Geoscience Reference
In-Depth Information
The drag force should be equal to the submerged weight in the terminal stage of
settling, yielding
a
1
a
2
gd
1
/
2
C
d
ρ
2
−
ρ
ρ
s
ω
=
(3.3)
s
3.1.2 Settling velocity of spherical particles
In the laminar (streamline) settling region (i.e., the particle Reynolds number
R
e
=
ω
1.0), Stokes derived the drag force on a spherical particle by solving the
Navier-Stokes equations without inertia terms. The derived drag coefficient is
s
d
/ν <
24
R
e
C
d
=
(3.4)
Inserting Eq. (3.4) into Eq. (3.3) leads to the Stokes law for the settling velocity of
spherical particles:
g
d
2
ν
18
ρ
1
−
ρ
ρ
s
ω
=
(3.5)
s
s
−
1
(meters per second) and m (meters), respectively.
Oseen (1927) solved the Navier-Stokes equations, including some inertia terms, and
obtained the following relation:
where
ω
s
and
d
are in m
·
1
16
R
e
24
R
e
3
C
d
=
+
(3.6)
Goldstein (1929) found a relatively complete solution of Oseen's approximation as
follows:
1
24
R
e
3
16
R
e
19
1280
R
e
+
71
20480
R
e
+···
C
d
=
+
−
(3.7)
Eq. (3.7) is valid for
R
e
up to 2.0. Beyond this range, the drag coefficient usually
has to be determined by experiments rather than theoretical solutions. Rouse (1938)
summarized the available experimental data and obtained the relation between
C
d
and
R
e
shown in Fig. 3.1, which can be used to determine
C
d
and, in turn, the settling
velocity of spherical particles.
Fig. 3.1 shows that when
R
e
1, 000 — i.e., in the turbulent settling region — the
drag coefficient is no longer related to the particle Reynolds number and has a value
of about 0.45, thus yielding
>
1.72
ρ
−
ρ
ρ
s
ω
=
gd
(3.8)
s
