Skip to main content
SpringerLink
Log in

A hyperbolic model for viscous Newtonian flows

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

Abstract

We discuss a pure hyperbolic alternative to the Navier–Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first-order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is interpreted as a system of material particles connected by bonds; the internal resistance to flow is interpreted as elastic stretching of the particle bonds; and a flow is a result of bond destructions and rearrangements of particles. Finally, we examine the model for simple shear flows, arbitrary incompressible and compressible flows of Newtonian fluids and demonstrate that Newton’s viscous law can be obtained in the framework of the developed hyperbolic theory as a steady-state limit. A basic relation between the viscosity coefficient, PSL time, and the shear sound velocity is also obtained.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Landau L.D., Lifshitz E.M.: Fluid Mechanics, Course of Theoretical Physics, vol. 6. Pergamon Press, New York (1966)

    Google Scholar 

  2. Godunov, S.K.: Elements of mechanics of continuous media. Nauka; (1978). (in Russian)

  3. Godunov S.K., Romenskii E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer/Plenum, Boston (2003)

    Book  MATH  Google Scholar 

  4. Frenkel J.: Kinetic Theory of Liquids. Dover, New York (1955)

    MATH  Google Scholar 

  5. Leonov A.I.: Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol. Acta 15(2), 85–98 (1976)

    Article  MATH  Google Scholar 

  6. Leonov A.I.: On a class of constitutive equations for viscoelastic liquids. J. Non-Newton. Fluid Mech. 25(1), 1–59 (1987)

    Article  MATH  Google Scholar 

  7. Tsinober A.: An Informal Conceptual Introduction to Turbulence. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  8. Yakhot, V., Chen, H.A., Staroselsky, I., Qian, Y., Shock, R., Kandasamy, S., et al.: A New Approach to Modelling Strongly Non-Equilibrium, Time-Dependent Turbulent Flow. Exa internal publication. Available from: http://www.exa.com/pdf/03_Yakhot_New_Approach.pdf (2001)

  9. Gomez H., Colominas I., Navarrina F., Paris J., Casteleiro M.: A hyperbolic theory for advection–diffusion problems: mathematical foundations and numerical modeling. Arch. Comput. Methods Eng. 17(2), 191–211 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jou D., Casas-Vazquez J., Lebon G.: Extended Irreversible Thermodynamics. 4th edn. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  11. Muller I., Ruggeri T.: Rational Extended Thermodynamics. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  12. Jordan P.M., Meyer M.R., Puri A.: Causal implications of viscous damping in compressible fluid flows. Phys. Rev. E 62(6), 7918 (2000)

    Article  ADS  Google Scholar 

  13. Romatschke P.: New developments in relativistic viscous hydrodynamics. Int. J. Mod. Phys. E 19(1), 1–53 (2010)

    Article  ADS  Google Scholar 

  14. Huovinen P., Molnar D.: Applicability of causal dissipative hydrodynamics to relativistic heavy ion collisions. Phys. Rev. C 79(1), 014906 (2009)

    Article  ADS  Google Scholar 

  15. Godunov, S.K.: Equations of Mathematical Physics. Nauka, Moscow (1979) (in Russian)

  16. Dubois F., Floch P.L.: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differ. Equ. 71(1), 93–122 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  18. Israel W., Stewart J.M.: Transient relativistic thermodynamics and kinetic theory. Ann. Phys. 118(2), 341–372 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  19. Bampi F., Morro A.: Viscous fluids with hidden variables and hyperbolic systems. Wave Motion 2(2), 153–157 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Geroch R.: Relativistic theories of dissipative fluids. J. Math. Phys. 36(8), 4226–4241 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Gomez H., Colominas I., Navarrina F., Casteleiro M.: A finite element formulation for a convection–diffusion equation based on Cattaneo’s law. Comput. Method Appl. Mech. Eng. 196(9), 1757–1766 (2007)

    Article  ADS  MATH  Google Scholar 

  22. Nishikawa H.: A first-order system approach for diffusion equation. II: Unification of advection and diffusion. J. Comput. Phys. 229(11), 3989–4016 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Montecinos G.I., Muller L.O., Toro E.F.: Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes. J. Comput. Phys. 266, 101–123 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157, 49–88 (1867)

