[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

On the example of two-phase continua experiencing stress-induced solid–fluid phase transitions, we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type, and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Euler L.: Principes généraux du mouvement des fluides. Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémories. 11 (1755)

  2. de Groot S.R., Mazur P.: Non-Equilibrium Thermodynamics. Dover, New York (1984)

    Google Scholar 

  3. Godunov S.K.: An interesting class of quasilinear systems. Sov. Math. Dokl. 2, 947–949 (1961)

    MATH  Google Scholar 

  4. Friedrichs K.O., Lax P.D.: Systems of conservation laws with a convex extension. Proc. Nat. Acad. Sci. U.S.A. 68, 1686–1688 (1971)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Godunov S.K.: Symmetric form of the magnetohydrodynamic equation. Chislennye Metody Mekhaniki Sploshnoi Sredy 3(1), 26–34 (1972)

    Google Scholar 

  6. Friedrichs K.O.: Conservation equations and the laws of motion in classical physics. Commun. Pure Appl. Math. 31, 123–131 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  7. Godunov S.K., Romenski E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer/Plenum Publishers, New York (2003)

    Book  MATH  Google Scholar 

  8. Ruggeri T.: The entropy principle: from continuum mechanics to hyperbolic systems of balance laws. Bollettino dell’Unione Matematica Italiana 8(B(1)), 1–20 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Clebsch, A.: Über die Integration der hydrodynamische Gleichungen. J. Reine Angew. Math. 56, 1–10 (1895)

  10. Dzyaloshinski I.E., Volovick G.E.: Poisson brackets in condense matter physics. Ann. Phys. 125, 67–97 (1980)

    Article  ADS  Google Scholar 

  11. Grmela M.: Particle and bracket formulations of kinetic equations. Contemp. Math. 28, 125–132 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grmela M.: Bracket formulation of dissipative fluid mechanics equations. Phys. Lett. A. 102, 355 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  13. Kaufman A.N.: Dissipative Hamiltonian systems: a unifying principle. Phys. Lett. A. 100, 419–422 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  14. Morrison P.J.: Bracket formulation for irreversible classical fields. Phys. Lett. A. 100, 423–427 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  15. Beris A.N., Edwards B.J.: Thermodynamics of Flowing Systems. Oxford University Press, Oxford (1994)

    Google Scholar 

  16. Grmela M., Oettinger H.C.: Dynamics and thermodynamics of complex fluids I: general formulation. Phys. Rev. E 56, 6620–6633 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  17. Oettinger H.C., Grmela M.: Dynamics and thermodynamics of complex fluids II: illustration of the general folmalism. Phys. Rev. E. 56, 6633–6650 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  18. Grmela M.: Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering. Adv. Chem. Eng. 39, 75–128 (2010)

    Article  Google Scholar 

  19. Grmela M.: Role of thermodynamics in multiscale physics. Comput. Math. Appl. 65(10), 1457–1470 (2013)

    Article  MathSciNet  Google Scholar 

  20. Oettinger H.C.: Beyond Equilibrium Thermodynamics. Wiley, New York (2005)

    Book  Google Scholar 

  21. Putz A.M.V., Burghelea T.I.: The solid–fluid transition in a yield stress shear thinning physical gel. Rheol. Acta 48, 673–689 (2009)

    Article  Google Scholar 

  22. Balmforth N.J., Frigaard I.A., Ovarlez G.: Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  23. Friedrichs K.O.: Symmetric positive linear differential equations. Pure Appl. Math. 11, 333–418 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  24. Godunov, S.K., Romensky, E.: Thermodynamics, conservation laws and symmetric forms of differential equations in mechanics of continuous media. In: Hafez, M., Oshima, K. (eds.) Computational Fluid Dynamics Review, pp. 19–31. Wiley, New York (1995)

  25. Godunov S.K., Mikhailova T.Y., Romenski E.I.: Systems of thermodynamically coordinated laws of conservation invariant under rotations. Sib. Math. J. 37, 690–705 (1996)

    Article  MATH  Google Scholar 

  26. Romensky E.: Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Math. Comput. Model. 28, 115–130 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Romensky E.I.: Thermodynamics and hyperbolic systems of balance laws in continuum mechanics. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications, pp. 745–761. Kluwer/Plenum, New York (2001)

    Chapter  Google Scholar 

  28. Callen H.B.: Thermodynamics. Wiley, New York (1960)

    MATH  Google Scholar 

  29. Cattaneo C.: Sulla conduzione del calore. Atti del Seminario Matematico e Fisico della Universita di Modena 3, 83–101 (1948)

    MathSciNet  Google Scholar 

  30. Jou D., Casas-Vàzquez J., Lebon G.: Extended Irreversible Thermodynamics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  31. Gouin H., Gavrilyuk S.: Hamilton’s principle and Rankine–Hugoniot conditions for general motions of mixtures. Meccanica 34, 39–47 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang H., Calderer M.C.: Incipient dynamics of swelling of gels. SIAM J. Appl. Math. 68(6), 1641–1664 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Trusdell C.: Rational Thermodynamics. Springer, New York (1984)

