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

Ahmed Elhanafy; Amr Guaily; Ahmed Elsaid

doi:doi:10.11648/j.mma.20160102.11

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

Ahmed Elhanafy, Amr Guaily, Ahmed Elsaid

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.

DOI	10.11648/j.mma.20160102.11
Published in	Mathematical Modelling and Applications (Volume 1, Issue 2, December 2016)
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), 2024. 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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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

Ahmed Elhanafy

Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, Mansoura, Egypt
Amr Guaily

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

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

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Plain Text BibTeX RIS

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

Copy | Download

@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}
}

Copy | Download

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  -

Copy | Download