Chapter 3
CHANNEL FLOW
3.1 Generalities
3.1.1
Preamble
Channel flow will be taken here to include plane Couette flow, and
thus some aspects of lubrication theory. An important application is
in ducting for ventilation. There is a strong parallel between pipe flow
and channel flow, extending to the techniques used, the laboratories
involved, and even the investigators. This parallel does not extend
to applications.
3.1.2
Equations and integrals
Channel flow is described by the equations of motion in rectangular
coordinates (
x, y, z
), with velocity components (
u, v, w
). The
mean flow is two-dimensional and rectilinear. Thus
v
=
w
= 0 and
∂/∂x
=
∂/∂z
= 0, except for the driving term
∂p/∂x
. The continuity
equation is automatically satisfied. The momentum equations in the
203
204
CHAPTER 3.
CHANNEL FLOW
appendix
1
become
0 =
−
∂p
∂x
+
∂
∂y
(
μ
∂u
∂y
−
ρ
u
′
v
′
)
;
(3.1)
0 =
−
∂p
∂y
−
∂
∂y
(
ρ
v
′
v
′
) ;
(3.2)
0 =
−
∂
∂z
(
ρ
w
′
w
′
)
.
(3.3)
As in pipe flow, the quantities in parentheses do not depend on
x
, so
that
∂
2
p
∂x
2
=
∂
2
p
∂x∂y
= 0
.
(3.4)
It follows that
∂p
∂x
= constant =
d
p
d
x
.
(3.5)
The second equation has the integral
p
+
ρ
v
′
v
′
=
p
w
=
x
d
p
d
x
+ constant
.
(3.6)
The shearing stress is conveniently defined as
τ
=
μ
∂u
∂y
−
ρ
u
′
v
′
(3.7)
and equation (3.1) has the integral
τ
−
τ
w
=
y
d
p
d
x
(3.8)
where the origin for
y
is taken (say) as the lower channel wall. If the
channel height is
h
, then with
τ
= 0 at
y
=
h/
2 and
τ
=
−
τ
w
at
y
=
h
,
τ
w
=
−
h
2
d
p
d
x
.
(3.9)
This expression can also be obtained by an overall force balance over
a length of the channel.
The third equation (3.3) needs a comment about sublayer vor-
tices; see section x. Note that no information is obtained about two
of the Reynolds normal stresses,
u
′
u
′
and
w
′
w
′
.
1
This appendix was not found.
3.1. GENERALITIES
205
3.1.3
Laminar flow
If the flow is laminar, the velocity profile is determined by a combi-
nation of equations (3.7) and (3.8),
μ
∂u
∂y
−
τ
w
=
y
d
p
d
x
(3.10)
with the integral
u
=
τ
w
μ
(
y
−
y
2
h
)
+ constant
.
(3.11)
With
u
= 0 at
y
= 0, the constant of integration is zero. With
u
=
u
c
at
y
=
h/
2,
u
c
=
τ
w
h
4
μ .
(3.12)
The velocity profile is the parabola
u
u
c
= 1
−
(
1
−
y
h/
2
)
2
.
(3.13)
The mean velocity is defined by
h
̃
u
=
h
∫
o
u
d
y .
(3.14)
Equation (3.13) then implies
̃
u
=
2
3
u
c
.
(3.15)
3.1.4
Development length
The same argument and the same numbers apply as for pipe flow.