Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 563547, 20 pages

http://dx.doi.org/10.1155/2015/563547

## Radiation and MHD Boundary Layer Stagnation-Point of Nanofluid Flow towards a Stretching Sheet Embedded in a Porous Medium: Analysis of Suction/Injection and Heat Generation/Absorption with Effect of the Slip Model

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11757, Egypt

Received 9 March 2015; Accepted 8 May 2015

Academic Editor: Gen Qi Xu

Copyright © 2015 Emad H. Aly. 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

In existence of the velocity slip model, suction/injection, and heat source/sink, the boundary layer flow near a stagnation-point over a heated stretching sheet in a porous medium saturated by a nanofluid, with effect of the thermal radiation and magnetic field, has been studied. The governing system of partial differential equations was transformed into a system of nonlinear ordinary equations using the appropriate similarity transforms. Then, the obtained system has been numerically solved by the Chebyshev pseudospectral differentiation matrix (ChPDM) approach. It was found that, at some special cases, the current results are in a very good agreement with those presented in the literature. In addition, the flow velocity, surface shear stress, temperature, and concentration are strongly influenced on applying the slip model, which is, therefore, extremely important to predict the flow characteristics accurately in the nanofluid mechanics. It was proved that this velocity slip condition is mandatory and should be taken into account in nanoscale research; otherwise, false results and a spurious physical sight are to be gained. Further, it was deduced that the influence of the stream velocity and shear stress reaches very rapidly the stable manner for both cases of the velocity ratio. However, when this ratio is equal to one, the skin friction coefficient, reduced Nusselt number, and reduced Sherwood number are constant and equal to zero, 0.721082, and 3.06155, respectively. Furthermore, it was proved that the reduced Nusselt number decreases with increase of Brownian motion and thermophoresis; has a very weak effect on increasing Lewis number; increases with increase of Prandtl number; and is higher in the cases of suction, velocity ratio > 1 and heat source in comparison with injection, velocity ratio < 1, and heat sink, respectively. Moreover, the reduced Sherwood number increases with increase of Brownian motion, thermophoresis, and Lewis number; decreases with increase of Prandtl number; is higher in the cases of suction and velocity ratio > 1 in comparison with injection and velocity ratio < 1, respectively; and is approximately the same in the heat source and heat sink cases. Finally, it was shown that the most effective region for radiation effect is .

#### 1. Introduction

*Applications on the Stretching Sheet Study*. The problem of flow and heat transfer in boundary layer over a stretching surface has attracted many researchers because of its numerous applications, for example, in metallurgical processes, such as drawing of continuous filaments through quiescent fluids, annealing and tinning of copper wires, glass blowing, manufacturing of plastic and rubber sheets, crystal growing, and continuous cooling and fiber spinning. In addition, there are wide-ranging applications in many engineering processes, such as polymer extrusion, wire drawing, continuous casting, manufacturing of foods and paper, glass fiber production, and stretching of plastic films. During the manufacture of these sheets, the melt issues from a slit and is subsequently stretched to achieve the desired thickness. The final product with the desired characteristics strictly depends upon the stretching rate, the rate of cooling in the process, and the process of stretching.

*Nanofluids and Their Applications*. Nanotechnology is nowadays considered as a significant factor which affects the industrial revolution of the current century. Nanofluids are nanometer-sized particles (diameter less than 50 nm) dispersed in a base fluid. Recent research on nanofluids showed that nanoparticles changed the fluid characteristics because thermal conductivity of these particles was higher than convectional fluids. It should be noticed that nanoparticles are of great scientific interest as they are effectively a bridge between bulk materials and atomic or molecular structures. These nanoparticles are typically made of metals, oxides, carbides, or carbon nanotubes. The common base fluids include water, ethylene glycol, toluene, and oil. Therefore, many researchers have focused on modeling the thermal conductivity and examined different types of nanofluids’ viscosity (see [1, 2]).

