Definition
An ideal gas – simple as it may seem – has a considerable internal structure which is revealed by the moment equations of the kinetic theory of gases. The internal structure makes itself felt in processes with steep gradients and rapid rates of change such as wave propagation and light scattering.
Extended thermodynamics writes all field equations as quasilinear first-order differential equations, and the entropy principle reduces those to symmetric hyperbolic equations which imply favorable physical and mathematical properties.
Introduction
Extended thermodynamics is an improved version of thermodynamics of irreversible processes which is based on equations of balance for all thermodynamic fields and on the entropy principle. The latter guarantees that the field equations of extended thermodynamics are symmetric hyperbolic if...
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Notes
- 1.
Inertial terms are ignored in the exploitation of the entropy inequality.
- 2.
Normally the summation over the repeated index α is understood. But here we write the sum explicitly so as to indicate that it extends over α = 5, 6…v -1 rather than over all α from 0 to v-1.
- 3.
The form of the collision term on the right-hand side of the Boltzmann equation is universally known as the Stoßzahlansatz. The German word has defied translation and is therefore routinely used in English-language texts as well. For its derivation and interpretation, the reader may consult any book on the kinetic theory, e.g., Waldmann (1958) or Chapman and Cowling (1936/1961).
- 4.
Round brackets enclosing indices denote symmetrization. Angular brackets will denote symmetrization and tracelessness.
- 5.
Atoms with a hypothetical repulsive power potential that falls off with an inverse fourth power. It is thought that such an interaction, while not perfect, is not bad for rarefied monatomic gases.
- 6.
M 0 is the ratio of the speed of the fluid into the shock structure and the ordinary speed of sound.
- 7.
Of course, cylindrical coordinates are appropriate for this case, so that q 1 and t <11> are contravariant components of the heat flux and the stress. t <11> does not vanish in the 13-moment theory, although it does in the Navier-Stokes theory.
References
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Müller, I. (2018). System of Symmetric Hyperbolic Equations, Extended Thermodynamics of Gases. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_61-1
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