Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

Faisal A. Fairag

doi:10.1016/j.cam.2006.07.028

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

Faisal A. Fairag^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity-pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.

Original language	English
Pages (from-to)	374-391
Number of pages	18
Journal	Journal of Computational and Applied Mathematics
Volume	206
Issue number	1
DOIs	https://doi.org/10.1016/j.cam.2006.07.028
State	Published - 1 Sep 2007

Bibliographical note

Funding Information:
This research work was undertaken while the author was on sabbatical leave from King Fahd University of Petroleum and Minerals. The author wishes to express his gratitude to the University for providing the financial support while on research attachment at Univesiti Tenaga Nasional, Malaysia. The author would like to express thanks to Prof. William J. Layton, Prof. Qiang Du and Dr. Songul Kaya for their valuable comments on this work.

Funding Information:
This work was supported by KFUPM under grant INT-278.

Keywords

Finite element method
Ladyzhenskaya model
Streamfunction formulation

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2006.07.028

Cite this

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title = "Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form",

abstract = "In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity-pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.",

keywords = "Finite element method, Ladyzhenskaya model, Streamfunction formulation",

author = "Fairag, \{Faisal A.\}",

note = "Funding Information: This research work was undertaken while the author was on sabbatical leave from King Fahd University of Petroleum and Minerals. The author wishes to express his gratitude to the University for providing the financial support while on research attachment at Univesiti Tenaga Nasional, Malaysia. The author would like to express thanks to Prof. William J. Layton, Prof. Qiang Du and Dr. Songul Kaya for their valuable comments on this work. Funding Information: This work was supported by KFUPM under grant INT-278. ",

year = "2007",

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doi = "10.1016/j.cam.2006.07.028",

language = "English",

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AU - Fairag, Faisal A.

N1 - Funding Information: This research work was undertaken while the author was on sabbatical leave from King Fahd University of Petroleum and Minerals. The author wishes to express his gratitude to the University for providing the financial support while on research attachment at Univesiti Tenaga Nasional, Malaysia. The author would like to express thanks to Prof. William J. Layton, Prof. Qiang Du and Dr. Songul Kaya for their valuable comments on this work. Funding Information: This work was supported by KFUPM under grant INT-278.

PY - 2007/9/1

Y1 - 2007/9/1

N2 - In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity-pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.

AB - In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity-pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier-Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.

KW - Finite element method

KW - Ladyzhenskaya model

KW - Streamfunction formulation

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DO - 10.1016/j.cam.2006.07.028

M3 - Article

AN - SCOPUS:34249283074

SN - 0377-0427

VL - 206

SP - 374

EP - 391

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 1

ER -

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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