# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003 [2] Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004 [3] Yusheng Jia, Weishi Liu, Mingji Zhang. Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022 [4] Marzia Bisi, Maria Groppi, Giampiero Spiga. Flame structure from a kinetic model for chemical reactions. Kinetic & Related Models, 2010, 3 (1) : 17-34. doi: 10.3934/krm.2010.3.17 [5] Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153 [6] Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga, Irina F. Potapenko. A general consistent BGK model for gas mixtures. Kinetic & Related Models, 2018, 11 (6) : 1377-1393. doi: 10.3934/krm.2018054 [7] Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji Zhang. Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064 [8] Mohamed Tij, Andrés Santos. Non-Newtonian Couette-Poiseuille flow of a dilute gas. Kinetic & Related Models, 2011, 4 (1) : 361-384. doi: 10.3934/krm.2011.4.361 [9] Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185 [10] Julien Coatléven, Claudio Altafini. A kinetic mechanism inducing oscillations in simple chemical reactions networks. Mathematical Biosciences & Engineering, 2010, 7 (2) : 301-312. doi: 10.3934/mbe.2010.7.301 [11] Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037 [12] Yuanchang Sun, Lisa M. Wingen, Barbara J. Finlayson-Pitts, Jack Xin. A semi-blind source separation method for differential optical absorption spectroscopy of atmospheric gas mixtures. Inverse Problems & Imaging, 2014, 8 (2) : 587-610. doi: 10.3934/ipi.2014.8.587 [13] Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic & Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255 [14] Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459 [15] C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872 [16] Congming Li, Eric S. Wright. Modeling chemical reactions in rivers: A three component reaction. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 377-384. doi: 10.3934/dcds.2001.7.373 [17] Arno F. Münster. Simulation of stationary chemical patterns and waves in ionic reactions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 35-46. doi: 10.3934/dcdsb.2002.2.35 [18] Thibaut Allemand. Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinetic & Related Models, 2009, 2 (2) : 379-402. doi: 10.3934/krm.2009.2.379 [19] Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks & Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 [20] Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks & Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295

2018 Impact Factor: 1.38

## Metrics

• HTML views (2)
• Cited by (1)

• on AIMS