Viscosity of a classical gas: The rare-collision versus the frequent-collision regime

The shear viscosity $\eta$ for a dilute classical gas of hard-sphere particles is calculated by solving the Boltzmann kinetic equation in terms of the weakly absorbed plane waves. For the rare-collision regime, the viscosity $\eta$ as a function of the equilibrium gas parameters—temperature $T$, particle number density $n$, particle mass $m$, and hard-core particle diameter $d$—is quite different from that of the frequent-collision regime, e.g., from the well-known result of Chapman and Enskog. An important property of the rare-collision regime is the dependence of $\eta$ on the external (“nonequilibrium”) parameter $\omega$, frequency of the sound plane wave, that is absent in the frequent-collision regime at leading order of the corresponding perturbation expansion. A transition from the frequent to the rare-collision regime takes place when the dimensionless parameter $n{d}^2{(T/m)}^{1/2}{\omega }^{-1}$ goes to zero. The scaled absorption coefficient for sound waves calculated in the rare and frequent-collision regimes is found to be in qualitative agreement with the experimental data.

Figure 1

The scaled absorption coefficient $\gamma /k_r$ [Eqs. (20) and (16)] (in units of the wave number $k_r$) as a function of the Knudsen parameter $\omega \tau$ for the FCR ($\omega \tau \ll 1$, blue line) and RCR ($\omega \tau \gg 1$, red line). Full dots are the experimental data from Ref. [39] and full squares are taken from Ref. [40] for ${}^{40}\mathrm{Ar}$ at normal conditions (see also Refs. [21, 29]).