Over the course of the next few days week, we’re going to derive this sucker:

What is it, exactly? An equation of course. But for what? The area of a triangle? A measure of how cool the author is? The erratic behaviour of your new girlfriend? An explanation of why men are such dogs?

Better. That equation — or more specifically — the set of three equations (there are actually three hidden in there) are collectively known as the Navier Stokes Equations — the fundamental equations governing the motion of practically all fluids (liquids and gases).

Unfortunately, although we can write out the equations pretty quickly and even derive them, solving them is a whole new can of worms. In their general form, they’re virtually impossible to solve analytically (on paper and pencil) and solving the equations using a computer is no walk in the park either — being the central focus of research in Computational Fluid Dynamics.

In fact, the Clay Mathematical Institute has long offered $1,000,000.00 to the first person who can answer a few basic, fundamental questions about these equations. Oh, and there is the small reward of appearing on the front page of every newspaper in the world, as well as gaining a good chunk of academic immortality.

Oh, and think of all the chicks you can pick up, too. This is the sort of thing babes go crazy over.

Trust me.

Unfortunately, unlike my previous excursions into the mathematical battlefield, I won’t be able to steer this towards everyone. There’s a fair amount of background material you’ll need to have in hand, and it would be a big mistake for me to try and write a self-contained series.

Here’s sort of an outline of the prerequisites:

1. Grade 12 Physics: Newton’s laws, vector arithmetic, and force diagrams.

2. Elementary Calculus: Differentiation, integration, differentials.

3. Vector and Multivariate Calculus: Partial derivatives, surface integrals, volume integrals, divergence

Basically, if you’re in Math, Physics, or Engineering, you should be fine. If you’re in Biology, Chemistry, Computer Science, or similar fields, you might need to Wikipedia some terms. If you’re in the Social Sciences or the Arts, you may…well, you may want to consider a career change (zing!).

I’m kidding. Really. Please, don’t hurt me.

Crates and Fluid Blobs

Let’s take a moment and reminisce about that stuffy old physics classroom you spent so much time in during high school. It’s June, stifling hot, there’s only one crappy fan in the class, and the dorky guy next to you won’t shut up (Hi Mom).

You were learning about Newton’s three laws, and the teacher would give a question about pushing crates.

There was always a crate, you know. Even if your teacher was sympathetic and drew in a toboggan to pull or a pile of textbooks to push, you knew it was really about a crate.

Yeeesh.

To analyze the motion of the crate, you applied Newton’s laws to balance certain equations involving the different forces involved — the applied force, the normal force, and the gravitational force. Sometimes, you’d use conservation of energy or conservation of momentum arguments to also find a balance between the different quantities.

For the most part, our analysis of fluids will be similar. But instead of looking at crates, we’ll be examining fluid elements — blobs of water if you will. We’ll apply conservation laws to these blobs of water, and just as we developed equations to describe the motion of crates being pushed, we’ll develop equations to describe the flow of a certain fluid.

Oh man, how wicked is that?