5 ms·

Summary: Navier-Stokes cannot translate to Boltzmann, because Navier-Stokes is incomplete... ...and even the best candidate to replace it fails at extremely low

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.

danbruc 11y ago
I never thought about this before reading the article but now it seems pretty obvious to me that both descriptions can not yield the same results under all circumstances. The Navier–Stokes equations are based on quantities like density and flow velocity which are only really meaningful if you have sufficiently many particles to average about. In consequence I am hardly surprised that one gets disagreeing results under extreme conditions like very low densities.
- semi-extrinsic 11y ago
  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. </grumpy-reviewer-mode>
  - tanderson92 11y ago
    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.
    - vanderZwan 11y ago
      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.
- darkmighty 11y ago
  I think it's not that obvious. For example, Maxwell's equations represent essentially a complete description of classical electrodynamics. Even though in most formulations it involves densities, it will work fine for point-like charges where the density is singular but integrable. Of course, when QM comes in the theory expands to quantum electrodynamics (QED), but Maxwell's equations match very well (exactly for linear media?) the quantum mechanical predictions when you average the probabilities. It sounds reasonable to expect a fluid theory that explains well macroscopic phenomena, even if the fluid really is granular, where we could hope the approximation would hold on average or something like that. But apparently it turns out that N-S doesn't model important qualitatively distinct phenomena.
pwnna 11y ago
I don't know if I'd call the theory "flawed". Navier-Stokes is a very successful set of equations that is capable of solving real problems as it sets the foundation for modern day computational fluid dynamics. Any subsequent theories must converge to Navier-Stokes at the regimes where Navier-Stokes is highly successful.
OopsCriticality 11y ago
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 theories that operate over ranges for which the governing assumptions for NS and GR no longer apply. Aesthetically, it would be lovely to have a master equation that works from the molecular scale up to the bulk scale. Practically, that master equation would likely reduce into the familiar forms that are presently used.
- vanderZwan 11y ago
  > 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.
gcb0 11y ago
flawed is what scientific journalism would say. everyone that deals with fluids know that the tools are mostly Balance Equations and such. i.e. good enough to predict everything mostly right. everyone know none of it is complete... or elegant.
tjradcliffe 11y ago
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).

danbruc 11y ago

I never thought about this before reading the article but now it seems pretty obvious to me that both descriptions can not yield the same results under all circumstances. The Navier–Stokes equations are based on quantities like density and flow velocity which are only really meaningful if you have sufficiently many particles to average about. In consequence I am hardly surprised that one gets disagreeing results under extreme conditions like very low densities.

semi-extrinsic 11y ago
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. </grumpy-reviewer-mode>
- tanderson92 11y ago
  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.
  - vanderZwan 11y ago
    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.
darkmighty 11y ago
I think it's not that obvious. For example, Maxwell's equations represent essentially a complete description of classical electrodynamics. Even though in most formulations it involves densities, it will work fine for point-like charges where the density is singular but integrable. Of course, when QM comes in the theory expands to quantum electrodynamics (QED), but Maxwell's equations match very well (exactly for linear media?) the quantum mechanical predictions when you average the probabilities. It sounds reasonable to expect a fluid theory that explains well macroscopic phenomena, even if the fluid really is granular, where we could hope the approximation would hold on average or something like that. But apparently it turns out that N-S doesn't model important qualitatively distinct phenomena.

pwnna 11y ago

I don't know if I'd call the theory "flawed". Navier-Stokes is a very successful set of equations that is capable of solving real problems as it sets the foundation for modern day computational fluid dynamics. Any subsequent theories must converge to Navier-Stokes at the regimes where Navier-Stokes is highly successful.

OopsCriticality 11y ago

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 theories that operate over ranges for which the governing assumptions for NS and GR no longer apply. Aesthetically, it would be lovely to have a master equation that works from the molecular scale up to the bulk scale. Practically, that master equation would likely reduce into the familiar forms that are presently used.

vanderZwan 11y ago
> 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.

gcb0 11y ago

flawed is what scientific journalism would say. everyone that deals with fluids know that the tools are mostly Balance Equations and such. i.e. good enough to predict everything mostly right. everyone know none of it is complete... or elegant.

tjradcliffe 11y ago

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).