# FeaturedB What makes the Navier Stokes equation so difficult?

1. Jan 16, 2018

### Staff: Mentor

2. Jan 17, 2018

In the link, I question whether there is a typo, and it should read $-\nabla P$ with a minus sign for the force per unit volume from a pressure gradient. In any case, a good article. $\\$ The Navier-Stokes equation and its solutions are rather complicated. As a graduate student, I saw one professor in a graduate course get stuck on a calculation he was attempting at the chalkboard with the Navier-Stokes equation. Technically, this professor was very astute, but on this calculation he was attempting, he was simply stuck. After 1/2 hour of struggling with it, he was totally exasperated with it. He actually told the class, "If anyone is disgusted they can leave". Everyone stayed, but the fact is, the Navier-Stokes equation is simply not a simple equation to try and solve.

3. Jan 17, 2018

### Staff: Mentor

The Navier Stokes equation is so hard to solve because it is non-linear. If the inertial terms were not present (either because of the geometry or because the inertial terms are negligible0, it would (and can) be much easier to solve. In turbulent flow, the inertial terms become even more problematic because the flow becomes non-time-steady, with many coupled spatial and time scales of variation. Turbulent flow continues to be a challenging issue (involving simplifying approximations) even today.

4. Jan 17, 2018

### Andy Resnick

The equation that's shown in the article is a very simplified form of the NS momentum equation- the left side has been simplified for the case of incompressible fluids (as stated), and the right side is only for Newtonian fluids in the presence of gravity. There are also no boundary conditions discussed- for example, multiphase fluid systems (surfaces across which material properties may be discontinuous) or the presence of solid boundary surfaces (the no-slip boundary condition is surprisingly problematic). Finally, there are 2 other equations that simultaneously hold- continuity equations for mass and energy. All 3 sets of equations must be satisfied.

Even for the 'simple' case of homogeneous, isothermal, incompressible fluids, the existence of weak solutions to the momentum equation has not yet been proven for ≥ 3 dimensions (AFAIK). Pierre-Louis Lions has done a lot of work in this area, most of it well outside of my expertise.

5. Jan 17, 2018

### Lord Crc

In real systems I assume the turbulence can't be significantly smaller than the interaction radius of the particles, nor can the particles travel faster than the speed of light due to relativity.

Does modifications that address the above make the problem more tractable?

6. Jan 17, 2018

### Staff: Mentor

Huh? Wr’re Talking about non-relativistic fluid mechanics, right?

7. Jan 17, 2018

### Lord Crc

I mentioned one non-relativistic and one relativistic effect which, from what I can gather, should prevent particles from actually reaching infinite speeds in real systems. The Quanta article highlighted those infinities as one of the core challenges in obtaining solutions to the NS equations, from what I understood. Maybe I read that wrong?

Anyway, I was just curious if either of these effects have been studied in relation to NS and if so what their impact was on the ability to solve the (presumably modified) equations.

Obviously both effects are at play in real fluids, though I would assume the relativistic corrections not to be relevant to most earth-bound fluid problems.

8. Jan 17, 2018

The Navier-Stokes equations are derived using relatively few assumptions, but one of them is that the fluid behaves as a continuum. In this case, that means that a dimensionless number called the Knudsen number is small. This number is the ratio of mean free path to the relevant scales of motion. In other words, the equations would cease to be valid long before you reach a length scale on the order of a single particle.

In practice, the smallest meaningful length scale in a given flow is what is called the Kolmogorov scale, which usually ranges anywhere from 10-7 to 10-3 meters for most Earth-bound problems depending on the situation. Basically, these are the smallest scales at which flow energy is dissipated, and anything much smaller than that doesn't really need to be resolved to capture the relevant physics.

The two major examples where the continuum assumption breaks down is in a rarefied flow (edge of space, for example) or micro-/nanofluids (where the length scales get so small that they approach the mean free path in magnitude).

