Physics for Geologists, Second edition

Fluids and fluid flow 121
where V is the terminal velocity of a small sphere of diameter d and mass den-
sity p, falling through static fluid of mass density p, and kinematic viscosity
v. The dimensionless numbers a and b must be determined experimentally.
When a = 1 and b = 18, we have Stokes' Law. Stokes' Law applies only to
a single small sphere and then only when its terminal velocity is small enough
for terms in v2 to be neglected. It should not be used for multiple grains set-
tling in a fluid. The point here is that when v2 is not negligible, kinetic energy
is not negligible and the relationship between V and p is no longer linear.
It has been found that Equation 12.7 applies reasonably well to multiple
grains, and grains large enough to give significant terms in v2 -that is, it can
be used for sediment settling provided the coefficients a and b are determined
for similar material.
It was said above that resistance to a moving object is proportional to the
square of its velocity, independent of shape. If an object is falling in air, it
cannot 'know' whether it is falling, or being suspended by a rising wind.
Winnowing
In many parts of the world, wheat is separated from the chaff by beating
the corn with a stick and then tossing the wheat and chaff in a breeze. The
grains fall back into the basket: the chaff blows away. The reason for this
is that the grains are aerodynamically better shaped than the chaff (like the
parachutist and his parachute).
Graded beds are commonly the result of turbidity currents and the settling
of the larger grains from it faster than the smaller in deep water. They are
commonly graded in three dimensions because as the velocity of the current
decreases so the ability of the water to suspend sediment decreases. In any
one position a bed may be graded from coarse to fine upwards; and in any
one bed, the grades may become finer away from the source.
In Chapter 2 we considered briefly the energy consequences of volcanic
eruptions that throw ash and dust into the air and found that the velocity
required to throw any mass to a height of 25 km was so improbable that
volcanic dust must have been taken to those heights by thermal convection
currents due to the volcano itself.
The velocity at which the material falls to Earth can be seen from
Equation 12.7 to be proportional to its size and to its density. In general,
the larger, denser fragments will fall back to Earth sooner than the smaller,
less dense. We must not expect Stokes' law to prevail because (a) some of
the terminal velocities may be large enough for the kinetic energy term in
v2 to be significant; and (b) there will be interactions between the mass of
fragments falling at different speeds.
When Krakatoa (NW end of Java) erupted in 1883, a cloud of fine
dust spread around the Earth and affected sunrises and sunsets for several
months - as happened after the eruption of Pinotubo in the Philippines in
Copyright 2002 by Richard E. Chapman