## Visco-inertial dissipation in a narrow tube

Profile of particle $\vec{v}$elocity in a cylinder
of radius $R$ $\mu$m at Hz ( $\delta_v$ : $\mu$m ).
$$\delta_v = \sqrt{\displaystyle\frac{\eta}{\rho_0 \omega} }$$
$$\beta = \frac{R}{\delta_v}$$

At low frequencies ($\beta$ of the order of 1), the flow exhibits a viscous regime (the velocity profile is parabolic, the boundary layer is of the order of magnitude of the tube / pore radius ).
At high frequencies (Reynolds number $\beta$ higher than 100), the flow exhibits an inertial regime (the velocity is almost the same in the cross-section, the boundary layer is small compared to the tube / pore radius).
A quick numerical application leads to a variation of $\delta_v$ between 350 and 11 $\mu$m in the audible frequency range : [20 – 20 000] Hz.
Almost the same result is obtained for the thermal boundary layer ($\delta_t$ goes from 410 to 13 $\mu$m in the audible frequency range).
In other words, if you want to take advantage of these visco-inertial & thermal effects, pores & pore connections should have dimensions of the order of 10 to ~500 $\mu$m.