THE EXASPERATING TENDENCY OF a poured liquid to cling to the outside of the container is known as the teapot effect. What causes it? One .immediately thinks of surface tension, but it turns out that another factor is more important: the pressure is higher at the outer surface of the fluid than it is at the inner surface, so that the liquid is pushed against the container.

At times, such as when one is pouring acid, one needs to forestall the effect so that the liquid will not run down the side of the container and onto the counter. The technique is to put a glass stirring rod across the top of the container. The liquid then runs along the rod and arrives unspilled in the container one is trying to fill.

A similar technique works for pouring cream or milk into coffee. You tend to pour slowly in order not to generate splashes, and so you inadvertently precipitate the teapot effect. You can avoid it by putting the blade of a table knife across the top of the dispenser.

The teapot effect can also be seen in fountains where the water runs slowly from a pool or a horizontal pipe. The water clings to the structure instead of arcing away from it. Some indoor water gardens rely on this effect. Water flows down a wall built with many projecting stones. Moss grows on the wall, which is kept wet by the teapot effect.

More often the effect is unwanted. When water runs slowly off the sill of a window, it can round the bottom edge of the sill and run into the juncture of the sill and the house wall. If the water seeps into the juncture, it can damage the wall. To avoid the problem a rectangular groove is cut into the underside of the sill parallel to the wall. The vertical sides of the groove are too steep for the stream to flow over, and so the water falls directly from the sill.

The teapot effect was named and investigated in 1956 by Markus Reiner of the Israel Technion-Institute of Technology. Initially he was curious about a seemingly unrelated phenomenon. If a cube of salt is put in fresh water with its top surface horizontal, it dissolves in an odd way. It shrinks in height, but it does not change in width or depth. The flow of the dissolved salt can be monitored by placing a small amount of potassium permanganate on the top surface to color the salty water. Since salt water is denser than fresh water, the flow should be across and off the top of the crystal. Instead the flow rounds the top edges and travels down the sides. Apparently the flow of salty water along the sides protects them from dissolution.

In another experiment an Erlenmeyer flask was submerged and inverted in a large container of fresh water. A stream of concentrated salty water was directed onto the bottom of the flask past a piece of permanganate. The stream flowed along the bottom, turned the edge and then continued down the sloping side instead of arcing away from the flask. After several centimeters it became unsteady, detached from the flask and proceeded downward.

In the next experiment the flask was placed right side up in a container of concentrated salty water. A stream of fresh water was directed at the bottom of the flask. The stream rounded and -then flowed along the sloping side of the flask instead of arcing upward off the bottom. After traveling for a few centimeters it oscillated and broke free, rising to the surface of the salty water.

Reiner experimented with other solids. He altered the slopes, produced streams of sinking cold water in a container of warm water and replaced the smoothly curving edge of the glass flask with a sharp edge of metal. Whatever he did, the streams still rounded the edge and adhered to the solid for at least a short distance.

Reiner also examined the flow of water poured from a teapot. Again one fluid (the water) flowed through another (the air) in the presence of a solid surface. If the flow was fast, the stream left the spout in an arc with no dribbling. If the flow was slow, the stream sometimes went along the spout and then broke off at the bottom of the pot. Sometimes the flow left the lip of the spout in a rearward arc. Flowing at an intermediate speed, the stream split in two; one part moved in an arc and the other ran down the spout.

An explanation of the teapot effect was published in 1957 by Joseph B. Keller of New York University. To follow the flow one constructs lines called streamlines that are everywhere tangent to the velocity vector of the water. If the water flows over a horizontal surface, the streamlines are all horizontal, indicating a uniform flow. If the water flows over and past a shallow obstacle, the lines crowd together, indicating an increase in the rate of flow.

The illustration below shows streamlines for flows over an edge of a square corner and of a thin plate. In both cases the lines crowd together, indicating an increase in the speed of the water. The lines nearest the solid surface crowd together the most, indicating the greatest increase in speed.

