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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?

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    $\begingroup$ Fluid mass conservation law. $\endgroup$ Jan 27, 2019 at 19:24
  • $\begingroup$ The continuity equation $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec v) = 0$ does not demand incompressibility of the fluid. In fact, this is a more general property as a result of the symmetry of a system via Noether's theorem. $\endgroup$ Jan 28, 2019 at 5:31
  • $\begingroup$ @GodotMisogi what symmetry gets you this continuity equation? And applied to what Lagrangian? $\endgroup$
    – Tabin
    Jan 31, 2023 at 2:05

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You have already accepted that the fluid is incompressible, and that means its volume cannot change. Since the mass of the liquid cannot change either, that means the density must be a constant.

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  • $\begingroup$ Ok, so why are some fluids incompressible, while others aren't? By what physical mechanism? $\endgroup$
    – Chillpadde
    Jul 27, 2022 at 19:12

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