This blog is a good start:
Using Mathematica to Simulate and Visualize Fluid Flow in a Box
It fully solves 2D problem of one moving boundary
and gets nice vertex flows:
There are detailed descriptions of proper equations and numerical discretization. You can generalize to 3D. I would look also in latest V10 functionality to see if anything can be used there to upgrade the methods of the blog - it was written before V10 came out. For instance, take a look at this Stokes Flow example which is a simplification of NS:
