# Why can't the Navier Stokes equations be derived from first principle physics?

At the 109th UCLA Faculty Research lecture, Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles [of physics]".

In physics there are lots of first principles, and so the first question is what set of first principles would one expect to derive the Navier Stokes equations?

And the second, and main question is why does a derivation fail? Are we missing some yet to be discovered set of first principles in this area of physics?

• I don't know about any first principles in physics other than experiments. I am looking forward to being enlightened though, what other people think physical first principles are supposed to be. Jun 29, 2015 at 3:50
• Are not the NS equations a moment of the BBGKY hierarchy? It's been a while since I looked at this, but I recall it going something like this: mpe.dimacs.rutgers.edu/2013/10/08/… Jun 29, 2015 at 5:27
• @CuriousOne: I'm skeptical that experiments can be first principles. An experiment is formulated with reference to a theory - for example they might be done by humans that have a notion of "particle", "mass", "location" which are theoretical physical concept which are more of less hard to define and not necessary. Maybe it works if "an experiment" is nothing than a sheet of numbers to you. As soon as you imply "this is a sheet with distances, measured in time intervals of five clock ticks", you're deep in theory land, speaking in context of a bunch of notions that people made up. Jun 29, 2015 at 7:51
• @NikolajK: Physics is science, it starts with observations and then proceeds from there. The technical requirements to convert observations and experiments into theory are not quite as trivial as many think and it's not a bad idea to have at least a little bit of an idea about them. On the other hand, since you don't seem to know the difference between faith and trust... well, let's leave it there. Jun 29, 2015 at 9:19
• @docscience: Found it: Around 46:20-49:50 and 51:22-53:30. Jun 29, 2015 at 18:37

None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be consistent with the known simpler principles but it has also to incorporate something new.

In this case you appear to have the impression that an attempt to derive the Navier-Stokes equations runs into some impassable hurdle and therefore fails, but this isn't the case. If you search for derivations of the Navier-Stokes equations you will find dozens of such articles, including (as usual) one on Wikipedia. But these are not derivations in the sense that mathematicians will derive theorems from some initial axioms because they require some extra assumptions, for example that the stress tensor is a linear function of the strain rates. I assume this is what Putterman means.

Later:

Phil H takes me to task in a comment, and he's right to do so. My first paragraph considerably overstates the case as the number of equations that introduce a fundamentally new principle are very small.

My answer was aimed at explaining why Putterman says the Navier-Stokes equations can't be derived but actually they can be, as can most equations. Physics is based on reductionism, and while I hesitate to venture into deep philosophical waters physicists basically mean by this that everything can be explained from a small number of basic principles. This is the reason we (some of us) believe that a theory of everything exists. If such a theory does exist then the Navier-Stokes equations could in principle, though not in practice, be derived from it.

Actually the Navier-Stokes equations could in principle be derived from a statistical mechanics treatment of fluids. They don't require any new principles (e.g. relativity or quantum mechanics) that aren't already included in a the theoretical treatment of ideal fluids. In practice they are not derivable because those derivations based on a continuum approach rather than a truly fundamental treatment.

• Disagree on the first paragraph - we use useful equations covering bulk behaviour which could be derived from simpler principles because it is easier to use (where the derived relations are sufficient). $PV = nRT$ is not fundamental, and it breaks down for anything too far from an ideal gas. But no-one would throw it away when considering gas behaviour for a combustion engine, for example, because starting from first principles would get you to the same, or worse result given an amount of resource effort. Jun 29, 2015 at 9:30
• The Chapman-Enskog equation is the link between statistical mechanics and the Navier-Stokes equations that you allude to at the end. Jun 29, 2015 at 23:54
• While I appreciate your clarification, I frankly think that your first sentence is completely ridiculous (no offense - I know that you have an extremely deep understanding of physics). Most non-introductory physics textbooks begin by listing all the fundamental postulates in the first chapter, and then the entire rest of the book is dedicated to deriving corollaries from those postulates. Would you say that no equation outside of that first chapter is ever "interesting" or "gives new information"? Jul 26, 2017 at 8:36
• It is not necessary to assume that the stress tensor is a linear function of the strain rate. It actually does not impact the general form of the NSE. The continuity assumption is essential.
– knia
May 27, 2020 at 8:25

They are derivable from classical mechanics using either the continuum or molecular points of view.

Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress and rate-of-strain are linearly related. One might regard this as an assumption but it can also be viewed as the first term in a power law expansion.

