Journal of Computational Engineering

Volume 2014 (2014), Article ID 345153, 17 pages

http://dx.doi.org/10.1155/2014/345153

## Heat and Mass Transfer in the Boundary Layer of Unsteady Viscous Nanofluid along a Vertical Stretching Sheet

Department of Mathematics, Osmania University, Hyderabad 500 007, India

Received 18 July 2014; Accepted 22 November 2014; Published 18 December 2014

Academic Editor: Fu-Yun Zhao

Copyright © 2014 Eshetu Haile and B. Shankar. 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

Heat and mass transfer in the boundary-layer flow of unsteady viscous nanofluid along a vertical stretching sheet in the presence of magnetic field, thermal radiation, heat generation, and chemical reaction are presented in this paper. The sheet is situated in the *xz*-plane and *y* is normal to the surface directing towards the positive *y*-axis. The sheet is continuously stretching in the positive *x*-axis and the external magnetic field is applied to the system parallel to the positive *y*-axis. With the help of similarity transformations, the partial differential equations are transformed into a couple of nonlinear ordinary differential equations. The new problem is then solved numerically by a finite-difference scheme known as the Keller-box method. Effects of the necessary parameters in the flow field are explicitly studied and briefly explained graphically and in tabular form. For the selected values of the pertinent parameters appearing in the governing equations, numerical results of velocity, temperature, concentration, skin friction coefficient, Nusselt number, and Sherwood number are obtained. The results are compared to the works of others (from previously published journals) and they are found in excellent agreement.

#### 1. Introduction

The flow over a stretching surface is an important problem in many engineering processes with applications in industries such as extrusion, melt-spinning, hot rolling, wire drawing, glass-fiber production, manufacture of plastic and rubber sheets, and cooling of a large metallic plate in a bath, which may be an electrolyte. In industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a finite distance away. The thin polymer sheet constitutes a continuously moving surface with a nonuniform velocity through an ambient fluid [1]. Bachok et al. [2] studied boundary layer flow of nanofluids over a moving surface in a flowing fluid and, moreover, a study on boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition was conducted by Makinde and Aziz [3]. Olanrewaju et al. [4] examined boundary layer flow of nanofluids over a moving surface in a flowing fluid in the presence of radiation. An analysis of mixed convection heat transfer from a vertical continuously stretching sheet has been presented by Chen [5].

In many practical situations the material moving in a quiescent fluid is due to the fluid flow induced by the motion of the solid material and/or by the thermal buoyancy. Therefore, the resulting flow and the thermal field are determined by these two mechanisms, that is, surface motion and thermal buoyancy. It is well known that the buoyancy force stemming from the heating or cooling of the continuous stretching sheet alters the flow and the thermal fields and thereby the heat transfer characteristics of the manufacturing processes [6]. Effects of thermal buoyancy on the flow and heat transfer over a stretching sheet were reported by many researchers. Chen and Strobel [7] investigated buoyancy effects in boundary layer adjacent to a continuous moving horizontal flat plate. Karwe and Jaluria [8] showed that the thermal buoyancy effects are more prominent when the plate moves vertically, that is, aligned with the gravity, than when it is horizontal. Ali [9] examined the buoyancy effect on the boundary layer induced by continuous surface stretched with rapidly decreasing velocities. Buoyancy driven heat and mass transfer over a stretching sheet in a porous medium with radiation and ohmic heating was studied by Dulal and Hiranmoy [10]. Abo-Eldahab and El Aziz [11] presented the problem of steady, laminar, hydromagnetic heat transfer by mixed convection over an inclined stretching surface in the presence of space—and temperature dependent heat generation or absorption effects. Ali and Al-Yousef [12, 13] investigated the problem of laminar mixed convection adjacent to a moving vertical surface with suction or injection. On the other hand, Khan et al. [14] studied the unsteady free convection boundary layer flow of a nanofluid along a stretching sheet with thermal radiation and viscous dissipation effects in the presence of a magnetic field.

