Abstract
We propose a 4-velocity unidimensional discrete Boltzmann model with two different speeds 2, 1 and two different masses 1, 2. With the three conservation laws of mass, momentum, and energy satisfied, we can introduce a nontrivial temperature. First, we determine the similarity shock waves satisfying physical properties: positivity, shock stability, inequalities of the subsonic and supersonic flows, increase or decrease of both mass and temperature across the shock. It results that either the speed of the shock front is higher than the speed 1 of the slow particles and the shocks are compressive or less than 1 and the shocks are rarefactive. We observe overshoots of the temperature, across the shock, with bumps higher and higher as the shock front speed increases. Second, we study the (1+1)-dimensional shock waves. They represent the superposition and collision of two compressive shocks traveling in opposite directions and we observe temperature overshoots for not too large times.
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R. Gatignol,Lectures Notes in Physics, No. 36 (1975); H. Cabannes,Transp. Theory Stat. Phys. 16:809 (1987); T. Platkowski and R. Illner,SIAM Rev. 30:212 (1988); D. d'Humiéres, inCellular Automata and Modeling of Complex Physical Systems, P. Manneville, N. Boccara, G. Vichniac, and R. Bidaux, eds. (Springer, 1990).
T. Carleman,Publ. Sci. Inst. Mittag-Leffler (Uppsala) (1957); J. McKean,Commun. Pure Appl. Math. 28:435 (1973); R. Illner,Math. Meth. Appl. Sci. 1:189 (1979); T. Ruijgrook and T. T. Wu,Physica 13A:401 (1982); N. Ianiro and J. L. Lebowitz,Found. Phys. 15:531 (1985).
A. V. Bobylev, Mathematical Congress Warsaw, Book Abstract, B29 (1983); J. Wick,Math. Meth. Appl. Sci. 6:513 (1984); T. Platkowski,J. Mec. Theor. Appl. 4:555 (1985); H. Cornille,J. Math. Phys. 28:1567 (1987),Lett, in Math. Phys. 19:211 (1990)
H. Cornille, inPartially Integrable Evolution Equations in Physic, R. Conte and N. Boccara, eds. (Kluwer Academic Publishers, 1990), p. 39.
Y.-H. Qian, D. d'Humières, and P. Lallemand,J. Stat. Phys., to appear.
T. Platkowski, inDiscrete Kinetic Theory, Lattice Gas Dynamics, R. Monaco, ed. (World Scientific, 1989), p. 248.
T. Platkowski,Transp. Theory Stat. Phys. 18:221 (1989).
H. Cornille and Y.-H. Qian,Compt. Rend. Acad. Sci. Paris 309:1883 (1989); H. Cornille,Inverse Methods in Action, P. C. Sabatier, ed. (Springer-Verlag, 1990), p. 414.
D. d'Humières et al.,Eur. Lett. 2:291 (1986); J. Nadigaet al., inProgress in Astronautics and Aeronautics, Vol. 116, J. Muntz, ed. (1988), p. 159; J. Chenet al, Physica 37D:42 (1989).
J. E. Broadwell, B. T. Nadiga, S. Chen, E. P. Muntz, J. L. Lebowitz, and H. McKean,J. Phys. Fluids 7:1243 (1964).
P. D. Lax,Commun. Pure Appl. Math. 10:537 (1957); G. B. Whitham,Commun. Pure Appl. Math. 12:113 (1959); R. Gatignol,Phys. Fluids 18:153 (1975);Transp. Theory Stat. Phys. 16:837 (1987); R. Monaco and R. Pandolfi Bianchi, inEuromech Colloquium No. 246, Hypersonic Aerodynamics of Spacecraft (Torino, April 1989).
H. Cornille,J. Phys. A 20:1973 (1987);Phys. Lett. A 125:253 (1987);J. Stat. Phys. 48:789 (1987).
H. Cabannes and D. H. Tiem,Complex Systems 1:574 (1987).
H. Cornille, inIII International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, Salice-Term 1988, G. Toscani, ed. (Springer-Verlag, to appear).
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Cornille, H., Qian, YH. Temperature overshoots for a 4-velocity unidimensional discrete Boltzmann model. J Stat Phys 61, 683–712 (1990). https://doi.org/10.1007/BF01027297
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DOI: https://doi.org/10.1007/BF01027297