Bubbles are a serious business. While they're beloved as a childhood pastime and a bathtub luxury, the physics behind the delicate, iridescent clusters remains remarkably complex.
Now mathematicians have pinned down the ephemeral physical processes that mark the life, and death, of these suds. Their findings, published this week in the journal Science, could prove useful to chemical engineers seeking to better understand all kinds of foams, from shaving cream to plastic insulation.
Bubble clusters, known as foams, appear in a variety of forms and play an important role in everyday life: in the washing machine, in ocean froth, on the head of a happy-hour beer. And they form the basis of products such as fire retardants, bicycle helmets and refrigerator insulation.
These deceptively commonplace materials are difficult to understand because their relevant parts work on different scales of space and time.
The fragile, filmy wall of a bubble is just micrometers thick, thinner than a human hair. The gas within that transparent shell is vast by comparison.
And while it takes tens or hundreds of seconds for gravity to drain the liquid out of bubbles, they burst at speeds of hundreds of centimeters per second and rearrange themselves in a series of steps that takes less than a second, according to UC Berkeley mathematicians Robert Saye and James Sethian.
Generally, when trying to understand and model such a system, researchers look at the smallest scales to get a detailed understanding of the forces at play.
But here, with more than one scale to consider, focusing on only the small details could have caused researchers to miss the forest for the trees.
Fortunately, the large and small scales involved in foams usually can be dealt with independently, Saye and Sethian explained in their report.
"The details at one space or time scale are not necessarily important at another scale," they explained.
So to describe the foam's behavior, the researchers produced separate equations for the different aspects of foam dynamics. One set of equations governed how the walls of bubbles thin out and then break as liquid drains from their fragile skins.
Other equations focused on how liquid flows at the junctions between bubble membranes. Still more addressed how the spheres shuffle their positions within the foam as other bubbles burst, contorting to conserve surface area.
The pair also developed equations to describe the way a sunset would look when reflected in bubbles, its golden tones spread out into rippling hues like an oil slick.
When they were done writing their equations, they used them to create a video of a cluster of bubbles popping out of existence, one by one.
Separating out each layer of equations made the task slightly easier — though it still took five days for supercomputers at the Department of Energy's National Energy Research Scientific Computing Center to churn through them and produce the simulation.
Taken together, the equations describe the dynamics of foam made up of hundreds of bubbles of various sizes, Saye and Sethian wrote in Science.
"It is just a first step along a road to understanding the kind of foams we find in industry, in chemical plants," said Denis Weaire, a physicist with Trinity College in Ireland who was not involved in the work.
"Today the chemical engineer faced with designing such [a] plant must rely on extrapolation from experience, and guesswork," Weaire said. "To do better we need realistic models. They could arise out of calculations like this."