# The Navier-Stokes Equations

A simple introduction to a million dollar problem

Summer, what a beautiful time. Everyone loves standing on the water’s edge watching the waves breaking on the shore. But how many of you have ever wondered about the extreme complexity of the movement of the water? It seems smooth and regular but as soon as a wave breaks on the sand it splits into hundreds of different currents and bubbles. It became completely unpredictable.This incredible complexity is regulated by the Navier-Stokes equations.

Most people are familiar with Newton’s second law of motion. It says that the force on an object is equal to its mass multiplied by the acceleration of the object.

This equation applies to every macroscopic object in the world. However, if you want to know the behavior of fluids you need something else.

What you want are the Navier-Stokes equations. They are used by engineers and physicists all over the world in many fields, which go from aircraft design to blood circulation. They are also very complicated to solve and that is why they are one of the seven *Millennium Prize Problem*. Solving one of these problems will win you a million dollars, and here is one of them.

As with any advanced equation, it can be intimidating, but the concepts they represent are not too complicated. We will explore what they mean piece by piece, to understand why they are so important.

# Introduction

Before we begin we have to make some assumptions.

First, we are working with Newtonian fluids. This is the simplest mathematical way to account for viscosity in a fluid. There are no real fluids that fall perfectly under this category but in most cases, air and water can be treated as a Newtonian fluid. The other very important hypothesis we’re making is that the fluid is incompressible. This means that its density, rho, is a constant.

## Conservation of mass

The first equation tells us that the mass of fluid we are working on is conserved. The fluid can change its shape but this equation tells us that the mass is the same form start to finish.

Now let’s talk math. The letter *u* represents the velocity of the fluid and it is a vector. It has three components, we can call them *u, v, w* and they represent the velocity of the fluid in the *x, y,* and *z-direction*. The greek letter nabla *∇* followed by a dot is the divergence operator. This means that we have to differentiate its components in every direction (in this case *x, y, z*).

The first derivative is telling us how the *x* component of the velocity is changing as we move on the *x-direction*. The same is true for the other two derivatives. Because this equation is equal to zero, it is showing us that the mass is conserved.

## Conservation of momentum

The second equation is actually a set of three differential equations. This is the one that can be described as Newton’s second law for the fluids. If we expand the expression we obtain a complex system.

Since it is a lot simpler to understand, we are going to concentrate on the conservation of momentum not expanded.

When we work with fluids we can say that mass and density are the same things (as long as their volume is the same). If we are considering two fluids, we can say that the denser one is the “heavier” one (e.g. mercury and water). In this case, the greek letter *ρ (rho)* is the density of the fluid.

We have the mass, now to go back to Newton’s second law we need the acceleration. That is exactly the result of the time derivative of the velocity vector.

Now, all we have left are the terms on the right side of the equal sign, and those represent all the forces that are applied to the fluid.

The first term *∇p* is the gradient of the pressure. It represents the difference in pressure in the space where the fluid is contained. For example, if there is a zone with low pressure and another with a higher pressure the fluid will move from the high-pressure section to the lower one. The gradient of *p* is saying all of that.

The second term* *describes the viscosity of the fluid. Think about two different liquids, like water and honey. When you pour a glass of water it falls quickly and easily. When you do the same thing with honey it is sticky and it pours very slowly. This is what that expression is saying.

The last term, *F*, is the easiest one because it represents all the external forces that may be applied to that fluid. Usually, the force we consider here is gravity.

This is it, all those fancy symbols and letter mean is just *force = mass x acceleration*.

# Use of the Navier-Stokes equations

Since solving these equations is so complicated, to use them we have to make a lot of approximations. Some examples are the Poiseuille and Couette flow. With a lot of assumptions, these two scientists were able to find a solution to the Navier-Stokes equations for a very specific application. However, if we want to use them for a more complicated task, like the weather forecasts, we need to do something else.

The most common way to use these equations is by transforming them, with the Reynolds average. This is an advanced mathematical process and the results are the Reynolds equations. They are usually called RANS (which means Reynolds averaged Navier-Stokes) equations.

These equations are used when the fluid is in a turbulent flow. They look almost exactly like the Navier-Stokes equations, except for the last term. This is called *Reynolds stress tensor*. It is the quantity that allows us to account for the turbulence in the fluid.

In the RANS equations, the quantities we use are averaged on a time interval. This interval has to be small enough to allow the observation of the phenomenon we’re studying. At the same time, it has to be big enough to make the turbulence effect lose importance.

With the right assumptions, these equations work. We know how to use them to make the F1 cars faster, to make a spacecraft go to the ISS, and to make the weather forecast. So you may be wondering why is the proof of these equations worth a million dollars?

# A million dollars prize

From a physics point of view, these equations are just Newton’s second law that works with fluids. When we make some assumptions and some simplifications, we can do amazing things with them.

The real problem is this. Solving these equations without approximations is just awfully complicated. It is so difficult that it hasn’t been proved that the solutions actually exist. And this is where the Millennium Prize comes in.

The official statement of the problem (*find it **here*) is:

Prove or give a counter-example of the following statement: in three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

What this means is that to win the prize you have to do three things:

- Prove that a solution exists
- The solution has to exist in every point of space
- The solution has to be smooth. This means that a little change in the initial conditions produces only a little variation of the result.

For an engineer is usually enough to know that these equations work, even if only to some level of approximation. Yet, for a mathematician is very important to know if the solutions exist and how they behave.

You may now be thinking that if they work as they are, spending time and energy searching for a proof is a total waste of time. Well, like many technological advances in the history of mankind the result may not seem very important. What’s important is the path to get there, which can bring new knowledge and improvements to our life.

Think about space missions. If humans never decided to go and walk on the moon we wouldn’t have many objects that are used to improve our life. MRI machines and pacemakers come from technologies developed for space exploration. Today doctors all over the world use them every day to save lives. The same thing applies here. The path to discovering the solution to the Navier-Stokes equations will help us to improve our understanding of fluids and more. It can lead us to new discoveries and will probably require the invention of new math. This can then be used to answer many other problems, to invent new technologies to improve our lives and make us better.a