Journal of Fluids

Volume 2015, Article ID 634186, 14 pages

http://dx.doi.org/10.1155/2015/634186

## Magnetohydrodynamic Mixed Convection Stagnation-Point Flow of a Power-Law Non-Newtonian Nanofluid towards a Stretching Surface with Radiation and Heat Source/Sink

Department of Mathematics, Osmania University, Hyderabad, Telangana 500007, India

Received 29 September 2014; Revised 28 January 2015; Accepted 29 January 2015

Academic Editor: Robert Spall

Copyright © 2015 Macha Madhu and Naikoti Kishan. 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

Two-dimensional MHD mixed convection boundary layer flow of heat and mass transfer stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface in the presence of thermal radiation and heat source/sink is investigated numerically. The non-Newtonian nanofluid model incorporates the effects of Brownian motion and thermophoresis. The basic transport equations are made dimensionless first and the complete nonlinear differential equations with associated boundary conditions are solved numerically by finite element method (FEM). The numerical calculations for velocity, temperature, and nanoparticles volume fraction profiles for different values of the physical parameters to display the interesting aspects of the solutions are presented graphically and discussed. The skin friction coefficient, the local Nusslet number and the Sherwood number are exhibited and examined. Our results are compatible with the existing results for a special case.

#### 1. Introduction

Many of the non-Newtonian fluids encountered in chemical engineering processes are known to follow the empirical Ostwald-de Waele power-law model. The concept of boundary layer was applied to power-law fluids by Schowalter [1]. Acrivos [2] investigated the boundary layer flows for such fluids in 1960; since then, a large number of related studies have been conducted because of their importance and presence of such fluids in chemicals, polymers, molten plastics, and others. The theory of non-Newtonian fluids offers mathematicians, engineers, and numerical specialists varied challenges in developing analytical and numerical solutions for the highly nonlinear governing equations. However, due to the practical significance of these non-Newtonian fluids, many authors have presented various non-Newtonian fluid models like Elbashbeshy et al. [3], Nadeem et al. [4], Nadeem et al. [5], Nadeem et al. [6], Nadeem and Akbar [7], Nadeem and Ali [8], Buongiorno [9], and Łukaszewicz [10]. Many interesting applications of non-Newtonian power-law fluids were presented by Shenoy [11]. Details of the behavior of non-Newtonian fluids for both steady and unsteady flow situations, along with mathematical models, are studied by Astarita and Marrucci [12], Bohme [13], and Kishan and Shashidar Reddy [14].

Nanotechnology has immense applications in industry since materials with sizes of nanometers exhibit unique physical and chemical properties. Fluids with nano-scaled particles interaction are called nanofluid. It represents the most relevant technological cutting edge currently being explored. Nanofluid heat transfer is an innovative technology which can be used to enhance heat transfer. Nanofluid is a suspension of solid nanoparticles (1–100 nm diameters) in conventional liquids like water, oil, and ethylene glycol. Depending on shape, size, and thermal properties of the solid nanoparticles, the thermal conductivity can be increased by about 40% with low concentration (1%–5% by volume) of solid nanoparticles in the mixture. The nanoparticles used in nanofluid are normally composed of metals, oxides, carbides, or carbon nanotubes. Water, ethylene glycol, and oil are common examples of base fluids. Nanofluids have their major applications in heat transfer, including microelectronics, fuel cells, pharmaceutical processes, and hybrid-powered engines, domestic refrigerator, chiller, nuclear reactor coolant, grinding, space technology, and boiler flue gas temperature reduction. They demonstrate enhanced thermal conductivity and convective heat transfer coefficient counterbalanced to the base fluid. Nanofluids have been the core of attention of many researchers for new production of heat transfer fluids in heat exchangers, plants, and automotive cooling significations, due to their enormous thermal characteristics, Nadeem et al. [15]. The nanofluid is stable; it introduces very little pressure drop, and it can pass through nanochannels (e.g., see Zhou [16]). The word nanofluid was coined by Choi [17]. Xuan and Li [18] pointed out that, at higher nanoparticle volume fractions, the viscosity increases sharply, which suppresses heat transfer enhancement in the nanofluid. Therefore, it is important to carefully select the proper nanoparticle volume fraction to achieve heat transfer enhancement. Buongiorno noted that the nanoparticles’ absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity (which he called the slip velocity). He considered in turn seven slip mechanisms: inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity settling.

