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Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations

Received: 18 October 2022     Accepted: 12 November 2022     Published: 22 November 2022
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Abstract

In fluid mechanics, the Navier-Stokes equations are non-linear partial differential equations that describe the motion of Newtonian fluids. A fluid can be a liquid or a gas. Therefore, the Navier-Stokes equation concerns many phenomena that surround us. The analytical resolution, the search for exact solutions of these equations modeling a fluid is difficult. But they often allow, by an approximate resolution, to propose a model of many phenomena, such as ocean currents and air mass movements in the atmosphere for meteorologists, the behavior of skyscrapers or bridges under the action of wind for architects and engineers, or that of airplanes, trains or high-speed cars for their design offices, as well as the flow of water in a pipe and many other flow phenomena of various fluids. In mathematics, nonlinearity complicates things. In physics, too, the difficulty arises. For this term nonlinearity has its translation in the complexity of the physical phenomena described. This difficulty of resolution partly affects the analyses or descriptions of the modeled phenomena. The objective of this work is the search for exact solutions of the Navier-Stokes equations in dimension 2 and in dimension 3. The method of the reduced differential transform is used to find the exact solutions of these Navier-Stokes equations in 2D and 3D. This method gives an algorithm that favors the rapid convergence of the problem to the exact solution sought. Besides the introduction, this article is structured as follows: the presentation of the method, its application on the two selected Navier-Stokes problems whose exact solutions are obtained with ease, then intervenes the conclusion of the whole work.

Published in Pure and Applied Mathematics Journal (Volume 11, Issue 6)
DOI 10.11648/j.pamj.20221106.11
Page(s) 96-101
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), 2022. Published by Science Publishing Group

Keywords

Navier-Stokes Equation 2D, Navier-Stokes Equation 3D, Exact Solution, Reduced Differential Transform Method

References
[1] Afzal Soomro M., & Hussain J. (2019). On Study of Generalized Novikov Equation by Reduced Differential Transform Method. Indian Journal of Sciences and Technology, Vol. 12. DOI: 10.17 485/ijst/2019, ISSN (online): 0974-5645, pp: 1-6.
[2] Al-Sawoor, Ann J., Al-Amr Mohammed O. (2013). Reduced Differential Transform Method for the Generalized Ito System. International Journal of Enhanced Research in Science Technology & Engineering. ISSN: 2319-7463, Vol. 2, Issue 11, novembre-2013, pp: 135-145.
[3] Amiroudine, S. and Battaglia, J. L., (2011). Mécanique des fluides. Dunod.
[4] Cheng, X., Wang, L., & Shen, S. (2022). On Analytical Solutions of the Conformable Time-Fractional Navier-Stokes Equation. Reports on Mathematical Physics, 89 (3), 335-358.
[5] Hesam, S., Nazemi, A., Haghbin A. (2012). Reduced Differential Transform Method for Solving the Fornberg-Witham Type Equation. International Journal of Nonlinear Science. Vol. 13, N°. 2, pp: 158-162.
[6] Jafari H., Jassim, H., K., Moshokoa S., P., Ariyan V., Tcchier F., (2016). Reduced differential transform method for partial differential equations within local fractional derivative operators. Advances in Mechanical Engineering. Vol. 8 (4). pp: 1–6.
[7] Keskin Y., Oturanc G., (2009). Reduced Differential Transform Method for Partial Differential Equations. International Journal of Nonlinear Sciences and Numerical Simulation. 10 (6), pp: 741-749.
[8] Keskin Y., (2010). Application of Reduced Differential Transformation Method for Solving Gas Dynamics Equation. Int. J. Contemp. Math. Sciences, Vol. 5, N° 22, pp: 1091-1096.
[9] Keskin Y., Oturanc G. (2010). Reduced Differential Transform Method for solving linear and Nonlinear wave Equations. Iranian Journal of Science & Technology. Transaction A, Vol. 34, N°A2, pp: 113-122.
[10] Mirzaee, F. (2011). Differential Transform Method for Solving Linear and Nonlinear Systems of ordinary Differential Equations. Applied Mathematical Sciences. Vol. 5, n°. 70, pp: 3465-3472.
[11] Mukhtar, S., Shah, R., & Noor, S. (2022). The Numerical Investigation of a Fractional-Order Multi-Dimensional Model of Navier-Stokes Equation via ovel Techniques. Symmetry. 14 (6), 1102.
[12] Nemah, E. M. (2020). Homotopy Transforms analysis method for solving fractional Navier-Stokes equations with applications. Iraqi Journal of science, 2048-2054.
[13] Prakash, A., Prakasha, D. G., & Veeresha, P. (2019). A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform. Nonlinear Engineering, 8 (1), 695-701.
[14] Rashidi, M. M., & Shahmohamadi, H. (2009). Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk. Communications in Nonlinear Science and Numerical Simulation, 14 (7), 2999-3006.
[15] Singh, B. K., & Kumar, P. (2018). FRDTM for numerical simulation of multi-dimensional, time-fractional model of Navier–Stokes equation. Ain Shams Engineering Journal, 9 (4), 827-834.
[16] Wellot, Y. A. S., & Nkaya, G. D. (2022). Combinate of Reduced Differential Transformation Method and Picard’s Principe. Applied and Computational Mathematics, 11 (4), 87-94.
Cite This Article
  • APA Style

