Non-Newtonian and Viscoelastic Fluids

  • Michel O. Deville
  • Thomas B. Gatski


The theory of non-Newtonian and viscoelastic fluids flourished in the second half of last century with the developments of (molten and dilute) polymers and the growth of materials science and engineering that generated many new products and applications. The mathematical setting of the constitutive equations required new tools from tensor analysis and algebra that have already been exposed in previous chapters. Here the concentration will be on the various constitutive relations that became dominant over time because of their generality and/or their physical relevance. Despite the major effort carried out by numericists over the last four decades, numerical simulations at high Weissenberg number values (and zero Reynolds number) are not always feasible and this poses deep questions and concerns about getting the appropriate models for those fluids. However the log-formulation introduced by Fattal and Kupferman (J Non-Newton Fluid Mech 123:281–285, 2004) has eased that difficulty. Nonetheless in problems where experimental data are available, numerical results based on non-Newtonian models may sometimes be in error by an order of magnitude. This situation is detrimental in the long run to engineering and should be tackled by all means: theory, experiments and computations.


Normal Stress Difference Orientation Tensor Dumbbell Model Brownian Force Giesekus Model 
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.


  1. Barnes HA, Hutton JF, Walters K (1987) An introduction to rheology. Wiley, New York Google Scholar
  2. Beris AN, Edwards BJ (1994) Thermodynamics of flowing systems with internal microstructure. Oxford University Press, New York Google Scholar
  3. Bingham EC (1916) An investigation of the laws of plastic flow. US Bur Stand Bull 13:309–353 Google Scholar
  4. Bird RB, Armstrong RC, Hassager O (1987a) Dynamics of polymeric liquids, vol 1: Fluid mechanics. Wiley, New York Google Scholar
  5. Bird RB, Curtiss CF, Armstrong RC, Hassager O (1987b) Dynamics of polymeric liquids, vol 2: Kinetic theory. Wiley, New York Google Scholar
  6. Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 16:99–127 CrossRefGoogle Scholar
  7. Chandrasekhar S (1943) Stochastic problems in physics and astronomy. Rev Mod Phys 15:1–89 CrossRefzbMATHMathSciNetGoogle Scholar
  8. Chilcott MD, Rallison JM (1988) Creeping flow of dilute polymer solutions past cylinders and spheres. J Non-Newton Fluid Mech 29:381–432 CrossRefzbMATHGoogle Scholar
  9. Coleman BD, Noll JM (1960) An approximation theorem for functionals, with applications in continuum mechanics. Arch Ration Mech Anal 6:355–370 CrossRefzbMATHMathSciNetGoogle Scholar
  10. Cross MM (1965) Rheology of non-Newtonian fluids: a new flow equation for pseudo-plastic systems. J Colloid Sci 20:417–437 CrossRefGoogle Scholar
  11. de Gennes PG (1971) Reptation of a polymer chain in the presence of fixed obstacles. J Chem Phys 55:572–579 CrossRefGoogle Scholar
  12. Doi M, Edwards SF (1978a) Dynamics of concentrated polymer systems, part 1. Brownian motion in the equilibrium state. J Chem Soc Faraday Trans 74:1789–1801 CrossRefGoogle Scholar
  13. Doi M, Edwards SF (1978b) Dynamics of concentrated polymer systems, part 2. Molecular motion under flow. J Chem Soc Faraday Trans 74:1802–1817 CrossRefGoogle Scholar
  14. Doi M, Edwards SF (1978c) Dynamics of concentrated polymer systems, part 3. The constitutive equation. J Chem Soc Faraday Trans 74:1818–1832 CrossRefGoogle Scholar
  15. Doi M, Edwards SF (1979) Dynamics of concentrated polymer systems, part 4. Rheological properties. J Chem Soc Faraday Trans 75:38–54 CrossRefGoogle Scholar
  16. Doi M, Edwards SF (1988) The theory of polymer dynamics. Oxford University Press, Oxford Google Scholar
