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Journal of Scientific Computing

, Volume 79, Issue 1, pp 369–388 | Cite as

Numerical Simulations of Viscoelastic Fluid Flows Past a Transverse Slot Using Least-Squares Finite Element Methods

  • Hsueh-Chen LeeEmail author
  • Hyesuk Lee
Article
  • 39 Downloads

Abstract

This paper presents a least-squares (LS) finite element method for linear Phan-Thien–Tanner (PTT) viscoelastic fluid flows. We consider stabilized weights in the LS method for the viscoelastic model and prove that the LS approximation converges to the linearized solutions of the linear PTT model; the convergence is at the optimal rate for the velocity in the \(H^{1}\)-norm and at suboptimal rates for the stress and pressure in the \(L^{2}\)-norm, respectively. For numerical experiments we first consider the flow through a planar channel to illustrate our theoretical results. The LS method is then applied to a flow through the slot channel with two depth ratios and the effects of physical parameters are discussed. Numerical solutions of the channel problem indicate that flow characteristics of the viscoelastic polymer solution are described by the results obtained using the method. Furthermore, we present the hole pressure for various Weissenberg numbers, and compare with that derived from the Higashitani–Pritchard (HP) theory.

Keywords

Least-squares The PTT model Hole pressure Normal-stress difference Transverse slot Weissenberg number 

Notes

Acknowledgements

The first author gratefully acknowledges the financial support provided in part by the Ministry of Science and Technology of Taiwan under Grant 107-2115-M-160-001-MY2. The second author is grateful for the financial support provided in part by the US National Science Foundation under Grant DMS-1418960.

References

  1. 1.
    Alves, M.A., Oliveira, P.J., Pinho, F.T.: Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J. Non-Newton. Fluid Mech. 110, 45–75 (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Azaiez, J., Guénette, R., Ait-Kadi, A.: Numerical simulation of viscoelastic flows through a planar contraction. J. Non-Newton. Fluid Mech. 62, 253–277 (1996)CrossRefGoogle Scholar
  3. 3.
    Bochev, P.B., Gunzburger, M.D.: Finite element methods of least-squares type. SIAM Rev. 40, 789–837 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cai, Z., Westphal, C.R.: An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type. J. Non-Newton. Fluid Mech. 159, 72–80 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cai, Z., Manteufeel, T.A., Mccormick, S.F.: First-order system least-squares for velocity–vorticity–pressure form of the Stokes equations, with application to linear elastically. Electron. Trans. Numer. Anal. 3, 150–159 (1995)MathSciNetGoogle Scholar
  6. 6.
    Chen, T.F., Cox, C.L., Lee, H.C., Tung, K.L.: Least-squares finite element methods for generalized Newtonian and viscoelastic flows. Appl. Numer. Math. 60, 1024–1040 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, T.F., Lee, H., Liu, C.C.: Numerical approximation of the Oldroyd-B model by the weighted least-squares/discontinuous Galerkin method. Numer. Methods Partial Diff. Equ. 29, 531–548 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coronado, O.M., Arora, D., Behr, M., Pasquali, M.: Four-field Galerkin/least-squares formulation for viscoelastic fluids. J. Non-Newton. Fluid Mech. 140, 132–144 (2006)CrossRefzbMATHGoogle Scholar
  9. 9.
    Higashitani, K., Pritchard, W.G.: A kinematic calculation of intrinsic errors in pressure measurements made with holes. Trans. Soc Rheol. 16, 688–696 (1972)Google Scholar
  10. 10.
    Huilgol, R.R., Phan-Thien, N.: Fluid Mechanics of Viscoelasticity: General Principles, Constitutive Modelling, Analytical and Numerical Techniques. Elsever, Amsterdam (1997)Google Scholar
  11. 11.
    Lee, H.C.: A nonlinear weighted least-squares finite element method for the Oldroyd-B viscoelastic flow. Appl. Math. Comput. 219, 421–434 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lee, H.C.: An adaptively refined least-squares finite element method for generalized Newtonian fluid flows using the Carreau model. SIAM J. Sci. Comput. 36, 193–218 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lee, H.C.: A nonlinear weighted least-squares finite element method for the Carreau–Yasuda non-Newtonian model. J. Math. Anal. Appl. 432, 844–861 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, H.C.: Numerical simulations of viscoelastic fluid flows using a least-squares finite element method based on von Mises stress criteria. Int. J. Appl. Phys. Math. 7, 157–164 (2017)CrossRefGoogle Scholar
  15. 15.
    Lee, H.C.: Adaptive weights for mass conservation in a least-squares finite element method. Int. J. Comput. Math. 95, 20–35 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Liu, J.L.: Exact a posteriori error analysis of the least-squares finite element method. Appl. Math. Comput. 116, 297–305 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Renardy, M., Hrusa, W.J., Nohel, A.: Mathematical Problems in Viscoelasticity. Wiley, New York (1987)zbMATHGoogle Scholar
  18. 18.
    Tanner, R.I., Pipkin, A.C.: Intrinsic errors in hole-pressure measurements. Trans. Soc. Rheol. 13, 471–484 (1969)CrossRefGoogle Scholar
  19. 19.
    Thien, N.P., Tanner, R.I.: A new constitutive equation derived from network theory. J. Math. Anal. Appl 2, 353–365 (1977)zbMATHGoogle Scholar
  20. 20.
    Wu, G.H., Lin, Y.M.: Creeping flow of a polymeric liquid passing over a transverse slot with viscous dissipation. Int. J. Heat Mass Transf. 45, 4703–4711 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Yin, H.J., Zhong, H.Y., Fu, C.Q., Lei, W.: Numerical simulations of viscoelastic flows through one slot channel. J. Hydrodyn. 19, 201–216 (2007)CrossRefGoogle Scholar
  22. 22.
    Zhou, S., Hou, L.: A weighted least-squares finite element method for Phan-Thien–Tanner viscoelastic fluid. J. Math. Anal. Appl. 436, 66–78 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.General Education CenterWenzao Ursuline University of LanguagesKaohsiungTaiwan
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

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