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 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.


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