In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity[1] [2] 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
u=u(x,t),
which gives the velocity of an element of fluid at a position
x
t.
The flow speed q is the length of the flow velocity vector[3]
q=\|u\|
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:
See main article: article.
The flow of a fluid is said to be steady if
u
| \partialu | |
| \partialt |
=0.
See main article: article and Incompressible flow.
If a fluid is incompressible the divergence of
u
\nabla ⋅ u=0.
That is, if
u
See main article: article and Irrotational flow.
A flow is irrotational if the curl of
u
\nabla x u=0.
That is, if
u
\Phi,
u=\nabla\Phi.
\Delta\Phi=0.
See main article: article and Vorticity.
The vorticity,
\omega
\omega=\nabla x u.
If the vorticity is zero, the flow is irrotational.
\phi
u=\nabla\phi.
The scalar field
\phi
In many engineering applications the local flow velocity
u
\bar{u}
| V |
A
| \bar{u}= |
| ||
| A |