An analysis of the shock structure in the 14 moment system of extended thermodynamics with first, second and third order closure based on the maximum entropy principle (MEP) is presented, as a follow up of a recent investigation of the shock structure in the 13 moment system with first and second MEP-based closure. It is seen that when adopting higher order closures, the strength of the subshock that appears in the shock structure profile for large enough Mach numbers is remarkably reduced with respect to what is found with the first order closure, and the overall profile of the shock structure solution is in better agreement with experimental results.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer, New York (1993)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998)
Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24, 271–292 (2012)
Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Monatomic rarefied gas as a singular limit of polyatomic gas in extended thermodynamics. Phys. Lett. A 377, 2136–2140 (2013)
Arima, T., Mentrelli, A., Ruggeri, T.: Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments. Ann. Phys. 345, 111–140 (2014)
Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Monatomic Gas. Springer, Basel (2015)
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)
Jaynes, E.T.: Information theory and statistical mechanics II. Phys. Rev. 108, 171–190 (1957)
Kogan, M.N.: Rarefied Gas Dynamics. Plenum Press, New York (1969)
Kapur, J.N.: Maximum Entropy Models in Science and Engineering. Wiley, New York (1989)
Boillat, G., Ruggeri, T.: Moment equations in the kinetic theory of gases and wave velocities. Contin. Mech. Thermodyn. 9, 205–212 (1997)
Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996)
Junk, M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93, 1143–1167 (1998)
Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 5, 285–292 (1998)
Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Thermodynamic theory of the shock wave structure in a rarefied polyatomic gas: beyond the Bethe–Teller theory. Phys. Rev. E 89, 013025 (2014)
Taniguchi, S., Arima, T., Ruggeri, T., Sugiyama, M.: Shock wave structure in rarefied polyatomic gases with large relaxation time for the dynamic pressure. J. Phys. Conf. Ser. 1035, 012009 (2018)
Ruggeri, T., Taniguchi, S.: Shock waves in hyperbolic systems of non-equilibrium thermodynamics. In: Berezovski, A., Soomere, T. (eds.) Applied Wave Mathematics II: Selected Topics in Solids, Fluids, and Mathematical Methods and Complexity. Mathematics of Planet Earth, vol. 6, pp. 167–186. Springer, Cham (2019)
Brini, F., Ruggeri, T.: Second-order approximation of extended thermodynamics of a monatomic gas and hyperbolicity region. Contin. Mech. Thermodyn. 32, 23–39 (2020)
Mentrelli, A., Ruggeri, T.: Shock structure in extended thermodynamics with second order maximum entropy principle closure (submitted)
Pennisi, S., Ruggeri, T.: Classical limit of relativistic moments associated with Boltzmann Chernikov equation: optimal choice of moments in classical theory (submitted)
Kremer, G.M.: Extended thermodynamics of ideal gases with 14 fields. Ann. I. H. P. Phys. Theor. 45, 419–440 (1986)
Brini, F., Ruggeri, T.: Entropy principle for the moment systems of degree associated to the Boltzmann equation. Critical derivatives and non controllable boundary data. Contin. Mech. Thermodyn. 14, 165–189 (2002)
Weiss, W.: Continuous shock structure in extended thermodynamics. Phys. Rev. E 52, R5760–R5763 (1995)
Ruggeri, T.: Shock waves in hyperbolic dissipative systems: non equilibrium gases. In: Proceedings of the Euromech Colloquium 270 (Reggio Calabria, Italy; September 1990) (1991)
Kosuge, S., Aoki, K.: Shock-wave structure for a polyatomic gas with large bulk viscosity. Phys. Rev. Fluids 3, 023401-1/42 (2018)
Kosuge, S., Kuo, H.-W., Aoki, K.: A kinetic model for a polyatomic gas with temperature-dependent specific heats and its application to shock-wave structure. J. Stat. Phys. 177, 209–251 (2019)
This work has been partially supported by GNFM/INdAM and by the Italian MIUR through the PRIN2017 project “Multiscale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions”(Project Number: 2017YBKNCE).
Conflict of interest
The author states that there is no conflict of interest.
It is my pleasure to dedicate this paper to Masaru Sugiyama and Giuseppe Toscani, who recently celebrated their 70th birthday. I have not been lucky enough to collaborate with Giuseppe Toscani yet, but having enjoyed several of his talks I have developed a true admiration for him. With Masaru Sensei Sugiyama I was more lucky, since I have had (and still have) the chance of collaborating with him in Nagoya and Bologna over the years. Not only have I learned a lot from him, but I also have the most sincere esteem of him. I sincerely wish to both of them all the best.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Mentrelli, A. Shock structure in the 14 moment system of extended thermodynamics with high order closure based on the maximum entropy principle. Ricerche mat (2020). https://doi.org/10.1007/s11587-020-00511-x
- Extended thermodynamics
- Shock structure
- Maximum entropy principle
Mathematics Subject Classification