By definition, the characteristic viscous length should be lower or equal to the thermal characteristic length:

\[
\Lambda \leq \Lambda'
\]
for fibrous or felts it is usually admitted that the static air-flow resistivity increases with mass density [Tar96]

The tortuosity for fibrous or felt materials can be approximated with (see e.g. Umnova et al. [UAL00])

\[
\alpha_{\infty} \simeq 1/\phi
\]
which is sometimes called Archie’s empirical law [Arc42]: $(1/\phi)^r$.
From a different reasoning, Tournat et al. [TPLJ04] derived the expression $\alpha_\infty = 1+(1-\phi)$ for sound flow perpendicular to the axis of parallel identical cylinders for large porosity values. Obviously, this last expression is a Taylor expansion around $\phi = 1$ up to the order 1 of the expression $\alpha_{\infty} \simeq 1/\phi$. From a graphical observation, it appears these two expressions for $\alpha_{\infty}$ diverge for $\phi \lesssim$ 0.96

The tortuosity of a stack of identical solid spheres is calculated following J. G. Berryman [Ber80] as:

\[
\alpha_{\infty} = 1 + \displaystyle\frac{1-\phi}{2\phi}
\]
In addition, one can check the value of

- • the viscous pore shape factor $M$ for models based on Johnson Koplik & Dashen works.
- • the thermal pore shape factor $M'$ for models based on Champoux-Allard-Lafarge works.

The pore shape factors $M$ and $M'$ should have orders of magnitude of the order around 1 for usual acoustical porous materials. $M=M'=1$ for straight cylindrical pores.

\[
\begin{align}
M &=& \displaystyle{\frac{8\alpha_{\infty}\eta}{\sigma\phi\Lambda^{2}}} \\
M' &=& \displaystyle{\frac{8k'_{0}}{\phi\Lambda'^{2}}}
\end{align}
\]
## Check consistency of your parameter values