Continuity equation

In fluid dynamics, the continuity equation is an expression of conservation of mass. In (vector) differential form, it is written as

${\partial \rho \over \partial t} + \nabla \cdot (\rho \vec{u}) = 0.$

where $\rho$ is density, $t$ is time, and $\vec{u}$ is fluid velocity. In cartesian tensor notation, it is written as

${\partial \rho \over \partial t} + {\partial \over \partial x_j}(\rho u_j) = 0.$

For incompressible flow, the density drops out, and the resulting equation is

${\partial u_j\over \partial x_j} = 0$

in tensor form or

$\nabla \cdot \vec{u} = 0$

in vector form. The left-hand side is the divergence of velocity, and it is sometimes said that an incompressible flow is divergence free.