Flow velocity

In continuum mechanics the macroscopic velocity,^[1][2] also flow velocity in fluid dynamics or drift velocity in electromagnetism, is a vector field which is used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.

Definition

The flow velocity u of a fluid is a vector field

${\displaystyle \mathbf {u} =\mathbf {u} (\mathbf {x} ,t),}$

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

The flow speed q is the length of the flow velocity vector^[3]

$q=||{\mathbf {u}}||$

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 $\mathbf{u}$ does not vary with time. That is if

${\frac {\partial {\mathbf {u}}}{\partial t}}=0.$

Incompressible flow

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

$\nabla \cdot {\mathbf {u}}=0.$

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

Irrotational flow

A flow is irrotational if the curl of $\mathbf {u}$ is zero:

$\nabla \times {\mathbf {u}}=0.$

That is, if $\mathbf {u}$ 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 ${\mathbf {u}}=\nabla \Phi .$ 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

$\omega =\nabla \times {\mathbf {u}}.$

Thus in irrotational flow the vorticity is zero.

The velocity potential

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

${\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .}$

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

See also

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

References

1. Duderstadt, James J.; Martin, William R. (1979). "Chapter 4:The derivation of continuum description from transport equations". In Wiley-Interscience Publications. Transport theory. New York. p. 218. ISBN 978-0471044925.
2. Freidberg, Jeffrey P. (2008). "Chapter 10:A self-consistent two-fluid model". In Cambridge University Press. Plasma Physics and Fusion Energy (1 ed.). Cambridge. p. 225. ISBN 978-0521733175.
3. Courant, R.; Friedrichs, K.O. (1999) [unabridged republication of the original edition of 1948]. Supersonic Flow and Shock Waves. Applied mathematical sciences (5th ed.). Springer-Verlag New York Inc. p. 24. ISBN 0387902325. OCLC 44071435.