Separating scales to model bursting bubbles

Bubbles of soap and other liquids have long been known to adopt the shape that minimizes their surface area. An isolated bubble is a sphere; bubbles in a foam or cluster meet so that their surfaces form 120° angles at junctions. But the shapes of the bubble surfaces in a cluster don’t succumb so easily to analytical description. The numerical techniques to treat arbitrary stable bubble geometries ¹ didn’t begin to mature until the early 1990s.

And that equilibrium picture is far from a complete physical description of a bubble cluster, an inherently nonequilibrium system. Under pressure gradients and gravity, fluid drains from the liquid films that constitute the bubble walls. When one of the films gets too thin, it ruptures. The remaining bubbles are left to rearrange into a new configuration, and the cycle begins again. Each of the processes affects the others, but they occur on such different time scales—ranging from a fraction of a millisecond to tens or hundreds of seconds—that re-creating them all in a single numerical simulation has been computationally prohibitive.

Now mathematicians Robert Saye and James Sethian (University of California, Berkeley) have created a framework for capturing the essential physics from the various scales while efficiently using computer resources. ² For each of the three processes—drainage, rupture, and rearrangement—they developed a separate numerical model with its own equations, simplifying assumptions, and characteristic time step. By treating each process in turn with the appropriate model, they can transmit the critical information from one scale to another and produce realistic simulations of large bubble clusters.

For small systems, such as two bubbles merging into one, the researchers can compare their numerical results with experiment, and they find excellent agreement. For larger systems, such as the simulated 27-bubble cluster shown in the figure, matching the initial conditions between simulation and experiment would be too difficult. But the simulations reproduce the qualitative features seen in real bubble systems, including rupture cascades in which the bursting of a small bubble induces several larger bubbles to burst in rapid succession.

The researchers anticipate that by modifying their models to include additional physics—evaporation, liquid–solid phase transitions, and so forth—they’ll be able to address a variety of “bubble problems” that have industrial and scientific applications. For example, solid plastic and metal foams, materials of interest for their light weight, are produced by hardening liquid foams. (See the article by John Banhart and Denis Weaire, Physics Today, July 2002, page 37 .) Simulations of the processes involved in their production may suggest new ways of controlling their properties.