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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Definition
[ edit ]
The flow velocity u of a fluid is a vector field
u
=
u
(
x
,
t
)
,
{\displaystyle \mathbf {u} =\mathbf {u} (\mathbf {x} ,t),}
which gives the velocity of an element of fluid at a position
x
{\displaystyle \mathbf {x} \,}
and time
t
.
{\displaystyle t.\,}
The flow speed q is the length of the flow velocity vector[3]
q
=
‖
u
‖
{\displaystyle q=\|\mathbf {u} \|}
and is a scalar field.
Uses
[ edit ]
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 ]
The flow of a fluid is said to be steady if
u
{\displaystyle \mathbf {u} }
does not vary with time. That is if
∂
u
∂
t
=
0.
{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}=0.}
Incompressible flow
[ edit ]
If a fluid is incompressible the divergence of
u
{\displaystyle \mathbf {u} }
is zero:
∇
⋅
u
=
0.
{\displaystyle \nabla \cdot \mathbf {u} =0.}
That is, if
u
{\displaystyle \mathbf {u} }
is a solenoidal vector field .
Irrotational flow
[ edit ]
A flow is irrotational if the curl of
u
{\displaystyle \mathbf {u} }
is zero:
∇
×
u
=
0.
{\displaystyle \nabla \times \mathbf {u} =0.}
That is, if
u
{\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
Φ
,
{\displaystyle \Phi ,}
with
u
=
∇
Φ
.
{\displaystyle \mathbf {u} =\nabla \Phi .}
If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero:
Δ
Φ
=
0.
{\displaystyle \Delta \Phi =0.}
Vorticity
[ edit ]
The vorticity ,
ω
{\displaystyle \omega }
, of a flow can be defined in terms of its flow velocity by
ω
=
∇
×
u
.
{\displaystyle \omega =\nabla \times \mathbf {u} .}
If the vorticity is zero, the flow is irrotational.
The velocity potential
[ edit ]
If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field
ϕ
{\displaystyle \phi }
such that
u
=
∇
ϕ
.
{\displaystyle \mathbf {u} =\nabla \mathbf {\phi } .}
The scalar field
ϕ
{\displaystyle \phi }
is called the velocity potential for the flow. (See Irrotational vector field .)
Bulk velocity
[ edit ]
In many engineering applications the local flow velocity
u
{\displaystyle \mathbf {u} }
vector field is not known in every point and the only accessible velocity is the bulk velocity or average flow velocity
u
¯
{\displaystyle {\bar {u}}}
(with the usual dimension of length per time), defined as the quotient between the volume flow rate
V
˙
{\displaystyle {\dot {V}}}
(with dimension of cubed length per time) and the cross sectional area
A
{\displaystyle A}
(with dimension of square length):
u
¯
=
V
˙
A
{\displaystyle {\bar {u}}={\frac {\dot {V}}{A}}}
.
See also
[ edit ]
Drift velocity
Enstrophy
Group velocity
Particle velocity
Pressure gradient
Strain rate
Strain-rate tensor
Stream function
Velocity potential
Vorticity
Wind velocity
References
[ edit ]
^ 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 )
^ 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 )
^ 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 .
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Flow_velocity&oldid=1198171278 "
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a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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