Navier-Stokes Solution Effects

[Silavite]

I understand that the Navier-Stokes equations are a set of equations that define the behavior of fluids and that a millennium prize is up for grabs for a complete solution to these equations.

What I don't understand is what effects such a solution and understanding of these equations would have. How could it affect practical things like aircraft design, building design, and combustion dynamics? Could such a solution be vital to understanding these things, or is it simply (albeit impressive) academic?  I know it's a bit presumptuous to ask for the effects a solution could have when we don't even know the nature of the solution, but this is something that I am quite curious about.

[KSK]

"Turbulence is the most important unsolved problem of classical physics."
-  Richard Feynman.

That's what a complete solution to the Navier-Stokes would provide - a way of properly calculating turbulence effects. I think.

[cfds]

Funnily enough, a complete solution is not required for the millennium prize. I f I remember correctly "any progress" is sufficient. Which says a lot about the problems with turbulence....

[Hannu2]

Practical significance would depend on solution. There are many numerical algorithms to calculate approximate solutions for Navier-Stokes equations. If general solution would be simple enough, it would be possible to decrease developing costs but it is not necessarily true. For example, there are some solutions for restricted 3 body problem, which are impractical because numerical algorithms give same accuracy with far less computer time (several order of magnitudes). However, proven existence of these solutions are very important piece of mathematics of such differential equations and in any case solutions of Navier and Stokes equations would be too. It is the reason of prize and possible practical applications would be bonus.

[Tullius]

First, I am a mathematician that looked up on wikipedia what Navier-Stokes and the question are about and not a physicist, but I think that is not a bad thing for answering the above question.

The Navier-Stokes equation, like the heat equation or most differential equations, can be solved numerically, i.e. approximately, and that to a rather high precision (given sufficiently large computing power). And that is also the most we can ask for, as explicit solutions generally can only be found for special cases.

So everything is good and your question is pointless?

Well, it is not so easy. If you want to approximate a solution numerically, it needs to exist and be sufficiently nice. And usually it is also very nice to have a unique solution. This is were the mathematics come into play.

For the heat equation, under minimal restrictions on your domain (the shape of your object) and the initial values (the initial distribution of the heat in your object), you can mathematically prove the existence and uniqueness of a bounded solution (boundedness is good, since it prevents the temperature from rising to infinity). So while mathematics is in general incapable of providing us with an explicit solution, it ensures us that the equation actually makes sense for physics and more importantly that our numerical approach to find a solution is sane (approximating a solution, which does not exist or is not unique, doesn't make sense).

For Navier-Stokes, we don't know under which conditions a solution exists: So trying to find an approximate solution might, under some circumstances, make no sense, since there is no exact solution to approximate. And to make matters worse, we don't know when this happens.

So everything you do with Navier-Stokes on the beaviour of fluids is based on the wild guess that, in the case you are looking at, a sufficiently nice solution exists.

Since the Navier-Stokes approach seems to work nicely in practice, people have come up with the hypothesis that there exist minimal assumptions for the existence of a sufficiently nice solution. If you prove that hypothesis, you would earn yourself a nice prize (the problem is one of the Millenium Problems) and would make the physicists and engineers happy, because you just told them that the approach they were taking all these years in approximating the behaviour fluids actually makes sense.

Edited April 4, 2017 by Tullius