Famous Fluid Equations Are Incomplete | Hacker News Reader

Famous Fluid Equations Are Incomplete

quantamagazine.org

103 pointsretupmoc0126 comments

If you are interested in some of the details:

The Navier-Stokes equations can be derived from the Boltzmann equation by applying a slight perturbation, expanding the result as a series, and taking the moments.

Taking the moments is essentially an integration, which comes with the implicit assumption that the system you're describing has sufficiently many particles. When running low on particles, this integration does not make sense. This is why the resulting equations do not apply at low densities.

The Navier-Stokes equations are the second order expansion of this procedure. The result of the first order expansion are the Euler equations.

This is called the Chapman-Enskog procedure. It's really quite illuminating when you see it for the first time. There's a great derivation in [1] if you can get your hand on it.

[1] http://www.uscibooks.com/shu3.htm

When I saw this derivation during a course Theoretical Astrophysics it was indeed very enlightening, what is interesting is that it easily generalises to magneto hydrodynamics and other more complicated situations (mixture of multiple different fluids, fluids that react with each other etc.). I believe Landau Lifshitz contains some of them.

The best commentary I have seen on the article comes from a coworker, who took the time to dissect why the conclusion from this article is not surprising:

The notion of a fluid is more generally related to the concept of a continuum which allows for the PDE description the Navier-Stokes equations offer. It is taken for granted that density or velocity are point-quantities in space, but there can be no such simplifying description in rarefied situations or more precisely when the Knudsen number is not small. Batchelor 1967 has a good discussion on this. In addition the notion of viscosity which relies on writing the deviatoric stress as proportional to the gradient in velocity relies on dropping the higher order terms in the velocity gradient Maclaurin series assuming they are small (which they usually are for very small Knudsen number).A Boltzmann-like description will always be more general because it is a pdf-based description which is really just fancy counting and doesn't have the Knudsen number limitation. Therefore calling the Navier-Stokes equations incomplete is a bit imprecise. It would be more accurate to say that the labels (fluid, material, continuum) are great simplifications which are incredibly useful when they apply.

> Therefore calling the Navier-Stokes equations incomplete is a bit imprecise.

Oh, those sloppy mathematicians... ;)

(for the non-physicists/mathematicians: a running gag between mathematicians and physicists is that the former accuse the latter of being sloppy, because the latter take a lot of mathematical liberties. Allegedly, in my old university there was a joint class between physics and mathematics (I never got that far to see for myself), and the professor would start the first lesson with "I brought barf bags for the mathematicians. You're going to need them." I even have a friend who switched from physics to maths because he claimed to be disgusted by the way physicists "proved" their "theorems". Luckily he mellowed out a bit after marrying an applied physicists - they even published a paper together.)

Haha, yea. From an engineering perspective... you can spend all day debating the philosophical implications of taking a derivative and have very interesting conversations, or you could just take the derivative because it's useful and go make things.

We had a professor in quantum optics who would quip before doing certain things (e.g. zeta function regularization) that "the following derivation is unsuitable for people with a preexisting heart condition and mathematicians".

Fluid dynamics is hard.

"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." - Werner Heisenberg

Summary: Navier-Stokes cannot translate to Boltzmann, because Navier-Stokes is incomplete... ...and even the best candidate to replace it fails at extremely low pressures.

This is very, very exciting, because it means our theoretical understanding of fluid dynamics is flawed.

Flawed theory often (usually?) leads to radical rethink and wildly different perspectives.

Declaring our theoretical understanding of fluid dynamics flawed because Navier-Stokes requires the continuum assumption strikes me as being similar to declaring General Relativity flawed because it fails to include quantum theory. Navier-Stokes and General Relativty are incomplete, yes, but they are remarkably accurate and useful over the range for which their governing assumptions hold—we have complementary theories that operate over ranges for which the governing assumptions for NS and GR no longer apply.

Aesthetically, it would be lovely to have a master equation that works from the molecular scale up to the bulk scale. Practically, that master equation would likely reduce into the familiar forms that are presently used.

> similar to declaring General Relativity flawed because it fails to include quantum theory.

Or similarly declaring classical mechanics flawed for not being accurate around light speed (just trying to give another, possibly more relatable analogy - although your GR/quantum theory is more appropriate because it also deals with macro/micro scale).

There's a reason we still teach Newton's laws in high school, aside from being easier to grasp: it's accurate enough for basically any practical situation.

If you're is not dealing with light speed (so.. particle accelerators and not much else), synchronised time measurements over long distances[0] or space stuff, you are not going to need it.

And even in the space case: when they taught us special relativity in our first year of physics, our professor was quick to point out that classical mechanics was accurate enough to calculate the trajectories of the Apollo missions.

