
Mathematicians Find Wrinkle in Famed Fluid Equations - digital55
https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/
======
pge
I'm struggling to understand the significance of this, at least as the N-S
equations are used in the real world. Many years ago, I interned with the Navy
writing fortran code for fluid dynamics simulations on submarine hulls, and
IIRC there were flow dynamics we observed consistently in the real world (e.g.
oscillating vortices) that were fundamentally inconsistent with the results
coming from our N-S calculations (which would say there could not be an
oscillation because it was a steady state flow). There was always a hand-
waving of "N-S is actually right; our computer models are just not fine-
grained enough." But at the same time, given the computational limits of our
grids(particularly at the time -25 years ago), it was understood and accepted
that N-S would yield only an approximation. That's only one data point, but it
certainly seemed to me that no one was relying on N-S as an accurate predictor
of motion (as you would a newtonian model of a ball rolling or something like
that), but rather just as a first order approximation. If that impression is
accurate, a result that says N-S isn't always accurate is kind of a statement
of the obvious. What am I missing?

~~~
sdfadfa
N-S aren't "right" though. The derivations make a lot of good assumptions that
breaks down in certain materials/situations.

What is right is its starting point on the conservation of momentum and
energy. Then it makes certain assumptions about the stress-tensor which are
not necessarily true. Meaning, you can derive the N-S from consv. of mass and
E and a certain stress tensor (ST), but its not derived from a universal ST.

~~~
btrettel
NS are a pretty good model for the underwater scenarios a Navy would be
interested in, so I think they can be called "right" here. The fluids are
regarded as "Newtonian" so the stress tensor model is good, and the density of
the fluid is high enough that the continuum approximation is good. The largest
source of error is likely the approximations made to model the turbulence, or
in other words, reduce the computational complexity while also reducing
accuracy.

------
mannykannot
"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.

~~~
FabHK
"I am an old man now, and when I die and go to heaven there are two matters on
which I hope for enlightenment. One is quantum electrodynamics, and the other
is the turbulent motion of fluids. And about the former I am rather
optimistic." \- Horace Lamb

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foota
Based off the abstract, there's one class of weak solutions, the Leray
solutions, which are known to exist. And now they've shown for a different
class of weak solutions to NS that the solutions are not unique. Is that
right?

~~~
boxfire
That's correct, while taking away the entropy inequality as well. The thing is
that very inequality is one of these laws of thermodynamics we so love, so I
kind of see this as hot air.

~~~
ianai
Aye either they have upturned a theory as august as general relativity or
they’ve thrown the baby out with the bath water somewhere.

~~~
pencilhappen
What's funny is the article says this about the Navier-Stokes equations: "The
equations work. They describe fluid flows as reliably as Newton’s equations
predict the future positions of the planets"

Newton's equations do not in fact reliably predict Mercury's orbit, and it
took GR to do it. Lazy journalist!

~~~
alephnil
The Navier-Stokes equations assume that the medium is continuous even at
infinitely small scales, which obviously not the case for natural fluids, that
are made of discrete atoms. Thus the equations are only correct at
sufficiently large scales. They work fine for describing the airflow around an
aeroplane, but not the airflow around the head of a hard drive, which is small
enough that the finite size of atoms must be taken into account. On a more
visible scale, you have Brownian motion of small particles, which can be seen
even in a low magnification microscope. The Navier-Stokes equations predict
that these effect does not exist. The equations are still useful approximation
in a lot of cases.

~~~
rusk
How are Navier Stokes the other way, on the macro scale? e.g. in the context
of meteorology. Asking because I had a discussion with somebody recently where
they claimed the fundamental flaw to the science underlying Climate Change
science is over-reliance on NS at macro scale as a way of predicting climate
behaviour, or something. I took it to be Baloney, but I'm wondering if there
is some strands of truth to it ...

~~~
qubex
As you intuit, there is no reason to presume that Navier-Stokes would be
unreliable at macro scales relevant to meteorology, simply because it is so
thoroughly tested in experimental settings and to such sensitivities that it
is known that all relevant factors are accounted for.

(Of course, why would one presume that if it _is_ inaccurate at planetary
scales, it biases observations towards the climate change narrative? It's just
the typical “God of the gaps” kind argument.)

~~~
rusk
“God of the gaps” yeah that’s pretty much what I thought!

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BucketSort
Btw this is work relating to one of the famed Millennial Problems -
[http://www.claymath.org/millennium-
problems/navier%E2%80%93s...](http://www.claymath.org/millennium-
problems/navier%E2%80%93stokes-equation)

~~~
FabHK
As the article pointed out.

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CamperBob2

       Using this approach, Buckmaster and Vicol 
       prove that these very weak solutions to the 
       Navier-Stokes equations are nonunique. They 
       demonstrate, for example, that if you start 
       with a completely calm fluid, like a glass 
       of water sitting still by your bedside, two 
       scenarios are possible. The first scenario 
       is the obvious one: The water starts still 
       and remains still forever. The second is 
       fantastical but mathematically permissible: 
       The water starts still, erupts in the middle
       of the night, then returns to stillness.
    

Also permissible, and also vanishingly unlikely, under quantum theory.

It's a stretch, but I wonder if it's reasonable to think of data points in the
vector field describing fluid motion as probabilities rather than definite
measurements. That might allow their behavior to be modeled with different
mathematical tools.

