# Mathematical expression

The mathematical definition of the static air flow resistivity, $\sigma$, is:

APMR

Acoustical Porous Material Recipes

The mathematical definition of the static air flow resistivity, $\sigma$, is:

This expression is the generalized Darcy's law. $\phi$ is the open porosity of the material, $\vec{v}$ the velocity of the fluid particles subjected to the pressure gradient $\vec{\nabla}p$ ($\phi\vec{v}$ is thus the fluid flow inside the porous material).

The definition of $\sigma$ is based on the assumption $\vec{v}$ and $\vec{\nabla}p$ are constants (i.e. do not exhibit a harmonic time dependence). For harmonic time dependencies of $\vec{v}$ and $\vec{\nabla}p$, $\sigma$ can be extended to the concept of dynamic air-flow resistivity.

Some authors prefer to use the static [viscous] permeability $k_0$ instead of the static air-flow resistivity. This static permeability, which have the dimension of a surface ($m^2$) is defined as: \[ k_0 = \frac{\eta}{\sigma} \]

where $\eta$ is the dynamic viscosity of air ($\sim$ 1.84 $\times$ $10^{-5}$ N.s.m$^{-2}$ at ambiant temperature and pressure conditions).

$k_0$ does not depend on the fluid property while $\sigma$ is specific to a fluid (hence its name: static air-flow resistivity).

The static air flow resistivity characterizes, partly, the visco-inertial effects at low frequencies (when the viscous boundary layer is of the order of magnitude of the the characteritic size of the pores).

The models by Delany-Bazley [DB70] and Delany-Bazley-Miki [Mik90] use only this parameter to described the behavior of fibrous acoustical materials.

For acoustical materials, the range of values for the static air flow resistivity is approximately [$10^3$ $10^6$] N.s.m$^{-4}$.

The resistivity can be directly measured. The [ISO 9053] standard is dedicated to its measurement.