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March  2015, 8(1): 79-116. doi: 10.3934/krm.2015.8.79

## Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion

 1 CMAP, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex 2 ZCAM, Tsinghua University, Beijing, 100084, China

Received  June 2014 Revised  September 2014 Published  December 2014

We analyze a mathematical model for the relaxation of translational and internal temperatures in a nonequilibrium gas. The system of partial differential equations---derived from the kinetic theory of gases---is recast in its natural entropic symmetric form as well as in a convenient hyperbolic-parabolic symmetric form. We investigate the Chapman-Enskog expansion in the fast relaxation limit and establish that the temperature difference becomes asymptotically proportional to the divergence of the velocity field. This asymptotic behavior yields the volume viscosity term of the limiting one-temperature fluid model.
Citation: Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
##### References:
 [1] G. Billet, V. Giovangigli and G. de Gassowski, Impact of volume viscosity on a shock-hydrogen bubble interaction, Comb. Theory Mod., 12 (2008), 221-248. doi: 10.1080/13647830701545875.  Google Scholar [2] D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pure Appl., 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.  Google Scholar [3] D. Bruno and V. Giovangigli, Relaxation of internal temperature and volume viscosity, Phys. Fluids, 23 (2011), 093104. doi: 10.1063/1.3640083.  Google Scholar [4] D. Bruno and V. Giovangigli, Erratum: "Relaxation of internal temperature and volume viscosity'' [Phys. Fluids 23, 093104 (2011)], Phys. Fluids, 25 (2013), 039902. doi: 10.1063/1.4795334.  Google Scholar [5] D. Bruno, F. Esposito and V. Giovangigli, Relaxation of rotational-vibrational energy and volume viscosity in $H-H_2$ mixtures, J. Chem. Physics, 138 (2013), 084302. Google Scholar [6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.  Google Scholar [7] G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.  Google Scholar [8] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar [9] G. Emanuel, Bulk viscosity of a dilute polyatomic gas, Phys. Fluids A, 2 (1990), 2252-2254. doi: 10.1063/1.857813.  Google Scholar [10] G. Emanuel, Effect of bulk viscosity on a hypersonic boundary layer, Phys. Fluids A, 4 (1992), 491-495. doi: 10.1063/1.858322.  Google Scholar [11] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, 24, Springer-Verlag, Berlin, 1994.  Google Scholar [12] A. Ern and V. Giovangigli, Volume viscosity of dilute polyatomic gas mixtures, Eur. J. Mech. B/Fluids, 14 (1995), 653-669. Google Scholar [13] A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport, Lin. Alg. App., 250 (1997), 289-315. doi: 10.1016/0024-3795(95)00502-1.  Google Scholar [14] A. Ern and V. Giovangigli, The Kinetic equilibrium regime, Physica-A, 260 (1998), 49-72. Google Scholar [15] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.  Google Scholar [16] J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam, 1972. Google Scholar [17] K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.  Google Scholar [18] V. Giovangigli, Multicomponent Flow Modeling, Birkhaüser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar [19] V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows, Math. Mod. Meth. App. Sci., 8 (1998), 251-297. doi: 10.1142/S0218202598000123.  Google Scholar [20] V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768. doi: 10.1002/mma.429.  Google Scholar [21] V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Phys. D, 241 (2012), 649-670. doi: 10.1016/j.physd.2011.12.002.  Google Scholar [22] V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids, Math. Mod. Meth. App. Sci., 23 (2013), 2193-2251. doi: 10.1142/S0218202513500309.  Google Scholar [23] V. Giovangigli and L. Matuszewski, Structure of entropies in dissipative multicomponent fluids, Kin. Rel. Mod., 6 (2013), 373-406. doi: 10.3934/krm.2013.6.373.  Google Scholar [24] V. Giovangigli and W.-A. Yong, Volume viscosity and fast internal energy relaxation: Convergence results,, (submitted for publication)., ().   Google Scholar [25] S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.  Google Scholar [26] R. E. Graves and B. Argrow, Bulk viscosity: Past to present, J. Therm. Heat Transfer, 13 (1999), 337-342. doi: 10.2514/2.6443.  Google Scholar [27] J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, New-York, Wiley, 1954. Google Scholar [28] T. J. R. Hughes, L. P. Franca and M. Mallet, 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 (1986), 223-234. doi: 10.1016/0045-7825(86)90127-1.  Google Scholar [29] S. M. Karim and L. Rosenhead, The second coefficient of viscosity of Liquids and gases, Rev. Mod. Phys, 24 (1952), 108-116. doi: 10.1103/RevModPhys.24.108.  Google Scholar [30] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University 1984. Google Scholar [31] S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservations laws and applications, Proc. Roy. Soc. Edinburgh, 106 (1987), 169-194. doi: 10.1017/S0308210500018308.  Google Scholar [32] S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tôhoku Math. J., 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.  Google Scholar [33] S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 174 (2004), 345-364. doi: 10.1007/s00205-004-0330-9.  Google Scholar [34] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-1054-2.  Google Scholar [35] C. Lattanzio and W.-A. Yong, Hyperbolic-parabolic singular limits for first-order nonlinear systems, Comm. Partial Diff. Equ., 26 (2001), 939-964. doi: 10.1081/PDE-100002384.  Google Scholar [36] T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Physics, 108 (1987), 153-175. doi: 10.1007/BF01210707.  Google Scholar [37] F. R. McCourt, J. J. Beenakker, W. E. Köhler and I. Kuscer, Non Equilibrium Phenomena in Polyatomic Gases, Volume I: Dilute Gases, Volume II: Cross Sections, Scattering and Rarefied Gases, Clarendon Press, Oxford, 1990. Google Scholar [38] E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flow, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.  Google Scholar [39] G. J. Prangsma, A. H. Alberga and J. J. M. Beenakker, Ultrasonic determination of the volume viscosity of N${}_2^{}$, CO, CH${}_4^{}$, and CD${}_4^{}$ between 77 and 300K, Physica, 64 (1973), 278-288. Google Scholar [40] D. Serre, Systèmes de Lois de Conservation I et II, Diderot Editeur, Art et Science, Paris, 1996.  Google Scholar [41] D. Serre, The Structure of Dissipative Viscous System of Conservation laws, Physica D, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.  Google Scholar [42] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.  Google Scholar [43] L. Tisza, Supersonic absorption and Stokes viscosity relation, Phys. Rev., 61 (1942), 531-536. doi: 10.1103/PhysRev.61.531.  Google Scholar [44] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.  Google Scholar [45] A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.), 87 (1972), 504-528.  Google Scholar [46] L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren molekülen, Zeitschr. Naturforschg., 17a (1962), 363-376.  Google Scholar [47] W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in the theory of shock waves, Progress in nonlinear differential equations and their applications, Birkhäuser Boston, 47 (2001), 259-305.  Google Scholar [48] W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Diff. Equ., 155 (1999), 89-132. doi: 10.1006/jdeq.1998.3584.  Google Scholar [49] W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws, Siam J. Appl. Math., 60 (2000), 1665-1675. doi: 10.1137/S0036139999352705.  Google Scholar [50] W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rat. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.  Google Scholar [51] W.-A. Yong, An interesting class of partial differential equations, J. Math. Physics, 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.  Google Scholar

