Spoons and other small objects that are either flat or slightly curved can also create sheets, but they may generate other shapes as well. A small circular obstacle can be made to send out sheets that fold back on themselves to form a symmetrical bell shape-a water bell. You may have seen similar sheets and bells in fountain displays. Sometimes the sheets in a fountain are so large that they are almost water sculptures. A1though other factors figure in shaping the sheets, it is primarily surface tension that holds them together against disintegration and folds them back on themselves to form the bells.

The easiest way for me to form a bell is with a screw cap from a soft-drink bottle. The flat top surface is about the right size for a good bell and I can wedge two fingers into the bottom of the cap for support. To avoid spoiling the bell with my hand I reach up into the stream of water from below, positioning the flat part of the cap in the stream at the right distance from the faucet. The sheets fold back on my forearm to form a large bell.

Water sheets can also be made from a stream of water projected upward, which fountains usually are. A small obstacle in the stream then sends sheets of water to the sides, either upward or downward depending on the shape of the obstacle. You can make a water bell in an upward stream, as in a downward one, by adjusting the speed at which the water strikes the obstacle, but the shape of the bell may not be the same. If the obstacle has a concave surface, the sheets are directed downward to form a bell of about the same shape as one formed under a faucet. If the obstacle has a convex surface, the sheets are initially directed upward. Then they may curve over and come down. The bell may have a top that is more saucerlike than the rounded tops you will usually get in the sink.

Water sheets have been studied since at least 1833, when Félix Savart wrote about them. By 1935 most of the principles determining the shapes of the sheets and bells were understood, although perhaps not in detail because of the difficulty of their mathematics. In the past 20 years or so interest in water sheets has revived because of the need to control fluid layers in such areas as spray painting. Particularly important is the way thin liquid films break up into drops. The maximum radius of a sheet in the sink is primarily determined by the surface tension of the water. The sheet disintegrates at the radius where the surface tension is no longer strong enough to hold the film together against the outward push of the moving water that is, against the inertial force of the fluid as it flows radially outward from the impact point.

Except for the bottle cap the best obstacles for my work with water bells were the flat bottom of a 20-milliliter plastic beaker and a curved watch glass about six centimeters in diameter. The beaker was just large enough for me to squeeze in two fingers, so that I could make water sheets uninterrupted by my hands. With the watch glass I had a curved surface to investigate, but my fingers invariably disturbed the water sheets.

I first played with a normal falling stream of water, creating big sheets and bells that ended up to a large extent on my clothes and the kitchen floor. When the flow rate was fairly high, the sheets from the beaker were deflected to the sides with such momentum that they reached the disintegration radius before they had a chance to curve over. With lower impact speeds the sheets did curve before disintegrating. I got horizontal sheets at high speeds and nearly perfect water bells at lower speeds. The watch glass was less successful. The concave side yielded short sheets that were almost immediately spoiled by turbulence. The convex side sent water downward initially and generated better sheets.

To obtain an upward stream of water I attached a rubber hose to the faucet, directed the other end upward and held the rig in place with a clamp attached to a ring stand. The arrangement had the advantage of producing the water sheets near the basin of the sink so that they were less likely to wet me and the kitchen. The beaker made large sheets and bells, depending on how I adjusted the flow rate of the water and how much distance I kept between the open end of the hose and the bottom of the beaker. The concave side of the watch glass gave similar results. The convex side sent the water sheets upward; by adjusting the rate of flow I could vary the shape of the sheet as it curved over at its maximum height and began to fall downward. The turnover was either smooth and gradual or sharp and abrupt. When I put a fork in the upward stream, several water sheets were created, one sheet through each space between the prongs, with all the sheets angled off in different directions. In addition the portion of the stream hitting the concave section of the fork produced part of a water bell. The combined effect was quite lovely.

A good description of water bells was written in 1953 by Frank L. Hopwood, who studied their shapes at different rates of flow. He made bells by directing water through a brass tube that was about 10 centimeters long and 2.1 centimeters in external diameter. A coaxial rod at the top of the tube had fixed to it either a short cylinder or a nut. The water had to pass between the rod and the walls of the tube and then go out through the narrow slit between the top of the tube and the obstacle on top. By screwing either the cylinder or the nut up or down on the rod Hopwood could adjust the opening of the slit.

