A Monatomic Ideal Gas—Prototype of a Continuous Medium with Microstructure

Abstract

A complete set of equations for the description of the properties of monatomic ideal gases is formed by the balance equations of all moments of the distribution function which satisfies the Boltzmann equation. In a manner of speaking these balance equations describe a continuum and the elements of its microstructure are the moments. There are infinitely many moments and for rapidly changing processes with steep gradients they are all needed. However, for slow and smooth processes the necessary set of balance equations may be cut off at some point and may then still be useful. The most drastic cut-off provides the Euler equations in which no dissipation occurs, so that its applicability is limited to isentropic flows. A less rigorous cut-off leads to the equations of Navier-Stokes and Fourier which permit the mathematical treatment of viscous flow and heat conduction. Those curtailed sets of balance equations have been studied at great length: Their study is the subject of ordinary thermodynamics. A still less drastic cut-off leads to Grad’s 13- and 14-moment equations. These provide some improvement upon both Euler and Navier-Stokes-Fourier. Thus they forbid rigid rotation of a gas in the presence of heat conduction. Even so, the Grad theory is not suitable for high frequency sound propagation and high frequency light scattering and for the study of shock structures. The study of Grad’s equations and of the balance equations for higher moments is the subject of extended thermodynamics. Mathematically speaking the set of balance equations of moments in the kinetic theory of gases provides a prototype of the hyperbolic balance laws of continuum physics. Indeed, the equations are hyperbolic and the entropy principle makes them symmetric hyperbolic. High frequency light scattering in monatomic gases proves the applicability and usefulness of extended thermodynamics, because it furnishes results that are in full agreementwith experiments at any frequency, while ordinary thermodynamics merely describes the low-frequency limit. In hyperbolic equations there is a competition between non-linearity and dissipation: Non-linearity attempts to steepen a field to a shock while dissipation smoothes it out. And if dissipation is big enough, no shock singularities will appear. In extended thermodynamics singularities can be prevented by adding more equations, hence more dissipative terms. The study of shock structures makes this evident. Hyperbolic equations have as many sound modes as there are equations and all their speeds are different from the ordinary sound speed. Extended thermodynamics proves that the highest sound speed increases monotonically as more equations are added. It is when a shock structure moves more rapidly than the highest sound speed that discontinuities appear in the theory. One may thus say that the flow is truely supersonic only when its speed is quicker than the highest characteristic speed. However, that will generally happen at Mach numbers well beyond Ma equal to 1. It is a clear sign that more equations are needed.