normal' helical flow in rivers.] In addition, the largest velocities will develop toward the outer bank of the river, where "erosion is necessarily stronger" than on the inner bank.

Secondary flow in a river, the result of reduced centrifugal forces at the bottom

I find the tea-leaf analogy an excellent way to explain the development of river meanders and point bars; just like tea leaves gather in the middle of the cup, sand grains are most likely to be left behind on the inner bank of a river bend. Yet Einstein's paper is usually not mentioned in papers discussing river meandering -- an interesting omission since a reference to Einstein always lends more weight and importance to one's paper (or blog post).

A recent and very interesting exception is a paper published in Sedimentology. It is titled "Fluvial and submarine morphodynamics of laminar and near-laminar flows: a synthesis" and points out how laminar flows can generate a wide range of depositional forms and structures, like channels, ripples, dunes, antidunes, alternate bars, multiple-row bars, meandering and braiding, forms that are often considered unequivocal signs of turbulent flow. [This issue of Sedimentology is open access, so do click on the link and check out the paper!]. As they start discussing meandering rivers and point bars, Lajeunesse et al. suggest that Einstein's teacup is extremely different dynamically from the Mississippi River, yet it can teach us about how point bars form:

A flow in a teacup with a Reynolds number of the order of 10² cannot possibly satisfy Reynolds similarity with the flow in the bend of, for example, the Mississippi River, for which the Reynolds number is of the order of 10⁷. Can teacups then be used to infer river morphodynamics? 

The answer is affirmative. When dynamical similarity is rigorously satisfied, the physics of the two flows are identical. However, even when dynamical similarity is not satisfied, it is possible for a pair of flows to be simply two different manifestations of the same phenomenon, both of which are described by a shared physical framework. Any given analogy must not be overplayed because the lack of complete dynamic similarity implies that different features of a phenomenon may be manifested with different relative strengths. This shared framework nevertheless allows laminar-flow morphodynamics to shed useful light on turbulent-flow analogues.

Apart from helping understand river meandering, the tea leaf paradox has inspired a gadget that separates red blood cells from blood plasma; and helps getting rid of trub (sediment remaining after fermentation) from beer.

That explains the 'beer' part of the title. And it is time to have one.

References

Einstein, A. (1926). Die Ursache der Meanderbildung der Flusslaufe und des sogenannten Baerschen Gesetzes Die Naturwissenschaften, 14 (11), 223-224 DOI: 10.1007/BF01510300

Lajeunesse, E., Malverti, L., Lancien, P., Armstrong, L., Metivier, F., Coleman, S., Smith, C., Davies, T., Cantelli, A., & Parker, G. (2010). Fluvial and submarine morphodynamics of laminar and near-laminar flows: a synthesis Sedimentology, 57 (1), 1-26 DOI: 10.1111/j.1365-3091.2009.01109.x