Civil Engineering Reference
In-Depth Information
Let
F
be the parameter defined by
j
ω
ω
d
D(ω)
D(
0
)
F
=
1
−
(5.120)
At low frequencies the diffusion skin depth is large around the pores in the microp-
orous medium and
F
is close to 1. At high frequencies, the diffusion skin depth decreases
and
F
tends to 0. At the macroscopic scale the bulk modulus of the fluid equivalent to
the porous medium is given by
−
1
1
K
p
(ω)
+
φ
p
)
φ
m
F(ω)
P
0
K
dp
(ω)
=
(
1
−
(5.121)
Illustration in the case of a slit material (from Olny and Boutin 2003)
The microporous panels of thickness 2
b
are separated with air-gaps of thickness 2
a
(see Figure 5.12). The mesoporosity
φ
p
b)
. The viscous permeability
q
of a
slit material of porosity
φ
, with slits of thickness 2
a
is obtained with Equations (4.27)
and (5.2)
=
a/(a
+
1
tanh
j
1
/
2
(a
√
2
/δ)
2
j
1
/
2
(a
√
2
/δ)
2
jφ
δ
q(ω)
=−
√
2
−
(5.122)
where
δ
=
(
2
η/(ωρ
0
))
1
/
2
(see Equation 4.58) is the viscous skin depth. The thermal
permeability has the same form, except that the viscous skin depth must be replaced by
the thermal skin depth
δ
=
δ/B,B
being the square root of the Prandtl number. The
x
3
, p
1
0
p
p
p
m
1
Pore
2
Microporous
domain
3
x
2
0
4
2
a
2
b
Figure 5.12
The slit medium and the pressure field in the microporous domain:(0)
b/δ
d
=
0
.
1, (1)
b/δ
d
=
1, (2)
b/δ
d
=
2, (3)
b/δ
d
=
5, (4)
b/δ
d
=
100 (Olny and Boutin
2003). Reprinted with permission from Olny, X., & Boutin, C. Acoustic wave prop-
agation in double porosity media.
J. Acoust. Soc. Amer.
114
, 73 - 89. Copyright 2003,
Acoustical Society of America.
