Civil Engineering Reference
In-Depth Information
l
2
r
2
r
1
l
1
Figure 5.A.1
A pore made up of an alternating sequence of cylinders.
The velocities are supposed to be constant in each cylinder. Then,
υ
1
and
υ
2
are
related by
υ
1
υ
2
=
S
2
S
1
(5.A.1)
If the fluid is nonviscous, the Newton equations in the cylinders are
∂p
1
∂x
=
jωρ
0
υ
1
−
(5.A.2)
∂p
2
∂x
=
−
jωρ
0
υ
2
(5.A.3)
where
p
1
and
p
2
are the pressures in cylinders 1 and 2 and
ρ
0
is the density of the fluid.
The macroscopic pressure derivative
∂p/∂x
and the macroscopic velocity
υ
are
∂p
∂x
=
∂p
1
∂x
l
1
l
1
+
∂p
2
∂x
l
2
l
1
+
l
2
+
(5.A.4)
l
2
l
1
S
1
υ
1
l
1
S
1
+
l
2
S
2
υ
2
l
1
S
1
+
υ
=
l
2
S
2
+
(5.A.5)
l
2
S
2
The quantities
υ
and
∂p/∂x
are linked by the following equation:
∂p
∂x
=
−
α
∞
ρ
0
jωυ
(5.A.6)
where
α
∞
is given by
[
l
1
S
1
+
l
2
S
2
][
l
2
S
1
+
l
1
S
2
]
(l
1
α
∞
=
(5.A.7)
+
l
2
)
2
S
1
S
2
Appendix 5.B: Calculation of the characteristic length
In a porous medium, the temperature at high frequencies is constant over the cross-section,
except for a small region close to the frame -air boundary. The boundary between the
frame and the air in a porous material is represented in Figure. 5.B.1.
