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A BGK Model for Gas Mixtures of Polyatomic Molecules Allowing for Slow and Fast Relaxation of the Temperatures

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Abstract

Kinetic models for polyatomic gases have two temperatures for the two different types of degrees of freedom, the translational and the internal energy degrees of freedom. Therefore, in the case of BGK models one expects two types of relaxations, a relaxation of the distribution function to a Maxwell distribution and a relaxation of the two temperatures to an equal value. The speed for the first type of relaxation may be faster or slower than the second type of relaxation. Models found in the literature often allow only for one of these two cases. We believe that a model should allow for both cases. That is why we derive a new multi-species polyatomic BGK model which allows for both regimes. For this new model we prove conservation properties, positivity of the temperatures, the H-theorem and characterize the equilibrium as a Maxwell distribution with equal temperatures. Moreover, we prove the convergence rate to equilibrium and that we can actually capture both regimes of the relaxation processes.

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References

  1. Andries, P., Le Tallec, P., Perlat, J., Perthame, B.: The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B 19, 813–830 (2000)

    Article  MathSciNet  Google Scholar 

  2. Bennoune, M., Lemou, M., Mieussens, L.: Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. J. Comput. Phys. 227, 3781–3803 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bernard, F., Iollo, A., Puppo, G.: Accurate asymptotic preserving boundary conditions for kinetic equations on Cartesian grids. J. Sci. Comput. 65, 735–766 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bernard, F., Iollo, A., Puppo, G.: Polyatomic Models for Rarefied Flows. Clarendon Press, Oxford (2017)

    MATH  Google Scholar 

  5. Bisi, M., Canizo, J.A., Lods, B.: Entropy dissipation estimates for the linear Boltzmann operator. Funct. Anal. 269(4), 1028–1069 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, Berlin (1975)

    MATH  Google Scholar 

  7. Cercignani, C.: Rarefied Gas Dynamics, From Basic Concepts to Actual Calculations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  8. Crestetto, A., Crouseilles, N., Lemou, M.: Kinetic/fluid micro–macro numerical schemes for Vlasov–Poisson–BGK equation using particles. Kinet. Relat. Models 5, 787–816 (2012)

    Article  MathSciNet  Google Scholar 

  9. Dimarco, G., Pareschi, L.: Numerical methods for kinetic equations. Acta Numer. 23, 369–520 (2014)

    Article  MathSciNet  Google Scholar 

  10. Filbet, F., Jin, S.: A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 20, 7625–7648 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  11. Klingenberg, C., Pirner, M., Puppo, G.: A consistent kinetic model for a two-component mixture with an application to plasma. Kinet. Relat. Models 10, 445–465 (2017)

    Article  MathSciNet  Google Scholar 

  12. Klingenberg, C., Pirner, M., Puppo, G.: A consistent kinetic model for a two-component mixture of polyatomic molecules (2017)

  13. Matthes, D.: Lecture Notes on the Course Entropy Methods and Related Functional. University of California, Berkeley (2006)

    Google Scholar 

  14. Park, S.J., Yun, S.-B.: Entropy production estimates for the polyatomic ellipsoidal BGK model. Appl. Math. Lett. 58, 26–33 (2016)

    Article  MathSciNet  Google Scholar 

  15. Park, S., Yun, S.: Entropy production estimates for the polyatomic ellipsoidal BGK model. Appl. Math. Lett. 58, 26–33 (2016)

    Article  MathSciNet  Google Scholar 

  16. Park, S., Yun, S.: Cauchy problem for the ellipsoidal BGK model for polyatomic particles (2017)

  17. Pieraccini, S., Puppo, G.: Implicit–explicit schemes for BGK kinetic equations. J. Sci. Comput. 32, 1–28 (2007)

    Article  MathSciNet  Google Scholar 

  18. Pirner, M.: Existence and uniqueness of mild solutions for a BGK model for gas mixtures of polyatomic molecules (2018)

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Acknowledgements

The author thanks Christian Klingenberg for helpful revisions that improved this paper. The author thanks him and Gabriella Puppo for many discussions on polyatomic modelling and multi-species kinetics.

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  1. University of Würzburg, Emil-Fischer-Str. 40, 97074, Würzburg, Germany

    Marlies Pirner

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  1. Marlies Pirner
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Correspondence to Marlies Pirner.

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Pirner, M. A BGK Model for Gas Mixtures of Polyatomic Molecules Allowing for Slow and Fast Relaxation of the Temperatures. J Stat Phys 173, 1660–1687 (2018). https://doi.org/10.1007/s10955-018-2158-y

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  • DOI: https://doi.org/10.1007/s10955-018-2158-y

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