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Famous Fluid Equations Are Incomplete

quantamagazine.org

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Re: Famous Fluid Equations Are Incomplete

#21

Post by vanderZwan »

tanderson92 wrote: post #19
semi-extrinsic wrote: post #8

Earlier quoted context omitted.

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.

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.

Re: Famous Fluid Equations Are Incomplete

#22

Post by vanderZwan »

OopsCriticality wrote: post #11
PeterWhittaker wrote: post #2

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 theori…

> 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.

Re: Famous Fluid Equations Are Incomplete

#23

Post by vanderZwan »

MyHypatia wrote: post #18

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…

> 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.)

Re: Famous Fluid Equations Are Incomplete

#24

Post by tjradcliffe »

PeterWhittaker wrote: post #2

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.

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).

Re: Famous Fluid Equations Are Incomplete

#27

Post by MyHypatia »

vanderZwan wrote: post #23
MyHypatia wrote: post #18

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…

> 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…

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.

Re: Famous Fluid Equations Are Incomplete

#28

Post by semi-extrinsic »

vanderZwan wrote: post #23
MyHypatia wrote: post #18

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…

> 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…

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".
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