Geology Reference
In-Depth Information
Fig. 13.6
Parabolic
streamline pattern,
resulting from
superposition of a linear
flow and a source
S
.Itis
clearly visible a stagnation
point close to
S
.This
pattern is often used as a
model of the interaction
between a mantle plume
and the large-scale
asthenospheric flow (Sleep
1987
,
1990
)
(Fig.
13.6
). The kinematic pattern at any given
time
t
may include
stagnation points
,thatis,
points where
v
D
0, either in the fluid interior
or along its boundary. Two or more streamlines
may meet at a stagnation point, as illustrated in
Fig.
13.6
. In general, streamlines can be closed
lines, extend to infinity, or terminate at stagnation
points. It should be noted that the path
r
D
r
(
t
)
of a fluid parcel does not coincide, in general,
with any streamline unless the motion is steady. A
flowfieldissaidtobe
two
-
dimensional
when the
velocity
v
(
r
,
t
) is at any point normal to some fixed
direction. In this instance, it is always possible to
choose a Cartesian reference frame such that
v
(
r
)
has components (
u
(
r
),0, (
r
)) at any point
r
.Two-
dimensional velocity fields in fluid materials have
some interesting properties that greatly simplify
the solution of the Navier-Stokes equations when
the fluid can be considered incompressible.
For a two-dimensional incompressible fluid,
the mass conservation Eq. (
13.8
) reduces to:
Fig. 13.7
Areal flux through a path between two points
A
and
B
The function § is termed the
stream func-
tion
and represents a valuable tool for solving a
number of problems in geodynamics. By (
13.63
),
we see that the stream function associated with
any two-dimensional flow is not unique, because
an arbitrary constant may be added to § that
generates the same velocity field (
u
, ). Now let
us consider an arbitrary path in the (
x
,
z
)plane
between two points
A
and
B
(Fig.
13.7
). At any
point along this curve, the versor normal to the
tangent vector
d
r
D
(
dx
,
dy
) is given by:
dy
dr
;
@
u
@x
C
@
@
z
D
0
dx
dr
(13.62)
n
D
(13.64)
This condition is clearly satisfied if
u
and
are
The areal flux across the path is defined as
the line integral of the normal component of the
velocity along the path. By (
13.63
)and(
13.64
),
this quantity is given by:
derivatives of a scalar field §
D
§(
x
,
z
,
t
):
@§
@
z
I D
@§
@x
u
D
(13.63)