Flow velocity

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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.


The flow velocity u of a fluid is a vector field


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

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

q = || \mathbf{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:

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:


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

Irrotational flow

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


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.


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


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


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

See also


  1. Duderstadt, James J., Martin, William R. (1979). "Chapter 4:The derivation of continuum description from transport equations". In Wiley-Interscience Publications (ed.). Transport theory. New York. p. 218. ISBN 978-0471044925.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  2. Freidberg, Jeffrey P. (2008). "Chapter 10:A self-consistent two-fluid model". In Cambridge University Press (ed.). Plasma Physics and Fusion Energy (1 ed.). Cambridge. p. 225. ISBN 978-0521733175.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>