NO ONE can accuse H. Norman Abramson of sitting in an ivory tower. “It is common everyday knowledge to each of us that any small container filled with liquid must be moved or carried very carefully to avoid spills,” the US engineer and fluid dynamicist wrote almost half a century ago. “Experience has taught us that the unrestrained free surface of the liquid has an alarming propensity to undergo rather large excursions, for even very small motions of the container.”
Hardly rocket science, you might think. The irony is, it was. Abramson’s words come from the introduction to a . At the time, the US space agency was aiming to catapult astronauts to the moon atop rockets filled with liquid propellants. Abramson’s magnum opus, entitled The Dynamic Behavior of Liquids in Moving Containers with Applications to Space Vehicle Technology, presented the sum of human knowledge on a phenomenon of life-or-death importance to the space race: sloshing.
It is a problem even now. Fuel sloshing has probably caused the failure of several rocket launches, and in 1998 a slosh-induced tumble cut short a course adjustment and set back by a year the Near Earth Asteroid Rendezvous (NEAR) mission to the asteroid 433 Eros. In 2005, the European Space Agency launched an entire mission, dubbed Sloshsat, to study fluid dynamics in microgravity. NASA’s own investigations continue.
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Back on Earth, we are still baffled by the burning question Abramson touched on. Why does hot coffee leap from its cup so readily, and can we do anything about it?
From space fuel to coffee, this much we know: sloshing results largely from the interplay of two things. One is a liquid’s inertia, which stops it following when its container changes speed suddenly. This causes waves. The other is resonance, in which these waves get bigger as a result of some external pulse, or forcing frequency, that roughly matches the liquid’s natural oscillation frequency. The same effect makes an unplucked guitar string vibrate if a sound with its natural pitch is played through a nearby speaker.
The carried coffee problem is a tougher bean to grind, because the forcing comes not from a single frequency but from a complex biomechanical process with a simple name: walking. “It makes the story more interesting, but characterising the problem more difficult,” says of the University of California, Santa Barbara. Walking’s intricate rhythms, with accelerations in all three dimensions, mean sloshing is better understood when it happens in a spacecraft than in a common cup of coffee.
It was the sight of mathematicians struggling with their mid-morning coffees at a conference in 2011 that convinced Krechetnikov and his student Hans Mayer that the problem merited more systematic study. Back in their lab, the pair rigged an electronic spill sensor to a standard, cylindrical mug filled with coffee, and filmed themselves as they carried it down the uncarpeted hallway outside Krechetnikov’s office. “I don’t have any special training to carry coffee, so I considered myself a good subject,” says Krechetnikov.
A frame-by-frame analysis of the films revealed two things. First, a high initial acceleration – starting out too quickly – can lead to instant spillage as the inertial waves send the coffee over the mug’s lip. No surprise there. If you’re less of a jackrabbit, the coffee’s surface merely wiggles at first. As you walk, however, the rhythm of your gait sets up a resonance in the liquid that amplifies those initial wiggles into ever-larger waves.
And herein lies the problem. Unlike a guitar string, liquid in a container has what physicists call a wide resonance well: even a rough match between the natural and forcing frequencies can cause resonance. Krechetnikov and Mayer found the natural oscillation frequency of coffee in standard-sized mugs to be between 2.6 and 4.3 hertz, depending on the mug’s height and diameter. The forward-backward component in most people’s gaits, meanwhile, supplies a forcing frequency between 1 and 2.5 hertz – a close enough match for resonance to cause slopping within seven to 10 steps. Random variation in real walking rhythms introduces higher-frequency noise that also contributes to resonance forcing. The best antidote seems to be simple concentration: on average, when cup-carriers focused on not spilling, they made it further slop-free ().
These findings were enough to earn Krechetnikov and Mayer this year’s . But what practical lessons can we draw to minimise spillages – preferably without the social dampener of obsessing about our drink?
The first is that a taller, narrower mug has a higher resonant frequency, taking it further away from the danger zone. The same is true for a mug that narrows a little from base to lip. A wider cup is more spill-prone for a second reason, says , a physicist at Boston College, Massachusetts. Whenever an initial acceleration or resonance causes the surface of the liquid to deflect from horizontal, the larger radius means that a wave of a given slope has become big enough by the time it reaches the edge of the cup to go overboard. That also explains why it is so hard to carry a bowl of soup without slopping it.
Baffled and stymied
Some relief might also be achieved by using a more flexible cup, which can absorb more of the energy of sloshing than a rigid mug. For those in search of a more engineered solution, Krechetnikov recommends a series of baffles or ridges around the inside of the container. These cause vortices in the liquid and break up any resonant tendencies.
Alcoholic drinks produce their own challenges to the slosh-prone. Beer is probably a bad choice. A standard UK pint glass is about 14.5 centimetres high and 9 centimetres in diameter at the lip, and should resonate at about 3.2 Hz by Krechetnikov’s reckoning, well within the danger zone. Some beer glasses, such as the , have a bulge near the top that may act like a baffle and stymie resonance, but Krechetnikov is reluctant to say without more research. Meanwhile, Herczynski recommends swapping to a thicker tipple such as eggnog. “The energy that’s supplied to the sloshing eggnog from your walking is dissipated through friction,” he says. “All things being equal, it should be safer to walk with.”
“All things being equal, eggnog should be a safer drink to walk with”
The most elegant of all solutions to spilled drinks, though, says Krechetnikov, might be to carry your drink on a tray suspended from a carrier ring, as servers often do in the Middle East. The set-up acts as a pendulum with its own, far slower frequency, which buffers the liquid from the rhythm of your walk. As far as solutions to Abramson’s problem go, you might call rock-it science.
