Navier-Stokes is hard?
Author: James Lowman

I was recently asked the question; “How can the Navier-Stokes equations both describe our observable world and not be known to always have solutions in 3D?

For those that don’t know, the Navier-Stokes (NS) equations represent a mathematical model that describes fluid. It is derived from quantities in physics that are conserved. Those are:

  • Conservation of mass
  • Conservation of momentum
  • Conservation of energy

The Navier-Stokes equations are yet to be considered solved. There is a prize association called “The Millennium Prize Problems” which offers one million dollars to anyone that can solve one of the seven problems they have listed, and the existence and smoothness of a solution to the Navier-Stokes equations is on that list.

So the question I was asked seems completely reasonable. How can these equations describe physical phenomena with knowledge of a solution? The answer touches on an interesting, and modern, intersection in science.

We use a number of assumption to simplify a NS system in order to approximate a solution. These approximations are usually handled by a computational fluid dynamics software package. They are high powered suites that allow for customized physics simulations. The assumptions most often used to simplify the NS equations are things like the fluids being in-compressible, the turbulence affects are averaged, or that the fluid is Newtonian. Suffice to say, even with a high powered computational engine, more often then not a scaled down and simplified version of the NS equations are all that we can easily approximate.

So if the solutions to the fluid problems are all approximate, how can we know if they are describing real physics? To compare a NS system solution to an actual physical fluid flow, we would need to perform an experiment and collect data on such a flow. This results in flawed data, because the act of introducing measurement tools into fluid domains involves disrupting the fluid. The best measurements of fluid dynamics themselves are approximations.

So what we have ended up with is approximate mathematical approximations of fluids approximating approximate experiments. Its not hard to see why the student’s question needed clarification.

The answer is: Because it works.

That’s a pretty hand-waved explanation, that no student should accept. But for anyone interested, the derivation of Navier-Stokes is a fun derivation that follows from those conservation laws that were listed, and experiments can be found in open source publications that show how much effort was placed on collecting data with minimal interference. The interested party might be able discern that experimental data matches model solutions (reasonably well), and we can start to believe that Navier-Stokes delivers on the promise of predicting real world physics.