Mass conservation
= Mass conservation
= Conservation of mass
{synonym}
<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> is
$$
\partial_t\rho+\nabla\cdot(\rho\mathbf v)=0,
$$
or 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>.