Journal of Engineering

Volume 2016 (2016), Article ID 5874864, 11 pages

http://dx.doi.org/10.1155/2016/5874864

## Stagnation Point Flow of Nanofluid over a Moving Plate with Convective Boundary Condition and Magnetohydrodynamics

^{1}Department of Mathematics, University of Peshawar, Peshawar 25120, Pakistan^{2}Department of Mathematics, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand^{3}Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand^{4}Department of Mathematics, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa

Received 2 November 2015; Accepted 3 May 2016

Academic Editor: Haruhiko Ohta

Copyright © 2016 Fazle Mabood 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

A theoretical investigation is carried out to examine the effects of volume fraction of nanoparticles, suction/injection, and convective heat and mass transfer parameters on MHD stagnation point flow of water-based nanofluids (Cu and Ag). The governing partial differential equations for the fluid flow, temperature, and concentration are reduced to a system of nonlinear ordinary differential equations. The derived similarity equations and corresponding boundary conditions are solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. To exhibit the effect of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fraction, skin friction factor, and local Nusselt and local Sherwood numbers, numerical results are presented in graphical and tabular forms. It is found that the friction factor and heat and mass transfer rates increase with magnetic field and suction/injection parameters.

#### 1. Introduction

In recent years, the requirements of modern technology have encouraged interest in fluid flow studies which involve interaction of several phenomena. One such study is stagnation point flow over a permeable surface which plays an important role in many engineering problems including petroleum industries, ground water flows, extrusion of a polymer sheet from a dye, and boundary layer control. The study of a stagnation point flow towards a solid surface in moving fluid is traced back to Hiemenz in 1911. He analyzed two-dimensional stagnation point flow on stationary plate using a similarity transformation. Consequently, many investigators have extended that idea to different aspects of the stagnation point flow problem. For instance, Chamkha [1] investigated the mixed convection MHD flow near the stagnation point of a vertical semi-infinite surface. Ramachandran et al. [2] studied the dual solution analysis for different range of the buoyancy parameter. Devi et al. [3] extended the work of [2] and obtained a similarity solution for unsteady case. Grosan and Pop [4] studied axisymmetric mixed convection nanofluid flow past a vertical cylinder, Shateyi and Makinde [5] investigated MHD stagnation point flow towards a radially stretching convectively heated disk, Ibrahim et al. [6] examined MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, and Mustafa et al. [7] reported stagnation point flow of a nanofluid towards a stretching sheet.

In modern metallurgy and metal-working process, the magnetohydrodynamic (MHD) flow of an electrically conducting fluid towards a stretching surface is significant. These processes include the fusing of metals with electrical furnace by using magnetic field and cooling the inner first wall of the nuclear reactor containment vessel where the hot plasma is isolated from the wall by applying a magnetic field. Ishak et al. [8] studied MHD mixed convection flow near the stagnation point on a vertical permeable surface. Ishak et al. [9] numerically investigated the MHD stagnation point flow over a stretching sheet by using Keller box method and observed that velocity at a point increases with strong magnetic field when the free stream velocity is greater than the stretching velocity. Furthermore, Mahapatra et al. [10] investigated MHD stagnation point flow of a non-Newtonian fluid towards a stretching sheet and observed that, for a given magnetic parameter, the skin friction increases with an increase in power law index. Some more related studies to stagnation point flow can be found in [11–18].

Choi and Eastman [19] was the first who introduced the theory of nanofluid. Since then it has been an active field of research for about two decades. Nanofluid is a single phase mixture of suspended nanometer sized solid particles and fibers in conventional base fluids. Water, ethylene glycol mixture, toluene, and so forth are commonly used as base fluids. Wang and Mujumdar [20] found that the addition of very small amount of nanoparticles to conventional heat transfer liquids enhanced the thermal conductivity of the fluid up to approximately two times. Nanofluids offer many diverse advantages in industrial applications such as microelectronics, biomedicine, transportation, fuel cell**, **and nuclear reactors [21]. The literature is comprehensively occupied by nanofluids studies [22–29].

In this study, our main objective is to analyze the effect of volume fraction of nanoparticles on MHD stagnation point flow towards a moving surface with convective heat and mass transfer parameters. The governing boundary layer equations have been transformed to a two-point boundary value problem using similarity variables. These similarity equations were solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. The effects of governing parameters on the dimensionless velocity, temperature, and particle concentration as well as on local skin friction, and Nusselt and Sherwood numbers have been investigated.

#### 2. Mathematical Formulation

We considered the two-dimensional, steady, and laminar boundary layer flow of water-based nanofluids containing two types of nanoparticles, Cu and Ag, over a moving flat plate. A uniform magnetic field of strength was applied in a direction normal to plane* y* = 0. We assumed that the plate was heated by a fluid with heat transfer coefficient and temperature . The concentration of the fluid at the plate surface was assumed to be while the uniform temperature and concentration far away from the plate were taken as and , respectively. Under these assumptions, the steady boundary layer equations governing the flow, heat, and mass transfer areThe boundary conditions for the velocity components, temperature, and nanoparticle fraction are defined asHere, and are the velocity components along the axes and , respectively. is the free stream velocity, is species diffusivity, is electrical conductivity, is concentration of the nanofluid, and are the kinematic viscosity, density, and thermal diffusivity of the nanofluid, respectively, which are expressed as follows [7, 29]:We introduce the following transformations:where is the dimensionless concentration and represents the stream function and is defined as , , so that (1) is satisfied identically. The governing equations are reduced by using (7) as follows:The transformed boundary conditions arewhere primes denote differentiation with respect to and the four parameters are defined asHere, is suction () and injection () parameter, is magnetic parameter, is Prandtl number, is Schmidt number, is convective heat transfer parameter, and is convective mass transfer parameter.

The quantities of practical interest, in this study, are the local skin friction coefficient and local Nusselt and Sherwood numbers which are defined aswhere is the shear stress at surface of the wall and and are the wall heat and mass fluxes, respectively. Using (7), we obtainwhere is the local Reynolds number.

#### 3. Numerical Solution

Using similarity transformation, the governing equations of the problem are reduced to a system of nonlinear, coupled ordinary differential equations (8) which are solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method for different values of parameters such as magnetic parameter, suction/injection parameter, nanoparticles volume fraction, and convective heat and mass transfer parameter for water-based nanofluids. The effects of these parameters on the dimensionless velocity, temperature, and concentration as well as skin friction and the rate of heat and mass transfer are investigated. Since the physical domain of considered problem is unbounded, whereas the computational domain has to be finite, in all our computations we have used the value of to ensure that all numerical solutions approach the asymptotic values correctly for all values of physical quantities considered in this study. For these numerical computations, the step size and convergence criteria are chosen to be 0.001 and 10^{−6}, respectively.

The physical properties of the fluid, water, and the nanoparticles are given in Table 1. To validate the obtained numerical solution, comparison has been made with previously published data from the literature for skin friction in Table 2 and they are found to be in a favorable agreement. Simultaneous effects of different parameters on friction factor and Nusselt and Sherwood numbers are presented in Table 3, while other parameter values have been kept preset. From Table 3, it is clear that all physical quantities of interest, that is, skin friction and Nusselt and Sherwood numbers, are the increasing functions of and .