A combined Chapman-Enskog and Grad method. III. Polyatomic gases in magnetic fields

Abstract

In this work we develop a kinetic theory of rarefied polyatomic gases consisting of spherical molecules with internal rotational energy in an external constant magnetic field. We assume that the energy of the internal rotational variable can be treated classically [1] but a similar procedure could also be applied if the energy of the internal variable were treated quantum-mechanically [2]. The standard methods used in kinetic theory of transport processes are the Chapman-Enskog [3] and the Grad method [4]. In the Chapman-Enskog method the deviation from equilibrium is found as a solution of an integral equation that follows from the Boltzmann equation. In the Grad’s method the deviation from equilibrium of the distribution function is written in terms of moments of the distribution function. The transport coefficients of the latter are then obtained from the field equations of the moments through an iteration scheme. An alternative method was proposed by Bezerra, Reinecke and Kremer [5] that combines the features of the Chapman-Enskog and Grad methods. It requires neither a solution of the integral equation nor the use of the field equations for the moments. As in Grad’s method the deviation of equilibrium of the distribution function is written in terms of the moments of the distribution function whereas the constitutive equations follow directly from the Boltzmann equation. The combined Chapman-Enskog and Grad method was applied to a monatomic gas and mixtures [5] and to ionized gases [6]. The aim of this work is to apply the combined Chapman-Enskog and Grad method to polyatomic gases and to determine the changes of thermal conductivity and shear viscosity coefficients in a presence of a constant magnetic field (Senftleben-Beenakker effect) [7]. The model of the rough spheres of Bryan [3] is used for molecular collisions; the principal feature of this model is the reversing of the relative velocity of the points which come into contact in a binary collision. The theory is based on thirty-seven scalar fields that are the moments of the distribution function. The influence of dominant polarizations ! Ck and ! [7] are considered and the thermal conductivity tensor and the shear viscosity tensor are obtained as functions of the external magnetic field. The inversion of a fourth-order tensor is used to determine the shear viscosity tensor. The results obtained agree with those that follow from the Chapman-Enskog method [8, 9].