The pain of the rib steak.

I'm going to say this is complete commitment, Thio.

Never your steaks and fluid mechanics.

Why did you do that?

No.

What are the Na'vi Estates Equations?

The 1st 1 grand dot U equals, not on the 2nd 1 row.

Do you buy the T equals minor scratch P plus mu Grand Square.

You rode like roast 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, the fluid in your head.

These guys tell you how it moves, how it behaves.

What's going on with that fluid so fluid?

I tend to think of it as something which changes shape to match the container.

That it is, of course, a liquid.

But also a gas, even some solids.

So, like ice.

Do you think about a glacier flowing down a mountain?

You watch those time lapse videos kind of looks like a river.

So ice in that sense behaves like a fluid.

It is changing its shape to fit the valley, the glacial valley, which is blowing it.

So the first wall, this is list.

You 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 hiding anything.

I'm not removing anything.

I would want the same massive fluid to still be there.

Masses conserved.

It's a pretty standard.

Lower physics makes a lot of sense.

This guy is just Newton's second law.

So this is just having goes that mass times Acceleration is force.

The top one tends to be the mass equation.

Conservation of mass or the in compress ability equation on the 2nd 1 is the momentum equation or the small one in the big one.

I quite like that.

Let's start with the little one.

So you here, This is our velocity.

So this is just speed with a direction.

It's a vector.

You might say you is equal to U V.

W.

So we've got a component in the ex direction, a component in the UAE direction, a component in the set direction.

How fast the water in the river is flowing.

How fast The air around a Formula One car is going thinking about aerodynamics, how fast the custody is blowing off your spoon, any sort of movement of the fluid that 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.

Now, Abla, we do love our Greek letters in maths so that the symbol NAMBLA is telling us what to do to our were lost to you.

So Noblet is ingredient.

It's a derivative telling us to differentiate our vector.

You in a particular way on the way it tells us to do that is we have our three components.

U V w What we're gonna do is we're just gonna do, do you buy the ex?

So differentiate the first bit of respect to its coordinate, which is exits the first corner of the X coordinate different shape, respect to act.

Then we're gonna add derivative of the 2nd 1 with respect to our second coordinate, do I?

And hopefully you've figured out the pattern we're then gonna add on DW by the set.

So this is the divergence off our velocity.

It's just three derivatives.

So it's saying, How does the X component of my velocity you?

How does that change as I move in the ex direction that how does V my y component change in the wind direction on?

How does the third component W in the said direction?

How does that change in the set direction?

So this is equal to zero equation number one.

This just tells us.

Mass is conserved now.

The 2nd 1 The Big Boy.

So Newton's second law in disguise.

So we're expecting force equals mass times acceleration.

Here we've got you our velocity.

And when we take a time derivative of velocity, that's exactly what acceleration is.

You're going into speed.

You increase your speed, you accelerated.

Your speed has changed.

With respect to time or you decrease your speed, you decelerated.

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 salt water, so salt water is heavier in that sense.

But it's 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 row 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 new to 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 1st 2 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 slide in grinding past one another.

There's internal force there, and then the 3rd 1 this 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.

Then you would nor normally you just replace F g.

Call it gravity.

That's your external force.

In most situations, if you want to go really fancy, you can put in electromagnetism, and we can sort of combine Naevia Stokes Maxwell's equations and get my Neto hydrodynamics, and that is how stars form and Galaxies form on that is just next level.

Italian Stokes was hard, like Neto, hydrodynamics like Try to model the growth of the sun.

It's hard as it sounds.

Internal Forces versus external forces.

So it's just a summer.

Various forces What are these individual forces?

So let's go get the 1st 1 this Noblet p or grant off p.

So this is very similar to our mass equation up here.

So the Grady into PR pressure.

Greedy int It's a vector representing the change in pressure when there is a difference in pressure.

The change in pressure, high pressure over here, low pressure.

Over here, the air moves from high pressure, too low pressure.

There's a greedy int.

There's a difference.

Pressure between two points that causes the fluid to move between or along.

That pressure Grady in.

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 that.

You can think of it as being in layers, and it's like those layers slide past one another.

They create a friction on How strong that friction is is the viscosity.

So air super thin air particles, they move around, they do that thing.

It's all good.

But if you got like honey sliding past all the bits of honey, it's very sticky.

It's very thick, much larger viscosity.

So that's your second internal force.

Where is the problem with all this?

This is why these equations also great because it's muss, 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 could possibly disagree with.

It's just Newton's second law movement.

Fluid around Mass is concerned, and this is why these equations work.

So they've been around since the 18 twenties 18 forties, when Naevia Stokes both worked on them on dhe.

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 could be used for almost anything you can think of involving the flu so they could be aerodynamics off 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, maximizing the way the drug is deposited to do with the flight, the pollution modeling climate, modelling, ocean modeling 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 $1,000,000.

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 satisfy three particular properties Versatile.

You want a solution to exist, you know, going equation you want, right?

I want to solve it.

That would be nice.

Second of all, you want a unique solution.

You know, if you didn't experiment with throwing a glass of water across the room and he did it again and it is something different, like a loop, the 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 and how I started my experiment and I want the result also just have a title 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?

We've also been taught that butterflies flapping their wings from course cyclones.

The butterfly effect, like a chain reaction, is one thing leads to another, leads to another.

But in the in the sense off, having an equation you in put something into your greatest 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 has some velocity our you some pressure, some viscosity import it.

Naevia Stokes tells you how that fluids gonna 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.

Give us that your tiny bit.

If you did that experiment, you 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 Naevia 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.

Import it.

We don't even know if the solution is gonna come out.

So how come these equations are being used by climate modelers and buy Formula One teams?

And all these things that you tell me a using nephew stacks?

It sounds like they using a really unreliable too, I wouldn't say is unreliable because it's based on such standard laws of physics.

Masses conserve 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 sort of the mathematical complexity of it the tricky bit is we don't know position always gonna 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 to use them.

And then another way is to what's called averaging.

So rather than having a velocity field to find 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 off the Naevia Stakes 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 going to 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 has patches off, let's say 10 kilometers squared bits of their 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 $1,000,000 prize to think about what we've done already.

So we have the equations.

They make complete sense with happy with them.

Masses conserve Newton's second law great, and then we also know that solutions exist on.

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 said and just have X and Y are two dimensions, we can do it.

We can show there's always a solution.

It's always well behaved.

It works.

Then when you go to three dimensions, for whatever reason, it's just not working.