# Do the incompressible Navier-Stokes Equations apply to incompressible fluids or incompressible flows?

As far as I understand an incompressible fluid is one where the density is constant and an incompressible flow is one where the material derivative of density is constant ($\frac{D\rho}{Dt}=0$). Both result in the same condition - that $\nabla \cdot \mathbf{u} = 0$ everywhere ($\mathbf{u}$ is the velocity field). Does this mean that the incompressible Navier stokes should apply both to incompressible fluids and more generally to incompressible flows?

This Wikipedia Stokes Navier Equations states:

The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:

The stress is Galileian invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient.

The fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $$V$$ is an isotropic tensor.

Incompressible flow does not imply that the fluid itself is incompressible. Under the right conditions, even compressible fluids can – to good approximation – be modelled as an incompressible flow.

Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the flow velocity.

Reading through the pretty comprehensive article might be of help.

Another site which deals with their application is: Navier Stokes equations and seems to cover both compressible and incompressible flows, allied with the equation of continuity.