Abstract
We consider molecular extended thermodynamics (molecular ET) of rarefied polyatomic gases with the system composed of two hierarchies of balance equations for the moments of a distribution function. The internal degrees of freedom of a molecule are properly taken into account in the distribution function. By the reasoning of physical relevance, the truncation orders of the two hierarchies are shown to be dependent on each other. And the two closure procedures based on the maximum entropy principle and on the entropy principle are also proved to be equivalent to each other.
Characteristic velocities of a hyperbolic system of balance equations for a polyatomic gas are compared to those obtained for a monatomic gas. The lower bound estimate for the maximum equilibrium characteristic velocity established for a monatomic gas is proved to be valid also for a rarefied polyatomic gas, that is, the estimate is independent of the degrees of freedom of a molecule. As a consequence, also for polyatomic gases, when the number of moments increases the maximum characteristic velocity becomes unbounded.
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Notes
- 1.
Upon inspection it can be seen that in the one-dimensional case, for any N and M, \(\{F_{i_1 i_2 \ldots i_A}\), \(0 \leqslant A \leqslant N\}\) is mapped into {F p,q, \(0 \leqslant p+2q \leqslant N\}\), and \(\{G_{lli_1 i_2 \ldots i_{A'}}\), \(0 \leqslant A' \leqslant M\}\) is mapped into \(\{G_{p',q'}\), \(0 \leqslant p'+2q' \leqslant M\}\).
References
G. Boillat, T. Ruggeri, Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn. 9, 205 (1997)
T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, A study of linear waves based on extended thermodynamics for rarefied polyatomic gases. Acta Appl. Math. 132, 15 (2014)
T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Monatomic rarefied gas as a singular limit of poyatomic gas in extended thermodynamics. Phys. Lett. A 377, 2136 (2013)
M. Pavić, T. Ruggeri, S. Simić, Maximum entropy principle for rarefied polyatomic gases. Physica A 392, 1302 (2013)
T. Arima, A. Mentrelli, T. Ruggeri, Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Ann. Phys. 345, 111 (2014)
G. Boillat, T. Ruggeri, On the evolution law of the weak discontinuities for hyperbolic quasi-linear systems. Wave Motion 1, 149 (1979)
T. Ruggeri, Stability and discontinuity waves for symmetric hyperbolic systems, in Non-Linear Wave Motion ed. by A. Jeffrey (Longman Press, New York, 1989), pp.148–161
A. Muracchini, T. Ruggeri, L. Seccia, Dispersion relation in the high frequency limit and non-linear wave stability for hyperbolic dissipative systems. Wave Motion 15, 143 (1992)
Z. Banach, W. Larecki, T. Ruggeri, Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues. Wave Motion 51, 955 (2014)
G. Boillat, T. Ruggeri, On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn. 10, 285 (1998)
S. Taniguchi, T. Arima, T. Ruggeri, M. Sugiyama, Effect of dynamic pressure on the shock wave structure in a rarefied polyatomic gas. Phys. Fluids 26, 016103 (2014)
T. Arima, T. Ruggeri, M. Sugiyama, S. Taniguchi, On the six-field model of fluids based on extended thermodynamics. Meccanica 49, 2181 (2014)
G. Boillat, T. Ruggeri, Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. 137, 305 (1997)
T. Ruggeri, Galilean invariance and entropy principle for systems of balance laws. The structure of the extended thermodynamics. Continuum Mech. Thermodyn. 1, 3 (1989)
S. Pennisi, T. Ruggeri, Classical limit of relativistic moments associated with Boltzmann-Chernikov equation: optimal choice of moments in classical theory. J. Stat. Phys. 179 231–246 (2020)
B. Rahimi, H. Struchtrup, Capturing non-equilibrium phenomena in rarefied polyatomic gases: a high-order macroscopic model. Phys. Fluids 26, 052001 (2014)
T. Arima, A. Mentrelli, T. Ruggeri, Extended thermodynamics of rarefied polyatomic gases and characteristic velocities. Rend. Lincei Mat. Appl. 25, 275 (2014)
V.M. Zhdanov, The kinetic theory of a polyatomic gas. Sov. Phys. JETP 26, 1187 (1968)
F.J. McCormack, Kinetic equations for polyatomic gases: the 17-moment approximation. Phys. Fluids 11, 2533 (1968)
L. Desvillettes, R. Monaco, F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions. Eur. J. Mech. B. Fluids 24, 219 (2005)
M.C. Carrisi, S. Pennisi, An 18 moments model for dense gases: entropy and Galilean relativity principles without expansions. Entropy 17, 214 (2015)
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Ruggeri, T., Sugiyama, M. (2021). Nesting Theory of Many Moments and Maximum Entropy Principle. In: Classical and Relativistic Rational Extended Thermodynamics of Gases. Springer, Cham. https://doi.org/10.1007/978-3-030-59144-1_9
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