Journal of Engineering

Volume 2016 (2016), Article ID 6752520, 8 pages

http://dx.doi.org/10.1155/2016/6752520

## Interaction of Magnetic Field and Nonlinear Convection in the Stagnation Point Flow over a Shrinking Sheet

Department of Mathematics, Central University of Himachal Pradesh, TAB, Shahpur, Kangra, Himachal Pradesh 176206, India

Received 27 November 2015; Revised 26 March 2016; Accepted 28 March 2016

Academic Editor: Yuanxin Zhou

Copyright © 2016 Rakesh Kumar and Shilpa Sood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The steady two-dimensional boundary layer stagnation point flow due to a shrinking sheet is analyzed. The combined effects of magnetic field and nonlinear convection are taken into account. The governing equations for the flow are modeled and then simplified using the similarity transformation and boundary layer approach. The numerical solution of the reduced equations is obtained by the second-order finite difference scheme also known as Keller box method. The influence of the pertinent parameters of the problem on velocity and temperature profiles, skin friction, and sheet temperature gradient are presented through the graphs and tables and discussed. The magnetic field and nonlinear convection parameters significantly enhance the solution range.

#### 1. Introduction

The analysis of the hydromagnetic flow over stretching or shrinking surfaces is very demanding due to wide range of its applications in industry, physics, and engineering sciences including bioengineering. The impact of magnetic field on the flow of an electrically conducting viscous fluid finds its applications in purification of crude oil, glass manufacturing, paper production, polymer sheets, MHD electrical power generation, magnetic material processing, and so forth [1]. Moreover, the final product relies on the rate of cooling, which is decided by the configuration of the boundary layer near the stretching/shrinking sheet. Chakrabarti and Gupta [2] investigated the hydromagnetic flow and heat transfer over a stretching surface. Zhang and Wang [3] presented a rigorous mathematical analysis to analyze the MHD flow of power law fluid over a stretching sheet. The axially symmetric stagnation point flow of an electrically conducting fluid under transverse magnetic field was examined by Kakutani [4]. An analysis for three-dimensional stagnation point flow over a stretching surface was made by Attia [5] considering magnetic field and heat generation. Ali et al. [6] extended the above paper by considering the induced magnetic field. Recently, Ali et al. [7] have reported on the effects of mixed convection parameter and magnetic field over a vertical stretching sheet in the neighborhood of the stagnation point. Mahapatra et al. [8] investigated the MHD stagnation point flow of power law fluid over a sheet which is stretching in its own plane with a velocity proportional to the distance from the stagnation point. Very recently, Khan et al. [9] analyzed the thermodiffusion effects on the MHD stagnation point flow of nanofluid over a stretching sheet. Hayat et al. [10] investigated the stagnation point flow on a non-Newtonian fluid over a stretching sheet. Some significant aspects of the MHD stagnation point flow over stretching surfaces can be found in Hayat et al. [11], Shateyi and Makinde [12], Ibrahim et al. [13], Mahapatra and Gupta [14], Ishak et al. [15], and so forth. In recent times, the researchers are attracted towards the flow over shrinking surfaces. These flows are different from the flow over stretching surfaces in many ways. In shrinking sheet problems, the surface of the sheet is stretched towards a slot and hence generating a velocity away from the sheet. Therefore, the generated vorticity does not remain within the boundary layer and the flow will be unlikely to exist [16]. Wang [17] confirmed that the solutions can be found only for small shrinking rates and multiple solutions may exist for two-dimensional cases. Mahapatra and Nandy [18] were the researchers who analyzed that if suitable suction or stagnation point is added, then the vorticity can be controlled and the similarity solution will exist. Moreover, the stagnation region is the region which faces the highest pressure, heat transfer, and rates of mass deposition [19]. Keeping this in mind, the buoyancy effects on the MHD stagnation point flow of nanofluid were discussed by Makinde et al. [20]. Zaimi et al. [21] examined the boundary layer flow and heat transfer for nanofluid over a nonlinearly stretching/shrinking sheet. Akbar et al. [22] obtained the dual numerical solutions for the MHD stagnation point flow of Carreau fluid towards a permeable shrinking sheet. Some contributions on the study of stagnation point flow over shrinking surfaces for various physical situations and different fluids appeared in literature. Some of them can be found in Yian et al. [23], Bachok et al. [24], Nadeem et al. [25], Bhattacharyya et al. [26], Van Gorder et al. [27], Mansur et al. [28], and Ishak et al. [29]. However, for viscous fluid flows with heat transfer, the influence of linear dependence of density on temperature, that is, free convection, is found to be highly substantial in applications relating to industrial manufacturing processes and, therefore, cannot be ignored. But when the temperature difference between the surface and the ambient fluid is substantially large, the nonlinear density temperature (NDT) variations in the buoyancy force term have a significant effect on the flow and heat transfer features. Vajravelu and Sastri [30] discussed the flow between two parallel plates by considering the quadratic density temperature (QDT) variation and showed that the flow and heat transfer rates are substantially affected by it. Bhargava and Agarwal [31] examined the fully developed free convection flow in circular pipe with nonlinear density temperature variations. The nonlinear convection effects on the flow past a flat porous plate have been reported by Vajravelu et al. [32]. The flow dynamics of shrinking surfaces is still unknown and many of its characteristics are yet to be investigated. Motivated by this, the aim is to target the interaction between magnetic field and nonlinear convection on the stagnation point flow over a shrinking sheet.

#### 2. Formulation of the Problem

Consider a steady two-dimensional mixed convection flow of an incompressible, electrically conducting, and viscous fluid over a shrinking sheet as shown in Figure 1. Here, we have considered the Cartesian coordinate system with fixed origin such that the -axis is along the direction of the shrinking surface and the -axis is taken normal to the sheet. A magnetic field of uniform strength is applied normal to shrinking sheet and magnetic Reynolds number is assumed to be small to neglect the induced magnetic field. Here, we have supposed that external fluid velocity is , where is the stagnation flow strength, and velocity of the sheet is , where represents shrinking of the sheet and represents stretching of the sheet. We have also assumed that fluid has an ambient temperature and as temperature of the sheet.