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Comment by pencilhappen | Hacker News Reader

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!

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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.

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 ...

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.)

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

Fluid equations assume that there's a single characteristic velocity for the atoms at each point in space. So, for physical systems where local velocity distributions are wide or even multi-modal, fluid equations won't capture the physics.

In plasma physics, Laser Wakefield dynamics is an example of a system that can't be modeled as a fluid.

I sort of doubt these considerations apply to the atmosphere, but this is one of the main heuristics for when you can't use a fluid equation.

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.

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?

I think part of it is just mathematical interest, and maybe part of it is hope for more efficient or otherwise better approximations for fluid behavior.

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

The Navier-Stokes equations do not in fact reliably predict the flow of all real-world fluids. The comparison to Newton's equations is perfectly apt. Good enough for most applications, but may be imprecise in some cases.

Sounds like a perfectly accurate statement to me, in both directions.

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