AIR BUBLES RISING through a viscous fluid sometimes take on curious shapes, combining and interacting in perplexing ways. Such behavior often goes unnoticed because it is hard to observe the bubbles in an ordinary container, particularly when the fluid is opaque. One way to enhance the visibility and at the same time somewhat reduce the complexity of the various goings-on is to place the fluid in a Hele-Shaw cell, a device named for Henry S. Hele-Shaw, the English engineer who devised it around the turn of the century.

The cell consists primarily of two transparent plates separated by a narrow gap. A thin spacer runs along the internal edges of the plates to maintain their separation and keep the fluid from leaking out. Air bubbles are introduced into the cell through a port along one of the edges. The fluid can be pushed or pulled through the cell by a pump connected to other ports. Alternatively, the cell can simply be propped up at a slant or mounted vertically so that gravity and buoyancy move the fluid and the bubbles.

The most obvious advantage of a Hele-Shaw cell is that the bubbles are always visible. Even if the fluid in bulk is opaque, a thin layer of it is transparent because it hardly absorbs any light. Moreover, the flow is effectively two-dimensional and so is easier to analyze than when it is three-dimensional in a wider container.

Hele-Shaw cells have been used extensively for investigating various features of bubble formation and fluid flow, but it was not until recently that some rather curious features of bubbles were noticed. In 1986 Tony Maxworthy of the University. of Southern California reported that when a series of bubbles were released individually in a viscous silicone oil, they tended to queue and then move up through the fluid like a wagon train. Often a "bubble stack," as Maxworthy called it, became unstable, the lead bubble splitting down its center. As the stack continued to ascend, the instability and splitting progressed down through the length of the stack, bubble by bubble, until there were two separate stacks.

In additional research Maxworthy and several other investigators studied the shapes air bubbles assume when they move through glycerin and other viscous fluids in a Hele-Shaw cell. You might think that surface tension would keep a bubble always circular or slightly oval, but actually a number of odd shapes happen to be stable.

I wondered if the bubble stacks and odd bubble shapes might be attainable in a homemade Hele-Shaw cell. From a local store specializing in plastic supplies I purchased two square acrylic plates that measured 15 inches on a side and were 3/4 inch thick. To separate the plates I used narrow, thin strips of Styrofoam that are sold in hardware stores for weatherproofing windows. The strips come with a sticky side that adheres well to the acrylic. I applied strips along three edges on the face of one plate. Along the fourth edge I applied two shorter strips so that a wide central hole was left between them to serve as an air vent.

I next laid the plate on paper towels and poured or squirted a fluid onto the surface inside the strips to cover about half of the area. Then I placed the second plate on the first one, taking care to align their edges. I squeezed the plates by tightening four C-clamps along their left and right sides, one near each corner. The plates were then separated by less than a millimeter, but the width varied across the cell because of the pressure of the fluid. The fluid filled most of the cell.

I planned to let gravity drive air bubbles through the fluid, and so I needed to mount the cell upright with the air vent at the top. Because some of the fluids I employed leaked slowly through the bottom Styrofoam strip, I worked over a sink. To support the cell I fashioned a bridge from four metersticks. They were taped in pairs with wide plastic tape, and the pairs were laid across the sink with a small separation between them. I balanced the cell on top of the two Y beam bridge, with the cell's bottom Styrofoam strip centered on the gap between the beams [see Figure 1]. On the other side of the sink I put a flood lamp clamped to a laboratory stand. To diffuse its light I suspended a wide sheet of white paper between the lamp and the cell. When I placed the cell on the bridge, often some air bubbles were already in the fluid, having been trapped when I clamped the plates. Each bubble was outlined with a distinct dark line, whereas the regions within the bubble and elsewhere in the cell were uniformly lit. The dark line was caused by refraction of the light as it passed through the shallow, curved surface of the bubble that straddled the gap between the plates. The refraction sent the light rays off to one side of my field of view, which left the border darker than the other regions. The bubbles were so stark that they resembled a penciled sketch.

To make additional bubbles I passed the needle of a syringe through the gap between the bridge beams and then carefully forced it up through the bottom Styrofoam strip in the cell. The syringe was filled with air, and pushing on the plunger injected a small amount of the air into the fluid, where it formed a bubble. The bubble gradually rose away from the syringe and traveled up through the cell. Injecting a large bubble was easy, but producing small ones was more difficult: the resistance of the fluid required a strong push on the plunger before the fluid yielded-with a sudden "give" that often produced a bubble larger than I wanted. I found that if I pulled back on the plunger before the bubble left the tip of the syringe, I could remove some of the air and thereby shrink the bubble.