Starting with a microscopic point of view, one can derive the Navier-Stokes equations from taking moments of the Boltzmann equation. In this approach, the linear relation between stress and rate-of-strain appears naturally as the first term in the Chapman-Enskog expansion.

Many undergraduate fluids textbooks include a derivation from the continuum point of view. The derivation from a molecular point of view is done in first-year graduate textbooks such as Introduction to Physical Gas Dynamics by Vincenti and Kruger.

• You don't need the conservation of energy to derive the NSE. The NSE are a transport equation for the momentum, and you can use them to derive a transport equation for the energy (just like in regular classical mechanics).
– knia
May 27, 2020 at 8:31

I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), but they are not accounted for by the Navier-Stokes (NS) equation, and it is not completely obvious where these effects are hidden in the standard derivations of the NS equation from kinetic theory.

Long time tails are related to fluctuations, and so are ultimately a reflection of the fact that any coarse grained description must depend on a scale, and that the most general theory of non-equilibrium correlation functions at long distances and long times must involve more than a deterministic, continuous partial differential equation such as the Navier-Stokes equation.

The role of noise terms has been studied by a number of people, beginning with Landau and Lifschitz. The basic conclusions are:

1) There is a systematic low energy (long time) theory of correlation functions, which involves a gradient expansion of the conserved currents, and averaging over noise terms fixed by fluctuation-dissipation relations. The Navier-Stokes approximation corresponds to linear derivatives in the stress tensor, and no noise terms. This is a consistent approximation in three dimensions (but not in two).

2) At higher order noise terms have to be included, and kinetic coefficients become scale dependent. The hydrodynamic equations require a cutoff, and the best we can hope for is that low energy (long time) predictions are cutoff independent order by order in the low energy expansion.

First question is what set of first principles would one expect to derive the Navier Stokes equations?

This question is Simple. In physics, a calculation is said to be from first principles, or ab initio, if it starts directly at the level of established laws of physics and does not make assumptions such as empirical model and fitting parameters.

And the second, and main question is why does a derivation fail?

The main Problem in Turbulence is it's scaling. We need to use empirical factors to correct the results if the dimensions of the system is changed. Due to the complexity of 3D flow equations, this key problem has lead to a situation where practically all problem solving efforts are trying to mathematically found the link between Measured data and the equations themselves.
This must fail because the current equations are WRONG. (Feynman, Key to Science)
This comes from the simple fact, that it disagrees with the experiment.

Are we missing some yet to be discovered set of first principles in this area of physics?

This obviously must be the case. And as the equations clearly can't be made any more simple as they already are, they must be too simple at the moment. Some aspect must be missing.

I have personally made an invention, which I have patented and also tested in full scale in a Lab. This invention was based on my idea about the cause of Turbulence; and I indeed really managed to make the [noisiest turbomachine ever]5 to a high efficiency and vibration free smoothly running machine.

I really killed the turbulence. This caused us to destroy a 3-hole pitot pipe in the Lab, due to a laminar-alike fluctuating flow. The flow just wasn't turbulent, as you would expect it to be.

The solution was based on the idea, that I need to prevent the fluid to "break on pieces" because of sudden shock. This meant in my thoughts that the viscous forces can't be transferred through the Fluid as there is intern surfaces, which can only interact with collision and friction.

Now, if you Look Navier Stokes Equations, you immediately notice that viscosity is just not handled that way. Though this idea is pretty simple, and I immediately found some predicting success from it. It’s just a mathematical horror to try add these aspects on the 3D Navier stokes equations. First we need a scale free limit and definition, which tells us when exactly we should calculate viscosity, and when friction and collision. Imagine a Kinetic gas theory, where you would have a particle velocity depended particle size?

But I actually found the way, and I was able to derive these modified Navier Stokes Equations in such a matter, that this "mess" can be handled statistically like in Kinetic gas theory. After I got the Idea of the path, it was just straight forward; The equations were relatively simple, and the mathematical results fitted perfectly to old Measurement data.

Just to verify that this works, I have also succesfully calculated few turbulent Pipeflow losses with this new model.

ANSWER; Yes. The Navier-Stokes are incomplete. Universal Continuous and Smooth solutions does not exist, as the aspect that fluid breaks in pieces due to the accelerating conditions which can be defined with Froude number $$\sqrt3$$. The energy dissipation of this breakdown can be treated statistically, and this provides perfect match to experimental data on all scales.