The Navier-Stokes equations can be solved exactly only in highly symmetric setups, like Couette flow, where fluid between a stationary plate and a moving plate develops a simple linear velocity profile because symmetry cancels out the equation's hardest nonlinear terms. For complex geometries like a Formula One car, no closed-form solution exists, so engineers instead use computational fluid dynamics: breaking space into a grid of cells and solving the equations numerically cell by cell to approximate velocity and pressure everywhere. This numerical approach is treated as a practical engineering success, producing usable results for drag, lift, and downforce, but it is explicitly distinguished from a mathematical proof of what solutions to the exact equations must look like.
- 01
In Couette flow, the no-slip condition pins fluid velocity to zero at the stationary plate and to the plate's speed at the moving plate, with a linear ramp in between.
- 02
Other exactly solvable cases mentioned include pipe flows and certain vortices and jets, where symmetry can be exploited to cancel nonlinear terms.
- 03
CFD results are described as a giant table of numbers rather than a formula, and simulations are inherently limited by finite grid size, finite time steps, and rounding.
- 04
The hosts note that simulating a shape well enough for engineering, illustrated with the meme of testing a cow's aerodynamics, is not the same achievement as resolving the mathematical existence and smoothness problem.
2,130 words · auto-generated from the episode video
1:33:36>> Yeah. Okay. So, we have a set of differential equations, and we'd like to have solutions to those differential equations. Now, what do we mean by that? Like, in a normal algebra problem, if I write something like x + 2 = 5, I know what x is. X is three, right? I can like do the thing. In a differential equation, what we're asking is something larger. What is the function that solves the differential equation? In our case, it would be what is the velocity field that solves this differential equation. Let's take a simple example for let's take for example um the derivative of y in terms of t is equal to some r * y. Meaning um I've got some quantity. The
1:34:16change of that quantity with respect to time is proportional to the quantity itself multiplied by let's say some rate. That is an exponential, right? The the solution to that is e to the power of RT. And this is how people get rich. This is uh this is compound interest, right? You start with some amount of money and then the rate tells you the interest over time. And the more money you have, the faster your money grows. And this is a this is a very simple differential equation that we can solve very accurately. Right? This is a toy equation and the answer is very nice. Um, Navier Stokes is less accommodating. Okay. The unknown
1:34:57is not how much money you have like a single one-dimensional variable. It's a three-dimensional velocity field, right? >> So, it's a number at every point in 3D space, but that number has a direction, right? It's a it's a length and it's got a direction that's also three-dimensional. So it's it's it's quite it's quite massive like this this equation >> make it's and it's so funny because now I that I understand that it makes my brain hurt trying to say we want to solve for this now given that level of complexity which is now very clear what that level of complexity is >> exactly like the the the fluid is interacting with itself and everything. So this velocity field has to solve for
1:35:37that entire big thing, right? We got to find all of the arrows everywhere, >> right? >> In some sense. Yes. Right. For all time cuz the arrows might be changing >> over time >> over time. And we need to we need to have an expression for all of that. Now there are certain special situations in which the Navier Stokes is actually exactly solvable. >> Okay, it's not true that the Navier Stokes is just unsolvable. You can't solve this equation. There are situations, there are certain boundary conditions where you can exactly solve the Navier Stokes equations. Um, let's go through one of them. It's called plain coet coete flow. I never know how to pronounce it. >> I think it might be coet. >> Is it coete?
1:36:18>> Let us know in the comments. >> Yeah. In any case, French. >> Yeah, it it's probably French, which is why I think it's coete. Um, here's what's happening. You've got two plates. Okay, those are the Navier Stokes equations that we're trying to solve. Yes, >> I've got two plates. One plate, let's say, is the the table >> um completely stationary. Another infinite plate is moving in the x direction in just one direction at a certain velocity. And in between two plates, you've got water. You've got some kind of fluid. Okay. What is the the velocity profile of the water in between these two plates? It's very simple, right? There's a stationary plate. There's a plate on top that is moving at a spec fixed velocity in one
1:36:59direction. and there's fluid in the middle. So, what is the is the velocity field going through? Now, we know from the no slip condition that the fluid that's right next to the table that's stationary is going to be stationary and the fluid that's on the plate that's moving is going to be moving with the plate. What happens in the middle? >> It's actually pretty simple. >> You solve for the Nav Stokes equations >> and you get this profile >> right >> near the bottom where it's stationary the velocity is zero. And as you approach the plate in the top, the velocity gradually increases linearly. And so it's just a function of the height, right? The higher you are, the closer you are to the top plate, the
1:37:39closer you are to the velocity. It's a linear relationship. Boom. >> Exactly solvable. >> That that makes total sense. And again, we've sort of created an exactly solvable simple >> Yeah. >> Uh use case because, you know, by having the bottom plate be stationary. >> Yeah. having the bottom even if the bottom plate is actually moving in the other direction totally fine as long as it's the same like you know the same velocity and there's so much symmetry here >> they're only moving it on one plane >> exactly it's only one coordinate that we have to worry about so a bunch of the derivatives over there just go to zero like there's so much symmetry and geometry here that it murders all of the difficult terms of that equation and
1:38:19you've got this steady profile and you're good to go. It's a recurring theme in fluid mechanics which is if you've got a nice symmetric distribution of stuff you can solve things >> fine you're fine. Yeah. This is called coete flow. You can also do pipe flows like there's a handful of vortices and jets where you can obtain beautiful closed form solutions based on the symmetry of the situation. And just to ask as a a brief side question, part of the other part of why those are so solvable is because we have such a good understanding of geometry as like like these other mathematical disciplines, we can apply some of those learnings and concepts in a way that make these easy or it's not it's not not relevant at
1:39:00all. >> Well, no, certainly I think I think I mean for that one for that one it's like pretty easy because you can just take derivatives and they go to zero and whatever, right? But there's other ones like there's certain vortices and jets where you can manipulate the symmetry to cancel out terms and things like that. You can you can use our knowledge of symmetry, >> right, >> to to to make like transformations that make these equations much easier to handle. The the subtle note I'm just trying to make here is is these things don't happen in a vacuum. Like our general mathematical understanding% >> has applications in subtle small ways all through. >> All through all through >> Yeah. all through. And so for that's that's for a very nice symmetric thing.
