SWIRLING FLOWS IN TEACUPS
by
Daniel T. Valentine


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In the food industry, of course, knowledge of brewing tea that includes the flavoring, dying and mixing of water is important. Tea and coffee are primary beverages of our culture. Yet, the preparation of tea is not only important to our sustenance, the effect of rotation on the fluid motions in a teacup is also of scientific and engineering interest. There are related problems in atmospheric and environmental sciences, and in manufacturing and aeronautical engineering that are better understood if we elect to investigate rotating flows in cylinders that are essentially teacups.

The rotational motion of the water in a stirred teacup is a type of vortical motion. Vortices are ubiquitous in nature, in our homes and in industry. Hurricanes and tornados are dramatic illustrations of naturally occurring vortices. Another common vortex is the drain vortex that we observe when we open the drain of a tub. Cyclone dust collectors in woodworking shops use swirling flows to separate the sawdust from the air. Chemical mixers in industry induce vortices as part of the stirring action.

Vortices and rotating flows have fascinated people for centuries. During the time of Aristotle, Empedocles suggested that the Earth was supported by a cosmic vortex. Since tea has also been prepared and drunk for centuries, it may not be so surprising that Empedocles used the motion of tea leaves that follow the motion induced by stirring tea to illustrate his conjecture of the vortex-supported Earth. Although his description of the "teacup phenomenon" was reported to support a conjecture of a higher purpose, his description of this phenomenon is remarkably accurate. Because he used the teacup phenomenon as an analogy, it must have been, in some sense, common knowledge at that time (around 350 BC). To describe this phenomenon in more than observational detail, the laws of Newton developed in the 17th century along with the mathematics developed in the 19th century are required. The equations describing the motion of a fluid were known by the end of the 19th century; however, it wasn't until the invention of the computer in the middle of this century that solutions of these equations were more readily accessible. At this turning point (1950-1960) interest in problems analogous to the teacup problem was rekindled in a rather vigorous way. This interest continues today.



Artist's conception of the secondary motion superimposed on the vortical motion in a stirred cup of tea.(Artwork is by Joe Briggs).


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When the tea leaves are rotating around the bottom of a cup, they spiral towards the center of the cup as they follow the motion of the water that was induced by stirring. They move towards the center after the spoon is removed and, hence, when the water in the cup begins to spin down towards a state of rest. The pressure near the side walls of the cup is higher than the pressure in the center when the water is rotating. This can be observed by the shape of the surface of the water which is concave from the viewpoint of the drinker. This pressure variation is required to create the centripetal acceleration that balances the centrifugal acceleration of the rotating liquid water. However, the water near the bottom of the cup cannot move as freely because the water adjacent to the bottom sticks to the bottom (that is, the water moves much more slowly near the bottom because of friction or viscous effects). The water touching the wall does not move at all (this is the no-slip boundary condition that occurs in flows of viscous fluids, for example, water). As a consequence of fluid friction, the angular momentum of the water near the bottom is not enough to oppose the effect of the radial pressure field created by the rotating water away from the bottom boundary layer; in fact the pressure variation is such as to push the water near the bottom of the cup towards the center. Because mass is conserved in this flow, the water that is caused to move towards the center of the cup turns upward towards the surface. Subsequently, it turns towards the side wall at the surface and finally moves down towards the bottom boundary layer replenishing the water that was originally there. This circulatory pattern of motion is the secondary motion (that can be viewed in a meridional plane). The primary motion is, of course, the circulatory motion initially induced by stirring with the spoon.


Instantaneous streamlines in a meridional plane in a stirred cup of tea with a drop of milk descending in the center (computer simulation).

At Clarkson University, research on rotating flows in cylindrical containers and related problems is being conducted. Professor Daniel T. Valentine with graduate student Kieth D. Miller (Physics of Fluids 6, 2702, 1994). have investigated the "spin-down" problem, which is the flow that ensues when the walls of a cylinder, initially rotating at a constant angular speed for a reasonably long time such that the fluid within is in solid- body rotation, is suddenly stopped. This is an idealization of the teacup problem. When the walls are instantaneously stopped, the fluid inside continues to rotate. Almost instantly, the fluid near the top and bottom walls moves towards the center of the cylinder. The predominant motion is confined to relatively thin layers known as Ekman boundary layers. The motion towards the center and subsequently along the axis of the cylinder induces a leading wave; rotating flows can support internal waves. Subsequent to the leading wave, ring vortices can occur along the side wall (Goertler vortices) along with a host of three-dimensional phenomena, all of which tend to enhance mixing. The overall secondary motion is as described in 350 BC: however, the details were not known at that time. The details are beautiful illustrations of the complexity of flow patterns that occur even in relatively simple geometries.

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Another interesting problem associated with the rotating flow in a stirred cup of tea arises if we take milk in our beverage. If, unlike the classical approach of pouring milk into the cup prior to adding the water, we pour a drop of milk into the tea at the center of the water surface shortly after removing the stirring spoon, the flow is disturbed in a very interesting way. Because of its greater density, the cold milk descends into the hot tea causing vorticity in the rotating tea to become entrained in the wake of the milk drop and stretched by the descent. A dimple may appear on the surface of the tea due to the low pressure associated with the increased angular speed of the entrained water as it tries to conserve its angular momentum (this increase in rotation rate is similar to the increase in rotation rate of a playground "merry-go-round" as the rider moves towards the center when the "merry-go-round" is rotating).

An added complication in the milk-drop problem is the buoyancy effect. If a fluid of lower density is placed into a fluid of higher density in a gravitational field, the lighter fluid tends to float above the heavier one due to buoyancy. In space travel this effect is reduced considerably.

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Swirling flows are ubiquitous in nature and in many of the devices designed to improve our standard of living. The simple pleasure of contemplating the motions in our teacup (or coffee cup), is a contemplation of swirling flows that have potential applications in areas such as crystal growth, chemical mixing, weather studies, the food industry and more.