Famous Fluid Equations Are Incomplete
quantamagazine.org
Famous Fluid Equations Are Incomplete
1–10 of 28 posts
Re: Famous Fluid Equations Are Incomplete
#2Summary: 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.
Re: Famous Fluid Equations Are Incomplete
#3If you are interested in some of the details:
The Navier-Stokes equations can be derived from the Boltzmann equation by applying a slight perturbation, expanding the result as a series, and taking the moments.
Taking the moments is essentially an integration, which comes with the implicit assumption that the system you're describing has sufficiently many particles. When running low on particles, this integration does not make sense. This is why the resulting equations do not apply at low densities.
The Navier-Stokes equations are the second order expansion of this procedure. The result of the first order expansion are the Euler equations.
This is called the Chapman-Enskog procedure. It's really quite illuminating when you see it for the first time. There's a great derivation in [1] if you can get your hand on it.
Re: Famous Fluid Equations Are Incomplete
#4Summary: 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.
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.
Re: Famous Fluid Equations Are Incomplete
#5"The terms in the series quickly become unruly, however; energy, instead of diminishing at shorter and shorter distances in the gas, seems to amplify."
This sounds a whole lot like the ultraviolet catastrophe. The solution there was quantization of energy packets and a statistical treatment of the fewer amount of packets that come through.
Re: Famous Fluid Equations Are Incomplete
#6Summary: 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.
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.
Re: Famous Fluid Equations Are Incomplete
#7> He began by rewriting the complicated Boltzmann equation as the sum of a series of decreasing terms. Theoretically, this chunky decomposition of the equation would be more easily recognizable as a different, but axiomatically equivalent, physical description of a gas — perhaps, a fluid description. The terms in the series quickly become unruly, however; energy, instead of diminishing at shorter and shorter distances in the gas, seems to amplify. This prevented Hilbert and others from summing up the series and interpreting it. Nonetheless, there was reason for optimism: The leading terms of the series looked like the Navier-Stokes equations when a gas becomes denser and more fluidlike. “So the physicists were happy, sort of," said Ilya Karlin, a physicist at ETH Zurich in Switzerland. “It’s in all the textbooks.”
This reminds me a lot of perturbation theory, a method used to solve the complicated equations of quantum field theory. The technique basically involves summing up a bunch of Feynman diagrams (of decreasing significance), and it has been used to calculate the value of the gyromagnetic ratio of an isolated electron to within 10 decimal places of its experimentally measured value (which is absolutely amazing, both from a theoretical and experimental standpoint).
However, what's peculiar about this summation is that it fails to converge. You would think that by adding up smaller and smaller terms, the series would eventually reach some limiting value, but that doesn't occur. So the most predictive theory that mankind has ever created (quantum electrodynamics) works only as long as you don't keep adding up more terms.
(*Technically speaking, this isn't a failure of QED, but of the method used to solve its equations. There are other solution techniques that don't have this problem.)
Re: Famous Fluid Equations Are Incomplete
#8Summary: 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.
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 unde…
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.
Re: Famous Fluid Equations Are Incomplete
#9The linked article is not related to the quest to determine whether the Navier-Stokes equations are capable of supporting Turing machine-like computation:
https://www.quantamagazine.org/20140224-a-fluid-new-path-in-...
Re: Famous Fluid Equations Are Incomplete
#10No wonder I had such a tough time in my Fluid Dynamics class. The material was incomplete! Do over!