MONITORING THE FLOW OF A fluid in three dimensions is difficult because the motion can be quite complex and one's view of the flow's interior is often obscured. Just before the turn of the century the English engineer Henry S. Hele-Shaw devised a way to simplify the flow by confining it to a thin layer. A Hele-Shaw cell consists of two parallel transparent plates separated by a narrow gap. Spacers between the plates keep the fluid in the cell. With this apparatus one can view and photograph a fluid that is either trapped in the cell or made to flow through it from end to end.

For half a century the device was merely a convenient way of observing fluid flow. Then, in about 1956, Sir Geoffrey Taylor, a British physicist renowned for his work in the physics of fluids, recognized that the cell could serve to model a flow problem important to the oil industry. In some oil fields oil is trapped in a porous material and hence cannot be pumped out directly, as a pool of oil can. One solution is to pump water through the material and so drive the oil to form a pool, from which it can be recovered. Taylor saw the problem on a grander scale. How do two fluids behave when one of them (such as water) with a low viscosity drives another (such as oil) that has a greater viscosity? In his analysis Taylor found that the interface between the two fluids can become unstable, generating wavelike shapes.

When the unstable nature of the interface was verified, flows within a Hele-Shaw cell drew fresh attention. Checking

various fluids, investigators discovered that the unstable interfaces give rise to many surprising shapes. Research done in the past two years has shown that under certain circumstances an interface develops a fractal appearance. The term fractal refers to a type of symmetry in which a pattern found at one magnification is present at other magnifications as well; such a pattern is said to be self-similar. Although the patterns seen in Hele-Shaw cells are fractal only for a limited range of magnifications, they have generated much interest in the study of unstable fluid interfaces.

I set out to reproduce some of the research by constructing a simple Hele-Shaw cell. Although the cell is often made from plate glass, I chose acrylic plates to make it easier to drill holes in the walls of the cell. The square plates, measuring 15 inches on a side and three-quarters of an inch in thickness, were bought from a local plastic supply house for about $40. Thinner (and less expensive) plates would serve, but a thickness of less than half an inch would probably allow the plates to buckle when they are mounted in the cell.

In some experiments I wanted to inject a fluid into the center of the cell. To accommodate the needle of a syringe I drilled a hole through the acrylic plate that was to be the top of the cell. The hole was an eighth of an inch in diameter. To facilitate the injection I also bent the tip of one of the needles so that it would slip slightly under the top plate.

I experimented with several types of spacers. The best spacers proved to be the narrow, thin strips of foam rubber that are sold for weatherproofiing windows. I chose strips tha were a quarter of an inch wide. After cutting the strips to fit the perimeter of the bottom plate, I stuck them in place with their sticky side down.

At one end of the cell I cut several narrow openings in the foam-rubber strip so that fluid could escape if the cell was full when additional fluid had to be added during an experiment. The plates were held together with six or more small C-clamps. To prepare for an experiment I removed the clamps and the top plate and then poured a liquid onto the bottom plate. If I wanted the entire cell to be full during the experiment, I poured enough liquid so that about half of the bottom plate was covered. When I added the top plate and squeezed the plates together with the clamps, the liquid would fill the cell, with any extra leaking through the holes in the spacer.

To avoid bubbles I usually poured the liquid onto the half of the bottom plate nearest to me and then laid the top plate down carefully, beginning at my side of the cell. Bubbles were nonetheless sometimes still present when the clamps were tightened. To remove them I stuffed a paper towel into the central hole and stood the cell on end. After the bubbles had migrated to the top of the cell, most of them popped when they reached the openings in the spacer.

When the clamps were fully tightened, the plates were separated by about a millimeter. The spacing is crucial to some of the experiments because the size of the patterns developed by the interface depends on it. Some of the patterns also depend on the left-to-right width of the cell: they may not appear if the cell is too narrow. This can present a problem in homemade cells, because you want the cell to be wide enough to promote the patterns but not so wide that the plates buckle.

To photograph the patterns I positioned a flood lamp to shine upward between two tables of equal height. Across the gap between the tables I taped a sheet of white drawing paper to diffuse the light and to protect the lamp from any spilled liquid. After filling a cell I placed it above the paper so that it rested, on its clamps, on the tables.

