Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation

Okey Oseloka Onyejekwe

doi:doi:10.11648/j.ijfmts.20170302.11

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Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation

Okey Oseloka Onyejekwe

Published in International Journal of Fluid Mechanics & Thermal Sciences (Volume 3, Issue 2)

Received: 12 December 2016 Accepted: 27 December 2016 Published: 24 April 2017

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Abstract

The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.

Published in	International Journal of Fluid Mechanics & Thermal Sciences (Volume 3, Issue 2)
DOI	10.11648/j.ijfmts.20170302.11
Page(s)	16-24
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), 2017. Published by Science Publishing Group

Keywords

Stagnation Point Flow, Heat Transfer, Prescribed Flux, Crank-Nicolson-Newton-Richtmeyer, Time Marching Scheme, Steady State, Prandtl Number, Stretching and Shrinking Sheet

References

[1]	A. Bejan, Convective heat transfer 2^nd ed. Wiley NY 1995.
[2]	K. Hiemenz, “Die grenschicht an einem in den gleichformingen flussgkeitsstrom eingetauchten ger-aden kreiszylinder, Dingl.”, Polytech. Journal vol. 32 pp. 321-410, 1911.
[3]	F. Homann, “Der einfluss grosset zahigkeit bei der stromung um den Zylinder and um die kugel”, Zeitschrift fur Angewandte Mathematic und mechanic, vol. 16 pp. 153-164, 1936.
[4]	T. Y. Na Computational methods in engineering boundary value problem, Academic Press, NY 1979.
[5]	L. J. Crane, “Flow past a stretching sheet”, Zeitschrift fur Angewandte Mathematik und Physik vol. 21 pp. 645-647, 1970.
[6]	P. Caragher and L. J. Crane, “Heat transfer n a continuous stretching sheet”, ZAMM vol. 62 pp. 564-577, 1985.
[7]	P. S. Gupta and A. S. Gupta, “Heat and mass transfer for a stretching sheet with uniform suction and blowing”, The Canadian Journal of Chemical Engineering, vol. 55 pp. 744-746, 1977.
[8]	T. C. Chiam, “Stagnation point flow towards a stretching sheet” J. Phys. Soc. Jpn. Vol. 63 pp. 2443-2455, 1994.
[9]	B. C. Sakiadis, “Boundary-layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow”, American institute of chemical engineers(AICHE) Journal vol. 7(1) pp. 26-28, 1961.
[10]	B. C. Sakiadis, “Boundary –layer behavior on continuous solid surfaces: II The boundary layer on a continuous flat surface”, AICHE J. vol. 7 pp. 221-225, 1961. Doi:10.1002/aic.690070211
[11]	T. R. Mahapatra and A. S. Gupta, “Heat transfer in stagnation-point flow towards a stretching sheet”, Heat and Mass Transfer, vol. 38 pp. 517-521, 2002.
[12]	P. Dulal and P. S. Hiremath, “Computational modelling of heat transfer over an unsteady stretching surface embedded in porous medium”, Mecccanica, vol. 45 pp. 415-524, 2009.
[13]	D. B. Ingham and I. Pop, Transport in porous media III Elsevier, Oxford 2005.
[14]	D. A. Nield and A. Bejan, Convection in porous media, 3^rd ed. Springer science and business media inc. N. York 2005.
[15]	B. K. Dutta, P. Roy A. S. Gupta,” Temperature field in flow over a stretching sheet with uniform heat flux”, Int. Comm. Heat and Mass Trans vol.28 pp. 1234-1237, 1985.
[16]	M. E. Ali and E. Magyari, “Unstaedy fluid and heat flow induced by a submerged stretching surface while its steady motion is slowing down gradually”, Int. Jnl. Heat and Mass Trans. vol. 50 pp. 188-195, 2007.
[17]	A. Ishak, R. Nazar, and I. Pop, “Magnetohydrodynamics stagnation point flow towards a stretching vertical sheet”, Magneto-hydrodynamics, vol. 42 pp.17-30, 2006.
[18]	H. I. Anderson, “MHD flow of viscoelastic fluid past a stretching surface”, Acta Mechanica,, vol. 95 pp. 227-230, 1992.
[19]	M. I. Char, “Heat and mass transfer in a hydromagnetic flow of the viscoelstic fluid past a stretching surface”, Physics of Fluids vol. 27 pp. 1915-1917, 1994.
[20]	T. R. Mahapatra and S. K. Nandy, “Momentum and heat transfer in MHD axisymmetric stagnation point flow over a shrinking sheet”, Jnl. Applied Fluid Mechanics, vol. 6 pp. 121-129, 2011.
[21]	O. D. Makinde and W. M. Charles, “Computational dynamics of hydromagnetic stagnation flow towards a stretching sheet”, Applied, Comput. Math. vol. 9 (2010) pp. 243-251, 2010
[22]	O. D. Makinde, “On MHD boundary –layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux”, Int. Jnl. Num. Mthds. For Heat and Fluid flow. Vol. 19 pp. 546-554, 2009.
[23]	K. Bhattacharyyya, “Heat transfer in boundary layer stagnation –point flow towards a shrinking sheet with non-uniform heat flux:”, Chin. Physics B vol. 222 pp. 07405-1-07405-5, 2013.
[24]	I. Mealey and J. H. Merkin, “Free convection boundary layers on a verical surface in a heat generating porous medium”, IMA Journal of Applied Mathematics vol. 73 pp. 231-253, 2008.
[25]	A. Postelnicu and I. Pop, “Similarity solution of free convection boundary layers over vertical and horizontal surfaces in porous media with internal heat generation”, Int. Comm. Heat and Mass Transfer, vol. 26 pp. 1183-1191, 1999.
[26]	D. C. Wilcox, Basic fluid mechanics 2^nd ed. DCW industries 2003.

