
Famous Fluid Equations Are Incomplete - retupmoc01
https://www.quantamagazine.org/20150721-famous-fluid-equations-are-incomplete/
======
vhffm
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](http://www.uscibooks.com/shu3.htm)

~~~
orbifold
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.

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MyHypatia
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 generally 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 discussion 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
imprecise. It would be more accurate to say that the labels (fluid,
material, continuum) are great simplifications which are incredibly useful
when they apply.

~~~
vanderZwan
> _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.)

~~~
MyHypatia
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.

------
habosa
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

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PeterWhittaker
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.

~~~
danbruc
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.

~~~
semi-extrinsic
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>

~~~
tanderson92
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.

~~~
vanderZwan
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.

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dnautics
"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.

------
Xcelerate
> 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.)

~~~
plus
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.

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GregBuchholz
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-...](https://www.quantamagazine.org/20140224-a-fluid-new-path-in-grand-
math-challenge/)

------
amelius
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...

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kunstmord
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).

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

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

