The Chapman-Enskog and Moment Methods

Abstract

The kinetic equation postulated in this work is not amenable to methods of closed analytical solutions in the case of physically realistic interaction potential models. It is, therefore, necessary to resort to an approximate method of solution. The most frequently used approximate methods of solution are the Chapman-Enskog method [1] and the Maxwell—Grad moment method [2,3]. Hilbert [4] developed a solution method for the Boltzmann equation as an application of his theory of linear integral equations. This Hilbert method was modified by Enskog [5]. Enskog’s method was later shown to yield the results identical with the Chapman method [6]. The Chapman method was based on the Maxwell transfer equations, which are related to the moment evolution equations originally proposed by James Clerk Maxwell [7]. The Enskog method is now known as the ChapmanEnskog method in the literature. The second method was later developed further by Grad [2], in particular, for the Boltzmann equation. The Chapman—Enskog method and the moment method give rise to identical linear transport coefficients although higher order results are not necessarily the same. The linear transport coefficients so calculated are extensively verified to be completely consistent with experimental results, and it is correct to state that the theory of linear transport processes in monatomic gases are well established and understood. However, nonlinear transport processes are a different matter, and it is necessary to investigate them in a thermodynamically consistent manner. Since the theory of nonlinear transport processes, which is an important point of interest in this work, can be built on the Chapman—Enskog solution for linear transport processes, it is useful to have some discussions on the method. We will also discuss the moment method since it is directly related to the nonequilibrium ensemble method which we would like to present in this work.