 Mass conservation

Mass conservation requires that a material region retain its mass when no mass-producing process is included. For mass density $\rho$ and velocity field $\mathbf{v}$, the local continuity equation isor equivalently $D\rho /Dt+\rho \nabla \cdot \mathbf{v}=0$. The second form says that a moving fluid element loses mass density as its volume expands, rather than losing mass.

A pipe narrowing in a steady flow gives a simple test: the mass flow through every cross-section is the same, so $\rho AU$ is constant for a uniform section profile. Constant mass density then makes $AU$ constant. In Lagrangian coordinates, the equivalent relation is $\rho J={\rho }_0$, where $J=\det F$ is the local volume ratio. A fixed observation region can gain mass through its boundary even though the material moving through it obeys mass conservation.