First Visualization Of Chaos In Three Dimensions

July 31, 1998

It's a curiosity in your coffee cup, a minor problem when baking in the kitchen and a $20 billion problem for the U.S. chemical industry: Why does it take so long for stirred liquids to mix?

Part of the answer is that regular stirring causes some streams to return to their starting point. The path the stream travels may be highly complex, but eventually it meets up with itself, forming "regular islands" -- essentially unmixed pockets of liquid woven intricately through the mixture.

Researchers who study fluid mechanics have had a hard time understanding mixing, which is also found in nature in convection in the earth's mantle, dispersion in the oceans and in weather patterns. Recently scientists have made progress by linking mixing to chaos, the study of order in systems where individual components are unpredictable. But until now they have only been able to devise experiments in two dimensions to study simple flat flows.

In the cover article of the July 31 issue of the journal Science, a team headed by Julio M. Ottino, professor and chair of chemical engineering at Northwestern's Robert R. McCormick School of Engineering and Applied Science, has provided the first experimental realization of a 3-D chaotic flow, showing all the intricacies of the regular islands and the chaotic regions in the flow.

"Mixing is one of the nicest applications of exploiting chaos," Ottino said. "In many cases, you would like to get rid of chaos, but in the case of mixing, we would often like to enhance it, and we would certainly like to understand it."

The experimental device Ottino's group created in Northwestern's Laboratory for Fluid Dynamics, Chaos, and Mixing, is a clear Plexiglas cylindrical tank filled with glycerin. Chaotic flow is created by a nearly vertical shaft spinning a nearly horizontal flat disk.

"If the impeller were perfectly vertical, nothing interesting would happen," Ottino explained, "so we destroy the symmetry by tilting it just a little."

Needles immersed in the whirling glycerin inject streams of fluorescent dyes that glow red, green and yellow when they pass through a laser beam. The beam cuts a clean slice through the tank from top to bottom; with the room lights off, the stream paths are cross-sectioned where they pass through this vertical plane.

The laser is the key to visualizing the chaos in this complicated 3-D system.

"A lot of the order comes to light when you slice it," Ottino said. "When you light the whole thing, it looks like a mess."

A flat circular dye stream would appear as just two dots where the stream crosses the sheet. "But when you tilt the impeller, instead of this part joining smoothly with that one, they just keep missing each other," Ottino explained. "There are these near-misses, and the dye path traces a doughnut region around the tank." These doughnuts eventually appear as two matched circular shapes on the laser sheet.

The stream may spiral around the doughnut as many as seven times before joining up with itself. It is an island if it rejoins itself; a "higher-period" island if it takes many revolutions to do so. Ottino's earlier, two-dimensional experiments, which were featured on the covers of Nature and Scientific American, were "period one," he said. "This is period seven."

The intricate structures of the higher-period islands remain nearly unchanged even after 12 hours of stirring. But there are other parts of the dye, Ottino said, that in theory will never go back to exactly where they started.

"Dye in those parts will wander aimlessly forever, in theory, without ever going back to its initial location," he said. Those parts receive the most efficient mixing. This is of great interest to chemical companies, because costly problems can arise when chemicals react much faster than they can be mixed.

Ottino said the applications that spin out of the new work for industry, geophysics and meteorology are impossible to predict.

"Everybody has their problems, and they might see their problems mapped in some way to this flow," he said. "That's the way it works, you just do this simple thing, which you understand, and people start seeing themselves reflected into this picture. There's no way for me to know in what ways they'll see themselves reflected."

Further experiments are now being done with lower viscosity fluids and higher flow rates. Other variables can also be added, such as spinning the tank in addition to the impeller.

Other authors on the Science report are graduate student Gerald O. Fountain and visiting professor Devang V. Khakhar.

The research was funded by the U.S. Department of Energy.

Northwestern University

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