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The rigorous analysis yields a slightly different constant, so finally
( gρ/ 4 μ )( a 2
r 2 ) .
v
=
(7.21)
This solution is sketched in Figure 7.10(b). As intuition would suggest, the velocity
is a maximum along the axis and falls off smoothly to zero at the sides.
Now let's look at what happens if the radius of the conduit is doubled. The
velocity at the axis, where r
( gρ/ 4 μ ) a 2 , which depends on the
square of the radius, so v a goes up by a factor of 4. However, if the radius doubles,
then the area of cross-section of the conduit, which is π a 2 , also goes up by a factor
of 4. Therefore, the volumetric flow rate - the volume of fluid that flows past a
given point per unit time - goes up by a factor of 4
=
0, is v a
=
16. A rigorous calculation
of the volumetric flow rate, φ , integrating the flow in concentric circles out from
the axis, yields
×
4
=
π
8 μ
a 4 .
φ
=
(7.22)
Thus doubling a increases φ by 2 4
16, as we already deduced.
If we look at this from the other way around, it implies that large changes in
the flow rate can be accommodated with only moderate changes in plume radius.
The estimates of plume flow rates based on hotspot swells that were discussed in
Section 7.2 vary by over a factor of 10 between Hawaii (the strongest) and some of
the weaker plumes for which estimates are still possible. Our analysis here implies
that the radii of these plumes will not differ by more than about a factor of 2. This
implies, for example, that the seismic detectability will not differ greatly among
these plumes.
We estimated earlier that the Hawaiian plume has a volumetric flow rate of
240 m 3 /s, or 7.4 km 3 /yr. We can use Eq. (7.22) and previous values of other quan-
tities to calculate the plume radius. First, assuming the plume is 300 C hotter
than ambient mantle, ρ
=
30 kg/m 3 . The viscosity of the upper mantle
is not very accurately determined, but a value of 3
=
ραT
=
10 20 Pa s is plausible. The
plume material would then be as much as 100 times less viscous, so we can assume
3
×
10 18 Pa s. Substitution of these values into Eq. (7.22) yields a
49 700 m
or 49.7 km. The agreement with our previous 'guesstimates' of a plume diameter
of about 100 km is not to be taken too seriously, because there is considerable
uncertainty in some of these numbers. On the other hand, Eq. (7.22) implies that
the radius is not too sensitive to these uncertainties. For example, if the viscosity
within the plume is 10 19 Pa s, the calculated radius is only increased by a factor of
1.3 to 65 km. We can therefore conclude with some assurance that plume diameters
of about 100 km are fluid-dynamically quite plausible.
×
=
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