May 10, 2013
Math Helps Crack The Mystery Behind Bubbles
[ Watch the Video: Evolution of a Bubble Cluster ]
redOrbit Staff & Wire Reports - Your Universe Online
The mathematicians applied their equations to create mesmerizing computer-generated movies showing the slow and sedate disappearance of wobbly foams one burst bubble at a time.
"This work has application in the mixing of foams, in industrial processes for making metal and plastic foams, and in modeling growing cell clusters," said James Sethian, a UC Berkeley professor of mathematics who leads the mathematics group at Lawrence Berkeley National Laboratory (LBNL).
"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," said Robert Saye, who will graduate from UC Berkeley this month with a PhD in applied mathematics.
"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."
Sethian and Saye 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, while another set dealt with the flow of liquid inside the junctions between the bubble membranes. A third set handled the wobbly rearrangement of bubbles after one pops, while a fourth set of equations were used to create a movie of the foam with a sunset reflected in the bubbles.
Solving the full set of equations of motion took five days using supercomputers at the LBNL's National Energy Research Scientific Computing Center (NERSC).
The researchers now plan to look at manufacturing processes for small-scale new materials.
"Foams were a good test that all the equations coupled together," Sethian said.
"While different problems are going to require different physics, chemistry and models, this sort of approach has applications to a wide range of problems."
Sethian and Saye will report their results in the May 10 issue of the journal Science.