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[edit]

Main article: 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[edit]

Main article: 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[edit]

Main article: 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.


Main article: Vorticity

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[edit]

Main article: Potential flow

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.)


  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. 

See also[edit]

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