Even Calgon can’t take you away from applied mathematics.

In a paper published Thursday in Science, UC Berkeley mathematicians James Sethian and Robert Saye reveal algorithms that describe air pockets surrounded by liquid membranes and joined at plateau junctions (translation: a cluster of bubbles).

It took several years to do all the math, Sethian told me by phone from his office in Berkeley. The final product: a nifty computer-generated soap bubble that’s indistinguishable from the real thing.

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Let’s start with the basics here. What’s a bubble?

A bubble is a pocket of air surrounded by a very thin membrane. And that membrane contains liquid.

Now if you imagine two bubbles, they’re going to connect together sharing a common, liquid wall. If you have three bubbles, a cluster, these bubbles share a membrane and they’re pulling on each other in an equilibrium.

So that’s not a real bubble we’re seeing up there?

It's a computer calculation, but it's not graphics. We’re solving the fundamental equations of motion that have to do with how bubbles drain, rupture, and then rearrange to a new position.

You plug that algorithm into a computer, and out comes the video. Cool! What’s that we see reflected in the bubbles?

We've taken a sunset scene and computed how this would be reflected off this bubble cluster.

Your paper says that there are three kinds of equations at work in this video. I might rephrase that as “three things that bubbles do.” What are they?

I’d call them three stages:

Say you have a cluster of bubbles, as if you filled your kitchen sink full of dishwashing liquid. It may look for a minute that’s nothing going on there. But in fact, the fluid in those thin films [the walls of the bubble] is draining down.

As the liquid drains, it pools into the shared edges. Those are called “plateau junctions,” named after a mathematician named Plateau.

One of the walls finally becomes too thin and it pops. And now it’s gone. So if you think about it, these bubbles are no longer in an equilibrium. They have to move to a new position. They’re settling into a nice new place where everyone is equally pulling on each other.

So, in brief, the life cycle of a bubble is…

They’re draining. Someone gets too thin and pops. That disturbs the equilibrium, so they all have to move to a new position, and then the cycle repeats itself.

Why can’t a cluster of liquid soap bubbles stay how it is forever?

Well in part, it’s gravity. Gravity will start pulling liquid down.

But -- and this gets a bit technical -- the junctions are also sucking liquid out of the membranes due to a version of capillary pressure. In other words, because of the curvature of these junctions, there’s a pressure differential that causing the fluid to be pooled, or flowed into these plateau junctions.

Why is this process of bubble clusters reshaping themselves so hard to calculate?

One challenge is that these three processes happen on very, very different time scales.

The draining can take place over hundreds of seconds. The pop happens at hundreds of meters per second. That’s flying. And then the process of rearranging itself, to reach that new equilibrium, happens on the order of a tenth of a second or so. So these are vastly different timescales.

And the same thing is true on physical scales: The thickness of that membrane is on the order of tens to hundreds of nanometers to reach that new equilibrium. It’s tiny. But a cluster of bubbles is big, inches across.

So you’re trying to compute the solution to a problem in which the timescales are varying by six or so orders of magnitude, and the space scales are varying.

You with me?

I am.

In mathematics, we call this a multi-scale problem. You have to find a way to decouple these scales from each other: to solve them separately, but then somehow let them talk to each other.

How long did it take to do this work?

We’ve been working on developing these algorithms for, I’d say, three to five years.

It took five days on a supercomputer called Hopper at Lawrence Berkeley Lab to do the calculation you’re seeing in this movie.

And finally, what could this possibly have to do with bike helmets?

This work can help show how to optimally design solid foams, like polyurethane.

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