Physics, or at least applied physics, is all about finding the sweet spot between good enough approximation vs ease of calculation, and knowing where you're "wrong" in case you need more accuracy.

[0] Because of the earth's rotation - handwave handwave something with inertial frames.

I don't know if I'd call the theory "flawed". Navier-Stokes is a very successful set of equations that is capable of solving real problems as it sets the foundation for modern day computational fluid dynamics. Any subsequent theories must converge to Navier-Stokes at the regimes where Navier-Stokes is highly successful.

I never thought about this before reading the article but now it seems pretty obvious to me that both descriptions can not yield the same results under all circumstances. The Navier–Stokes equations are based on quantities like density and flow velocity which are only really meaningful if you have sufficiently many particles to average about. In consequence I am hardly surprised that one gets disagreeing results under extreme conditions like very low densities.

I'm also quite surprised that this article tries to spin it as very novel. We've known this for literally a hundred years. Moreover, there's no mention of the pioneers in the field - Chapman, Engskog, Burnett, Knudsen, etc - much to my dismay.

The recommendation is for major revisions including a detailed literature review.

</grumpy-reviewer-mode>

I was also dismayed when they referred to KdV (Korteweg de Vries) theory as a "relatively unheralded" theory. KdV theory is an incredibly well known and thoroughly studied area of Mathematics.

Well, those two statements aren't necessarily mutually exclusive, because it can still be relatively unheralded. But only because every physicist knows of Navier-Stokes.

I think it's not that obvious. For example, Maxwell's equations represent essentially a complete description of classical electrodynamics. Even though in most formulations it involves densities, it will work fine for point-like charges where the density is singular but integrable. Of course, when QM comes in the theory expands to quantum electrodynamics (QED), but Maxwell's equations match very well (exactly for linear media?) the quantum mechanical predictions when you average the probabilities.

It sounds reasonable to expect a fluid theory that explains well macroscopic phenomena, even if the fluid really is granular, where we could hope the approximation would hold on average or something like that. But apparently it turns out that N-S doesn't model important qualitatively distinct phenomena.

This is fairly boring because the "incompleteness" is built in to the Navier-Stokes equation from the off, and we (physicists) have been well-aware of it for over a century.

Solutions to the Navier-Stokes equation are turbulent on all scales, but reality is only turbulent down to the atomic scale. This matters practically in rarefied gas dynamics, but it matters formally--that is, to mathematicians--no matter what.

Also, the Navier-Stokes equation is typically solved with extremely simple boundary conditions, but reality has surface tension and whatnot.

Ergo: the Navier-Stokes equation is an incomplete description of reality. This is not news. There may be some news in the generalized understanding of how to turn the atomic-level Boltzmann equation into an appropriate macroscopic equation, but the incompleteness of the Navier-Stokes equation is just not all that interesting.

This is fairly usual in physics: the mathematical language we use to describe reality is in most cases approximate, and leaves out various (physically insignificant) terms, as well as including (physically impossible) solutions (waves that propagate backward in time, etc).

flawed is what scientific journalism would say.

everyone that deals with fluids know that the tools are mostly Balance Equations and such. i.e. good enough to predict everything mostly right.

everyone know none of it is complete... or elegant.

"The terms in the series quickly become unruly, however; energy, instead of diminishing at shorter and shorter distances in the gas, seems to amplify."

This sounds a whole lot like the ultraviolet catastrophe. The solution there was quantization of energy packets and a statistical treatment of the fewer amount of packets that come through.

> He began by rewriting the complicated Boltzmann equation as the sum of a series of decreasing terms. Theoretically, this chunky decomposition of the equation would be more easily recognizable as a different, but axiomatically equivalent, physical description of a gas — perhaps, a fluid description. The terms in the series quickly become unruly, however; energy, instead of diminishing at shorter and shorter distances in the gas, seems to amplify. This prevented Hilbert and others from summing up the series and interpreting it. Nonetheless, there was reason for optimism: The leading terms of the series looked like the Navier-Stokes equations when a gas becomes denser and more fluidlike. “So the physicists were happy, sort of," said Ilya Karlin, a physicist at ETH Zurich in Switzerland. “It’s in all the textbooks.”

This reminds me a lot of perturbation theory, a method used to solve the complicated equations of quantum field theory. The technique basically involves summing up a bunch of Feynman diagrams (of decreasing significance), and it has been used to calculate the value of the gyromagnetic ratio of an isolated electron to within 10 decimal places of its experimentally measured value (which is absolutely amazing, both from a theoretical and experimental standpoint).

