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 |
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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. |
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Copyright © The Author(s), 2022. Published by Science Publishing Group |
Newtonian Fluid, Navier-Stokes Equations, Axisymmetric Problem, Weak Solution, Numerical Solution, Finite Element Method, Weighted Sobolev Space
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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
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
@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} }
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 -