Rheologica Acta

, Volume 49, Issue 6, pp 543–554

Polymeric liquids in extension: fluid mechanics or rheometry?

  • Ole Hassager
  • Jose Manuel Roman Marin
  • Kaijia Yu
  • Henrik Koblitz Rasmussen
Original Contribution


We use a transient 3D free surface finite element method to simulate flow of entangled polymer fluids in the dual cylinder wind-up extensional rheometer. The constitutive equations are K-BKZ integral representations of the Doi–Edwards models with and without the independent alignment approximation (IA). It is demonstrated that the actual kinematics in this rheometer is a mixture of planar and uniaxial extension. Moreover, the ratio of planar to uniaxial deformation is highly dependent upon whether IA is invoked. Without IA, the flow has a tendency toward planar extension, while it tends to be more uniaxial with IA invoked. As a second illustration of the techniques, we simulate the phenomenon of delayed rupture after rapid extension of entangled polymer systems. It is demonstrated that this phenomenon can be explained on the basis of the Doi–Edwards model in terms of a Considere-type instability after chain stretch relaxation.


Extensional flow Rotational rheometer Delayed rupture K-BKZ FEM 


  1. Anna SL, McKinley GH, Nguyen DA, Sridhar T (2001) An interlaboratory comparison of measurements from filament-stretching rheometers using common test fluids. J Rheol 45:83–114CrossRefADSGoogle Scholar
  2. Bach A, Rasmussen HK, Longin P-Y, Hassager O (2002) Growth of non-axisymmetric disturbances of the free surface in the filament stretching rheometer: experiments and simulation. J Non-Newton Fluid Mech 108:163–186CrossRefMATHGoogle Scholar
  3. Bach A, Almdal K, Rasmussen HK, Hassager O (2003) Elongational viscosity of narrow molar mass distribution polystyrene. Macromolecules 36:5174–5179CrossRefADSGoogle Scholar
  4. Baumgaertel M, Schausberger A, Winter H (1990) The relaxation of polymers with linear flexible chains of uniform length. Rheol Acta 29:400–408CrossRefGoogle Scholar
  5. Bent J, Hutchings LR, Richards RW, Gough T, Spares R, Coates PD, Grillo I, Harlen OG, Read DJ, Graham RS, Likhtman AE, Groves DJ, Nicholson TM, McLeish TCB (2003) Neutron-mapping polymer flow: scattering, flow visualization, and molecular theory. Science 301(5640):1691–1695CrossRefPubMedADSGoogle Scholar
  6. Bernstein B, Kearsley EA, Zapas LJ (1963) Study of stress relaxation with finite strain. Trans Soc Rheol 7:391–410CrossRefGoogle Scholar
  7. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymer liquids: fluid mechanics. Wiley, New YorkGoogle Scholar
  8. Currie PK (1982) Constitutive equations for polymer melts predicted by the Doi–Edwards and Curtiss–Bird kinetic theory models. J Non-Newton Fluid Mech 11:53–68CrossRefMATHGoogle Scholar
  9. Doi M, Edwards SF (1978a) Dynamics of concentrated polymer systems. Part 2.—molecular motion under flow. J Chem Soc Faraday Trans 2(74):1802–1817Google Scholar
  10. Doi M, Edwards SF (1978b) Dynamics of concentrated polymer systems. Part 3.—the constitutive equation. J Chem Soc Faraday Trans 2(74):1818–1832Google Scholar
  11. Doi M, Edwards SF (1986) The theory of polymer dynamics. Oxford University Press, New YorkGoogle Scholar
  12. Hassager O, Hansen R (2010) Constitutive equations for the Doi–Edwards model without independent alignment. Rheol Acta. doi:10.1007/s00397-010-0434-0 Google Scholar
  13. Hassell DG, Hoyle D, Auhl D, Harlen O, Mackley MR, McLeish TCB (2009) Effect of branching in cross-slot flow: the formation of “W cusps”. Rheol Acta 48:551–561CrossRefGoogle Scholar