To photograph the bubbles I magnified them by mounting on my 35-millimeter camera three lenses from a close-up kit. During the photography I switched off the room lights so that only the diffused light from the flood lamp illuminated the cell. The viscosity of the fluid was usually high enough, and the confines of the cell narrow enough, so that the bubbles moved sedately, allowing me to bring them into focus and take a photograph before they shifted appreciably. (The slow migration of the bubbles also meant that each test of a fluid in the cell required at least an hour of observation and sometimes much more.) I had some film that was colorbalanced for the flood lamp, but often I used film that was easier to find but was not balanced; it yielded photographs with an unnatural tint, as may be evident in some of the accompanying illustrations.

For the tests I collected a variety of fluids. I found glycerin in a drugstore and then explored the shelves there and in several food stores for other appropriate materials. When a fluid was packaged in a transparent container, I inverted the container and watched as an air bubble rose through the fluid. If the bubble moved slowly, that meant the fluid was viscous, and I added it to my collection.

Before describing my attempts to replicate Maxworthy's bubble stacks, let me say a bit more about what he observed with silicone oil. When his bubbles overtook one another to form a stack, each bubble was separated from its neighbor by a thin membrane of oil. Occasionally a membrane failed: two bubbles coalesced, and their sides wrestled into a new, wider bubble.

If the stack did not coalesce fully, it sometimes became unstable along the topmost bubble's leading edge, which is normally convex upward. Once the instability began to develop, the edge straightened and then began to sink into the bubble. The instability arose from the fact that a dense fluid (the oil) overlay a less dense fluid (the air) along the edge. If surface tension along the edge kept the edge upwardly convex for a time, it kept the oil from sinking into the bubble. As the bubble advanced through the oil, however, tiny waves played along the leading edge. Sometimes one of the waves grew large enough to upset the stable curvature of the edge, and then the edge began to collapse. A collapse could also be initiated if the edge happened on a small bubble that was almost stationary. As the edge approached the small bubble, the fluid pressure between them increased and could become large enough to initiate a collapse.

If the collapse extended to the trailing edge of the top bubble, the bubble split into two parts. The collapse and splitting then marched through the rest of the stack, separating the bubbles into two stacks. The larger stack experienced more buoyancy and rose faster than the smaller one. As the tail of the larger stack passed the head of the smaller one, the latter was entrained and forced to join up at the tail of the larger stack. One time Maxworthy saw a stack in which the collapse and splitting began at the upper edge of the second bubble in the stack. The splitting moved down through the stack, but the two stacks it produced were held side by side because of their attachment to the unaffected topmost bubble.

I went through my collection of fluids to see if I could generate bubble stacks, hoping also to see the tendency toward splitting. Air bubbles in com syrup, glycerin and several shampoos failed to form stacks. The bubbles did encroach on one another, but within seconds after two bubbles met, the thin membrane between them popped and the bubbles coalesced.

I had better luck with some other shampoos. The best was a product called Shower Gel, which consists of propylene glycol, glycerin and soluble collagen, along with several other ingredients. The series of photographs in Figure 2 shows how one stack behaved. The top bubble in the complex stack was initially unstable but then restabilized on the left side of the stack. Meanwhile, the instability passed to the second bubble, which collapsed and split. The bifurcation of the bubbles then traveled down through the stack until finally a stack of two bubbles remained on the right side. The longer stack on the left side then began to split at the top. As it rose, the short stack was pulled into its tail and finally underwent collapse and splitting. As I watched this play of bubbles in the partial darkness of the room, I thought of a band of trilobites playing a game of tag.

Individual large bubbles also displayed instability, sometimes collapsing without any visible provocation. At other times a collapse was initiated by a small bubble that a large bubble happened to meet during its ascent. Usually when a stack encountered a small bubble, the stack would push the bubble off to one side and slide past it. The small bubble would then force an indentation in the side of the stack closest to it. Occasionally the indentation was so extensive that the side of the stack collapsed, and its bubbles began to split.

A stack tends to avoid small, approximately stationary bubbles, but it is attracted to somewhat larger and more mobile bubbles. In several cases I watched a large stack deviate from its upward path to overtake individual bubbles off to one side; the stack would actually pick up speed and chase after the other bubbles. During the chase some bubbles in the stack extended and thinned, their left and right sides becoming straight or even sinking inward.

To make the wakes of individual bubbles more apparent, I opened up the cell, squirted common food coloring in a band across the Shower Gel and then refastened the cell. The procedure introduced air bubbles in the cell. When they rose up through the food coloring, their wakes became visible. When a bubble began in the clear gel, it left a clear trail as it passed through the colored region; when a bubble began in the colored region, it left a colored trail as it traveled through the clear region; the photograph at the left shows this effect. Usually the length of one of these trails was several times the diameter of the bubble that created it.

