boundary layer flow and heat transfer due to an exponentially shrinking sheet with variable magnetic...

www.ijsret.org 617 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882 Volume 4, Issue 6, June 2015 Boundary Layer Flow and Heat Transfer due to an Exponentially Shrinking Sheet with Variable Magnetic Field Rakesh Kumar Department of Mathematics, Central University of Himachal Pradesh, India E-Mail: [email protected] ABSTRACT An analysis is carried out to study the effects of variable magnetic field on the flow field and heat transfer over an exponentially shrinking sheet. Using the exponential similarity transformations, the governing equations are transformed into self similar nonlinear ordinary differential equations. The transformed differential equations are solved using a power series expansion to obtain a closed form solution. The numerical results are depicted through graphs to illustrate the influence of pertinent parameters of the problem. It is found that suction parameter and magnetic field have substantial effect on velocity and temperature profiles. Keywords: Exponentially shrinking sheet, variable magnetic field, Heat transfer. I. INTRODUCTION The study of boundary layer flow and heat transfer over a stretching/shrinking surface has attracted the attention of several researchers due to its various applications in the fields of technology and industry. The extraction and manufacturing of polymer sheets, hot rolling and glass-fiber production, paper production, cooling of metallic sheets or electronic chips, processing of magnetic materials, MHD electrical power generation and purification of crude oil are some examples for the application of these flows (Fang et al.[1]) . In shrinking sheet problems, the surface is stretched towards a slot, causing a velocity away from the sheet. The physical grounds reveals that vorticity flow over the shrinking sheet does not remain confined and the flow is unlikely to exist (Lok et al.[2]). Mahapatra and Nandy [3] in their stability analysis found that an adequate suction or stagnation flow is required to confine the vorticity with in the boundary layer. Wang [4] during his investigatio n of liquid film behaviour on an unsteady stretching sheet observed this unusual flow due to shrinking surface. Miklavcic and Wang [5] obtained the existence and uniqueness condi tions for the similarity solution of viscous fluid over shrinking surfaces and showed that the behaviour of fluid depends on the extern ally imposed ma ss suction. A new of Blasius solution over shrinking shee t was reported by Fang et al. [6]. Hayat et al. [7] analyzed the three dimensional rotating flow due to a shrinking sheet. Fang [8] also presented a solution for the boundary layer flow over a shrinking sheet with power law velocity . Fang and Zhang [9] obtained an analytical solution for the heat transfer over a shrinking sheet. VanGorder and Vajravelu [10] discussed about the multiple solutions for the MHD flow over a stretching or shrinking sheet. The dual and triple solutions for MHD slip flow of non-Newtonian fluid over a shrinking surface were obtained by Turkyilmazoglu [11]. Further, the dual solutions for unsteady stagnation flow over a shrinking sheet were investigated by Bhattacharyya [12]. In the above research paper, the authors consider either linear or non-linear sheets, and there is lesser number of research papers available in literature for flows over exponentially shrinking surfaces. Moreover, the dynamics of fluid flows over shrinking surfaces is still unknown. Magyari and Keller [13] are assumed to be the first one to study the boundary flow over an exponentially stretching sheet. In this sequence, the other paper on the boundary layer flow and heat transfer over an exponentially shrinking sheet was reported by Bhattacharyya [14]. Bachok et al. [15] considers an exponentially stretching/shrinking sheet in nanofluids to study the stagnation point flow and corresponding heat transfer. The stagnation point flow and heat transfer over an exponentially shrinking sheet was investigated by Bhattacharyya and Vajravelu [16] and was extended by Rohni et al. [17] by considering the effects of free convection and mass suction. Bhattacharyya et al. [18] in his paper found that a strong magnetic field has predominant effect on the flow field and due to this; the similarity solution of the flow over a shrinking sheet is always unique. In some of the above said papers the uniform strength of magnetic field was assumed. But in general, its non-linear characteristics can be utilized to predict and control the behaviour of fluid flows over shrinking surfaces. Motivated by this, the objective of the present study is to analyze the effect of variable magnetic field on the flow and heat transfer of a viscous fluid over an exponentially shrinking sheet.

boundary layer flow and heat transfer due to an exponentially shrinking sheet with variable magnetic...

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