Forced convective heat transfer can be enhanced effectively by using nanofluids, a type of fluid adding different suspending nanoparticles into the conventional base liquid (Pak and Cho [19], Wen and Ding [20], and Ding et al. [21]). However, the characteristics of nanofluids and the mechanism of the enhancement of the forced convective heat transfer of nanofluids are still not clear. Recently, nanofluids have attracted much attention since anomalously large enhancements in effective thermal conductivities were reported over a decade ago (Masuda et al. [22] and Keblinski et al. [23]). Subsequent studies by various groups have reported that nanofluids also have other desirable properties and behaviours such as enhanced wetting and spreading (Wasan and Nikolov [24] and Chengara et al. [25]), as well as increased critical heat fluxes under boiling condition (You et al. [26]).

Boundary layer flow and heat transfer over a continuously stretched surface have received considerable attention in recent years. This is because of the various possible engineering and metallurgical applications such as hot rolling, metal and plastic extrusion, wire drawing, glass fibre production, continuous casting, crystal growing, and paper production. Crane [27] was the first to investigate the boundary layer flow caused by a stretching sheet moving with linearly varying velocity from a fixed point whilst the heat transfer aspect of the problem was investigated by Carragher and Crane [28] under the conditions that the temperature difference between the surface and the ambient fluid was proportional to the power of the distance from a fixed point. The behaviour of non-Newtonian nanofluids could be useful in evaluating the possibility of heat transfer enhancement in various processes of these industries. Several investigators have studied non-Newtonian nanofluid transport in various geometries under various boundary conditions in porous or nonporous media. Ellahi et al. [29] have elaborated that non-Newtonian nanofluids have potential roles in physiological transport as biological solutions and also in polymer melts, paints, and so forth.

The effect of heat source/sink is very important in cooling process industries. Effects of heat source/sink on the boundary layer flow over a stretching sheet were studied by Cortell [30], Abel et al. [31], Dessie and Kishan [32], Tufail et al. [33], Elbashbeshy and Bazid [34], and Chen [35]. In fact, heat source/sink concepts in fluids have relevance in problems dealing with chemical reactions geonuclear repositions and these concerned with dissociating fluids. Transport phenomena associated with magnetohydrodynamics arise in physics, geophysics, astrophysics, and many branches of chemical engineering which include crystal magnetic damping control, hydromagnetic chromatography, conducting flow in trickle-bed reactors, and enhanced magnetic filtration control (Prasad et al. [36]).

In the present paper, our aim is to investigate the effects of MHD, heat source/sink, and thermal radiation on heat and mass transfer by mixed convection boundary layer stagnation point flow of non-Newtonian power-law fluid towards a stretching surface with a nanofluid. The effects of Brownian motion and thermophoresis are included for the nanofluid. Numerical solutions of the boundary layer equations are obtained and a discussion is provided for several values of the nanofluid parameters governing the problem. The dimensionless profiles of velocity, temperature, and nanoparticle volume fraction as well as the skin friction coefficient, local Nusselt number, and Sherwood number for the different flow parameters have been discussed.

