Geology Reference
In-Depth Information
Fig. 13.3
Couette flow (
a
) and Poiseuille flow (
b
)
Furthermore, for a stationary incompressible flow
it results: @
v
x
/@
x
D
@
v
x
/@
t
D
0. Accordingly, the
Navier-Stokes equations reduce to the following
simple equations:
8
<
as
Poiseuille
-
Couette flows
and have been used
by several authors, especially in recent years, to
model flows in the asthenosphere (e.g., Schubert
and Turcotte
1972
; Parmentier and Oliver
1979
;
Conrad et al.
2010
; Höink and Lenardic
2010
;
Höink et al.
2011
; Natarov and Conrad
2012
).
When @
p
/@
x
¤
0, the flow is said to be
pressure
driven
, whereas for @
p
/@
x
D
0, the magnitude
of
v
x
decreases linearly with depth (Fig.
13.3
a)
and we say that the flow is a
Couette flow
.
Finally, when a non-zero horizontal pressure gra-
dient exists but
v
0
D
0, then the velocity profile
is parabolic and symmetric with respect to the
plane
z
D
h
/2. This kind of pressure-driven flow is
called a
Poiseuille flow
(Fig.
13.3
b) and is always
in the direction of
decreasing
pressure.
By (
13.46
), we note that the effect of an in-
creasing viscosity is a tendency towards a Couette
flow for any assigned pressure gradient:
@p
@x
C
ǜ
@
2
v
x
0
D
@
z
2
(13.43)
@p
@
z
C
¡
.
z
/g
:
0
D
The second equation follows from the hypoth-
esis that the velocity field is horizontal. It simply
says that the variations of pressure with depth are
hydrostatic, thereby @
p
/@
x
is independent from
z
and the pressure field has the form:
Z
z
p.x;
z
/
D
f.x/
C
g
¡.
z
/d
z
(13.44)
0
Therefore, the first equation can be rewritten
as an ordinary second-order differential equation:
1
v
0
as ǜ
!1
z
h
v
x
.
z
/
(13.47)
d
2
v
x
d
z
2
D
1
ǜ
@p
@x
(13.45)
It is useful to calculate the net areal flux per
unit area through a vertical cross-section in the
asthenosphere. This quantity coincides with the
average horizontal velocity in the asthenospheric
channel and is given by:
Finally, using the no-slip boundary conditions:
v
x
(0)
D
v
0
and
v
x
(
h
)
D
0, we have the following
simple solution:
v
0
h
C
z
C
v
0
1
2ǜ
@p
@x
z
2
h
2ǜ
@p
@x
Z
v
x
.
z
/
D
h
h
2
12ǜ
1
h
@p
@x
C
1
2
v
0
(13.48)
ˆ
a
D
v
x
.
z
/d
z
D
(13.46)
0
The solution (
13.46
) states that a velocity
profile through the asthenosphere has a parabolic
shape, granted that the assumptions done are
valid. This kind of flow and its variants are known
Assuming
h
D
300 km, an average viscosity
ǜ
D
10
20
Pa s, and a pressure gradient of 10 kPa
km
1
, we have that the pressure-driven contribu-
