Edit: as Markus Lohmayer points out, this is really only Euler’s equation; Navier Stokes also includes viscosity which I have not mentioned yet.
Edit 2: I think that the final equation that I got for momentum conservation may actually be wrong, because momentum is carried by the fluid in a different way than quantities like mass or entropy. So there’s still a significant amount of deconfusion that needs to happen; I haven’t cracked this nut yet!
Introduction
The NavierStokes equation describes the evolution of a fluid flowing through space. It is important for many applications, and it is famously hard to solve. I have felt for a while that I should probably get a basic understanding of what it is saying, partially because it’s important in DECAPODES, and partially out of a general feeling that it is an important part of physics to know, and partially because it’s in the first chapter of Beyond Equilibrium Thermodynamics, which I have been trying to read.
I have been so conditioned to think about everything in terms of exterior algebra (as a result of this) that I had to put NavierStokes in exterior algebra terms to make sense of it. (This has been done before; this is just recording my personal process of doing that).
This post is a result of this digestion process; hopefully it will be useful to others who are similarly exterior algebrapilled.
Geometric setup
The first difficulty in the NavierStokes equation is to figure out the different variables, and the types of those variables. There are a couple ways of doing this; this is the way of organizing the variables that made the most sense to me.
First of all, we work on an underlying Reimannian manifold X. We will use the musical isomorphims \sharp : T^\ast X \rightleftarrows T X : \flat coming from this Reimannian structure, and by abuse of notation we will also use this for turning vector fields (elements of \Gamma (T X)) into 1forms (elements of \Omega^1 X = \Gamma (T^\ast X)) and viceversa.
Also coming from the Reimannian structure, we have the Hodge star \star \colon \Omega^k X \to \Omega^{nk} X, where n is the dimension of X.
We will denote vectorvalued kforms with values in the vector bundle E by
Specifically, sections of E are elements of \Omega^0(X,E), so the musical isomorphisms can be seen as going between \Omega^1 X and \Omega^0(X, TX).
Types of variables
We can now talk about the variables involved in NavierStokes. All of these are implicitly also functions of time.

\rho \in \Omega^n(X) is the mass nform; integrating \rho over a volume V gives the total mass of the fluid in V. \star \rho is the mass density 0form.

v \in \Omega^0(X, TX) is the vectorvalued velocity 0form, which describes the “flow field” of the system. We assume that all of the particles in the infinitesimal neighborhood of x are flowing with velocity v(x).

\Phi \in \Omega^{n1}(X) is the fluid flux n1 form, given by the equation
\Phi = \star (v^{\flat})Integrating this over a surface gives the flux of fluid through that surface. Determining where in NavierStokes I wanted to use v, and where I wanted to use \Phi was half the battle in figuring out what was going on.

M \in \Omega^n(X,TX) is the vectorvalued momentum 0form, given by the equation
M = \rho v 
\pi \in \Omega^{n1}(X,TX) is the vectorvalued force n1 form, which when integrated over a surface gives the force exerted on that surface by the interaction of molecules. In general, \pi may be some function of the other variables, but for now we let it stay abstract.
Conservation equations
NavierStokes is formulated by considering two conservation equations: one for mass, and the other for momentum. We start with the one for mass.
Mass conservation
We start by considering mass conservation over a volume element V \subset X. The conservation law
states that the change in the mass inside V is equal to the flux of mass through the boundary of V, which is the flux in fluid multiplied by the fluid density. We can then apply Stoke’s law to get
Then, noticing that V was arbitrary, we see that we must have
It is typical to condense this further by introducing a “material derivative”, defined on an nform \alpha by
where we have extended the traditional notation \frac{D}{Dt} to emphasize the dependence on v, via \Phi = \star (v^\flat). We can interpret this as “the change in the extensive quantity of the fluid \alpha”. This is the change in \alpha that’s not accounted for by the movement of the fluid. With this, we can write the mass conservation law as
Now we consider the equation for momentum
Momentum conservation
The conservation law for momentum looks very similar to the one for mass, except it is a vector equation rather than a scalar equation.
Before we interpret this, note that this actually doesn’t make sense, because M is a tangentvector valued nform, and we can’t add together tangent vectors to form the integral. However, as long as V is sufficiently small, we can trivialize the tangent bundle to make this work, and we only care about small V because we are turning this into a differential equation. At some point, however, I would like someone to tell me how to think about momentum conservation over a large V.
With this in mind, we interpret the equation as saying that the total momentum in V changes in two ways. One is by fluid with certain momentum moving through the boundary, bringing its momentum with it. The other is by forces exerted on V by the molecules around V.
We can apply the same trick as we did for mass conservation to turn this into a differential equation. We apply Stoke’s law to get
Then we argue that because this holds for every V, we have
We can finish this by simplifying with the material derivative, and we get
Conclusion
In the end, the laws are very simple! We have mass: \frac{D_v}{D_v t} \rho = 0, and momentum: \frac{D_v}{D_v t}M = \mathrm{d}\pi.
There are a variety of ways of spicing things up from here.
 Expand \pi out into some function of the variables.
 Consider a multicomponent fluid, with masses \rho_1,\ldots,\rho_k. Then there are some additional terms for how the components mix, and perhaps react.
 Consider an external force, perhaps coming from gravity or electromagnetism, which imparts momentum change to the terms
 Derive laws for energy conservation and entropy production from the equations momentum and mass conservation, which is what happens next in Beyond Equilibrium Thermodynamics
However, these will have to wait for another time.