  25. Cattaneo, C.: Sulla conduzione del calore. Atti. Semin. Mat. Fis. Univ. Modena. 3 (1948)

  26. Joseph D.D.: Fluid Dynamics of Viscoelastic Liquids. Vol. 84 of Applied Mathematical Sciences. Springer, Berlin (1990)

    Book  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Wilkins M.L.: Calculation of Elastic–Plastic Flow. California Univ. Livermore Radiation LAB, Livermore (1963)

    Google Scholar 

  29. Romenskii E.I.: Hypoelastic form of equations in nonlinear elasticity theory. J. Appl. Mech. Techn. Phys. 15(2), 255–259 (1974)

    Article  ADS  Google Scholar 

  30. Trangenstein J.A., Colella P.: A higher-order Godunov method for modelling finite deformation in elastic–plastic solids. Commun. Pure Appl. Math. XLIV, 41–100 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gavrilyuk S.L., Favrie N., Saurel R.: Modelling wave dynamics of compressible elastic materials. J. Comput. Phys. 227, 2941–2969 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Peshkov, I., Grmela, M., Romenski, E.: Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions. Continuum Mech. Thermodyn. (2014). doi:10.1007/s00161-014-0386-1

  33. Besseling, J.F.: A thermodynamic approach to rheology. In: Parkus, H., Sedov, L.I. (eds.) Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. IUTAM Symposia, pp. 16–53. Springer, Vienna (1968)

  34. Grmela, M., Ottinger, H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56(6), 6620 (1997)

  35. Ottinger H.C., Grmela M.: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E 56(6), 6633 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  36. Grmela M.: Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering. Adv. Chem. Eng. 39, 75–129 (2010)

    Article  Google Scholar 

  37. Godunov S.K., Romenskii E.I.: Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates. J. Appl. Mech. Tech. Phys. 13(6), 868–884 (1972)

    Article  ADS  Google Scholar 

  38. Eckart C.: The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Phys. Rev. 73(4), 373–382 (1948)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Rubin M.B.: An elastic–viscoplastic model exhibiting continuity of solid and fluid states. Int. J. Eng. Sci. 25(9), 1175–1191 (1987)

    Article  MATH  Google Scholar 

  40. Rubin M.B., Yarin A.L.: On the relationship between phenomenological models for elastic-viscoplastic metals and polymeric liquids. J. Non-Newton. Fluid Mech. 50(1), 79–88 (1993)

    Article  MATH  Google Scholar 

  41. Godunov S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk. SSSR 139(3), 521–523 (1961)

    MathSciNet  MATH  Google Scholar 

  42. Romensky E.I.: Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Math. Comput. Model. 28(10), 115–130 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Romensky, E.I.: Thermodynamics and hyperbolic systems of balance laws in continuum mechanics. In: Toro, E.F. (ed.) Godunov Methods, pp. 745–761. Springer, Berlin (2001)

  44. 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 

  45. Godunov, S.K., Romenskii, E.I.: Elements of mechanics of continuous media. Nauchnaya Kniga (1998) (in Russian)

  46. Miller G.H., Colella P.: A high-order Eulerian Godunov method for elastic–plastic flow in solids. J. Comput. Phys. 167(1), 131–176 (2001)

    Article  ADS  MATH  Google Scholar 

  47. Barton P.T., Drikakis D., Romenski E.I.: An Eulerian finite-volume scheme for large elastoplastic deformations in solids. Int. J. Numer. Method. Eng. 81(4), 453–484 (2010)

    MathSciNet  MATH  Google Scholar 

  48. Godunov S.K., Peshkov I.M.: Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium. Comput. Math. Math. Phys. 50(8), 1409–1426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Favrie N., Gavrilyuk S.L.: Diffuse interface model for compressible fluid–compressible elastic–plastic solid interaction. J. Comput. Phys. 231, 2695–2723 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Barton P., Romenski E.: On computational modelling of strain-hardening material dynamics. Commun. Comput. Phys. 11(5), 1525–1546 (2012)

