# Mathematics of popping bubbles in a foam

Bubble baths and soapy dishwater, the refreshing head on a beer, and the luscious froth on a cappuccino: All are foams, beautiful yet ephemeral as the bubbles pop one by one. Now, two researchers from the Department of Energy’s (DOE’s) Lawrence Berkeley National Laboratory (Berkeley Lab) and the University of California, Berkeley have described mathematically the successive stages in the complex evolution and disappearance of foamy bubbles, a feat that could help in modeling industrial processes in which liquids mix or in the formation of solid foams such as those used to cushion bicycle helmets.

Applying these equations, they used supercomputers at DOE’s National Energy Research Scientific Computing Center (NERSC) to create mesmerizing computer-generated visualization showing the slow and sedate disappearance of wobbly foams one burst bubble at a time.

The applied mathematicians, James Sethian and Robert Saye, report their findings in *Science*.

“This work has application in the mixing of foams, in industrial processes for making metal and plastic foams, and in modeling growing cell clusters,” says Sethian. “These techniques, which rely on solving a set of linked partial differential equations, can be used to track the motion of a large number of interfaces connected together, where the physics and chemistry determine the surface dynamics.”

The problem with describing foams mathematically has been that the evolution of a bubble cluster a few inches across depends on what’s happening in the extremely thin walls of each bubble, which are thinner than a human hair.

“Modeling the vastly different scales in a foam is a challenge, since it is computationally impractical to consider only the smallest space and time scales,” Saye says. “Instead, we developed a scale-separated approach that identifies the important physics taking place in each of the distinct scales, which are then coupled together in a consistent manner.”

Saye and Sethian discovered a way to treat different aspects of the foam with different sets of equations that worked for clusters of hundreds of bubbles. One set of equations described the gravitational draining of liquid from the bubble walls, which thin out until they rupture. Another set of equations dealt with the flow of liquid inside the junctions between the membranes. A third set handled the wobbly rearrangement of bubbles after one pops. Using a fourth set of equations, the mathematicians solved the physics of a sunset reflected in the bubbles, taking account of thin film interference within the bubble membranes, which can create rainbow hues like an oil slick on wet pavement. Then they used NERSC’s Hopper Systems to solve the full set of equations of motion.

“Solving the full set of equations on a desktop computer would be time-consuming. Instead, we used massively parallel computers at NERSC and computed our results in a matter of days,” says Saye.

“Foams were a good test that all the equations coupled together,” Sethian says. “While different problems are going to require different physics, chemistry and models, this sort of approach has applications to a wide range of problems.”

The mathematicians next plan to look at manufacturing processes for small-scale new materials.

“DOE’s longstanding support for core basic applied mathematics has been instrumental in providing the opportunities to develop the mathematics and algorithms behind this work,” says Sethian.