Effects of external gravitational field on highly rarefied gases: analysis based on stochastic soft-sphere collision models

Abstract

This study examines the effects of an external gravitational field on highly rarefied gases in the transitional-flow regime near the free-molecular-flow regime. In our theoretical study, we rederive the classical kinetic theory for an ideal gas in terms of the kinetics of the constituent particles to account for the effect of particle acceleration by an external gravitational field. Subsequently, we derive an extended expression for the virial pressure equation as a generic description of the dynamics under an external gravitational field. We employ the soft-sphere model for the following reasons: In highly rarefied gases, short-range and instantaneous collisional interactions are dominant. Thus, by expanding the asymmetric two-body potential in the virial pressure equation and retaining only the contribution of the short-range interaction, we can obtain a soft-sphere model that represents the interaction in the collision direction as a harmonic oscillation. In the absence of dissipation, the soft-sphere model has been confirmed to reproduce fully elastic collisions. In our collision simulations, we define two parameters. The first parameter represents the collision probability between each pair of approaching particles, and the second represents the ratio of the magnitude of the external potential energy to the total kinetic energy of the particles. The behavior of the system is analyzed by varying the values of these two parameters. Our analysis shows that if the external potential energy is sufficiently small (1%–5%) compared with the total kinetic energy, then a pressure difference emerges between the walls. However, the system retains the properties of equilibrium statistical mechanics, as indicated by the Maxwell–Boltzmann (MB) distribution. In conclusion, highly rarefied gases obey the MB distribution even when placed under weak gravitational fields.