Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane

Edris Ghonoodi; Davood Domeiri Ganji

doi:doi:10.11648/j.engmath.20210502.11

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Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane

Edris Ghonoodi, Davood Domeiri Ganji

Published in Engineering Mathematics (Volume 5, Issue 2)

Received: 18 May 2021 Accepted: 10 July 2021 Published: 15 July 2021

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Abstract

In this study, two highly accurate and simple analytical methods (known as semi exact solutions), the variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied for illustrating transient condition of viscous fluid flow over oscillating plane and also oscillating viscous fluid flow over stationary plane. The flow of an incompressible viscous fluid, caused by the oscillation of a flat wall and also the flow of an oscillating fluid flow over stationary wall are considered by Navier-Stokes equations and are subjected to the behavior of fluid flow in boundary layer at transient condition. The main purpose of this article is to solve transient Navier-Stokes first and second equations in new mathematical solving method which is called semi exact solutions where in each case, the velocity of viscous fluid is determined as a function of time and also vertical distance from plane in boundary layer at transient condition. Results reveal the boundary layer thickness and also the transient fluid flow velocity in boundary layer and even more it shows that the (VIM) and (ADM) methods are very effective and accurate in comparison with the exact solution results. The results demonstrate the velocity of fluid in boundary layer as a function of displacement and time and it is shown that in different time, the value of velocity obtained by “VIM” and “ADM” solving methods is almost equal to velocity which is derived from exact or numerical solutions. So the main background and reason of applying the mentioned methods is to verify the accuracy of “VIM” and “ADM” in solving different fluid mechanics equations especially Navier-Stokes equations.

Published in	Engineering Mathematics (Volume 5, Issue 2)
DOI	10.11648/j.engmath.20210502.11
Page(s)	13-21
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), 2021. Published by Science Publishing Group

Keywords

Analytical Methods, VIM, ADM, Viscous Flow, Oscillating Wall, Stationary Wall, Transient Condition, Stokes Equation

References

[1]	Incompressible Flow, Fourth Edition. Ronald L. Panton (2013).
[2]	A note on an unsteady flow of a viscous fluid due to an oscillating plane wall, M. Emin Erdogan (2000).
[3]	The Modified Variational Iteration Method for Solving Linear and Nonlinear Ordinary Differential Equations, Fariborzi Araghi, Gholizadeh Dogaheh, Sayadi (2011).
[4]	Davood Domairry Ganji & Seyed Hashemi Kachapi, Nonlinear Equations Analytical Methods and Applications, Springer, (2012).
[5]	Adomian G. A review of the decomposition method in applied mathematics. J Math Anal Appl (1988), 501–544.
[6]	A. M. Wazwaz A new approach to the nonlinear advection problem: An application of the decomposition method, Applied Mathematics and Computation, 72 (1995), 175-181.
[7]	Luo XG. A two-step Adomian Decomposition Method. Appl Math Comput (2005), 570–583.
[8]	VISCOUSFLUID FLOW, Tasos C. Papanastasiou Georgios C. Georgiou, (2000).
[9]	C. Mamaloukas, Numerical solution of the flow of a viscous conducting fluid produced by an oscillating plane wall subjected to a transvers magnetic field (2004), 1-8.
[10]	Fluid Mechanics, SG2214, HT2010, (2010), 1-4.
[11]	ON A UNIQUE NONLINEAR OSCILLATOR, P. M. MATHEWS and M. LAKSHMANAN, (1974).
[12]	Progress in nonlinear science, Davood Domairry Ganji & Seyed Hashemi Kachapi, (2011).
[13]	Viscous fluid flow inside an oscillating cylinder and its extension to Stokes’ second problem, Physics of Fluids 32, 043601 (2020).
[14]	Boundary Layer over a Flat Plate, P. P. Puttkammer, (2013)
[15]	On the Study of Oscillating Viscous Flows by Using the Adomian-Padé Approximation, Chi-Min Liu, (2015).