Convective heat transfer in nanofluids is a topic of major contemporary interest both in applied sciences and engineering. Choi [3] may be the first author to use the term “nanofluid” in his seminal paper presented in 1995 at the ASME Winter Annual Meeting, where it was reported that one of the promising nanofluids applications is heat transfer enhancement. In [4], Choi et al. showed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. Masuda et al. [5], Lee et al. [6], Xuan and Li [7], and Xuan and Roetzel [8] stated that, with low nanoparticles concentrations (1–5 Vol%), the thermal conductivity of the suspensions can increase more than 20%. Regarding, some interest has been given to the study of this type of flow and some useful results have been recently introduced by many authors; see, for example, [9–13]. Aly and Ebaid [14] presented a comprehensive study of convective transport in nanofluids. Buongiorno [15] developed an analytical model for convective transport in nanofluids considering the Brownian diffusion and thermophoresis. He developed an explanation for abnormal convective heat transfer enhancement observed in nanofluids and also showed that Brownian diffusion and thermophoresis were the most important nanoparticle/base-fluid slip mechanisms.

Many of the publications on nanofluids are about the understanding of their behaviors so that they can be utilized where straight heat transfer enhancement is paramount as in many industrial applications, nuclear reactors, transportation, electronics, and biomedicine and food. Examples include nanofluid adhesive: electronics cooling, vehicle cooling, transformer cooling, super powerful and small computers cooling, and electronic devices cooling; medical applications: cancer therapy and safer surgery by cooling and process industries; and materials and chemicals: detergency, food and drink, oil and gas, paper and printing, and textiles. Ultra high-performance cooling is necessary for many industrial technologies. However, poor thermal conductivity is a drawback in developing energy-efficient heat transfer fluids necessary for ultra high-performance cooling.

*Investigations on the Stagnation-Point*. In 1911, Hiemenz was the pioneer to analyze two-dimensional stagnation-point flow on stationary plate using a similarity transformation to reduce the Navier-Stokes equations to nonlinear ordinary differential equations. Due to its wide range of applications in cooling of electronic devices by fans, cooling of nuclear reactors during emergency shutdown, solar central receivers exposed to wind currents, and many hydrodynamic processes in engineering application, the flow near the stagnation-point has attracted the attention of many investigators for more than a century.

The idea of stagnation-point flow of a nanofluid is extended and studied recently. Accordingly, Mahapatra and Gupta [16] and Ishak et al. [17] studied numerically boundary layer and magnetohydrodynamics stagnation-point flow towards a stretching sheet. Their analysis showed that velocity at a point increases with an increase in the magnetic field when the free stream velocity is greater than the stretching velocity. Layek et al. [18] have reported heat and mass transfer boundary layer stagnation-point flow of an incompressible viscous fluid towards a heated porous stretching sheet embedded in a porous medium subject to suction/blowing with internal heat generation or absorption. The governing boundary layer equations were transformed by scaling group of transformation into a system of ordinary differential equations. Moreover, Bachok et al. [19] expanded the dimension of the problem of stagnation-point flow and heat transfer to three-dimensional stagnation-point flow in nanofluids. Mustafa et al. [20] studied the stagnation-point flow of a nanofluid towards a stretching sheet using homotopy analysis method. To follow up some deducing properties in [18], Hamad and Ferdows [21] studied heat and mass transfer for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/injection. They deduced that the inclusion of nanoparticles into the base fluid of this problem changes the flow pattern, where suction tends to stabilize the boundary layer flow and blowing can reduce the friction drag [18].

