Skip to main content
SpringerLink
Log in
Book cover

From Particle Systems to Partial Differential Equations II pp 67–132Cite as

Dissipative Reactive Fluid Models from the Kinetic Theory

  • Conference paper
  • First Online:

  • 1045 Accesses

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 129))

Abstract

We present a kinetic framework describing mixtures of polyatomic species undergoing chemical reactions. Using a generalized Chapman-Enskog expansion, we derive the corresponding macroscopic fluid model. The hyperbolic-parabolic structure of the resulting system of partial differential equations is investigated and closely related to the underlying kinetic framework. We also discuss the Cauchy problem for smooth solutions as well as numerical algorithms for the evaluation of multicomponent transport coefficients using structural properties derived from the kinetic theory.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

Applications

  1. Rosner, D.E.: Transport Processes in Chemically Reacting Flow Systems. Butterworths, Boston (1986)

    Google Scholar 

  2. Roenigk, K.F., Jensen, K.F.: Low pressure CVD of silicon nitride. J. Electrochem. Soc. 134, 1777–1785 (1987)

    Google Scholar 

  3. Ern, A., Giovangigli, V., Smooke, M.: Numerical study of a three-dimensional chemical vapor deposition reactor with detailed chemistry. J. Comp. Phys. 126, 21–39 (1996)

    MATH  Google Scholar 

  4. Anderson Jr, J.D.: Hypersonics and High Temperature Gas Dynamics. McGraw-Hill Book Company, New York (1989)

    Google Scholar 

  5. Laboudigue, B., Giovangigli, V., Candel, S.: Numerical solution of a free-boundary problem in hypersonic flow theory: nonequilibrium viscous shock layers. J. Comp. Phys. 102, 297–309 (1992)

    MATH  Google Scholar 

  6. Williams, F.A.: Combustion Theory, 2nd edn. The Benjamin/Cummings Pub. Co., Inc., Menlo Park (1985)

    Google Scholar 

  7. Ern, A., Giovangigli, V.: Thermal diffusion effects in hydrogen/air and methane/air flames. Comb. Theor. Mod. 2, 349–372 (1998)

    MATH  Google Scholar 

  8. Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion. Edwards, Philadelphia (2005)

    Google Scholar 

  9. Giovangigli, V.: Multicomponent Flow Modeling. Birkhäuser, Boston (1999)

    MATH  Google Scholar 

Kinetic theory

  1. Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Wiley, New York (1954)

    MATH  Google Scholar 

  2. Waldmann, L.: Transporterscheinungen in Gasen von mittlerem Druck. Handbuch der Physik 12, 295–514 (1958)

    MathSciNet  Google Scholar 

  3. Waldmann, L., Trübenbacher, E.: Formale Kinetische Theorie von Gasgemischen aus Anregbaren Molekülen. Zeitschr. Naturforschg. 17(a), 363–376 (1962)

    Google Scholar 

  4. Van de Ree, J.: On the definition of the diffusion coefficients in reacting gases. Physica 36, 118–126 (1967)

    Google Scholar 

  5. Curtiss, C.F.: Symmetric gaseous diffusion coefficients. J. Chem. Phys. 49, 2917–2919 (1968)

    Google Scholar 

  6. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1970)

    Google Scholar 

  7. Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North Holland Pub. Co., Amsterdam (1972)

    Google Scholar 

  8. Cercignani, C.: The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, vol. 67. Springer, Berlin (1988)

    Google Scholar 

  9. Grunfeld, C.: On a class of Kinetic equations for reacting gas mixtures with multiple collisions. C. R. Acad. Sci. Paris 316(Série I), 953–958 (1993)

    Google Scholar 

  10. Ern, A., Giovangigli, V.: Multicomponent Transport Algorithms, Lecture Notes in Physics, New Series “Monographs”, m 24 (1994)

    Google Scholar 

Reactive Kinetic theory

  1. Prigogine, I., Xhrouet, E.: On the perturbation of Maxwell distribution function by chemical reactions in gases. Physica 15, 913–932 (1949)

    MATH  Google Scholar 

  2. Prigogine, I., Mathieu, M.: Sur la Perturbation de la Distribution de Maxwell par des Réactions Chimiques en Phase Gazeuse. Physica 16, 51–64 (1950)

