From the Princeton Weekly Bulletin, November 3, 1997
New natural law in fluid dynamics
By JoAnn Gutin
Archimedes was lying in his bathtub, and Newton was
sitting in an orchard. But Assistant Professor of Chemical
Engineering Sandra Troian had her Eureka! moment in front of a
computer in Room A 316 of the E-Quad on a balmy summer evening in
1996. Her discovery, which reveals new details of how liquids behave
under stress, is so basic to the field of fluid dynamics that the
journal Nature called it "a universal law."
And although it's been 15 months since that night and a month since her paper describing the finding appeared in Nature, Troian is still thrilled. "I'm amazed," she says, "at the simplicity and beauty of the result," which reveals the existence of a general boundary condition for liquid flow at solid surfaces.
Microscopic behavior
Troian's speciality is interfacial fluid dynamics, the science of why liquids behave as they do. Scientists in this area can tell you, for instance, exactly why a drop of water will spread on the kitchen counter but bead on a newly waxed car. If pressed, they could write an equation to tell you the precise moment when the catsup should hit the burger or when Alaska's North Slope oil will reach the tanker at Valdez.
For the past half-dozen years Troian's energy has gone into studying the microscopic behavior of liquids near solids; specifically, to understand what happens at the molecular level when a thin film of liquid is in the act of spreading on a surface. This phenomenon is increasingly important in the era of miniaturization and nanotechnology; to mention only one application, tiny motors need a layer of lubricant only a couple of molecules thick.
But early on in Troian's exploration of the subject she realized that the mathematical models people were using to study the spread of fluid weren't refined enough to be useful to her. "There's a well-known mathematical singularity that occurs when you try to describe the spreading of a droplet on a solid surface," she explains. "Various utilitarian boundary conditions have been used to get around this problem, but they're all phenomenological and unsatisfactory."
Utilitarian solutions don't make Troian happy. Ever since she first discovered physics, as the only girl in her physics class at Classical High School in Providence, R.I., she's been drawn to the elegance of the patterns created by physical forces: the crown-shaped splash of a drop falling into a glass of milk or the sinuous lines of windswept sand dunes. One reason she found a year she spent as a post-doctoral associate at the Collège de France so satisfying, she says, was the Gallic approach to physics: "They're interested in the fundamental questions, of course, but they often choose to work on beautiful phenomena, too."
The walls of Troian's office are testament to her aesthetic tendency: they're decorated with examples of natural symmetry, including soap froths, splashing drops and rupturing films. Though many of these patterns are familiar, complete mathematical description of them has in some cases been elusive. Troian refers to these images of solved and unsolved problems as "The Gallery of the Beauty and the Beast." "The Beast," she says, " is the mathematics of free surface flows."
2,000 molecules, 40 simulations
The utilitarian models of fluid flow survived not because physicists like rough-and-ready explanations, but because in most cases they worked. And while there were a few phenomena that didn't quite fit, "People just assumed this was a problem that could never be solved. They couldn't see any way you were going to get at this information without a way of probing motion at the molecular scale, " Troian says. Nevertheless, she decided she could attack the problem with computer simulations, because "computers have become so powerful, and these days people have good code" for creating cyber-liquids and cyber-surfaces.
So Troian and Peter Thompson, a former professor of physics and mechanical engineering at Duke University who now heads a local software firm, the Celerity Group, got to work. They devised a computer program that would show them what happened when they sandwiched a liquid between two solid plates and moved the top plate at different speeds.
After designing the surface and the liquid, which consists of about 2,000 molecules, "You move the upper plate and just keep checking how all the molecules are responding to the induced stress," Troian explains. The simplest of these cyber-experiments took a few hours of number-crunching on her SGI INDY computer; the most complex took a couple of days.
After about 40 simulations, Troian figured out a way to collapse all the computerized experiments onto one master curve. On the day in July 1996 that she set the computer to the task of collapsing the data into one common form, Troian and Thompson had gone out to dinner, but they decided to stop back by the lab to see how things were going.
"So we walked into the lab, brought up the file and saw this amazing plot appear. We just gasped!" The data all fell into one smooth curve, suggesting that a single equation could be used to describe all the solid-liquid inter-actions they had simulated. "And then we thought, 'This is so simple and so elegant--could it be right?'"
Slip rate approaches infinity
Before Troian and Thompson's finding, physicists have assumed that the last few molecular layers of a fluid would either remain stationary even as the fluid moved, or would move a distance of a few molecules, in direct proportion to the amount of shear. This condition was, if not exactly dogma, at least something everyone agreed to assume; it was a mainstay of the utilitarian models. But Troian's experiments revealed that if you increase the shear rate--the speed at which the top plate moves--the liquid at the interface starts to slip. And in an entirely unexpected finding, at very high shear rates the amount of slip approaches infinity.
Besides providing a simple and elegant solution to the old boundary-condition problem, this finding could have significant real-world applications. "Knowing that liquids can undergo lots of slip," Troian says, "somebody might be able to work backwards" and invent, for instance, a liquid that wouldn't adhere to the inner surfaces of a machine and thus wouldn't degrade. "It could be a really amazing lubricant."
And who knows? It might be beautiful, too.
Troian, who earned her PhD at Cornell in 1987, joined the Princeton faculty in 1993 after three years as a senior physicist at Exxon Research. She is a recipient of both Research Initiation and Faculty Career awards from the National Science Foundation.
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