Recently, a good physical explanation of nanofluid stagnation-point flow and radiation heat transfer over a stretching sheet with first-order velocity slip and temperature jump in a porous medium has been introduced by Zheng et al. [22]. Further, Ibrahim et al. [23] analyzed magnetohydrodynamics stagnation-point flow towards a stretching sheet in nanofluid. They found that the magnitude of the skin friction coefficient, the reduced Nusselt number, and reduced Sherwood number are all increasing with the magnetic parameter when the velocity ratio exceeds 1, and when this ratio is less than 1 they all decrease. In addition, the heat transfer rate at the surface increases with velocity ratio and Prandtl number. Likewise, the mass transfer rate at the surface increases with an increase in both Lewis number and velocity ratio. Very recently, Jalilpour et al. [24] studied the MHD boundary layer flow of a nanofluid towards a stretching surface with suction or blowing and surface heat flux. They found that the magnitude of the reduced Nusselt number decreases with an increase in magnetic number, thermophoresis parameter, and Lewis number. Further, the reduced Sherwood number decreases with increasing magnetic number and thermophoresis parameter and increases with increasing Lewis number. Furthermore, the reduced Nusselt number and reduced Sherwood number are higher when suction is induced.

*Importance of the Slip Model in Nano Studies*. In 1997, Thompson and Troian [25] proved that the slip velocity is related to the slip length, the shear rate at the wall, and a critical shear rate at which the slip length diverges. In addition, wall slip readily occurs for an array of complex fluids such as foams, emulsions, suspensions, and polymer solutions. Further, the fluids that exhibit boundary slip have important technological applications such as in the polishing of artificial heart valves and internal cavities. Later, Majumder et al. [26] showed experimentally that nanofluidic flow usually exhibits partial slip against the solid surface, which can be characterized by the so-called slip length (around 3.4–68 micrometers for different liquids). They proved that the classical no-slip condition is no longer valid for fluid flows at the micro- and nanoscale and, instead, a certain degree of tangential slip must be allowed. Hence, effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet at constant wall temperature has been studied by Noghrehabadi et al. [27] to extend the work done by Khan and Pop [28]. In addition, Nandeppanavar et al. [29] have tabulated the literature of the first-order slip, and this resulted in making Fang et al. [30] considered only the effect of the second-order slip on the flow on a shrinking sheet. Therefore, they may be the first researchers to investigate the analysis of the second-order slip flow and heat transfer over a stretching sheet using Wu’s slip model [31]. Further, Turkyilmazoglu [32] has analytically studied the heat and mass transfer of magnetohydrodynamic second-order slip flow. He has mentioned that there exists a unique solution for any combination of the considered parameters if the stretching sheet is considered. Furthermore, Roşca and Pop [33] investigated the steady flow and heat transfer over a vertical permeable stretching/shrinking sheet with second-order slip. This very important study showed clearly that the second-order slip flow model is necessary to predict the flow characteristics accurately.

To prove the point of view in [33], Aly and Ebaid [14] have recently investigated five nanofluids flow over an isothermal stretching sheet with effect of Wu’s slip model [31]. They showed that increase of the slips slows down the velocity, increases the temperature with an impressive effect in the injection case, and decreases the local skin friction and the reduced Nusselt number with significant effects. Recently, Aly and Vajravelu [34] examined the effects of the second-order slip parameter for the nano boundary layer flow over two-dimensional and axisymmetric stretching surfaces in the presence of a transverse magnetic field in a porous medium. They found that this parameter affects considerably the flow characteristics for increasing values of the magnetic parameter and decreasing values of the porosity parameter. In addition, the presence of the magnetic field, permeability, first and second slip factors lead to a decrease in the nano boundary layer thickness. Very recently, Aly and Hassan [35] investigated the dynamic effects including the Brownian motion and thermophoresis, the developments of the second-order slip velocity on the boundary layer flow and heat transfer over a stretching surface in the presence of nanoparticle fractions. They proved that the second-order slip parameter influences strongly the flow velocity and surface shear stress on the stretching sheet and also the reduced Nusselt and the reduced Sherwood numbers.

