June  2017, 10(2): 513-539. doi: 10.3934/krm.2017020

On modified simple reacting spheres kinetic model for chemically reactive gases

1. 

Department of Mathematics, California State University, Northridge, California 91330, USA

2. 

Centre of Mathematics, University of Minho, 4710-057 Braga, Portugal

* Corresponding author: Jacek Polewczak

Received  December 2015 Revised  May 2016 Published  November 2016

We consider the modified simple reacting spheres (MSRS) kinetic model that, in addition to the conservation of energy and momentum, also preserves the angular momentum in the collisional processes. In contrast to the line-of-center models or chemical reactive models considered in [23], in the MSRS (SRS) kinetic models, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the MSRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard spheres-like. We consider a four component mixture $A$, $B$, $A^*$, $B^*$, in which the chemical reactions are of the type $A+B\rightleftharpoons A^*+B^*$, with $A^*$ and $B^*$ being distinct species from $A$ and $B$. We provide fundamental physical and mathematical properties of the MSRS model, concerning the consistency of the model, the entropy inequality for the reactive system, the characterization of the equilibrium solutions, the macroscopic setting of the model and the spatially homogeneous evolution. Moreover, we show that the MSRS kinetic model reduces to the previously considered SRS model (e.g., [21], [27]) if the reduced masses of the reacting pairs are the same before and after collisions, and state in the Appendix the more important properties of the SRS system.

Citation: Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic & Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020
References:
[1]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.  Google Scholar

[2]

F. Carvalho, J. Polewczak and A. J. Soares, On the kinetic systems for simple reacting spheres: modeling and linearized equations, in From Particle Systems and Partial Differential Equations Ⅰ, Springer Proceedings in Mathematics & Statistics, 75 (2014), 251-267. doi: 10.1007/978-3-642-54271-8_12.  Google Scholar

[3]

F. Carvalho, J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres: an application to coloring processes, in From Particle Systems and Partial Differential Equations Ⅱ, Springer Proceedings in Mathematics & Statistics, 129 (2015), 153-172. doi: 10.1007/978-3-319-16637-7_4.  Google Scholar

[4] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.  Google Scholar
[5]

A. S. Cukrowski, The role of products and a reverse reaction in analysis of nonequilibrium effects in a bimolecular chemical reaction in a dilute gas, Physica A, 275 (2000), 134-151.  doi: 10.1016/S0378-4371(99)00410-0.  Google Scholar

[6]

J. S. Dahler and L. Qin, Nonequilibrium Statistical Mechanics of Chemically Reactive Fluids, J. Chem. Physics, 118 (2003), 8396-8404.  doi: 10.1063/1.1565331.  Google Scholar

[7]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

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A. Ern and V. Giovangigli, Kinetic theory of reactive gas mixtures with application to combustion, Transp. Theory Stat. Phys., 32 (2003), 657-677.  doi: 10.1081/TT-120025071.  Google Scholar

[9]

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

[10] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999.  doi: 10.1007/978-1-4612-1580-6.  Google Scholar
[11]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Physics, 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.  Google Scholar

[12]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.  Google Scholar

[14]

G. M. Kremer, M. P. Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow Phys. Fluids 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.  Google Scholar

[15]

G. M. KremerM. P. Bianchi and A. J. Soares, Analysis of the trend to equilibrium of a chemically reacting system, J. Phys. A: Math. Theor., 40 (2007), 2553-2571.  doi: 10.1088/1751-8113/40/10/020.  Google Scholar

[16]

J. Polewczak, The kinetic theory of simple reacting spheres: Ⅰ. Global existence result in a dilute-gas case, J. Stat. Physics, 100 (2000), 327-362.  doi: 10.1023/A:1018608216136.  Google Scholar

[17]

J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres Ⅰ, in 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove. AIP Conference Proceedings, 1333 (2011), 117-122. doi: 10.1063/1.3562635.  Google Scholar

[18]

R. D. Present, On the velocity distribution in a chemically reacting gas, J. Chem. Phys., 31 (1959), 747-750.  doi: 10.1063/1.1730456.  Google Scholar

[19]

I. Prigogine and M. Mahieu, Sur lapPerturbation de la distribution de Maxwell par des réactions chimiques en phase gazeuse, Physica, XVI (1950), 51-64.   Google Scholar

[20]

I. Prigogine and E. Xhrouet, On the perturbation of Maxwell distribution function by chemical reaction in gases, Physica, 15 (1949), 913-932.  doi: 10.1016/0031-8914(49)90057-9.  Google Scholar

[21]

L. Qin and J. S. Dahler, The kinetic theory of a simple, chemically reactive fluid: Scattering functions and relaxation processes, J. Chem. Physics, 103 (1995), 725-750.  doi: 10.1063/1.470106.  Google Scholar

[22]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28.  doi: 10.1063/1.1731889.  Google Scholar

[23]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[24]

