Civil Engineering Reference
In-Depth Information
Both parameters
ρ
and
K
only depend on the flow resistivity in the previous equations.
At high frequencies, when
ω
tends to infinity, the viscous skin depth tends to zero and
the velocity is the velocity without viscosity, except in a small layer close to the surface
of the cylinders. When the viscous skin depth is very thin compared with the smallest
lateral dimension of the pores, the velocity distribution close to the surface of the pores
is the same as for a plane surface. Le
q
be the distance from the surface to a point close
to the surface. In Figure 4.3
q
=
a
−
x
1
. The mean velocity component in the direction
x
3
is given by
S
{
1
−
exp[
−
q(
1
+
j)/δ
]
}
d
S
S
∂p
∂x
3
jωρ
0
v
3
=−
(4.103)
where d
S
is the infinitesimal area related to d
q
,and
S
is the area of the cross-section.
This expression is valid close to the surface, where the plane approximation is valid
and d
S
l
d
q
,
l
being the perimeter of the pore, and far from the surface, where the
exponential is negligible. The contribution of the exponential function to the integral is
=
δ
1
+
−
exp[
−
q(
1
+
j)/δ
]d
S
=
j
l
(4.104)
S
where
l
is the perimeter of the pore. This leads to the following approximation for the
effective density
ρ
0
1
δ
2
/j
l
2
S
ρ
=
+
(4.105)
which can be written in terms of the hydraulic radius ¯
r
=
2
S/l
ρ
0
1
+
2
/j
δ
¯
r
ρ
=
(4.106)
From Equation (4.46), the related bulk modulus is given by
γP
0
K
=
1
)/
1
(4.107)
+
√
2
/j
δl
2
SB
γ
−
(γ
−
which can be rewritten
γP
0
1
+
−
1
)
2
/j
δ
¯
rB
K
=
(γ
(4.108)
Both the high-frequency limits of the effective density and the bulk modulus depend
only on the hydraulic radius.
4.7
The Biot model for rigid framed materials
4.7.1
G
s
Biot (1956) has pointed out that
G
s
(s
)
is very similar to
G
c
(s)
if
s
is taken as
Similarity between
G
c
and
4
3
s
s
=
(4.109)
