Fluids' Dynamic

If you're ever poured a colored liquid into clear water you've seen now it first billows outward before diffusing throughout the container. And you've probably noticed how the amount of liquid poured and how fast it's poured affects the shape of the billowing. Although the action seems to produce similar results from similar amounts and speeds, it would seem impossible to predict with any great accuracy how the two currents would interact.

But believe it or not, there are mathematical equations that describe those changes to a degree that scientists can often predict not just something as simple as two liquids in one container but the interactions of ocean currents and airflows in the atmosphere. They're called the Navier-Stokes equations and they've been around for almost two hundred years. Claude-Louis Navier and George Gabriel Stokes didn't work as a team to develop them, but their development of how to apply Newton's laws of motion to elastic materials linked up and were collected under their names. Navier is one of the 72 names inscribed on the Eiffel Tower and Stokes held the Lucasian Chair in Mathematics at Cambridge -- a job also held by Isaac Newton, Paul Dirac and Stephen Hawking, among others.

Navier-Stokes equations help meteorologists forecast weather changes. Air behaves like a very, very thin fluid so the equations can predict some of its motions. Oceanographers predict changes in sea currents depending on the temperature or relative strength of some motion in the water. Both groups will use computers to build models of likely air or water behavior given starting conditions. Because new factors can change conditions in an instant, those predictions are not necessarily as precise or accurate as they would be in computer simulations.

As their name indicates, the Navier-Stokes equations are mathematical operations. They have proven more than adequate to describing the physical world in which we live. This means that physicists, as well as oceanographers, meteorologists and other scientists who work with fluids are quite satisfied with them. Mathematicians, on the other hand, aren't. Mathematicians deal with equations that may or may not apply to "real world" situations; either way they focus on the numbers and such involved as abstract concepts instead of physical things.

And the mathematicians think that the Naver-Stokes equations may have a problem or two when they are handled outside of their real-world contexts. Under certain conditions, the equations describe two possible states for a fluid at the same time, which is a no-no (unless you're doing quantum mechanics, but that's another beastie). The example in the story at Quanta magazine is of a perfectly still glass of water. When the Navier-Stokes equations are turned loose on it under certain parameters, then you have a glass of water that either stayed still all night or at some time spontaneously erupted in the glass and then returned to its still state. Ghost Hunters and similar shows notwithstanding, that sort of thing doesn't happen. But even if it did, the Navier-Stokes equations should tell an observer which one it was rather than coming up with both answers at the same time.

If the math crowd does figure out that the Navier-Stokes equations are flawed, they probably won't get abandoned. After all, Albert Einstein showed that Newton's own Laws of Motion got a little wrinkly when things were either very fast or very small, but we still use Newton's understanding most of the time. Things rarely move that fast and even though we know the very very small is real, its fuzziness doesn't translate to everyday-sized objects. So the physicists, meteorologists, oceanographers and others will probably keep using them (although the meteorologists on TV will usually choose whichever model allows them to monger the most fear).

The possible dichotomy does provoke interesting possibilities. One of the things that Einstein did with his theories of relativity was explain a kink in Mercury's orbit that plain ol' Newtonian physics couldn't. Could the mathematical inadequacy of the Navier-Stokes equations prompt some new world-flipping paradigm shift? Who knows? But it will be fun to watch.