Mathematicians solve a foamy problem

Two researchers from University of California Berkeley have now described mathematically the successive stages in the complex life of foamy bubbles – with implications as diverse as plastics and cell biology.

Applying their equations, the team created computer-generated movies that portray the slow and sedate disappearance of wobbly foams one bubble-burst at a time.

“This work has applications in the mixing of foams, in industrial processes for making metal and plastic foams, and in modelling growing cell clusters,” said James A. Sethian, a UC Berkeley Professor of Mathematics. “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.”

Describing foams mathematically is problematic because the evolution of a bubble cluster a few inches across depends on what’s happening in the extremely thin walls of each bubble.

“Modelling 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 I. Saye, a Mathematics PhD student at Berkeley. “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.”

The pair discovered a new way of treating different escapes of the foam with different equations that worked for clusters of hundreds of bubbles. The equations took five days to solve using supercomputers at the LBNL’s National Energy Research Scientific Computer Centre (NERSC).

One equation set described the gravitational draining of liquid from the bubble walls, which thin out until they rupture. In another set of equations, the flow of liquid inside the junctions between the bubble membranes was dealt with.

“Foams were a good test that all the equations coupled together,” said Sethian. “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 equations are reported in Science

Reference: Multiscale Modelling of Membrane Rerrangement, Drainage and Rupture in Evolving Foams