Journal of Engineering

Volume 2015 (2015), Article ID 452592, 7 pages

http://dx.doi.org/10.1155/2015/452592

**Flow over Exponentially Stretching Sheet through Porous Medium with Heat Source/Sink**

^{1}Department of Mathematics, KES, Bhubaneswar 751002, India^{2}Department of Mathematics, I.T.E.R., Siksha ‘O’ Anusandhan University, Bhubaneswar, Odisha 751030, India^{3}Department of Mathematics, Ravenshaw University, Bhubaneswar 753002, India

Received 2 September 2015; Accepted 4 November 2015

Academic Editor: Oronzio Manca

Copyright © 2015 I. Swain et al. 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

An attempt has been made to study the heat and mass transfer effect in a boundary layer MHD flow of an electrically conducting viscous fluid subject to transverse magnetic field on an exponentially stretching sheet through porous medium. The effect of thermal radiation and heat source/sink has also been discussed in this paper. The governing nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations and then solved numerically using a fourth-order Runge-Kutta method with a shooting technique. Graphical results are displayed for nondimensional velocity, temperature, and concentration profiles while numerical values of the skin friction local Nusselt number and Sherwood number are presented in tabular form for various values of parameters controlling the flow system.

#### 1. Introduction

The magnetohydrodynamics (MHD) heat and mass transfer from different geometry embedded in a porous medium are of interest for engineering and geographical applications such as geothermal reservoirs, thermal insulation, cooling of nuclear reactors, and enhanced oil recovery. Many chemical engineering processes like metallurgical and polymer extrusion processes involve cooling of molten liquid being stretched into a cooling system; the fluid mechanical properties of the penultimate product depend mainly upon the cooling liquid used and the rate of stretching. Some polymer fluids like polyethylene oxide and polyisobutylene solution in cetane, having better electromagnetic properties, are normally used as cooling liquid as their flow can be regulated by external magnetic fields in order to improve the quality of final product. Sakiadis [1, 2] investigated the boundary layer flow induced by a moving plate in a quiescent ambient fluid. Thereafter, various aspects of the problem have been investigated by many authors such as Fang [3], Fang and Lee [4], and White [5].

Buoyancy is also of importance in an environment where differences between heat and air temperatures can give rise to complicated flow patterns [6]. Furthermore, magnetohydrodynamic (MHD) has attracted the attention of a large number of scholars due to its diverse applications. Chamkha and Abdul-Rahim Khaled [7] have investigated the effects of magnetic field on natural convection flow past a vertical surface. Makinde [8] and Makinde et al. [9] have studied mass diffusion effects on natural convection flow past a flat plate. A comprehensive account of the boundary layers flow over a vertical plate embedded in a porous medium can be found in Kim and Vafai [10] and Liao and Pop [11].

It is well known that fluids such as water, mineral oil, and ethylene glycol for conventional heat transfer are poor conductors of heat compared to most solids. An innovative way of improving the heat transfer in fluids by suspending small solid particles in the fluids was introduced by Choi [12]. This new kind of fluids is named nanofluids which is a suspension of solid nanoparticles of diameter 1–100 nm in conventional heat transfer basic fluids such as water, oil, or ethylene glycol. It is believed that these fluids increase the heat transfer performance of the base fluid enormously. This characteristic feature of nanofluids is to enhance the thermal conductivity which is more useful to meet today’s cooling rate requirements. A comprehensive survey of convective transport was presented by Buongiorno [13] by pointing out various facts concerning nanofluids. Similarity solution to heat and mass transfer analysis on MHD 3D water-based nanofluid was investigated by Baag and Mishra [14]. The study of magnetohydrodynamic (MHD) flow has many important industrial, technological, and geothermal applications such as high temperature plasmas, cooling of nuclear reactors, MHD accelerators and power generation systems, and liquid metal fluids. Magnetic nanofluids have colloidal suspensions containing magnetizable nanoparticles which have both the fluid and magnetic properties as well as thermal properties. Vajravelu and Rollins [15] analyzed heat transfer in an electrically conducting fluid over a stretching surface taking into account the magnetic field. Tripathy et al. [16] studied chemical reaction effect on MHD free convective surface over a moving vertical plane through porous medium. Mishra et al. [17] investigated the flow of heat and mass transfer on MHD free convection in a micropolar fluid with heat source.

Sparrow and Abraham [18] have investigated a new buoyancy model replacing the standard pseudo density difference for internal natural convection in gases. Sparrow and Abraham [19] used the relative velocity model where only one of the participating media is in motion. The steady laminar flow and heat transfer characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot by Al-Sanea [20]. Soundalgekar and Ramana Murty [21] have discussed the effects of power law surface temperature variation on the heat transfer from a continuous moving surface with constant surface velocity. More recently, Cortell [22] extended the work of Afzal et al. [23] by taking viscous dissipation effect in the energy balance. The effects of transpiration on the flow and heat transfer over a moving permeable surface in a parallel stream are analyzed by Ishak et al. [24]. The development of the boundary layer on a fixed or moving surface parallel to a uniform free stream in presence of surface heat flux has been investigated by Ishak et al. [25]. Patil et al. [26] have examined the role of internal heat generation or absorption effects on the flow and heat transfer over a moving vertical plate. In this study, authors have considered the steady flow and heat transfer characteristics. Unsteady mixed convection flows do not necessarily possess similarity solutions in many practical applications. The unsteadiness and nonsimilarity in such flows may be due to the free stream velocity or due to the curvature of the body or due to the surface mass transfer or even possibly due to all these effects. Because of the mathematical difficulties involved in obtaining nonsimilar solutions for such problems, many investigators have confined their studies to either steady nonsimilar flows or unsteady semisimilar or self-similar flows.

In the present study we proposed to investigate the effect of heat source/sink on the free convection flow of a viscous incompressible electrically conducting fluid on a vertical plate with variable wall temperature and concentration. The effect of pertinent parameters is presented in both graphical and tabular form. It is noticed that the results obtained will not only provide useful information for applications, but also serve as a complement to Mabood et al. [27].

#### 2. Mathematical Formulation

Consider a steady, laminar, incompressible, two-dimensional free convective heat and mass transfer along a semi-infinite vertical plate embedded in a doubly stratified, electrically conducting micropolar fluid. Choose the coordinate system such that the -axis is along the vertical plate and the -axis normal to the plate. The physical model and coordinate system are shown in Figure 1. The plate is maintained at temperature and concentration . The temperature and the mass concentration of the ambient medium are assumed to be linearly stratified in the forms and , respectively, where and are constants and varied to alter the intensity of stratification in the medium and and are the beginning ambient temperature and concentration at , respectively. A uniform magnetic field of magnitude is applied normal to the plate. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected in comparison with the applied magnetic field.