Teacupa

Teacupa, a tea-sprite of the Paradox Parlour, stirs a giant cup, and when the spin slows the leaves gather in the middle instead of being flung out.

The Physics

In a stirred cup, the swirl is held in balance by a pressure that rises toward the wall (the surface is higher there). Near the bottom, friction slows the swirl, but the pressure difference set by the surface is still there. So near the floor the inward pressure push wins, and water flows toward the centre, rises up the middle and returns outward near the top. This secondary flow carries the sunken leaves to the centre, where they settle.

∂p/∂r = ρ v_θ²/r

• p: pressure (Pa)
• r: distance from the axis (m)
• ρ: density of the water (kg/m³)
• v_θ: swirl (azimuthal) speed (m/s)

Albert Einstein explained this in a short 1926 paper in Die Naturwissenschaften on the causes of river meanders and Baer's law. The same secondary flow in river bends scours the outer bank and builds up the inner one. The friction layer at the floor is a form of Ekman boundary layer.

Honest caveat: The leaves do not go in because of any inward force on them. Centrifugal effects alone would push denser leaves outward. They go in because the bottom flow drags them there, and they stay because the upflow at the centre is too weak to lift them. The tidy picture holds for gentle spin-down; at high stirring speeds the flow is turbulent.

On the sticker

Teacupa stirs her glowing cup with a spoon, lit amber from below. Printed text: TEACUPA with the line LEAVES GO IN. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.