    MATH  Google Scholar 

  3. Takayanagi, K.: On the theory of chemically reacting gas. Prog. Theor. Phys. VI, 486–497 (1951)

    Google Scholar 

  4. Present, R.D.: On the velocity distribution in a chemically reacting gas. J. Chem. Phys. 31, 747–750 (1959)

    Google Scholar 

  5. Ludwig, G., Heil, M.: Boundary Layer Theory with Dissociation and Lonization, vol. VI, pp. 39–118. Academic Press, New York (1960) (In Advances in Applied Mechanics )

    Google Scholar 

  6. Ross, J., Mazur, P.: Some deductions from a formal statistical mechanical theory of chemical kinetics. J. Chem. Phys. 35, 19–28 (1961)

    Google Scholar 

  7. Shizgal, B., Karplus, M.: Nonequilibrium contributions to the rate of reaction. I. Perturbation of the velocity distribution function. J. Chem. Phys. 52, 4262–4278 (1970)

    MathSciNet  Google Scholar 

  8. Shizgal, B., Karplus, M.: Nonequilibrium contributions to the rate of reaction. II. Isolated multicomponent systems. J. Chem. Phys. 54, 4345–4356 (1971)

    Google Scholar 

  9. Shizgal, B., Karplus, M.: Nonequilibrium contributions to the rate of reaction. III. Isothermal multicomponent systems. J. Chem. Phys. 54, 4357–4362 (1971)

    Google Scholar 

  10. Vallander, S., Nagnibeda, E., Rydalevskaya, M.: Some Questions of the Theory of the Chemically Reacting Gas Mixture. Leningrad University Press, Leningrad (1977) (English translation U.S. Air Force FASTC-ID (RS) TO-0608-93)

    Google Scholar 

  11. Alexeev, B.V.: Mathematical Kinetics of Reacting Gases. Nauka, Moskow (1982)

    Google Scholar 

  12. Alexeev, B.V., Chikhaoui, A., Grushin, I.T.: Application of the generalized Chapman-Enskog method to the transport-coefficient calculation in a reacting gas mixture. Phys. Rev. E 49, 2809–2825 (1994)

    Google Scholar 

  13. Chikhaoui, A., Dudon, J.P., Kustova, E.V., Nagnibeda, E.A.: Transport properties in reacting mixture of polyatomic gases. Physica A 247, 526–552 (1997)

    Google Scholar 

  14. Ern, A., Giovangigli, V.: The Kinetic equilibrium regime. Physica-A 260, 49–72 (1998)

    Google Scholar 

  15. Monaco, R., Pandolfi Bianchi, M., Soares, A.J.: BGK-type models in strong reaction and kinetic chemical equilibrium regimes. J. Phys. A 38, 10413 (2005)

    Google Scholar 

Thermodynamic nonequilibrium

  1. Kogan, M.: Rarefied Gas Dynamics. Nauka, Moskow (1967) (English translation Plenum Press, New York (1969))

    Google Scholar 

  2. Brun, R.: Transport et Relaxation dans les Ecoulements Gazeux. Masson, Paris (1986)

    Google Scholar 

  3. Zdhanov, V., Alievskiy, M.: Relaxation and Transport i Molecular Gases. Nauka, Moskow (1989)

    Google Scholar 

  4. Bruno, D., Capitelli, M., Longo, S.: Direct simulation of non-equilibrium kinetics under shock conditions in nitrogen. Chem. Phys. Lett. 360, 31–37 (2002)

    Google Scholar 

  5. Zhdanov, V.M.: Transport Processes in Multicomponent Plasmas. Taylor and Francis, London (2002)

    Google Scholar 

  6. Nagnibeda, E.A., Kustova, E.A.: Kinetic Theory of Transport and Relaxation Processes in Non-Equilibrium Reacting Flows. Saint-Petersburg University Press, Russian (2003)

    Google Scholar 

  7. Colonna, G., Armenise, I., Bruno, D., Capitelli, M.: Reduction of state-to-state kinetics to macroscopic models in hypersonic flows. J. Thermophys. Heat Transf. 20, 477–486 (2006)

