A scenario in which the Euler fluid equations fail | Hacker News Reader

A scenario in which the Euler fluid equations fail

quantamagazine.org

62 pointsdigital5512 comments

To cut through the not particularly helpful analogies, the core result (which the article finally mentioned near the end) seems to be finding a situation where the Euler equations work, but only for a finite time before hitting a singularity. Previously, the only known cases either hit a singularity immediately or never hit one, i.e. every case either works forever or doesn't work at all.

I wonder if the illustration clip comes from this contraption by SmarterEveryDay: https://www.youtube.com/watch?v=EVbdbVhzcM4

The attribution is from Shutterstock, so unlikely unless Destin uploaded it there or someone stole his work.

Off topic, but if you're interested in fluid simulations and want to learn more about how they are actually made, John D Anderson's book on Computational Fluid Dynamics is worth working through.

It's been quite some time but I remember going through it in Matlab to make small, numerical simulations in the book.

The problem with fluids is that it's a very complex branch of physics with many gotchas and many difficulties that you must navigate through, often blindly, to get a reasonable approximation to a solution.

The fluids specialists I worked with on the past almost always needed actual physical tests to prove out their engineered designs, rework simulations, etc.

Does this lead to anything useful in the Navier-Stokes existence problem?

I'll guess no, or that would have been the bigger story.

Oh, true. And the article implies mathematicians were generally expecting this result. But AFAIK there's not a strong consensus on which way the Navier-Stokes problem will go. So probably unrelated techniques.

The constant attempt to break this down into easier analogies make it more confusing..

> They should be able to describe the motion of any fluid under any circumstances — and for more than two centuries, they have.

The Euler equations neglect viscosity.

> They also assume that fluids are “incompressible,” meaning that under the rules of the Euler equations, you can’t squeeze a fluid into a smaller space than the one it already occupies.

The Euler equations are compressible flow equations.

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These two mistakes could have been corrected had the author read the first sentence of the wikipedia page about the Euler equations.

The TL;DR is that there are two papers that show that under many assumptions, simplified versions of the Euler equations (which are already a big simplification), blow up.

https://arxiv.org/abs/1904.04795 https://arxiv.org/abs/1910.14071

Don’t be silly. You’re leaping from nitpicking to a blanket assumption that the author is completely clueless.

Elsewhere in the article: The Euler equations are not a literal description of a real-world fluid. They include several nonphysical assumptions.

The author does in fact know about the subject.

Further down in this article, Kevin Hartnett wrote:

> The Euler equations are not a literal description of a real-world fluid. They include several nonphysical assumptions. For example, the equations only work if internal currents within a fluid don’t generate friction as they move past each other. They also assume that fluids are “incompressible,” meaning that under the rules of the Euler equations, you can’t squeeze a fluid into a smaller space than the one it already occupies. > [...] > These unnatural provisos led the mathematician and physicist John von Neumann to quip that the equations model “dry water.” To model the motion of a more realistic fluid with internal friction (or viscosity), researchers use the Navier-Stokes equations instead.

So it seems they know what they are writing about, at least more than you supposed.

Thanks for toning down your attacks on the author! Although when you completely rewrite your post after a couple of replies, it’s polite to note that it’s been edited.

Your criticism is still incorrect. It’s true that there are both compressible and incompressible forms of “the” Euler equations, but the papers you linked to are very clear that they’re referring to the incompressible equations. The abstract of the more recent one begins:

We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation.

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