The million dollar equation (Navier-Stokes equations)

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In this video, I introduce the Navier-Stokes equations and talk a little bit about its chaotic behavior. Make sure to like and subscribe!

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[vcubingx]

Hey all, I removed a part of the video that had some misinformation, hence the "jump" from one section to another. I made a large error in what I was conveying, so here's a correction by viewer Jay Raut:

From what I understand (and don't quote me, its been a while since I've dealt with fluid dynamics), the problem with the Navier-Stokes equations is the issue of them being ill-conditioned. By that I mean that a small change in the input does not result in a small change in the outcome. This is important since with any system, a small input change should always yield a small output change, otherwise the reliability of the solver is questionable (the results should be reproducible, and near infinitesimal changes should not result in drastically different answers).

Now while the the equations are basically glorified F=ma equations, which means that they are most likely the correct DE that describe the underlying physics, the problem lies in the fact that we simply don't understand or appreciate them enough. Also, remember that the real underlying physics is much more complicated. We can break down the problem to its core where we consider all the fundamental forces of the universe and the quantum effects between each particle in the fluid. But, this is meaningless because we want a meaningful compressed description of the physics, similar to how Newton's laws of gravity are a simpler version of Einstein's.

I've solved the Navier-Stokes equations by hand in undergraduate classes for simple problems, and in these cases the equations are very well behaved. The solving process is actually very logical to the point where you realise that all you are doing is Solving F=ma.

The problem comes down to turbulence, and the fact that the simple Navier-Stokes model do not capture this phenomenon at all. There have been very complicated proposals to the NS equations which take turbulence into account, but these are loosely based on analytical physics and more empirical solutions. Introducing this does not only create a more accurate solution, but employing some numerical trick also make the solution very stable.

Also, there is also the problem of the DE itself. Its not simple to solve, and the numerical methods we usually employ to obtain approximation, are exactly that: approximations.

So if you read the problem statement more carefully, you will realise that there is no straight forward problem that has to be solved. It's like the people didn't know what to set as the problem itself, which has become the problem. To essentially solve the millennium problem, you would need to come up with some form of proof that the NS equations are truly the underlying physics of a fluid (or not). Remember I mentioned the problem of ill-conditioning? Well even if that is true, that does not mean that the NS is BS, and the turbulence modelling tricks can make the solution very stable. However, these tricks are sometimes based on nothing more than: 'it works'. This is not progressive work and that is what the millennium prize tries to address.
So answering the question in terms of your words, we don't know if the solution (real) is smooth. because of which we don't know if using tricks to make our modeled solutions smooth is the correct thing to do to obtain meaningful answers either. And upon finding out whether or not it is, we'd also like to know why? Essentially: solve turbulence, because nobody knows wtf is going on.

A second mistake is that isothermal refers to no loss or gain in TEMPERATURE and not heat.

Sorry about that, I definitely got a bit confused when typing up the script.

I'm considering making a follow-up video as to what was wrong with the video and explain what we are actually solving.

[raealawlh]

Isothermal refers to a constant temperature process. A process during which no heat escapes is known as adiabatic process.

[raresmircea]

Kids today that have a natural inclination for maths live in the golden age of learning

[andrewfischer-garbutt]

"In terms of divergance we have no divergance." - Gru

[rafaellisboa]

I love the navier-stokes equations, I'd definitely watch a continuation of this. Good job man I like your channel very much

[carlos]

This is the best overview of the Navier-Stokes equations that I have seen. The intuitive explanations were very helpful. Thanks!

[KakoriGames]

Some already pointed out mistakes, some key information left out, but overall a nice video. Having tried myself, I know how difficult it is to make videos like these with Manim, so congrats. Also, nice to see more people doing videos on math subjects.

[명성현-sm]

I'd love to see more in-depth coverage of Navie Stokes and fluid mechanics. The video helped me understand so much. thank you

[josemanuelmedeltorrero]

That is not the definition of smoothness, smoothness means that it is infinitely differentiable. (Whatever that means) It comes to the study of functions on smooth manifolds, hence smooth functions. The pendulum, for example, I’m not sure that it’s solution has a closed form, but Banach Fixed point theorem assures us that there is a solution!! And it is smooth!!! Now, you can ask then, what would it mean to not be smooth? Well for example the absolute value is not smooth since it is not differentiable at 0. But more than that, experiments on turbulence have shown that turbulence in fluids looks like a fractal!!! And let me tell you, fractals are not smooth in general!! In my opinion turbulence shows us that there is a loss of structure (again, whatever that means).

[mathOgenius]

I also know manim a lot.. but how do we create and show particles in that vector field?.. can you please tell me ?

[swastikbiswas]

This channel will be having 1M subscriber in 3-4 years .. I got this after solving Navier Stokes equation

[ashwinidixit]

Great work I always use RANS (Raynolds average navier Stokes equation) but never had this much clarity of it.

[mattieohya]

When looking at Navier-Stokes the fundamental properties you are looking at are bulk properties and are impossible to define as a individual atoms. The infinitesimals are assuming a continuous fluid where there are no such things as particles. Think of density in the context of a particle, outside of the arbitrary area that defines that particle the density would be 0 and thus the system wouldn't be continuous.

Rarefied gas dynamics is the feild of fluid mechanics where a gas is treated as a random assortment of molecules. And uses a variety of methods to figure out fluid flow when molecules are so far apart these bulk properties break down.

[lukamitrovic]

So when I solve it, will it be navier - stonks?

[x_gosie]

I will be happy if you make a series about the 7-millennium problems, with this kind of visual representation.💕😍

[metelicgunz]

Navier-Stokes one of the best ways to scare prospective engineering students.

[drpkmath]

Pretty amazing video graphics! Good work!

[nadiyayasmeen]

Ah yes. The beautiful Navier-Stokes equations

[vivekt]

2:27 We are not describing the behavior of individual molecules of fluid through Navier Stokes equation. In fact, the velocity of individual molecules can be much higher than the flow velocity. Kinetics theory of fluids deals with that topic. In deriving the Navier Stokes equation, we rather treat treat the fluid to be a continuum.

[alexismisselyn]

When they say prove the solutions are smooth, does it mean that the solutions are smooth but we can’t prove it?

As you said we can’t predict weather too many days ahead, so that means the solutions are chaotic but we haven’t proven that either?

Can chaotic solutions be smooth?