Grad Equations: Derivation and Analysis

Abstract

In certain physical problems, the Navier–Stokes equations are insufficient to describe the flow physics accurately, making it necessary to take recourse to higher order continuum transport equations. Grad pioneered the moment method by which successive approximations of the microscopic Boltzmann equation can be obtained. Grad proposed to expand the particle distribution function f(c, x, t) in terms of orthogonal Hermite polynomials, with an infinite set of Hermite coefficients being equivalent to the particle distribution function f itself. The number of Hermite coefficients to be considered in the expansion depends upon the degree of accuracy desired. The first few Hermite coefficients are the state variables, stress tensor, and heat flux vector. These variables are not expressed in terms of other thermodynamic variables but are on par with them, and satisfy their own differential equation. This distinct approach of Grad along with the resulting equations is expounded in this chapter. The linearized form of the Grad equations is applied to two practical problems and the obtained solutions are compared against that obtained through the Navier–Stokes equations. The Grad equations can be used to obtain the Navier–Stokes equations and Cattaneo’s equation as discussed here. The differences as well as the similarities between the Grad approach and the Chapman–Enskog approach (introduced in Chap. 5) are also brought forward.