Environmental Engineering Reference
In-Depth Information
Here, the first term represents the convective momentum flux, the second term
is the pressure tensor with pressure
p
, and the third term is the viscous stress tensor.
Substitution in Equation (3.2) yields
=
!
∂
ρ
!!
+
pI
!
r
ρ
−
τ
ρ
ð
Eq
:
3
:
37
Þ
+
∂
t
Example 3.8 Determination of the velocity profile for liquid flow
in a wide rectangular duct
Figure 3.4 shows a schematic of the situation. The liquid height is H. As a result
of a pressure drop, a noncompressible liquid flows steadily (no time dependency)
through the wide rectangular duct. Assume the pressure drop over the duct to be
constant, d
p
/d
x
. The fluid is to be considered Newtonian, which means
d
v
d
y
τ
y
=
−
η
Derive an expression for the velocity profile in the flow direction (x) as a func-
tion of the liquid depth direction (y).
Solution
In order to determine a velocity profile, one needs to set up a momentum balance.
Consider the rectangular duct to be infinitely broad. Then the momentum balance
over a differential volume
w
dx
dy can be written as
ρ
v
x
wdyv
x
−
ρ
v
x+dx
wdyv
x+dx
−
τ
y
wdx +
τ
y
+
dy
wdx +
p
x
wdy
−
p
x+dx
wdy = 0
The first two terms go to zero, as mass flow is constant and the medium is
incompressible. For dx
!
0, one can write
d
dy
=+
d
p
τ
y
dx
Liquid surface
z
y
τ
y
+dy
x
y
+dy
p
x
+d
x
p
x
H
y
τ
y
x+dx
x
FIGURE 3.4
Schematic of the rectangular duct with liquid flow; width (
w
)in
z
-direction
is infinite.
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