Take a bowl of clear viscous oil. Poke a hole in its bottom so the oil can drip slowly out, drop by drop by drop. Point a high-speed video lens at it, film 12,000 frames per second, and you'll get something like Linda Smolka's movie.

"It's kind of cool," she said, showing me a seconds-long video clip that seemed to define the concept *boing!*

She slowed it down until the steps became apparent: the bulge of oil out the hole, the inexorable lengthening of a bulbous filament, the sudden extrusion of a second, even finer thread.

"In the beginning the drop just hangs there, then the thread gets longer, the drop becomes more spherical. It pinches off at the drop first, then at the top of the thread," she explained.

Those actions leave a bit of the oily filament abandoned, strung out between orifice and drop, unattached to either. It's this detached bit that goes *boing!*, collapsing into itself to form a second , smaller drop.

A graduate student in mathematics, Smolka is studying the equations that describe the complicated creation of such drops.

"Think of an ink-jet printer," she explained when I visited her lab. "In the 1970s, when IBM came out with their first ink-jet printers, the problem their engineers had was to control a drop of ink —to make it go from the orifice to the paper without splattering, without making satellite drops. So they did a bunch of studies, which are proprietary, not public knowledge, and the printers we have today come from their knowledge of drops."

Another example is fiber spinning, such as the making of Kevlar or Lycra. These materials are made from fluids that develop into fibers. It's the opposite of the drop problem: Here, you don't want the threads to break.

Pesticide sprayers also need to control their drops: too small and the chemical will diffuse into the air instead of falling to the crop.

"I found out another great application at a mathematical modelling conference," Smolka continued. "There was a problem at Kodak with how they make their film, how they apply the emulsions to the hard backing of the film. They have a big vat with a rectangular slit cut out of the bottom so a sheet of fluid flows out —it's like a water fountain with water flowing over a wall. It's called curtain coating. But they could never get a totally flat sheet. It curled up at the ends. Surface tension was balli ng up the fluid, and Kodak didn't want that."

Knowing the math behind how a drop forms will help engineers to design better ways of controlling such flows, of directing how —or if —a viscous fluid like oil or ink will disintegrate into drops.

The William G. Pritchard Fluid Mechanics Laboratory in Penn State's math department is a high-ceilinged room fashioned out of the McAllister Building's lobby, its odd elongated shape filled with the ordinary lab paraphernalia of equipment benches, computer terminals, and electronics racks. Then there's the wave tank, like a 45-foot-long aquarium, its edges bright yellow, running thng length of the outside wall. As Smolka and I talked last April, Diane Henderson, associate professor of mathematics, was strenuously cleaning it in preparation for her next experiment. I was reminded of the first time I came to the Pritchard Lab, in 1988, and found Bill Pritchard dipping a wire loop into a bucket of soapy water to form dodecahedral soap bubbles: These are not paper-and-pencil mathematicians.

*Mathematics is about developing new ideas, about looking for theoretical structures in the world*, Pritchard had said then. The point of the lab he established with colleagues Jerry Bona and Ridgway Scott was to build bridges from mathematics to the rest of the world: to understand Nature through the applications of mathematics. The lab focused on the "continuous media" —water, oil, air —media whose motion is described by equations in which the variables are continuous. Although these equations, called the Navier-Stokes equations, are more than a century old, Pritchard explained, "there's very little we can say about the solutions to those equations." Approaching old problems in a new way, Pritchard hoped to develop simpler equations that could be solved.

"You say, I think what's happening is the following mechanism," he explained. "You're trying to isolate the essential mechanism, to identify that abstract, critical thing that controls Nature." After Pritchard died in 1994, at the age of 52, the lab was renamed in his honor.

Henderson joined Penn State in 1991, having earned a doctorate in physical oceanography from the Scripps Institution at the University of California, San Diego. Her mathematical research focuses on the equations that describe ocean waves: work that has practical applications as far-ranging as affecting the interpretation of satellite data from the ocean surface to explaining how an oil spill will spread.

Smolka's background is in civil engineering. After college, at Princeton, she consulted for a computer firm, worked on bridge projects in New York, and taught math at a private school before deciding to go on to graduate school. Learning of her interest in the practical applications of mathematics, a mathematician at the University of Colorado at Boulder suggested she see Henderson at Penn State. "Diane told me about this lab," Smolka remembered. "She wanted somebody to pick up on the drops project." Smolka enrolled in the mathematics department in 1995.

"On the Pinch-Off of a Pendant Drop" was not a project that fit into Henderson's wave research, but one that she and Pritchard had begun before he died. As he had mentioned in 1988, he was interested in "problems an engineer knows can't be solved." Drops easily qualified.

