Special issues on moment methods in kinetic gas theory

Abstract

This two-volume special issue of ‘Continuum Mechanics and Thermodynamics’ is the result of an international meeting on moment methods in kinetic gas theory that took place in November 2008 at ETH Zurich [1]. There have been over 30 participants from 10 different countries attending 13 invited lectures on various developments in moment methods. The workshop was hosted jointly by the ‘Research Institute for Mathematics (FIM)’ and the ‘Seminar for Applied Mathematics (SAM)’ at ETH Zurich, as well as supported by the ‘Swiss National Science Foundation (SNF)’. In recent years, moment methods in kinetic gas theory proved to be a successful tool to extend classical fluid dynamics for applications in microor rarefied gas flows. The classical equations of compressible fluid dynamics, known as the system ofNavier–Stokes and Fourier, loose their validity in extreme physical processes involving rarefaction or micro-scales. The lack of sufficient particle collisions is expressed in larger values of the Knudsen number K n—the ratio between the mean free path and an observation scale—and leads to thermal non-equilibrium which can only be described using refined mathematical models. Kinetic gas theory based on a statistical description of the gas provides a valid framework to model processes in a rarefied regime or at small scales [2,3]. The main variable used to describe the gas is the distribution function or probability density of the particle velocities. However, in situations of intermediate non-equilibrium this detailed statistical approach still yields a far too complex description of the gas. In these situations it is desirable to have a continuum model based on partial differential equations for the fluid mechanical field variables. The classical fluid dynamics equations can be derived fromBoltzmann’s equation as the first order contribution in a Chapman–Enskog expansion [4], an asymptotic expansion in the Knudsen number. It seemed natural to look for higher order contributions as an extension for the laws of Navier–Stokes and Fourier. However, the hope for useful equations was dashed by the observation of Bobylev [5] stating that these higher order models are inherently unstable in the general case. In the 1940s Harold Grad [6,7] developed an alternative to the Chapman–Enskog expansion by producing moment equations from Boltzmann’s equation. The fundamental idea of moment approximations is to replace the variable of a high dimensional distribution function f (x, t, c) for the velocity c ∈ R3 by a finite setmoment variables {Fn (x, t)}n=0,1,2,... at each (x, t). The working hypothesis is that at (x, t) a finite number of moments are sufficient to encode the details of f as a function of c. The first moments are the conservative variables of gas dynamics, the density, the momentum density and total energy density. Obviously, the motivation is that