Continuity equation

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: $\nabla {\vec {v}}=0$.