Although the Navier-Stokes equations, the mathematics describing a drop —and a jet, a wave, and many other forms of fluid flow —date from the 1880s, it wasn't until the invention of strobe photography by Harold Edgerton in the 1930s that scientists could actually see a drop form and begin to break down what happens when the drop pinches off from the thinning flow. Pritchard and Henderson thought a new imaging method —high-speed digital video —could give a similar boost to the field in the 1990s, since it would allow them to both see and measure the advent of a drop.

While they were devising their experimental setup, a new theory of drops was published in *Physical Review Letters* in 1993 by a young scholar, Jens Eggers, then at the University of Chicago. Eggers had taken the Navier-Stokes equations, simplified them by making various assumptions based on observation, and claimed to have found a solution to how a drop pinches off.

"Those equations allowed him to predict, for instance," Smolka explained, "the velocities of the two tips as they retreat once the drop itself has pinched off" —in essence, to quantify the *boing!*

His work inspired physicist Sidney Nagel and his colleagues at the University of Chicago, X.D. Shi and Michael Brenner, who tested Eggers's equations with computer simulations and actual experiments. Their drops made the cover of the 8 July 1994 issue of *Science*.

"They did experiments like Diane's and mine, except using still photography and a quick strobe light," said Smolka. "They could only get one picture from each experiment, one for each drop, so they couldn't take any temporal measurements: They couldn't take the measurements that would really check Eggers's predictions quantitatively."

The oil Smolka used was clear silicone, the stuff made famous by the breast implant debate. It sat in an open bowl-shaped reservoir; a valve determined how fast it dripped out. "I usually set it to have 30 seconds between drops. That's a slow drip. All this fancy stuff" —she motioned to a thicket of pipes and poles —"allows me to move the experiment very finely to get it in view of the camera." The digital camera —"if you unscrew the lens you'll see a computer chip" —was set on its side to accommodate as many pixels as possible on the chip's longer axis. When the picture appeared on the video screen, however, it seemed that the drops were "falling" from right to left, until Smolka picked up the screen and turned *it* on its side. Then comes the bulge, the bulbous thread, the *boing!*

"To go to the American Physical Sciences meeting and be able to show a movie of drops —to show a movie of something that happens that fast —well, it wows them." She smiled and admitted it wowed her too. "Every time I'm recording or taking images," she said, "I'm wowed. *Look*, I say to myself, *something new*."

The movie, though, isn't all. Because the camera is digital, Smolka can read the data gathered by any of the chip's 238 by 192 pixels. She can download that information to a PC, convert it into shades of gray, and using a standard photography software package, change its brightness, contrast , size —whatever she needs in order to see the action more clearly. "I can manually locate the edge of the tip of the thread," she said, giving a for-instance, "and find its velocity."

"We'll never be able to know where the drop pinched," she said , a bit sadly. "It's between two frames. I can estimate it to one twelve-thousandth of a second, but not closer than that. But since I have this camera and can take a lot of pictures —12,000 every second —I can measure the displacement —how much the thread moves from one position to the next —over time. What I do is find the position of the tip of the thread from one frame to the next. If I subtract their positions, I've essentially measured how far the tip has moved.

"Now in a car, you think of velocity in terms of miles per hour. I can take my displacement, which equals a length, and divide by the framing rate, which is a time: one twelve-thousandth of a second.

"Here's an example." She got out her lab notebook and flipped to a page. "The velocity here was 3,117 millimeters per second."

When Smolka plotted the velocities of her drops, she saw a series of curves: the falling oil sped up and slowed down depending on whether the thread was in its thin or its bulbous stage.

"This is actually the exciting part of our work," Smolka said. "I gave a talk at the American Physical Sciences meeting and Brenner" —who in the meantime had moved from the University of Chicago to M.I.T. —"was there. He was the reviewer for a paper Diane and I had submitted and he was very excited when he saw our pictures. He wanted to explain to me what I could do with this work."

Smolka had graphed her data first on a linear scale, resulting in curves that showed that Eggers's theory had "overpredicted the velocities. We found they weren't as fast as he had predicted." When she sent her results to Eggers, who had by then moved to a university in Germany, he thought the difference must be due to air drag. "If we could do our experiments in a vacuum, he said, the results would follow his theory.

"This is where Brenner came in." Following Brenner's suggestion, Smolka replotted some of her data —the velocities of each thread tip at the moment a drop pinched off —on a logarithmic scale. As Brenner had pointed out, Egger used a power law equation to describe the tip velocities, and the solutions to such an equation, when graphed on a logarithmic scale, will form a straight line.

Smolka took out another graph. "The dashed line is Eggers's prediction. The circles are my data. And you can fit a straight line to that data. It confirms that pinch-off does follow a power law like Eggers predicted.

