How does the tap drip and why should you care?

Engineers at Purdue University have developed a new mathematical method that speeds up the time it takes to calculate the behaviour of how drops form as they come out of a nozzle or tap.

Research that would take months with conventional techniques now can be performed within hours, said Osman Basaran, a professor of Chemical Engineering at the University.

The findings have potentially broad applications, from improved inkjet printers to more precise pharmaceutical research.

One problem plaguing many industrial processes in which a liquid is sprayed through a nozzle is a phenomenon called 'period doubling,' in which the droplets coming out of the 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 liquid emerges from a tap and evolves into a drop, the changing pressures and velocities in each section are computed. Then, the long-term behaviour 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.

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.

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.