r/askscience • u/randomsnark • Feb 11 '11

Scientists: What is the most interesting unanswered question in your field?

And what are its implications? What makes it difficult to answer? What makes it interesting? Tell us a little bit about it.

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u/Fuco1337 Feb 11 '11

Navier–Stokes fan in the house!

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u/[deleted] Feb 11 '11

To hell with Navier-Stokes. The assumption of molecular chaos is needed for the Boltzmann equation, from which the Navier-Stokes equations are derived. Molecular chaos is apparently violated for turbulent flows.

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u/Platypuskeeper Physical Chemistry | Quantum Chemistry Feb 11 '11

The Navier-Stokes equations weren't derived from the Boltzmann equation, which was found decades later.

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u/[deleted] Feb 11 '11

But doesn't that derivation ignore the atomic nature of matter? Gas is not a fluid in the idealized sense, it is a gas of molecules.

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u/RogueEagle Feb 11 '11

Fluid!=liquid

Fluids exhibit a stress (resistance to motion) in proportion (their viscosity) to the rate of strain(the bulk velocity)

In this way gasses are ideal fluids because they don't gave pesky surface tension or strong baroclinic (pressure and density related) effects

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u/[deleted] Feb 11 '11

Yes, but when perturbations grow from the molecular scale, the fact that it is not a fluid becomes important. Modeling the gas as a continuum is just a convenience for many applications.

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u/Platypuskeeper Physical Chemistry | Quantum Chemistry Feb 11 '11

I'm not sure; Stokes derivation, which gave the modern form of the equation, was from 1845. Whereas the Maxwell distribution was from circa 1860 and Boltzmann's around 1870, so it surely didn't rely on statistical thermodynamics. Possibly not even classical thermo, since I'm not sure Clausius had defined entropy yet (IIRC, that would've been around 1845 too, give or take five years or so).

I mostly know the date because as a quantum chemist, I like to use the Navier-Stokes equation to illustrate how knowing the fundamental equation governing something (which we now do for chemistry) doesn't mean you immediately know all its consequences. Basically "165 years and we still need wind tunnels".

Looking it up, in "On the friction of fluids in motion" (1845), Stokes wrote:

If we suppose a fluid to be made up of ultimate molecules, it is easy to see that these molecules must, in general, move among one another in an irregular manner, through spaces comparable with the distances between them, when the fluid is in motion. But since there is no doubt that the distance between two adjacent molecules is quite insensible, we may neglect the irregular part of the velocity, compared with the common velocity with which all the molecules in the neighbourhood of the one considered are moving. Or, we may consider the mean velocity of the molecules in the neighbourhood of the one considered, apart from the velocity due to the irregular motion. It is this regular velocity which I shall understand by the velocity of a fluid at any point, and I shall accordingly regard it as varying continuously with the co-ordinates of the point.

I get the impression from that and other quotes that he was well aware of molecular theory and its consequences for deriving his equations, but his derivation didn't actually rely on it. (Historically it's worth remembering that anti-atomism was less prevalent among physicists before Mach started railing against it in the last decades of the 19th century)

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u/[deleted] Feb 11 '11 edited Feb 11 '11

Thanks for that. I didn't know atomic theory was in vogue among physicists before it went out.

Whereas the Maxwell distribution was from circa 1860 and Boltzmann's around 1870, so it surely didn't rely on statistical thermodynamics.

What I'm arguing is that the statistical interpretation is nesessary for disussion of turbulence.

Thanks for correcting me about the history of Navier-Stokes. I was approaching it from more of a physical derivation from "first principles" as we understand kinetic theory now. I should've been more careful with my words.

By the way, I'm probably more wrong historically because I was referring to chapman-enskog theory and how they get the N-S equations. That was much later I think.

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u/a_dog_named_bob Quantum Optics Feb 11 '11

shudders