I first investigated experiments that were reported in 1958 by Taylor and Philip G. Saffman of the University of Cambridge. Imagine a Hele-Shaw cell filled on one side with a fluid of low viscosity and on the other side with a fluid of high viscosity; there is a straight interface between them. If the low-viscosity fluid is forced against the high-viscosity one so that the interface moves slowly, the interface remains straight. If the speed exceeds a critical value, however, the interface grows unstable, breaking up into wavelike shapes that become more pronounced with time. The interface soon resembles a glove: fingers of the low-viscosity fluid extend into the high-viscosity fluid. The instability and the growth of fingers stem from the difference between the viscosities of the two fluids. Interfacial tension (or surface tension, if one of the fluids is air) along the interface opposes the instability, because it attempts to reduce the surface area of the interface. Still, if the interface moves fast enough, the fingers form and grow.

Saffman demonstrated the appearance of fingers with a vertical cell in which air overlay glycerin. The. air served as the low-viscosity fluid. When additional air was forced into the top of the cell, glycerin was forced out of the bottom. The speed of the air-glycerin interface was high enough so that the interface developed waves, which grew into fingers of air stretching down into the glycerin. Surprisingly, one finger soon dominated the pattern, with the other fingers quickly becoming frozen in length. The width of the dominant finger eventually was half the width of the cell.

I could replicate Saffman's experiment only crudely. To do so I filled part of my Hele-Shaw cell with corn oil and then stood it on end to drain the oil to the bottom. When the oil. had settled, I laid the cell down on the table and partially opened the middle clamp on the side nearest me. The resulting sudden expansion of the space between the plates forced the oil to flow toward me, breaking up the oil-air interface into a series of short fingers. I also produced fingers of air in another experiment. With the oil placed in the cell so that it surrounded the central hole, I inserted 11' the tip of an eye-dropper tube into the hole and drew oil rapidly out of the cell into the tube. When the perimeter of the pool of oil raced toward the hole, fingers of air formed.

Earlier this year Tony Maxworthy of the University of Southern California reported a related experiment. He partially filled a cell with silicone oil, rotating the cell to wet the entire interior. Then he stood the cell on end. When the oil had drained to the bottom and the oil-air interface was straight,. he turned the cell over so that the oil was above the air. In some trials the final orientation of the cell was vertical; in others the cell was at various angles to the horizontal.

According to Taylor's analysis, when the more viscous fluid is driven into the less viscous one, the interface is unstable only if it moves at a speed lower than a certain critical value. For each angle Maxworthy tried, the interface should move fast enough to be stable, and yet in each trial fingers of air reached up into the descending oil. Apparently when the cell was initially rotated into position for a trial, the interface began moving at a low speed, allowing the fingers to form. Once the speed exceeded the critical value, the stabilization of the interface tended to promote the growth of a few of the fingers while arresting the growth of the others. Eventually one finger dominated.

When the cell was tilted from the horizontal by only a few degrees, the spacing between the fingers was wide and the dominant finger developed only slowly. When the cell was vertical, the spacing between fingers was narrower and the dominant finger grew rapidly; in addition the fingers split into smaller fingers, a process known as bifurcation. The splitting is due to the high speed of the interface when the cell is vertical and to the narrowness of the gap between the plates.

The experiment is easy to do with a homemade cell. After placing the cell on end to drain corn oil to the bottom of the cell (the central hole was plugged), I inverted it. Waves and then fingers formed quickly. When the fingers of air began to bifurcate, I laid the cell flat and photographed the fingers.

I repeated the experiment with other liquids. One liquid was Slime, a gel sold in toy stores. Slime is a member of a peculiar class of liquids that is designated non-Newtonian. The viscosity of such liquids changes when the liquid is sheared or stressed; in Slime's case the viscosity increases. To the Slime I added an equal volume of water; I stirred the mixture and partially filled the cell with it. When I inverted the cell, the interface quickly developed fingers of air that were interspaced with narrow fingers of the mixture. The head of each liquid finger was a blob wider than the rest of the finger. Thick molasses yielded similar liquid fingers, except that the blobs were more pronounced and the fingers tended to pinch off after a while, releasing the blobs.

In another series of experiments I injected dyed water into various liquids in the cell. The liquids included glycerin, corn oil and molasses. The dye was methyl violet, which I mixed with the water. I filled the syringe with the dyed water, taking care not to draw air into the instrument. Then I injected the dyed water into the central hole (using the bent needle) or through the foam spacer along the perimeter (using a straight needle).

