In fluid mechanics, an incompressible flow is a fluid flow in which the divergence of velocity is zero. It is an idealization used to simplify analysis. In reality, all fluids are compressible to some extent.

The Partial differential equation for incompressible fluids is:

$\{\backslash nabla\; \backslash cdot\; \backslash mathbf\{u\}\; =\; 0\}$.

From the equation of continuity

$\{\backslash partial\; \backslash rho\; \backslash over\; \backslash partial\; t\}\; +\; \backslash nabla\; \backslash cdot\; (\backslash rho\; \backslash mathbf\{u\})\; =\; 0$,

i.e.

$\{\backslash frac\{D\backslash rho\}\{Dt\}\}\; +\; \backslash rho\backslash nabla\backslash cdot\; \backslash mathbf\{u\}\; =\; 0$,

and the fact that

$\{\backslash rho\; >\; 0\}$,

we see that a fluid is incompressible if and only if

$\{\backslash frac\{D\backslash rho\}\{Dt\}\}\; =\; 0$,

that is, the mass density is constant following the fluid.

Relation to solenoidal field

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An incompressible flow is described by a velocity field which is solenoidal. But a solenoidal field, besides having a zero divergence, also has the additional connotation of having non-zero curl (i.e., rotational component).

Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the velocity field is actually Laplacian.

Fluid mechanics

flusso incomprimibile | Inkompressibles Fluid