Properties of Tensor Hermite Polynomials

TL;DR

This paper explores the properties of 3-D Tensor Hermite Polynomials, focusing on their scaling, translation, and rotation, and applies these to gas dynamics and the Boltzmann Transport Equation.

Contribution

It provides a detailed analysis of Tensor Hermite Polynomial properties and derives a temperature-based criterion for their application in gas mixture modeling.

Findings

Derived rotation relations for order 6 Hermite Tensors

Established a temperature criterion for binary gas mixtures

Analyzed scaling effects related to particle velocities

Abstract

A description of Orthogonal Tensor Hermite Polynomials in 3-D is presented. These polynomials, as introduced by Grad in 1949 [1], can be used to obtain a series solution to the Boltzmann Transport Equation. The properties that are explored are scaling, translation and rotation. Order 6 Hermite Tensors are studied while obtaining the rotation relations. From the scaling of the independent variables of particle velocities, a criterion on temperature is obtained which implies that the equation can be applied to binary gas mixtures only if the temperature of the hotter constituent is less than four times that of the cooler one. This criterion and other properties of the tensor hermite polynomials obtained in this paper can be used to study gas dynamics in the thermosphere.