Classical Fluids via Quantum Mechanics

The subject of this post is probably a bit too technical to interest many readers, but I’ve been meaning to post something about it for a while and seem to have an hour or so to spare this morning so here goes. This is going to be a battle with the clunky WordPress latex widget too so please bear with me if it’s a little difficult to read.
The topic something I came across a while ago when thinking about the way the evolution of the matter distribution in cosmology is described in terms of fluid mechanics, but what I’m going to say is not at all specific to cosmology, and perhaps isn’t all that well known, so it might be of some interest to readers with a general physics background.
Consider a fluid with density . The velocity of the fluid at any point is . The evolution of such a fluid can be described by the continuity equation:

and the Euler equation

in which is the pressure and is a potential describing other forces on the fluid (in a cosmological context, this would include its self-gravity). To keep things as simple as possible, consider a pressureless fluid (as ...