(Brady: Dude, this is the first for Numberphile)
On the ribs as well. The pain of the ribs, like
there's - I'm gonna say this is a complete commitment to,
to Navier-Stokes and fluid mechanics.
(Brady: Why did you do that?)
Well...
What are the Navier-Stokes eqations?
The first one: grad dot u = 0
And the second one
Rho, Du by Dt = - grad p + Mu grad squared u + rhoF
Like most mathematical equations perhaps, they look quite scary,
but they're actually universal laws of physics.
So this is why these guys model every single fluid we have on Earth.
And I mean every fluid.
Put a fluid in your head,
these guys tell you how it moves, how it behaves,
what's going on with that fluid.
So a fluid, I tend to think of it as something which changes shape
to match the container that it is in.
Of course a liquid, but also a gas, even some solids.
So like ice, you think about a glacier flowing down a mountain,
you watch those time lapse videos, it kinda looks like a river.
So ice in that sense behaves like a fluid,
it is changing its shape to fit the valley, the glacier valley, which it's flowing in.
So the first one
this is literally just saying
that mass is conserved.
So this is just saying I have some blob of fluid,
it moves around, you know, with some velocity and maybe changes shape,
I'm not adding anything, I'm not removing anything
I would want the same mass of fluid
to still be there.
Mass is conserved. It's a
pretty standard law of physics, makes a lot of sense.
This guy is just Newton's second law.
So this is just telling us that
mass times acceleration is force.
The top one tends to be the mass equation, conservation of mass,
or the incompressibility equation.
And the second one is the momentum equation - or
the small one and the big one. I quite like that.
Let's start with the little one.
So u, here, this is our velocity.
So this is just speed with a direction.
It's a vector. You might say u is equal to u, v, w.
So we've got a component in an x-direction, a component in a y-direction, a component in a z-direction.
How fast the water in a river is flowing, how fast the air around the Formula 1 car is going.
Thinking about a aerodynamics, how fast the custard is flowing off your spoon.
Any sort of movement of the fluid, the speed and that motion
is going to be encapsulated by the velocity, that's the key thing
about its motion.
We've then got this other symbol here, nabla,
we do love our Greek letters in maths. So the symbol nabla is,
is telling us what to do to our velocity u.
So nabla is a gradient, it's a derivative.
It's telling us to differentiate our vector u in a particular way.
And a way it tells us to do that
is we have our three components - u, v, w
and what we're gonna do is we're just gonna do
du by dx, so differentiate the first bit with respect to its coordinate, which is x.
It's the first coordinate, x coordinate, differentiate with respect to x.
Then we're gonna add derivative of the second one with respect to our second coordinate - dy.
And hopefully you figured out the pattern. We're then gonna add on dw by dz.
So this is the divergence of our velocity, it's just three derivatives.
So it's saying how does the x component of my velocity u,
how does that change as I move in the x direction?
Then how does v, my y component, change in the y direction? And how does the third component w, in the z direction,
how does that change in the z direction? So this is equal to zero, equation number one. This just tells us mass is conserved.
Now, the second one, the big boy. So Newton's
second law in disguise. So
we're expecting force equals mass times acceleration.
Here we've got u, our velocity, and when we take a time derivative of velocity
that's exactly what acceleration is. You're going at a speed, you increase your speed,
you've accelerated. Your speed has changed with respect to time - or you decrease your speed, you decelerate it.
So that's what the first term is going to describe and then we also need the mass. You think of
mass in this situation to be a density.
So it's how, you know, fresh water has a smaller density than saltwater.
So saltwater is heavier in that sense, but it's, you sort of, mass and density are the same thing when it comes to fluids,
that's how you you sort of work with those things. So rho, this Greek letter here, that is going to be our density.
So this is our mass in that situation. So this is mass times acceleration.
That's our Newton's second law of the left-hand side. And then all of this
sort of stuff going on over here, these are just all the forces.
So what we've got, we have the first two terms. These guys are what we call the internal forces.
So this is the force between all of those fluid particles hitting into each other, crashing
sliding, grinding past one another - there's internal force there. And then the third one - it's just capital F,
this is a bit of a cheat because we just say this is our external force F.
So this could be,
gravity is the standard thing you would, normally you just replace F with G,
call it gravity. That's your external force in most situations.
