Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation

Rachid Ghenji; Mohamed El Hatri

doi:doi:10.11648/j.ajam.20221004.14

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Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation

Rachid Ghenji, Mohamed El Hatri

Published in American Journal of Applied Mathematics (Volume 10, Issue 4)

Received: 26 June 2022 Accepted: 26 July 2022 Published: 17 August 2022

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Abstract

In this work, we are interested in the mathematical study of the flow of a Newtonian Navier-Stokes fluid, coupled to the energy equation, in a domain with axial symmetry. The study consists first of all in reducing this problem, which is posed in a domain in dimension three (3-D), to a problem whose spatial domain is in dimension two, using the transformation of Cartesian coordinates in cylindrical coordinates, assuming that the problem data does not depend on the angle of rotation. The problem thus obtained is a so-called axially symmetric problem presenting a degeneracy on the axis of symmetry, hence the interest of this study. The study of this problem is the subject of the first part of this article which deals with the existence and uniqueness of the weak solution of the problem in a Sobolev space with appropriate weight. The results of this part have already been published by the same authors that we recall here with some slight modifications in order to facilitate the reading and understanding of the second part of the article. In this second part, we approach the existence and the unicity of the numerical solution of the posed problem. It is obtained using the Lagrange finite element method whose polinomial space is of degree one. The study in question highlights the necessary algebraic relations between the different physical parameters of the problem to which the flow in question obeys.

Published in	American Journal of Applied Mathematics (Volume 10, Issue 4)
DOI	10.11648/j.ajam.20221004.14
Page(s)	141-159
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

Newtonian Fluid, Navier-Stokes Equations, Axisymmetric Problem, Weak Solution, Numerical Solution, Finite Element Method, Weighted Sobolev Space

References

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[2]	R. Agroum, Discrétisation spectrale des équations de Naviers-Stokes couplées avec l’équation de la chaleur, Thèse de Doctorat de l’Université de Pierre et Marie- Curie, Paris VI, 2014.
[3]	H. Atamni, M. El Hatri, N. Popivanov, Polynomial Approximation in Weighted Sobole space. Comptes Rendus de l’Académie Bulgare des Sciences, V. 54, N◦3, 2001.
[4]	H-O. Bae, Regularity of the 3D Navier-Stokes equations with viewpoint of 2D flow, J. Diff. Eq., V. 264, issue 7, 2018, p. 4889-4900.
[5]	A. Benaderrahmane, Etude Numérique de l’application des Nanofluides dans l’amélioration du Transfert Thermique dans les Capteurs Solaires, http://hdl.handle.net/123456789/1928, 2017.
[6]	C. Bernadi, B. Métivet, B. Pernaud-Thomas, Couplage des équations de Navier-Stokes et de la chaleur: le modèle et son approximation par éléments finis, M2AN, vol. 29, n◦7,1995, p. 871 − 921.
[7]	P.G. Ciarlet, The finite Element Methode of Elliptic Problem. North-Holland, 1976.
[8]	E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, 1993.
[9]	M-C. El Ja¨ı, F-Z. Chalqi, a modified model for parabolic trough solar receiver, Americain Journal of Engineerin Research (AJER), volume 02, Issue 05, pp 200-211; 2013.
[10]	M. El Hatri, Estimation d’Erreur Optimale et de type Superconvergence de la Méthode des Eléments Finis pour un Problème aux Limites Dégénérés, M2AN, Tome 21, n◦EndExpansion1 (1987), p. 27 − 61.
[11]	M. El Hatri, R. Ghenji1 and N. Popivanov, Axisymmetric problem of non-stationary Navier- Stokes equations coupled with the heat equation . AIP Conference Proceedings 2333, 120005 (2021); https://doi.org/10.1063/5.0041937
[12]	A.A. Hachicha, Numerical modelling of a parabolic trough solar collector, Thesis, Universitat Politècnica de Catalunya, 2013, pp.99.
[13]	A. Kufner, O. John, S. Fuˇ cik, Function Spaces, Academia, Prague, 1977.
[14]	B. Mercier and G. Raugel, Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et série de Fourier en θ. RAIRO, Numerical Analysis (VoL. 16, n◦4, p. 405-4061).
[15]	J. Neustupa, Axysimetrical flow of Navier-Stokes fluid in the whole space with non-zero angular velocity component, Math. Bohemica, N◦2, 2001, pp. 469-481.
[16]	C. Pérez, J-M. Thomas, S. Blancher, R. Creff, The coupled steady Navier-Stokes/Energy equations problem with temperature-dependent viscosity in open channel flows, Part 1: Physical model and Analysis of the continuous problem, 2014.
[17]	Q. Sylvain, Les centrales Solaires à Concentration, Faculté des Scienses Appliquées, Université de Liège, 2007.
[18]	R. Temam, Navier-Stokes equations, Theory and numerical analysis, Elsevier-North Holland.1979.
[19]	F.M White, Viscous Fluid Flow, Third Edition, Ed. McGraw Hill, 2006.
[20]	Z.Zhang, Apointwiseregularitycriterionforaxysymetric Navier-Stokes system, J. Math Anal. and Appl. 461 (2018) 1-6.

