Rarefied gas dynamics and its applications to vacuum technology

Abstract

Basic concepts of rarefied gas dynamics are given in a concise form. Some problems of rarefied gas flows are considered, namely, calculations of velocity slip and temperature jump coefficients, gas flow through a tube due to pressure and temperature gradients, and gas flow through a thin orifice. Results on the two last problems are given over the whole range of gas rarefaction. A methodology for modelling the Holweck pump is described. An extensive list of publications on these topics is given. 1 Brief history of rarefied gas dynamics Rarefied gas dynamics is based on the kinetic approach to gas flows. In 1859 Maxwell [1] abandoned the idea that all gaseous molecules move with the same speed and introduced the statistical approach to gaseous medium, namely, he introduced the velocity distribution function and obtained its expression in the equilibrium state. Thus Maxwell gave the origin to the kinetic theory of gases. Then, in 1872 Boltzmann [2] deduced the kinetic equation which determines the evolution of the distribution function for gaseous systems being out of equilibrium. In 1909 Knudsen [3], measuring a flow rate through a tube, detected a deviation from the Poiseuille formula at a low pressure. Such a deviation was explained by the fact that at a certain pressure the gas is not a continuous medium and the Poiseuille formula is not valid anymore. A description of such a flow required the development of a new approach based on the kinetic theory of gases. This can be considered as the beginning of rarefied gas dynamics. Later, advances were made by Hilbert [4], Enskog [5] and Chapman [6] to solve the Boltzmann equation analytically via an expansion of the distribution function with respect to the Knudsen number. The main result of this solution was a relation of the transport coefficients to the intermolecular interaction potential, but no numerical calculation of rarefied gas flows could be realized at that time. In 1954 the so-called model equations [7,8] were proposed to reduce the computational efforts in calculations of rarefied gas flows. Using these models it was possible to obtain numerical results on rarefied gas flows in the transition regime. Thus in 1960 a numerical investigation of rarefied gas flows began in its systematic form. For a long time, it was possible to solve only the model equations. Practically, all classical problems of gas dynamics (Poiseuille flow, Couette flow, heat transfer between two plates, flow past a sphere, etc.) were solved over the whole range of gas rarefaction by applying the model equations. In 1989 first results based on the exact Boltzmann equation were reported, see, for example, Ref. [9]. However, even using the powerful computers available nowadays, a numerical calculation based on the Boltzmann equation itself is still a very hard task, which requires great computational efforts. Thus, the model equations continue to be a main tool in practical calculations. Below, the main concepts of rarefied gas dynamics and some examples of its application will be given. In the last section, the main results of rarefied gas dynamics that could be applied to vacuum technology are listed.