Flow velocity

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Short description: Vector field which is used to mathematically describe the motion of a continuum

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 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).


The flow velocity u of a fluid is a vector field

[math]\displaystyle{ \mathbf{u}=\mathbf{u}(\mathbf{x},t), }[/math]

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

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

[math]\displaystyle{ q = \| \mathbf{u} \| }[/math]

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 [math]\displaystyle{ \mathbf{u} }[/math] does not vary with time. That is if

[math]\displaystyle{ \frac{\partial \mathbf{u}}{\partial t}=0. }[/math]

Incompressible flow

If a fluid is incompressible the divergence of [math]\displaystyle{ \mathbf{u} }[/math] is zero:

[math]\displaystyle{ \nabla\cdot\mathbf{u}=0. }[/math]

That is, if [math]\displaystyle{ \mathbf{u} }[/math] is a solenoidal vector field.

Irrotational flow

A flow is irrotational if the curl of [math]\displaystyle{ \mathbf{u} }[/math] is zero:

[math]\displaystyle{ \nabla\times\mathbf{u}=0. }[/math]

That is, if [math]\displaystyle{ \mathbf{u} }[/math] 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 [math]\displaystyle{ \Phi, }[/math] with [math]\displaystyle{ \mathbf{u}=\nabla\Phi. }[/math] If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: [math]\displaystyle{ \Delta\Phi=0. }[/math]


The vorticity, [math]\displaystyle{ \omega }[/math], of a flow can be defined in terms of its flow velocity by

[math]\displaystyle{ \omega=\nabla\times\mathbf{u}. }[/math]

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 [math]\displaystyle{ \phi }[/math] such that

[math]\displaystyle{ \mathbf{u}=\nabla\mathbf{\phi}. }[/math]

The scalar field [math]\displaystyle{ \phi }[/math] is called the velocity potential for the flow. (See Irrotational vector field.)

Bulk velocity

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

[math]\displaystyle{ u_{\rm{}av}=\frac{\dot{V}}{A} }[/math].

See also


  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). Supersonic Flow and Shock Waves. Applied mathematical sciences (5th ed.). Springer-Verlag New York Inc. pp. 24. ISBN 0387902325. OCLC 44071435. https://archive.org/details/supersonicflowsh0000cour/page/24.