And if you want to go really fancy
you can put in electromagnetism. And we can sort of combine
Navier-Stokes with Maxwell's equations and get magnetohydrodynamics.
And that is how stars form and galaxies form and that is just next-level. Navier-Stokes was hard
magnetohydrodynamics, like trying to model the growth of the Sun.
It's as hard 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.
So this is very similar to our mass equation up here. So the gradient of p, our pressure gradient, is a vector
representing the change in pressure. When there is a difference in pressure, a change in pressure, high pressure over here, low pressure over here,
the air moves, from high pressure to low pressure. If there's a gradient there's a difference in pressure between two points
that causes the fluid to move between or along that pressure gradient.
So that is creating a force. So then the final internal force - this guy. So this is viscosity. Kind of what it's made up,
so you've got all the, you can think of it as being in layers and it's slid-- those layers slide past one another, they create a friction.
And how strong that friction is, is is the viscosity. So air, super thin, air particles
they move around, they do their thing. It's all good. But if you've got like, honey
sliding past other bits of honey, it's very sticky. It's very thick much larger viscosity. So that's your second internal force.
(Brady: Where's the problem with all this?)
This is why these equations are so great. Because it's mass is conserved,
it's Newton's second law,
it's all just, makes complete sense.
There is nothing we have said here, hopefully, that anybody can possibly disagree with. It's just Newton's second law, move
my fluid around, mass is conserved. And this is why these equations work. So they've been around since the 1820s,
the 1840s where Navier and Stokes both worked on them and
this is why they work, this is why we keep studying them. But the problem is
we don't actually know if they always have solutions. They can be used for almost anything you can think of involving a fluid.
So they could be
aerodynamics of a Formula One car, they could be designing new aircraft to go faster than the speed of sound,
this could be blood flow around the body for drug delivery,
maximising the way the drug is deposited to do with the blood flow.
This could be pollution modelling, climate modelling, ocean modelling - or anything involving a fluid. It has to satisfy these equations.
So these are always your starting point, but then the problem is, and this is why you get the million dollars,
we just don't really understand them
mathematically. In the sense that when you have a set of equations,
as a mathematician you want this, that set of equations
to satisfy three particular properties. First of all, you want a solution to exist.
You know? Got equation, you want to go, right, I want to be able to solve it. That would be nice.
Second of all you want a unique solution, you know
if you did an experiment with throwing a glass of water across the room and then you did it again, and it did something different,
you know, it did like a loop-de-loop in the air. It makes no sense. And then the tricky bit is you want smooth
solutions.
Well-behaved
solutions. Like, I've made a tiny little change in how I started my experiment
and I want the result to also just have a tiny little change. Quantifying tiny, but not to blow up to infinity.
That wouldn't make sense because I've changed something so, so small - why have I got an entirely different solution?
(Brady: We've also been taught that butterflies flapping their wings can cause cyclones)
The butterfly effect - it's like a chain reaction. It's one thing leads to another leads to another.
But in the in the sense of having an equation,
you input something into your equation - it's like a function machine. You input some initial condition, it outputs
what's gonna happen next, predicts almost like the future. So you start with your fluid, it has some velocity,
our u. Some pressure, some viscosity, input it - Navier-Stokes tells you how that fluid is going to move.
You know, you can do these experiments so
you know what's gonna happen and you want the equations to give you that result.
If you just tweak your velocity a tiny bit, if you did that experiment you'd 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 a solution exists all the time.
So given an initial condition - here's a velocity, here's a pressure, input it. We don't even know if a solution's going to come out.
(Brady: So how come these equations are being used by)
(climate modellers and by Formula One teams and all these things that you tell me are using Navier-Stokes. It sounds like they're using a really unreliable tool.)
I wouldn't say it's unreliable, because it's based on
such standard laws of physics: mass is conserved, Newton's second law. So I think everyone's happy that that makes complete sense.
We've got that bit, right.
There's no reason for that not to be correct.
But then the the sort of the mathematical complexity of it, the tricky bit, is
we don't know if a solution is always going to exist and so we kind of find ways to
cheat.
So we might make
simplifications, we might make assumptions to reduce some of these terms, or to remove time from the problem. Or you can
get ways around it by making
assumptions and making simplifications.
