Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect

Bahaa Saleh; Yousef Abdel-Rahim

doi:doi:10.11648/j.mma.20160101.13

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Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect

Bahaa Saleh, Yousef Abdel-Rahim

Published in Mathematical Modelling and Applications (Volume 1, Issue 1)

Received: 3 September 2016 Accepted: 8 October 2016 Published: 17 October 2016

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Abstract

The paper has presented and discussed a single generalized algebraic formulation for magneto-hydrodynamic (MHD) flow over an isothermal exponentially stretching sheet under an exponential magnetic field over a range of a magnetic parameter (M), 0≤M≤1.0 and has analyzed relative weights of different terms in the governing equation. Solution methodology is based on minimization of the residual of the governing equation and results are in perfect agreement with other previously published works. Wall shear stress has been formulated as single algebraic equation of M. Inside flow region, shear stress is maximum at the wall and suffers an exponential decrease in vicinity of sheet at similarity variable (η), η≤4.0, where 1^st and 3^rd terms in the governing equation are the most dominant terms. Within the vicinity of the sheet, the velocity has suffered an exponential decrease that became steeper with the increase of M, signifying a retardation effect of the magnetic field. Beyond η=4.0 the flow region is almost stagnant. The analysis shows that high nonlinearity of the governing equation has led to an oscillatory nature especially in the vicinity of the sheet, which becomes more damped at higher values of M. In the range, 0≤η≤0.25, the 2nd nonlinear term in the equation can be neglected, while in the range, 0.25≤η≤0.75, the 4th term can be neglected. In the range, 0.75≤η≤1.0 both the 3rd and 4th terms of the equation can be neglected. Although neglecting any term of the governing equation will be at the sacrifice of the accuracy of the solution, yet the 2nd term, which is nonlinear, can be totally deleted from the equation at a sacrifice of about 10% of the accuracy of the solution.

Published in	Mathematical Modelling and Applications (Volume 1, Issue 1)
DOI	10.11648/j.mma.20160101.13
Page(s)	13-19
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

MHD Boundary Layer Flow, Stretching Sheet, Magnetic Field, Shear Stress

References

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APA Style

Bahaa Saleh, Yousef Abdel-Rahim. (2016). Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Mathematical Modelling and Applications, 1(1), 13-19. https://doi.org/10.11648/j.mma.20160101.13

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ACS Style

Bahaa Saleh; Yousef Abdel-Rahim. Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Math. Model. Appl. 2016, 1(1), 13-19. doi: 10.11648/j.mma.20160101.13

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AMA Style

Bahaa Saleh, Yousef Abdel-Rahim. Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Math Model Appl. 2016;1(1):13-19. doi: 10.11648/j.mma.20160101.13

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@article{10.11648/j.mma.20160101.13,
  author = {Bahaa Saleh and Yousef Abdel-Rahim},
  title = {Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect},
  journal = {Mathematical Modelling and Applications},
  volume = {1},
  number = {1},
  pages = {13-19},
  doi = {10.11648/j.mma.20160101.13},
  url = {https://doi.org/10.11648/j.mma.20160101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160101.13},
  abstract = {The paper has presented and discussed a single generalized algebraic formulation for magneto-hydrodynamic (MHD) flow over an isothermal exponentially stretching sheet under an exponential magnetic field over a range of a magnetic parameter (M), 0≤M≤1.0 and has analyzed relative weights of different terms in the governing equation. Solution methodology is based on minimization of the residual of the governing equation and results are in perfect agreement with other previously published works. Wall shear stress has been formulated as single algebraic equation of M. Inside flow region, shear stress is maximum at the wall and suffers an exponential decrease in vicinity of sheet at similarity variable (η), η≤4.0, where 1st and 3rd terms in the governing equation are the most dominant terms. Within the vicinity of the sheet, the velocity has suffered an exponential decrease that became steeper with the increase of M, signifying a retardation effect of the magnetic field. Beyond η=4.0 the flow region is almost stagnant. The analysis shows that high nonlinearity of the governing equation has led to an oscillatory nature especially in the vicinity of the sheet, which becomes more damped at higher values of M. In the range, 0≤η≤0.25, the 2nd nonlinear term in the equation can be neglected, while in the range, 0.25≤η≤0.75, the 4th term can be neglected. In the range, 0.75≤η≤1.0 both the 3rd and 4th terms of the equation can be neglected. Although neglecting any term of the governing equation will be at the sacrifice of the accuracy of the solution, yet the 2nd term, which is nonlinear, can be totally deleted from the equation at a sacrifice of about 10% of the accuracy of the solution.},
 year = {2016}
}

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TY  - JOUR
T1  - Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect
AU  - Bahaa Saleh
AU  - Yousef Abdel-Rahim
Y1  - 2016/10/17
PY  - 2016
N1  - https://doi.org/10.11648/j.mma.20160101.13
DO  - 10.11648/j.mma.20160101.13
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 13
EP  - 19
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20160101.13
AB  - The paper has presented and discussed a single generalized algebraic formulation for magneto-hydrodynamic (MHD) flow over an isothermal exponentially stretching sheet under an exponential magnetic field over a range of a magnetic parameter (M), 0≤M≤1.0 and has analyzed relative weights of different terms in the governing equation. Solution methodology is based on minimization of the residual of the governing equation and results are in perfect agreement with other previously published works. Wall shear stress has been formulated as single algebraic equation of M. Inside flow region, shear stress is maximum at the wall and suffers an exponential decrease in vicinity of sheet at similarity variable (η), η≤4.0, where 1st and 3rd terms in the governing equation are the most dominant terms. Within the vicinity of the sheet, the velocity has suffered an exponential decrease that became steeper with the increase of M, signifying a retardation effect of the magnetic field. Beyond η=4.0 the flow region is almost stagnant. The analysis shows that high nonlinearity of the governing equation has led to an oscillatory nature especially in the vicinity of the sheet, which becomes more damped at higher values of M. In the range, 0≤η≤0.25, the 2nd nonlinear term in the equation can be neglected, while in the range, 0.25≤η≤0.75, the 4th term can be neglected. In the range, 0.75≤η≤1.0 both the 3rd and 4th terms of the equation can be neglected. Although neglecting any term of the governing equation will be at the sacrifice of the accuracy of the solution, yet the 2nd term, which is nonlinear, can be totally deleted from the equation at a sacrifice of about 10% of the accuracy of the solution.
VL  - 1
IS  - 1
ER  -