However, what's peculiar about this summation is that it fails to converge. You would think that by adding up smaller and smaller terms, the series would eventually reach some limiting value, but that doesn't occur. So the most predictive theory that mankind has ever created (quantum electrodynamics) works only as long as you don't keep adding up more terms.

(*Technically speaking, this isn't a failure of QED, but of the method used to solve its equations. There are other solution techniques that don't have this problem.)

The issue isn't that an infinite sum of tiny terms don't converge -- the issue is that individual terms of perturbation theory diverge. An example can be found in J. Chem. Phys. 112, 2000, 9736-9748 "Divergence in Moller--Plesset Theory: A Simple Explanation Based on a Two-State Model" DOI 10.1063/1.481611 (Note that this is specifically in reference to Moller--Plesset Perturbation Theory, but the divergence is a general phenomenon)

I'm not saying that all perturbation theories diverge. Moller--Plesset perturbation theory doesn't even always diverge. But the divergence behaviour is not in the form of an infinite sum of tiny terms being infinite, but rather the individual terms of the perturbation theory increasing without bound (and oscillating sign).

Also note that it is possible for truncations of perturbation theory to diverge with increasing order, but for the infinite sum of all (divergent) PT terms to converge and be finite.

The linked article is not related to the quest to determine whether the Navier-Stokes equations are capable of supporting Turing machine-like computation:

https://www.quantamagazine.org/20140224-a-fluid-new-path-in-...

If one says "X equations are incomplete", that means that there is more than one solution to X. However, somehow I suspect that is not what is meant here...

Some thoughts: expansion-in-series-based methods (including Hilbert's, which is not used in practice) and the Chapman-Enskog method work only for moderately rarefied gas flows (where you can neglect higher-order collisions; this can be derived explicitly using the BBGKY hierarchy). Also, since the Chapman-Enskog method is asymptotic, it is not guaranteed that higher-order equations (inviscid Euler equations being the zero-order equations and Navier-Stokes equations being the first-order equations) will provide an accurate description of flows. Indeed, the second-order equations (Burnett and super-Burnett equations) seem to fail in some cases, while providing more correct results in others. But given the complexity of the equations themselves and the complexity of the boundary conditions, no one really uses them. The cool thing about the Chapman-Enskog method is that it gives a closed set of equations, so you don't need empirical models for heat conductivity, viscosity, etc.

That's the first point – that methods depending on series decomposition might never guarantee a solution that's accurate in all cases. There are also moment-based methods (Grad's method, for example, being one of the most famous), which have additional equations for parts of the stress tensor (I think; never really read much about them). The second point is that the equations correspond to conservation laws: mass, linear momentum, energy. The equation corresponding to the conservation of angular momentum is usually neglected: the terms related to internal angular momenta of particles are considered to cancel each other out (which seems logical, since unless there's some magnetization happening, the particles will be chaotically oriented and the average of the angular momentum will be 0), and in that case, the equation is satisfied since it just follows from the equation corresponding to the conservation of linear momentum. However, there's been some research recently on whether this equation can actually be neglected and what implications it carries, whether it's connected to turbulence or some other effects.

The third point is that in high-altitude hypersonic flows, there are far more complex effects going on in flows that just simple collisions between particles – there are transitions of internal energy (which is a quantity described by quantum mechanics), chemical reactions (dissociation, exchange reactions), and this all complicates the Navier-Stokes equations – additional terms appear (bulk viscosity, relaxation terms, relaxation pressure). And correct modelling of these terms requires solving large linear systems with quite complex coefficients, and to complicate things further, for many of the processes mentioned, there aren't any easy or even correct models (to take into account dissociation, for example, you need to know the cross-section of the reaction for each vibrational level of each molecular species involved in the flow), since these models are either computed via quantum mechanics (which takes enormous amounts of computational power) or are obtained experimentally (which limits the range of conditions under which the results are obtained).

DSMC methods have being increasingly popular as of late, but of course, they can't provide theoretical results, while it is possible to observe some interesting effects even in theory using the Chapman-Enskog method.

So the problem is not only getting more "correct" equations, it's also being able to correctly model everything that goes into the equations we currently have, and then being able to solve them (for a simple flow of a N2/N mixture, if you use a detailed description of the flow, you get a system of 51 PDEs). And in engineering applications drastically over-simplified models are often used, and yet it's not like every high-altitude air/space-craft has burned to a crisp because of this. While new, "more correct" equations are interesting, of course, there's enough work to be done with the current ones.

Source: I do theoretical research and numeric computations of rarefied gas flows for a living (at the Saint-Petersburg State University).

No wonder I had such a tough time in my Fluid Dynamics class. The material was incomplete! Do over!

lol -- I should go back and demand a recount for all those exams I sweated through...

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