Skip to main content
SpringerLink
Log in

Finite element simulation of three-dimensional viscoelastic flow at high Weissenberg number based on the log-conformation formulation

  • Published:
Mechanics of Time-Dependent Materials Aims and scope Submit manuscript

Abstract

Viscoelasticity is an important characteristic of many complex fluids such as polymer solutions and melts. Understanding the viscoelastic behavior of such complex fluids presents mathematical, modeling and computational challenges, particularly in the case of fluids affected by elastic turbulence at high Weissenberg number. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with log-conformation formulation transformation is employed to improve computational stability at high Weissenberg number. The momentum equation is calculated after introducing an ellipticity factor and the constitutive equation is calculated based on the logarithm of the conformation tensor. The finite element-finite difference formulations of governing equations are derived. The planar contraction as a representative benchmark problem is used to test the robustness of the numerical method to predict real flow patterns of viscoelastic fluids at different Weissenberg numbers. The simulation results predicted with differential constitutive models based on the logarithm of the conformation tensor agree well with Quinzani’s experimental results. Both the stability and the accuracy are improved compared with traditional calculation method. The numerical methodology proposed in the study can well predict complex flow patterns of viscoelastic fluids at high Weissenberg number.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Abbreviations

\(\boldsymbol{c}\) :

Conformation tensor

\(\boldsymbol{d}\) :

Deformation rate tensor

\(H\) :

Spring elastic constant

\(\boldsymbol{I}\) :

Kronecker delta

\(k_{B}\) :

Boltzmann constant

\(\boldsymbol{L}\) :

Velocity gradient tensor

\(p\) :

Hydrostatic pressure

\(\boldsymbol{Q}\) :

End-to-end vector

\(\boldsymbol{u}\) :

Velocity vector

\(\nabla\) :

Hamilton differential operator

\(\eta_{n}\) :

Newtonian-contribution viscosity

\(\bar{\eta} \) :

Reference viscosity

\(\eta_{v}\) :

Viscoelastic-contribution viscosity

\(\lambda_{p}\) :

Penalty factor

\(\lambda\) :

Relaxation time

\(\rho\) :

Material density

\(\boldsymbol{\sigma} \) :

Cauchy stress tensor

\(\boldsymbol{\tau} \) :

Extra stress tensor

References

  • Afonso, A., Oliveira, P.J., Pinho, F.T., Alves, M.A.: The log-conformation tensor approach in the finite-volume method framework. J. Non-Newton. Fluid Mech. 157, 55–65 (2009)

    Article  Google Scholar 

  • Azaiez, J., Guénette, R., Aït-Kadi, A.: Numerical simulation of viscoelastic flows through a planar contraction. J. Non-Newton. Fluid Mech. 62, 253–277 (1996)

    Article  Google Scholar 

  • Balci, N., Thomases, B., Renardy, M., Doering, C.R.: Symmetric factorization of the conformation tensor in viscoelastic fluid models. J. Non-Newton. Fluid Mech. 166, 546–553 (2011)

    Article  Google Scholar 

  • Comminal, R., Hattel, J.H., Alves, M.A., Spangenberg, J.: Vortex behavior of the Oldroyd-B fluid in the 4-1 planar contraction simulated with the streamfunction-log-conformation formulation. J. Non-Newton. Fluid Mech. 237, 1–15 (2016)

    Article  MathSciNet  Google Scholar 

  • Coronado, O.M., Arora, D., Behr, M., Pasquali, M.: A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation. J. Non-Newton. Fluid Mech. 147, 189–199 (2007)

    Article  Google Scholar 

  • Fattal, R., Kupferman, R.: Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newton. Fluid Mech. 123, 281–285 (2004)

    Article  Google Scholar 

  • Fattal, R., Kupferman, R.: Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation. J. Non-Newton. Fluid Mech. 126(1), 23–37 (2005)

    Article  Google Scholar 

  • Guenette, R., Fortin, M.: A new mixed finite method for computing viscoelastic flows. J. Non-Newton. Fluid Mech. 60, 27–52 (1995)

    Article  Google Scholar 

  • Hill, R.: Aspect of invariance in solid mechanics. Adv. Appl. Mech. 18, 1–75 (1978)

    MathSciNet  MATH  Google Scholar 

  • Hulsena, M.A., Fattal, R., Kupferman, R.: Flow of viscoelastic fluids past a cylinder at high Weissenberg number: Stabilized simulations using matrix logarithms. J. Non-Newton. Fluid Mech. 127, 27–39 (2005)