So that's one way to use them. And then another way is to,
what's called averaging. So rather than having a velocity field defined everywhere,
you can say "well what if I just take a big circle of fluid, take the average velocity in that circle,
and I want to know how that changes and how that behaves?"
That we can do, so that's called averaging, Reynolds averaging of the Navier-Stokes equations
and that is the kind of stuff that we do in climate modelling. Because you can't - the computer power alone to model every
particle in the atmosphere, like, it's gonna take longer than the life of the earth to run that on current computers.
So you just say "well I'll just model the atmosphere as patches of let's say ten kilometers squared bits
of air, and as long as I know the average speed in that 10 kilometers squared I'm happy" So it's all about the averaging. But,
mathematically, it should in theory be able to be solved for each individual bit
and that's where we struggle.
I think the best way to figure out
what you need to do to get the million-dollar prize is to think about what we've done already.
So we have the equations. They make complete sense; we're happy with them. Mass is conserved, Newton's second law, great, and then
we also know that solutions exist and they are well behaved in two dimensions.
Unfortunately, we don't live in a two-dimensional world,
we have three dimensions. So if we were to ignore z and just have x and y, our two dimensions, we can do it,
we can show there's always a solution. It's always well behaved, it works.
And then when you go to three dimensions for whatever reason, it's just not working, we can't do it.
We have shown that weak solutions exist. So these are sort of rather than being full solutions,
they're similar to the averaging solutions, not exactly, but like some form of solution we can get.
We can get solutions when the initial velocity is really small.
We restrict to say we're only going to move at tiny speeds to start with then we can show solutions always exist.
We can show solutions always exist for finite time,
so, so up to some, you know time equals 100 or something we can get solutions exist as well.
But we just cannot get that they exist for three dimensions for all of time for all possible
Initial conditions. (Brady: So Tom will the person who wins the Millennium prize be the person who explains why that's the case?)
(Like of course, you can't do it in three dimensions, and this is why. Or could they possibly say)
("Yes, you can do it in three dimensions you silly-billy. There was something you didn't notice")
(Or what what will that person do or what are they being asked to do?)
Yeah, so the the wording of the Millennium Problem is fantastic. It just says further our understanding
of the Navier-Stokes equations.
It's, it's the most vague of statements but there are some attempts that qualify or quantify what that means.
It could be a case of "show that a solution always exists in three dimensions for all possible initial conditions" or
it could be a case of as you said say "well, of course
it's going to blow up in three
dimensions and it's going to go to infinity and there's we can't expect solutions to exist" So you can do it both ways.
The most recent progress in 2016 by Terence Tao, he actually showed that for the averaged
Navier-Stokes equations, that you do get blow up in finite time. The way he's approached that could be a way of showing that
the full 3D equations could also blow up in finite time.
So that would be that we don't always expect a
solution. So we don't really know which way it's going to go. I think one of the key things is about understanding
turbulence. So turbulence is this
chaotic, random
motion of air,
particles of water; think of two waves
crashing into each other in the ocean. That is almost as random and chaotic a situation as you can get. If you do that again
you're not going to expect the same thing. It's, it's so so difficult to model and to understand. And
fundamentally fluids - they just are turbulent.
Not every fluid but like air, water, rapids; that's all, it's all turbulent.
And that's sort of I think where the key problem lies, because when we plug our data into our computers the computer
averages things because it can't solve for the turbulence. It can't solve for all those really small little bits and all those little interactions.
So it says, I'm going to take this big square and average the velocity or average the length scale and
that's the way that we do it and it works but it's not
mathematically
understanding the equations.
Practically we are happy with Navier-Stokes. We can use them, the equations for basically anything we want - it works,
it's great, it looks amazing. It allows us to design all of these amazing aircraft to fly into space all of this stuff,
it works. But it's just from a mathematics point of view, we just don't have that proof.
It's a classic case of maths wanting to know for certain,
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 it might not.
(Brady: If I gave you the dream computer,)
(that doesn't exist, and all the possible inputs you need at the start, 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 bit of data, any initial condition you want.
But then the computer may say "oh, but the solution goes to infinity,
the solution blows up". And of course in a real situation we cannot have
a water particle, a piece of air, moving at infinite speed.
