2 and
ν
=
1
.
0
×
10
−
6
m
2
/
s. Hence,
Re
=
(
a
/
6
×
10
−
5
)
3
, where
a
is measured in meters. Thus, expression (9.128), which is strictly speaking only valid
when Re
≪
1, but which turns out to be approximately valid for all Reynolds numbers less than unity, only holds for
sand grains whose radii are less than about 60 microns. Such grains fall through water at approximately 8
×
10
−
3
m
/
s.
For the case of a droplet of water falling through air at 20
◦
C and atmospheric pressure, we have
ρ/ρ
=
780 and
ν
=
1
.
5
×
10
−
5
m
2
/
s. Hence, Re
=
(
a
/
4
×
10
−
5
)
3
, where
a
is measured in meters. Thus, expression (9.128) only holds
for water droplets whose radii are less than about 40 microns. Such droplets fall through air at approximately 0
.
2 m
/
s.
At large values of
r
/
a
, Equations (9.105), (9.106), and (9.112) yield
v
r
(
r
,θ
)
=
−
V
cos
θ
+
3
2
V
cos
θ
a
r
+
O
parenleftbigg
a
r
parenrightbigg
2
,
(9.130)
v
θ
(
r
,θ
)
=
V
sin
θ
−
3
4
V
sin
θ
a
r
+
O
parenleftbigg
a
r
parenrightbigg
2
.
(9.131)
It follows that
[
ρ
(
v
· ∇
)
v
]
r
=
ρ
v
r
∂v
r
∂
r
+
v
θ
r
∂v
r
∂θ
−
v
2
θ
r
∼
ρ
V
2
a
r
2
,
(9.132)
and
(
μ
∇
2
v
)
r
∼
μ
∂
2
v
r
∂
r
2
∼
μ
V a
r
3
.
(9.133)
Hence,
[
ρ
(
v
· ∇
)
v
]
r
(
μ
∇
2
v
)
r
∼
ρ
V r
μ
∼
Re
r
a
,
(9.134)
where Re is the Reynolds number of the flow in the immediate vicinity of the sphere. [See Equation (9.94).] Now, our
analysis is based on the assumption that advective inertia is negligible with respect to viscosity. However, as is clear
from the above expression for the ratio of inertia to viscosity within the fluid, even if this ratio is much less than unity
close to the sphere—in other words, if Re
≪
1—it inevitably becomes much greater than unity far from the sphere:
i.e.
, for
r
≫
a
/
Re. In other words, inertia always dominates viscosity, and our Stokes flow solution therefore breaks
down, at su
ffi
ciently large
r
/
a
.
9.11
Axisymmetric Stokes Flow In and Around a Fluid Sphere
Suppose that the solid sphere discussed in the previous section is replaced by a spherical fluid drop of radius
a
. Let
the drop move through the surrounding fluid at the constant velocity
V
e
z
. Obviously, the fluid from which the drop is
composed must be immiscible with the surrounding fluid. Let us transform to a frame of reference in which the drop
is stationary, and centered at the origin. Assuming that the Reynolds numbers immediately outside and inside the drop
are both much less than unity, and making use of the previous analysis, the most general expressions for the stream
function outside and inside the drop are
ψ
(
r
,θ
)
=
sin
2
θ
parenleftbigg
A
r
+
B r
+
C r
2
+
D r
4
parenrightbigg
,
(9.135)
and
ψ
(
r
,θ
)
=
sin
2
θ
A
r
+
Br
+
C r
2
+
D r
4
,
(9.136)
respectively. Here,
A
,
B
,
C
,
etc.
are arbitrary constants. Likewise, the previous analysis also allows us to deduce that
v
r
(
r
,θ
)
=
−
2 cos
θ
parenleftbigg
A
r
3
+
B
r
+
C
+
D r
2
parenrightbigg
,
(9.137)
v
θ
(
r
,θ
)
=
sin
θ
parenleftbigg
−
A
r
3
+
B
r
+
2
C
+
4
D r
2
parenrightbigg
,
(9.138)

186
FLUID MECHANICS
−
2
−
1
0
1
2
z/a
−
2
−
1
0
1
2
x/a
Figure 9.7:
Contours of the stream function in the x-z plane for Stokes flow in and around a fluid sphere.

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- Fluid Dynamics, Fluid Mechanics, stress tensor, Fluid Motion