I've noticed a serious lack of examples for students learning fluid mechanics, and I started trying to come up with examples. In the case of incompressible fluids (constant density) the only thing required is a velocity field, and that comes from a velocity potential.
But what about compressible fluids? Is there any straightforward way to generate a density function to be compatible (in terms of satisfying the continuity equation) with a given velocity field? For that matter, can any vector field be a velocity field if you use an appropriate density function? Are the conservation equations themselves the only constraints on a model/example for fluid flow?