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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 and 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.

[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.

[3]

D. Bruno and V. Giovangigli, Relaxation of internal temperature and volume viscosity, Phys. Fluids, 23 (2011), 093104. doi: 10.1063/1.3640083.

[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.

[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.

[6]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.

[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.

[8]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.

[9]

G. Emanuel, Bulk viscosity of a dilute polyatomic gas, Phys. Fluids A, 2 (1990), 2252-2254. doi: 10.1063/1.857813.

[10]

G. Emanuel, Effect of bulk viscosity on a hypersonic boundary layer, Phys. Fluids A, 4 (1992), 491-495. doi: 10.1063/1.858322.

[11]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, 24, Springer-Verlag, Berlin, 1994.

[12]

A. Ern and V. Giovangigli, Volume viscosity of dilute polyatomic gas mixtures, Eur. J. Mech. B/Fluids, 14 (1995), 653-669.

[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.

[14]

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

[15]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[16]

J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam, 1972.

[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.

[18]

V. Giovangigli, Multicomponent Flow Modeling, Birkhaüser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.

[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.

[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.

[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.

[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.

[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.

[24]

V. Giovangigli and W.-A. Yong, Volume viscosity and fast internal energy relaxation: Convergence results,, (submitted for publication)., (). 

[25]

S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.

[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.

[27]

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

[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.

[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.

[30]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University 1984.

[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.

[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.

[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.

[34]

J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-1054-2.

[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.

[36]

T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Physics, 108 (1987), 153-175. doi: 10.1007/BF01210707.

[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.

[38]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flow, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.

[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.

[40]

D. Serre, Systèmes de Lois de Conservation I et II, Diderot Editeur, Art et Science, Paris, 1996.

[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.

[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.

[43]

L. Tisza, Supersonic absorption and Stokes viscosity relation, Phys. Rev., 61 (1942), 531-536. doi: 10.1103/PhysRev.61.531.

[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.

[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.

[46]

L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren molekülen, Zeitschr. Naturforschg., 17a (1962), 363-376.

[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.

[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.

[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.

[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.

[51]

W.-A. Yong, An interesting class of partial differential equations, J. Math. Physics, 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.

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.

[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.

[3]

D. Bruno and V. Giovangigli, Relaxation of internal temperature and volume viscosity, Phys. Fluids, 23 (2011), 093104. doi: 10.1063/1.3640083.

[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.

[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.

[6]

S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1970.

[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.

[8]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.

[9]

G. Emanuel, Bulk viscosity of a dilute polyatomic gas, Phys. Fluids A, 2 (1990), 2252-2254. doi: 10.1063/1.857813.

[10]

G. Emanuel, Effect of bulk viscosity on a hypersonic boundary layer, Phys. Fluids A, 4 (1992), 491-495. doi: 10.1063/1.858322.

[11]

A. Ern and V. Giovangigli, Multicomponent Transport Algorithms, Lecture Notes in Physics Monographs, 24, Springer-Verlag, Berlin, 1994.

[12]

A. Ern and V. Giovangigli, Volume viscosity of dilute polyatomic gas mixtures, Eur. J. Mech. B/Fluids, 14 (1995), 653-669.

[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.

[14]

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

[15]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004.

[16]

J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, North Holland, Amsterdam, 1972.

[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.

[18]

V. Giovangigli, Multicomponent Flow Modeling, Birkhaüser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.

[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.

[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.

[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.

[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.

[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.

[24]

V. Giovangigli and W.-A. Yong, Volume viscosity and fast internal energy relaxation: Convergence results,, (submitted for publication)., (). 

[25]

S. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523.

[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.

[27]

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

[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.

[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.

[30]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral Thesis, Kyoto University 1984.

[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.

[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.

[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.

[34]

J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-1054-2.

[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.

[36]

T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Physics, 108 (1987), 153-175. doi: 10.1007/BF01210707.

[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.

[38]

E. Nagnibeda and E. Kustova, Non-Equilibrium Reacting Gas Flow, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01390-4.

[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.

[40]

D. Serre, Systèmes de Lois de Conservation I et II, Diderot Editeur, Art et Science, Paris, 1996.

[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.

[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.

[43]

L. Tisza, Supersonic absorption and Stokes viscosity relation, Phys. Rev., 61 (1942), 531-536. doi: 10.1103/PhysRev.61.531.

[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.

[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.

[46]

L. Waldmann and E. Trübenbacher, Formale kinetische Theorie von Gasgemischen aus anregbaren molekülen, Zeitschr. Naturforschg., 17a (1962), 363-376.

[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.

[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.

[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.

[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.

[51]

W.-A. Yong, An interesting class of partial differential equations, J. Math. Physics, 49 (2008), 033503, 21pp. doi: 10.1063/1.2884710.

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