  17. Ericksen JL (1960) Transversely isotropic fluids. Kolloid Z 173:117–122 CrossRefGoogle Scholar
  18. Fattal R, Kupferman R (2004) Constitutive laws for the matrix-logarithm formulation of the conformation tensor. J Non-Newton Fluid Mech 123:281–285 CrossRefzbMATHGoogle Scholar
  19. Gatski TB, Speziale CG (1993) On algebraic stress models for complex turbulent flows. J Fluid Mech 254:59–78 CrossRefzbMATHMathSciNetGoogle Scholar
  20. Giesekus H (1966) Die elastizität von flüssgkeiten. Rheol Acta 5:29–35 CrossRefGoogle Scholar
  21. Giesekus H (1982) A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J Non-Newton Fluid Mech 11:69–109 CrossRefzbMATHGoogle Scholar
  22. Gordon RJ, Schowalter WR (1972) Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions. Trans Soc Rheol 16:79–97 CrossRefzbMATHGoogle Scholar
  23. Hasimoto H, Sano O (1980) Stokeslet and eddies in creeping flow. Annu Rev Fluid Mech 12:335–363 CrossRefMathSciNetGoogle Scholar
  24. Herschel WH, Bulkley R (1926) Konsistenzmessungen von gummi-benzollösungen. Kolloid Z 39:291–300 CrossRefGoogle Scholar
  25. Hulsen MA, van Heel APG, van den Brule BHAA (1997) Simulation of viscoelastic flows using Brownian configuration fields. J Non-Newton Fluid Mech 70:79–101 CrossRefGoogle Scholar
  26. Ianniruberto G, Marrucci G (1996) On compatibility of the Cox–Merz rule with the model of Doi and Edwards. J Non-Newton Fluid Mech 65:241–246 CrossRefGoogle Scholar
  27. Ianniruberto G, Marrucci G (2000) Convective orientational renewal in entangled polymers. J Non-Newton Fluid Mech 95:363–374 CrossRefzbMATHGoogle Scholar
  28. Ianniruberto G, Marrucci G (2001) A simple constitutive equation for entangled polymers with chain stretch. J Rheol 45:1305–1318 CrossRefGoogle Scholar
  29. Itô K (1951) On stochastic differential equations. Mem Am Math Soc 4:1–51 Google Scholar
  30. James DF (2009) Boger fluids. In: Annual review of fluid mechanics. Annual Reviews, New York, pp 129–142 Google Scholar
  31. Jeffreys H (1976) The earth: its origin, history and physical constitution, 6th edn. Cambridge University Press, Cambridge Google Scholar
  32. Jongen T, Gatski TB (1999) A unified analysis of planar homogeneous turbulence using single-point closure equations. J Fluid Mech 399:117–150 CrossRefzbMATHGoogle Scholar
  33. Jongen T, Gatski TB (2005) Tensor representations and solutions of constitutive equations for viscoelastic fluids. Int J Eng Sci 43:556–588 CrossRefzbMATHMathSciNetGoogle Scholar
  34. Joseph DD (1990) Fluid dynamics of viscoelastic liquids. Springer, Berlin zbMATHGoogle Scholar
  35. Keunings R (2004) Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory. In: Rheology reviews 2004. British Society of Rheology, London, pp 67–98 Google Scholar
  36. Larson RG (1988) Constitutive equations for polymer melts and solutions. Butterworth, Boston Google Scholar
  37. Larson RJ (1983) Convection and diffusion of polymer network strands. J Non-Newton Fluid Mech 13:279–308 CrossRefzbMATHGoogle Scholar
  38. Likhtman AE, Graham RS (2003) Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. J Non-Newton Fluid Mech 114:1–12 CrossRefzbMATHGoogle Scholar
  39. Lozinski A, Owens RG, Phillips TN (2011) The Langevin and Fokker-Planck equations in polymer rheology. In: Ciarlet PG, Glowinski R, Xu J (eds) Numerical methods for non-Newtonian fluids. Handbook of numerical analysis. Elsevier, Amsterdam, pp 211–303 CrossRefGoogle Scholar
  40. Marrucci G, Greco F, Ianniruberto G (2001) Integral and differential constitutive equations for entangled polymers with simple versions of CCR and force balance on entanglements. Rheol Acta 40:98–103 CrossRefGoogle Scholar
  41. McLeish TCB, Larson RG (1998) Molecular constitutive equations for a class of branched polymers: the pom-pom polymer. J Rheol 42:81–110 CrossRefGoogle Scholar
  42. Milner ST, McLeish TCB, Likhtman AE (2001) Microscopic theory of convective constraint release. J Rheol 45:539–563 CrossRefGoogle Scholar
