Incompressible vs. Compressible Navier-Stokes

The Navier-Stokes equations are a family of systems. The difference between incompressible and compressible flow isn't cosmetic. It changes the unknowns, the mathematics, and the open problems. The physical split: density that changes vs. density that doesn't "Incompressible vs compressible" boils down to density. Does it stay constant, or does it change? Try it. Fill a syringe with water and push the plunger. The water moves, but under everyday conditions it doesn't compress noticeably. Water resists compression so strongly that treating it as incompressible is an excellent approximation. Incompressible. Now fill that syringe with air and seal the end. Push the plunger in and you'll feel the air give way, the same mass of air now packed into less volume as it compresses under your thumb. That's compressible flow. In incompressible flow, the density $\rho$ is constant throughout the fluid, and every tiny parcel keeps its volume as it moves through space. Compressible flow is different.