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 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
).
Definition
[
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]
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.
Uses
[
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]
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
[
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]
The flow of a fluid is said to be
steady
if
does not vary with time. That is if
Incompressible flow
[
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]
If a fluid is incompressible the
divergence
of
is zero:
That is, if
is a
solenoidal vector field
.
Irrotational flow
[
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]
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:
Vorticity
[
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]
The
vorticity
,
, of a flow can be defined in terms of its flow velocity by
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
such that
The scalar field
is called the
velocity potential
for the flow. (See
Irrotational vector field
.)
Bulk velocity
[
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]
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
(with the usual dimension of length per time), defined as the quotient between the
volume flow rate
(with dimension of cubed length per time) and the cross sectional area
(with dimension of square length):
- .
See also
[
edit
]
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
.