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In case of incompressible flow, divergence of velocity field is zero so we can say that the velocity field is solenoidal. However, given that a flow field is solenoidal, is it necessarily incompressible?

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TD;DR: No
Compressibility is a characteristic of the fluid (or maybe also the way you describe a problem).

$\frac{\partial \rho}{\partial p} \neq 0 $

This means the a pressure variation will also cause a variation in density.
In most cases water and air at low velocities can be treated incompressible because the fraction above is very close to Zero. This means that the fluid’s density does not change (much) when pressure is applied. Hence, in order to see compressibility effects one needs a pressure variation.

Solenoidality is a characteristic of a vector-field. It is a phenomenological description of the (e.g. velocity or pressure) flow-field. But this only means that a particular flow field is divergence free. It does not (necessarily) state anything about the physical characteristics of the fluid.

This means a compressible fluid can be solenoidal if the boundary conditions and forces create one.

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  • $\begingroup$ I don't know anything about solenoidal fields from from physical perspective except for the fact that divergence of such a field is zero. Will you please elaborate your explanation for me? Thanks. $\endgroup$ Jan 22 '18 at 5:53
  • $\begingroup$ Sure, imagine you have a flow field with no divergence (like some bulk-flow at high MachNumber). This flow will not show any characteristics of compressibility. Here (simplification) a compressible simulation/calculation of the velocity field (but not density) will give very similar answers to the one you would get with a incompressible simulation. But compressibility effects will show up once you have some curved streamlines. $\endgroup$
    – rul30
    Jan 22 '18 at 6:06

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