As for relativistic effects, I believe relativistic fluid mechanics can still use some form of the Navier-Stokes equations provided the flow can be adequately modeled as a continuum and also that some allowance is made for the curvature of spacetime. I don't have any real experience there, though, so I will let someone else comment in depth if they know more.

9. Jan 18, 2018

### Lord Crc

So what I'm struggling to see is how these infinities show up, especially if the minimum length scales are so large. Or is it an indication that you're breaking the continuum condition?

In any case, thank you for the informative reply.

10. Jan 18, 2018

### Andy Resnick

Turbulence is definitionally a classical phenomenon with all of the attendant 'simplifications' (continuum, etc). The infinities occur when the flow is perturbed- in laminar flow, any perturbation is spatially and temporally confined and the flow reaches some equilibrium configuration. In turbulent flow, any perturbation 'blows up' in space and time- the flow is always unsteady, there is no equilibrium, and while there may be places where, for example, the velocity is formally infinite (non-zero vorticity), the 'infinities' refer to the lack of an steady-state solution.

The field is wide-open: there's little in the way of turbulent heat transport and I know of no investigations into turbulent plasmas or magnetic fluids (other than Chandrasekhar's book on stability).

Depending on your background, you may find these references interesting:

https://www.amazon.com/Turbulence-Structures-Dynamical-Cambridge-Monographs/dp/0511622708
https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.56.223
http://rspa.royalsocietypublishing.org/content/164/917/192

11. Jan 18, 2018

These are all very active areas of research. For example, Los Alamos and Lawrence Livermore National Laboratories both have very active research communities dedicated to turbulent plasmas, which necessarily also includes the influence of electromagnetic effects. In essence, this is what the National Ignition Facility is studying on a very massive scale.

Turbulent heat transport is a very important and active field within high-speed aeronautics due to its importance in the design of thermal protection systems on spacecraft, missiles, etc. It's also quite important in mechanical engineering where efficient design of heat exchangers or efficient combustion is important.

Chandrasekhar's book is about the stability of fluid flows, i.e. how instabilities grow to eventually break down to turbulence. It says little about the actual behavior of turbulence. Most texts on that subject treat the problem (largely) statistically (e.g. Statistical Fluid Mechanics Vol I&II by Monin and Yaglom, Turbulent Flows by Pope, Turbulence by Davidson, or A First Course in Turbulence by Lumley).

12. Jan 19, 2018

### Staff: Mentor

In a related article on Navier Stokes, mathematicians find some extreme conditions where the equations appear to fail:

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/

13. Jan 20, 2018

### Auto-Didact

14. Jan 20, 2018

I am not a mathematician, more of a mechanician, but to me, the article sounded like they basically just proved that, if you start out with too coarse of a grid for your solution, then the solution when you refine that grid might be non-unique. That seems almost obvious to me, so perhaps I am missing something.

15. Jan 21, 2018

### Somali_Physicist

Quick question, is this turbulence found In the electromagnetic world?

16. Jan 22, 2018

### WWGD

17. Jan 22, 2018

Now, again, I am not a mathematician working on fluid mechanics but a mechanician/engineer, but to the best of my knowledge his work isn't really taken seriously. He is an electrical engineer who basically had a symbolic math programming language (probably Mathematica or something similar) spit out an answer, it didn't really adhere to the requirements of the Clay problem, and it was published in the first ever volume of an obscure journal and also on arXiv.

18. Jan 23, 2018

### Zel'dovich

Indeed, due to the mass conservation, the left side of the equation is also valid for compressible fluids. It is in the right side where you see the incompressibility, since the stress tensor would have additional terms for compressible fluids .

19. Jan 23, 2018

### Tom Kunich

I have a problem with the idea that there is always a solution. I cannot see that turbulent flow can ever be predicted in small scale.

20. Jan 23, 2018

### Andy Resnick

I don't think so- density was factored out of the material derivative: D/Dt (ρv) →ρ Dv/Dt. If the density is a constant, you have an incompressible fluid.