That just makes no physical sense. But the computer is saying that's the solution. So there's some disagreement there between the physical
practicality of what can happen,
and what the computer is outputting. And that would suggest that there's something missing in our mathematical understanding.
(Brady: So are you telling me that you can input)
(finite numbers into the Navier-Stokes equation, like realistic numbers that would apply to something in the real world, and)
(Navier-Stokes spits out)
(impossible things like, "oh, yeah. Your river is gonna be flowing at infinity miles per hour" and things like that?)
Exactly, yeah, so that there's there's a really— (Brady: Well then the equations are wrong!)
But, but it's conservation of mass and it's Newton's second law. How can it be wrong?
This is the, this is the sort of paradox.
It's, it's everything we've done here makes complete sense. Because these, these two laws: mass is conserved, Newton's second law,
this will hold for any - like this works for a solid. It's not just fluid mechanics,
you do the same thing with solid mechanics. You get slightly different internal forces, but it's the same. It's the same set up.
So, so there's it doesn't seem like that bit should be wrong.
You know, it's, it's conservation of mass, its force is mass times acceleration. There's no reason to think that that's incorrect
so something, somewhere is going wrong in the
the mathematical
understanding. There's an example about the flow of a fluid around a right-angled corner.
(Brady: A bit like a canal?) - Basically a canal but we've got a really
sharp right angle on 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 canal do I have an infinite velocity canal?
I'm going to guess I don't. So,
it's, it's that. There's, there's no - like everywhere else it works perfectly,
but at this little point it's saying my velocity is infinite,
but it quite clearly isn't. That's it in a nutshell. The equations work for all possible practical uses.
So if you were building this canal you'd be completely happy. You would know the speed everywhere, except one single tiny, tiny point.
But then, as a mathematician, you're like, "But why is it infinity? Why is it infinity there? I want to know." And, and that's it. That's,
we're obviously missing that mathematical understanding.
There's some difference there between the physics of it and the equations, we just do not understand it. (Brady: So it's kind of like how)
(division, like just simple division, works and I use it every day.)
(It's just there's this is one glitch where if you divide by zero, it kind of goes a bit weird.)
(But that doesn't matter because I never actually need to divide by zero.) - Exactly. There are always ways around it.
There are approximations, there are, you know, you know it's a very large number so you just leave it at that.
(Brady: So this is something that doesn't really matter, but it matters to you?)
It doesn't matter to any
application of Navier-Stokes, but it does matter to
maths. And it's the classic case of "alright if we solve it
we have no practical uses" but to actually figure out what's going on here,
the maths we're going to uncover,
it's gonna be brand new. It's gonna be stuff
we've never looked at before, thinking of the problem in entirely new ways and no doubt
we will discover new maths; leads to all kinds of other incredible new advances in all this stuff to do with fluids.
It's just, you know, all of these things that we use Navier-Stokes for;
pollution modeling, climate modeling, blood flow,
aerodynamics - all of these things. If we really understand the equations those things are just gonna get better.
(Brady: Tell me how much you love these equations.)
Umm, well, they are my favourite equations. And just, just to emphasise the level of favouritism of these equations - the full
Navier-Stokes equations, as written down on our piece of paper.
(Brady: Hang on which way around are we?)
If we're talking this, if - yeah. So those should be exactly the ones I've written down.
(Brady: Oh there we go.) - The little guy and the big guy. (Brady: Dude)
(This is a first for Numberphile.) - On the ribs as well, the pain of the ribs.
Like, that's, I'm gonna say this is complete commitment to, to Navier-Stokes and fluid mechanics. (Brady: Why did you do that?)
Well, I,
ny PhD was studying fluid mechanics and these equations just - they just model it.
So I spent four years of my life
trying to - not necessarily understand them but studying them and using them, and it just felt like, it felt like the right thing to get
to be honest. (Brady: What did what did the artist say when you - )
He had a lot of fun with it. So he's actually,
he's quite an intelligent guy. He's got a load of like physics formulas and he has a portrait of Einstein tattooed on his back.
So he's really into his physics and his maths himself. So he was saying "Oh, I've never seen these what are they?"
So I spent the sort of two hours of being tattooed pretty much doing what we just did and talking about Navier-Stokes.