  14. James DF, Walters K (1994) A critical appraisal of available methods for the measurement of extensional properties of mobile systems. In: Collyer AA (ed) Techniques of rheological measurement. Elsevier, New York, pp 33–53Google Scholar
  15. Kaye A (1962) College of aeronautics. Cranheld, note no. 134Google Scholar
  16. Luap C, Muller C, Schweizer T, Venerus DC (2005) Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution. Rheol Acta 45:83–91CrossRefGoogle Scholar
  17. Lyhne A, Rasmussen HK, Hassager O (2009) Simulation of elastic rupture in extension of entangled monodisperse polymer melts. Phys Rev Lett 102:138301CrossRefPubMedADSGoogle Scholar
  18. Marín JMR, Rasmussen HK (2009) Lagrangian finite-element method for the simulation of K-BKZ fluids with third order accuracy. J Non-Newton Fluid Mech 156:177–188CrossRefGoogle Scholar
  19. Marucci G, Grizzuti N (1986) The Doi–Edwards model in slow flows. Predictions of the Weissenberg effect. J Non-Newton Fluid Mech 21:319–328CrossRefGoogle Scholar
  20. Marucci G, Grizzuti N (1988) Fast flow of concentrated polymers: predictions of the tube model on chain stretching. Gazz Chim Ital 118:179–185Google Scholar
  21. McKinley GH, Sridhar T (2002) Filament-stretching rheometry of complex fluids. Annu Rev Fluid Mech 34:375–415CrossRefMathSciNetADSGoogle Scholar
  22. McLeish TCB, Allgaier J, Bick DK, Bishko G, Biswas P, Blackwell R, Blottière B, Clarke N, Gibbs B, Groves DJ, Hakiki A, Heenan RK, Johnson JM, Kant R, Read DJ, Young RN (1999) Dynamics of entangled H-polymers: theory, rheology, and neutron-scattering. Macromolecules 32:6734–6758CrossRefADSGoogle Scholar
  23. Nielsen JK, Hassager O, Rasmussen HK, McKinley GH (2009) Observing the chain stretch transition in a highly entangled polyisoprene melt using transient extensional rheometry. J Rheol 53:1327–1346CrossRefADSGoogle Scholar
  24. Pearson DS, Kiss AD, Fetters LJ, Doi M (1989) Flow-induced birefringence of concentrated polyisoprene solutions. J Rheol 33:517–535CrossRefGoogle Scholar
  25. Petrie CJS (1979) Elongational flows. Pitman, LondonMATHGoogle Scholar
  26. Sentmanat ML (2003) Dual wind-up extensional rheometer. US patent no.6 578 431Google Scholar
  27. Sridhar T (1990) An overview of the project M1. J Non-Newton Fluid Mech 35:85–92CrossRefGoogle Scholar
  28. Tanner RI, Walters K (1998) Rheology: an historical perspective. Elsevier, AmsterdamMATHGoogle Scholar
  29. Urakawa O, Takahashi M, Masuda T, Ebrahimi NG (1995) Damping functions and chain relaxation in uniaxial and biaxial extensions: comparison with the Doi–Edwards theory. Macromolecules 28:7196–7201CrossRefADSGoogle Scholar
  30. Wang Y, Wang SQ (2008) From elastic deformation to terminal flow of a monodisperse entangled melt in uniaxial extension. J Rheol 52:1275–1290CrossRefADSGoogle Scholar
  31. Wang Y, Boukany P, Wang SQ, Wang X (2007) Elastic breakup in uniaxial extension of entangled polymer melts. Phys Rev Lett 99:237801CrossRefPubMedADSGoogle Scholar
  32. Yu K, Marín JMR, Rasmussen HK, Hassager O (2010) 3D modeling of dual wind-up extensional rheometers. J Non-Newton Fluid Mech 165:14–23. doi:10.1016/j.jnnfm.2009.08.006 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Ole Hassager
    • 1
  • Jose Manuel Roman Marin
    • 1
  • Kaijia Yu
    • 1
  • Henrik Koblitz Rasmussen
    • 1
  1. 1.Department of Chemical and Biochemical EngineeringTechnical University of DenmarkKongens LyngbyDenmark

Personalised recommendations