The control-volume conservation laws of fluid mechanics can also be written locally as differential equations for the continuum fields.

Conservation of mass

For density $\rho$ and velocity field $\mathbf v$,

$$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf v)=0.$$

This is the continuity equation. It says that a local increase in mass density must be supplied by net mass flow into the surrounding region.

For incompressible flow of constant density, it reduces to

$$\nabla\cdot\mathbf v=0.$$

Conservation of momentum

Newton's second law applies to fluid parcels as well as rigid bodies. The acceleration of a parcel is its material derivative,

$$\frac{D\mathbf v}{Dt}.$$

Pressure, viscous stresses and body forces such as gravity provide the forces that change momentum.

From integral to local descriptions

Control-volume equations describe balances over finite regions. Differential conservation laws describe the same balances point by point.

This local form is the starting point for deriving the partial differential equations solved in computational fluid dynamics.