Application of the Method of Generating Functions to the Derivation of Grad’s N-Moment Equations for a Granular Gas

S. H. Noskowicz; D. Serero

Mathematical Modelling of Natural Phenomena (2011)

  • Volume: 6, Issue: 4, page 151-174
  • ISSN: 0973-5348

Abstract

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A computer aided method using symbolic computations that enables the calculation of the source terms (Boltzmann) in Grad’s method of moments is presented. The method is extremely powerful, easy to program and allows the derivation of balance equations to very high moments (limited only by computer resources). For sake of demonstration the method is applied to a simple case: the one-dimensional stationary granular gas under gravity. The method should find applications in the field of rarefied gases, as well. Questions of convergence, closure are beyond the scope of this article.

How to cite

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Noskowicz, S. H., and Serero, D.. "Application of the Method of Generating Functions to the Derivation of Grad’s N-Moment Equations for a Granular Gas." Mathematical Modelling of Natural Phenomena 6.4 (2011): 151-174. <http://eudml.org/doc/222186>.

@article{Noskowicz2011,
abstract = {A computer aided method using symbolic computations that enables the calculation of the source terms (Boltzmann) in Grad’s method of moments is presented. The method is extremely powerful, easy to program and allows the derivation of balance equations to very high moments (limited only by computer resources). For sake of demonstration the method is applied to a simple case: the one-dimensional stationary granular gas under gravity. The method should find applications in the field of rarefied gases, as well. Questions of convergence, closure are beyond the scope of this article.},
author = {Noskowicz, S. H., Serero, D.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Grad’s method of moments; granular gas; generating function; computer aided; Grad's method of moments},
language = {eng},
month = {7},
number = {4},
pages = {151-174},
publisher = {EDP Sciences},
title = {Application of the Method of Generating Functions to the Derivation of Grad’s N-Moment Equations for a Granular Gas},
url = {http://eudml.org/doc/222186},
volume = {6},
year = {2011},
}

TY - JOUR
AU - Noskowicz, S. H.
AU - Serero, D.
TI - Application of the Method of Generating Functions to the Derivation of Grad’s N-Moment Equations for a Granular Gas
JO - Mathematical Modelling of Natural Phenomena
DA - 2011/7//
PB - EDP Sciences
VL - 6
IS - 4
SP - 151
EP - 174
AB - A computer aided method using symbolic computations that enables the calculation of the source terms (Boltzmann) in Grad’s method of moments is presented. The method is extremely powerful, easy to program and allows the derivation of balance equations to very high moments (limited only by computer resources). For sake of demonstration the method is applied to a simple case: the one-dimensional stationary granular gas under gravity. The method should find applications in the field of rarefied gases, as well. Questions of convergence, closure are beyond the scope of this article.
LA - eng
KW - Grad’s method of moments; granular gas; generating function; computer aided; Grad's method of moments
UR - http://eudml.org/doc/222186
ER -

