# Mathematical expression

The mathematical definition of the static permeability, $k_0$, is:

$k_0 = \lim_{\omega \rightarrow 0} k(\omega)$

where the dynamic permeability $k(\omega)$ is defined as:

$\phi\vec{v} = -\frac{k(\omega)}{\eta}\vec{\nabla} p$

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

The static air-flow resistivity can be deduced from the static permeability: $\sigma = \frac{\eta}{k_0}$

# Physical description

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

For acoustical materials, the range of values for the static permeability is approximately [$10^{-11}$ $10^{-8}$] m$^{2}$. In addition, it can be shown $k_0$ should always be smaller than or equal to the static thermal permeability $k'_0$:

$k_0 \Big(\equiv \displaystyle\frac{\eta}{\sigma}\Big) \le k'_0$

The static permeability can be deduced from the measurement of the static air-flow resistivty. (The [ISO 9053] standard is dedicated to the measurement of $\sigma$).