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

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 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.

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), 2016. Published by Science Publishing Group

Keywords

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

References
[1] Sakiadis BC. Boundary layer behavior on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axisymmetric flow. AIChE J 1961; 5: 26-28.
[2] Erickson LE, Fan LT, Fox VG. Heat and mass transfer on a moving continuous flat sheet with suction or injection. Int. Eng. Chem. 1966; 5: 19-25.
[3] Lawrence J. Crane. Flow past a stretching sheet. Zeitschrift für angewandte Mathematik und Physik ZAMP 1970; 21(4): 645-647.
[4] Magyari E, Keller B. Exact Solutions for self-similar boundary-layer flows induced by permeable stretching walls. Eur. J. Mech. -B/Fluids 2000; 19: 109-122.
[5] Elbashbeshy EMA. Heat transfer over an exponentially stretching continuous surface with suction. Arch. Mech. 2001; 53 (6): 643-651.
[6] Mukhopadhyay S. Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation. Ain Shams Engineering Journal 2013; 4: 485-491.
[7] Kumaran V, Banerjee AK, Kumar AV, Vajravelu K. MHD flow past a stretching permeable sheet. Appl. Math. Comput. 2009; 210, : 26-32.
[8] Gupta PS, Gupta AS. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 1977; 55: 744-746.
[9] Pavlov KB. Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnitnaya Gidrodinamika; (USSR) 1974; 4: 146-147.
[10] Kumaran V, Ramanaiah G. A note on the flow over a stretching sheet. Acta Mech. 1996; 116: 229-233.
[11] Liao SJ. A new branch of solution of boundary layer flows over an impermeable stretched sheet. Int. J. Heat Mass Transfer 2005; 48: 2529-2539.
[12] Mabood F, Khan WA, Ismail AI. MHD flow over exponential radiating stretching sheet using homotopy analysis method. J of King Saud Univ. - Eng. Sci. 2014: article in press.
[13] Liao SJ. A new branch of solution of boundary-layer flows over a permeable stretching sheet. Int. J. Nonlinear Mech. 2007; 42: 819-830.
[14] Abbasbandy S. Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass Transfer 2007; 34: 380-387.
[15] Sajid M. Hayat T. Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet. Int. Commun. Heat Mass Transfer 2008; 35: 347-356.
[16] Rashidi MM, Ali M, Freidoonimehr N, Nazari F. Parametric analysis and optimization of entropy generation in unsteady MHD flow over a stretching rotating disk using artificial neural network and particle swarm optimization algorithm. Energy 2013; 55: 497-510.
[17] Rashidi MM, Ali M, Freidoonimehr N, Hossenin A, Anwar BO, Hung TK. Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica 2014; 49: 469-482.
[18] Tamizharasi R, Kumaran V. Pressure in MHD/Brinkman flow past a stretching sheet. Commun Nonlinear Sci Numer Simulat 2011; 16: 4671-4681.
[19] Kumaran V., Tamizharasi R. Brinkman flow past a stretching sheet. Transp Porous Med. 2011; 87: 541-60.
[20] Sahoo B. Effects of slip, viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet. Appl Math Mech-Engl Ed 2010; 31(2): 159-73.
[21] Kumaran V, Banerjee AK, Vanavkumar A, Vajravelu K. MHD flow past a stretching permeable sheet. Appl Math Comput 2009; 210: 26-32.
[22] Pantokratoras A. Flow adjacent to a stretching permeable sheet in a Darcy–Brinkman porous medium. Transp Porous Med 2009; 80: 223-7.
[23] Sajid M, Ali N, Javed T, Abbas Z. Stretching a curved surface in a viscous fluid. Chin. Phys. Lett. 2010; 27: 024703.
[24] Hsiao KL. Heat and mass convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipation. Commun. Nonlinear Sci. Numer. Simul. 2010; 15: 1803-1812.
[25] Abbas Z, Wang Y, Hayat T, Oberlack M. Mixed convection in the stagnation point flow of aMaxwell fluid towards a vertical stretching surface. Nonlinear Anal.: Real World Appl. 2010; 11: 3218-3228.
[26] Sajid M, Ahmed B, Abbas Z. Steady mixed convection stagnation point flow of MHD Oldroyd-B fluid over a stretching sheet. Journal of the Egyptian Mathematical Society 2014; article in press.
[27] Tiegang F, Zhang Ji, Shanshan Yao. A new family of unsteady boundary layers over a stretching surface. Applied Mathematics and Computation 2010; 217: 3747-3755.
[28] Kumaran V, Banerjee AK, Vanav Kumar A, Vajravelu K. MHD flow past a stretching permeable sheet. Applied Mathematics and Computation 2009; 210: 26-32.
[29] Gowdara MP, Bijjanal JG. Unsteady flow and heat transfer of a fluid-particle suspension over an exponentially stretching sheet. Ain Shams Engineering Journal 2014; 5: 613-624.
[30] Ishak, A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Sains Malaysiana 2011; 40(4): 391-5.
[31] Abdel-Rahim YM, Rahman MM. Laminar semi-porous channel electrically conducting flow under magnetic field. 10th Int. Conf. on Heat Transfer, Fluid Mechanics and Thermodynamics. HEFAT2014. Orlando, Florida, 14-16 July 2014.
[32] Abdel-Rahim YM, Abou Al-Sood MM, Ahmed M. Modeling of laminar viscous flow in a semi-porous channel under uniform magnetic field, ASME- IMECE 2011-62883, November 11- 17, Denver, CO, USA., 2011, Technical Publication#IMECE2011-62883.
Cite This Article
  • 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
  • Mechanical Engineering Department, College of Engineering, Taif University, Taif, Saudi Arabia

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

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