Civil Engineering Reference
In-Depth Information
X
3
0
0
v
3
v(x
1
)
X
1
−
a
a
M
2a
Figure 4.3
Flow in a slit limited by the planes
x
1
=
aand
x
1
=−
a
.
In this equation,
l
is equal to
l
=
(j ωρ
o
/η)
1
/
2
(4.22)
As with the case of
l
, both determinations of the square root in Equation (4.22) give
identical results. The mean velocity is equal to
a
−
a
υ
3
d
x
1
2
a
υ
3
=
(4.23)
Upon integrating, we find
1
−
tanh
(s
j
1
/
2
)
s
j
1
/
2
1
jωρ
o
∂p
∂x
3
υ
3
=−
(4.24)
In this equation,
s
is equal to
ωρ
o
a
2
η
1
/
2
s
=
(4.25)
The effective density of the air in the slit is defined by rewriting Equation (4.24) as
∂p
∂x
3
jωρυ
3
=−
(4.26)
where
ρ
o
1
tanh
(s
j
1
/
2
)
s
j
1
/
2
ρ
=
−
(4.27)
On the other hand, the true density
ρ
o
can be used in the Newton equation
tanh
(s
j
1
/
2
)(s
j
1
/
2
)
∂p
∂x
3
=
jωρ
o
υ
3
+
jωρ
o
υ
3
−
(4.28)
1
−
[tanh
(s
j
1
/
2
)(s
j
1
/
2
)
]
