In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is scalar, the flow speed. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall).

The flow velocity u of a fluid is a vector field

which gives the velocity of an element of fluid at a position ${\displaystyle \mathbf {x} \,}$ and time ${\displaystyle t.\,}$

The flow speed q is the length of the flow velocity vector

and is a scalar field.

The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:

The flow of a fluid is said to be steady if ${\displaystyle \mathbf {u} }$ does not vary with time. That is if

If a fluid is incompressible the divergence of ${\displaystyle \mathbf {u} }$ is zero: