Details

Srboljub Simić^[a]

On the macroscopic/kinetic closure of balance laws

Pages: 223-243
Received: 16 May 2023
Accepted in revised form: 28 March 2024.
Mathematics Subject Classification: 76P05, 82B40, 82C40.
Keywords: Extended thermodynamics, moment method, closure problem.
Authors address:
[a]: University of Novi Sad, Department of Mathematics and Informatics, Trg Dositeja Obradovi&cgrave;a 4, Novi Sad, 21000, Serbia

The author gratefully acknowledge the financial support of the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Grants No. 451-03-66/2024-03/200125 & 451-03-65/2024-03/200125).

Abstract: This paper presents a review of the results which illustrate the interplay of macroscopic (continuum) approach and kinetic theory of gases in solving the closure problem. Continuum approach to the closure problem is based upon entropy principle, and it is limited since phenomenological coefficients cannot be explicitly determined. On the other hand, kinetic theory provides closed systems of equations as approximate solutions to the Boltzmann equation, but it is mainly limited to rarefied gases. Combined closure procedure, reviewed in this paper, proposes a systematic matching procedure in which advantages of both approaches are taken into account. It is illustrated by the classical examples of Navier-Stokes-Fourier system and 13 moments model, but also with novel applications to the multi-temperature mixture of Euler fluids.

References

[1]: T. Arima, T. Ruggeri, M. Sugiyama and S. Taniguchi, Galilean Invariance and Entropy Principle for a System of Balance Laws of Mixture Type, Rend. Lincei Mat. Appl. 28 (2017), 495-513. DOI
[2]: V. Artale, F. Conforto, G. Martalò and A. Ricciardello, Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monoatomic gases, Ric. Mat. 68 (2019), 485-502. DOI
[3]: E. S. Benilov and M. S. Benilov, The Enskog-Vlasov equation: a kinetic model describing gas, liquid, and solid, J. Stat. Mech. (2019), 103205. DOI
[4]: M. Bisi, G. Martalò and G. Spiga, Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions, EPL (Europhysics Letters) 95 (2011), 55002. DOI
[5]: M. Bisi, G. Martalò and G. Spiga, Shock wave structure of multi-temperature Euler equations from kinetic theory for a binary mixture, Acta Appl. Math. 132 (2014), 95-105. DOI
[6]: M. Bisi, M. Groppi, A. Macaluso and G. Martalò, Shock wave structure of multi-temperature Grad 10-moment equations for a binary gas mixture, EPL (Europhys. Letters) 133 (2021), 54001. DOI
[7]: G. Boillat and T. Ruggeri, Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions, Arch. Rational Mech. Anal. 137 (1997), 305-320. DOI
[8]: G. Boillat and T. Ruggeri, Moment equations in the kinetic theory of gases and wave velocities, Contin. Mech. Thermodyn. 9 (1997), 205-212. DOI
[9]: S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases, Cambridge University Press, Cambridge, 1991. Zbmath
[10]: G. Dimarco, R. Loubère, J. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, J. Comput. Phys. 353 (2018), 46-81. DOI | MR
[11]: H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2 (1949), 331-407. DOI | MR
[12]: S. R. De Groot and P. Mazur, Non-equilibrium thermodynamics, Dover Publications, New York, 2011. MR (ed. 1992)
[13]: L. Desvillettes, R. Monaco and 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 (2005), 219-236. DOI
[14]: W. Dreyer, Maximisation of the entropy in non-equilibrium, J. Phys. A Math. Gen. 20 (1987), 6505-6517. DOI
[15]: A. Frezzotti and P. Barbante, Kinetic theory aspects of non-equilibrium liquid-vapor flows, Mech. Eng. Rev. 4 (2017), 16-00540. DOI
[16]: V. Giovangigli, Kinetic derivation of Cahn-Hilliard fluid models, Physical Review E 104 (2021), 054109. DOI
[17]: M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, Cambridge, 2010.
[18]: E. Ikenberry and C. Truesdell, On the Pressures and the Flux of Energy in a Gas according to Maxwell's Kinetic Theory, I, J. Rational Mech. Anal. 5 (1956), 1-54. MR
[19]: M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969.
[20]: S. Kosuge, K. Aoki and S. Takata, Shock-wave structure for a binary gas mixture: Finite-difference analysis of the Boltzmann equation for hard sphere molecules, Eur. J. Mech. B Fluids 20 (2001), 87-126. DOI
[21]: G. M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases, Springer-Verlag, Berlin, 2010. MR
[22]: C. D. Levermore, Moment Closure Hierarchies for Kinetic Theories, J. Stat. Phys. 83 (1996), 1021-1065. DOI
[23]: I.-S. Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Rational Mech. Anal. 46 (1972), 131-148. DOI
[24]: D. Madjarević and S. Simić, Shock structure in helium-argon mixture - A comparison of hyperbolic multi-temperature model with experiment, EPL (Europhysics Letters) 102 (2013), 44002. DOI
[25]: D. Madjarević, T. Ruggeri and S. Simić, Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures, Physics of Fluids 26 (2014), 106102. DOI
[26]: D. Madjarević and S. Simić, Entropy growth and entropy production rate in binary mixture shock waves, Phys. Rev. E 100 (2019), 023119. DOI
[27]: D. Madjarević, M. Pavić-Čolić and S. Simić, Shock Structure and Relaxation in the Multi-Component Mixture of Euler Fluids, Symmetry 13 (2021), 955. DOI
[28]: I. Müller, On the entropy inequality, Arch. Rational Mech. Anal. 26 (1967), 118-141. DOI
[29]: I. Müller, A Thermodynamic Theory of Mixtures of Fluids, Arch. Rational Mech. Anal. 28 (1968), 1-39. DOI
[30]: I. Müller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, 1998. MR
[31]: L. Pareschi and G. Russo, Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator, SIAM Journal on Numerical Analysis, 37 (2000), 1217-1245. DOI
[32]: L. Pareschi and T. Rey, Moment Preserving Fourier-Galerkin Spectral Methods and Application to the Boltzmann Equation, SIAM Journal on Numerical Analysis 60 (2022), 3216-3240. DOI
[33]: M. Pavić-Čolić, D. Madjarević and S. Simić, Polyatomic gases with dynamic pressure: Kinetic non-linear closure and the shock structure, Int. J. Non Linear Mech. 92 (2017), 160-175. DOI
[34]: M. Pavić-Čolić, Multi-velocity and multi-temperature model of the mixture of polyatomic gases issuing from kinetic theory, Phys. Lett. A 383 (2019), 2829-2835. DOI
[35]: M. Pavić-Čolić and S. Simić, Six-Field Theory for a Polyatomic Gas Mixture: Extended Thermodynamics and Kinetic Models, Fluids 7 (2022), 381. DOI
[36]: T. Ruggeri, Galilean Invariance and Entropy Principle for Systems of Balance Laws. The Structure of the Extended Thermodynamics, Continuum Mech. Thermodyn. 1 (1989), 3-20. DOI
[37]: T. Ruggeri and S. Simić, On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single- and multi-temperature models, Mathematical Methods in the Applied Sciences 30 (2007), 827-849. DOI
[38]: T. Ruggeri and A. Strumia, Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincaré Sect. A (N.S.) 34 (1981), 65-84. Numdam
[39]: T. Ruggeri and M. Sugiyama, Rational Extended Thermodynamics beyond the Monatomic Gas, Springer, Cham, 2015. DOI
[40]: T. Ruggeri and M. Sugiyama, Classical and Relativistic Rational Extended Thermodynamics of Gases, Springer, Cham, 2021. DOI
[41]: S. Simić, M. Pavić-Čolić and D. Madjarević, Non-equilibrium mixtures of gases: Modelling and computation, Riv. Mat. Univ. Parma 6 (2015),135-214. MR | Article
[42]: S. Simić and D. Madjarević, Shock structure and entropy growth in a gaseous binary mixture with viscous and thermal dissipation, Wave Motion 100 (2021), 102661. DOI
[43]: C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969. MR