Solutions of the moment hierarchy in the kinetic theory of Maxwell models

TL;DR

This paper reviews solutions to the moment hierarchy in Maxwell models of kinetic theory, highlighting specific flow states where the hierarchy can be exactly solved despite general coupling challenges.

Contribution

It provides an overview of exact solutions for certain flow states in Maxwell models, illustrating cases where the moment hierarchy is recursively solvable.

Findings

Exact solutions for planar Fourier flow with and without gravity

Solutions for planar Couette and force-driven Poiseuille flows

Analysis of uniform shear flow and its non-Newtonian properties

Abstract

In the Maxwell interaction model the collision rate is independent of the relative velocity of the colliding pair and, as a consequence, the collisional moments are bilinear combinations of velocity moments of the same or lower order. In general, however, the drift term of the Boltzmann equation couples moments of a given order to moments of a higher order, thus preventing the solvability of the moment hierarchy, unless approximate closures are introduced. On the other hand, there exist a number of states where the moment hierarchy can be recursively solved, the solution generally exposing non-Newtonian properties. The aim of this paper is to present an overview of results pertaining to some of those states, namely the planar Fourier flow (without and with a constant gravity field), the planar Couette flow, the force-driven Poiseuille flow, and the uniform shear flow.