    Article  Google Scholar 

  • Jafari, A., Fiétier, N., Deville, M.O.: A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Comput. Fluids 39, 1425–1438 (2010)

    Article  MathSciNet  Google Scholar 

  • Jafari, A., Fiétier, N., Deville, M.O.: Simulation of viscoelastic fluids in a 2D abrupt contraction by spectral element method. Int. J. Numer. Methods Fluids 78, 217–232 (2015)

    Article  MathSciNet  Google Scholar 

  • Knechtges, P., Behr, M., Elgeti, S.: Fully-implicit log-conformation formulation of constitutive laws. J. Non-Newton. Fluid Mech. 214, 78–87 (2014)

    Article  Google Scholar 

  • Kwack, J.H., Masud, A., Rajagopal, K.R.: Stabilized mixed three-field formulation for a generalized incompressible Oldroyd-B model. Int. J. Numer. Methods Fluids 83, 704–734 (2017)

    Article  Google Scholar 

  • Li, D.Y., Zhang, H., Cheng, J.P., Li, X.B., Li, F.C., Qian, S., Joo, S.W.: Numerical simulation of heat transfer enhancement by elastic turbulence in a curvy channel. Microfluid. Nanofluid. 21(25), 1–16 (2017)

    Google Scholar 

  • Medvid’ová, M.L., Notsu, H., She, B.: Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid. Int. J. Numer. Methods Fluids 81, 523–557 (2016)

    Article  MathSciNet  Google Scholar 

  • Mu, Y., Zhao, G.Q., Chen, A.B., Wu, X.H.: Modeling and simulation of three-dimensional extrusion swelling of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models. Int. J. Numer. Methods Fluids 72, 846–863 (2013)

    Article  MathSciNet  Google Scholar 

  • Palhares Junior, I.L., Oishi, C.M., Afonso, A.M., Alves, M.A., Pinho, F.T.: Numerical study of the square-root conformation tensor formulation for confined and free-surface viscoelastic fluid flows. Adv. Model. Simul. Eng. Sci. 3, 2–32 (2016)

    Article  Google Scholar 

  • Pan, T.W., Hao, J., Glowinski, R.: On the simulation of a time-dependent cavity flow of an Oldroyd-B fluid. Int. J. Numer. Methods Fluids 60, 791–808 (2009)

    Article  MathSciNet  Google Scholar 

  • Quinzani, L.M., Armstrong, R.C., Brown, R.A.: Birefringence and laser-Doppler velocimetry (LDV) studies of viscoelastic flow through a planar contraction. J. Non-Newton. Fluid Mech. 52, 1–36 (1994)

    Article  Google Scholar 

  • Saramito, P.: On a modified non-singular log-conformation formulation for Johnson-Segalman viscoelastic fluids. J. Non-Newton. Fluid Mech. 211, 16–30 (2014)

    Article  Google Scholar 

  • Wang, X., Li, X.: Log-conformation-based pressure-stabilized fractional step algorithm for viscoelastic flows. Chin. J. Comput. Phys. 28(6), 853–860 (2011)

    Google Scholar 

Download references

Acknowledgements

This work is financially supported by the National Natural Science Foundation of China (No. 51675308, No. 51205231), the Natural Science Foundation of Shandong Province (No. ZR2012EEQ001), the Key Research and Development Program of Shandong Province (No. 2018GGX103014) and the Joint Funds of the Ministry of Education of China (No. 6141A02011705).

Author information

Authors and Affiliations

  1. Key Laboratory for Liquid–Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong, 250061, China

    Yue Mu, Guoqun Zhao, Yujia Cui, Jiejie Feng & Foufei Ren

  2. Engineering Research Center for Mould & Die Technologies, Shandong University, Jinan, Shandong, 250061, China

    Yue Mu, Anbiao Chen, Guoqun Zhao, Yujia Cui, Jiejie Feng & Foufei Ren

Authors
  1. Yue Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anbiao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guoqun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yujia Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jiejie Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Foufei Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yue Mu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, Y., Chen, A., Zhao, G. et al. Finite element simulation of three-dimensional viscoelastic flow at high Weissenberg number based on the log-conformation formulation. Mech Time-Depend Mater 23, 477–495 (2019). https://doi.org/10.1007/s11043-018-9401-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11043-018-9401-4

Keywords

Search

Navigation