    Google Scholar 

  8. Capitelli, M., Armenise, I., Bruno, D., Cacciatore, M., Celiberto, R., Colonna, G., de Pascale, O., Diomede, P., Esposito, F., Gorse, C., Hassouni, K., Laricchiuta, A., Longo, S., Pagano, D., Pietanza, D., Rutigliano, M.: Non-equilibrium plasma kinetics: a state-to-state approach. Plasma Sourc. Sci. Tech. 16, S30–S44 (2007)

    Google Scholar 

  9. Nagnibeda, E., Kustova, E.: Non-equilibrium Reacting Gas Flows. Springer, Berlin (2009)

    MATH  Google Scholar 

  10. Bruno, D., Giovangigli, V.: Relaxation of internal temperature and volume viscosity. Phys. Fluids 23, 093104 (2011)

    Google Scholar 

  11. Bruno, D., Giovangigli, V.: Erratum: Relaxation of internal temperature and volume viscosity. Phys. Fluids 25, 039902 (2013)

    Google Scholar 

  12. Bruno, D., Esposito, F., Giovangigli, V.: Relaxation of rotational-vibrational energy and volume viscosity in H-H\(_2\) mixtures. J. Chem. Phys. 138, 084302 (2013)

    Google Scholar 

Mathematical kinetic theory

  1. DiPerna, R.J., Lions, P.L.: On the global existence for boltzmann equations: global existence and weak stability. Ann. Math. 130, 321–366 (1989)

    MATH  MathSciNet  Google Scholar 

  2. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, vol. 106. Springer, Berlin (1994)

    Google Scholar 

  3. Grunfeld, C., Georgescu, E.: On a class of Kinetic equations for reacting gas mixtures. Mat. Fiz. Analiz. Geom. 2, 408–435 (1995)

    Google Scholar 

  4. Lions, P.-L., Masmoudi, N.: From Boltzmann equation to the Navier-Stokes and Euler equations I. Arch. Ration. Mech. Anal. 158, 173–193 (2001)

    MATH  MathSciNet  Google Scholar 

  5. Golse, F., Levermore, C.D.: The Stokes-Fourier and acoustic limits for the Boltzmann equation. Commun. Pure Appl. Math. 55, 336–393 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Saint-Raymond, L.: Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal. 166, 47–80 (2003)

    Google Scholar 

  7. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Golse, F., Saint-Raymond, L.: Hydrodynamic limits for the Boltzmann equation. Riv. Mat. Univ. Parma 7, 1–144 (2005)

    Google Scholar 

  9. Saint-Raymond, L.: Hydrodynamic limits of the Boltzmann equation. In: Lecture Notes in Mathematics 1971. Springer, Berlin-Heidelberg (2009)

    Google Scholar 

  10. Brull, S., Pavan, V., Schneider, J.: Derivation of a BGK model for mixtures. Eur. J. Mech. B/Fluids 33, 74–86 (2012)

    Google Scholar 

Macroscopic theories

  1. Marcelin, M.R.: Sur la Mécanique des Phénomènes Irréversibles. C. R. Acad. Sci. Paris 1052–1055 (1910)

    Google Scholar 

  2. Meixner, J.: Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemisch reagierenden, dissoziierenden und anregbaren Komponenten. Ann. Phys. 43, 244–270 (1943)

    MathSciNet  Google Scholar 

  3. Prigogine, I.: Etude thermodynamique des phénomènes irréversibles. Dunod, Paris (1947)

    Google Scholar 

  4. Irving, J.H., Kirkwood, J.G.: The statistical mechanics of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)

    MathSciNet  Google Scholar 

  5. Bearman, R.J., Kirkwood, J.G.: The statistical mechanics of transport processes. XI. Equations of transport in multicomponent systems. J. Chem. Phys. 28, 136–145 (1958)

    MathSciNet  Google Scholar 

  6. de Groot, S.R., Mazur, P.: Non-equilibrium thermodynamics. Dover Publications.Inc, New York (1984)

    Google Scholar 

  7. Gorban, A.N., Bykov, V.I., Yablonski, G.S.: Essays on Chemical Relaxation (in Russian). Nauka, Novosibirsk (1986)

    Google Scholar 

  8. Keizer, J.: Statistical Thermodynamics of Nonequilibrium Processes. Springer, New York (1987)

    Google Scholar 

Multicomponent transport algorithms

  1. Hirschfelder, J.O., Curtiss, C.F.: Theory of propagation of flames. Part I: general equations. Proc. Comb. Inst. 3,121–127 (1949)

    Google Scholar 

  2. Williams, F.A.: Elementary derivation of the multicomponent diffusion equation. Am. J. Phys. 26, 467–469 (1958)

    Google Scholar 

  3. Dixon-Lewis, G.: Flame structure and flame reaction Kinetics, II. Transport phenomena in multicomponent systems. Proc. Roy. Soc. A 307, 111–135 (1968)

    Google Scholar 

  4. Oran, E.S., Boris, J.P.: Detailed modeling of combustion systems. Prog. Energy Combust. Sci. 7, 1–72 (1981)

    Google Scholar 

  5. Giovangigli, V.: Mass conservation and singular multicomponent diffusion algorithms. IMPACT Comput. Sci. Eng. 2, 73–97 (1990)

    Google Scholar 

  6. Giovangigli, V.: Convergent Iterative methods for multicomponent diffusion. IMPACT Comput. Sci. Eng. 3, 244–276 (1991)

    MATH  MathSciNet  Google Scholar 

  7. Ern, A., Giovangigli, V.: Thermal conduction and thermal diffusion in dilute polyatomic gas mixtures. Physica-A 214, 526–546 (1995)

    Google Scholar 

  8. Ern, A., Giovangigli, V.: The structure of transport linear systems in dilute isotropic gas mixtures. Phys. Rev. E 53, 485–492 (1996)

    MathSciNet  Google Scholar 

  9. Ern, A., Giovangigli, V.: Optimized transport algorithms for flame codes. Comb. Sci. Tech. 118, 387–395 (1996)

    Google Scholar 

  10. Ern, A., Giovangigli, V.: Projected iterative algorithms with application to multicomponent transport. Linear Algebra Appl. 250, 289–315 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Giovangigli, V.: Multicomponent transport algorithms for partially ionized plasmas. J. Comp. Phys. 229, 4117–4142 (2010)

    MATH  MathSciNet  Google Scholar 

Impact of multicomponent transport

  1. Hancock, R.D., Schauer, F.R., Lucht, R.P., Katta, V.R., Hsu, K.Y.: Thermal diffusion effects and vortex-flame interactions in hydrogen jet diffusion flames. Proc. Combust. Inst. 26, 1087–1093 (1996)

    Google Scholar 

  2. Ern, A., Giovangigli, V.: Impact of multicomponent transport on planar and counterflow hydrogen/air and methane/air flames. Comb. Sci. Tech. 149, 157–181 (1999)

    Google Scholar 

  3. Bendahklia, R., Giovangigli, V., Rosner, D.: Soret effects in laminar counterflow spray diffusion flames. Comb. Theory Mod. 6, 1–17 (2002)

    Google Scholar 

  4. Hilbert, R., Tap, F., El-Rabii, H., Thevenin, D.: Impact of detailed chemistry and transport models on turbulent combustion simulations. Prog. Ener. Comb. Sci. 30, 61–117 (2004)

    Google Scholar 

  5. Galkin, V.S., Rusakov, S.V.: On the theory of bulk viscosity and relaxation pressure. Appl. Math. Mech. 69, 943–954 (2005)

    MathSciNet  Google Scholar 

  6. Elizarova, T., Kholkov, A., Montero, S.: Numerical simulaition of shock waves structure in nitrogen. Phys. Fluids 19, 068102 (2007)

    Google Scholar 

  7. Billet, G., Giovangigli, V., de Gassowski, G.: Impact of volume viscosity on a shock/hydrogen bubble interaction. Comb. Theor. Mod. 12, 221–248 (2008)

    MATH  Google Scholar 

  8. Dworkin, S., Smooke, M.D., Giovangigli, V.: The impact of detailed multicoponent transport and thermal diffusion effects on soot formation in ethylene/air flames. Proc. Comb. Inst. 32, 1165–1172 (2009)

    Google Scholar 

  9. Kustova, E.: On the role of bulk viscosity and relaxation pressure in nonequilibrium flows. In: Abe (ed.) 26th International Symposiuù on Rarefied Gas Dynamic, AIP Conference Proceedings 1024, pp. 807–812 (2009)