*Aim of the Present Work*. The purpose of the present work is therefore to analyze the boundary layer stagnation-point flow towards a stretching sheet in a porous medium saturated by a nanofluid in suction/injection cases and in the presence of radiation, magnetic field, and heat generation/absorption with effect of the velocity slip boundary condition. The mathematical formulation of the current model is introduced in Section 2, where proper similarity transforms are then applied to gain a system of nonlinear ordinary differential equations which represents the flow through the boundary layer. The resulting system is numerically solved by the Chebyshev pseudospectral differentiation matrix (ChPDM) in Section 3 and is then discussed via tables and figures in Section 4, besides comparison with already published results.

#### 2. The Mathematical Modelling

##### 2.1. Description of the Problem

In this research, we consider a steady laminar two-dimensional flow of a viscous incompressible electrically conducting nanofluid near a stagnation-point over a heated stretching sheet in a porous medium, which is saturated by a nanofluid coinciding with the plane , where the flow region is confined to . It is assumed that two equal and opposite forces are applied along the -axis regarding the surface is stretched where the position of the origin is fixed [18]. In addition, the nanofluid is investigated under the effect of thermal radiation and a uniform magnetic field of strength is applied in the positive direction of -axis. However, the magnetic Reynolds number is assumed to be small, that the induced magnetic field is negligible [24]. Further, the velocities of the external flow and stretching sheet are and , respectively, where and are positive constants.

##### 2.2. Basic Equations

In the presence of heat source/sink and with neglecting the viscous dissipation, the basic conservation of mass, momentum, thermal energy, and nanoparticles equations for nanofluids, which match the above physical model, can be written in the Cartesian coordinates and assubject to the following boundary conditions:where and are the velocity components along the axes and , respectively, is the temperature, and is the volume fraction of nanoparticles. In addition, is the density; is the kinematic viscosity, where is the dynamic viscosity; is the specific heat; is the thermal diffusivity, where is the thermal conductivity; and and are the Brownian diffusion and thermophoresis diffusion coefficients, respectively. Further, is the permeability of the porous medium, is the heat generation or absorption coefficient, is the radiative heat flux, is the electrical conductivity, is the ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid (see Khan and Pop [28], Nield and Kuznetsov [36]), and expresses the slip velocity model. It should be noted that and denote the wall and ambient values, respectively, of the parameters, while refers to the fixed parameter for particles.

Regarding the approximation of Rosseland for radiation (see [37, 38]), the radiative heat flux is simplified aswhere and are the Stefen-Boltzmann constant and mean absorption coefficient, respectively. It is assumed that the temperature changes within the flow such that the term may be expressed as a linear function of the temperature. Hence, on expanding in a Taylor series about and neglecting the higher order terms, one obtainsTherefore, (3) reduces to

##### 2.3. Velocity Slip Model

As mentioned in Section 1, for the continuum modeling of fluidic transport, the assumption of no-slip boundary condition is no longer assumed, and a certain degree of tangential slip must be allowed. In the present work, we consider Wu’s slip model which is given by (see [14, 29–31])where is an arbitrary Knudsen number, , is the momentum accommodation, and is the molecular mean free path. Based on the definition of , it is noticed that, for any given value of , we have . Thus the molecular mean free path is always positive. Therefore, the value of should be negative and hence the second term in the right-hand side of (9) is a positive number.

##### 2.4. Similarity Transforms

The appropriate similarity transforms of (1), (2), (8), and (3) with the boundary conditions (5) are introduced as follows [23]:The stream function is defined in the usual way asto identically satisfy (1). On substituting (10) into (2), (4), and (8), the following ordinary differential equations are obtained:subject to the following boundary conditions:where the governing nondimensional parameters are defined as follows: is the magnetic field, is the permeability, is the velocity ratio, is the thermal radiation, is the heat source (when (for ), is the Brownian motion, and is the thermophoresis. Further, and are Prandtl and Lewis numbers, respectively, is the suction when and injection for , , and are the first-order velocity slip and second-order velocity slip, respectively.