    Google Scholar 

  51. Barton P.T., Deiterding R., Meiron D., Pullin D.: Eulerian adaptive finite-difference method for high-velocity impact and penetration problems. J. Comput. Phys. 240, 76–99 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  52. Kulikovskii A.G., Pogorelov N.V., Semenov A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems. CRC Press, Boca Raton (2000)

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  54. Godunov S.K., Demchuk A.F., Kozin N.S., Mali V.I.: Interpolation formulas for Maxwell viscosity of certain metals as a function of shear-strain intensity and temperature. J. Appl. Mech. Tech. Phys. 15(4), 526–529 (1974)

    Article  ADS  Google Scholar 

  55. Godunov S.K., Kozin N.S.: Shock structure in a viscoelastic medium with a nonlinear dependence of the Maxwellian viscosity on the parameters of the material. J. Appl. Mech. Tech. Phys. 15(5), 666–671 (1974)

    Article  ADS  Google Scholar 

  56. Godunov S.K., Denisenko V.V., Kozin N.S., Kuz’mina N.K.: Use of relaxation viscoelastic model in calculating uniaxial homogeneous strains and refining the interpolation equations for Maxwellian viscosity. J. Appl. Mech. Tech. Phys. 16(5), 811–814 (1975)

    Article  ADS  Google Scholar 

  57. Ndanou S., Favrie N., Gavrilyuk S.: Criterion of hyperbolicity in hyperelasticity in the case of the stored energy in separable form. J. Elast. 115(1), 1–25 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Joseph D.D., Renardy M., Saut J.C.: Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Ration. Mech. Anal. 87(3), 213–251 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  59. Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  60. Rutkevich I.M.: On the thermodynamic interpretation of the evolutionary conditions of the equations of the mechanics of finitely deformable viscoelastic media of maxwell type. J. Appl. Math. Mech. 36(2), 283–295 (1972)

    Article  Google Scholar 

  61. Dupret F., Marchal J.M.: Loss of evolution in the flow of viscoelastic fluids. J. Non-Newton. Fluid Mech. 20, 143–171 (1986)

    Article  MATH  Google Scholar 

  62. Trangenstein J.A., Colella P.: A higher-order Godunov method for modeling finite deformation in elastic-plastic solids. Commun. Pure Appl. Math. 44(1), 41–100 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  63. Radhakrishnan, K., Hindmarsh, A.C.: Description and use of LSODE, the Livermore solver for ordinary differential equations. National Aeronautics and Space Administration (1993)

  64. MATLAB and Statistics Toolbox Release. The MathWorks, Inc., Natick, Massachusetts, United States (2012)

  65. Sadovskii V.M., Sadovskaya O.V.: On the acoustic approximation of thermomechanical description of a liquid crystal. Phys. Mesomech. 16(4), 312–318 (2013)

    Article  Google Scholar 

  66. Shliomis, M.I.: Hydrodynamics of a liquid with intrinsic rotation. Sov. J. Exp. Theor. Phys. 24, 173–177 (1967). Available from: http://www.jetp.ac.ru/cgi-bin/dn/e_024_01_0173.pdf

  67. Berezin Y.A., Trofimov V.M.: A model of non-equilibrium turbulence with an asymmetric stress. Application to the problems of thermal convection. Contin. Mech. Thermodyn. 7(4), 415–437 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. Saurel R., Metayer O.L., Massoni J., Gavrilyuk S.: Shock jump relations for multiphase mixtures with stiff mechanical relaxation. Shock Waves 16(3), 209–232 (2007)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IUSTI UMR 7343, CNRS, Aix Marseille Université, Marseille, France

    Ilya Peshkov

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Evgeniy Romenski

  3. Novosibirsk State University, Novosibirsk, Russia

    Evgeniy Romenski

Authors
  1. Ilya Peshkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. 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 Victor Eremeyev, Peter Schiavone and Francesco dell’Isola.

I. Peshkov: 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., Romenski, E. A hyperbolic model for viscous Newtonian flows. Continuum Mech. Thermodyn. 28, 85–104 (2016). https://doi.org/10.1007/s00161-014-0401-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0401-6

Keywords

Search

Navigation