    Google Scholar 

  10. Armenise, I., Kustova, E.: On different contributions to the heat flux and diffusion in nonequilibrium flows. Chem. Phys. 428, 90–104 (2014)

    Google Scholar 

Hyperbolic-parabolic systems

  1. Godunov, S.: An interesting class of quasilinear systems. Sov. Math. Dokl 2, 947–949 (1961)

    MATH  Google Scholar 

  2. Lady\(\tilde{\text{ z }}\)enskaja, O.A., Solonikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, Rhode Island (1968)

    Google Scholar 

  3. Friedrichs, K.O., Lax, P.D.: Systems of conservation laws with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686–1688 (1971)

    MATH  MathSciNet  Google Scholar 

  4. Kawashima, S.: Systems of Hyperbolic-Parabolic Composite type, with Application to the Equations of Magnetohydrodynamics. In: Doctoral Thesis, Kyoto University (1984)

    Google Scholar 

  5. Hughes, T.J.R., Franca, L.P., Mallet, M.: A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. Appl. Mech. Eng. 54, 223–234 (1986)

    MATH  MathSciNet  Google Scholar 

  6. Ruggeri, T.: Thermodynamics and Symmetric Hyperbolic Systems, pp. 167–183. Rend. Sem. Mat. Univ. Torino, Hyperbolic Equations (1987)

    Google Scholar 

  7. Kawashima, S., Shizuta, Y.: On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tôhoku Math. J. 40, 449–464 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Chen, G.Q., Levermore, C.D., Liu, T.P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)

    MATH  MathSciNet  Google Scholar 

  9. Serre, D.: Systèmes de Lois de Conservation I and II. Diderot Editeur. Art et Science, Paris (1996)

    Google Scholar 

  10. Giovangigli, V., Massot, M.: Asymptotic stability of equilibrium states for multicomponent reactive flows. Math. Mod. Meth. Appl. Sci. 8, 251–297 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Dafermos, C.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  12. Giovangigli, V., Graille, B.: Asymptotic stability of equilibrium states for ambipolar plasmas. Math. Mod. Meth. Appl. Sci. 14, 1361–1399 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Kawashima, S., Yong, W.-A.: Dissipative structure and entropy for hyperbolic systems of conservation laws. Arch. Ration. Mech. Anal. 174, 345–364 (2004)

    MATH  MathSciNet  Google Scholar 

  14. Kawashima, S., Yong, W.A.: Decay estimates for hyperbolic balance laws. J. Anal. Appl. 28, 1–33 (2009)

    MATH  MathSciNet  Google Scholar 

  15. Serre, D.: The structure of dissipative viscous system of conservation laws. Physica D 239, 1381–1386 (2010)

    MATH  MathSciNet  Google Scholar 

  16. Giovangigli, V., Matuszewski, L.: Mathematical modeling of supercritical multicomponent reactive fluids. Math. Mod. Meth. Appl. Sci. 23, 2193–2251 (2013)

    MATH  MathSciNet  Google Scholar 

  17. Dafermos, C.M.: Hyperbolic systems of balance laws with weak dissipation II. J. Hyp. Diff. Equat. 10, 173–179 (2013)

    MATH  MathSciNet  Google Scholar 

  18. Giovangigli, V., Matuszewski, L.: Structure of entropies in dissipative multicomponent fluids. Kin. Relat. Mod. 6, 373–406 (2013)

    MATH  MathSciNet  Google Scholar 

  19. Giovangigli, V., Yong, W.A.: Volume viscosity and fast internal energy relaxation: symmetrization and Chapman-Enskog expansion. Kin. Relat. Mod. (in press)

    Google Scholar 

Cauchy problem

  1. Vol’Pert, A.I., Hudjaev, S.I.: On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sbornik. 16, 517–544 (1972)

    Google Scholar 

  2. Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)

    MATH  MathSciNet  Google Scholar 

  3. Kawashima, S.: Large-time behavior of solutions to hyperbolic-parabolic systems of conservations laws and applications. Proc. Roy. Soc. Edinb. 106A, 169–1944 (1987)

    MathSciNet  Google Scholar 

  4. Giovangigli, V., Massot, M.: The local Cauchy problem for multicomponent reactive flows in full vibrational nonequilibrium. Math. Meth. Appl. Sci. 21, 1415–1439 (1998)

