The Navier-Stokes Equations: Part I

 This post is dedicated to BK. I've procrastinated this one because it's a huge topic and one quite important to me, learned from good teachers.

Fluid mechanics is an interesting an important area to study in its own right: everything in the universe behaves in the same way when it gets hot enough. But for me it also served as the example model for ideas that crop up in later areas from electromagnetism to the method of characteristics for first order non-linear PDEs. It's the perfect playground for vector calculus. Instead of trying to demonstrate the sheer breadth of the field, I present a quote from David Tong whose lecture notes serve as a great resource.

Fluid mechanics explains how oil flows through pipes and how the motion of the atmosphere manifests itself in the climate, and how many decades of focussing on the former has resulted in an urgent and desperate need to better understand the latter.

 The main character of fluid mechanics is the Navier-Stokes equation. (Sometimes pluralised as it as a vector equation so is really 3 equations) While fairly simple to state, they are devilishly difficult to solve in generality and it is not even known whether smooth initial conditions always lead to well defined and well behaved solutions. This is the target of the Millenium Prize, offering $1,000,000 for an answer to the above question. I reiterate: a million dollars is being offered not for a solution to the equations but just to verify whether a solution always exists. (currently I'm not very interested in the 7 Millenium problems but don't be surprised if this is retitled Millenium Problems Part I: Navier-Stokes)

Before we examine the constituent parts, lets be clear what we're actually describing. Our unknown is the velocity field which tells us that in which direction a rubber duck thrown into the fluid (at a specific point) will travel. More specifically, at each point in space x the velocity field assigns to that point a velocity u(x) which we can view as an arrow pointing in the direction that a fluid particle at that point will move in. Problematically, I just referred to a fluid particle. But a velocity field (actually any vector field) has to be defined at all points in space. So we have to impose an assumption: we suppose that we are zoomed out enough that the fluid looks like it has no gaps. And this is good enough! So rarely are we working on a scale where the individual interactions between molectules contribute to the fluids flow.

[[An aside: FC suggested a post on the biggest result in maths that relies on an approximation. After some thought I've concluded that every mathematical result is equally valid as each is logically consistent and all fit into the cogent framework we can all be proud of. The approximation only enters when we make claims about those mathematical statements in fact being physical ones. As far as I can tell Newton's theory of gravity or Einstein's theory of relativity claim to be perfect descriptions so I would have to say that the continuum hypothesis (i.e. the assumption that there are no gaps in the fluid) is the biggest such approximation. Very open to discussion on this one though.]]

We've got a velocity field in some initial configuration: now we want to ask how it changes over time. Rigid body mechanics will tell you to wack a time derivative on it and call it a day but this is a question of perspective: the derivative with respect to time (and hence a partial derivative fixing position) measures what happens if I sit at a fixed point in the fluid and look at the fluid particles moving around me. Imagine I'm a sensor attached to a river bed. This is known as the Eulerian perspective (like Euler didn't have enough named after him). What turns out to be more fruitful is the Lagrangian perspective: We imagine we're a fish being carried along in the river and measure how the velocity field changes around us. This is described by something called the material derivative, denoted Du/Dt as opposed to du/dt. The difference between the Eulerian and Lagrangian perspective is more important than the explicit expression of the material derivative. An important example of this is incompressibility: if we think about the density of a fluid at each point, this can change from point to point. For example the density of water in the ocean varies with depth (due to temperature changes). But an individual packet of water moving about in the ocean can't change its density: this would be equivalent to it expanding or contracting. So in an incompressible fluid density is "materially conserved" i.e. the material derivative is 0.

This accidentally became much longer than intended so will have to be 2 parts.