show all references

##### References:
 [1] G. Billet, V. Giovangigli and G. de Gassowski, Impact of volume viscosity on a shock-hydrogen bubble interaction, Comb. Theory Mod., 12 (2008), 221-248. doi: 10.1080/13647830701545875.  Google Scholar [2] D. Bresch and B. Desjardins, On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids, J. Math. Pure Appl., 87 (2007), 57-90. doi: 10.1016/j.matpur.2006.11.001.  Google Scholar [3] D. Bruno and V. Giovangigli, Relaxation of internal temperature and volume viscosity, Phys. Fluids, 23 (2011), 093104. doi: 10.1063/1.3640083.  Google Scholar [4] D. Bruno and V. Giovangigli, Erratum: "Relaxation of internal temperature and volume viscosity'' [Phys. Fluids 23, 093104 (2011)], Phys. Fluids, 25 (2013), 039902. doi: 10.1063/1.4795334.  Google Scholar [5] D. Bruno, F. Esposito and V. Giovangigli, Relaxation of rotational-vibrational energy and volume viscosity in $H-H_2$ mixtures, J. Chem. Physics, 138 (2013), 084302. Google Scholar [6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.  Google Scholar [7] G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830. doi: 10.1002/cpa.3160470602.  Google Scholar [8] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar [9] G. Emanuel, Bulk viscosity of a dilute polyatomic gas, Phys. Fluids A, 2 (1990), 2252-2254. doi: 10.1063/1.857813.  Google Scholar [10] G. Emanuel, Effect of bulk viscosity on a hypersonic boundary layer, Phys. Fluids A, 4 (1992), 491-495. doi: 10.1063/1.858322.  Google Scholar [11] A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, 24, Springer-Verlag, Berlin, 1994.  Google Scholar [12] A. Ern and V. Giovangigli, Volume viscosity of dilute polyatomic gas mixtures, Eur. J. Mech. B/Fluids, 14 (1995), 653-669. Google Scholar [13] A. Ern and V. Giovangigli, Projected iterative algorithms with application to multicomponent transport, Lin. Alg. App., 250 (1997), 289-315. doi: 10.1016/0024-3795(95)00502-1.  Google Scholar [14] A. Ern and V. Giovangigli, The Kinetic equilibrium regime, Physica-A, 260 (1998), 49-72. Google Scholar [15] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.  Google Scholar [16] J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam, 1972. Google Scholar [17] K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686-1688. doi: 10.1073/pnas.68.8.1686.  Google Scholar [18] V. Giovangigli, Multicomponent Flow Modeling, Birkhaüser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.  Google Scholar [19] V. Giovangigli and M. Massot, Asymptotic stability of equilibrium states for multicomponent reactive flows, Math. Mod. Meth. App. Sci., 8 (1998), 251-297. doi: 10.1142/S0218202598000123.  Google Scholar [20] V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768. doi: 10.1002/mma.429.  Google Scholar [21] V. Giovangigli and L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Phys. D, 241 (2012), 649-670. doi: 10.1016/j.physd.2011.12.002.  Google Scholar [22] V. Giovangigli and L. Matuszewski, Mathematical modeling of supercritical multicomponent reactive fluids, Math. Mod. Meth. App. Sci., 23 (2013), 2193-2251. doi: 10.1142/S0218202513500309.  Google Scholar [23] V. Giovangigli and L. Matuszewski, Structure of entropies in dissipative multicomponent fluids, Kin. Rel. Mod., 6 (2013), 373-406. doi: 10.3934/krm.2013.6.373.  Google Scholar [24] V. Giovangigli and W.-A. Yong, Volume viscosity and fast internal energy relaxation: Convergence results,, (submitted for publication)., ().   Google Scholar [25] S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.  Google Scholar [26] R. E. Graves and B. Argrow, Bulk viscosity: Past to present, J. Therm. Heat Transfer, 13 (1999), 337-342. doi: 10.2514/2.6443.  