  43. Mompean G (2002) On predicting abrupt contraction flows with differential and algebraic viscoelastic models. Comput Fluids 31:935–956 CrossRefzbMATHGoogle Scholar
  44. Mompean G, Thais L (2007) Assessment of a general equilibrium assumption for development of algebraic viscoelastic models. J Non-Newton Fluid Mech 145:41–51 CrossRefzbMATHGoogle Scholar
  45. Mompean G, Jongen T, Deville MO, Gatski TB (1998) On algebraic extra-stress models for the simulation of viscoelastic flows. J Non-Newton Fluid Mech 79:261–281 CrossRefzbMATHGoogle Scholar
  46. Mompean G, Thompson RL, Souza Mendes PR (2003) A general transformation procedure for differential viscoelastic models. J Non-Newton Fluid Mech 111:151–174 CrossRefzbMATHGoogle Scholar
  47. Mompean G, Thais L, Tomé MF, Castelo A (2011) Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models. Comput Fluids 44:68–78 CrossRefGoogle Scholar
  48. Nelson E (2001) Dynamical theories of brownian motion. Princeton University Press, Princeton. Google Scholar
  49. Öttinger H-C (1996) Stochastic processes in polymeric fluids: tools and examples for developing simulation algorithms. Springer, Berlin zbMATHGoogle Scholar
  50. Owens RG, Phillips TN (2002) Computational rheology. Imperial College Press, London CrossRefzbMATHGoogle Scholar
  51. Peterlin A (1966) Hydrodynamics of macromolecules in a velocity field with longitudinal gradient. J Polym Sci, Polym Phys 4:287–291 Google Scholar
  52. Peters GWM, Baaijens FPT (1997) Modelling of non-isothermal viscoelastic flows. J Non-Newton Fluid Mech 68:205–224 CrossRefGoogle Scholar
  53. Phan-Thien N (2002) Understanding viscoelasticity basics of rheology. Springer, Berlin zbMATHGoogle Scholar
  54. Phan-Thien N, Tanner RI (1977) A new constitutive equation derived from network theory. J Non-Newton Fluid Mech 2:353–365 CrossRefGoogle Scholar
  55. Pope SB (1975) A more general effective viscosity hypothesis. J Fluid Mech 72:331–340 CrossRefzbMATHGoogle Scholar
  56. Rubio P, Wagner MH (1999) Letter to the editor: A note added to Molecular constitutive equations or a class of branched polymers: the pom-pom polymer (J. Rheol., 42:81–110, 1998). J Rheol 43:1709–1710 CrossRefGoogle Scholar
  57. Tanner RI (1989) Engineering rheology. Clarendon, Oxford Google Scholar
  58. Trouton FT (1906) On the coefficient of viscous traction and its relation to that of viscosity. Proc R Soc Lond Ser A 77:426–440 Google Scholar
  59. Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids. Birkhäuser, Boston CrossRefzbMATHGoogle Scholar
  60. Verbeeten WMH, Peters GWM, Baaijens FPT (2001) Differential constitutive equations for polymer melts: the extended pom-pom model. J Rheol 45:823–843 CrossRefGoogle Scholar
  61. Wang C-C (1965) A representation theorem for the constitutive equation of a simple material in motions with constant stretch history. Arch Ration Mech Anal 20:329–340 zbMATHGoogle Scholar
  62. Wang C-C (1970) A new representation theorem for isotropic functions. Arch Ration Mech Anal 36:166–223 CrossRefzbMATHGoogle Scholar
  63. Warner HR (1972) Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind Eng Chem Fundam 11:379–387 CrossRefMathSciNetGoogle Scholar
  64. Wilson HJ (2011) Graduate lectures on polymeric fluids. University College of London, London. Google Scholar
  65. Yamamoto M (1956) Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions. J Phys Soc Jpn 11:413–421 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Mechanical EngineeringSwiss Federal Institute of Technology, EPFLLausanneSwitzerland
  2. 2.Institute PPRIMECNRS-Université de Poitiers-ENSMAFuturoscope ChasseneuilFrance
  3. 3.Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric SciencesOld Dominion UniversityNorfolkUSA

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