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Author Information

Bahaa Saleh

Mechanical Engineering Department, College of Engineering, Taif University, Taif, Saudi Arabia; Mechanical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt
Yousef Abdel-Rahim

Mechanical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt

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APA Style

Bahaa Saleh, Yousef Abdel-Rahim. (2016). Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Mathematical Modelling and Applications, 1(1), 13-19. https://doi.org/10.11648/j.mma.20160101.13

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ACS Style

Bahaa Saleh; Yousef Abdel-Rahim. Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Math. Model. Appl. 2016, 1(1), 13-19. doi: 10.11648/j.mma.20160101.13

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AMA Style

Bahaa Saleh, Yousef Abdel-Rahim. Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect. Math Model Appl. 2016;1(1):13-19. doi: 10.11648/j.mma.20160101.13

Copy | Download

@article{10.11648/j.mma.20160101.13,
  author = {Bahaa Saleh and Yousef Abdel-Rahim},
  title = {Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect},
  journal = {Mathematical Modelling and Applications},
  volume = {1},
  number = {1},
  pages = {13-19},
  doi = {10.11648/j.mma.20160101.13},
  url = {https://doi.org/10.11648/j.mma.20160101.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20160101.13},
  abstract = {The paper has presented and discussed a single generalized algebraic formulation for magneto-hydrodynamic (MHD) flow over an isothermal exponentially stretching sheet under an exponential magnetic field over a range of a magnetic parameter (M), 0≤M≤1.0 and has analyzed relative weights of different terms in the governing equation. Solution methodology is based on minimization of the residual of the governing equation and results are in perfect agreement with other previously published works. Wall shear stress has been formulated as single algebraic equation of M. Inside flow region, shear stress is maximum at the wall and suffers an exponential decrease in vicinity of sheet at similarity variable (η), η≤4.0, where 1st and 3rd terms in the governing equation are the most dominant terms. Within the vicinity of the sheet, the velocity has suffered an exponential decrease that became steeper with the increase of M, signifying a retardation effect of the magnetic field. Beyond η=4.0 the flow region is almost stagnant. The analysis shows that high nonlinearity of the governing equation has led to an oscillatory nature especially in the vicinity of the sheet, which becomes more damped at higher values of M. In the range, 0≤η≤0.25, the 2nd nonlinear term in the equation can be neglected, while in the range, 0.25≤η≤0.75, the 4th term can be neglected. In the range, 0.75≤η≤1.0 both the 3rd and 4th terms of the equation can be neglected. Although neglecting any term of the governing equation will be at the sacrifice of the accuracy of the solution, yet the 2nd term, which is nonlinear, can be totally deleted from the equation at a sacrifice of about 10% of the accuracy of the solution.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Analysis of a Generalized Formulation of MHD Isothermal Flow over Exponentially Stretching Sheet Under Variable Magnetic Effect
AU  - Bahaa Saleh
AU  - Yousef Abdel-Rahim
Y1  - 2016/10/17
PY  - 2016
N1  - https://doi.org/10.11648/j.mma.20160101.13
DO  - 10.11648/j.mma.20160101.13
T2  - Mathematical Modelling and Applications
JF  - Mathematical Modelling and Applications
JO  - Mathematical Modelling and Applications
SP  - 13
EP  - 19
PB  - Science Publishing Group
SN  - 2575-1794
UR  - https://doi.org/10.11648/j.mma.20160101.13
AB  - The paper has presented and discussed a single generalized algebraic formulation for magneto-hydrodynamic (MHD) flow over an isothermal exponentially stretching sheet under an exponential magnetic field over a range of a magnetic parameter (M), 0≤M≤1.0 and has analyzed relative weights of different terms in the governing equation. Solution methodology is based on minimization of the residual of the governing equation and results are in perfect agreement with other previously published works. Wall shear stress has been formulated as single algebraic equation of M. Inside flow region, shear stress is maximum at the wall and suffers an exponential decrease in vicinity of sheet at similarity variable (η), η≤4.0, where 1st and 3rd terms in the governing equation are the most dominant terms. Within the vicinity of the sheet, the velocity has suffered an exponential decrease that became steeper with the increase of M, signifying a retardation effect of the magnetic field. Beyond η=4.0 the flow region is almost stagnant. The analysis shows that high nonlinearity of the governing equation has led to an oscillatory nature especially in the vicinity of the sheet, which becomes more damped at higher values of M. In the range, 0≤η≤0.25, the 2nd nonlinear term in the equation can be neglected, while in the range, 0.25≤η≤0.75, the 4th term can be neglected. In the range, 0.75≤η≤1.0 both the 3rd and 4th terms of the equation can be neglected. Although neglecting any term of the governing equation will be at the sacrifice of the accuracy of the solution, yet the 2nd term, which is nonlinear, can be totally deleted from the equation at a sacrifice of about 10% of the accuracy of the solution.
VL  - 1
IS  - 1
ER  -

Copy | Download