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# Grad's Method

DOI link for Grad's Method

Grad's Method book

# Grad's Method

DOI link for Grad's Method

Grad's Method book

## ABSTRACT

When treating the slightly non-equilibrium gas states, it would be natural to choose the local Maxwellian as the distribution function of the zero approximation. After writing it down in the Cartesian system of coordinates and forming a number of polynomials out of the ordered full series of monomials 1, er, ere•, ... , where ( = ( mf kT) 112 c is the dimensionless velocity (for simplicity the species index a is omitted for the time being), we can easily find [4) that the orthogonalization procedure for the series terms with the weight function w (1(1) = (211r3' 2 exp ( -e2 /2) results in the tensorial Hermite polynomials in the expansion defined as

where the Hermite polynomials meet the orthogonality conditions

Taking into account (4.1.3) and with the help of (4.1.2), we can readily obtain the following relations between the expansion coefficients a~1 ••• r .. and the moments of the distribution function

(4.1.5)

Grad (1,2] was the first to suggest the expansion off in the tensorial Hermite polynomials (4.1.2). A variation of the original Grad's procedure, which in particular helps us to follow the relation of the procedure and the expansion used in the Cbapman-Enskog-Burnett theory (5,6) comes in the application of the irreducible Hermite polynomials H:!:~.r .. {E) instead of the polynomials H~ ... r,.. {E). The introduclion of these polynomials is easily justified if for the distribution function of the zero approximation the variables are taken in the spheri-

cal (e,e,rp) rather than Cartesian system of coordinates [7]. The orthogonalization procedure with respect to the series em' em+2 ' em+'' ... (0 ~ m ~ oo) with the Gaussian weight function over e2 /2 brings about the polynomials L!:a (e2 /2), which are directly related to the~ nine polynomials .S:.+t/2 (e2 /2) used in the Chapman-Enskog-Burnett theory

where

In particular, at any m

Sl ( 2)- 3 2 m+l/2 u - m + 2 - u .