    Yanick Alain Servais Wellot. (2022). Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations. Pure and Applied Mathematics Journal, 11(6), 96-101. https://doi.org/10.11648/j.pamj.20221106.11

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    ACS Style

    Yanick Alain Servais Wellot. Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations. Pure Appl. Math. J. 2022, 11(6), 96-101. doi: 10.11648/j.pamj.20221106.11

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    AMA Style

    Yanick Alain Servais Wellot. Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations. Pure Appl Math J. 2022;11(6):96-101. doi: 10.11648/j.pamj.20221106.11

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  • @article{10.11648/j.pamj.20221106.11,
      author = {Yanick Alain Servais Wellot},
      title = {Application of the Reduced Differential Transform Method to Solve the Navier-Stokes Equations},
      journal = {Pure and Applied Mathematics Journal},
      volume = {11},
      number = {6},
      pages = {96-101},
      doi = {10.11648/j.pamj.20221106.11},
      url = {https://doi.org/10.11648/j.pamj.20221106.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20221106.11},
      abstract = {In fluid mechanics, the Navier-Stokes equations are non-linear partial differential equations that describe the motion of Newtonian fluids. A fluid can be a liquid or a gas. Therefore, the Navier-Stokes equation concerns many phenomena that surround us. The analytical resolution, the search for exact solutions of these equations modeling a fluid is difficult. But they often allow, by an approximate resolution, to propose a model of many phenomena, such as ocean currents and air mass movements in the atmosphere for meteorologists, the behavior of skyscrapers or bridges under the action of wind for architects and engineers, or that of airplanes, trains or high-speed cars for their design offices, as well as the flow of water in a pipe and many other flow phenomena of various fluids. In mathematics, nonlinearity complicates things. In physics, too, the difficulty arises. For this term nonlinearity has its translation in the complexity of the physical phenomena described. This difficulty of resolution partly affects the analyses or descriptions of the modeled phenomena. The objective of this work is the search for exact solutions of the Navier-Stokes equations in dimension 2 and in dimension 3. The method of the reduced differential transform is used to find the exact solutions of these Navier-Stokes equations in 2D and 3D. This method gives an algorithm that favors the rapid convergence of the problem to the exact solution sought. Besides the introduction, this article is structured as follows: the presentation of the method, its application on the two selected Navier-Stokes problems whose exact solutions are obtained with ease, then intervenes the conclusion of the whole work.},
     year = {2022}
    }
    

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    AU  - Yanick Alain Servais Wellot
    Y1  - 2022/11/22
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    AB  - In fluid mechanics, the Navier-Stokes equations are non-linear partial differential equations that describe the motion of Newtonian fluids. A fluid can be a liquid or a gas. Therefore, the Navier-Stokes equation concerns many phenomena that surround us. The analytical resolution, the search for exact solutions of these equations modeling a fluid is difficult. But they often allow, by an approximate resolution, to propose a model of many phenomena, such as ocean currents and air mass movements in the atmosphere for meteorologists, the behavior of skyscrapers or bridges under the action of wind for architects and engineers, or that of airplanes, trains or high-speed cars for their design offices, as well as the flow of water in a pipe and many other flow phenomena of various fluids. In mathematics, nonlinearity complicates things. In physics, too, the difficulty arises. For this term nonlinearity has its translation in the complexity of the physical phenomena described. This difficulty of resolution partly affects the analyses or descriptions of the modeled phenomena. The objective of this work is the search for exact solutions of the Navier-Stokes equations in dimension 2 and in dimension 3. The method of the reduced differential transform is used to find the exact solutions of these Navier-Stokes equations in 2D and 3D. This method gives an algorithm that favors the rapid convergence of the problem to the exact solution sought. Besides the introduction, this article is structured as follows: the presentation of the method, its application on the two selected Navier-Stokes problems whose exact solutions are obtained with ease, then intervenes the conclusion of the whole work.
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Author Information
  • Department of Exacts Sciences, Teachers Training College, Marien Ngouabi University, Brazzaville, Congo

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