Mechanics of Time-Dependent Materials

, Volume 23, Issue 4, pp 477–495 | Cite as

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

  • Yue MuEmail author
  • Anbiao Chen
  • Guoqun Zhao
  • Yujia Cui
  • Jiejie Feng
  • Foufei Ren


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.


Viscoelastic flow HWNP Log-conformation formulation Finite element method 



Conformation tensor


Deformation rate tensor


Spring elastic constant


Kronecker delta


Boltzmann constant


Velocity gradient tensor


Hydrostatic pressure


End-to-end vector


Velocity vector


Hamilton differential operator


Newtonian-contribution viscosity

\(\bar{\eta} \)

Reference viscosity


Viscoelastic-contribution viscosity


Penalty factor


Relaxation time


Material density

\(\boldsymbol{\sigma} \)

Cauchy stress tensor

\(\boldsymbol{\tau} \)

Extra stress tensor



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).


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Yue Mu
    • 1
    • 2
    Email author
  • Anbiao Chen
    • 2
  • Guoqun Zhao
    • 1
    • 2
  • Yujia Cui
    • 1
    • 2
  • Jiejie Feng
    • 1
    • 2
  • Foufei Ren
    • 1
    • 2
  1. 1.Key Laboratory for Liquid–Solid Structural Evolution and Processing of Materials (Ministry of Education)Shandong UniversityJinanChina
  2. 2.Engineering Research Center for Mould & Die TechnologiesShandong UniversityJinanChina

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