~~~
eru
> Also permissible, and also vanishingly unlikely, under quantum theory.

Navier-Stokes has nothing to do with Quantum Mechanics..

~~~
snakeboy
You missed the point of their comment.

The responders' point was that Quantum Mechanics is a different framework for
modeling physical phenomena which takes a probabilistic framework, and so if
fluid were to be modeled in a similar framework, you could work more naturally
with these "vanishingly unlikely" events.

~~~
eru
Navier-Stokes is only incidentally related to real life. It's pure math.

~~~
Gibbon1
Navier Stokes is probably Turing Complete.

~~~
eru
Probably, yes. But the gymnastics required to make it Turing complete are
unlikely to be in the regime that describes real fluids.

~~~
Gibbon1
In the sixties some people experimented with fluidic logic gates.

~~~
eru
Those probably relied on friction?

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davidw
Totally unfamiliar with the math and physics, but I recall the name from one
of the earlier parts of Cryptonimicon.

~~~
qubex
Navier & Stokes is the name of the scientists who indecently formulated the
equation that bears their name that describes the motion of most fluids. In
_Cryptonomicon_ the main character is given a simple question about a boat on
a river during his admission to the army, deploys heavy-duty fluid dynamics to
give a non-obvious answer, and consequentially gets classified as a moron and
relegated to menial duties (which suit him just fine, as I recall).

~~~
davidw
Yeah it was a pretty funny scene - it's supposed to be this simple math
question, but the ... "Asperger"... he might be called these days - character,
Lawrence Waterhouse, really digs into the problem. He ends up discovering
something that he submits to a math journal for publication, but the army
folks think he's an idiot.

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tulasinandan
The author explaining the paper:
[https://www.youtube.com/watch?v=F71SRP3MZcw](https://www.youtube.com/watch?v=F71SRP3MZcw)

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phkahler
This makes senses to me. If you run a simulation at too low a resolution you
run into ambiguity. By refining the grid you can resolve that ambiguity.
Turbulent flow is chaotic, so this makes perfect sense to me. What this may
lead to is a method of determining criteria for adaptive grid refinement. But
I thought that already existed, so maybe just an improvement over what's out
there.

~~~
reikonomusha
This doesn’t have to do with numerical error introduced by discretization.
This has to do with uniqueness of solutions.

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ericbowman
That is fascinating. Is there any sort of immediate real-world impact (like to
weather forecasting)?

~~~
labster
Immediate? Certainly not. Weather models do have incomplete information, but
the equations are approximated by Taylor series (1st to 3rd order, depending
on the model, last time I checked).

It's possible that this can explain some of the differences between models or
ensemble runs... but you have to realize that most of the error comes from
incomplete data in the initial and boundary conditions. Looking for weather
model effects is like looking for relativistic effects in automobiles.

~~~
m_mueller
Almost. Most models use some variations of Runge-Kutta approximation, to third
order usually.

Edit: typos

~~~
tanderson92
Convergence, or at least consistency and order, of Runge-Kutta methods is
shown via Taylor series, so I don't see the problem or why the GP is only
"Almost" correct.

~~~
thethirdone
The Runge-Kutta methods are not a Taylor series though. So "almost" is apt.

~~~
labster
Geez guys, I chose a term that I thought most people would be more likely to
understand. I realize that some of us have worked a lot more on numerical
methods, but other hackers never made it past Calculus 2.

~~~
thethirdone
I wouldn't have corrected you. I think your comment was fine.

> I realize that some of us have worked a lot more on numerical methods, but
> other hackers never made it past Calculus 2.

While that didn't occur to me, I do appreciate trying to keep things
accessible.

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Glauc
I wonder what the interplay of the material's EOS has with constraining these
solutions. Note I only had time to skim the article.

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bitL
Is there any online course I can take that would teach me Navier-Stokes?

~~~
derjames
I would recommend this text book: Transport Phenomena. Bird, Steward and
Lightfoot. The first part is related to the motion of fluids/transfer of
momentum ( The one you are interested ). The second part is the transfer of
energy and the third part is related to the transfer of mass.

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aperrien
Does anyone have a link to this particular paper?

~~~
yorwba
The link was given in the article:
[https://arxiv.org/abs/1709.10033](https://arxiv.org/abs/1709.10033)

~~~
aperrien
Thanks, I missed that somehow.

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epx
Luce Irigaray knew it already!