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APA Style

Okey Oseloka Onyejekwe. (2017). Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. International Journal of Fluid Mechanics & Thermal Sciences, 3(2), 16-24. https://doi.org/10.11648/j.ijfmts.20170302.11

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ACS Style

Okey Oseloka Onyejekwe. Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. Int. J. Fluid Mech. Therm. Sci. 2017, 3(2), 16-24. doi: 10.11648/j.ijfmts.20170302.11

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AMA Style

Okey Oseloka Onyejekwe. Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. Int J Fluid Mech Therm Sci. 2017;3(2):16-24. doi: 10.11648/j.ijfmts.20170302.11

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@article{10.11648/j.ijfmts.20170302.11,
  author = {Okey Oseloka Onyejekwe},
  title = {Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation},
  journal = {International Journal of Fluid Mechanics & Thermal Sciences},
  volume = {3},
  number = {2},
  pages = {16-24},
  doi = {10.11648/j.ijfmts.20170302.11},
  url = {https://doi.org/10.11648/j.ijfmts.20170302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijfmts.20170302.11},
  abstract = {The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.},
 year = {2017}
}

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TY  - JOUR
T1  - Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation
AU  - Okey Oseloka Onyejekwe
Y1  - 2017/04/24
PY  - 2017
N1  - https://doi.org/10.11648/j.ijfmts.20170302.11
DO  - 10.11648/j.ijfmts.20170302.11
T2  - International Journal of Fluid Mechanics & Thermal Sciences
JF  - International Journal of Fluid Mechanics & Thermal Sciences
JO  - International Journal of Fluid Mechanics & Thermal Sciences
SP  - 16
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PB  - Science Publishing Group
SN  - 2469-8113
UR  - https://doi.org/10.11648/j.ijfmts.20170302.11
AB  - The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.
VL  - 3
IS  - 2
ER  -

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Author Information

Okey Oseloka Onyejekwe

Computational Science Program, Addis Ababa University, Arat Kilo Campus, Addis Ababa, Ethiopia

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APA Style

Okey Oseloka Onyejekwe. (2017). Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. International Journal of Fluid Mechanics & Thermal Sciences, 3(2), 16-24. https://doi.org/10.11648/j.ijfmts.20170302.11

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ACS Style

Okey Oseloka Onyejekwe. Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. Int. J. Fluid Mech. Therm. Sci. 2017, 3(2), 16-24. doi: 10.11648/j.ijfmts.20170302.11

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AMA Style

Okey Oseloka Onyejekwe. Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation. Int J Fluid Mech Therm Sci. 2017;3(2):16-24. doi: 10.11648/j.ijfmts.20170302.11

Copy | Download

@article{10.11648/j.ijfmts.20170302.11,
  author = {Okey Oseloka Onyejekwe},
  title = {Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation},
  journal = {International Journal of Fluid Mechanics & Thermal Sciences},
  volume = {3},
  number = {2},
  pages = {16-24},
  doi = {10.11648/j.ijfmts.20170302.11},
  url = {https://doi.org/10.11648/j.ijfmts.20170302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijfmts.20170302.11},
  abstract = {The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Unsteady Formulations for Stagnation Point Flow Towards a Stretching and Shrinking Sheet with Prescribed Surface Heat Flux and Viscous Dissipation
AU  - Okey Oseloka Onyejekwe
Y1  - 2017/04/24
PY  - 2017
N1  - https://doi.org/10.11648/j.ijfmts.20170302.11
DO  - 10.11648/j.ijfmts.20170302.11
T2  - International Journal of Fluid Mechanics & Thermal Sciences
JF  - International Journal of Fluid Mechanics & Thermal Sciences
JO  - International Journal of Fluid Mechanics & Thermal Sciences
SP  - 16
EP  - 24
PB  - Science Publishing Group
SN  - 2469-8113
UR  - https://doi.org/10.11648/j.ijfmts.20170302.11
AB  - The unsteady stagnation point flow and heat transfer with prescribed flux towards a stretching and shrinking sheet with viscous dissipation is studied. Similarity transformation is adopted to initially convert the governing differential equations into nonlinear ordinary differential equations. The two-point boundary value ordinary differential equations (ODE) are subsequently converted into partial differential equations by introducing a time-marching scheme. A Crank-Nicolson Newton-Richtmeyer scheme is employed to discretize the resulting equations. Initial guesses are made for the dependent variables and the solution advanced in time until temporal variations of the scalar profile are diminished and the steady-state solutions satisfy the similarity equations. A variation of the heat flux at one of the boundaries produced noticeable variations in the temperature field that can be related to the magnitude of the Prandtl number and velocity ratio parameter.
VL  - 3
IS  - 2
ER  -

Copy | Download