No abstract

It is shown analytically and numerically on the basis of the kinetic theory that the heat-conduction equation is not suitable for describing the temperature field of a gas in the continuum limit around bodies at rest in a closed domain or in an infinite domain without flow at infinity, where the flow vanishes in this limit. The behavior of the temperature field is first discussed by asymptotic analysis of the time-independent boundary-value problem of the Boltzmann equation for small Knudsen numbers. Then, simple examples are studied numerically: as the Knudsen number of the system approaches zero, the temperature field obtained by the kinetic equation approaches that obtained by the asymptotic theory and not that of the heat-conduction equation, although the velocity of the gas vanishes.

Steady evaporating flows from a cylindrical condensed phase in an infinite expanse of its vapor gas are investigated numerically on the basis of the Boltzmann–Krook–Welander equation. Not only the mass flow rate and the energy flow rate from the cylinder, but also the local variables of the gas over the whole flow field are obtained for a wide range of the Knudsen number and the pressure ratio, which is defined by the pressure at infinity divided by the saturation gas pressure at the temperature of the condensed phase. The acceleration of gas flows, especially to a supersonic flow, near the cylinder and the deceleration to the stationary state at infinity are clarified. The discontinuity of the velocity distribution function in the gas, a typical behavior of a gas around a convex body, is analyzed accurately with the difference scheme devised for this purpose, and its relation to the S layer [Phys. Fluids 16, 1422 (1973)] is discussed.

The two-dimensional Bénard problem for a rarefied gas in a rectangular domain is studied numerically on the basis of kinetic theory. To be more specific, the instability of a stationary stratified gas is investigated by a finite-difference analysis of the Boltzmann–Krook–Welander equation (the so-called BGK model) with the diffuse reflection condition on the top (cooled) and bottom (heated) boundaries and the specular reflection condition on the side boundaries. The study is a continuation of the previous paper by the present authors with H. Motohashi [in Rarefied Gas Dynamics, edited by J. Harvey and G. Lord (Oxford U.P., Oxford, 1995), Vol. 1, p. 135], where the parameter range for which a steady convection exists was clarified, and some examples of steady flow patterns as well as those of flow bifurcation were presented. In the present paper, the attention is focused on the formation of steady flow patterns and the stability of an array of rolls. In particular, the effects of the initial condition and the aspect ratio of the domain on the types of steady convection as well as on the process of its formation and the stability of the double-roll solutions are investigated closely. The existence range of a steady convection is also discussed with some new examples of flow patterns supplementing the previous paper.

A semi-infinite expanse of a gas in contact with its plane condensed phase is considered. The steady gas flow condensing on or evaporating from the condensed phase is investigated numerically on the basis of the Boltzmann–Krook–Welander equation in the case where there is a gas motion along the condensed phase. First, it is shown by a time-dependent analysis that the steady evaporating flow is possible only when there is no gas motion along the condensed phase. Then, with the aid of time-dependent and time-independent analyses, the effect of gas motion along the condensed phase on the steady condensing flow is clarified. That is, the relation of the parameters at infinity and of the condensed phase that allows a steady flow, together with the accurate profiles of the steady solutions, is established. The present results of the half-space problem complete the boundary condition of the compressible Euler equation required in describing steady gas flows around the condensed phase of a smooth shape in the continuum flow limit.