    MATH  MathSciNet  Google Scholar 

  5. Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    Google Scholar 

  6. Yong, W.-A.: Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172, 247–266 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Feireisl, E., Novotný, A.: The low mach number limit for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 186, 77–107 (2007)

    MATH  MathSciNet  Google Scholar 

  8. Feireisl, E., Petzeltoá, H., Trivisa, K.: Multicomponent reactive flows: global-in-time existence for large data. Commun. Pure Appl. Anal. 7, 1017–1047 (2008)

    MATH  MathSciNet  Google Scholar 

  9. Yong, W.A.: An interesting class of partial differential equations. J. Math. Phys. 49, 033503 (2008)

    MathSciNet  Google Scholar 

  10. Kwong, Y.S., Trivisa, K.: Stability and large-time behavior for multicomponent reactive flows. Nonlinearity 22, 2443–2471 (2009)

    MathSciNet  Google Scholar 

  11. Hopf, D.: Asymptotic behavior of solutions to a model for the flow of a reacting fluid. Arch. Ration. Mech. Anal. 196, 951–979 (2010)

    MathSciNet  Google Scholar 

  12. Giovangigli, V., Massot, M.: Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry. Math. Meth. Appl. Sci. 27, 739–768 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Beauchard, K., Zuazua, E.: Large time asymptotics for partially dissipative hyperbolic systems. Arch. Ration. Mech. Anal. 199, 177–227 (2011)

    MATH  MathSciNet  Google Scholar 

  14. Giovangigli, V., Yong, W.A.: Volume Viscosity and Fast Internal Energy Relaxation: Error Estimates (submitted for publication)

    Google Scholar 

Miscelaneous

  1. Giovangigli, V.: Plane flames with multicomponent transport and complex chemistry. Math. Mod. Meth. Appl. Sci. 9, 337–378 (1999)

    MATH  MathSciNet  Google Scholar 

  2. Giovangigli, V., Tran, B.: Mathematical analysis of a Saint-Venant model with variable temperature. Math. Mod. Meth. Appl. Sci. 20, 1–47 (2010)

    MathSciNet  Google Scholar 

  3. Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of a viscous and heat-conductive fluids. J. Math. Kyoto Univ. 200, 67–104 (1980)

    MathSciNet  Google Scholar 

  4. Johnson, C., Szepessy, A., Hansbo, P.: On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comput. 54, 107–129 (1990)

    MATH  MathSciNet  Google Scholar 

  5. Giovangigli, V.: Higher order entropies. Arch. Ration. Mech. Anal. 187, 221–285 (2008)

    MATH  MathSciNet  Google Scholar 

  6. Giovangigli, V.: Higher order entropies for compressible fluid models. Math. Mod. Meth. Appl. Sci. 19, 67–125 (2009)

    MATH  MathSciNet  Google Scholar 

  7. Evans, L.C.: A survey of entropy methods for partial differential equations. Bull. AMS 41, 409–438 (2004)

    MATH  Google Scholar 

  8. Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pure Appl. 87, 57–90 (2007)

    MATH  MathSciNet  Google Scholar 

  9. Alazard, T.: Low mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180, 1–73 (2006)

    MATH  MathSciNet  Google Scholar 

  10. Feireisl, E., Novotnỳ, A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser, Basel (2009)

    MATH  Google Scholar 

  11. Laurent, F., Massot, M.: Multi-fluid modeling of laminar poly-dispersed spray flames: origin, assumptions and comparison of the sectional and sampling methods. Comb. Theor. Mod. 5, 537–572 (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CMAP-CNRS, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Vincent Giovangigli

Authors
  1. Vincent Giovangigli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Giovangigli .

Editor information

Editors and Affiliations

  1. Centre of Mathematics, University of Minho, Braga, Portugal

    Patrícia Gonçalves

  2. Centre of Mathematics, University of Minho, Braga, Portugal

    Ana Jacinta Soares

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Giovangigli, V. (2015). Dissipative Reactive Fluid Models from the Kinetic Theory. In: Gonçalves, P., Soares, A. (eds) From Particle Systems to Partial Differential Equations II. Springer Proceedings in Mathematics & Statistics, vol 129. Springer, Cham. https://doi.org/10.1007/978-3-319-16637-7_2

Download citation

Publish with us

Policies and ethics

Search

Navigation