Navier-Stokes made easy

February 19th, 2025

This page is still under construction.

In high school physics, you learned about Newton’s Laws, one of which is \(\sum\textbf{F} = m\textbf{a}\). It allowed us to figure out how particles fly through the air or how the planets revolve around our Sun. Just to tease apart this humble equation: the left-hand side is a sum of forces. The net force doesn’t act directly on a particle’s position. Rather, it effects its acceleration (with mass as a scaling term. 10N of force causes a less massive particle to accelerate more; In this relationship, mass is referred to as the inertial term). Acceleration is then related to velocity which is then related to the position of the particle through integration (insert curve or figures).

Believe it or not, Newton’s 2nd Law can also be applied to describe how fluid flows. After all, fluids, like water moving in a river, is made out of mass, is subject to forces, and most importantly moves. It’s all a matter of framing the problem as high school physics did for single, point-mass particles.

In the end, it all boils down to fundamental conservation laws:

1. Conservation of mass
2. Conservation of momentum
3. Conservation of energy

Setting up the variables

What is the inertial term for fluids? Obviously, it’s the mass of the water. But we immediately run into a problem that wasn’t present when we talked about point-particles. With fluids, we now have to consider volume. So the amount of mass depends on the volume of water. Does this mean the equations of motion have to be re-solved every time we consider a different volume of water?

Not quite. Instead of thinking about the inertial term of water as its mass, we can instead normalize it with respect to its volume. Mass/volume. If you guessed that this ratio of mass/volume is constant for a given material, you’d be right. The density of a fluid is an intrinsic property that doesn’t change with the size of the system!

So instead of thinking about the momentum of a single particle, we can instead consider the momentum density of a fluid.

Conservation of mass

We’ve all have heard that mass cannot be created or destroyed. It can only change form. (For a moment, I want the particle physicists in the crowd to calm down — why are you even here anyways???) This principle is called Conservation of Mass, and it applies to fluids as well.

Remember that when discussing fluids, we tend to talk in terms of density instead of mass. Moreover, given that we’re considering a constant volume in space, the density and mass are directly proportional to each other!

Consider a spot in a body of fluid that’s flowing around. There’s mass there all right, and that means there’s density too. Mass can leave this spot or enter this spot. The rate of change of density at this spot depends on the relative rates of mass leaving and entering.

So let’s discuss the change in density over time. Given a space, the value of density depends on where you are in the space and when you make the measurement (after all, density can change over time, even for a given spot!). In a 3D space, this means density is a function of 4 variables: \(x\), \(y\), \(z\), and \(t\): \(\rho(x, y, z, t)\). When we talk about the change in density over time, we consider one spot in space, so we hold \(x\), \(y\), and \(z\) constant. We express this in math with a partial derivative:

Consider whether the density changes in the following scenarios:

1. A trickle of current flows in from the left and that same trickle of a current flows out on the right.
2. A big current flows in from the left and that same big current flows out on the right.
3. A trickle flows in form the left and a big current flows out on the right.
4. A super dense fluid (oatmeal?) trickles in from the left and big, very not-dense fluid (watery oatmeal?) flows out on the right

Whether or not the density at the spot changes depends not only on the relative flows in and out, but also the densities of the stuff moving in and out respectively. Instead of just considering the net movement of \(u\), we’re interested in the net movement of density.

There is a very neat math notation for talking about this “moving in and out” business. It’s called the divergence operator, and we apply it to vector fields like so: \(\nabla \cdot \textbf{u}\).

What’s a vector field again? To review this, we’ll start with the idea of scalar fields. A scalar is just a plain ol’ quantity… something like temperature, for example. Now consider a space — the room you’re sitting in is perfect! At every given position in your room (on surfaces, within furniture, in the air), we can assign a temperature. Here we have described a scalar field (the field being your room, I guess). Graduating to the next level, instead of assigning a scalar at every point in space, we instead assign directional vectors. We’d end up with a vector field. This is the type of structure we need to describe flows in a space. The vector can represent the velocity of the fluid at that point in space. The velocity vector at one point in space can be different at another point.

Going back to the divergence operator, we first return to our little spot. Notice how this spot is embedded within a vector field? If we add up the vectors going out of the spot minus the vectors going into the spot, we get the divergence of the velocity field evaluated at the given position.

Consider the following examples to get better intuition with divergence:

• Imagine if all the vectors were pointing inwards. It’s like you packed a handful of snow to make a dense snowball. This divergence would be negative.
• Imagine if all the vectors were pointing outwards. It’s like you and your friends pulled apart an 8-slice pizza from each side. This divergence would be positive.
• Imagine if some vectors were going in and some were going out. It’s like you …?

Notice in all cases, there is “surrounding movement” and that is the velocity vector field at the point of interest.

Believe it or not, we have all the pieces we need to talk describe the conservation of mass for fluids. Ready? Drumroll please…

In Cases 1 and 2, we compressed (or decompressed the pizza?) in question. If we disallow our medium to be compressible, we force the divergence to be 0. This means that any movement out needs to be balanced by movement in and vice versa.

Conservation of momentum

Email Kevin to finish this.