Hence,
C
f
=
f
x
f
y
=
f
x
f
y
=
4
3
d
0
l
H
(
k
)
,
(10.77)
where
H
(
k
)
=
k
ln
1
+
k
1
−
k
−
3
k
2
ln
1
+
k
1
−
k
−
2
k
.
(10.78)
The function
H
(
k
) is a monotonically decreasing function of
k
in the range 0
<
k
<
1.
In fact,
H
(
k
→
0)
→
3
/
(4
k
), whereas
H
(
k
→
1)
→
1. Thus, if
k
∼ O
(1) [i.e., if
(
d
1
−
d
2
)
/
d
1
∼ O
(1)] then
C
f
∼ O
(
d
0
/
l
)
1. In other words, the e
ff
ective coe
ﬃ
cient
of friction between two solid bodies in relative motion that are separated by a thin
fluid layer is independent of the fluid viscosity, and much less than unity. This result
is significant because the coe
ﬃ
cient of friction between two solid bodies in relative
motion that are in direct contact with one another is typical of order unity. Hence,
the presence of a thin lubricating layer does indeed lead to a large reduction in the
frictional drag acting between the bodies.
Incompressible Viscous Flow
299
10.8
Stokes Flow
Steady flow in which the viscous force density in the fluid greatly exceeds the ad
vective inertia per unit volume is generally known as
Stokes flow
, in honor of George
Stokes (1819–1903). Because, by definition, the Reynolds number of a fluid is the
typical ratio of the advective inertia per unit volume to the viscous force density (see
Section 1.16), Stokes flow implies Reynolds numbers that are much less than unity.
In the time independent, low Reynolds number limit, Equations (10.1) and (10.2)
reduce to
∇ ·
v
=
0
,
(10.79)
0
=
−∇
P
+
µ
∇
2
v
.
(10.80)
It follows from these equations that
∇
P
=
µ
∇
2
v
=
−
µ
∇ ×
(
∇ ×
v
)
=
−
µ
∇ ×
ω
,
(10.81)
where
ω
=
∇ ×
v
, and use has been made of Equation (A.177). Taking the curl of
this expression, we obtain
∇
2
ω
=
0
,
(10.82)
which is the governing equation for Stokes flow. Here, use has been made of Equa
tions (A.173), (A.176), and (A.177).
10.9
Axisymmetric Stokes Flow
Let
r
,
θ
,
ϕ
be standard spherical coordinates. Consider axisymmetric Stokes flow
such that
v
(
r
)
=
v
r
(
r
, θ
)
e
r
+
v
θ
(
r
, θ
)
e
θ
.
(10.83)
According to Equations (A.175) and (A.176), we can automatically satisfy the in
compressibility constraint (10.79) by writing (see Section 7.4)
v
=
∇
ϕ
× ∇
ψ,
(10.84)
where
ψ
(
r
, θ
) is the Stokes stream function (i.e.,
v
· ∇
ψ
=
0). It follows that
v
r
(
r
, θ
)
=
−
1
r
2
sin
θ
∂ψ
∂θ
,
(10.85)
v
θ
(
r
, θ
)
=
1
r
sin
θ
∂ψ
∂
r
.
(10.86)
300
Theoretical Fluid Mechanics
Moreover, according to Section C.4,
ω
r
=
ω
θ
=
0, and
ω
ϕ
(
r
, θ
)
=
1
r
∂
(
r
v
θ
)
∂
r
−
1
r
∂v
r
∂θ
=
L
(
ψ
)
r
sin
θ
,
(10.87)
where (see Section 7.4)
L
=
∂
2
∂
r
2
+
sin
θ
r
2
∂
∂θ
1
sin
θ
∂
∂θ
.
(10.88)
Hence, given that
∇
ϕ

=
1
/
(
r
sin
θ
), we can write
ω
=
∇ ×
v
=
L
(
ψ
)
∇
ϕ.
(10.89)
It follows from Equations (A.176) and (A.178) that
∇ ×
ω
=
∇
ϕ
× ∇
[
−L
(
ψ
)]
.
(10.90)
Thus, by analogy with Equations (10.84) and (10.89), and making use of Equa
tions (A.173) and (A.177), we obtain
∇ ×
(
∇ ×
ω
)
=
−∇
2
ω
=
−L
2
(
ψ
)
∇
ϕ.
(10.91)
Equation (10.82) implies that
L
2
(
ψ
)
=
0
,
(10.92)
which is the governing equation for axisymmetric Stokes flow. In addition, Equa
tions (10.81) and (10.90) yield
∇
P
=
µ
∇
ϕ
× ∇
[
L
(
ψ
)]
.
(10.93)
10.10
Axisymmetric Stokes Flow Around a Solid Sphere
Consider a solid sphere of radius
a
that is moving under gravity at the constant ver
tical velocity
V
e
z
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 Fall '06
 MOHAMEDATTIA
 Fluid Dynamics, The Land, stress tensor, Fluid Motion, theoretical fluid mechanics