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From Extended Thermodynamics to Granular Materials

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Mathematical Models of Granular Matter

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1937))

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Taking into account the analogy with the kinetic approach of rarefied gases, we present a brief review of some recent results obtained in Rational Extended Thermodynamics, suggesting that this theory could be useful in modeling granular materials.

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References

  1. Haff P.K.: Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech., 134, 401 (1983)

    Article  MATH  Google Scholar 

  2. Capriz G.: Elementary preamble to a theory of granular gases. Rend. Sem. Mat. Univ. Padova, 110, 179–198 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Capriz G., Mullenger G.: Dynamics of granular fluids. Rend. Sem. Mat. Univ. Padova, 111, 247–264 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Capriz G.: Pseudofluids. In: Capriz G., Mariano P.M. (ed) Material substructures in complex bodies: from atomic level to continuum. Elsevier (2006) in print

    Google Scholar 

  5. Cercignani C., Illner R., Pulvirenti M.: The mathematical theory of dilute gases. Springer Series in Applied Mathematical Sciences, 106. Springer, Berlin (1994)

    MATH  Google Scholar 

  6. Benedetto D., Caglioti E., Pulvirenti M.: A kinetic equation for granular media. Model. Math. Anal. Numer., 31, 615 (1997)

    MATH  MathSciNet  Google Scholar 

  7. Bobylev A.V., Carrillo J.A., Gamba I.: On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Stat. Phys., 98, 743–773 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Müller I., Ruggeri T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy 37, 2nd ed. Springer, New York (1998)

    MATH  Google Scholar 

  9. Grad H.: On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2, 331–407 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jenkins J.T., Richman M.W.: Grad’s 13–moments system for a dense gas of inelastic spheres. Arch. Ration. Mech. Anal., 87, 355–377 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bisi M., Spiga G., Toscani G.: Grad’s equations and hydrodynamics for weakly inelastic granular flows. Phys. Fluids, 16, 12, 4235–4247 (2004)

    Article  MathSciNet  Google Scholar 

  12. Boillat G., Ruggeri T.: Moment Equations in the Kinetic Theory of Gases and Wave Velocities. Cont. Mech. Thermodyn. 9, 205–212 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Junk M.: Domain of definition of Levermores Five Moments System. J. Stat. Phys., 93, 1143–1167 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ruggeri T.: On the non-linear closure problem of moment equation. In: Ciancio V, Donato A, Oliveri F, Rionero S (eds) Proceedings WASCOM 99 10th Conference on Waves and Stability in Continuous Media, pp. 434–443. World Scientific (2001)

    Google Scholar 

  15. Brini F., Ruggeri T: Entropy Principle for the Moment Systems of degree alpha associated to the Boltzmann Equation. Critical Derivatives and Non Controllable Boundary Data. In coll. con F. Brini. Cont. Mech. Thermodyn. 14 (2), 165–189 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dreyer W.: Maximization of the entropy in non-equilibrium. J. Phys. A: Math. Gen., 20 (1987)

    Google Scholar 

  17. Levermore C.D.: Moment Closure Hierarchies for Kinetic Theories. J. Stat. Phys. 83, 1021 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kogan M.N.: in Proc. 5th Symp. on Rarefied Gas Dynamics, 1, suppl. 4, Academic Press, New York, 359–368 (1967)

    Google Scholar 

  19. Jaynes E.T.: Information Theory and Statistical Mechanics. Phys. Rev. 106, 620 (1957)

    Article  MathSciNet  Google Scholar 

  20. Kapur J.N.: Maximum entropy models in science and engineering. John Wiley, New York (1989)

    MATH  Google Scholar 

  21. Weiss W.: Zur Hierarchie der Erweiterten Thermodynamik. Dissertation, TU Berlin (1990)

    Google Scholar 

  22. Boillat G., Ruggeri T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rat. Mech. Anal., 137, 305–320 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  23. Boillat G., Ruggeri T.: Maximum Wave Velocity in the Moments System of a Relativistic Gas. In coll. con G. Boillat. Cont. Mech. Thermodyn. 11, pp. 107–111 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Boillat G., Ruggeri T.: Relativistic Gas: Moment Equations and Maximum Wave Velocity. In coll. con G. Boillat. J. Math. Phys., 40, No. 12, pp. 6399–6404 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shizuta Y., Kawashima S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J., 14, 249–275 (1985)

    MATH  MathSciNet  Google Scholar 

  26. Kawashima S.: Large-time behavior of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edimburgh, 106A, 169 (1987)

    MathSciNet  Google Scholar 

  27. Boillat G., Ruggeri T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn., 10, 285 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hanouzet B., Natalini R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Rat. Mech. Anal., 169, 89 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yong W.A.: Entropy and global existence for hyperbolic balance laws. Arch. Rat. Mech. Anal., 172, 247 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ruggeri T., Serre D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quarterly of Applied Math., 62, 163–179 (2004)

    MATH  MathSciNet  Google Scholar 

  31. Boillat G.: Sur l’existence et la recherche déquations de conservation supplémentaires pour les Systémes Hyperboliques. C. R. Acad. Sc. Paris, 278A, 909 (1974). CIME Course, Recent Mathematical Methods in Nonlinear Wave Propagation. In: Ruggeri T. (ed) Lecture Notes in Mathematics, 1640, 103–152, Springer (1995)

    Google Scholar 

  32. Ruggeri T., Strumia A.: Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré, 34A 65–84 (1981)

    MathSciNet  Google Scholar 

  33. Ruggeri T.: Global existance of smooth solutions and stability of the constant state for dissipative hyperbolic systems with applications to extended thermodynamics, in Trends and Applications of Mathematics to Mechanics, STAMM 2002. p. 215, Springer-Verlag (2005)

    Google Scholar 

  34. Ruggeri T.: Some Recent Mathematical Results in Mixtures Theory of Euler Fluids, in Proceedings WASCOM 2003, R. Monaco, S. Pennisi, S. Rionero and T. Ruggeri (Eds.), p. 441 World Scientific-Singapore

    Google Scholar 

  35. Ruggeri T.: Global Existence, Stability and Non Linear Wave Propagation in Binary Mixtures of Euler Fluids, in New Trends in Mathematical Physics. Convegno in onore di S. Rionero, Napoli 2003. p. 205 World Scientific (2005)

    Google Scholar 

  36. Zeng Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal. 150, no. 3, 225 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lou J., Ruggeri T.: Acceleration waves and weaker Shizuta-Kawashima condition. Rendiconti del Circolo Matematico di Palermo, Serie II, 78, 305 (2006)

    Google Scholar 

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Ruggeri, T. (2008). From Extended Thermodynamics to Granular Materials. In: Capriz, G., Mariano, P.M., Giovine, P. (eds) Mathematical Models of Granular Matter. Lecture Notes in Mathematics, vol 1937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78277-3_5

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