Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T16:07:21.492Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  18 July 2011

S. H. Noskowicz  and
D. Serero
S. H. Noskowicz*
Affiliation:
School of Mechanical Engineering, Faculty of engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
D. Serero
Affiliation:
Institute for Multiscale Simulation, Universität Erlangen-Nürnberg, Nägelsbachstraße 49b, 91052 Erlangen, Germany
*
⋆Corresponding author. E-mail: [email protected] and [email protected]
Get access

Abstract

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

Keywords

Grad’s method of momentsgranular gasgenerating functioncomputer aided
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 4: Granular hydrodynamics , 2011 , pp. 151 - 174
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beylich, A. E.. Solving the kinetic equation for all Knudsen numbers. Phys. Fluids 12 (2000), 444–465. CrossRefGoogle Scholar
G. A. Bird. Molecular gas dynamics and the direct simulation theory of gas flows. Oxford University Press, 1994.
Bisi, M., Spiga, G., and Toscani, G.. Grad’s equations and hydrodynamics for weakly inelastic flows. Phys. Fluids 16 (2004), 4235–4247. CrossRefGoogle Scholar
Bobylev, A. V.. The Chapman-Enskog and Grad methods for solving the Boltzmann equation. Sov. Phys., dokl 27 (1982), 29–31. Google Scholar
Brey, J. J., Dufty, J.W., Kim, C. S. and Santos, A.. Hydrodynamics for granular flows at low density. Phys. Rev. E 58 (1997), 4638–4653. CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, W.-J, and Moreno., F.. Hydrodynamics of an open vibrated system. Phys. Rev. E 63 (2001), 061305. CrossRefGoogle ScholarPubMed
N.V. Briliantov and T. Pöschel. Kinetic theory of granular gases. Oxford University Press, Oxford, 2004.
Campbell, C. S.. Rapid granular flows. Annu. Rev. Fluid Mech. 22 (1990), 57–92. CrossRefGoogle Scholar
C. Cercignani. Theory and application of the Boltzmann equation. Scottish Acad. Press, Edinburgh and London, 1975.
S. Chapman and T. G. Cowling. The mathematical theory of nonuniform gases. Cambridge University Press, Cambridge, 1970.
García-Colin, L., Velasco, R. M., and Uribe, F. J.. Inconsistency in the moment’s method for solving the Boltzmann equation. J. Non-Equilib. Thermodyn. 29 (2004), 257–277. CrossRefGoogle Scholar
Garzó, V. and Dufty, J. W.. Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59 (1998), 5895–5911. CrossRefGoogle ScholarPubMed
Goldhirsch, I.. Rapid granular flows. Annu. Rev. Fluid Mech. 35 (2003), 267–293. CrossRefGoogle Scholar
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).
Goldshtein, A. and Shapiro, M.. Mechanics of collisional motion of granular materials, part 1: general hydrodynamic equations. J. Fluid Mech. 282 (1995), 75–114. CrossRefGoogle Scholar
Grad, H.. On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2 (1949), 331–407. CrossRefGoogle Scholar
Ivchenko, I. N., Loyalka, S. K., and Thompson, R.V.. The polynomial expansion method for boundary value problems of transport in rarefied gases. Z. angew. Math. Phys. 49 (1998), 955–966. CrossRefGoogle Scholar
Jenkins, J. T. and Richman, M. W.. Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rational. Mech. Anal. 28 (2001), 355–377. Google Scholar
M. N. Kogan. Rarefied gas dynamics. Plenum, New York, 1969.
Levermore, C. D. and Morokoff, W.J.. The gaussian moment closure for gas dynamics. SIAM J.App. Math. 59 (1998), 72–96. CrossRefGoogle Scholar
Mintzer, D.. Generalized orthogonal polynomial solutions of the Boltzmann equation. Phys. Fluids 8 (1965), 1076–1090. CrossRefGoogle Scholar
Nagai, R., Honma, H., Maeno, K., and Sakurai, A.. Shock wave solution of the Boltzmann kinetic equation in a 13-moment approximation. Shock Waves 13 (2003), 213–220. CrossRefGoogle Scholar
Noskowicz, S. H., Bar-Lev, O., Serero, D., and Goldhirsch, I.. Computer-aided kinetic theory and granular gases. Europhys. Lett. 79 (2007), 60001. CrossRefGoogle Scholar
Ohr, Y. G.. Improvement of the grad 13 moment method for strong shock waves. Phys. Fluids 13 (2001), 2105–2114. CrossRefGoogle Scholar
Ramirez, R., Risso, D., Soto, R., and Cordero, P.. Hydrodynamic theory for granular gases. Phys. Rev. E 62 (2000), 2521–2530. CrossRefGoogle ScholarPubMed
Rosenau, P., Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A 40 (1989), 7193–7196. CrossRefGoogle ScholarPubMed
Sela, N. and Goldhirsch, I.. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361 (1998), 41–74. CrossRefGoogle Scholar
Soto, R.. Granular systems on a vibrating wall: the kinetic boundary condition. Phys. Rev. E 69 (2004), 61305–61310. CrossRefGoogle ScholarPubMed
Struchtrup, H. and Torrilhon, M.. Regularization of Grad’s 13 momemt equations: derivation and linear analysis. Phys. Fluids 15 (2003), 2668–2680. CrossRefGoogle Scholar
Thatcher, T., Zheng, Y., and Struchtrup, H.. Boundary conditions for Grad’s 13 moment equations. Progress in Computational Fuid Dynamics 8 (2008), 69–83. CrossRefGoogle Scholar