ENG  Vol.6 No.12 , November 2014
Analytic Solution of MHD Stagnation Point Flow over a Stretching Permeable Surface with Effects of Viscous Dissipation and Joule Heating
Abstract
A mathematical analysis is investigated to obtain an analytic solution of magneto hydrodynamic stagnation-point flow towards permeable stretching surface with viscous dissipation and joule heating. In the presence of uniform suction, a transverse magnetic field normal to the surface is applied when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. The governing nonlinear momentum and energy equations are solved by homotopy analysis method (HAM) to obtain the complete analytic solution and a good agreement is found. The convergence region shows the validity of the HAM solutions. It is observed that the velocity at a point increases/decreases more with increase in the magnetic parameter when the free stream velocity is greater/less than the stretching velocity in presence of suction. An interesting result of the analysis is that, in the presence of suction parameter, the temperature increases with the increase in magnetic parameter at a certain distance from the stretching surface. Near stagnation point on the surface, heat flows from the surface to the fluid and far from the stagnation-point, heat flows from the fluid to surface due to combining effect of ohmic dissipation and strain energy inside the boundary layer.

Cite this paper
Panigrahi, S. and Reza, M. (2014) Analytic Solution of MHD Stagnation Point Flow over a Stretching Permeable Surface with Effects of Viscous Dissipation and Joule Heating. Engineering, 6, 827-840. doi: 10.4236/eng.2014.612077.
References

[1]   Hiemenz, K. (1911) Die grenzschicht an einem in den gleichformingen flussigkeitsstrom eingetauchten geraden kreiszylinder. Dingler’s Polytechnic Journal, 326, 321-410.

[2]   Homann, F. (1943) Der einfluss grosser zahighkeit bei der stromung um den zylinder und um die kugel. Zeitschrift für Angewandte Mathematik und Mechanik, 16, 153-164.
http://dx.doi.org/10.1002/zamm.19360160304

[3]   Crane, L.J. (1970) Flow Past a Stretching Plate. Zeitschrift für Angewandte Mathematik und Physik, 21, 645-647.
http://dx.doi.org/10.1007/BF01587695

[4]   Dutta, B.K., Roy, P. and Gupta, A.S. (1985) Temperature Field in Flow over a Stretching Surface with Uniform Heat Flux. Int. Comm. International Communications in Heat and Mass Transfer, 12, 89-94.
http://dx.doi.org/10.1016/0735-1933(85)90010-7

[5]   Mahapatra, T.R. and Gupta, A.S. (2002) Heat Transfer in Stagnation-Point Flow towards a Stretching Sheet. Heat and Mass Transfer, 38, 517-521.
http://dx.doi.org/10.1007/s002310100215

[6]   Mahapatra, T.R. and Gupta, A.S. (2004) Stagnation Point Flow of a Viscoelastic Fluid. The International Journal of Non-Linear Mechanics, 39, 811-820.
http://dx.doi.org/10.1016/S0020-7462(03)00044-1

[7]   Hassanien, I.A. (2002) Similarity Solutions for Flow and Heat Transfer of a Viscoelastic Fluid over a Stretching Sheet Extruded in a Cross Cooling Stream. Zeitschrift für Angewandte Mathematik und Mechanik, 82, 409-419.
http://dx.doi.org/10.1002/1521-4001(200206)82:6<409::AID-ZAMM409>3.0.CO;2-1

[8]   Nazar, R., Amin, N., Filip, D. and Pop, I. (2004) Stagnation Point Flow of a Micropolar Fluid towards a Stretching Sheet. International Journal of Non-Linear Mechanics, 39, 1227-1235.
http://dx.doi.org/10.1016/j.ijnonlinmec.2003.08.007

[9]   Schlichting, H. and Bussmann, K. (1943) Exakte Losungen fur die Laminare Grenzchicht mitAbsaugung und Ausblasen. Schriften der Deutschen Akademie der Luftfahrtforschung B, 7, 25-69.

[10]   Ariel, P.D. (1994) Stagnation Point Flow with Suction: An Approximate Solution. Journal of Applied Mechanics, 61, 976-978.
http://dx.doi.org/10.1115/1.2901589

[11]   Weidman, P.D. and Mahalingam, S. (1997) Axisymmetric Stagnation Point Flow Impinging on a Transversely Oscillating Plate with Suction. Journal of Engineering Mathematics, 31, 305-318.
http://dx.doi.org/10.1023/A:1004211515780

[12]   Attia, H.A. (2003) Hydromagnetic Stagnation Point Flow with Heat Transfer over a Permeable Surface. Arabian Journal for Science and Engineering, 28, 107-112.

[13]   Attia, H.A. (2003) Homann Magnetic Flow and Heat Transfer with Uniform Suction or Injection. Canadian Journal of Physics, 81, 1223-1230.
http://dx.doi.org/10.1139/p03-061

[14]   Attia, H.A. and Seddeek, M.A. (2007) On the Effectiveness of Uniform Suction or Injection on Two Dimensional Stagnation-Point Flow towards a Stretching Surface with Heat Generation. Chemical Engineering Communications, 194, 553-564.
http://dx.doi.org/10.1080/00986440600992537

[15]   Pavlov, K.B. (1974) Magneto Hydrodynamic Flow of an Incompressible Viscous Fluid Caused by the Deformation of a Plane Surface. Magnitnaya Gidrodinamika, 4, 146-147.

[16]   Chakrabarti, A. and Gupta, A.S. (1979) Hydromagnetic Flow and Heat Transfer over a Stretching Sheet. Quarterly of Applied Mathematics, 37, 73-78.

[17]   Andersson, H.I. (1992) MHD Flow of a Viscoelastic Fluid Past a Stretching Surface. Acta Mechanica, 95, 227-230.
http://dx.doi.org/10.1007/BF01170814

[18]   Liao, S.J. (1992) The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai.

[19]   Hilton, P.J. (1953) An Introduction to Homotopy Theory. Cambridge University Press, Cambridge.

[20]   Grigolyuk, E.E. and Shalashilin, V.I. (1991) Problems of Nonlinear Deformation: The Continuation Method Applied to Nonlinear Problems in Solid Mechanics. Kluwer Academic Publishers, Kluwer.
http://dx.doi.org/10.1007/978-94-011-3776-8

[21]   Mahapatra, T.R. and Gupta, A.S. (2001) Magnetohydrodynamic Stagnation-Point Flow towards a Stretching Sheet. Acta Mechanica, 152, 191-196.
http://dx.doi.org/10.1007/BF01176953

[22]   Mahapatra, T.R., Dholey, S. and Gupta, A.S. (2007) Momentum and Heat Transfer in Magneto Hydrodynamic Stagnation-Point Flow of a Viscoelastic Fluid towards a Stretching Surface. Meccanica, 42, 263-272.
http://dx.doi.org/10.1007/s11012-006-9040-8

 
 
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