A moving fluid is described by a velocity field that assigns a velocity vector to each point in the flow:

$$\mathbf v=\mathbf v(\mathbf r,t).$$

Steady and unsteady flow

A flow is steady when the field measured at each fixed point does not change with time. Fluid particles can still move through the field; steady does not mean stationary.

Streamlines

A streamline is a curve everywhere tangent to the local velocity vector. In steady flow, particles follow fixed streamlines.

Streamlines provide a geometric picture of the direction of motion without requiring every particle to be tracked separately.

Volumetric flow rate

The volumetric flow rate through a surface is the volume crossing it per unit time. For uniform velocity perpendicular to area $A$,

$$Q=Av.$$

More generally, only the velocity component normal to the surface contributes to flow through it.

Mass flow rate

For uniform density,

$$\dot m=\rho Q=\rho Av.$$

Flow rate converts a local velocity description into the total amount of fluid passing through a chosen cross-section.