# 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: \[ \sigma = \frac{\eta}{k_0} \]