I believe the trails are left by fluid that is slowly discharged from thin layers next to the bubble. Consider a vertical section that extends from plate to plate and through a bubble [see Figure 4]. The air in the bubble is separated from each plate by an intermediate layer of fluid. The layers are so thin that during the bubble's ascent the fluid can pass through them only slowly. As fresh fluid enters the top part of each layer, the fluid in the bottom part of the layer is discharged below the bubble Now consider a vertical section that is parallel to the plates and extends through the bubble. As the fluid passes the left and right sides of the ascending bubble, it converges just below the bubble. The convergence compresses the fluid that is discharged from the thin layers.

Suppose that the bubble travels from the clear gel into the colored gel. The fluid coming around the left and right sides is then colored, but the gradual discharge from the thin layers remains clear for a while. The flow of colored gel into the region below the bubble compresses the discharge into a thin clear trail that lies along the path taken by the bubble. A similar story accounts for the colored trail a bubble leaves when it travels from colored gel into clear gel.

Several investigators have found that an air bubble traveling through a viscous fluid can take on slightly different shapes depending on circumstances. The tug-of-war between surface tension, buoyancy and fluid pressure tends to keep a bubble circular or elliptical, but it can instead be flattened on both the leading and trailing edges or be shaped like a pear, with the leading edge noticeably wider than the trailing edge. The flat and pear shapes are typically seen when the cell is horizontal and the fluid is forced through it.

Last year Anne R. Kopf-Sill and George M. Homsy of Stanford University reviewed the theory behind these bubble shapes and added observations of new shapes that were even curiouser. They found bubbles that had tails-some of them short and rounded and others much longer than the diameter of the main part of the bubble, with a small circle at the tip. Kopf-Sill and Homsy also were the first to sight a shape, which had been predicted from theory, in which the trailing edge is flat and the leading edge is slightly sunken.

I tested the fluids in my collection, looking for the various shapes that had been reported. Very small bubbles in glycerin, pancake syrup, corn oil, Shower Gel and two other shampoos called Agree and Prell were circular. Somewhat larger bubbles were ellipses whose long axis was vertical.

In one case, working with Prell, I spotted what might have been an example of a bubble with a sunken leading edge, but my attention had been focused on other bubbles at the time, and so I cannot be sure that the bubble had not simply been stretched to one side and distorted by a larger bubble that had passed it. Large bubbles in syrup, corn oil and glycerin often developed short tails once they got going. The tails were persistent but wiggled slightly. Strung across them was a faint border that seemed to complete the generally oval boundary of the bubble.

I concentrated on the bubbles in glycerin. When one bubble overtook another, the membrane between them lasted only seconds before the air in the bubbles broke through. What remained in the breached region were shallow, curved ridges of glycerin that clung to both plates. As the newly formed bubble ascended, the ridges descended and their left and right ends distorted the bubble noticeably, rendering it pear-shaped.

Sometimes a tiny spot of glycerin appeared inside a bubble. It may have been caught up on an imperfection in one of the plates, or it could have been clinging to some contamination in the cell. When the spot reached the lower edge of the bubble, it shoved the edge outward. As the tail of the bubble neared the height of the spot, the tail extended over to it, as if it were being wagged.

Although I usually worked with the cell standing upright, I sometimes wanted to slow the ascent of the bubbles by tilting the cell. To do so, I laid a large sheet of white paper on a tabletop, placed the lower end of the cell on the paper and propped the upper end up on two boxes. I angled the lamp so that its light scattered from the paper up through the cell. The lower end of the cell was at the very edge of the table, and so I could still inject it with air with a syringe.

If you build your own Hele-Shaw cell, you might test other viscous fluids such as honey, an ungelled gelatin dessert or motor oils. (A warning about the particularly viscous motor oils: they not only make a mess but also are extremely difficult to clean out of the cell when you have finished with them.) If you cannot locate a syringe for your work, you can entrap air bubbles in a fluid simply by lifting up the top plate and then clamping it down again before the fluid has a chance to level out.

Bibliography

BUBBLE FORMATION, MOTION AND INTER-ACTION IN A HELE-SHAW CELL. T. Maxworthy in Journal of Fluid Mechanics, Vol. 173, pages 95-114; December, 1986.

STABILITY OF BUBBLES IN A HELE-SHAW CELL. S. Tanveer and P. G. Saffman in Physics of Fluids, Vol. 30, No. 9, pages 2624-2635; September, 1987.

THE AMATEUR SCIENTIST. Jearl Walker in Scientific American, Vol. 257, No. 5, pages 134-138; November, 1987.

BUBBLE MOTION IN A HELE-SHAW CELL. Anne R. Kopf-Sill and G. M. Homsy in Physics of Fluids, Vol. 31, No.1, pages 18-26; January, 1988.

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