It should be noted that, when , the current study reduces to the classical problem of flow and heat transfer due to a stretching surface in a viscous fluid. In this case, the boundary value problem for becomes ill-posed without physical significance. Further, as special cases, when and with , system (12) to (16) is reduced to that one obtained by Ibrahim et al. [23], with , to the system introduced by Hamad and Ferdows [21], and with to that given by Jalilpour et al. [24] (on dropping the prescribed surface heat flux). Furthermore, when , the present model matches that one obtained by Zheng et al. [22] (with absence of the temperature jump).

##### 2.5. Quantities of Practical Interest

In this type of study, it should be noted that the quantities of practical interest are the skin friction coefficient , local Nusselt number , and local Sherwood number , which are defined aswhere , , and are the skin friction (or shear stress), heat flux from the surface, and mass flux, respectively, which are given by

Therefore, on using (11) in (17) and (18), we obtainwhere is the local Reynolds number based on the stretching velocity, and are referred as reduced Nusselt number and reduced Sherwood number as mentioned by Khan and Pop [28] and then by many others; see, for example, Vajravelu et al. [39], Hamad and Ferdows [21], and Aly and Ebaid [14].

#### 3. Numerical Approach

In order to solve the coupled ordinary differential equations (12)–(14), which are third order in and second order in both and , with the boundary conditions (15) and (16), we apply the Chebyshev pseudospectral differentiation matrix (ChPDM) technique. Hence, these equations become

The resulting nonlinear equations (20)–(22) are associated with the boundary conditions equations (23) and (24) that contain equations which are solved using Newton method. The computer program of the numerical method was executed in MATHEMATICA running on a PC. For more details of transforming the domain of the problem to the Chebyshev one, that is, , definition of the associated collocation points () and the matrix entries , where th is the derivative of any selected function, the reader is advised to see [35]. Recently, this approach has been applied in nano boundary layer study ([34, 35]), and further, Aly [40] and Aly and Sayed [41] have used it for investigating the nanofluids flow.

#### 4. Results and Discussion

In this work, suction/injection and heat source/sink flow near a stagnation-point over a heated stretching sheet in a porous medium saturated by a nanofluid, with effect of thermal radiation and magnetic field in the presence of the velocity slip model, have been studied. As presented in Section 2, the governing system of partial differential equations has been transformed into a system of nonlinear ordinary equations using similarity transformations. It was found that the solution depends on magnetic (), permeability (), and radiation () parameters; Prandtl () and Lewis () numbers; Brownian motion () and thermophoresis () coefficients; heat source/sink (); suction/injection (); first-order () and second-order () slip factors. Then, the obtained system has been numerically solved by the Chebyshev pseudospectral differentiation matrix (ChPDM) as briefly introduced in Section 3.

It should be noted that the current velocity slip model affects directly equation of the stream function , as shown in the boundary condition (15), and therefore its inflow involves also for the temperature and concentration (volume fraction of nanoparticles). In the next subsection, the importance of applying this slip model on investigating system (12)–(14) is to be studied. For implementation of this model, comparison with already published results is done in some special cases for different changes of the selected parameters, where the dotted curves refer to the present results.

##### 4.1. Implementation of the Present Physical Model

###### 4.1.1. Special Case 1: When

Figure 1 illustrates influence of the velocity ratio on the stream velocity profile when , , , , (solid curves as in Figure in [23]), and , . This graph shows that when , the stream velocity increases (decreases) uniformly over the entire domain and the boundary layer thickness decreases as values of increase. However, as the slip model presents, the influence of the stream velocity reaches very rapidly the stable manner at , achieving the boundary condition at infinity for both cases of . This is because, as expected for the fluid flows at nanoscales, the shear stress at the wall decreases with an increase in the first- and second-order slip parameters (see [10, 34, 35]). Hence, the stream velocity becomes slower and, however, temperature and concentration (volume fraction of nanoparticles) increase, as shown in the next three figures.