Non-equilibrium relativistic two-phase flow with heat exchange

Giambò, Sebastiano; Muscianisi, Giuseppa

doi:10.1090/S0033-569X-2012-01272-9

Authors: Sebastiano Giambò and Giuseppa Muscianisi
Journal: Quart. Appl. Math. 70 (2012), 773-786
MSC (2010): Primary 76T10, 76D33, 83A05
DOI: https://doi.org/10.1090/S0033-569X-2012-01272-9
Published electronically: July 18, 2012
MathSciNet review: 3052090
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We consider two different systems of hyperbolic balance laws governing relativistic two-phase flows with general equations of state. The phases are assumed to be immiscible. In the first one the thermal equilibrium is attained through a relaxation procedure consistent with the second law of thermodynamics, while the second is a fully relaxed model in which the two phases reach an equal temperature instantaneously.

Finally, we deduce and compare the wave velocities for both the models: the hydrodynamical velocity of the relaxed system can never exceed that of the relaxation system.

References

Rémi Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi-conservative approach, J. Comput. Phys. 125 (1996), no. 1, 150–160. MR 1381808, DOI https://doi.org/10.1006/jcph.1996.0085
Grégoire Allaire, Sébastien Clerc, and Samuel Kokh, A five-equation model for the simulation of interfaces between compressible fluids, J. Comput. Phys. 181 (2002), no. 2, 577–616. MR 1927402, DOI https://doi.org/10.1006/jcph.2002.7143
Nikolai Andrianov, Richard Saurel, and Gerald Warnecke, A simple method for compressible multiphase mixtures and interfaces, Internat. J. Numer. Methods Fluids 41 (2003), no. 2, 109–131. MR 1950175, DOI https://doi.org/10.1002/fld.424

A. M. Anile, Relativistic fluids and magneto-fluids, Cambridge University Press, Cambridge, 1989.

Selçuk Ş. Bayin, Anisotropic fluid spheres in general relativity, Phys. Rev. D (3) 26 (1982), no. 6, 1262–1274. MR 675034, DOI https://doi.org/10.1103/PhysRevD.26.1262

Mahdy Cissoko, Wavefronts in a relativistic cosmic two-component fluid, Gen. Relativity Gravitation 30 (1998), no. 4, 521–534. MR 1618979, DOI https://doi.org/10.1023/A%3A1018857805768

Mahdy Cissoko, Wave fronts in a mixture of two relativistic perfect fluids flowing with two distinct four-velocities, Phys. Rev. D (3) 63 (2001), no. 8, 083516, 12. MR 1831145, DOI https://doi.org/10.1103/PhysRevD.63.083516

A. A. Coley and B. O. J. Tupper, Imperfect fluid cosmologies with thermodynamics: some exact solutions, Astrophys. J. 280 (1984), no. 1, 26–33. MR 743301, DOI https://doi.org/10.1086/161964

A. A. Coley and B. O. J. Tupper, Two-fluid cosmological models, J. Math. Phys. 27 (1986), no. 1, 406–416. MR 816462, DOI https://doi.org/10.1063/1.527347

Vincent Deledicque and Miltiadis V. Papalexandris, A conservative approximation to compressible two-phase flow models in the stiff mechanical relaxation limit, J. Comput. Phys. 227 (2008), no. 21, 9241–9270. MR 2463206, DOI https://doi.org/10.1016/j.jcp.2007.12.011

Ken Dunn, Two-fluid cosmological models in Gödel-type spacetimes, Gen. Relativity Gravitation 21 (1989), no. 2, 137–147. MR 990147, DOI https://doi.org/10.1007/BF00761083

Joan Josep Ferrando, Juan Antonio Morales, and Miquel Portilla, Two-perfect fluid interpretation of an energy tensor, Gen. Relativity Gravitation 22 (1990), no. 9, 1021–1032. MR 1067616, DOI https://doi.org/10.1007/BF00757813

Tore Flåtten, Alexandre Morin, and Svend Tollak Munkejord, Wave propagation in multicomponent flow models, SIAM J. Appl. Math. 70 (2010), no. 8, 2861–2882. MR 2735107, DOI https://doi.org/10.1137/090777700

V. E. Fortov and I. V. Lomonosov, Equations of state of matter at high energy densities, The Open Plasma Physics Journal 3 (2010), 122–130.

Sebastiano Giambò and Serena Giambò, A seven equation model for relativistic two fluid flows—I, ROMAI J. 5 (2009), no. 2, 59–70. MR 2721255

Sebastiano Giambò and Serena Giambò, A seven equation model for relativistic two fluid flows-II, ROMAI J. 6 (2010), no. 1, 95–105. MR 2739286

---, A model for relativistic fluid mixture with different pressure laws, EPL 90 (2010), 54004.

E. N. Glass and J. P. Krisch, Two-fluid atmosphere for relativistic stars, Classical Quantum Gravity 16 (1999), no. 4, 1175–1184. MR 1696147, DOI https://doi.org/10.1088/0264-9381/16/4/007

Yu. B. Ivanov, V. N. Russkikh, and V. D. Toneev, Relativistic heavy-ion collisions within three-fluid hydrodynamics: hadronic scenario, Phys. Rev. C 73 (2006), 044904.