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APA Style

Rachid Ghenji, Mohamed El Hatri. (2022). Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. American Journal of Applied Mathematics, 10(4), 141-159. https://doi.org/10.11648/j.ajam.20221004.14

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ACS Style

Rachid Ghenji; Mohamed El Hatri. Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. Am. J. Appl. Math. 2022, 10(4), 141-159. doi: 10.11648/j.ajam.20221004.14

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AMA Style

Rachid Ghenji, Mohamed El Hatri. Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. Am J Appl Math. 2022;10(4):141-159. doi: 10.11648/j.ajam.20221004.14

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@article{10.11648/j.ajam.20221004.14,
  author = {Rachid Ghenji and Mohamed El Hatri},
  title = {Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {4},
  pages = {141-159},
  doi = {10.11648/j.ajam.20221004.14},
  url = {https://doi.org/10.11648/j.ajam.20221004.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221004.14},
  abstract = {In this work, we are interested in the mathematical study of the flow of a Newtonian Navier-Stokes fluid, coupled to the energy equation, in a domain with axial symmetry. The study consists first of all in reducing this problem, which is posed in a domain in dimension three (3-D), to a problem whose spatial domain is in dimension two, using the transformation of Cartesian coordinates in cylindrical coordinates, assuming that the problem data does not depend on the angle of rotation. The problem thus obtained is a so-called axially symmetric problem presenting a degeneracy on the axis of symmetry, hence the interest of this study. The study of this problem is the subject of the first part of this article which deals with the existence and uniqueness of the weak solution of the problem in a Sobolev space with appropriate weight. The results of this part have already been published by the same authors that we recall here with some slight modifications in order to facilitate the reading and understanding of the second part of the article. In this second part, we approach the existence and the unicity of the numerical solution of the posed problem. It is obtained using the Lagrange finite element method whose polinomial space is of degree one. The study in question highlights the necessary algebraic relations between the different physical parameters of the problem to which the flow in question obeys.},
 year = {2022}
}

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TY  - JOUR
T1  - Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation
AU  - Rachid Ghenji
AU  - Mohamed El Hatri
Y1  - 2022/08/17
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221004.14
DO  - 10.11648/j.ajam.20221004.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 141
EP  - 159
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221004.14
AB  - In this work, we are interested in the mathematical study of the flow of a Newtonian Navier-Stokes fluid, coupled to the energy equation, in a domain with axial symmetry. The study consists first of all in reducing this problem, which is posed in a domain in dimension three (3-D), to a problem whose spatial domain is in dimension two, using the transformation of Cartesian coordinates in cylindrical coordinates, assuming that the problem data does not depend on the angle of rotation. The problem thus obtained is a so-called axially symmetric problem presenting a degeneracy on the axis of symmetry, hence the interest of this study. The study of this problem is the subject of the first part of this article which deals with the existence and uniqueness of the weak solution of the problem in a Sobolev space with appropriate weight. The results of this part have already been published by the same authors that we recall here with some slight modifications in order to facilitate the reading and understanding of the second part of the article. In this second part, we approach the existence and the unicity of the numerical solution of the posed problem. It is obtained using the Lagrange finite element method whose polinomial space is of degree one. The study in question highlights the necessary algebraic relations between the different physical parameters of the problem to which the flow in question obeys.
VL  - 10
IS  - 4
ER  -

