Monday, 13 April 2009

Albert Einstein, Tea, and the Wall of Death

You may have noticed that when you stir a mug of tea, the tea leaves at the bottom congregate in the middle.

This is potentially disturbing behaviour – see, when you stir the tea, centrifugal forces throw the tea and the tea leaves against the outer wall of the mug, as if there's an outward-pointing gravitational field (the "wall of death" effect) [*]. If the tea leaves were denser than the tea, we'd expect them to be thrown harder and to "sink outwards" and collect at the outer wall, while if the the leaves were lighter than the tea, we'd expect them to be floating around at the top of the mug, rather than sitting at the bottom. So our two main options for where the tea leaves ought to end up, based on density, seem to be either "bottom of mug, around the edge", or "top of mug". In real life, the tea leaves decide to do something else.

So what gives?

Albert Einstein published the answer in a paper entitled "The cause of the formation of meanders in the courses of rivers and of the so-called Baer's Law" in 1926, which was published in Die Naturwissenschaften, complete with a diagram of a cross-section of a cup of tea.

What's happening here (said Einstein) is that the tea leaves are sinking because they are denser than the surrounding tea, but they're also being swept to the centre of the mug by a vortex circulation pattern.
The stirred tea doesn't just rotate within the mug as a simple solid cylinder. There's frictional dragging associated with the sides of the mug, and with the bottom of the mug. The side-wall dragging effect is almost the same at all heights in the tea [**], but the base-dragging effect means that as the stirred tea slows through friction with the mug, the tea at the bottom has always lost a little more speed, because of the additional source of friction. It's always rotating a little more slowly than the rest of the tea. So the centrifugal forces in the "slower" layer of tea at the bottom of the mug push outwards less strongly than those in the faster-rotating tea at the top, and as a result, the tea at the top of the mug "wins the battle" and pushes the tea at the bottom of the mug back on itself, inwards to the centre.

So the rotational speed differential induces a vortex circulation pattern in the tea. The rotating tea at the top surface moves outwards to the edge of the mug, and then crawls down the side-walls in a spiral until it reaches the bottom. Then it moves inwards towards the centre, and finally forms a rising column of tea in the centre of the mug until it returns to its starting point, and does the whole thing all over again (since the average rotation speed has slowed even more since the last circulation cycle).

The current is usually powerful enough to scrape the tea leaves inwards towards the centre of the mug, but its usually not quite strong enough to lift them back up to the surface, so the leaves tend to collect as a little curve-sided cone-shaped pile in the centre. Which you can see if you have a glass mug, and don't add milk.

Tea, Einstein, vortex. Problem solved.


* If we're doing this as a Newtonian calculation we say that the tea is riding up the sides of the mug because it's attempting to move in a straight line and is being thwarted by the crockery wall, whereas if we apply Mach's Principle, and/or the general principle of relativity, it's equally legitimate to say that the tea itself isn't rotating, but the outside universe rotating around it creates a special sort of radial gravitational field that draws the tea outwards from the rotation axis. In the first calculation the tea pushes outwards against the walls because of its inertial mass, in the second the outward effect is gravitational, and the tea is drawn outward by the effect of its gravitational mass. So under a general theory of relativity, the inertial and gravitational masses of a body can't be separated, because the inertial and gravitational descriptions are interchangeable.

** Okay, so the side-wall dragging effect is different at different heights too, because the base-dragging effect gives the tea different rotational velocities at different heights. This often happens in physics, we define two variables that're
supposed to be independent, and then we find that in practice they interact and cross-breed and twist and twirl around each other in exotic ways that're much more difficult to model. We often try to ignore these additional levels of complexity in the hope that they won't upset our final results too much. Sometimes we're right.

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