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Mathematicians Find Wrinkle in Famed Fluid Equations

quantamagazine.org

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Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#101

Post by davidw »

qubex wrote: post #97
davidw wrote: post #96

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

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 s…

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#102

Post by btrettel »

ericbowman wrote: post #13

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

Fluid dynamicist here. At the risk of looking like a heretic, I am willing to say that I don't think the NS Millennium Prize problem and related things will end up being very useful practically.

I can't see how a definitive answer to the question will result in better turbulence modeling, which is what matters from a practical point of view. If it turns out that the solutions are not unique then we could probably find an additional condition to add (e.g., the entropy condition) to make the solutions unique. If the solutions are unique, bounded, etc. then that's great and it would have no impact practically speaking aside from perhaps helping the reputation NS has for accuracy. Some people seem to think that solving the NS Millennium Prize problem would likely lead to a solution for the turbulence problem, but as I said, I can't see how. I'd be interested if anyone could explain this belief better.

There may be other benefits. I've found papers that find bounds on different fluid dynamics quantities to be interesting, and the motivation for these studies are the NS problem from what I understand. Unfortunately the results from these papers tend to be less useful than bounds I can derive specifically for applications myself.

(In a nutshell the turbulence problem is that NS has far too high a computational cost/complexity to be used in practical simulations. So cheaper approximations to NS are used, which you can cladsify as "turbulence models". How steep the drop-off in accuracy is as you reduce complexity is an open question. My opinion is that fluids probably require high computational cost for accuracy a-priori. Things like correlations from experiments can get around this as you are using pre-computed results, and that may be what we should go for in my philosophy.)

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#103

Post by jerf »

ericbowman wrote: post #13

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

In addition to the other fine replies, weather forecasting's inaccuracies are dominated by the lack of information, then by lack of processing power. Lack of closed-form solutions to NS or better solutions rates quite a ways down the list of issues it has, or put another way, even if we had a magic box that completely accurately solved NS for weather forecasting, it would not get that much accurate. (My suspicion is that it would literally be measured in "minutes" more accurate, rather than the "days" you'd like, but I concede I can't prove that... but bear in mind that it may well be the cases that it would be milliseconds more accurate or something, not just that I could be wrong about it being "days", as the errors in the initial data compound over the course of the simulation no matter what math you throw at the problem.)

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#104

Post by monochromatic »

incidentnormal wrote: post #78
eadmund wrote: post #75

Earlier quoted context omitted.

What I got from the article is that there's a possibility of a chaotic result: it may not be possible to get computed results which are arbitrarily close to the real-world results. I was reminded a bit of the famous Lorenz system (originally for weather prediction, IIRC), which turns out to be chaotic under some conditions.

I think it's not a chaotic result as such (as the Lorenz weather prediction models you're referencing are) - which would mean large differences in outcome from very small differences in input, but a case where multiple different outcomes can arise from identical inputs.

Right. It’s already known that NS solutions can be chaotic. It would be a big deal if they turned out to be non-unique.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#105

Post by monochromatic »

sundarurfriend wrote: post #57
alephnil wrote: post #49

Earlier quoted context omitted.

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 b…

If the equations are only useful approximations anyway, why are edge case breakdowns so important or surprising like the article seems to indicate? If they are as imprecise as Newton's laws, why are mathematicians looking for "unfailing" precision from them? Or is the article simply wrong in the initial few paragraphs?

The Clay problems are pure math problems. Any approximate application they have to the real world is just an accident.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#107

Post by btrettel »

rusk wrote: post #54
alephnil wrote: post #49

Earlier quoted context omitted.

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 b…

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 ...

Who you were talking to doesn't seem to know what they were talking about.

alephnil mentioned a real problem, but the solution in that case is to not use NS. From a practical standpoint NS is a good model of fluids in many instances because there is a certain minimum scale of motion due to viscosity (the Kolmogorov scale) and this usually is much larger than the size of the atoms or molecules. If this is true then a continuous approximation is fine. No present climate simulation can afford to compute everything down to that scale, so a low pass filter is applied to filter out the small scales and turn their effect on the large scales into a single term that can be modelled. This turbulence modeling approach is called large eddy simulation (LES), and it relies on the fact that outside of certain special cases (e.g., major chemical reactions) the small scales have a universal behavior. (Kolmogorov was the first to propose that the small scales are universal back in 1941.) This approach works pretty well usually. If the person you were talking to said the small scale model was wrong, I'd give them more credit, but this approach is generally the most accurate moderate cost turbulence modeling approach.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#108

Post by snakeboy »

im3w1l wrote: post #39
gjulian wrote: post #35

Earlier quoted context omitted.

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 st…

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

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

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#109

Post by im3w1l »

snakeboy wrote: post #108
im3w1l wrote: post #39

Earlier quoted context omitted.

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

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

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

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#110

Post by derjames »

bitL wrote: post #106

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

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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