    Book  Google Scholar 

  34. Romenski, E.: Conservative formulation for compressible fluid flow through elastic porous media. In: Vazquez-Cendon, E., Hidalgo, A., Garcia-Navarro, P., Cea, L. (eds.) Numerical Methods for Hyperbolic Equations, pp. 193–199. Taylor & Francis Group, London (2013)

  35. Gavrilyuk S.: Multiphase flow modeling via Hamilton’s principle. CISM Courses Lect. 535, 163–210 (2011)

    MathSciNet  Google Scholar 

  36. Favrie N., Gavrilyuk S.L., Saurel R.: Solid–fluid diffuse interface model in cases of extreme deformations. J. Comput. Phys. 228(16), 6037–6077 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. Marsden J. E., Ratiu T., Weinstein A.: Semidirect products and reduction in mechanics. Trans. Am. Math. Soc. 281, 147–177 (1984)

    Article  Google Scholar 

  38. Hamiltonian R.S.: Fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256 (1988)

    Article  Google Scholar 

  39. Miller G.H., Colella P.: A high-order Eulerian Godunov method for elastic-plastic flow in solids. J. Comput. Phys. 167, 137–176 (2001)

    Article  ADS  Google Scholar 

  40. Romenski E., Drikakis D., Toro E.: Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput. 42, 68–95 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  41. Godunov S.K., Romenskii E.I.: Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates. J. Appl. Mech. Tech. Phys. 13, 868–884 (1972)

    Article  ADS  Google Scholar 

  42. Godunov S.K.: Elements of Continuum Mechanics. Nauka, Moscow (1978)

    Google Scholar 

  43. Grmela M.: Fluctuation in extended mass-action-law dynamics. Phys. D 24, 976–986 (2012)

    Article  MathSciNet  Google Scholar 

  44. Romenski E., Toro E.F.: Compressible two-phase flows: two-pressure models and numerical methods. Comput. Fluid Dyn. 13, 403–416 (2004)

    Google Scholar 

  45. Steinmann P.: Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int. J. Eng. Sci. 34(15), 1717–1735 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  46. Powell K.G., Roe P.L., Linde T.J., Gombosi T.I., DeZeeuw D.L.: A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  47. Munz C.D., Omnes P., Schnaider R., Sonnendrücker E., Voss U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161, 484–511 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  48. Babii D.P., Godunov S.K., Zhukov V.T., Feodoritova O.B.: On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics. Comput. Math. Math. Phys. 47(3), 427–441 (2007)

    Article  MathSciNet  Google Scholar 

  49. Bouchbinder E., Langer J.S., Procaccia I.: Athermal shear-transformation-zone theory of amorphous plastic deformation. I. Basic principles. Phys. Rev. E. 75, 036107 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  50. Kosevich A.M.: Dynamical theory of dislocations. Physics-Uspekhi. 7(6), 837–854 (1965)

    Article  MathSciNet  Google Scholar 

  51. Kosevich, A.M.: Crystal dislocations and the theory of elasticity. In: Nabarro, F.R.N. Dislocations in Solids, pp. 33–141. North-Holland, Amsterdam (1979)

  52. Malygin G.A.: Dislocation self-organization processes and crystal plasticity. Physics-Uspekhi. 42(9), 887 (1999)

    Article  ADS  Google Scholar 

  53. Toro E.F.: Riemann Solvers and Numerical Methods in Fluid Dynamics. Springer, Berlin (1999)

    Book  Google Scholar 

  54. Godunov S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 47, 271–306 (1959)

    MathSciNet  Google Scholar 

  55. LeVeque R.J.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  56. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (1999)

  57. Moyers-Gonzales M., Burghelea T.I., Mak J.: Linear stability analysis for plane-Poiseuille flow of an elastoviscoplastic fluid with internal microstructure for large Reynolds numbers. J. Non-Newton. Fluid Mech. 166, 515–531 (2011)

    Article  Google Scholar 

  58. El Afif A., Grmela M.: Non-Fickian mass transport in polymers. J. Rheol. 46(3), 591–628 (2002)

    Article  ADS  Google Scholar 

  59. Godunov S.K., Peshkov I.M.: Thermodynamically consistent nonlinear model of an elastoplastic Maxwell medium. Comput. Math. Math. Phys. 50, 1409–1426 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Polytechnique de Montreal, Montreal, Canada

    Ilya Peshkov & Miroslav Grmela

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Evgeniy Romenski

Authors
  1. Ilya Peshkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miroslav Grmela
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Evgeniy Romenski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilya Peshkov.

Additional information

Communicated by Andreas Öchsner.

Ilya Peshkov is on leave from Sobolev Institute of Mathematics, Novosibirsk, Russia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peshkov, I., Grmela, M. & Romenski, E. Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions. Continuum Mech. Thermodyn. 27, 905–940 (2015). https://doi.org/10.1007/s00161-014-0386-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0386-1

Keywords

Search

Navigation