This might come as a bit of a shock to some, since the simplest 3d example is a 3-torus, and normal folks don't think about water flowing around in a 3-torus! We are saying goodbye to ordinary reality and embracing the joys of math here. Can you add a sentence or 2 to calm the reader?
It's technically a bit more tricky, but perfectly fine, to work with a compact Riemannian manifold with boundary. Then we can study water in a bottle.
I wasn't thinking about readers who would like to think in three dimensions, I only thought about one and two dimensional examples. So that's clearly a hole in the exposition.
With regard to
Can you add a sentence or 2 to calm the reader?
Could you give me a heads-up when you are done with your first sweep? I'd like to avoid racing conditions :-)