References

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  1. A. E. Beylich. Solving the kinetic equation for all Knudsen numbers. Phys. Fluids12 (2000), 444–465.  Zbl1149.76318
  2. G. A. Bird. Molecular gas dynamics and the direct simulation theory of gas flows. Oxford University Press, 1994.  
  3. M. Bisi, G. Spiga, and G. Toscani. Grad’s equations and hydrodynamics for weakly inelastic flows. Phys. Fluids16 (2004), 4235–4247.  Zbl1187.76049
  4. A. V. Bobylev. The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys., dokl27 (1982), 29–31.  
  5. J. J. Brey, J.W. Dufty, C. S. Kim and A. Santos. Hydrodynamics for granular flows at low density. Phys. Rev. E58 (1997), 4638–4653.  
  6. J. J. Brey, W.-J Ruiz-Montero, and F. Moreno.. Hydrodynamics of an open vibrated system. Phys. Rev. E63 (2001), 061305.  
  7. N.V. Briliantov and T. Pöschel. Kinetic theory of granular gases. Oxford University Press, Oxford, 2004.  
  8. C. S. Campbell. Rapid granular flows. Annu. Rev. Fluid Mech.22 (1990), 57–92.  
  9. C. Cercignani. Theory and application of the Boltzmann equation. Scottish Acad. Press, Edinburgh and London, 1975.  Zbl0403.76065
  10. S. Chapman and T. G. Cowling. The mathematical theory of nonuniform gases. Cambridge University Press, Cambridge, 1970.  Zbl0049.26102
  11. L. García-Colin, R. M. Velasco, and F. J. Uribe. Inconsistency in the moment’s method for solving the Boltzmann equation. J. Non-Equilib. Thermodyn.29 (2004), 257–277.  Zbl1111.82045
  12. V. Garzó and J. W. Dufty. Dense fluid transport for inelastic hard spheres. Phys. Rev. E59 (1998), 5895–5911.  
  13. I. Goldhirsch. Rapid granular flows. Annu. Rev. Fluid Mech.35 (2003), 267–293.  Zbl1125.76406
  14. S. H. Noskowicz, D. Serero, and O. Bar-Lev. Generating functions and kinetic theory: a computer aided method. Application: constitutive relations for granular gases up to moderate densities. in preparation (2011).  
  15. A. Goldshtein and M. Shapiro. Mechanics of collisional motion of granular materials, part 1: general hydrodynamic equations. J. Fluid Mech.282 (1995), 75–114.  Zbl0881.76010
  16. H. Grad. On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths2 (1949), 331–407.  Zbl0037.13104
  17. I. N. Ivchenko, S. K. Loyalka, and R.V. Thompson. The polynomial expansion method for boundary value problems of transport in rarefied gases. Z. angew. Math. Phys.49 (1998), 955–966.  Zbl0928.76093
  18. J. T. Jenkins and M. W. Richman. Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rational. Mech. Anal.28 (2001), 355–377.  Zbl0617.76085
  19. M. N. Kogan. Rarefied gas dynamics. Plenum, New York, 1969.  
  20. C. D. Levermore and W.J. Morokoff. The gaussian moment closure for gas dynamics. SIAM J.App. Math.59 (1998), 72–96.  Zbl0922.76273
  21. D. Mintzer. Generalized orthogonal polynomial solutions of the Boltzmann equation. Phys. Fluids8 (1965), 1076–1090.  Zbl0131.45106
  22. R. Nagai, H. Honma, K. Maeno, and A. Sakurai. Shock wave solution of the Boltzmann kinetic equation in a 13-moment approximation. Shock Waves13 (2003), 213–220.  Zbl1063.76087
  23. S. H. Noskowicz, O. Bar-Lev, D. Serero, and I. Goldhirsch. Computer-aided kinetic theory and granular gases. Europhys. Lett.79 (2007), 60001.  
  24. Y. G. Ohr. Improvement of the grad 13 moment method for strong shock waves. Phys. Fluids13 (2001), 2105–2114.  Zbl1184.76404
  25. R. Ramirez, D. Risso, R. Soto, and P. Cordero. Hydrodynamic theory for granular gases. Phys. Rev. E62 (2000), 2521–2530.  
  26. P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A40 (1989), 7193–7196.  
  27. N. Sela and I. Goldhirsch. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech.361 (1998), 41–74.  Zbl0927.76008
  28. R. Soto. Granular systems on a vibrating wall: the kinetic boundary condition. Phys. Rev. E69 (2004), 61305–61310.  
  29. H. Struchtrup and M. Torrilhon. Regularization of Grad’s 13 momemt equations: derivation and linear analysis. Phys. Fluids15 (2003), 2668–2680.  Zbl1186.76504
  30. T. Thatcher, Y. Zheng, and H. Struchtrup. Boundary conditions for Grad’s 13 moment equations. Progress in Computational Fuid Dynamics8 (2008), 69–83.  Zbl1173.76046

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