Why Tea Leaves Settle in the Center of a Mug
As part of explainer training we completed an example worksheet with a very simple question on it concerning the spinning tea leaf exhibit. “Why do tea leaves settle in the center of the cup?” Several of us pondered this for several minutes and came up with some adequate sounding answers, but I sensed I still hadn’t grasped what was really going on. After an entire week of quizzing physics students, searching the internet, and staring obsessively and abysmally into the bottom of tea cups, I finally found a satisfactory explanation, which, like any good know-it-all, I am posting here for your reading pleasure. By the way, this problem is commonly referred to as the “tea leaf paradox,” and it remained unsolved for decades until Einstein lent his genius to resolving the problem. It may be a bit unfair to be grading 5th graders on that.
“When a person stirs a liquid in a cup, they create a primary flow of the liquid that involves a constant rotation around the center of the cup. Almost like the rotation of a solid body. In order to go around in circles, which involves acceleration toward the center of rotation, the liquid requires a centripetal force on each unit mass. A pressure gradient directed outward through the rotating fluid, and terminating at the solid wall of the cup, supplies this force; and a person can see the effect of this pressure gradient by noticing that the rotating fluid free-surface is shaped like a parabola with the highest fluid surface out near the cup rim. This state of affairs is not in equilibrium, unfortunately, and there develops a secondary flow of liquid which is the real focus of our attention in explaining the behavior of the tea leaves.
This secondary flow of liquid depends on the free surface of liquid in the cup, and friction (viscosity) of the liquid at the cup surface. It occurs in the following way. Viscosity prevents any relative motion between the liquid and the cup at the point of contact. In effect the fluid is “stuck” to the solid surface of the cup, and must come to rest there. A fluid dynamicist would say that there is fluid in a thin “boundary layer” near the solid cup surface in which fluid behaves differently than it does in the bulk of the primary flow. Fluid is also at rest along a vertical line at the exact center of rotation near the center of the cup. But there is a greater height of fluid near the cup wall than there is at cup center, and what results is a pressure gradient that sends fluid along the floor of the cup inward. This disturbs the basic solid rotation of liquid throughout the cup.
Thus a secondary flow now develops that consists of fluid that flows inward at the bottom of the cup, rises at the center of the cup (center of rotation actually), flows outward near the surface, and finally downward along the walls. This secondary flow will sweep loose material, like tea leaves, toward the center of the bottom of the cup. A fluid dynamicist would say that there is a “point of stagnation” on the bottom of the cup near the center, and once materials reach this stagnation point they remain there. Eventually this secondary flow will de-spin the liquid, flatten the surface, and bring everything to a state of rest.”
copied from Kevin Kilty writing for The Citizen Scientist;helical flow this explanation can be found at: http://www.sas.org/tcs/weeklyIssues_2005/2005-07-01/backscatter/index.html