B. Shizgal and M. Karplus, Nonequilibrium contributions to the rate of reaction. Ⅰ. Perturbation of the velocity distribution function, J. Chem. Phys., 52 (1969), 4262-4278.   Google Scholar

[25]

A. W. SilvaG. M. Alves and G. M. Kremer, Transport phenomena in a reactive quaternary gas mixture, Physica A, 374 (2007), 533-548.  doi: 10.1016/j.physa.2006.07.039.  Google Scholar

[26]

H. Van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.  doi: 10.1016/0031-8914(73)90372-8.  Google Scholar

[27]

N. Xystris and J. S. Dahler, Kinetic theory of simple reacting spheres, J. Chem. Phys., 68 (1978), 387-401.  doi: 10.1063/1.435772.  Google Scholar

show all references

References:
[1]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.  Google Scholar

[2]

F. Carvalho, J. Polewczak and A. J. Soares, On the kinetic systems for simple reacting spheres: modeling and linearized equations, in From Particle Systems and Partial Differential Equations Ⅰ, Springer Proceedings in Mathematics & Statistics, 75 (2014), 251-267. doi: 10.1007/978-3-642-54271-8_12.  Google Scholar

[3]

F. Carvalho, J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres: an application to coloring processes, in From Particle Systems and Partial Differential Equations Ⅱ, Springer Proceedings in Mathematics & Statistics, 129 (2015), 153-172. doi: 10.1007/978-3-319-16637-7_4.  Google Scholar

[4] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.  Google Scholar
[5]

A. S. Cukrowski, The role of products and a reverse reaction in analysis of nonequilibrium effects in a bimolecular chemical reaction in a dilute gas, Physica A, 275 (2000), 134-151.  doi: 10.1016/S0378-4371(99)00410-0.  Google Scholar

[6]

J. S. Dahler and L. Qin, Nonequilibrium Statistical Mechanics of Chemically Reactive Fluids, J. Chem. Physics, 118 (2003), 8396-8404.  doi: 10.1063/1.1565331.  Google Scholar

[7]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.  Google Scholar

[8]

A. Ern and V. Giovangigli, Kinetic theory of reactive gas mixtures with application to combustion, Transp. Theory Stat. Phys., 32 (2003), 657-677.  doi: 10.1081/TT-120025071.  Google Scholar

[9]

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

[10] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999.  doi: 10.1007/978-1-4612-1580-6.  Google Scholar
[11]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Physics, 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.  Google Scholar

[12]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.   Google Scholar

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.  Google Scholar

[14]

G. M. Kremer, M. P. Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow Phys. Fluids 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.  Google Scholar

[15]

G. M. KremerM. P. Bianchi and A. J. Soares, Analysis of the trend to equilibrium of a chemically reacting system, J. Phys. A: Math. Theor., 40 (2007), 2553-2571.  doi: 10.1088/1751-8113/40/10/020.  Google Scholar

[16]

J. Polewczak, The kinetic theory of simple reacting spheres: Ⅰ. Global existence result in a dilute-gas case, J. Stat. Physics, 100 (2000), 327-362.  doi: 10.1023/A:1018608216136.  Google Scholar

[17]

J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres Ⅰ, in 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove. AIP Conference Proceedings, 1333 (2011), 117-122. doi: 10.1063/1.3562635.  Google Scholar

[18]

R. D. Present, On the velocity distribution in a chemically reacting gas, J. Chem. Phys., 31 (1959), 747-750.  doi: 10.1063/1.1730456.  Google Scholar

[19]

I. Prigogine and M. Mahieu, Sur lapPerturbation de la distribution de Maxwell par des réactions chimiques en phase gazeuse, Physica, XVI (1950), 51-64.   Google Scholar

[20]

I. Prigogine and E. Xhrouet, On the perturbation of Maxwell distribution function by chemical reaction in gases, Physica, 15 (1949), 913-932.  doi: 10.1016/0031-8914(49)90057-9.  Google Scholar

[21]

L. Qin and J. S. Dahler, The kinetic theory of a simple, chemically reactive fluid: Scattering functions and relaxation processes, J. Chem. Physics, 103 (1995), 725-750.  doi: 10.1063/1.470106.  Google Scholar

[22]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28.  doi: 10.1063/1.1731889.  Google Scholar

[23]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.  Google Scholar

[24]

B. Shizgal and M. Karplus, Nonequilibrium contributions to the rate of reaction. Ⅰ. Perturbation of the velocity distribution function, J. Chem. Phys., 52 (1969), 4262-4278.   Google Scholar

[25]

A. W. SilvaG. M. Alves and G. M. Kremer, Transport phenomena in a reactive quaternary gas mixture, Physica A, 374 (2007), 533-548.  doi: 10.1016/j.physa.2006.07.039.  Google Scholar

[26]

H. Van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.  doi: 10.1016/0031-8914(73)90372-8.  Google Scholar

[27]

N. Xystris and J. S. Dahler, Kinetic theory of simple reacting spheres, J. Chem. Phys., 68 (1978), 387-401.  doi: 10.1063/1.435772.  Google Scholar

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