Geology Reference
In-Depth Information
z
(a)
(b)
v
r
δ
z
B
R
Δρ
v
r
a
-a
a
Figure 7.10. (a) A parcel of fluid in a conduit of buoyant fluid. Coordinates are
r
and
z
. The conduit has radius
a
and the fluid parcel has radius
r
and height δ
z
.
The fluid has a density deficit
ρ
relative to fluid outside the conduit. The fluid
parcel has a buoyancy force
B
and is resisted by a viscous resistance force
R
.
(b) The variation of velocity across the conduit.
sides. We can look at the balance of the buoyancy force and the viscous resisting
force on this parcel. The buoyancy force is
gρ
π
r
2
δ
z,
B
=−
(7.19)
where the minus sign is because a negative density difference should yield a
force in the upward, positive direction. To get the viscous resistance, we need a
representative velocity gradient. If the parcel is moving upwards at velocity
v
,then
v/r
will be suitable. The resisting viscous stress is then
μv/r
. This stress acts over
the surface of the parcel, which we take to be a vertical cylinder. Its surface area is
its circumference, 2π
r
, times its height δ
z
. Thus the resisting force is
R
=
2π
r
δ
zμv/r
=
2πδ
zμv.
(7.20)
Balancing these forces then yields
(
gρ/
2
μ
)
r
2
.
v
=−
Although this analysis is a bit rough, a more rigorous analysis, given in
Dynamic
Earth
[1], yields the same result. However, this result is not complete, because it
says that the velocity is zero on the axis of the conduit and non-zero and negative at
the edge, where,
r
a
. We want a solution in which the velocity is zero at the edge,
so we must add a constant velocity (which doesn't change the resisting viscous
force) equal to (
gρ/
2
μ
)
a
2
,soweget
=
(
gρ/
2
μ
)(
a
2
r
2
)
.
v
=
−
