Civil Engineering Reference
In-Depth Information
This last equation describes the static measurement of flow resistivity as represented
in Figure 2.4.
At the first-order approximation in 1/
ω
, Equations (4.67) and (4.69) yield
4
3
ρ
0
+
σφ
jω
ρ
=
(4.85)
γP
0
K
=
1
)/
4
(4.86)
σφ
jB
2
ωρ
0
γ
−
(γ
−
3
+
In the high-frequency range, Equations (4.61) and (4.63) yield
ρ
0
1
+
2
/j
δ
R
ρ
=
(4.87)
γP
0
K
=
1
)/
1
(4.88)
+
√
2
/j
δ
BR
γ
−
(γ
−
where
2
/
j
=
1
−
jand
δ
is the viscous skin depth given by Equation (4.58).
4.6.2 Effective density and bulk modulus in slits
For the material represented in Figure 4.8, the circular pores are now replaced by slits.
Equation (4.76) becomes
p
2
−
p
1
2
na υe
=
p
2
−
p
1
φ υe
σ
=
(4.89)
Using Equation (4.74) at
ω
=
0gives
a
2
3
η
∂p
∂x
υ
=
−
(4.90)
The flow resistivity
σ
is given by
3
η
φa
2
σ
=
(4.91)
Using Equation (4.91), Equation (4.25) can be rewritten as
3
ωρ
o
φσ
1
/
2
s
=
(4.92)
Equations (4.27) and (4.92) can be used to calculate
ρ
and
K
. The description of the
viscous force in the Newton equation (4.28) can be modified in the following way; the
quantity
ρ
o
ω/s
is given by
ρ
o
ω/s
=
σφs
/
3
(4.93)
