Flow velocity
The flow velocity in fluid dynamics is a vector field used to mathematically describe the motion of a continuum. In general, the flow velocity field is

(1) 
which gives the velocity of a microscopic element of fluid at a position and a time . Here, is a vector.
Steady flow
The flow of a fluid is said to be steady if does not vary with time. That is if

(2) 
Incompressible flow
If a fluid is incompressible the divergence of is zero

(3) 
Average velocity
The average fluid velocity (defined as the volumetric flow crossing a surface divided by the area of the surface) is

(4) 
For instance, the average velocity of flow in a pipe of radius in the axial direction is

(5) 
It should be noted that average velocity is a scalar. For convenience, the brackets are often not used. Higher order averages of velocity (raised to the power can be found as

(6) 
Velocity in multicomponent mixture
If a system is comprised of two or more components, both a mass average velocity and a molar average velocity can be defined. Here,

(7) 
where is the mass fraction of species and is the velocity of species . The summation is over all the species that make up the system. For the molar average velocity,

(8) 
where is the molar fraction of species and is the average velocity of species .