Velocity Field

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 the flow speed and is a scalar.
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).

Definition

The flow velocity ''u'' of a fluid is a vector field

:$\mathbf=\mathbf(\mathbf,t),$

which gives the velocity of an '' element of fluid'' at a position $\mathbf\,$ and time $t.\,$

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

:$q = \, \mathbf \,$

and is a scalar field.

Uses

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:

Steady flow

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

:$\frac=0.$

Incompressible flow

If a fluid is incompressible the divergence of $\mathbf$ is zero:

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

That is, if $\mathbf$ is a solenoidal vector field.

Irrotational flow

A flow is ''irrotational'' if the curl of $\mathbf$ is zero:

:$\nabla\times\mathbf=0.$

That is, if $\mathbf$ is an irrotational vector field.

A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential $\Phi,$ with $\mathbf=\nabla\Phi.$ If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: $\Delta\Phi=0.$

Vorticity

The ''vorticity'', $\omega$, of a flow can be defined in terms of its flow velocity by

:$\omega=\nabla\times\mathbf.$

If the vorticity is zero, the flow is irrotational.

The velocity potential

If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field $\phi$ such that

:$\mathbf=\nabla\mathbf.$

The scalar field $\phi$ is called the velocity potential for the flow. (See Irrotational vector field.)

Bulk velocity

In many engineering applications the local flow velocity $\mathbf$ vector field is not known in every point and the only accessible velocity is the bulk velocity (or average flow velocity) $U$ which is the ratio between the volume flow rate $\dot$ and the cross sectional area $A$, given by

:$u_=\frac$.

See also

*Velocity gradient
* Velocity potential
*Drift velocity
*Group velocity
*Particle velocity
*Vorticity
*Enstrophy
*Strain rate
*Stream function
*Pressure gradient
*Wind velocity

References

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Fluid dynamics
Continuum mechanics
Vector calculus
Velocity
Spatial gradient
Vector physical quantities