Journal of Engineering

Volume 2015, Article ID 382061, 10 pages

http://dx.doi.org/10.1155/2015/382061

## Numerical Study of the Influence of Heat Source on Stagnation Point Flow towards a Stretching Surface of a Jeffrey Nanoliquid

^{1}Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, Shimoga, Karnataka 577 451, India^{2}Department of Mathematics, SEA College of Engineering and Technology, K. R. Puram, Bengaluru, Karnataka 560 049, India

Received 5 August 2014; Accepted 27 February 2015

Academic Editor: Oronzio Manca

Copyright © 2015 G. K. Ramesh. 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 analysis is carried out to study the flow of Jeffrey fluid near a stagnation point towards a permeable stretching sheet. In particular, we investigate the effect of temperature dependent internal heat generation or absorption in the presence of nanoparticles. The governing system of partial differential equations is transformed into ordinary differential equations, which are then solved numerically using the fourth-fifth-order Runge-Kutta-Fehlberg method. Comparisons with previously published work on special cases of the problem are performed and found to be in excellent agreement. The results of the governing parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.

#### 1. Introduction

Investigations on boundary layer flow and heat transfer of non-Newtonian fluids are increasing substantially due to the large number of practical applications in industrial and manufacturing processes. Examples of such applications are drilling muds, plastic polymers, optical fibers, hot rolling paper production, metal spinning, cooling of metallic plates in cooling baths, and many others. In the past, investigators proposed different non-Newtonian models because a single model cannot predict all the features of non-Newtonian materials. There is one subclass of non-Newtonian fluids known as Jeffrey fluid which has attracted much attention from the researchers in view of its simplicity. This fluid model is capable of describing the characteristics of relaxation and retardation times [1, 2]. The investigation of flow due to a stretching sheet has been intentional because of this flow’s various industrial applications, such as in the manufacturing of polymer sheets, filaments, and wires. During the manufacturing process, the moving sheet is assumed to stretch on its own plane, and the stretched surface interacts with the ambient fluid both mechanically and thermally. Stretching and shrinking can occur in a variety of materials each having a different strength, stretching transparency, and luster. Initially, Sakiadis [3] introduced the concept of a boundary layer flow over a stretching surface. Crane [4] modified the idea introduced by Sakiadis and extended this idea for both linear and exponentially stretching sheets.

Flow in the neighborhood of a stagnation point in a plane was first studied by Hiemenz [5] and Mahapatra et al. [6–8] investigated the magnetohydrodynamic stagnation point flow towards a stretching sheet; they have shown that the velocity at a point decreases/increases with increase in the magnetic field when the free stream velocity is less/greater than the stretching velocity. Also they have studied the temperature distribution when the surface at constant temperature and constant heat fluxes. Further they have extended their work on power-law fluid and discussed the unique solutions of stagnation point flow of a power-law fluid towards a stretching surface. The study of heat source/sink effects on heat transfer is very important because its effects are crucial in controlling the heat transfer. Postelnicu et al. [9] examined the effect of variable viscosity on forced convection flow past a horizontal flat plate in a porous medium with internal heat generation, but in heat generation part they considered only space dependent heat source. Abel et al. [10] analyzed the non-Newtonian viscoelastic boundary layer flow of Walter’s liquid B past a stretching sheet, taking account of nonuniform heat source.

Aforementioned studies were primarily concerned with the laminar flow of a clear fluid. In the recent past a new class of fluids, namely, nanofluids, has attracted the attention of the science and engineering community because of the many possible industrial applications of these fluids. Nanotechnology is an emerging science that is finding extensive use in industry due to the unique chemical and physical properties that the nanosized materials possess. These fluids are colloidal suspensions, typically metals, oxides, carbides, or carbon nanotubes in a base fluid. The term nanofluid was coined by Choi [11] in his seminal paper presented in 1995 at the ASME Winter Annual Meeting. It refers to fluids containing a dispersion of submicronic solid particles (nanoparticles) with typical length on the order of 1–50 nm. Kuznetsov and Nield [12] analytically studied the natural convective boundary layer flow of a nanofluid past a vertical plate. In a recent paper Khan and Pop [13] studied for the first time the problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid. Mustafa et al. [14] investigated the stagnation point flow of a nanofluid towards a stretching surface using homotopy analysis method. Alsaedi et al. [15] examined the influence of heat generation/absorption on the stagnation point flow of nanofluid towards a linear stretching surface. Rahman et al. [16] investigated the dynamics of the natural convection boundary layer flow of water based nanofluids over a wedge in the presence of a transverse magnetic field with internal heat generation or absorption with the help of Matlab software. Nandy and Mahapatra [17] analyzed the effects of velocity slip and heat generation/absorption on magnetohydrodynamic stagnation point flow and heat transfer over a stretching/shrinking surface and obtained the solution numerically using fourth-order Runge-Kutta method with the help of shooting technique. Different from a stretching sheet, it was found that the solutions for a shrinking sheet are nonunique. Makinde et al. [18] studied the combined effects of buoyancy force, convective heating, Brownian motion, and thermophoresis on the stagnation point flow and heat transfer of an electrically conducting nanofluid towards a stretching sheet under the influence of magnetic field. Effect of magnetic field on stagnation point flow and heat transfer due to nanofluid towards a stretching sheet have been investigated by Ibrahim et al. [19]. Recently Ramesh and Gireesha [20] investigated the boundary layer flow of Maxwell fluid over a stretching surface with convective boundary condition in the presence of nanoparticles and heat source/sink effect. Nadeem et al. [21, 22] reported the numerical solutions of non-Newtonian nanofluid flow over a stretching sheet using the Jeffrey fluid model. Further they obtained the analytic solution for nonorthogonal stagnation point flow of a non-Newtonian nanofluid towards a stretching surface with heat transfer; here they use the second-grade model. Nadeem et al. [22] studied the natural convection boundary layer flow over a downward-pointing vertical cone in a porous medium saturated with a non-Newtonian nanofluid in the presence of heat generation or absorption and they used power-law model. Some interesting recent investigations related to the topic are presented in [23–25].

In this paper, we study the behaviour of the stagnation point flow towards a stretching sheet with the effects of heat source/sink and suction in the presence of nanoparticles. Similarity transforms are presented for this problem, and nondimensionalized equations are addressed numerically. Graphical results for various values of the parameters are presented to gain thorough insight towards the physics of the problem. To the best of my knowledge, this problem has not been studied before.

#### 2. Mathematical Analysis

Consider the flow of an incompressible Jeffrey fluid in the region driven by a stretching surface located at with a fixed stagnation point at . The stretching velocity and the ambient fluid velocity are assumed to vary linearly from the stagnation point; that is, and , where and are constant as shown in Figure 1. The problem under consideration is governed by the following boundary layer equations of Jeffrey fluid and nanoparticles and heat generation or absorption are given by Nadeem et al. [21]:where and are the velocity components along the - and -axes, respectively. Further, , , , , , and are, respectively, the thermal diffusivity, density of the base fluid, density of the particles, kinematic viscosity of the fluid, fluid temperature, and ambient fluid temperature. and are ratios of relaxation to retardation times and retardation time, is the dimensional heat generation/absorption coefficient, is the Brownian diffusion coefficient, is the thermophoresis diffusion coefficient, and is the specific heat at constant pressure. Here is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the ordinary fluid and is the nanoparticle volume fraction.