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uptownfunk
So will they get the million?

~~~
emileokada
No. They've shown non-uniqueness of solutions of a strictly weaker problem.
They have yet to say anything about the actual Navier-Stokes problem.

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westurner
Navier-Stokes equations:
[https://en.wikipedia.org/wiki/Navier–Stokes_equations](https://en.wikipedia.org/wiki/Navier–Stokes_equations)

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floatingatoll
It's disappointed to see that their reaction to non-unique results is "must be
broken", rather than "we've rediscovered the quantum physics uncertainty
principle in fluid mechanics".

~~~
snakeboy
The Uncertainty Principle is a very well established result which is quite
generalized mathematically beyond Heisenberg's result. This has nothing to do
with that principle.

~~~
floatingatoll
If you weaken the vector density until a point that you can generate multiple
possible outputs from a given weakened input, then what level of weakening
guarantees _no_ unique outputs from the equations?

These equations were designed for a system in which every 'atom' (vector) has
a perfectly knowable 'spin', and they begin to produce unexpected results as
the uncertainty is dialed up through weakening.

It's just a shame that the considerations stop at "therefore we broke the
equations" as opposed to "gee, that looks familiar". What's the Navier-Stokes
equivalent of the diffraction experiment? What does the interference pattern
of two vector fields even _look_ like? Why aren't they trying to study the
interference patterns of the one input, two outputs scenario?

I get that this is all "obviously pointless" to others, but no one I've asked
can actually explain _why_ these comparisons are unacceptable. You completely
dismiss it without any explanation other than "science is well-established",
as if somehow that's meaningful.

So, yeah, disappointment.

~~~
gjulian
Not the GP, but you have to take into account that these are studies done by
mathematicians, not physicists, so there are to points to consider.

First, weakening is not related to uncertainty, at least not in the normal
sense that I think you refer to. It is not related to the physical solution
itself but to our own rules of what we consider a solution. If instead of
vector fields we were working with animals, the strong solution would be "we
need this animal to be a duck" and the weakened one is "we need this animal to
quack when we poke it with a stick". So it is not similar to the quantum
uncertainty principle nor anything like it.

Second, mathematicians are interested only in these specific equations.
Breaking the equations means that they do not model correctly the real world,
and finding those new equations is the job of physicists. Maybe there are
other conditions on the solutions, or maybe the relaxation they did allows for
non-physical solutions. In fact, they do not prove that those solutions
satisfy the energy inequality, so it might be possible that all but one of
those non-unique solutions are only possible if you allow fluids to magically
gain energy out of nowhere (which obviously conflicts with thermodynamic
laws).

~~~
im3w1l
Reminds me of the Banach-Tarski paradox, where non-smoothness breaks
conservation.

~~~
snakeboy
Can you elaborate on what you mean here?

Isn't B-T just a consequence of accepting the axiom of choice and performing
some pathological decompositions of a ball? What do you mean by:

>non-smoothness breaks conservation

~~~
im3w1l
Normally translations and rotations conserve volume. But not when you have
these extremely non-smooth pieces.

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justaman
I'm not going to pretend I understand anything about fluid dynamics.

It brings to mind how people say "Bumble bee's defy the laws of physics to
fly".

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throwwit
Isn’t Navier-Stokes smoothness just a reframing of the three-body problem?
Karl Sundman solved that one in 1909, and n-body was generalized in 1990 by
Qiudong Wang.

Edit- to answer my own question:
[http://www.scholarpedia.org/article/N-body_simulations_(grav...](http://www.scholarpedia.org/article/N-body_simulations_\(gravitational\)#Mean_Field_Approach:_analogies_and_differences_with_fluid_dynamics)

~~~
qubex
I'm sorry if it seems I am persecuting you and putting all your ideas down,
but I really don't see how this can be true either.

The 3- or N-body problem is about point-particles interacting gravitationally
at nonzero distance according to an inverse square law. Navier-Stokes is, at
root and in the limit, about elastic collisions about infinitesimal corpuscles
that transfer momentum between each other.

Again, I can see the analogy “lots of things interacting”, but they have quite
little in common beyond that.

~~~
throwwit
Just the inverse n-vectors applied (regardless of space and field strength or
possibly just instantaneous field strength). Always good to be called out tho.

~~~
qubex
I don't understand what the first sentence of your reply is supposed to mean,
but the second sentence is a very mature response and belies your wisdom: yes,
science and rational thinking is all about putting ideas out there and
rejoicing when somebody helps you etch away at those that are not compatible
with reality, so that only plausible ones remain.

~~~
throwwit
Well what I meant was shrinking the n body system to a point then extending
that to a field. But it’s beyond me how the math works to invert those same
force vectors to an impulse.

~~~
throwwit
Apologies if I’m cargo-culting math... just positing.

~~~
qubex
[https://xkcd.com/451/](https://xkcd.com/451/)

;)

