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 .