The million dollar equation (Navier-Stokes equations)

Consider this lake flowing. If I gave you the velocity and pressure at every single point, would you be able to predict how the lake flows over time?

This question points to the Navia Stokes equations, a set of equations that can describe any fluid you can think of from water to air to honey to so on. You

might have heard of these equations because they're a part of the Millennium Prize problems. A collection of seven incredibly important and difficult mathematical problems. A correct

solution to any of these seven problems is awarded a million dollar cash prize.

Today, let's talk about this particular problem. The existence and smoothness of

problem. The existence and smoothness of the Navia Stokes equation. The Navia

Stokes equations are probably one of the most important partial differential equations for fluid mechanics. They're

used to forecast the weather, model airplanes and rockets, and predict water currents. Despite their extensive usage,

currents. Despite their extensive usage, there's still not that much we know about these equations, mathematically speaking. But first, let's talk about

speaking. But first, let's talk about what these equations actually are. For

the purpose of this video, I'm going to make a few assumptions about the fluid we're talking about. The first

assumption is that the fluid is Newtonian. There was a really good

Newtonian. There was a really good answer on Stack Exchange that explained what it meant for a fluid to be Newtonian, but let me just summarize it for you. When we call a fluid Newtonian,

for you. When we call a fluid Newtonian, it means that the rate at which we apply some sheer stress has no effect on its viscosity. An example from this answer

viscosity. An example from this answer on stack exchange is ketchup. When we

have difficulty removing ketchup from a bottle, something we all do is hit the bottom of the bottle and the ketchup comes out with ease. Looking at

ketchup's sheer rate versus viscosity graph, we can see that hitting the bottle, which increases the shear rate, decreases its viscosity, which lets the fluid out.

The second assumption is that the fluid is incompressible. It means exactly what

is incompressible. It means exactly what you think. If I compress the fluid that

you think. If I compress the fluid that is at pressure, there's no considerable variation in the volume of the fluid.

The last assumption is that the fluid is isothermal. This just means that as the

isothermal. This just means that as the fluid flows, there's no loss or gain of heat. And this takes us to the Navia

heat. And this takes us to the Navia Stokes equations. As daunting as they

Stokes equations. As daunting as they may seem, they're actually based on quite well-known physics properties.

Note that when we talk about fluids, we typically talk about an infinite decimal volume of fluid. And so we're trying to describe the motion of each individual molecule rather than the fluid as a whole.

This first equation right here tells us that mass is conserved within the fluid.

The operator here is called the divergence of a vector field. The vector

field in this case u is the velocity vector field of the fluid.

A vector field is what you get when you assign every single point in space to a vector. Vector fields can describe many

vector. Vector fields can describe many things from fluids to electric fields to gravitational fields and so on. The

divergence of a vector field is an operator that tells us how much a point tends to divert vectors away from it.

Consider this vector field. It appears

that vectors seem to be moving away from the origin. This indicates positive

the origin. This indicates positive divergence.

Similarly, this vector field seems to have vectors that flow into the origin.

This indicates negative divergence.

Numerically, we write the divergence as a dotproduct between the gradient vector and its vector field. This fantastic

video by 31bound explains why. So, if

you're interested, check it out.

In terms of fluids, the divergence of a vector field indicates how much or how little a point acts as a source of a fluid. If we imagine water in some area,

fluid. If we imagine water in some area, it's impossible for the water to simply disappear. It could either change forms

disappear. It could either change forms but mass is never destroyed and thus the divergence across the fluid has to be zero. And hence the first Navia Stokes

zero. And hence the first Navia Stokes equation.

The second equation is just a rewritten version of Newton's second law. If you

remember, Newton's second law tells us that the sum of forces acting on a body can be written as its mass time the acceleration.

Considering this for a single molecule of a fluid, let's see how we can derive the second Navia Stokes equation. First

of all, since we're considering each individual point, let's replace mass with density. The mathematical reason

with density. The mathematical reason behind doing this is that to consider each individual point, we have to divide by volume. And mass by volume is equal

by volume. And mass by volume is equal to density. Next, let's consider

to density. Next, let's consider acceleration.

Well, we have the velocity vector field U. And we know that the acceleration is

U. And we know that the acceleration is just the derivative of the velocity vector field. So we can replace a with

vector field. So we can replace a with du by dt. A quick side note, you might see this part written as d u by del t plus u.grad u. This is essentially the

plus u.grad u. This is essentially the same thing. When we expand du by dt, we

same thing. When we expand du by dt, we use the chain rule which gives us this other expression.

Now we need to consider what are all the forces acting on this molecule of a fluid.

We can break this down into internal forces which are forces exerted by the molecule of the fluid itself and external forces which are forces exerted by some external object. The first

internal force we take into account is pressure. Consider drinking from a

pressure. Consider drinking from a straw. Sucking on the end of a straw

straw. Sucking on the end of a straw creates an area of low pressure which forces the drink to move from the area of high pressure which is below to the area of lower pressure. This force is

dependent on the change in pressure or the pressure gradient which we can write as grad of P.

The second force we factor in is friction or viscosity. If I pour water into a cup, the water moves quite fast.

Whereas if I'm pouring honey, it moves quite slow. This is because there's a

quite slow. This is because there's a lot more friction between the molecules of honey compared to the molecules of water. The mathematical way of writing

water. The mathematical way of writing this force for Newtonian fluids is this expression right here. I'm not going to go into depth as to how this equation arises, but I've left a few resources

you can use to read up about this.

And finally, to account for the external forces, we just denote any external force by the letter F.

It's pretty common that gravity is the only external force. So, we sometimes write row * G instead of F.

And this is why the Navia Stokes equations can model any fluid. There's

simply two fundamental laws of physics written out for fluids. So why do these equations have a million-doll prize attached to it? Well, to answer that, let's just look at the actual problem

from the Clay Math Institute. It goes as follows. To give reasonable leeway to

follows. To give reasonable leeway to solvers while retaining the heart of the problem, we ask for a proof of one of the following four statements.

The first two questions ask for a smooth solution to the Navia Stokes equations.

Let's break down what this means. When

we say that a solution is smooth, mathematically it means that the solution is differentiable. Chaotic

doesn't mean that it's random. It simply

means that if I slightly change the initial condition, it results in a large change in the outcome. And this is where the question gets incredibly difficult.

Sure, the Navia Stokes equations lets us take some state of a fluid and let us predict what will happen in the future.

But the chaotic nature of the fluids with turbulence makes it incredibly hard to make long-term predictions. This is

why we can't predict the weather for more than 7 days.

This is also why we can't predict when we're going to have turbulence on an airplane.

Despite this, the Na'vi Stokes equations are incredibly useful. They're used

extensively to model aerodynamic objects. For example, a plane or a car.

objects. For example, a plane or a car.

Like I mentioned before, the weather forecast also relies on the Naver Stokes equations. It still remains one of the

equations. It still remains one of the most important partial differential equations ever. Thanks for watching.

equations ever. Thanks for watching.

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