The tube was mounted vertically in a large basin that was filled with the falling water. An overflow pipe drained the additional water into a nearby sink. The flow of water from the faucet was controlled by a stopcock. With this setup Hopwood's water bells joined with the surface of the water in the basin, giving the bells a closed interior.

Hopwood created either round-top or saucer-top bells by adjusting the height of the slit above the water level in the basin, the width of the slit and the rate of flow. Some of the bells with saucerlike tops had sides that either bent inward or were straight but slanted. In some instances the edge at the point of turnover was quite sharp. At times the edge formed cusps, and Hopwood saw small drops being thrown over the edge. When he made a large bell and then slowly decreased the flow rate of the water, the bell generated a series of these stable shapes (sides either straight or concave) in addition to unstable ones as it shrank. When it had become quite small, it could assume the shape of a cylinder with straight vertical sides.

As Hopwood increased the size of a water bell he punctured the side with his finger and found that the bell immediately doubled in size. Apparently the interior air pressure was lower than the atmospheric pressure outside the bell. If he punctured the side while he was decreasing the flow rate and the bell size, the result was a slight decrease in the size, implying that the pressure conditions were different.

To further investigate the air pressure inside a bell Hopwood slowly blew RR bubbles into it by allowing them to rise through the water in the basin and into its interior. If the bell was initially round on top and gently curved, the extra air pressure inside increased the diameter near the base; the diameter at the top contracted. The sides began to curve inward, a rim rose at the top (giving the top a saucer shape) and eventually a fold or ditch formed along the perimeter near the base. These changes continued until the extra pressure inside the bell was enough to blow air out from under the base. Then the bell shrank to its original shape.

You can achieve similar results by running a flexible straw through the water in the basin and up into the bell. Gently blow air into the bell. You could substitute for the straw a small rubber hose that has a squeeze bulb at one end. With either setup you could also investigate the changes brought about by removing air from the interior of the bell. Does a bell with a round top simply shrink, showing no other change in shape?

Hopwood suggested that two water bells, one inside the other, can be made with two slotted tubes that are on the same axial rod but are attached to independent water supplies. I made two incomplete water bells more easily with my curved watch glass. The hose from my faucet ran to a Y hose connector. One arm of the Y was connected to the hose I had already mounted vertically with a clamp and a ring stand. The other arm was connected to a hose that was similarly mounted on the ring stand but was higher up and pointed downward. I put clamps on each hose so that I could control the amount of water directed to each. When I turned on the water supply, I placed the watch glass in the stream. I could adjust the flow rate from each hose, the distance between orifices and the distance between the obstacle and each orifice.

The top and bottom surfaces of the watch glass each produced a water sheet. Usually the sheets merged at the rim of the watch glass. With some fiddling I could get the concave top to send upward a water sheet that was not affected by the water sheet sent down by the convex bottom. Except for the area where I held the watch glass I had two water bells well separated from each other.

Twisting the watch glass made the two sheets touch and then merge. Separating them again took some work: the surface tension tended to keep them joined. Sometimes I could separate the sheets only in a small area on one side of the watch glass. Then a layer of air would be trapped between the two separated sheets. Nearby there might be large and beautiful distortions of the sheets with gentle curves up and down. I replaced the watch glass with a fork and found similar distortions of the multiple sheets wherever they merged.

The shape of a bell is determined not only by the difference in pressure between the inside and the outside but also by gravity, the viscous flow of the air inside the bell and the pull of surface tension. The effect of surface tension is twofold: a pull up and down along the meridian of the bell and a horizontal. pull along the circumference. If the bell were spherical, the two pulls would be the same, but the different radii of curvature in the circumferential and meridional directions result in different pull.

Whether you investigate water bells and sheets casually or seriously you are sure to have a lot of fun with them. You may want to substitute new shapes for generating the figures or to create water sculptures with multiple and merging water bells. Still another way to making curved water sheets is by letting water pour over an edge, as in a waterfall. A shopping mall in Chicago has a stepped waterfall at least one story high, and each step displays curved water sheets. A shape resembling a bell can also be made by allowing water to pour down a vertical cylinder so that the sheets leaving the bottom of the cylinder join at a point below it.