The study of MHD boundary layer flow on a continuous stretching sheet has attracted considerable attention during the last few decades due to its numerous applications in industrial manufacturing processes. In particular, the metallurgical processes such as drawing, annealing, and tinning of copper wires involve cooling of continuous strips or filaments by drawing them through a quiescent fluid. Controlling the rate of cooling in these processes can affect the properties of the final product. Thus, rate of cooling can be greatly controlled by the use of electrically conducting fluid and the application of the magnetic field [15]. Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate was studied by Hamad et al. [16]. Effects of a thin gray fluid on MHD free convective flow near a vertical plate with ramped wall temperature under small magnetic Reynolds number [17] and free convective oscillatory flow and mass transfer past a porous plate in the presence of radiation of an optically thin fluid [18] have been studied. Moreover, MHD Flow and heat transfer over stretching/shrinking sheets with external magnetic field, viscous dissipation, and joule effects were studied by Jafar et al. [19].

Radiative heat transfer in which heat is transmitted from one point to another without heating the intervening medium has been found very important in the design of reliable equipment, nuclear plants, gas turbines, and various propulsion devices for aircraft, missiles, satellites, and space vehicles. Also, the effects of thermal radiation on the forced and free convection flows are important in the content of space technology and processes involving high temperature [10]. Influence of thermal radiation, viscous dissipation, and hall current on MHD convection flow over a stretched vertical flat plate was studied by Gnaneswara Reddy [20]. Vajravelu and Hadjinicolaou [21] examined heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. Hady et al. [22] analyzed the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation. Postelnicu [23] studied the influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Shakhaoath Khan et al. [24] examined possessions of chemical reaction on MHD heat and mass transfer nanofluid flow on a continuously moving surface. Besides this, effects of chemical reactions, heat and mass transfer on nonlinear magnetohydrodynamic boundary layer flow over a wedge with a porous medium in the presence of ohmic heating, and viscous dissipation were studied [25].

These days, because of the numerous applications of nanofluids in science and technology, a comprehensive study on heat and mass transfer in the boundary layer of unsteady viscous nanofluid in the presence of different fluid properties is indispensable. The paper entitled “Heat and mass transfer in the boundary layer of unsteady viscous nanofluid along a vertical stretching sheet” in the presence of thermal radiation, viscous dissipation, and chemical reaction is considered. Yet, the entitled paper has not been reported. Accordingly, we extended the works of Vendabai and Sarojamma [28] by incorporating viscous dissipation and chemical reaction terms in the energy and concentration equations, respectively, for more physical implications. The governing equations are reduced to a couple of nonlinear ODEs using similarity transformations; the resulting equations are solved numerically by using the Keller box. Effects of the pertinent parameters involved in the governing equations on* velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number* are briefly explained.

#### 2. Formulation of the Problem

Unsteady two-dimensional boundary layer flow and heat transfer of a nanofluid along a stretching sheet coinciding with the plane are considered. The Cartesian coordinate system has its origin located at the leading edge of the sheet with the positive -axis extending along the sheet in the upward direction, while the -axis is measured normal to the surface of the sheet and is positive in the direction from the sheet to the fluid. A schematic representation of the physical model and coordinates system is shown in Figure 1. For , it is assumed that the fluid and heat flows are steady but the unsteady scenario starts at . We assume that the sheet is being stretched with the velocity along the -axis, keeping the origin fixed. Let and be the temperature and concentration of the sheet whereas let and be the ambient temperature and concentration, respectively. An external variable magnetic field is applied along the positive -direction. The induced magnetic field is sufficiently weak to ignore magnetic induction effects; that is, magnetic Reynolds number is small. The charge density, external electrical field effects, and polarization voltage are ignored. Using Oberbeck Boussinesq approximation for buoyancy-driven flows, following Buongiorno [6, 15, 28–30], the governing equations for the flow become The boundary conditions associated to the differential equations are where is suction/injection velocity corresponds to suction velocity), , , is the radiative heat flux, and the heat generation coefficient is defined by , where represents the heat source if and the heat sink if .