On the Mathematical Modelling of a Compressible Viscoelastic Fluid

  • P. C. Bollada
  • T. N. PhillipsEmail author


Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.


Poisson Bracket Bulk Viscosity Cauchy Stress Maxwell Model Augmented Pressure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Belblidia F., Keshtiban I.J., Webster M.F.: Stabilised computations for viscoelastic flows under compressible implementations. J. Non-Newtonian Fluid Mech. 134, 56–76 (2006)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beris A.N., Edwards B.J.: Thermodynamics of Flowing Systems. Oxford University Press, New York (1994)Google Scholar
  3. 3.
    Bollada P., Phillips T.N.: On the effects of a compressible viscous lubricant on the load-bearing capacity of a journal bearing. Int. J. Numer. Meth. Fluids 55, 1091–1120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dressler M., Edwards B.J., Öttinger H.C.: Macroscopic thermodynamics of flowing polymeric liquids. Rheologica Acta 38(2), 117–136 (1999)CrossRefGoogle Scholar
  5. 5.
    Edwards B.J., Beris A.N.: Remarks concerning compressible viscoelastic fluid models. J. Non-Newtonian Fluid Mech. 36, 411–417 (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Grmela M.: Bracket formulation of dissipative fluid mechanics equations. Phys. Lett. A 102, 355–358 (1984)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Grmela M., Öttinger H.-C.: Dynamics and thermodynamics of complex fluids I. Development of the GENERIC formalism. Phys. Rev. E 56, 620–6632 (1997)Google Scholar
  8. 8.
    Kaufman A.N.: Dissipative Hamiltonian systems: a unifying principle. Phys. Lett. A 100, 419–422 (1984)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Keshtiban I.J., Belblidia F., Webster M.F.: Computation of incompressible and weakly-compressible viscoelastic liquids flow: finite element/volume schemes. J. Non-Newtonian Fluid Mech. 126, 123–143 (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    Marsden J.E., Hughes T.J.R.: Mathematical Foundations of Elasticity. Dover Publications, New York (1994)Google Scholar
  11. 11.
    Morrison P.J.: Bracket formulation for irreversible classical fields. Phys. Lett. A 100, 423–427 (1984)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Oldroyd J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Oliveira P.J.: A traceless stress tensor formulation for viscoelastic fluid flow. J. Non-Newtonian Fluid Mech. 95, 55–65 (2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    Öttinger H.-C., Grmela M.: Dynamics and thermodynamics of complex fluids II. Illustrations of the GENERIC formalism. Phys. Rev. E 56, 6633–6655 (1997)Google Scholar
  15. 15.
    Owens R.G., Phillips T.N.: Computational Rheology. Imperial College Press, London (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    Phan-Thien N.: Understanding Viscoelasticity. Springer, Berlin (2002)zbMATHGoogle Scholar
  17. 17.
    Temam R., Miranville A.: Mathematical Modelling in Continuum Mechanics. Cambridge University Press, Cambridge (2000)Google Scholar
  18. 18.
    Truesdell C.: Continuum Mechanics I: The Mechanical Foundations of Elasticity and Fluid Systems. International Science Review Series. Gordon and Breach, Inc, New York (1969)Google Scholar
  19. 19.
    Webster M.F., Keshtiban I.J., Belblidia F.: Computation of weakly-compressible highly viscous liquid flows. Eng. Comput. 21, 777–804 (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    White J.L., Metzner A.B.: Constitutive equations for viscoelastic fluids with application to rapid external flows.. AIChE J. 11(2), 324–330 (1965)CrossRefGoogle Scholar
  21. 21.
    Wilmanski K.: Continuum Thermodynamics. Series on Advances in Mathematics for Applied Sciences, vol 77. World Scientific (2009)Google Scholar
  22. 22.
    Woods, L.C.: The Thermodynamics of Fluid Systems. Oxford Engineering, Science Series 2. Oxford University Press, 1986Google Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK

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