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A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows

Received: 10 September 2016     Accepted: 14 October 2016     Published: 21 October 2016
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Abstract

The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.

Published in Mathematical Modelling and Applications (Volume 1, Issue 2)
DOI 10.11648/j.mma.20160102.11
Page(s) 26-35
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Unsteady Incompressible Viscoelastic Flow, Oldroyd-B Model, Pressure Stabilization Technique, The DEVSS Method, Galerkin Least Squares

References
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Cite This Article
  • APA Style

    Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. (2016). A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Mathematical Modelling and Applications, 1(2), 26-35. https://doi.org/10.11648/j.mma.20160102.11

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    ACS Style

    Ahmed Elhanafy; Amr Guaily; Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math. Model. Appl. 2016, 1(2), 26-35. doi: 10.11648/j.mma.20160102.11

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    AMA Style

    Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid. A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows. Math Model Appl. 2016;1(2):26-35. doi: 10.11648/j.mma.20160102.11

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  • @article{10.11648/j.mma.20160102.11,
      author = {Ahmed Elhanafy and Amr Guaily and Ahmed Elsaid},
      title = {A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows},
      journal = {Mathematical Modelling and Applications},
      volume = {1},
      number = {2},
      pages = {26-35},
      doi = {10.11648/j.mma.20160102.11},
      url = {https://doi.org/10.11648/j.mma.20160102.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160102.11},
      abstract = {The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.},
     year = {2016}
    }
    

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  • TY  - JOUR
    T1  - A Hybrid Stabilized Finite Element/Finite Difference Method for Unsteady Viscoelastic Flows
    AU  - Ahmed Elhanafy
    AU  - Amr Guaily
    AU  - Ahmed Elsaid
    Y1  - 2016/10/21
    PY  - 2016
    N1  - https://doi.org/10.11648/j.mma.20160102.11
    DO  - 10.11648/j.mma.20160102.11
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
    SP  - 26
    EP  - 35
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20160102.11
    AB  - The Oldroyd-B constitutive equation is used for the numerical simulation of unsteady incompressible viscoelastic flows. A novelty treatment is presented for the incompressibility constraint of the incompressible viscoelastic flow by using the modified continuity equation which allows using equal-order interpolation polynomials for all variables. The proposed technique circumvents the so-called LBB compatibility condition without pressure checkerboard and the solution instabilities with less computational costs compared with the traditional techniques. The discrete elastic-viscous stress-splitting method (DEVSS) is used to treat the instabilities resulting from the numerical simulation of viscoelastic flows. Two benchmark problems are simulated, namely, the flow through a channel with a bump and the flow inside a square cavity. Solutions are obtained for different Weissenberg number values and the results are compared with the published works.
    VL  - 1
    IS  - 2
    ER  - 

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Author Information
  • Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt

  • Engineering Mathematics & Physics Department, Faculty of Engineering, Cairo University, Giza, Egypt

  • Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt

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