Study on an N-Parallel FENE-P Constitutive Model Based on Multiple Relaxation Times for Viscoelastic Fluid

  • Jingfa Li
  • Bo YuEmail author
  • Shuyu SunEmail author
  • Dongliang Sun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10862)


An N-parallel FENE-P constitutive model based on multiple relaxation times is proposed in this paper, which aims at accurately describing the apparent viscosity of viscoelastic fluid. The establishment of N-parallel FENE-P constitutive model and the numerical approach to calculate the apparent viscosity are presented in detail, respectively. To validate the performance of the proposed constitutive model, it is compared with the conventional FENE-P constitutive model (It only has single relaxation time) in estimating the apparent viscosity of two common viscoelastic fluids: polymer and surfactant solutions. The comparative results indicate the N-parallel FENE-P constitutive model can represent the apparent viscosity of polymer solutions more accurate than the traditional model in the whole range of shear rate (0.1 s−1–1000 s−1), and the advantage is more noteworthy especially when the shear rate is higher (10 s−1–1000 s−1). Despite both the proposed model and the traditional model can’t capture the interesting shear thickening behavior of surfactant solutions, the proposed constitutive model still possesses advantage over the traditional one in depicting the apparent viscosity and first normal stress difference. In addition, the N-parallel FENE-P constitutive model demonstrates a better applicability and favorable adjustability of the model parameters.


FENE-P constitutive model N-parallel Viscoelastic fluids Multiple relaxation times Apparent viscosity 



The authors thank for support of National Natural Science Foundation of China (No. 51636006), project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges under Beijing Municipality (No. IDHT20170507), National Key R&D Program of China (Grant No. 2016YFE0204200) and the Program of Great Wall Scholar (CIT&TCD20180313).


  1. 1.
    Toms, B.A.: Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers. In: Proceedings of the 1st International Rheology Congress, II, Part 2, pp. 135–142. North Holland Publish Co., Netherlands (1949)Google Scholar
  2. 2.
    Renardy, M., Renardy, Y.: Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Nonnewton. Fluid Mech. 22, 23–33 (1986)CrossRefGoogle Scholar
  3. 3.
    Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. Roy. Soc. A 200, 523–541 (1950)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Oliveria, P.G.: Alternative derivation of differential constitutive equations of the Oldroyd-B type. J. Nonnewton. Fluid Mech. 160, 40–46 (2009)CrossRefGoogle Scholar
  5. 5.
    Giesekus, H.: A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Nonnewton. Fluid Mech. 11, 69–109 (1982)CrossRefGoogle Scholar
  6. 6.
    Bird, R.B., Dotson, P.J., Johnson, N.L.: Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Nonnewton. Fluid Mech. 7(2–3), 213–235 (1980)CrossRefGoogle Scholar
  7. 7.
    Everaers, R., Sukumaran, S.K., Grest, G.S., Svaneborg, C., Sivasubramanian, A., Kremer, K.: Rheology and microscopic topology of entangled polymeric liquids. Sciences 303(5659), 823–826 (2004)CrossRefGoogle Scholar
  8. 8.
    Ezrahi, S., Tuval, E., Aserin, A.: Properties, main applications and perspectives of worm micelles. Adv. Coll. Interface. Sci. 128–130, 77–102 (2006)CrossRefGoogle Scholar
  9. 9.
    Peterlin, A.: Streaming birefringence of soft linear macromolecules with finite chain length. Polymer 2, 257–264 (1961)CrossRefGoogle Scholar
  10. 10.
    Wei, J.J., Yao, Z.Q.: Rheological characteristic of drag reducing surfactant solution. J. Chem. Ind. Eng. (Chinese) 58(2), 0335–0340 (2007)Google Scholar
  11. 11.
    Ptasinski, P.K., Nieuwstadt, F.T.M., Van Den Brule, B.H.A.A., Hulsen, M.A.: Experiments in turbulent pipe flow with polymer additives at maximum drag reduction. Flow Turbul. Combust. 66, 159–182 (2001)CrossRefGoogle Scholar
  12. 12.
    Hashmet, M.R., Onur, M., Tan, I.M.: Empirical correlations for viscosity of polyacrylamide solutions with the effects of concentration, molecular weight and degree of hydrolysis of polymer. J. Appl. Sci. 14(10), 1000–1007 (2014)CrossRefGoogle Scholar
  13. 13.
    Purnode, B., Crochet, M.J.: Polymer solution characterization with the FENE-P model. J. Nonnewton. Fluid Mech. 77, 1–20 (1998)CrossRefGoogle Scholar
  14. 14.
    Zhang, H.X., Wang, D.Z., Gu, W.G., Chen, H.P.: Effects of temperature and concentration on rheological characteristics of surfactant additive solutions. J. Hydrodyn. 20(5), 603–610 (2008)CrossRefGoogle Scholar
  15. 15.
    Qi, Y.Y., Littrell, K., Thiyagarajan, P., Talmon, Y., Schmidt, J., Lin, Z.Q.: Small-angle neutron scattering study of shearing effects on drag-reducing surfactant solutions. J. Colloid Interface Sci. 337, 218–226 (2009)CrossRefGoogle Scholar
  16. 16.
    Galindo-Rosalesa, F.J., Rubio-Hernández, F.J., Sevilla, A.: An apparent viscosity function for shear thickening fluids. J. Nonnewton. Fluid Mech. 166, 321–325 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical Engineering, Beijing Key Laboratory of Pipeline Critical Technology and Equipment for Deepwater Oil and Gas DevelopmentBeijing Institute of Petrochemical TechnologyBeijingChina
  2. 2.Computational Transport Phenomena Laboratory, Division of Physical Science and EngineeringKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations