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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