When the dyed water is injected centrally, it spreads quickly to form a beautiful pattern with lobes that bifurcate repeatedly; when the water is injected through the side of the cell, it forms a pattern that is similar but more like a fern. Lincoln Paterson of Geomechanics in Victoria, Australia, has recently studied the structures that form when dyed water is injected into glycerin. He found that the size of the smallest features on the lobes is approximately four times the spacing between the plates of the cell. The dependence is due to the nature of the waves that develop on the interface between the water and the glycerin. The most unstable wave, which dominates the production of the lobes, has a wavelength that is about four times the spacing between the plates. Water and glycerin are miscible, but their diffusion into each other is sufficiently slow so that the patterns form and then last long enough to be photographed.

My best patterns appeared when I stood the cell on one end and injected the dyed water through the foam spacer at the other end. After the water spreads, some of it falls through the glycerin, undergoing numerous bifurcations.

Paterson also studied what happens when air is injected into glycerin from a central hole. In this case the fluids (air and glycerin) are immiscible. According to theory, an interface between immiscible fluids is unstable only if the wavelengths of the waves at the interface exceed a lower limit. Initially the air bubble is a circle whose circumference is too small to support any wave with a wavelength greater than the lower limit. Only when the bubble grows larger does the interface become unstable. Then it blossoms into a flowerlike pattern with bifurcating lobes. I was able to produce such patterns by inserting the eye-dropper tube into the central hole of my cell and blowing air through it. The resulting patterns are difficult to photograph in glycerin, which is clear; they are more pronounced in colored molasses or various dark syrups.

In 1985 Johann Nittmann and Gerard Daccord of Dowell Schlumberger in France and H. Eugene Stanley of Boston University reported that they had produced fractal-like fingers by injecting dyed water into an aqueous polymer solution. The fingers resembled the limbs of a bare tree, with short, narrow stems that jutted from main stems. The polymer solution was a polysaccharide with a high molecular mass. Since the solution was aqueous, there was essentially no interfacial tension between the solution and the injected water. Without that stabilizing factor the interface underwent abrupt changes in direction, giving rise to the fractal pattern.

Polysaccharides include cellulose, glycogen, chitin and starch. Attempting to duplicate the fractal fingers, I investigated a solution of water and common cornstarch. The solution is non-Newtonian, its viscosity increasing under stress. When the solution was thick, I was not able to inject the dyed water into it. I was more successful with moderate cornstarch concentrations: the dyed water created fernlike patterns that had the same thick branches and wide bifurcations I had seen in previous experiments. As I diluted the cornstarch solution more, the patterns became less interesting, until finally there was a simple ellipse. Various other starches yielded similar results.

I fared better with a mixture of water and cellulose. (The cellulose is available from biological-supply houses.) When the concentration of cellulose is moderate, the injected dyed water forms fingers with abrupt turns-something like the patterns Nittmann, Daccord and Stanley photographed but not as narrow.

Casting about my kitchen for other materials, I decided to test a common gelatin dessert Jell-O brand). After it gelled, I spooned a small amount into the cell and gradually clamped the plates together, allowing the gelatin to ooze between them. Then I injected dyed water through one edge. After a momentary hesitation, the water burst through the gelatin, creating an elegant fractal pattern.

I have only touched on the possibilities of the research that can be carried out with a Hele-Shaw cell. You may want to try your hand with other liquids. Perhaps you will find other common substances that yield fractal patterns. Since the field is far from having been fully explored, you may find entirely new patterns.

Bibliography

FINGERING WITH MISCIBLE FLUIDS IN A HELE SHAW CELL. Lincoln Paterson in The Physics of Fluids, Vol. 28, No. 1, pages 26-30; January, 1985.

FRACTAL GROWTH OF VISCOUS FINGERS: QUANTITATIVE CHARACTERIZATION OF A FLUID INSTABILITY PHENOMENON. Johann Nittmann, Gerard Daccord and H. Eugene Stanley in Nature, Vol. 314, No. 6007, pages 141-144; March 14, 1985.

THE NONLINEAR GROWTH OF A GRAVTATIONALLY UNSTABLE INTERFACE IN A HELE-SHAW CELL. T. Maxworthy in Journal of Fluid Mechanics, Vol. 177, pages 207-232; April, 1987.

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