Navier-Stokes Equations - Numberphile

(Brady: Dude, this is a first for Numberphile) In the ribs too. The rib pain, like there's... I'm going to say it's a total commitment to Navier-Stokes and fluid mechanics. (Brady: Why did you do that?) Well... What are the Navier-Stokes

equations

? The first: grad dot u = 0 And the second Rho, Du times Dt = - grad p + Mu grad square u + rhoF Like most mathematical

equations

, they may seem quite scary, but they are actually universal laws of physics. That's why these guys model each of the fluids we have on Earth. And I mean all fluids. Put a fluid on your head, these guys tell you how it moves, how it behaves, what's happening with that fluid.

So I tend to think of a fluid as something that changes shape to adapt to the container it's in. Of course, it is a liquid, but also a gas, even some solids. So like ice, you think of a glacier flowing down the mountain, you watch those time-lapse videos, it looks like a river. So in that sense, the ice behaves like a fluid, it is changing its shape to adapt to the valley, the glacier valley, that it flows through. So the first is to literally say that the mass is preserved. So this just says I have a drop of fluid, it moves, you know, with a certain speed and maybe changes shape, I'm not adding anything, I'm not taking anything away.

I would like the same mass of fluid to stay still. be there. The dough is preserved. It's a pretty standard law of physics and it makes a lot of sense. This guy is just Newton's second law. So this just tells us that mass times acceleration is force. The upper one tends to be the mass equation, the conservation of mass or the incompressibility equation. And the second is the momentum equation, or the small and the large. I quite like that. Let's start with the small one. So here, this is our speed. So this is just speed with one direction.

It is a vector. You could say that u is equal to u, v, w. So we have a component in the x direction, a component in the y direction, a component in the z direction. At what speed does the water flow in a river, at what speed does the air circulate around the Formula 1 car. Thinking about aerodynamics, how quickly does the cream come out of the spoon. Any type of movement of the fluid, the speed and that movement will be encapsulated by the speed, that is the key to its movement. Then we have this other symbol here, nabla, we love our Greek letters in math.

So the nabla symbol tells us what to do with our velocity u. So nabla is a gradient, it is a derivative. It's telling us to differentiate our vector u in a particular way. And one way it tells us to do that is we have our three components: u, v, w and what we're going to do is just do du times dx, so we differentiate the first bit with respect to its coordinate, which is x. It is the first coordinate, x coordinate, derived with respect to x. Then we will add the derivative of the second with respect to our second coordinate: dy.

And I hope you have discovered the pattern. Then we will add dw by dz. So this is the divergence of our velocity, it's just three derivatives. So it says: how does the x component of my velocity u change as I move in the x direction? So how does v, my y component, change in the y direction? And how does the third component w, in the z direction, change in the z direction? So this is equal to zero, equation number one. This simply tells us that mass is conserved. Now, the second one, the big boy. So Newton's second law in disguise.

So we expect force to be equal to mass times acceleration. Here we have u, our velocity, and when we take a time derivative of the velocity, that's exactly what acceleration is. You go at a speed, you increase your speed, you have accelerated. Your speed has changed with respect to time; or you slow down, you slow it down. That's what the first term will describe and then we also need the mass. You think of mass in this situation as a density. That's how, you know, fresh water has a lower density than salt water. So salt water is heavier in that sense, but in a way, mass and density are the same when it comes to fluids, that's how you work with those things.

So rho, this Greek letter here, will be our density. So this is our mass in that situation. So this is mass times acceleration. That's Newton's second law on the left side. And then all these kinds of things that happen here, these are just all the forces. So what we have are the first two terms. These types are what we call internal forces. So this is the force between all those fluid particles colliding with each other, colliding, sliding and rubbing against each other - there is an internal force there. And then the third one: It's just capital F, this is a little bit misleading because we're just saying this is our external force F.

So this could be, gravity is the standard thing you would do, normally you just replace F with G, call it gravity. That is your external strength in most situations. And if you want something really fancy, you can incorporate electromagnetism. And we can combine Navier-Stokes with Maxwell's equations and obtain magnetohydrodynamics. And that's how stars and galaxies form, and that's the next level. Navier-Stokes was about tough magnetohydrodynamics, like trying to model the growth of the Sun. It's as difficult as it sounds. Internal forces versus external forces. So it's just a sum of various forces. What are these individual forces?

So let's look at the first one, this nabla p or grad of p. This is very similar to our mass equation up here. So the gradient of p, our pressure gradient, is a vector that represents the change in pressure. When there is a pressure difference, a pressure change, high pressure here, low pressure here, the air moves, from high pressure to low pressure. If there is a gradient, there is a pressure difference between two points that causes fluid to move between or along that pressure gradient. So that is creating a force. So, the final internal force: this guy.

So this is viscosity. Pretty much what it's made up of, so you have everything, you can think of it as being in layers and it slides... those layers slide over each other, they create a friction. And how strong that friction is, is the viscosity. So the air, super thin air particles, move around and do their thing. Everything is alright. But if you have honey sliding around other pieces of honey, it's very sticky. It is very thick and has a much higher viscosity. So that is your second internal force. (Brady: Where's the problem with all this?) That's why these equations are so cool.

Because your mass is conserved, it's Newton's second law, everything is fair, it makes a lot of sense. Hopefully there is nothing we have said here that anyone can disagree with. It is simply Newton's second law, move my fluid and mass is conserved. And that's why these equations work. They've been around since the 1820s and 1840s where Navier and Stokes worked on them and that's why they work, that's why we continue to study them. But the problem is that we don't really know if they always have solutions. They can be used for almost anything you can imagine involving a fluid.

So, it could be the aerodynamics of a Formula One car, it could be designing new airplanes to go faster than the speed of sound, this could be blood flow around the body for drug delivery, maximizing the way it is deposited. the medicine in the blood. flow. This could be pollution models, climate models, ocean models, or anything involving a fluid. You have to satisfy these equations. These are always the starting point, but the problem is, that's why you get the million dollars, we just don't understand them mathematically. In the sense that when you have a set of equations, as a mathematician you want this or that set of equations to satisfy three particular properties.

First of all, you want a solution to exist. Know? I have the equation, do you want to go, sure, I want to be able to solve it. That would be fine. Second, you want a unique solution. You know, if you did an experiment by throwing a glass of water across the room and then you did it again, and it did something different, you know, it was like a loop loop. in the air. Has no sense. And then the complicated thing is that you want fluid solutions. Solutions with good behavior. For example, I made a small change to the way I started my experiment and I want the result to have a small change as well.

Quantify tiny, but don't fly to infinity. That wouldn't make sense because I changed something so, so small. Why do I have a completely different solution? (Brady: We have also been taught that the fluttering of butterflies can cause cyclones.) The butterfly effect is like a chain reaction. It's that one thing leads to the other and leads to the other. But in the sense of having an equation, you input something into your equation, it's like a function machine. You enter an initial condition, generate what will happen next, predict almost like the future. So you start with your fluid, it has a certain velocity, our u.

A little pressure, some viscosity, enter it: Navier-Stokes tells you how that fluid will move. You know, you can do these experiments to find out what's going to happen and you want the equations to give you that result. If you modify your speed a little, if you did that experiment you would get almost the same result. So you want that small change in the starting point to lead to a small change in the solution. And this is what we don't know about Navier-Stokes. We don't even know if there is a solution all the time. So, given an initial condition: here's a velocity, here's a pressure, enter it.

We don't even know if a solution is going to emerge. (Brady: So how come these equations are being used by) (climate modelers and Formula One teams and all these things you tell me are using Navier-Stokes? It seems like they're using a really unreliable tool) . I'm not saying it's unreliable, because it's based on standard physical laws: mass is conserved, Newton's second law. So I think everyone is happy that that makes a lot of sense. We have that part, right? There is no reason why that should not be correct. But then the kind of mathematical complexity, the complicated thing, is that we don't know if there's always going to be a solution, and therefore we find ways to cheat.