1:39:40Formula 1 cars not very symmetric. I mean there's some symmetry I guess left and right, right? So sure you can see that it's symmetric that like you can flip the the profile, right? But a Formula 1 car is insane. The geometry is extremely complicated. The wheels rotate. The ground moves relative to the car. There's thin boundary layers on the bodywork entirely. Air is accelerated beneath the floor. The vortices are deliberately generated around these waves to cause downforce. Um the nonlinear term does not vanish in the other one. The nonlinear term actually vanishes >> just because of the geometry of the thing here. This is the entire problem, right? There's no known formula that you
1:40:22can write out for what a Formula 1 car the the velocity and pressure at every point around a Formula 1 car is going to use. So instead what engineers do even in this visualization over here what they're doing is using a computer to computationally solve the Navier Stokes equation around this boundary. Okay. And this is called computational fluid dynamics. If there's any aerospace engineers or hydrodnamic engineers in the audience, leave one in the comments and tell us your favorite uh computational fluid dynamics program. There's a bunch of programs out there. This is one that I just picked up like on GitHub. And they always flaunt the
1:41:02number of cells that they can um break down their space into because you know imagine a Formula 1 car in a wind tunnel, right? That's some like volume. >> In order to solve these differential equations, you have to break up that volume into tiny little grid boxes. And then you can solve the Navier Stokes equations for each of the grid boxes. And you just can put them all together. >> Yeah. You can be like, "Oh, the velocity is here, but the velocity next to it is something." The the velocity to the right of and left of it is something. So, I can calculate a derivative that way, right? I can be like, "Oh, the difference divided by the grid spacing is my spatial derivative." >> Yeah. >> Right. And and I can I can do this computationally one at a time, and the
1:41:42more cells I have, the the better granularity and the more accuracy I have to the real thing. This this makes me think back to our hypersonics episode where we effectively discussed the same thing. >> Yeah. >> And and then sort of in addition to all of the challenges we've talked about >> uh when you start getting going from supersonic to hypersonic things get even weirder. >> Yeah. >> And we sort of discussed why it gets weirder and how we try to deal with that with these approximations that we've sort of discussed here. Again, Navier Stokes is not unsolved in all cases. We found clever ways. >> Yeah. To like approximate
1:42:23>> Yeah. a solution, right? And for engineers, this is good enough. The point they're like, honestly, like Yeah. >> Open AAI didn't solve anything. We've been modeling a Nav Stokes since you were before you were born. >> Yeah. For the Ferrari engineers, it's not good enough because apparently they still can't do it. Like, guys, figure it out. You have Lewis Hamilton and Charlotte Clair. Like guys, come please. I've been waiting for so long. Anyways, the result is not a closed form solution. Okay, it's this enormous table of numbers that tells you what the what the velocity is at every given point. And you can put any arbitrary shape into it. This is like a meme now in aerodynamics for some reason. Like all these aerospace
1:43:03engineers like put a cow, a CAD model of a cow into their CFD simulations and they're like, "H how aerodynamic is a cow?" I've seen so many versions of this meme for some reason. I don't know why, but a cow is a very non-erodynamic thing that people are trying to figure out, you know, what is the lift on a cow as it goes through a wind tunnel. In any case, the idea is any arbitrary shape, you can put it into a simulation >> and you can turn the simulation one after the other and computationally solve the Navier Stokes equations. >> For all practical purposes, that's totally fine. It's an engineering solution. >> It's not a physics understanding.
1:43:45>> Yeah. >> Andor mathematical understanding. And it's it's fine that it's an applied phys. >> It's a spectacular practical achievement. I mean, this is how we get >> airplanes. This is how we get fighter jets. This is how we get turbines. This is how we get wind uh wind farms for like power. Mhm. >> Um it's but it's not the same thing as the exact equations and what do those equations look like, right? What what is the solution to this thing look like for any arbitrary boundary condition? >> Um a simulation has a finite grid, right? It advances in finite time steps. It rounds numbers at some point. And so
1:44:26the problem is engineers are pretty happy taking Navier Stokes equations putting it into a CAD like putting CAD model in there using computational fluid dynamics they are good to go they'll tell you what the drag is they'll tell you what the lift is what the down force is on a formula 1 car and so on and so forth >> and unless you're making who hypersonics it's fine >> it's fine mathematicians on the other hand they're asking well what about like the existential problem of like I've got differential equations. Um, are the