Q:  Why tea leaves stay in the centre of a cup of swirling fluid.
Explanation 1:
Coriolis force, which drives atmospheric circulation around regions of high and low pressure, is a fine example of a fictitious force. But it is not the explanation for the beautiful tea leaf effect. That honor goes to pressure. As fluid swirls in a container, centrifugal force causes a rise in hydrostatic pressure from the center to the outer edge. Within the boundary layer at the stationary bottom surface, however, fluid angular velocity is reduced, and centrifugal force is insufficient to counteract the pressure gradient. Driven by this pressure gradient, fluid flows inward, then upward at the center, resulting in a toroidal secondary circulation pattern. Tea leaves are carried along for the ride, but as the flow slows down, the upward velocity in the center becomes insufficient to lift the leaves, and they pile up in a neat cone at the center of the bottom surface. Thinking of the system in terms of a rotating frame of reference does not change this explanation.
Explanation 2:
A better explanation of the “tealeaves” effect is the difference between the outward forces at the top and bottom of the cup.
The forces push the rotating body of liquid outwards against the cup walls, at both the top and bottom, but because of friction effects with the cup’s base, the liquid at the bottom of the cup rotates more slowly, and is pushed outwards more weakly. So the liquid at the top of the cup “wins”. Liquid moves outward at the top of the cup, moves downward alongside the walls, and forces itself inward at the bottom of the teacup, while the liquid that was originally at the bottom gets forced inwards and upwards as a rising column of liquid at the centre of the cup. The tealeaves get caught up in this vortex flow and dragged to the centre of the cup, but the upward current isn’t strong enough to completely lift them, so they collect as a pile in the centre.
Einstein wrote a nice paper describing and explaining the effect.
Explanation 3:
Nevertheless, analyzing a situation in terms of fictitious forces may, in fact, be the most effective way to understand what is actually going on. Take a stirred cup of tea, a charming example of a consequence of the Coriolis force. If a few tea leaves are present in the cup, they end up in a pile at the center of the bottom surface (and not along the edge, as one might expect, as a result of the also fictitious centrifugal force). If you imagine yourself rotating around in sync with the stirred fluid, most of the fluid would appear to be at rest while the cup would be counter-rotating around you. That rotating cup drags some adjacent fluid along with it. Meanwhile, near the bottom, the Coriolis force on that dragged fluid pushes it toward the center of the cup, carrying the leaves along with it.

Adapted From: http://www.scientificamerican.com/article.cfm?id=what-is-a-fictitious-force

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