Extended thermodynamics - the physics and Mathematics of the hyperbolic equations of Thermodynamics

Abstract

Extended thermodynamics grew out of the desire to replace the par­abolic equations of ordinary thermodynamics by hyperbolic equations, so as to obtain finite speeds of propagation as part of a systematic thermodynamic theory [1].Subtle changes to ordinary thermodynamics had to be made in order to obtain a systematic hyperbolic extended thermodynamics:

• the principle of local equilibrium had to be abandoned

• the entropy flux was no longer equal to heat flux divided by temperatur

• the principle of material objectivity was recognized as approximate [2].

The original form of extended thermodynamics extends (sic) the list of 5 clas­sical independent fields — density, velocity and temperature — by 8 additional ones, the densities of momentum flux and energy flux. The theory thus has 13 independent fields and, if applied to monatomic gases, it strongly resembles Grad’s theory of 13 moments in the kinetic theory of gases. And it exhibits five finite characteristic speeds, viz.V = yAfter the close relationship between extended thermodynamics and the mo­ment methods of the kinetic theory was recognized, it became plausible to investigate theories with more than 13 moments. Thus appeared the theories ET14 [3], — which came first — and then ET20, ET21, ET26, ET35, etc., up to very large numbers of hundreds, thousands and tenths of thousands of moments. Such theories imply many waves, each with their own speeds of propagation and it could be shown that the maximum speed increases with the number of moments. Thus for 8436 moments we have 342 longitudinal waves and the maximum speed is.

It can be shown that tends to infinity with the number of moments [4], or toc— the speed of light, — in the relativistic case [5].

The application of extended thermodynamics to high-frequency sound prop­agation [6] and to light scattering [6], [7] in rarefied gases makes it clear that extended thermodynamics is needed for processes with steep gradients and rapid changes; steepness and rapidity being measured in terms of mean free paths and mean times of free flight.

Also these applications reveal the role of extended thermodynamics as a the­ory of theories, in which the number of moments is the only parameter of adjustment. There is convergence in the sense that, starting from a certain number of moments, higher numbers do not change the result appreciably. For sound propagation and light scattering such converged results fit the ex­perimental data perfectly. A certain disappointment lies in the fact that the necessary number of moments is usually very high, — far beyond 13, or 14 —, in cases where the customary parabolic theory of Navier-Stokes-Fourier fails. Recent studies of the shock wave structure [8] and of shock-tube experiments [9] confirm that many moments are needed for quantitatively good results at high Mach numbers. They also show that the many waves of extended thermodynamics cooperate, — through absorption and dispersion — to develop into only three essential modes of propagation: the shock structure, the contact wave and the rarefaction wave.

A problem arises with boundary conditions for higher moments, which obvi­ously cannot be controlled.The new minimax principle [10] may help in this situation; sofar it was demonstrated to be useful for a relatively small number of moments in simple problems of heat conduction and one-dimensional flow. At the same time extended thermodynamics sheds a new light on the long standing question whether the kinetic energy of the particle of a gas is a good measure for its temperature [11].

A full account of extended thermodynamics until 1998 is given in the mono­graph by Müller & Ruggeri [12].