So we could make simplifications, we could make assumptions to reduce some of these terms or take time away from the problem. Or you can work around it by making assumptions and simplifications. That's one way to use them. And then another way is what is called averaging. So instead of having a velocity field defined everywhere, you can say "well, what if I take a large circle of fluid, I take the average velocity on that circle, and I want to know how it changes and how it behaves?" We can do that, it's called averaging, Reynolds averaging of the Navier-Stokes equations and that's the kind of thing we do in climate modeling.

Because you can't, computer power alone to model every particle in the atmosphere, it will take longer than the life of the Earth to execute that on today's computers. So you just say "well, I'll model the atmosphere as patches of, say, ten square kilometer chunks of air, and as long as I know the average velocity over those 10 square kilometers, I'm happy." So, it's about averaging. . But mathematically, in theory it should be able to be solved for each individual bit and that's where we have difficulties. I think the best way to know what needs to be done to get the million dollar prize is to think about what we've already done.

So we have the equations. They make a lot of sense; We are happy with them. Mass is conserved, Newton's second law, great, and then we also know that solutions exist and that they behave well in two dimensions. Unfortunately, we do not live in a two-dimensional world, we have three dimensions. So if we ignore z and just have x and y, our two dimensions, we can do it, we can show that there is always a solution. He always behaves well, it works. And then when you go to three dimensions, for whatever reason, it just doesn't work, we can't do it.

We have shown that weak solutions exist. So instead of being complete solutions, they are similar to average solutions, not exactly, but they are kind of a solution that we can get. We can get solutions when the initial velocity is really small. We are just saying that we will only move at small speeds to begin with and then we can prove that solutions always exist. We can show that solutions always exist for a finite time, so up to a certain point, you know time is equal to 100 or something, we can get that solutions exist as well. But we simply cannot understand them to exist in three dimensions all the time for all possible initial conditions. (Brady: So, Tom, the person who wins the Millennium Prize will be the person to explain why that's the case?) (Of course, you can't do it in three dimensions, and here's why.

Or could theysay it?) ("Yes, you can do it in three dimensions, silly. There was something you didn't notice.") (Or what will that person do or what will he or she be asked to do?) Yeah, so the wording of the Millennium Problem is fantastic. It simply improves our understanding of the Navier-Stokes equations. It's the vaguest statement, but there are some attempts to qualify or quantify what that means. It could be a case of "prove that there always exists a three-dimensional solution to all possible initial conditions" or it could be a case where, as you said, "well, of course it's going to explode in three dimensions and it's going to Ve to infinity and we can't wait. "There are solutions," so you can do it both ways.

Terence Tao's most recent progress in 2016 actually showed that, for the averaged Navier-Stokes equations, it explodes in time. finite. The way you approached it could be a way of showing that full 3D equations could also explode in finite time, so we don't always wait for a solution, so we don't really know where it will go. The key is to understand. turbulence. So turbulence is this chaotic, random movement of water particles; think of two waves crashing into each other in the ocean. That's about as random and chaotic a situation as you can imagine. You will expect the same.

It is very difficult to model and understand. And fundamentally fluids are turbulent, but like air, water, rapids, that's it. everything is turbulent. And I think that's where the key problem lies, because when we plug our data into our computers, the computer averages things out because it can't resolve the turbulence. You can't figure out all those little details and all those little interactions. So you say, I'm going to take this big square and average the velocity or average the length scale and that's the way we do it and it works, but you don't understand the equations mathematically.

We are practically happy with Navier-Stokes. We can use them, the equations for basically anything we want: it works, it's cool, it looks amazing. It allows us to design all these incredible planes to fly into space, all of this, it works. But from a mathematical point of view, we simply don't have that proof. It's a classic case of math wanting to know for sure, rather than just knowing that it works in every case we can think of, we don't have that proof that it will always work; or an explanation of why this might not be the case. (Brady: If I gave you the computer of your dreams) (which doesn't exist, and all the possible inputs you need at the beginning, this equation would get you there). (It would just take, you know, a lot of power.) - You make a good point.