When a portion of a stream of water increases in speed, its kinetic energy increases also. Suppose the flow is not significantly downward, so that one can rule out gravity as the cause of the increase in energy. Then the energy must come from the pressure in the liquid, which can be regarded as a form of potential energy. When the speed of a portion increases, the pressure at that point decreases. The total energy of the portion, that is, the sum of its kinetic energy and the energy associated with the pressure, remains constant.

Since the total energy of a streamline is constant, the increase in speed of the portion at an edge necessitates a decrease in the pressure there. The pressure of the water on the free surface of the stream is kept constant because of the continuous push by the air. Hence when the stream reaches the edge, a difference in pressure exists across its depth, with atmospheric pressure on the free surface and a reduced pressure at the edge. It is this pressure difference that can force the stream to flow over the edge.

A stream of water carries momentum because it has both speed and mass. When you pour water rapidly, the momentum is too great for the pressure difference at the edge of the container to make the stream flow over the edge. The stream arcs from the container in the normal way. With less momentum (slower flow) the pressure difference is sufficient to make the stream turn.

Consider the stream after it has rounded the edge of a plate. Keller discovered that the stream can flow along the underside of the plate. Again the streamlines are all horizontal, indicating that the velocity vectors of the water elements across the depth are parallel and the same size. There is still a pressure difference across the depth. At the bottom of the stream (on the free surface) atmospheric pressure prevails; at the top of the stream (next to the plate) the pressure is lower.

To understand this pressure difference consider a stationary pool of water. Atmospheric pressure pushes on the surface.

Below the free surface the pressure must be higher because of the weight of the water. The additional pressure increases with depth.

When a stream flows along the underside of a horizontal plate, the pressure must be higher at the bottom of the stream than it is at the top because of the weight of the water between the two points. Since the bottom is at atmospheric pressure, the top must be at less than atmospheric pressure. The stream is forced against the underside of the plate by the pressure difference.

The stream is not stable, so that it cannot flow indefinitely under the plate. Small perturbations quickly grow large enough to detach the stream from the plate. Keller found the length of travel to be several centimeters.

If the plate slants downward, the stream runs farther before falling away. Since the pull of gravity is then not perpendicular to the stream, its force is weaker. Moreover, the oscillations brought about by small perturbations to the stream do not grow as quickly. This is the type of flow one sees along the spout of a teapot in the teapot effect.

The stream might also be able to flow upward along a slightly slanted surface. Within a few millimeters, however, it will be slowed by gravity. Then it either collects in drops that fall or oscillates enough to detach from the surface.

The secret to the proper design of a milk dispenser lies in the shape of the spout's edge. If the milk can round the edge and then flow horizontally or downward, the teapot effect will result. If the edge is shaped so that the milk rounding it would have to flow upward at a sharp angle, the dispenser will be free of the teapot effect even at the lowest rates of pouring. With an improperly designed spout one's only solution is to position a knife across the lip. The milk flows along the vertical knife more readily than it would flow along the slanted or horizontal spout.

A related example of clinging flow is the Coanda effect, named after Henri Coanda, the Romanian engineer who discovered it. Sometimes the name is applied to the teapot effect, but usually it serves to describe two separate phenomena. When a jet of fluid (either a gas or a liquid) emerges from a slot and passes a shallow step, it can attach itself to the wall at the base of the step. The attachment results from an entrainment of the ambient fluid between the jet and the step. I do not believe this phenomenon is a factor in the teapot effect.

Sometimes the term Coanda effect refers to the attachment of a fluid stream to a continuously curved surface. Suppose a horizontal jet of water is directed at the side of a beach ball. As the surface of the ball begins to curve away from the jet it creates a partial vacuum at the adjacent water surface. Since the free surface of the jet remains at atmospheric pressure, the pressure difference across the jet forces the jet against the ball. This phenomenon is quite similar to the teapot effect except for the partial vacuum.

I did an experiment similar to the one Reiner had done with salt. In a container of warm water I put an ice cube tilted so that as water melted from the top surface it would run down that surface. (The freshly melted water sinks, because it is denser than the ambient warm water.) On the top surface I deposited a drop of food coloring so that I could monitor the flow. The stream rounded the downward edge of the cube and ran down the side for a few millimeters before it detached and fell to the bottom of the container.

