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Continuum Mechanics and Thermodynamics
Published in: Continuum Mechanics and Thermodynamics 1/2021

16-06-2020 | Original Article

A model system of mixed ionized gas dynamics

Author: Fumioki Asakura

Published in: Continuum Mechanics and Thermodynamics | Issue 1/2021

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Abstract

The aim of this paper is to study a one-dimensional model system of equations for ionized gas dynamics at high temperature where the gas is a mixture of two kinds of monatomic gas. In addition to the mass density, pressure, temperature and particle velocity, degrees of ionization of both gases are also involved. The equations of motion are derived from conservation of mass, momentum and energy, which are supplemented by the first and second law of thermodynamics, thermodynamic equations and Saha’s ionization equations by assuming that the local thermal equilibrium is attained. The equations constitute a strictly hyperbolic system, which guarantees that the initial value problem is well-posed locally in time for sufficiently smooth initial data. However the geometric properties of the system are rather complicated: In particular, we prove the existence of a region where convexity (genuine nonlinearity) fails for forward and backward characteristic fields. Also we study thermodynamic properties of shock waves by detailed analyses of the Hugoniot locus, which are used in a mathematical study of existence and uniqueness of solutions to the shock tube problem.
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Appendix
Available only for authorised users
Footnotes
1
An English translation of [10] is available upon request to F. Asakura.
 
2
This is the compatibility condition for \(dg = vdp - SdT\, (g: \text {Gibbs function}).\)
 
3
We notice that equation \(\eta _t = 0\) is equivalent to \(\left( \rho S\right) _t + \left( \rho uS\right) _x = 0\) in Eulerian coordinates.
 
4
For the sake of convenience, we adopt the notation \(\alpha _{\mathrm{A}}^{\pm }\) instead of \(\alpha _{\mathrm{A \pm }}.\)
 
5
\( \varGamma = \frac{v\left( \frac{\partial p}{\partial \alpha _{\mathrm{A}}}\frac{\partial v}{\partial T} - \frac{\partial v}{\partial \alpha _{\mathrm{A}}} \frac{\partial p}{\partial T}\right) }{T\left( \frac{\partial S}{\partial \alpha _{\mathrm{A}}}\frac{\partial v}{\partial T} - \frac{\partial v}{\partial \alpha _{\mathrm{A}}} \frac{\partial S}{\partial T}\right) }\)
 
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Metadata
Title
A model system of mixed ionized gas dynamics
Author
Fumioki Asakura
Publication date
16-06-2020
Publisher
Springer Berlin Heidelberg
Published in
Continuum Mechanics and Thermodynamics / Issue 1/2021
Print ISSN: 0935-1175
Electronic ISSN: 1432-0959
DOI
https://doi.org/10.1007/s00161-020-00899-9

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