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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

#81

Post by qubex »

R0b0t1 wrote: post #77
qubex wrote: post #64

Earlier quoted context omitted.

If you are really interested, somebody has bothered to make a special-relativity & quantum-indeterminacy compatible formulation of Navier-Stokes suitable for calculating shockwaves in extremely dense mediums such as the neutronium neutron stars are thought to be made of (where the speed of sound is comparable to the speed of light in a vacuum, hence the relativity, and the matter is degenerate and in states of superp…

Sir, would you please post a link to this code?

It wasn’t my code to link to and I probably last saw it in 1996, back in the era of dot-matrix printouts on green-and-white continuous paper (on a Windows NT 3.5 machine running FORTRAN PowerStation, if you care to commiserate). I think I can dig up the paper where the equation was published, though.

EDIT: This isn't the paper I had in mind, and the equation presented is ‘merely’ relativistic, but it gives you a feel for the beast: https://arxiv.org/pdf/astro-ph/0402502.pdf (see section C).

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#82

Post by qubex »

throwwit wrote: post #79
qubex wrote: post #70

Earlier quoted context omitted.

Ah OK I get it now. I’m an applied mathematician and a macroeconomist that studied turbulence in financial markets and crashes thereof. I can assure you that the dynamics are pretty distinct. In economics wealth is not a conserved quantity whereas in physics energy and momentum are.

I think bitcoin’s fixed-limit and deterministic transactions makes it a uniquely closed & conserved system (regardless of deflation and lost wallets). The discrepancies in valuation among exchanges & localities seem to show a relativistic (to borrow a physics term) quality. [I’m however not really into bitcoin or an economist though]

> bitcoin's fixed limit

That's what makes it unsuitable for being a currency.

> a uniquely closed & conserved system

Nope. Bitcoin may or may not be (lost wallets, as you point out, is one way in which it is not). But the ‘system’ is the economy, because money is moving in and out of bitcoin because it can be exchanged for other assets (goods, services, or other currencies when doing conversions).

So no. I'm not trying to be condescending, I'm just trying to nip this apparently valid but flawed analogy in the bud. The only commonality is the word ‘turbulence’ which is being used as a label for two entirely different phenomena that have some similitude and points of contact but are largely distinct and unrelatable.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#83

Post by qubex »

foota wrote: post #11
boxfire wrote: post #8

Earlier quoted context omitted.

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.

Isn't that possible though if the fluid isn't a closed system? (which it probably isn't)

All of the above arguments obtain if the fluid is a closed system and is only acting as a result of the forces it itself is exerting upon itself, subject to the various boundary conditions around it.

Of course if you open the system various outcomes are possible depending on what the external influences do to it (for example, the two solutions mentioned for the still water become entirely plausible if there's somebody roaming around who might put a lighter under the glass and cause the water in it to boil, but that isn't the point of the exercise).

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#84

Post by dkarl »

pge wrote: post #74

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 (w…

The beginning of the article does a horrible job of explaining the big question. The Millennium Prize problem[1] is a math problem: whether the Navier-Stokes equations always have a solution with certain properties. Some possible answers to the problem would mean there are some physical situations where they don't produce any prediction at all about what might happen next, or they produce multiple predictions, or they produce physically implausible predictions. (If you have a strong math background, the official problem description might be interesting to you; it's a bit beyond me.[2])

The article does get around to explaining it better if you keep going.

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

Numerical methods for solving the Navier-Stokes equations are approximate and therefore diverge from the correct solution. The same is true for a ball rolling down an incline, but the inaccuracies are smaller than you would ever care about in the real world. What your colleagues were saying about the Navier-Stokes equations is that the numerical error was often large enough that the calculated solutions were known to diverge from mathematical reality in significant ways, and therefore seeing them diverge from physical reality was consistent with physical reality and mathematical reality being the same.

[1] http://www.claymath.org/millennium-problems

[2] http://www.claymath.org/sites/default/files/navierstokes.pdf

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#85

Post by qubex »

throwwit wrote: post #47

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...

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#86

Post by throwwit »

qubex wrote: post #85
throwwit wrote: post #47

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...

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 se…

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#87

Post by ekelsen »

pge wrote: post #74

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 (w…

I think it's likely you were solving the Reynold's Averaged NS - https://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80... and dropping the turbulence term to get what you call "steady state".

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#88

Post by qubex »

throwwit wrote: post #86
qubex wrote: post #85

Earlier quoted context omitted.

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 se…

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.

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.

Re: Mathematicians Find Wrinkle in Famed Fluid Equations

#90

Post by throwwit »

qubex wrote: post #88
throwwit wrote: post #86

Earlier quoted context omitted.

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.

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.

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.
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