Applied Mathematics and Mechanics >
Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet
Received date: 2013-07-23
Revised date: 2013-12-27
Online published: 2014-07-01
The present article investigates the dual nature of the solution of the magnetohydrodynamic (MHD) stagnation-point flow of a Prandtl fluid model towards a shrinking surface. The self-similar nonlinear ordinary differential equations are solved numerically by the shooting method. It is found that the dual solutions of the flow exist for certain values of the velocity ratio parameter. The special case of the first branch solutions (the classical Newtonian fluid model) is compared with the present numerical results of stretching flow. The results are found to be in good agreement. It is also shown that the boundary layer thickness for the second solution is thicker than that for the first solution.
N. S. AKBAR;Z. H. KHAN;R. U. HAQ;S. NADEEM . Dual solutions in MHD stagnation-point flow of Prandtl fluid impinging on shrinking sheet[J]. Applied Mathematics and Mechanics, 2014 , 35(7) : 813 -820 . DOI: 10.1007/s10483-014-1836-9
[1] Hiemenz, K. Die Grenzschicht an einem in den gleichförmigen Flüessigkeitsstrom eingetauchten geraden Kreiszylinder. Dinglers Polytechnisches Journal, 321, 321-410 (1911)
[2] Sakiadis, B. C. Boundary layer behavior on continuous solid flat surfaces. Journal of AICHE, 7, 26-28 (1961)
[3] Crane, L. Flow past a stretching plate. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 21, 645-647 (1970)
[4] Chiam, T. C. Stagnation-point flow towards a stretching plate. Journal of the Physical Society of Japan, 63, 2443-2444 (1994)
[5] Wang, C. Y. Free convection on a vertical stretching surface. Journal of Applied Mathematics and Mechanics (ZAMM), 69, 418-420 (1989)
[6] Nazar, R., Amin, N., Filip, D., and Pop, I. Unsteady boundary layer flow in the region of the stagnation-point on a stretching sheet. International Journal of Engineering Science, 42, 1241- 1253 (2004)
[7] Ishak, A., Nazar, R., and Pop, I. Magnetohydrodynamic stagnation point flow towards a stretching vertical sheet. Magnetohydrodynamic, 42, 17-30 (2006)
[8] Sadeghy, K., Hajibeygib, H., and Taghavia, S. M. Stagnation-point flow of upper-convected Maxwell fluids. International Journal of Non-Linear Mechanics, 41, 1242-1247 (2006)
[9] Attia, H. A. Axisymmetric stagnation point flow towards a stretching surface in the presence of a uniform magnetic field with heat generation. Tamkang Journal of Science and Engineering, 1, 11-16 (2007)
[10] Kumari, M. and Nath, G. Steady mixed convection stagnation-point flow of upper convected Maxwell fluids with magnetic field. International Journal of Non-Linear Mechanics, 44, 1048- 1055 (2009)
[11] Klemp, J. B. and Acrivos, A. A moving-wall boundary layer with reverse flow. Journal of Fluid Mechanics, 76, 363-381 (1976)
[12] Riley, N. and Weidman, P. D. Multiple solutions of the Falkner-Skan equation for flow past a stretching boundary. SIAM Journal on Applied Mathematics, 49, 1350-1358 (1989)
[13] Ingham, D. B. Singular and non-unique solutions of the boundary-layer equations for the flow due to free convection near a continuously moving vertical plate. Zeitschrift für Angewandte Mathe- matik und Physik (ZAMP), 37, 559-572 (1986)
[14] Ridha, A. and Curie, M. Aiding flows non-unique similarity solutions of mixed-convection boundary-layer equations. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 47, 341- 352 (1996)
[15] Mahapatra, T. R., Nandy, S. K., Vajravelu, K., and van Gorder, R. A. Stability analysis of the dual solutions for stagnation-point flow over a non-linearly stretching surface. Meccanica, 47, 1623-1632 (2012)
[16] Makinde, O. D., Khan, W. A., and Khan, Z. H. Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet. Interna- tional Journal of Heat and Mass Transfer, 62, 526-533 (2013)
[17] Akbar, N. S., Nadeem, S., and Lee, C. Peristaltic flow of a Prandtl fluid model in an asymmetric channel. International Journal of Physical Sciences, 7, 687-695 (2012)
[18] Haverkort, J. W. and Peeters, T. W. J. Magnetohydrodynamics of insulating spheres. Magneto- hydrodynamics, 45, 111-126 (2009)
[19] Haverkort, J. W. and Peeters, T. W. J. Magnetohydrodynamic effects on insulating bubbles and inclusions in the continuous casting of steel. Metallurgical and Materials Transactions B, 41, 1240-1246 (2010)
[20] Lok, Y. Y., Amin, N., and Pop, I. Non-orthogonal stagnation point towards a stretching sheet. International Journal of Non-Linear Mechanics, 41, 622-627 (2006)
[21] Wang, C. Y. Stagnation flow towards a shrinking sheet. International Journal of Non-Linear Mechanics, 43(5), 377-382 (2008)
[22] Mahapatra, T. R. and Nandy, S. K. Stability of dual solutions in stagnation-point flow and heat transfer over a porous shrinking sheet with thermal radiation. Meccanica, 48, 23-32 (2013)
[23] Merkin, J. H. On dual solutions occurring in mixed convection in a porous medium. Journal of Engineering Mathematics, 20, 171-179 (1985)
[24] Weidman, P. D., Kubitschek, D. G., and Davis, A. M. J. The effect of transpiration on selfsimilar boundary layer flow over moving surfaces. International Journal of Engineering Science, 44, 730-737 (2006)
[25] Paullet, J. and Weidman, P. Analysis of stagnation point flow towards a stretching sheet. Inter- national Journal of Non-Linear Mechanics, 42(9), 1084-1091 (2007)
[26] Harris, S. D., Ingham, D. B., and Pop, I.Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Transport in Porous Media, 77, 267-285 (2009)
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