I next did several experiments on the flow of water from a clear plastic container used in the preparation of gravy. Normally after cooking meat I pour the drippings and grease from the pan into the container, wait until the grease separates into a layer on top of the drippings and then pour off the drippings For this purpose the container has a pouring spout extending from the bottom so that the bottom fluid rather than the top fluid is poured.

In my experiments I clamped the container to a ring stand mounted on the basin of my kitchen sink. Water ran from the faucet into the container and then through the pouring spout. I could adjust the angle of the container. In addition I could vary the rate at which water entered the container and thus the rate at which it left the spout.

I first tilted the container 45 degrees and set the water flowing at a moderately high rate. The water arced forward from the spout in the familiar way. As I decreased the flow rate the stream receded toward the container. Soon the rate was low enough so that the water rounded the edge of the spout slightly and arced rearward toward the container. When I decreased the flow more, the stream rounded the edge and flowed down the surface of the spout, eventually falling at the bottom. The attached stream was quite stable.

I squirted a few drops of food coloring into the container in order to follow the flow. Most of the flow was near the upper part of the spout. The water near the lower part hardly moved. When the water poured over the edge, it mixed too fast for me to distinguish speeds there or farther along the stream.

When the teapot effect was established, the water flowing along the underside of the spout changed shapes Near the edge it had a narrow, deep hump. Several millimeters down the side of the spout it suddenly changed to a wide, shallow flow that was maintained until the stream separated from the container. As I slowly increase flow the hump grew larger and began to oscillate. Drops were frequently thrown clear of the spout, but most of the water descended along the underside of the spout. I probed the flow with a needle at the edge of the spout. The hump and the rest of the inverted flow of water remained stable.

Then I increased the flow rate slightly. The hump grew into a loop that was nearly circular in cross section and extended a centimeter or more down the underside of the spout, changing then into a wide, shallow stream. Between the loop and the spout was a thin film of water. When I probed the film with a needle, I could detect no changes in the loop or the shallow stream below it. The water in the film seemed to be stagnant. At the lower end of the loop were stationary ripples. As I increased the flow rate again slightly these ripples grew more pronounced and the loop suddenly pulled away from the spout, forming the normal curved arc.

Similar stationary ripples can be seen in another thin stream of water. Hold a flat surface in a narrow stream from a faucet. Raise the surface toward the faucet. When it is close enough, the ripples appear. When it is closer still, the ripples become more pronounced, creating large oscillations in the diameter of the stream.

Lord Rayleigh once investigated such waves. They are stationary because they travel up the stream as quickly as the stream falls. Apparently the same kind of stationary wave is created in the thin stream forming the loop along the spout of a container. The impact of the loop on the spout creates the waves.

When the flow rate was low enough to create a hump rather than a loop, the stream occasionally oscillated enough to become detached from the spout. By rubbing my finger through the upper end of the spout I could easily make the stream reattach itself to the spout. If the flow rate was high enough to create a loop, this trick did not work.

The hump seems similar to one I can produce in a horizontal flow of water. When the stream meets an obstacle that is shallower than the stream, the water develops a hump over the obstacle or just downstream from it. The obstruction squeezes together the streamlines of the flow as the speed of flow increases. The speed can become supercritical, meaning that the water flows faster than waves can travel over water. At the downstream side of the hump the flow becomes subcritical. Often turbulence develops there.

Something comparable seems to happen in the water rounding the edge of a spout. The edge is an obstacle that squeezes the streamlines together. The free surface of the stream develops a hump that appears just below the edge of the spout. At the low side of the hump the water slows, spreading out in a shallow layer that continues down the underside of the spout. If the flow is fast enough, the low side of the hump is turbulent, throwing off drops. The rest of the stream responds to the pressure difference described by Keller and is forced against the spout.

