ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stagnation Point Solution Due to a Continuously Stretching Surface with Applied Magnetic Field Using HAM

Journal of Applied Nonlinear Dynamics 4(2) (2015) 141--152 | DOI:10.5890/JAND.2015.06.004

Rajeswari Seshadri; Shankar Rao Munjam

Department of Mathematics, Ramanujan School of Mathematical Sciences, Podicherry University Pondicherry-605 014, INDIA

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Abstract

In the present study, we consider the steady two-dimensional stagnation point flow due to a stretching surface in an ambient fluid. The fluid is viscous, incompressible and electrically conducting near the stagnation region on a stretching surface. A uniform magnetic field of strength B is applied in the positive y.direction normal to the stretching surface. The equations governing the flow are solved using Homotopy Analysis Method (HAM). The flow variables are computed in the form of a series with its coefficients containing the parameters such as the magnetic field and the ratio of stretching velocity so that the effect of these parameters on the flow variables can be computed without much computational effort. The optimal values of the convergence control parameters and the the averaged squared residual errors are computed for the flow variables. The software Mathematica is used to perform all the semi-analytical calculations. It was observed that the effect of ratio of stretching velocity V0 on flow velocity is very significant in the sense that zero skin friction occurs at V0 = 0.5. and the trend in the variation of velocity profiles as well as skin friction coefficients are just opposite for V0 < 0.5 and V0 > 0.5.

Acknowledgments

The author Shankar Rao Munjam gratefully acknowledge UGC- Rajiv Gandhi National Fellowship (RGN-SRF), Government of India for providing fiancial assistance.

References

1.  [1] Sakiadis, BC. (1961), Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface, AIChE Journal, 7(2), 221-225.
2.  [2] Crane, L.J. (1970), Flow past a stretching plate, Zeitschrift f angewandte Mathematik und Physik ZAMP, 21(4), 645-647.
3.  [3] Hiemenz, H. (1911), Die grenzschicht an einem in dengleich formizen flussigkeitsstrom eigetanch geraden kreiszlinder, Dingl Polytech. J, 326, 321-328.
4.  [4] Kumari, M. and Nath, G. (1999), Flow and heat transfer in a stagnation-point flow over a stretching sheet with a magnetic field, Mechanics Research Communications, 26(4), 469-478.
5.  [5] Kumari, M. and Nath, G. (1999), Development of flow and heat transfer of a viscous fluid in the stagnation- point region of a three-dimensional body with a magnetic field, Acta Mechanica, 135(1-2), 1-12.
6.  [6] Chiam, T.C. (1994), Stagnation Point flow towards a Stretching Plate, Journal of the Physical Society of Japan.
7.  [7] Mahapatra, T.R., and Gupta, A.S. (2001), Magnetohydrodynamic stagnation-point flow towards a stretching sheet, Acta Mechanica, 152(1-4), 191-196.
8.  [8] Mahapatra, T.R., Nandy, S.K., and Gupta, A. S. (2009), Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface, Applied Mathematics and Com- putation, 215(5), 1696-1710.
9.  [9] Mahapatra, T.R., Nandy, S.K., and Gupta, A.S. (2010), Dual solution of MHD stagnation-point flow towards a stretching surface, Engineering, (2), 299-305.
10.  [10] Ali, F.M., Nazar, R., Arifin, N.M., and Pop, I. (2011), MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field, Applied Mathematics and Mechanics, 32(4), 409-418.
11.  [11] Ali, F.M., Nazar, R., Arifin, N.M., and Pop, I. (2014), Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field, Applied Mathematics and Mechanics, 35(2), 155-166.
12.  [12] Ishak, A., Nazar, R., and Pop, I. (2007), Mixed convection on the stagnation point flow toward a vertical, continuously stretching sheet, Journal of Heat Transfer, 129(8), 1087-1090.
13.  [13] Ishak, A., Nazar, R., Arifin, N.M., Ali, F.M., and Pop, I. (2011), MHD stagnation-point flow towards a stretching sheet with prescribed surface heat flux, Sains Malaysiana, 40(10), 1193-1199.
14.  [14] Sharma, P.R., and Singh, G. (2009), Effects of variable thermal conductivity and heat source/sink on MHD flow near a stagnation point on a linearly stretching sheet, Journal of Applied fluid mechanics, 2(1), 13-21.
15.  [15] Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC press, Boca Raton: Chapman and Hall.
16.  [16] Liao, S.J. (2010), An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15(8), 2003-2016.
17.  [17] Liao, S.J. (2012), Homotopy Analysis Method in Nonlinear Differential Equations, Higher Education Press & Springer, Beijing & Heidelberg/New York/.
18.  [18] Oahimire, J.I., and Olajuwon, B.I. (2013), Hydromagnetic Flow Near a Stagnation Point on a Stretching Sheet with Variable Thermal Conductivity and Heat Source/Sink, International Journal of Applied Science and Engineering, 11(3), 331-341.
19.  [19] Rasekh, A., Farzaneh-Gord, M., Varedi, S.R., and Ganji, D.D. (2013), Analytical solution for magnetohydro- dynamic stagnation point flow and heat transfer over a permeable stretching sheet with chemical reaction, Journal of Theoretical and Applied Mechanics, 51(3), 675-686.
20.  [20] Kazem, S., Sanaeikia, A., Ahmadvand, M., and Saberi, H. (2011), An RBF solution to a stagnation point flow towards a stretching surface with heat generation, In Computational Science and Engineering (CSE), IEEE 14th International Conference, 239-244.
21.  [21] Cheng, J. and Dai, S. (2010), A uniformly valid series solution to the unsteady stagnation-point flow towards an impulsively stretching surface, Science China Physics, Mechanics and Astronomy, 53(3), 521-526.
22.  [22] Sinha, A., and Misra, J.C. (2014), Effect of Induced Magnetic Field on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer on a Stretching Sheet, Journal of Heat Transfer, 136(11), 112701.
23.  [23] Zhu, J., Zheng, L.C., and Zhang, X.X. (2009), Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis, Applied Mathematics and Mechanics, 30(4), 463-474.
24.  [24] Qi, D., and Hong-Qing, Z. (2009), Analytic solution for magnetohydrodynamic stagnation point flow towards a stretching sheet, Chinese Physics Letters, 26(10), 104701.
25.  [25] Akbar., N.S, Nadeem., S., Rizwan, UL Haq., and Shiwei Ye. (2014), MHD stagnation point flow of Carreau fluid toward a permeable shrinking sheet: Dual solutions, Ain Shams Engineering Journal, 5, 1233-1239.
26.  [26] Yadav, R.S. and Sharma, P.R. (2014), Effects of Heat Source/Sink on Stagnation Point Flow over A Stretching Sheet, International Journal of Engineering Research and Technology, 3(5), 1-8.
27.  [27] Malvandi, A., Hedayati, F., and Ganji, D.D. (2014), Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet, Powder Technology, 253, 377-384.
28.  [28] Ibrahim, W., and Makinde, O.D. (2015), Double-Diffusive in Mixed Convection and MHD Stagnation Point Flow of Nanofluid Over a Stretching Sheet, Journal of Nanofluids, 4(1), 28-37.
29.  [29] Hayat, T., Asad, S., Mustafa, M., and Alsaedi, A. (2015), MHD stagnation-point flow of Jeffrey fluid over a convectively heated stretching sheet, Computers & Fluids, 108, 179-185.
30.  [30] Hayat, T., Ali, S., Awais, M., and Alhuthali, M.S. (2015), Newtonian heating in stagnation point flow of Burgers fluid, Applied Mathematics and Mechanics, 36(1), 61-68.