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Flow velocity - Wikipedia

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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 scalar, the flow speed. 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

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

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

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

{\displaystyle 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:

The flow of a fluid is said to be steady if {\displaystyle \mathbf {u} } does not vary with time. That is if

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

Incompressible flow

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

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

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

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

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

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

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

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

If the vorticity is zero, the flow is irrotational.

The velocity potential

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

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

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

In many engineering applications the local flow velocity {\displaystyle \mathbf {u} } vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity {\displaystyle {\bar {u}}} (with the usual dimension of length per time), defined as the quotient between the volume flow rate {\displaystyle {\dot {V}}} (with dimension of cubed length per time) and the cross sectional area {\displaystyle A} (with dimension of square length):

{\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}}.
  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.{{cite book}}: CS1 maint: location missing publisher (link)
  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.{{cite book}}: CS1 maint: location missing publisher (link)
  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.