<s>
Streamlines	B-Application
,	I-Application
streaklines	I-Application
and	I-Application
pathlines	I-Application
are	O
field	B-Application
lines	I-Application
in	O
a	O
fluid	O
flow	O
.	O
</s>
<s>
Streamlines	B-Application
are	O
a	O
family	O
of	O
curves	O
whose	O
tangent	O
vectors	O
constitute	O
the	O
velocity	O
vector	O
field	O
of	O
the	O
flow	O
.	O
</s>
<s>
Streaklines	B-Application
are	O
the	O
loci	O
of	O
points	O
of	O
all	O
the	O
fluid	O
particles	O
that	O
have	O
passed	O
continuously	O
through	O
a	O
particular	O
spatial	O
point	O
in	O
the	O
past	O
.	O
</s>
<s>
Dye	O
steadily	O
injected	O
into	O
the	O
fluid	O
at	O
a	O
fixed	O
point	O
extends	O
along	O
a	O
streakline	B-Application
.	O
</s>
<s>
Pathlines	B-Application
are	O
the	O
trajectories	B-Application
that	O
individual	O
fluid	O
particles	O
follow	O
.	O
</s>
<s>
The	O
direction	O
the	O
path	O
takes	O
will	O
be	O
determined	O
by	O
the	O
streamlines	B-Application
of	O
the	O
fluid	O
at	O
each	O
moment	O
in	O
time	O
.	O
</s>
<s>
By	O
definition	O
,	O
different	O
streamlines	B-Application
at	O
the	O
same	O
instant	O
in	O
a	O
flow	O
do	O
not	O
intersect	O
,	O
because	O
a	O
fluid	O
particle	O
cannot	O
have	O
two	O
different	O
velocities	O
at	O
the	O
same	O
point	O
.	O
</s>
<s>
However	O
,	O
pathlines	B-Application
are	O
allowed	O
to	O
intersect	O
themselves	O
or	O
other	O
pathlines	B-Application
(	O
except	O
the	O
starting	O
and	O
end	O
points	O
of	O
the	O
different	O
pathlines	B-Application
,	O
which	O
need	O
to	O
be	O
distinct	O
)	O
.	O
</s>
<s>
Streaklines	B-Application
can	O
also	O
intersect	O
themselves	O
and	O
other	O
streaklines	B-Application
.	O
</s>
<s>
Streamlines	B-Application
and	O
timelines	O
provide	O
a	O
snapshot	O
of	O
some	O
flowfield	O
characteristics	O
,	O
whereas	O
streaklines	B-Application
and	O
pathlines	B-Application
depend	O
on	O
the	O
-history	O
of	O
the	O
flow	O
.	O
</s>
<s>
However	O
,	O
often	O
sequences	O
of	O
timelines	O
(	O
and	O
streaklines	B-Application
)	O
at	O
different	O
instants	O
—	O
being	O
presented	O
either	O
in	O
a	O
single	O
image	O
or	O
with	O
a	O
video	O
stream	O
—	O
may	O
be	O
used	O
to	O
provide	O
insight	O
in	O
the	O
flow	O
and	O
its	O
history	O
.	O
</s>
<s>
If	O
a	O
line	O
,	O
curve	O
or	O
closed	O
curve	O
is	O
used	O
as	O
start	O
point	O
for	O
a	O
continuous	O
set	O
of	O
streamlines	B-Application
,	O
the	O
result	O
is	O
a	O
stream	B-Application
surface	I-Application
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
closed	O
curve	O
in	O
a	O
steady	O
flow	O
,	O
fluid	O
that	O
is	O
inside	O
a	O
stream	B-Application
surface	I-Application
must	O
remain	O
forever	O
within	O
that	O
same	O
stream	B-Application
surface	I-Application
,	O
because	O
the	O
streamlines	B-Application
are	O
tangent	O
to	O
the	O
flow	O
velocity	O
.	O
</s>
<s>
A	O
scalar	O
function	O
whose	O
contour	O
lines	O
define	O
the	O
streamlines	B-Application
is	O
known	O
as	O
the	O
stream	O
function	O
.	O
</s>
<s>
Dye	O
line	O
may	O
refer	O
either	O
to	O
a	O
streakline	B-Application
:	O
dye	O
released	O
gradually	O
from	O
a	O
fixed	O
location	O
during	O
time	O
;	O
or	O
it	O
may	O
refer	O
to	O
a	O
timeline	O
:	O
a	O
line	O
of	O
dye	O
applied	O
instantaneously	O
at	O
a	O
certain	O
moment	O
in	O
time	O
,	O
and	O
observed	O
at	O
a	O
later	O
instant	O
.	O
</s>
<s>
where	O
""	O
denotes	O
the	O
vector	O
cross	O
product	O
and	O
is	O
the	O
parametric	O
representation	O
of	O
just	O
one	O
streamline	B-Application
at	O
one	O
moment	O
in	O
time	O
.	O
</s>
<s>
Here	O
is	O
a	O
variable	O
which	O
parametrizes	O
the	O
curve	O
Streamlines	B-Application
are	O
calculated	O
instantaneously	O
,	O
meaning	O
that	O
at	O
one	O
instance	O
of	O
time	O
they	O
are	O
calculated	O
throughout	O
the	O
fluid	O
from	O
the	O
instantaneous	O
flow	O
velocity	O
field	O
.	O
</s>
<s>
A	O
streamtube	O
consists	O
of	O
a	O
bundle	O
of	O
streamlines	B-Application
,	O
much	O
like	O
communication	O
cable	O
.	O
</s>
<s>
The	O
equation	O