If you photograph water bells made in your sink, you may find that the camera picks up much turbulence you did not see. The only way to eliminate this drawback is to run the water first through a number of thin, narrow tubes and several sheets of wire gauze.

Water sheets and bells can also be created by having two laminar water jets collide either head on or at a glancing angle.

Under the right conditions the water spreads into a sheet that can be either flat (out to the disintegration radius) or curved. By adjusting the flow rate and the diameter of each stream you can force curvature in either direction: the sheet curves away from the stream that has the greater momentum. The essentially flat sheets appear when the momentum of the streams is the same. At a suitable imbalance in the momentum the curved sheet folds back on a source to form a water bell.

This method of making water sheets has been employed ever since Savart first wrote about water bells. He calculated that the diameter of the sheet or bell is proportional to the pressure of the water creating the streams and is also proportional to the square of the diameters of the two openings through which the streams emerge. Experimental measurements put the disintegration radius just short of Savart's formula.

Extensive studies of colliding streams were done by J. H. Lienhard and J. C. P. Huang in the 1960's. They arranged for the streams to emerge from a large container of water some one foot to two feet deep. The depth could be adjusted by an overflow device that maintained a constant level of water. Two outlets on the side of the container consisted of tubes perpendicular to its wall. Precisely drilled holes were made in the projecting tubes so that when the container was filled with water, streams emerged from the holes toward each other. The arrangement was stable and had the advantage of making the speed of the streams easy to control by adjusting the height of the water in the container. The holes were also easily replaced with other holes of different diameters.

The water sheets can be made more readily but with far less control with my setup in which the water from a faucet is run through a Y connector and then the two streams are directed toward each other. I have little control over the diameter of a stream except by replacing the hose, but I can control the rate of flow by means of hose clamps. With this system water bells are easy to make.

The aspect of water sheets that most interests modern investigators is how the sheets disintegrate at their edges. The most intensive work on the subject was done by Lienhard and Huang on sheets they made with colliding streams. Some components of the disintegration can be seen clearly only with high-speed photographs or motion pictures, but others are visible in my homemade water sheets.

Lienhard and Huang investigated the disintegration in terms of a factor called the Weber number, which is a dimensionless number employed in studies of fluids. It is calculated by multiplying three quantities-the density and the diameter of the fluid stream and the square of its speed-and dividing the result by the surface tension. In general only the speed and the diameter can be varied. with my kitchen water sheets the speed is varied by the water valve, and since a falling stream narrows, a diameter can be chosen at any point along the stream.

At low Weber numbers (values ranging from 100 to 500) the sheets are stable and have a nearly circular perimeter, with beads of water forming along the perimeter. High-speed photography reveals that tiny beads form on the edge and move along the perimeter, coalescing into larger beads that finally break away as drops. You can see the larger beads and the drops without any photographic aid, but the motion of the beads along the perimeter is too rapid to be perceived.

At somewhat larger Weber numbers (500 to 3,000) the perimeter of the sheets forms noticeable cusps that result from curved waves initiated near the impact point of the streams. The waves propagate outward over the sheet to set up the embroidery pattern along the perimeter. Sheets with the largest stable radii are in the range of Weber numbers from about 1,000 to 2,000. At the top end of the range the cusps diminish and the perimeter is again circular. Under the same conditions large waves can be seen moving radially outward from the collision point to the edge of the sheet. The waves are termed antisymmetrical because the fluid on opposite sides of the sheets moves in opposite directions, so that at some places along the wave the sheet thins and at other places it thickens. I can improve the visibility of these waves and the ones forming the cusps at lower Weber numbers by shining a high-frequency stroboscopic light on the water sheets. The antisymmetrical waves become larger in amplitude at still larger Weber numbers. By then the breakup of the edge is readily apparent because the sheet begins to flap like a flag in the wind.

With my simple sink setup for making streams collide I could change the angle of the streams to get a more glancing collision. As the two streams were angled closer together on one side the sheet formed on that side became steadily

smaller because progressively more of the water was being thrown to the other side by the collision. When the angle between the two streams was reduced to 60 degrees or less, the sheet formed between them became quite short and the sheet on the other side grew considerably. The longer sheet was shaped like a leaf, being pointed at the far tip.

