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