Public Release: 

Mathematics of dripping faucet key to industry, research

Purdue University

WEST LAFAYETTE, Ind. -- Purdue engineers are the first to figure out the mathematics behind a problem plaguing machines that emit drops of liquid from a nozzle, findings that have potentially broad applications, from improved inkjet printers to more precise pharmaceutical research.

The new mathematical method drastically speeds up the time it takes to calculate the behavior of how drops form as they come out of a nozzle or faucet. Research that would take months with conventional techniques now can be performed within hours, said Osman Basaran, a professor of chemical engineering.

A research paper about the work appeared in the Dec. 18 issue of Physical Review Letters, a journal published by The American Physical Society.

Learning more about the behavior of drop formation is critical to improving many industrial processes in which a liquid is sprayed through a nozzle. The quality of inkjet printers could be improved by better controlling ink flow. Companies could more efficiently manufacture photographic films, adhesive tapes and analytical devices that are made by applying DNA-laden liquids onto "gene chips." The field of combinatorial chemistry, in which thousands of experiments are carried out simultaneously to speed up the discovery of new drugs and other products, depends on equipment that rapidly deposits precisely the same quantity of liquid over and over again into separate vessels.

One problem plaguing such processes is a phenomenon called "period doubling," in which the droplets coming out of a nozzle are not always the same size. Instead, every other drop, or every fourth drop, is the same size, depending on how fast the liquid is being sprayed. The Purdue engineers were the first to compute the mathematics behind period doubling.

The engineers also discovered that the formation of droplets changes dramatically, depending on whether the flow is being increased or decreased. For example, water droplets can have entirely different characteristics in two faucets that have exactly the same flow rate, depending on whether those faucets are being turned up or down. That phenomenon, called hysteresis, is critical to the quality control of various products, such as adhesive tapes or photographic films, in which the amount of liquid deposited onto a solid surface must remain consistent to prevent waste. Fluctuations in the performance of pumps and other equipment can cause the flow rate to increase and decrease with no warning, resulting in hysteresis.

The mathematical method works by computing the quickly changing pressures and velocities of the fluid in evolving drops. Each drop is broken into sections shaped like rectangles or squares. As the liquid emerges from the faucet and evolves into a drop, the changing pressures and velocities in each section are computed. Then, the long-term behavior of drop formation is simulated by tracking several hundred drops in sequence at the same flow rate.

"This requires solving about 50,000 equations simultaneously," Basaran said. "Think about it: A few equations are hard enough to solve -- 50,000 is quite taxing."

Conventional methods are far more time consuming because they can study only a single drop at a given flow rate.

"You might compute about 100 drops in a row," Basaran said. "That would have taken about 100 days. We figured out a simpler way of doing it that would take us minutes to do one drop, so we can do a few hundred drops in a couple of hours or less."

The research combines chemical engineering, physics and mathematics to solve "free boundary problems," which are especially difficult because they deal with objects that change shape. In contrast, "fixed boundary problems," such as water flowing through a pipe, are much easier because the pipe's boundaries do not change.

The Purdue engineers not only created the mathematical equations, they backed up their new algorithms with hard data. The researchers used a high-speed camera to capture how drops evolve as they come out of a nozzle and a laser to precisely count the frequency of drops.

"A lot of people who do computations don't do experiments, and a lot of people who do experiments don't do computations," Basaran said. "We are lucky that we can do both, and I think that's why we are more successful in this work than some other people."

The findings might be used to improve various manufacturing processes, and Basaran's work has attracted the attention of several companies.

The research paper, "Theoretical analysis of a dripping faucet," was written by Basaran; former doctoral student Bala Ambravaneswaran, who now works for the GE Plastics division of the General Electric Co.; and former undergraduate student Scott D. Phillips, who is now a graduate student at the Massachusetts Institute of Technology.

Future research will attempt to compute how drops form in more complex fluids, instead of water or alcohols. Many fluids important to industry contain particles, soaps and polymers, which makes drop formation even more difficult to understand and simulate.

Basaran said the findings would have been impossible without an expensive high-speed camera that was purchased with funds provided by the U.S. Department of Energy for unrelated "basic research," which is carried out with no specific practical applications in mind.

"People should understand the benefits of basic research and how it results in discoveries that were not predicted," he said.

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Writer: Emil Venere, (765) 494-4709, venere@purdue.edu.
Source: Osman Basaran, (765) 494-4061, obasaran@ecn.purdue.edu.

Related Web site: Osman Basaran: http://ChE.www.ecn.purdue.edu/ChE/Fac_Staff/Fac.Staff/obasaran/

NOTE TO JOURNALISTS: An electronic or a hard copy of the research paper referred to in this news release is available from Emil Venere, (765) 494-4709, venere@purdue.edu.

ABSTRACT
Theoretical Analysis of a Dripping Faucet
Bala Ambravaneswaran, Scott D. Phillips , and Osman A. Basaran
School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283
While previous studies of continuous emission of drops from a faucet have shown the richness of the system's nonlinear response, a theory of dripping has heretofore been lacking. Long-time behavior of dripping is simulated computationally by tracking the formation of up to several hundred drops in a sequence, rather than the usual single drop, at a given flow rate Q and verified by experiments. As Q increases, the system evolves from a period-1 system through a number of period doubling (halving) bifurcations as dripping ultimately gives way to jetting. That hysteresis can occur is also demonstrated.

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