# 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 (2011)

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

## Access Full Article

top## Abstract

top## How to cite

topNoskowicz, 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

top- A. E. Beylich. Solving the kinetic equation for all Knudsen numbers. Phys. Fluids12 (2000), 444–465. Zbl1149.76318
- G. A. Bird. Molecular gas dynamics and the direct simulation theory of gas flows. Oxford University Press, 1994.
- M. Bisi, G. Spiga, and G. Toscani. Grad’s equations and hydrodynamics for weakly inelastic flows. Phys. Fluids16 (2004), 4235–4247. Zbl1187.76049
- A. V. Bobylev. The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys., dokl27 (1982), 29–31.
- 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.
- J. J. Brey, W.-J Ruiz-Montero, and F. Moreno.. Hydrodynamics of an open vibrated system. Phys. Rev. E63 (2001), 061305.
- N.V. Briliantov and T. Pöschel. Kinetic theory of granular gases. Oxford University Press, Oxford, 2004.
- C. S. Campbell. Rapid granular flows. Annu. Rev. Fluid Mech.22 (1990), 57–92.
- C. Cercignani. Theory and application of the Boltzmann equation. Scottish Acad. Press, Edinburgh and London, 1975. Zbl0403.76065
- S. Chapman and T. G. Cowling. The mathematical theory of nonuniform gases. Cambridge University Press, Cambridge, 1970. Zbl0049.26102
- 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
- V. Garzó and J. W. Dufty. Dense fluid transport for inelastic hard spheres. Phys. Rev. E59 (1998), 5895–5911.
- I. Goldhirsch. Rapid granular flows. Annu. Rev. Fluid Mech.35 (2003), 267–293. Zbl1125.76406
- 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).
- 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
- H. Grad. On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths2 (1949), 331–407. Zbl0037.13104
- 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
- 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
- M. N. Kogan. Rarefied gas dynamics. Plenum, New York, 1969.
- C. D. Levermore and W.J. Morokoff. The gaussian moment closure for gas dynamics. SIAM J.App. Math.59 (1998), 72–96. Zbl0922.76273
- D. Mintzer. Generalized orthogonal polynomial solutions of the Boltzmann equation. Phys. Fluids8 (1965), 1076–1090. Zbl0131.45106
- 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
- S. H. Noskowicz, O. Bar-Lev, D. Serero, and I. Goldhirsch. Computer-aided kinetic theory and granular gases. Europhys. Lett.79 (2007), 60001.
- Y. G. Ohr. Improvement of the grad 13 moment method for strong shock waves. Phys. Fluids13 (2001), 2105–2114. Zbl1184.76404
- R. Ramirez, D. Risso, R. Soto, and P. Cordero. Hydrodynamic theory for granular gases. Phys. Rev. E62 (2000), 2521–2530.
- P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A40 (1989), 7193–7196.
- 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
- R. Soto. Granular systems on a vibrating wall: the kinetic boundary condition. Phys. Rev. E69 (2004), 61305–61310.
- H. Struchtrup and M. Torrilhon. Regularization of Grad’s 13 momemt equations: derivation and linear analysis. Phys. Fluids15 (2003), 2668–2680. Zbl1186.76504
- 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

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.