A. Krasiński, Inhomogeneous cosmological models, Cambridge University Press, Cambridge, 1997.

M. Krause and P. Alexander, Simulations of multi-phase turbulence in jet cocoons, Mon. Not. R. Astron. Soc. 376 (2007), 465–478.

J. P. Krisch and L. L. Smalley, Two fluid acoustic modes and inhomogeneous cosmologies, Classical Quantum Gravity 10 (1993), no. 12, 2615–2623. MR 1252288

F. Lagoutière, Modélisation mathématique et résolution numérique de problèmes de fluides compressible plusieurs constituants, Ph.D. thesis, Université de Paris VI, 2000.

Patrício S. Letelier, Anisotropic fluids with two-perfect-fluid components, Phys. Rev. D (3) 22 (1980), no. 4, 807–813. MR 583095, DOI https://doi.org/10.1103/PhysRevD.22.807

Patrício S. Letelier, Solitary waves of matter in general relativity, Phys. Rev. D (3) 26 (1982), no. 10, 2623–2631. MR 686726, DOI https://doi.org/10.1103/PhysRevD.26.2623

Patricio S. Letelier and Paulo S. C. Alencar, Anisotropic fluids with multifluid components, Phys. Rev. D (3) 34 (1986), no. 2, 343–351. MR 848075, DOI https://doi.org/10.1103/PhysRevD.34.343

A. Lichnerowicz, Ondes des choc, ondes infinitésimales et rayons en hydrodynamique et magnétohydrodynamique relativistes, Relativistic fluid dynamics Centro Internaz. Mat. Estivo (C.I.M.E.), I Ciclo, Bressanone, 1970) Edizioni Cremonese, Rome, 1971, pp. 87–203 (French). MR 0297295

H. Lund and T. Flåtten, Equilibrium conditions and sound velocities in two-phase flows, SIAM Annual Meeting 2010 (Pittsburg, Pennsylvania, USA), 2010.

S. Müller, P. Helluy, and J. Ballmann, Numerical simulation of cavitation bubbles by compressible two-phase fluids, Int. J. Fluid Mech. (2009).

Angelo Murrone and Hervé Guillard, A five equation reduced model for compressible two phase flow problems, J. Comput. Phys. 202 (2005), no. 2, 664–698. MR 2145395, DOI https://doi.org/10.1016/j.jcp.2004.07.019

F. Petitpas, J. Massoni, R. Saurel, E. Lapebie, and L. Munier, Diffuse interface models for high speed cavitating underwater systems, Int. J. Multiphase Flows 35 (2009), 747–759.

F. Petitpas, R. Saurel, E. Franquet, and A. Chinnayya, Modelling detonation waves in condensed materials: multiphase CJ conditions and multidimensional computations, Shock waves 19 (2009), 377–401.

E. Romenski and E. F. Toro, Compressible two–phase flows: Two–pressure models and numerical methods, Comp. Fluid Dyn. J. 13 (2004), 403–416.

R. Saurel, O. Le Metayer, J. Massoni, and S. Gavrilyuk, Shock jump relations for multiphase mixtures with stiff mechanical properties, Shock waves 16 (2007), 209–232.

Richard Saurel, Fabien Petitpas, and Remi Abgrall, Modelling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607 (2008), 313–350. MR 2436919, DOI https://doi.org/10.1017/S0022112008002061

Richard Saurel, Fabien Petitpas, and Ray A. Berry, Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures, J. Comput. Phys. 228 (2009), no. 5, 1678–1712. MR 2494233, DOI https://doi.org/10.1016/j.jcp.2008.11.002

R. F. Sistero, Relativistic cosmological models with pressure, Astrophys. Space Sci. 12 (1971), 484.

V. D. Toneev, Yu. B. Ivanov, E. G. Nikonov, W. Nörenberg, and V. N. Russkikh, Three-fluid simulations of relativistic heavy-ion collisions, Physics of Particles and Nuclei Letters 2(2) (2005), 43–49.

A. I. Voropinov and M. A. Podurets, Relativistic gravitational collapse of a cool white dwarf with allowance for the neutronization kinetics, Astronomy Letters 27 (2001), no. 6, 366–375.

Jeroen Wackers and Barry Koren, A fully conservative model for compressible two-fluid flow, Internat. J. Numer. Methods Fluids 47 (2005), no. 10-11, 1337–1343. 8th ICFD Conference on Numerical Methods for Fluid Dynamics. Part 2. MR 2128761, DOI https://doi.org/10.1002/fld.911

Rubén D. Zárate and Hernando Quevedo, Thermodynamic scheme of inhomogeneous perfect fluid mixtures, Classical Quantum Gravity 21 (2004), no. 1, 197–205. MR 2026686, DOI https://doi.org/10.1088/0264-9381/21/1/013

D. Zeidan, E. Romenski, A. Slaouti, and E. F. Toro, Numerical study of wave propagation in compressible two-phase flow, Internat. J. Numer. Methods Fluids 54 (2007), no. 4, 393–417. MR 2314750, DOI https://doi.org/10.1002/fld.1404

Ali Zein, Maren Hantke, and Gerald Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comput. Phys. 229 (2010), no. 8, 2964–2998. MR 2595804, DOI https://doi.org/10.1016/j.jcp.2009.12.026

W. Zimdahl, D. Pavon, and R. Maartens, Reheating and causal thermodynamics, Phys. Rev. D 55 (1997), 4861–4688.