# 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 ${\displaystyle \Omega }$ is ${\displaystyle M}$, by definition of mass density ${\displaystyle \rho }$:

${\displaystyle M=\int _{\Omega }\rho dV.}$

The net loss of matter in this region must be caused by an outward flow ${\displaystyle \rho {\vec {v}}}$ across its boundary:

${\displaystyle {\frac {\partial M}{\partial t}}=-\int _{\partial \Omega }\rho {\vec {v}}\cdot d{\vec {S}}.}$

According to Gauss theorem,

${\displaystyle \int _{\partial \Omega }\rho {\vec {v}}\cdot d{\vec {S}}=\int _{\Omega }\nabla (\rho {\vec {v}})dV.}$

Since the region is a general one, and it does not change with time, the resulting equation is

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla (\rho {\vec {v}})=0.}$

As a direct consequence an incompressible fluid, with constant ${\displaystyle \rho }$, implies a solenoidal velocity field: ${\displaystyle \nabla {\vec {v}}=0}$.