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Edris Ghonoodi, Davood Domeiri Ganji. (2021). Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Engineering Mathematics, 5(2), 13-21. https://doi.org/10.11648/j.engmath.20210502.11

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ACS Style

Edris Ghonoodi; Davood Domeiri Ganji. Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Eng. Math. 2021, 5(2), 13-21. doi: 10.11648/j.engmath.20210502.11

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AMA Style

Edris Ghonoodi, Davood Domeiri Ganji. Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Eng Math. 2021;5(2):13-21. doi: 10.11648/j.engmath.20210502.11

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@article{10.11648/j.engmath.20210502.11,
  author = {Edris Ghonoodi and Davood Domeiri Ganji},
  title = {Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {2},
  pages = {13-21},
  doi = {10.11648/j.engmath.20210502.11},
  url = {https://doi.org/10.11648/j.engmath.20210502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210502.11},
  abstract = {In this study, two highly accurate and simple analytical methods (known as semi exact solutions), the variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied for illustrating transient condition of viscous fluid flow over oscillating plane and also oscillating viscous fluid flow over stationary plane. The flow of an incompressible viscous fluid, caused by the oscillation of a flat wall and also the flow of an oscillating fluid flow over stationary wall are considered by Navier-Stokes equations and are subjected to the behavior of fluid flow in boundary layer at transient condition. The main purpose of this article is to solve transient Navier-Stokes first and second equations in new mathematical solving method which is called semi exact solutions where in each case, the velocity of viscous fluid is determined as a function of time and also vertical distance from plane in boundary layer at transient condition. Results reveal the boundary layer thickness and also the transient fluid flow velocity in boundary layer and even more it shows that the (VIM) and (ADM) methods are very effective and accurate in comparison with the exact solution results. The results demonstrate the velocity of fluid in boundary layer as a function of displacement and time and it is shown that in different time, the value of velocity obtained by “VIM” and “ADM” solving methods is almost equal to velocity which is derived from exact or numerical solutions. So the main background and reason of applying the mentioned methods is to verify the accuracy of “VIM” and “ADM” in solving different fluid mechanics equations especially Navier-Stokes equations.},
 year = {2021}
}

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TY  - JOUR
T1  - Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane
AU  - Edris Ghonoodi
AU  - Davood Domeiri Ganji
Y1  - 2021/07/15
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210502.11
DO  - 10.11648/j.engmath.20210502.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 13
EP  - 21
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210502.11
AB  - In this study, two highly accurate and simple analytical methods (known as semi exact solutions), the variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied for illustrating transient condition of viscous fluid flow over oscillating plane and also oscillating viscous fluid flow over stationary plane. The flow of an incompressible viscous fluid, caused by the oscillation of a flat wall and also the flow of an oscillating fluid flow over stationary wall are considered by Navier-Stokes equations and are subjected to the behavior of fluid flow in boundary layer at transient condition. The main purpose of this article is to solve transient Navier-Stokes first and second equations in new mathematical solving method which is called semi exact solutions where in each case, the velocity of viscous fluid is determined as a function of time and also vertical distance from plane in boundary layer at transient condition. Results reveal the boundary layer thickness and also the transient fluid flow velocity in boundary layer and even more it shows that the (VIM) and (ADM) methods are very effective and accurate in comparison with the exact solution results. The results demonstrate the velocity of fluid in boundary layer as a function of displacement and time and it is shown that in different time, the value of velocity obtained by “VIM” and “ADM” solving methods is almost equal to velocity which is derived from exact or numerical solutions. So the main background and reason of applying the mentioned methods is to verify the accuracy of “VIM” and “ADM” in solving different fluid mechanics equations especially Navier-Stokes equations.
VL  - 5
IS  - 2
ER  -

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Author Information

Edris Ghonoodi

Departement of Mechanical Engineering, Babol University of Technology, Babol, Iran
Davood Domeiri Ganji

