Geology Reference
In-Depth Information
Fig. 13.8
Basic flow
fields: uniform flow
(
top-left
), source monopole
(
top-right
), vortex flow
(
middle-left
), sink (
middle
right
), and doublet flow
(
bottom
)
radial and constant along circles about the source
or the sink. In a
vortex flow
the streamlines
are concentric circles about a point (Fig.
13.8
).
Finally,
doublet flows
result from the combination
of a source
S
C
and a sink
S
with equal strength.
In this case, the streamlines are circles passing
through
S
C
and
S
, as illustrated in Fig.
13.8
.It
is a simple exercise to find the stream function
associated with these basic flows. In the case of a
uniform flow in the
x
direction, we have:
that the source emits fluid isotropically at a steady
volumetric flowrate
Q
. Therefore, for any circle
C
(
r
) about the source we must have:
I
Q
D
v
d`
D
2 r
u
r
(13.76)
C.r/
Accordingly, the velocity field in polar coordi-
nates will be given by:
Q
2 r
I
u
™
D
0
u
r
D
(13.77)
@§
@
z
D
u
I
@§
@x
D
0
(13.73)
Finally, by (
13.72
)wehaveforthestream
function:
Integrating these equations gives:
Q
2
™
§
D
vz
(13.74)
§
D
(13.78)
In polar coordinates, this expression assumes
the form:
In the case of a sink flow, (
13.78
) holds with
a strength
Q
<0. Now let us turn to the vortex
flows, in which the radial pressure gradient is
always zero. Clearly,
u
r
D
0 for this class of
flows, and the velocity then depends only on the
distance from the vortex center. Therefore, in this
§
.r;™/
D
v
r sin ™
(13.75)
Let us consider now source flows. In this case,
the velocity
v
D
v
(
r
)
D
u
r
and we can assume