Returning to the water sheets produced by a stream directed onto an obstacle, I followed some work conducted by Sir Geoffrey Taylor (who has written several important papers on the dynamics of water sheets) by introducing a perturbation on the obstacle. With a knife I made a radial ridge on the bottom of one of my small plastic beakers. When the beaker is made to create water sheets in the middle range of Weber numbers, the ridge sends out curved waves similar to the ones that make the cusped perimeters in that range.

You might like to continue examining water sheets formed by obstacles or the collisions of streams. If you want to correlate the speed of the fluid with a sheet or a sheet's breakup mechanism, you will need to construct a fluid supply with a controlled pressure, as Lienhard and Huang have done. You may also want to investigate fluids other than water. Some of the non-Newtonian fluids I discussed in this department for November, 1978, may be particularly interesting, although you will have to avoid the ones with a viscosity so large that it prevents a flow adequate to produce a fluid sheet.

Roughly seven years ago Elizabeth Wood (known for her work with crystals and for her enchanting book Science for the Airplane Passenger) told me of another common sink phenomenon. With the water from the faucet flowing fairly slowly, hold the flat of a knife in the stream near the faucet. You may have to adjust the distance or the flow rate to see the effect, but usually the short length of the stream between the faucet and the knife exhibits ripples. The wavelength of the ripples depends on the speed of the water and the length of the falling stream.

For all seven years I occasionally searched for an explanation of the ripples. The answer finally came as Lienhard and I were discussing water bells for this article. He made an offhand remark about the effect and then pointed out that he had published the explanation in 1968 in the paper listed in the bibliography for this issue [below]. The explanation stemmed from work Lord Rayleigh had done in 1878 and 1879 on waves propagating along a liquid cylinder. What you see in the falling stream are the waves generated by the impact of the stream on the knife. They move upward along the water stream with the same speed at which the stream falls, and the result is that they look stationary to you.

To investigate these waves Lienhard arranged for a can full of water to leak through a small central hole onto the surface of water in a full martini glass. The diameter of the hole was .026 inch. The distance the water fell was both short and constant, but because the water level in the can slowly decreased, the pressure on the hole and the speed of the water also slowly decreased.

As the speed of the water draining from the can diminished, the correlation between the speed of the stream and the speed of the waves meant that the wavelength of the ripples had to increase. The wavelength finally reached an upper limit. Rayleigh showed that the water column would be unstable for any wave with a length exceeding the circumference of the column. The longest wavelength that can result in a stable wave in the falling stream is therefore equal to the circumference of the stream. When the speed of the water is low enough to result in instability, the ripples sometimes break up into droplets, sending some of them skimming across the surface of the water in the martini glass. (How such drops float above the bulk liquid was discussed in this department for June, 1978.) The greatest instability is in a wave with a wavelength about 4.5 times the diameter of the stream. It is therefore a larger wavelength than the largest one that gives a stable wave. Such a wave generates swellings in the stream that quickly grow larger until the stream breaks up into drops.

If you have access to a camera with a closeup lens, you can photograph the rippled stream as water slowly drains from a can. The wavelength of the ripples should be approximately inversely proportional to the distance between the water level in the can and the water level in the martini glass. With several closeup photographs you can measure the distance between adjacent ripples and plot their separation against the distance between the water levels. At the very least it is a good way to spend a hot summer afternoon.

Bibliography

WATER BELLS. F. L. Hopwood in The Proceedings of the Physical Society, Section B, Vol. 65, Part 1, No. 385B, pages 2-5; January 1, 1952.

CAPILLARY ACTION IN SMALL JETS IMPINGING ON LIQUID SURFACES. John H. Lienhard in Transactions of the American Society of Mechanical Engineers, Series D, Journal of Basic Engineering, Vol. 90, pages 137-138; March,1968.

THE BREAK-UP OF AXISYMMETRIC LIQUID SHEETS. J. C. P. Huang in Journal of Fluid Mechanics, Vol. 43, Part 2, pages 305-319; August 28, 1970.

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