#### 2. Mathematical Formulation

Consider steady, laminar, heat, and mass transfer by mixed convection, boundary layer stagnation-point flow of an electrically conducting, optically dense, and non-Newtonian power-law fluid obeying the Ostwald-de Waele model (see [37]) past a heated or cooled stretching vertical surface in the presence of thermal radiation. It is assumed that the stretching velocity is given by , and the velocity distribution in frictionless potential flow in the neighborhood of the stagnation point at is given by . We assumed that the uniform wall temperature and nanoparticles volume fraction are higher than those of their full stream values , . A uniform magnetic field is applied in the -direction normal to the flow direction. The magnetic Reynolds number is assumed to be small so that the induced magnetic field is neglected. In addition, the Hall effect and the electric field are assumed to be negligible. The small magnetic Reynolds number assumption uncouples the Navier-Stokes equations from Maxwell’s equations. All physical properties are assumed to be constant except for the density in the buoyancy force term. By invoking all of the boundary layer, Boussinesq and Rosseland diffusion approximations, the governing equations for this investigation can be written as The associated boundary conditions are , , , and are the and components of velocity, temperature, and nanoparticle volume fraction, respectively. , , , , , , , and are the gravitational acceleration, fluid density, thermal diffusivity, Brownian diffusion coefficient, thermophoretic diffusion coefficient, magnetic field, coefficient of thermal expansion, and coefficient of concentration of expansion, respectively. The term is assumed to be the amount of heat generated or absorbed per unit volume as a coefficient constant, which may take on either positive or negative value. When the wall temperature exceeds the free stream temperature , the source term and heat sink when . We have when (the ratio of free stream velocity and stretching velocity) which gives the shear stress as where is the consistency coefficient and is the power-law fluid. It needs to be mentioned that, for the non-Newtonian power-law model, the case of is associated with shear-thinning fluids (pseudoplastic fluids); corresponds to Newtonian fluids and applies to the case of shear thickening (dilatant).

Using the Rosseland approximation for radiation, the radiative heat flux is simplified as where and are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. We assume that the temperature differences within the flow, such as the term , may be expressed as a linear function of temperature. Hence, expanding in a Taylor series about a free stream temperature and neglecting higher-order terms, we get Using (7) and (8) in the last term of (3), we obtain In order to reduce the governing equations into a system of ordinary differential equations, the following dimensionless parameters are introduced It is worth mentioning that the continuity equation (4) is identically satisfied from our choice of the stream function with and . Substituting the dimensionless parameters into (2)–(4) gives The transformed boundary conditions areThe nine parameters appearing in (11) are defined as follows: where is the local Reynolds number based on the stretching velocity and is the thermal conductivity. It should be noted that corresponds to an assisting flow (heated plate), corresponds to an opposing flow (cooled plate), and yields forced convection flow.

The skin friction coefficient at the wall is given by the local Nusselt number is given by and the local Sherwood number is given by

#### 3. Method of Solution

##### 3.1. Finite Element Method

The finite element method is a powerful technique for solving ordinary or partial differential equations. The basic concept of FEM is that the whole domain is divided into smaller elements of finite dimensions called finite elements. This method is the most versatile numerical technique in engineering analysis and has been employed to study diverse problems in heat transfer, fluid mechanics, rigid body dynamics, solid mechanics, chemical processing, electrical systems, acoustics, and many other fields. The steps involved in the finite element analysis are as follows:(i)discretization of the domain into elements,(ii)derivation of element equations,(iii)assembly of element equations,(iv)imposition of boundary conditions,(v)solution of assembled equations.To solve the system of simultaneous nonlinear differential equations (11), with the boundary conditions (12a) and (12b), we assume that the system of (11) then reduced to and the corresponding boundary conditions now becomeFor computational purposes, the has been shifted to , without any loss of generality. The domain is divided into a set of 100 line elements, each element having two nodes.

##### 3.2. Variational Formulation

The variational form associated with (17)-(18) over a typical linear element is given by where , , , and are weight functions and may be viewed as the variation in , , , and , respectively.

##### 3.3. Finite Element Formulation

The finite element model from (20) by substituting finite element approximations of the form with , , where are the shape functions for a two-noded linear element and are taken as

The finite element model of the equations thus formed is given by where and are defined as where

Each element matrix is of the order . The whole domain is divided into 100 linear elements of equal size; after assembly of all the elements equations, we obtain a matrix of the order . This system of equations as obtained after assembly of the element equations is nonlinear. Therefore, an iterative scheme must be utilized in the solution. After imposing the boundary conditions only a system of 397 equations remains for the solution, which is solved by the Gauss elimination method maintaining an accuracy of 0.0005.

#### 4. Result and Discussion

The numerical solutions of governing equations (11) with boundary conditions (12a) and (12b) have been solved by using the variational finite element method. To validate our results, the numerical computations of these skin friction coefficients, Nusselt number, and Sherwood number which are, respectively, proportional to , , and are presented in tabular form and one result is compared with Mahapatra et al. [38]. The validation of present results has been verified with the classical case of Newtonian fluid and there is a good agreement between present and Mahapatra et al. (Table 1) [38].