# Visco-inertial effects

In 1987, Johnson Koplik and Dashen [JKD87] proposed a semi-phenomenological model to describe the complex density of an acoustical porous material with a motionless skeleton having arbitrary pore shapes. This expression is:

APMR

Acoustical Porous Material Recipes

In 1987, Johnson Koplik and Dashen [JKD87] proposed a semi-phenomenological model to describe the complex density of an acoustical porous material with a motionless skeleton having arbitrary pore shapes. This expression is:

\[
\widetilde{\rho}(\omega) = \frac{ \alpha_{\infty} \rho_{0} }{ \phi }
\Bigg[ 1 + \frac{ \sigma \phi }{ j \omega \rho_{0} \alpha_{\infty}} \sqrt{ 1 + j\frac{ 4 \alpha_{\infty}^{2} \eta \rho_{0} \omega }{ \sigma^{2} \Lambda^{2} \phi^{2}} }\Bigg]
\]

4 parameters are involved in the calculation of this dynamic density: the open porosity $\phi$, the static air flow resistivity $\sigma$, the high frequency limit of the tortuosity $\alpha_{\infty}$ and the viscous characteristic length $\Lambda$.

As reported in the section limitations, the low frequency limit of the real part of the dynamic mass density expression is not exact.

In 1991, Champoux and Allard [CA91] introduced an expression for the dynamic bulk modulus for the same kind of porous material based on the previous work by Johnson et al.

\[
\widetilde{K}(\omega) =
\displaystyle{\frac{\gamma P_{0}/\phi}
{\gamma - (\gamma-1)
\left[ 1-j\displaystyle{\frac{8\kappa}
{\Lambda'^{2}C_{p}\rho_{0}\omega}}
\sqrt{ 1+j
\displaystyle{\frac{\Lambda'^{2}C_{p}\rho_{0}\omega}
{16\kappa}}
} \, \right]^{-1} }}
\]

2 parameters are invloved in the calculation of this dynamic bulk modulus: the open porosity $\phi$ and the thermal characteristic length $\Lambda'$.

The expression of $\widetilde{\rho}$ given by Johnson, Koplik & Dashen does not describe the exact behavior of the dynamic mass density as $\omega$ tends to zero. The real part of the mass density (or the imaginary part of the dynamic permeability is not correct.

Similarly, the expression of $\widetilde{K}$ given by Champoux & Allard is not correct at low frequencies. [I know I have to develop this part. Give me some time to do so.]

Pride, Morgan & Gangi [PMG93] have proposed a modified expression of the dynamic density. However, the expression given by Pride, Morgan & Gangi and further modified by D. Lafarge [Laf93] is rarely used as the new required parameters need developments to be characterized.

Changes to the JCA model by Pride et al. and D. Lafarge are detailed in the Johnson-Champoux-Allard-Pride-Lafarge model.

Four parameters are required to compute $\widetilde{\rho}$ and describe the visco-inertial effects: $\phi$, $\sigma = \eta/k_0$ at low frequencies, $\alpha_{\infty}$ and $\Lambda$ at high and medium frequencies. In the mean time, only two parameters are used to compute $\widetilde{K}$ and describe the thermal effects: $\phi$ the thermal characteristic length $\Lambda'$. This last parameter is used to represent the medium and high frequency range of thermal effects.

This observation of a non-symmetry between the description of the visco-inertial and thermal effects leads Lafarge et al. [LLAT97] to introduce a new parameter, the static thermal permeability $k'_{0}$, in order to describe the low frequency behavior of thermal effects.

The new model for $\widetilde{K}$ ($\widetilde{\rho}$ remains unchanged) introduced by Lafarge et al. is detailed in the Johnson-Champoux-Allard-Lafarge model dedicated page.