"Now the slope of the line tells you the power, and the place that it intersects the number 1 on the graph gives you the coefficient of the equation. I got a coefficient of 3.7 and a power of 0.42. That's a pretty good experimental match-up to Eggers's prediction, which was 8.7 and 0.5."

But additional experiments showed that the power Smolka measured, by graphing her data, wasn't always so near to Eggers's suggested one-half: sometimes it was below one, sometimes substantially above. This meant that while the *boing!* did follow a power law, the power could not be considered a universal constant, as Eggers had thought.

A paper Brenner referred Smolka to, by Joseph Keller at Stanford and Michael Miksis at New York University, provided a clue as to why not. "Keller and Miksis developed a scaling argument," Smolka said, "relating shape to velocity. If you think of it like a pipe, just to give you a simple analogy, you can see that the thinner a pipe is, the faster the flow will be. And on the long skinny threads we get very fast retreat."

Applying the Keller-Miksis idea, Smolka found she could match the shape of the thread's tip with the power she had arrived at based on her measurements. "If the power is less than one-half, you'll get a rounded shape. If the power is less than one but greater than one-half, you'll get a kind of conical tip.

"We were kind of backtracking," she explained. "Knowing what the velocity was, we checked the shape. But here's the up-shot: If we take Eggers's theory, Eggers found there's just one power, one half, and that it's a universal constant. We find that it's not universal. The power of the tip's velocity depends on the tip's shape and the length of the thread. That's the correct way to look at this data."

Smolka completed her master's thesis detailing this idea, and she and Henderson submitted a paper to the journal, *Physics of Fluids*. It was accepted in July.

At that time Harvey Segur, an applied mathematician at the University of Colorado, Boulder —and the man who had originally put Smolka in touch with Henderson —was on sabbatical in Japan, visiting the physics department at the University of Tokyo. There he met with Miki Wadati, who was interested in drops. Segur had with him a preprint of Smolka and Henderson's paper, in which they had included a series of early photographs taken while Pritchard was helping devise their high-speed video set-up.

"Harvey saw our paper," Smolka said, "and he looked at this picture and he got an idea for solving the Navier-Stokes equation. He saw that there was some time interval during which the thread —he calls it a filament —was uniform in width. He used that observation to solve the Navier-Stokes equations and explain how the filament decreases over time.

"It's a really simple formula," she said, writing it on the board. "Basically it's saying that *h*, which is the radius of the filament, goes like one over the square root of time. Which makes it a power law, again. A square root is just a power of one-half."

Segur and Wadati sent their result to Henderson, who "was just waiting to tell me," Smolka remembered —waiting until Smolka had finished her master's thesis. "Harvey asked her if we would do the experiments to test the theory. She wanted to see if he was in the ballpark first. She did a rough estimate, measured from her old videos, and it looked as if the thread was following Harvey's equation. Then I did some experiments. I showed for two fluids that the thread does decrease like one over the square root of time.

"That means it's definitely worth pursuing this theory." To share their idea with the rest of the research community, Segur, Wadati, and Smolka submitted a paper to the *Journal of Fluid Mechanics*.

"I'll have to do many more experiments to build people's confidence," Smolka acknowledged. "It's a dissertation topic. Probably two to three years' worth of experiments. But if it works out it'll be really cool. It's a pretty revolutionary theory. It might be able to answer the question of when will the drop separate from the orifice? How long will the thread get before the drop pinches off? It might also be able to answer the question of when will the thread become unstable, that is, when will you get more than one drop?

"Compared to Eggers's theory, this is just a different way of looking at the problem. Eggers was trying to solve what happens close to the moment of pinch-off. You've got a thread that's getting thinner and thinner, the fluid is flowing faster and faster, the pressure is going through the roof—He's trying to solve a problem where the variables are going to infinity.

"This theory, on the other hand, is looking at the gross behavior of the drop.

"To solve the Navier-Stokes equations you have to make assumptions. The one that Harvey Segur and Miki Wadati made was that the thread's radius was uniform for a certain time period. That's reasonable, because we see that in our videos. But as they solve the equation they have to make more assumptions. So their solutions have to be tested. If the experiments concur, then okay, those assumptions were good ones. If not, you have to look back at the beginning and say, I missed something."

*Linda Smolka is a Ph.D. student in the department of mathematics, Eberly College of Science, 118 McAllister Building, University Park, PA 16802; 814-863-0516; smolka@math.psu.edu. Her adviser, Diane Henderson, Ph.D., is associate professor of mathematics,106 McAllister Bldg.; 865-3712; dmh@math.psu.edu. Their research is funded by the National Science Foundation, the David and Lucille Packard Foundation, and the Alfred P. Sloan Foundation.*