When the flow is only slightly faster, the pressure difference at the edge is insufficient to force the stream fully over the edge. The water, which is al most thrown free, is pulled together by surface tension to form a stream with an approximately circular cross section. The stream runs downward and forms the loop, but it is not thrown free of the spout because of a thin film of water that remains between it and the spout. The pull of this film brings the loop back to the spout, where the stream then slows and spreads out. The loop is quite unstable. Once it breaks free I cannot reestablish it with my finger because I cannot create the thin film needed to reattach it. As soon as I remove my finger the vigor of the flow drives the loop forward into the familiar curved arc.

Some people think that coating the underside of a teapot spout with butter overcomes the teapot effect. I tried this trick (using margarine) without success. After removing the margarine I adjusted the container so that the spout was approximately horizontal. If the flow was slow, the stream rounded the edge of the spout and traveled along its underside for one or two centimeters. Then it detached as a rearward arc. Just below the edge of the spout the stream again had a narrow hump that extended for a few millimeters. Thereafter the stream was wide and shallow.

When the flow rate was very low, the water on the underside of the spout would surge. In the region of detachment the water slowed enough to form hanging drops. With each surge enough water was added to a drop to make it fall. The surge first pushed the drop along the spout; after the drop detached, the remaining water retreated toward the open end of the spout until the next surge drove it forward again.

I also set the container so that the open end of the spout was almost horizontal. Low flow rates enabled the water to round the edge of the spout, form a hump and then become a wide, shallow stream. As I increased the flow a loop developed, grew large and then broke free of the spout. Not all the water followed the loop. Part of it still flowed down the side of the spout. Between the free stream and the attached one was a short, narrow film on which I could see stationary waves of small wavelength. The water that breaks free of the spout must have enough momentum to counteract the pressure difference across the depth of the stream at the edge of the spout. Apparently some of the water loses much of its momentum at the edge. It can then flow down the underside of the spout in the manner described by Keller.

I also experimented with a U tube made with side arms. I stoppered one of the open ends of the device and put the other end in a slow flow from the faucet. Water emerged from the side arm on the stoppered side. With the flow adjusted carefully the stream left the side arm in a rearward arc and then reattached itself at a low point on the U. Usually this arrangement was unstable because the stream would immediately pull up and cling to the full underside of the side arm and the tube.

I was curious about how far an inverted stream of water would flow if the surface of an object was tilted at a moderate angle. In my backyard I set up a seven-foot length of flat metal tilted from the vertical by about 45 degrees. At the upper end of its underside I attached a garden hose. As I adjusted the flow of water I produced an inverted stream that traveled the length of the metal. Sometimes the stream rounded the side edge of the metal and then flowed along its upper side. Sometimes the stream (either inverted or on top of the metal) slowly oscillated like a snake.

I did similar experiments with slender rods of metal, glass and plastic held at a moderate angle. I adjusted the stream from the faucet until it was narrower than the rod. When the stream struck the rod off center, it traveled along the underside of the rod and then spiraled around it several times. It never traveled upward in relation to the basin, but at some points it did travel horizontally.

The spiraling results from the curvature of the rod. Since the stream is forced to follow the curvature because of the pressure difference across its depth, it initially circles the underside of the rod and then the top side. This motion continues down the length of the rod. The spiraling disappears when I replace the narrow rod with a wider one held at the same angle. Then the stream attempts to spiral but ends up looping back to the underside of the rod, whence it descends as an inverted flow.

In a final experiment I placed a round measuring cup in the stream from the faucet. The circular cross section was in a vertical plane, so that the stream struck on one side. The stream clung to the side, following the curve until it was below the cup. Then it began to travel upward from the basin for a short distance before it detached. Apparently the pressure difference responsible for the inverted flow of the stream is strong enough to make the stream actually flow uphill before gravity slows it enough to pull it free of the cup.

Bibliography

THE TEAPOT EFFECT...A PROBLEM. Markus Reiner in Physics Today, Vol. 9, No. 9, pages 16-20; September, 1956.

TEAPOT EFFECT. Joseph B. Keller, Journal of Applied Physics, Vol. 28, No. 8, pages 859-864; August, 1957.

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