Influence of a drag force on linear transport in low-density gases. Stability analysis

Abstract

The transport coefficients of a dilute classical gas in the presence of a drag force proportional to the velocity of the particle are determined from the Boltzmann equation. The viscous drag force could model the friction of solid particles with a surrounding fluid (interstitial gas phase). First, when the drag force is the only external action on the state of the system, the Boltzmann equation admits a Maxwellian solution $f_0(\mathbf{v},t)$ with a time-dependent temperature. Then, the Boltzmann equation is solved by means of the Chapman-Enskog expansion around the local version of the distribution $f_0$ to obtain the relevant transport coefficients of the system: the shear viscosity $\eta$, the thermal conductivity $\kappa$, and a new transport coefficient $\mu$ (which is also present in granular gases) relating the heat flux with the density gradient. The results indicate that while $\eta$ is not affected by the drag force, the impact of this force on the transport coefficients $\kappa$ and $\mu$ may be significant. Finally, a stability analysis of the linear hydrodynamic equations with respect to the time-dependent equilibrium state is performed, showing that the onset of instability is associated with the transversal shear mode that could be unstable for wave numbers smaller than a certain critical wave number.