Continuity equation
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The continuity equation expresses the conservation of mass. It is a direct consequence of Gauss theorem.
If the mass enclosed in a region \(\Omega\) is \(M\), by definition of mass density \(\rho\):
\[M=\int_\Omega \rho dV .\]
The net loss of matter in this region must be caused by an outward flow \(\rho \vec{v}\) across its boundary:
\[\frac{\partial M}{\partial t}= - \int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} .\]
According to Gauss theorem,
\[\int_{\partial\Omega} \rho \vec{v} \cdot d\vec{S} = \int_\Omega \nabla( \rho \vec{v} ) dV .\]
Since the region is a general one, and it does not change with time, the resulting equation is
\[ \frac{\partial \rho}{\partial t} + \nabla (\rho \vec{v}) =0 .\]
As a direct consequence an incompressible fluid, with constant \(\rho\), implies a solenoidal velocity field: \( \nabla \vec{v} =0 \).
