Sound propagation in porous materials having a rigid frame - Propagation of Sound in Porous Media - page 79

l 2

r 2

r 1

l 1

Figure 5.3 A pore made up of an alternating sequence of cylinders.

As far as the macroscopic velocity is considered, the nonviscous fluid must be replaced
by a fluid of density ρ 0 α ∞ . The value of the tortuosity is an intrinsic property of the
porous frame that depends on the micro-geometry. When the saturating fluid is vis-
cous the effective density must tend to α ∞ ρ 0 when the viscous skin depth tend to zero
and the viscosity effects become negligible. For the material of Sections 4.9 and 4.10,
α ∞ = 1 / cos 2 ϕ in the direction perpendicular to the surface of the layer. A tortuosity
larger than 1 is due to the dispersion of the microscopic velocity in Equation (5.21). This
dispersion can be created by variations of the diameter of the pores. Let us consider, for
instance, a material with identical pores parallel to the direction of propagation, made up
of alternating cylinders represented in Figure 5.3, with lengths l 1 and l 2 and cross-sections
S 1 and S 2 , respectively. Even if the fluid is nonviscous, the description of the inertial
forces and the evaluation of α ∞ by Equation (5.21) are very complicated at the junction
of the two cylinders. A simple approximation is obtained by assuming constant veloci-
ties in each cylinder. It is shown in Appendix 5.A that with this approximation, α ∞ is

given by

(l 1 S 2 + l 2 S 1 )(l 1 S 1 + l 2 S 2 )

α ∞ =

(5.23)

l 2 ) 2 S 1 S 2 (l 1 +

The evaluation of tortuosity can be performed from resistivity measurements, as indi-
cated in Section 4.9. Another method simultaneously allowing the measurement of other
high-frequency parameters is described at the end of Section 5.3.5.

5.3.2 Viscous characteristic length
For materials with cylindrical pores, as indicated by Equations (4.107) and (4.155), the
high-frequency behaviour of ρ and K depends on tortuosity and on the hydraulic radius.
The viscous characteristic length defined by Johnson et al . (1986) replaces the hydraulic
radius for more general micro-geometries. Johnson et al . have defined the characteristic
dimension by

A υ i ( r w ) d A

2
=

V υ i ( r ) d V

(5.24)

For a static flow of nonviscous fluid in the porous structure, υ i ( r w ) is the velocity of
the fluid on the pore surface and the integral in the numerator is performed over the pore