Flow velocity

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).

DefinitionEdit

The flow velocity u of a fluid is a vector field

 

which gives the velocity of an element of fluid at a position   and time  

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

 

and is a scalar field.

UsesEdit

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 flowEdit

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

 

Incompressible flowEdit

If a fluid is incompressible the divergence of   is zero:

 

That is, if   is a solenoidal vector field.

Irrotational flowEdit

A flow is irrotational if the curl of   is zero:

 

That is, if   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   with   If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero:  

VorticityEdit

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

 

Thus in irrotational flow the vorticity is zero.

The velocity potentialEdit

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

 

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

Bulk velocityEdit

In many engineering applications the local flow velocity   vector field is not known in every point and the only accessible velocity is the bulk velocity (or average flow velocity)   which is the ratio between the volume flow rate   by

 

where   is the cross sectional area.

See alsoEdit

ReferencesEdit

  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.
  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.
  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. pp. 24. ISBN 0387902325. OCLC 44071435.