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

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## 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.

## Keywords

Viscoelastic flow HWNP Log-conformation formulation Finite element method## Nomenclature

- \(\boldsymbol{c}\)
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

## Notes

### 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).

## 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) CrossRefGoogle 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) CrossRefGoogle 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) CrossRefGoogle 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) MathSciNetCrossRefGoogle 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) CrossRefGoogle Scholar - Fattal, R., Kupferman, R.: Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newton. Fluid Mech.
**123**, 281–285 (2004) CrossRefGoogle 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) CrossRefGoogle Scholar - Guenette, R., Fortin, M.: A new mixed finite method for computing viscoelastic flows. J. Non-Newton. Fluid Mech.
**60**, 27–52 (1995) CrossRefGoogle Scholar - Hill, R.: Aspect of invariance in solid mechanics. Adv. Appl. Mech.
**18**, 1–75 (1978) MathSciNetzbMATHGoogle 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) CrossRefGoogle 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) MathSciNetCrossRefGoogle 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) MathSciNetCrossRefGoogle Scholar - Knechtges, P., Behr, M., Elgeti, S.: Fully-implicit log-conformation formulation of constitutive laws. J. Non-Newton. Fluid Mech.
**214**, 78–87 (2014) CrossRefGoogle 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) CrossRefGoogle 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) MathSciNetCrossRefGoogle 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) MathSciNetCrossRefGoogle 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) CrossRefGoogle 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) MathSciNetCrossRefGoogle 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) CrossRefGoogle 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) CrossRefGoogle 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