of	O
motion	O
of	O
a	O
fluid	O
on	O
a	O
streamline	B-Application
for	O
a	O
flow	O
in	O
a	O
vertical	O
plane	O
is	O
:	O
</s>
<s>
The	O
flow	O
velocity	O
in	O
the	O
direction	O
of	O
the	O
streamline	B-Application
is	O
denoted	O
by	O
.	O
</s>
<s>
is	O
the	O
radius	O
of	O
curvature	O
of	O
the	O
streamline	B-Application
.	O
</s>
<s>
is	O
the	O
pressure	O
gradient	O
and	O
the	O
velocity	O
gradient	O
along	O
the	O
streamline	B-Application
.	O
</s>
<s>
Streaklines	B-Application
can	O
be	O
expressed	O
as	O
,	O
</s>
<s>
The	O
parameter	O
,	O
parametrizes	O
the	O
streakline	B-Application
and	O
,	O
where	O
is	O
a	O
time	O
of	O
interest	O
.	O
</s>
<s>
In	O
steady	O
flow	O
(	O
when	O
the	O
velocity	O
vector-field	O
does	O
not	O
change	O
with	O
time	O
)	O
,	O
the	O
streamlines	B-Application
,	O
pathlines	B-Application
,	O
and	O
streaklines	B-Application
coincide	O
.	O
</s>
<s>
This	O
is	O
because	O
when	O
a	O
particle	O
on	O
a	O
streamline	B-Application
reaches	O
a	O
point	O
,	O
,	O
further	O
on	O
that	O
streamline	B-Application
the	O
equations	O
governing	O
the	O
flow	O
will	O
send	O
it	O
in	O
a	O
certain	O
direction	O
.	O
</s>
<s>
This	O
is	O
useful	O
,	O
because	O
it	O
is	O
usually	O
very	O
difficult	O
to	O
look	O
at	O
streamlines	B-Application
in	O
an	O
experiment	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
flow	O
is	O
steady	O
,	O
one	O
can	O
use	O
streaklines	B-Application
to	O
describe	O
the	O
streamline	B-Application
pattern	O
.	O
</s>
<s>
Streamlines	B-Application
are	O
frame-dependent	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
streamlines	B-Application
observed	O
in	O
one	O
inertial	O
reference	O
frame	O
are	O
different	O
from	O
those	O
observed	O
in	O
another	O
inertial	O
reference	O
frame	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
streamlines	B-Application
in	O
the	O
air	O
around	O
an	O
aircraft	O
wing	O
are	O
defined	O
differently	O
for	O
the	O
passengers	O
in	O
the	O
aircraft	O
than	O
for	O
an	O
observer	O
on	O
the	O
ground	O
.	O
</s>
<s>
In	O
the	O
aircraft	O
example	O
,	O
the	O
observer	O
on	O
the	O
ground	O
will	O
observe	O
unsteady	O
flow	O
,	O
and	O
the	O
observers	O
in	O
the	O
aircraft	O
will	O
observe	O
steady	O
flow	O
,	O
with	O
constant	O
streamlines	B-Application
.	O
</s>
<s>
When	O
possible	O
,	O
fluid	O
dynamicists	O
try	O
to	O
find	O
a	O
reference	O
frame	O
in	O
which	O
the	O
flow	O
is	O
steady	O
,	O
so	O
that	O
they	O
can	O
use	O
experimental	O
methods	O
of	O
creating	O
streaklines	B-Application
to	O
identify	O
the	O
streamlines	B-Application
.	O
</s>
<s>
Knowledge	O
of	O
the	O
streamlines	B-Application
can	O
be	O
useful	O
in	O
fluid	O
dynamics	O
.	O
</s>
<s>
The	O
curvature	O
of	O
a	O
streamline	B-Application
is	O
related	O
to	O
the	O
pressure	O
gradient	O
acting	O
perpendicular	O
to	O
the	O
streamline	B-Application
.	O
</s>
<s>
The	O
center	O
of	O
curvature	O
of	O
the	O
streamline	B-Application
lies	O
in	O
the	O
direction	O
of	O
decreasing	O
radial	O
pressure	O
.	O
</s>
<s>
The	O
magnitude	O
of	O
the	O
radial	O
pressure	O
gradient	O
can	O
be	O
calculated	O
directly	O
from	O
the	O
density	O
of	O
the	O
fluid	O
,	O
the	O
curvature	O
of	O
the	O
streamline	B-Application
and	O
the	O
local	O
velocity	O
.	O
</s>
<s>
Dye	O
can	O
be	O
used	O
in	O
water	O
,	O
or	O
smoke	O
in	O
air	O
,	O
in	O
order	O
to	O
see	O
streaklines	B-Application
,	O
from	O
which	O
pathlines	B-Application
can	O
be	O
calculated	O
.	O
</s>
<s>
Streaklines	B-Application
are	O
identical	O
to	O
streamlines	B-Application
for	O
steady	O
flow	O
.	O
</s>
<s>
Streamlined	O
objects	O
and	O
organisms	O
,	O
like	O
airfoils	O
,	O
streamliners	O
,	O
cars	O
and	O
dolphins	B-Application
are	O
often	O
aesthetically	O
pleasing	O
to	O
the	O
eye	O
.	O
</s>
<s>
The	O
Streamline	B-Application
Moderne	O
style	O
,	O
a	O
1930s	O
and	O
1940s	O
offshoot	O
of	O
Art	O
Deco	O
,	O
brought	O
flowing	O
lines	O
to	O
architecture	O
and	O
design	O
of	O
the	O
era	O
.	O
</s>