You can import any data, any initial conditions you want. But then the computer can say "oh, but the solution goes to infinity, the solution explodes." And of course, in a real situation we cannot have a particle of water, a piece of air, moving at infinite speed. That just doesn't make physical sense. But the computer says that's the solution. So there is some disagreement between the physical practicality of what can happen and what the computer is generating. And that would suggest that something is missing in our mathematical understanding. (Brady: So, you're telling me you can enter) (finite numbers in the Navier-Stokes equation, like realistic numbers that would apply to something in the real world, and) (Navier-Stokes spits) (impossible things like, "Oh, yeah.

Your river will flow at infinite miles per hour" and stuff like that?) Exactly, yeah, so there's a really— (Brady: Okay, so the equations are wrong!) But, but it's conservation. of mass and Newton's second law. How can it be wrong? This is the kind of paradox. It's, it's that everything we've done here makes a lot of sense. Because these, these two laws: mass is conserved, Newton's second law, this will hold for anyone, this is how it works for a solid. It's not just fluid mechanics, it's the same thing with solid mechanics. You get slightly different internal forces, but it's the same.

It's the same configuration. So, it doesn't seem like that part should be wrong. You know, it's conservation of mass, its force is mass times acceleration. There is no reason to think that is incorrect, so something, somewhere, is failing in mathematical understanding. There is an example about the flow of a fluid around a right angle corner. (Brady: Kinda like a canal?) - Basically a canal, but we have a very sharp right angle in the corner. Now you solve Navier-Stokes, this says that at this point, this right-angle corner, I have infinite velocity. If I build this channel, will I have an infinite speed channel?

I'm going to assume not. So, it's, it's that. There is, there isn't, like everywhere else it works perfectly, but at this little point it says my speed is infinite, but it clearly isn't. That's all in a nutshell. The equations work for all possible practical uses. So if you were building this channel, you would be completely happy. You would know the speed everywhere except one small, very small point. But then, as a mathematician, you think: "But why is there infinity? Why is there infinity there? I want to know." And that's it. That is, we obviously lack that mathematical understanding.

There is some difference between physics and equations, we just don't understand it. (Brady: So it's kind of like) (division, like simple division, works and I use it every day). (It's just that there's a glitch where, if you divide by zero, it gets a little weird.) (But that doesn't matter because I never actually need to divide by zero.) - Exactly. There are always ways to avoid it. There are approximations, you know, you know it's a very big number, so leave it at that. (Brady: So this is something that doesn't really matter, but does it matter to you?) It doesn't matter for any application of Navier-Stokes, but it does matter for mathematics.

And it's the classic case of "okay, if we solve it we have no practical uses," but to really figure out what's going on here, the mathematics that we're going to figure out is going to be completely new. It will be things we have never seen before, thinking about the problem in completely new ways and we will undoubtedly discover new mathematics; It leads to all kinds of incredible new advances in all this fluid stuff. It's just that we use Navier-Stokes for all these things; Pollution modeling, climate modeling, blood flow, aerodynamics – all these things. If we really understand the equations, those things will improve. (Brady: Tell me how much you like these equations.) Umm, well, those are my favorite equations.

And just, just to emphasize the level of favoritism of these equations: the complete Navier-Stokes equations, as written on our piece of paper. (Brady: Wait, which way are we?) If we're talking about this, yeah... yeah. So those should be exactly the ones I've written. (Brady: Oh, here we go.) - The little one and the big one. (Brady: Dude) (This is a first for Numberphile.) - In the ribs too, rib pain. I mean, I'm going to say that this is a total commitment to Navier-Stokes and fluid mechanics. (Brady: Why did you do that?) Well, I, my Ph.D., was studying fluid mechanics and these equations just model it.

So I spent four years of my life trying to, not necessarily understand them, but studying them and using them, and I felt it was the right thing to do to be honest. (Brady: What did the artist say when...?) He had a lot of fun with that. So he's actually a pretty smart guy. He has a lot of physics formulas and has a portrait of Einstein tattooed on his back. So he himself is very interested in physics and mathematics. Then he'd say, "Oh, I've never seen these, what are they?" So I spent two hours tattooing pretty much doing what we just did and talking about Navier-Stokes.