Google Scholar [27] J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids, New-York, Wiley, 1954. Google Scholar [28] T. J. R. Hughes, L. P. Franca and M. Mallet, 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 (1986), 223-234. doi: 10.1016/0045-7825(86)90127-1.  Google Scholar [29] S. M. Karim and L. Rosenhead, The second coefficient of viscosity of Liquids and gases, Rev. Mod. Phys, 24 (1952), 108-116. doi: 10.1103/RevModPhys.24.108.  Google Scholar [30] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University 1984. Google Scholar [31] S. Kawashima, Large-time behavior of solutions to hyperbolic-parabolic systems of conservations laws and applications, Proc. Roy. Soc. Edinburgh, 106 (1987), 169-194. doi: 10.1017/S0308210500018308.  Google Scholar [32] S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws, Tôhoku Math. J., 40 (1988), 449-464. doi: 10.2748/tmj/1178227986.  Google Scholar [33] S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 174 (2004), 345-364. doi: 10.1007/s00205-004-0330-9.  Google Scholar [34] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-1054-2.  Google Scholar [35] C. Lattanzio and W.-A. Yong, Hyperbolic-parabolic singular limits for first-order nonlinear systems, Comm. Partial Diff. Equ., 26 (2001), 939-964. doi: 10.1081/PDE-100002384.  Google Scholar [36] T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Physics, 108 (1987), 153-175. doi: 10.1007/BF01210707.  Google Scholar [37] F. R. McCourt, J. J. Beenakker, W. E. Köhler and I. Kuscer, Non Equilibrium Phenomena in Polyatomic Gases, Volume I: Dilute Gases, Volume II: Cross Sections, Scattering and Rarefied Gases, Clarendon Press, Oxford, 1990. Google Scholar [38] E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flow, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.  Google Scholar [39] G. J. Prangsma, A. H. Alberga and J. J. M. Beenakker, Ultrasonic determination of the volume viscosity of N${}_2^{}$, CO, CH${}_4^{}$, and CD${}_4^{}$ between 77 and 300K, Physica, 64 (1973), 278-288. Google Scholar [40] D. Serre, Systèmes de Lois de Conservation I et II, Diderot Editeur, Art et Science, Paris, 1996.  Google Scholar [41] D. Serre, The Structure of Dissipative Viscous System of Conservation laws, Physica D, 239 (2010), 1381-1386. doi: 10.1016/j.physd.2009.03.014.  Google Scholar [42] Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.  Google Scholar [43] L. Tisza, Supersonic absorption and Stokes viscosity relation, Phys. Rev., 61 (1942), 531-536. doi: 10.1103/PhysRev.61.531.  Google Scholar [44] T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.  Google Scholar [45] A. I. Vol'pert and S. I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. (N.S.), 87 (1972), 504-528.  Google Scholar [46] L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren molekülen, Zeitschr. Naturforschg., 17a (1962), 363-376.  Google Scholar [47] W.-A. Yong, Basic aspects of hyperbolic relaxation systems, in Advances in the theory of shock waves, Progress in nonlinear differential equations and their applications, Birkhäuser Boston, 47 (2001), 259-305.  Google Scholar [48] W.-A. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Diff. Equ., 155 (1999), 89-132. doi: 10.1006/jdeq.1998.3584.  Google Scholar [49] W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws, Siam J. Appl. Math., 60 (2000), 1665-1675. doi: 10.1137/S0036139999352705.  Google Scholar [50] W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rat. Mech. Anal., 172 (2004), 247-266. doi: 10.1007/s00205-003-0304-3.  Google Scholar [51] W.-A. Yong, An interesting class of partial differential equations, J. Math. Physics, 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.  Google Scholar
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