Continuity equation

A continuity equation expresses local conservation by balancing density changes against flux divergence. For mass density $\rho$ transported with fluid velocity $\mathbf{u}$ and no mass sources, it is ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{u})=0$.

Expanding the divergence gives $D\rho /Dt+\rho \nabla \cdot \mathbf{u}=0$, where $D/Dt$ is the material derivative. A fluid parcel can change density when the surrounding flow compresses or expands it. For constant nonzero density, the equation reduces to the incompressibility condition $\nabla \cdot \mathbf{u}=0$.

In a steady one-dimensional pipe flow, integrating across a section gives constant mass flow rate $\rho uA$, under the usual uniform-section approximation. Velocity need not remain constant when the density or channel area changes; the continuity equation constrains their product.

The continuity equation is a fundamental principle in fluid dynamics and other fields that describes the transport of some quantity (such as mass, energy, or charge) in a system. It expresses the idea that, in a closed system, the rate at which a quantity enters a volume must equal the rate at which it leaves that volume, plus any accumulation of that quantity within the volume.