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Author Information

Rachid Ghenji

Higher School of Technology, Mohamed Ben Abdellah University, Fes, Morocco
Mohamed El Hatri

Higher School of Technology, Mohamed Ben Abdellah University, Fes, Morocco

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Plain Text BibTeX RIS

APA Style

Rachid Ghenji, Mohamed El Hatri. (2022). Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. American Journal of Applied Mathematics, 10(4), 141-159. https://doi.org/10.11648/j.ajam.20221004.14

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ACS Style

Rachid Ghenji; Mohamed El Hatri. Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. Am. J. Appl. Math. 2022, 10(4), 141-159. doi: 10.11648/j.ajam.20221004.14

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AMA Style

Rachid Ghenji, Mohamed El Hatri. Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation. Am J Appl Math. 2022;10(4):141-159. doi: 10.11648/j.ajam.20221004.14

Copy | Download

@article{10.11648/j.ajam.20221004.14,
  author = {Rachid Ghenji and Mohamed El Hatri},
  title = {Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {4},
  pages = {141-159},
  doi = {10.11648/j.ajam.20221004.14},
  url = {https://doi.org/10.11648/j.ajam.20221004.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221004.14},
  abstract = {In this work, we are interested in the mathematical study of the flow of a Newtonian Navier-Stokes fluid, coupled to the energy equation, in a domain with axial symmetry. The study consists first of all in reducing this problem, which is posed in a domain in dimension three (3-D), to a problem whose spatial domain is in dimension two, using the transformation of Cartesian coordinates in cylindrical coordinates, assuming that the problem data does not depend on the angle of rotation. The problem thus obtained is a so-called axially symmetric problem presenting a degeneracy on the axis of symmetry, hence the interest of this study. The study of this problem is the subject of the first part of this article which deals with the existence and uniqueness of the weak solution of the problem in a Sobolev space with appropriate weight. The results of this part have already been published by the same authors that we recall here with some slight modifications in order to facilitate the reading and understanding of the second part of the article. In this second part, we approach the existence and the unicity of the numerical solution of the posed problem. It is obtained using the Lagrange finite element method whose polinomial space is of degree one. The study in question highlights the necessary algebraic relations between the different physical parameters of the problem to which the flow in question obeys.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Axisymmetric Problem of Stationnary Navier-Stokes Equations Coupled with the Heat Equation
AU  - Rachid Ghenji
AU  - Mohamed El Hatri
Y1  - 2022/08/17
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221004.14
DO  - 10.11648/j.ajam.20221004.14
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 141
EP  - 159
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221004.14
AB  - In this work, we are interested in the mathematical study of the flow of a Newtonian Navier-Stokes fluid, coupled to the energy equation, in a domain with axial symmetry. The study consists first of all in reducing this problem, which is posed in a domain in dimension three (3-D), to a problem whose spatial domain is in dimension two, using the transformation of Cartesian coordinates in cylindrical coordinates, assuming that the problem data does not depend on the angle of rotation. The problem thus obtained is a so-called axially symmetric problem presenting a degeneracy on the axis of symmetry, hence the interest of this study. The study of this problem is the subject of the first part of this article which deals with the existence and uniqueness of the weak solution of the problem in a Sobolev space with appropriate weight. The results of this part have already been published by the same authors that we recall here with some slight modifications in order to facilitate the reading and understanding of the second part of the article. In this second part, we approach the existence and the unicity of the numerical solution of the posed problem. It is obtained using the Lagrange finite element method whose polinomial space is of degree one. The study in question highlights the necessary algebraic relations between the different physical parameters of the problem to which the flow in question obeys.
VL  - 10
IS  - 4
ER  -

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