Departement of Mechanical Engineering, Babol University of Technology, Babol, Iran

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APA Style

Edris Ghonoodi, Davood Domeiri Ganji. (2021). Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Engineering Mathematics, 5(2), 13-21. https://doi.org/10.11648/j.engmath.20210502.11

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ACS Style

Edris Ghonoodi; Davood Domeiri Ganji. Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Eng. Math. 2021, 5(2), 13-21. doi: 10.11648/j.engmath.20210502.11

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AMA Style

Edris Ghonoodi, Davood Domeiri Ganji. Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane. Eng Math. 2021;5(2):13-21. doi: 10.11648/j.engmath.20210502.11

Copy | Download

@article{10.11648/j.engmath.20210502.11,
  author = {Edris Ghonoodi and Davood Domeiri Ganji},
  title = {Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {2},
  pages = {13-21},
  doi = {10.11648/j.engmath.20210502.11},
  url = {https://doi.org/10.11648/j.engmath.20210502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210502.11},
  abstract = {In this study, two highly accurate and simple analytical methods (known as semi exact solutions), the variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied for illustrating transient condition of viscous fluid flow over oscillating plane and also oscillating viscous fluid flow over stationary plane. The flow of an incompressible viscous fluid, caused by the oscillation of a flat wall and also the flow of an oscillating fluid flow over stationary wall are considered by Navier-Stokes equations and are subjected to the behavior of fluid flow in boundary layer at transient condition. The main purpose of this article is to solve transient Navier-Stokes first and second equations in new mathematical solving method which is called semi exact solutions where in each case, the velocity of viscous fluid is determined as a function of time and also vertical distance from plane in boundary layer at transient condition. Results reveal the boundary layer thickness and also the transient fluid flow velocity in boundary layer and even more it shows that the (VIM) and (ADM) methods are very effective and accurate in comparison with the exact solution results. The results demonstrate the velocity of fluid in boundary layer as a function of displacement and time and it is shown that in different time, the value of velocity obtained by “VIM” and “ADM” solving methods is almost equal to velocity which is derived from exact or numerical solutions. So the main background and reason of applying the mentioned methods is to verify the accuracy of “VIM” and “ADM” in solving different fluid mechanics equations especially Navier-Stokes equations.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Application of Analytical Methods About Equations of Stokes for Transient Condition in Flow Over Oscillating Plane and Oscillating Flow Over Stationary Plane
AU  - Edris Ghonoodi
AU  - Davood Domeiri Ganji
Y1  - 2021/07/15
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210502.11
DO  - 10.11648/j.engmath.20210502.11
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 13
EP  - 21
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210502.11
AB  - In this study, two highly accurate and simple analytical methods (known as semi exact solutions), the variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied for illustrating transient condition of viscous fluid flow over oscillating plane and also oscillating viscous fluid flow over stationary plane. The flow of an incompressible viscous fluid, caused by the oscillation of a flat wall and also the flow of an oscillating fluid flow over stationary wall are considered by Navier-Stokes equations and are subjected to the behavior of fluid flow in boundary layer at transient condition. The main purpose of this article is to solve transient Navier-Stokes first and second equations in new mathematical solving method which is called semi exact solutions where in each case, the velocity of viscous fluid is determined as a function of time and also vertical distance from plane in boundary layer at transient condition. Results reveal the boundary layer thickness and also the transient fluid flow velocity in boundary layer and even more it shows that the (VIM) and (ADM) methods are very effective and accurate in comparison with the exact solution results. The results demonstrate the velocity of fluid in boundary layer as a function of displacement and time and it is shown that in different time, the value of velocity obtained by “VIM” and “ADM” solving methods is almost equal to velocity which is derived from exact or numerical solutions. So the main background and reason of applying the mentioned methods is to verify the accuracy of “VIM” and “ADM” in solving different fluid mechanics equations especially Navier-Stokes equations.
VL  - 5
IS  - 2
ER  -

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