The Navier-Stokes equations consist of the momentum equation and the continuity equation. Consider the incompressible versions for the purpose of this question.
Continuity is always talked about as a restriction or a constraint, while the momentum equation is always talked about as a modified version of Newton's second law.
The latter is pretty acceptable to me as being physically motivated. The former is a little disappointing, as it is argued for by insisting on a constant density.
My question is why isn’t this explained by some force keeping the fluid together? Why can’t this be part of the momentum equation as some sort of internal force that keeps things together? Why must it be its own equation?