Non-alignment stagnation-point flow of a nanofluid past a permeable stretching/shrinking sheet: Buongiorno’s model

The paper deals with a stagnation-point boundary layer flow towards a permeable stretching/shrinking sheet in a nanofluid where the flow and the sheet are not aligned. We used the Buongiorno model that is based on the Brownian diffusion and thermophoresis to describe the nanofluid in this problem. The main purpose of the present paper is to examine whether the non-alignment function has the effect on the problem considered when the fluid suction and injection are imposed. It is interesting to note that the non-alignment function can ruin the symmetry of the flows and prominent in the shrinking sheet. The fluid suction will reduce the impact of the non-alignment function of the stagnation flow and the stretching/shrinking sheet but at the same time increasing the velocity profiles and the shear stress at the surface. Furthermore, the effects of the pertinent parameters such as the Brownian motion, thermophoresis, Lewis number and the suction/injection on the flow and heat transfer characteristics are also taken into consideration. The numerical results are shown in the tables and the figures. It is worth mentioning that dual solutions are found to exist for the shrinking sheet.

enhancement may be due to the following items; the use of nanoparticles to increase the thermal conductivity of the base fluid and chaotic motion of very small particles increases the turbulence in the fluid and thus expediting the process of energy exchange. Further, Lee et al. 8 gave an overview on the thermal conductivity data, mechanisms and models in nanofluid by several research groups. They found that the inconsistency findings for each experimental thermal conductivity data is due to differences in sample quality, thermal conductivity dependence on many factors and differences in the measurement uncertainties. Therefore, they suggested the use of quality nanofluid, reference samples and equipment in a study. In addition, the issues involved in the mechanism to explain the thermal conductivity in nanofluid occur because of the lack of knowledge about the basic concepts of science to the mechanism. They also found that nanofluid model consisting of a combination of static and dynamic mechanism is seen to be more effective in describing the events in nanofluid. Nanofluid research is very important because of its use in various areas such as in the industrial cooling, electronics, electrical and many others. This field is not only carried out in the laboratory for academic purposes, but there are researchers who apply the nanofluid studies on the real devices to enhance the heat transfer performance of the devices 5 .
Formulations model for heat transfer by convection in the nanofluid have been proposed by many researchers. Among the famous is the model by Buongiorno 9 which takes into account the Brownian motion and thermophoresis effect. Mathematical model introduced by Buongiorno were used in the studies by Nield and Kuznetsov 10 , Corcione et al. 11 , Tham et al. 12 and recently by Garoosi et al. 13 . Generally, the findings of these studies found that the Brownian motion and thermophoresis parameters affect the boundary layer and the heat transfer in the nanofluid. Not only that, the Buongiorno model has been used to study the nanofluid past a stretching/shrinking sheet. Example of such study is the one by Rahman et al. 14 that also considered the permeable surface of the sheet and with the second order slip velocity. They also employ a new boundary condition for the nanoparticles volume fraction at the surface of the shrinking sheet. Mustafa et al. 15 also used the Buongiorno model to study the stagnation flow towards the stretching sheet using the homotopy analysis method (HAM). The research in the stretching or shrinking sheet is worth studying because it is crucial for the industrial applications such as the aerodynamic extrusion of plastic sheets, condensation process of metallic plate in a cooling bath and glass 16 , wire drawing and hot rolling 17 .
In this present study, we investigated the stagnation flow of nanofluid towards a permeable stretching/ shrinking sheet where the flow and the sheet are not aligned. The non-alignment function is proposed by Wang 4 and to the best of our knowledge, we are the first to consider the non-alignment function in the nanofluid that past a permeable stretching/shrinking sheet using Buongiorno's model. According to Wang 4 , the non-alignment of the stagnation flow and the stretching/shrinking sheet can destroy the symmetry and complicates the flow field. The study by Wang 4 has been extended by many researchers in various physical conditions such as a recent study by Najib et al. 18 . It should be mentioned that the governing system of ordinary differential equations are solved using the BVP4C function in Matlab. On the other hand, it is worth mentioning to this end that there are several other papers in the literature on symmetry breaking in flows or general analysis of flows [19][20][21] .

Problem Formulation
Consider the steady flow of a viscous nanofluid in the region y > 0 driven by a permeable stretching/ shrinking surface located at y = 0 as shown in Fig. 1, where x, y and z are the Cartesian coordinates measured along the plate, normal to it and in the transversal direction, respectively. Let (u, v, w) be the velocity components in the directions x, y and z, respectively.
Following Wang 4 , we assume that for a two-dimensional stagnation point, the stretching/shrinking velocities of the surface are = ( + ) u b x c w and w = w 0 , where b > 0 is the stretching rate (shrinking rate if b < 0), − c is the location of the stretching origin and w 0 is the mass flux velocity with w 0 < 0 for suction and w 0 > 0 for injection or withdraw flow. It is also assumed that the fluid velocities outside the boundary layer (or the velocity of the inviscid nanofluid) are u e = ax and w e = − az, where a > 0 is the strength of the stagnation flow. Further, it is assumed that the uniform temperature and the uniform nanoparticle volume fraction of the plate are T w and C w , respectively, while those of the ambient fluid are T ∞ and C ∞ . It should be mentioned here that the stretching axis and the stagnation flow are not aligned 14 . Under the boundary layer approximations, the basic equations of the problem under consideration are, see Miklavčič and Wang 22 and Kuznetsov and Nield 23 , where T is the temperature of the nanofluid, C is the nanoparticle volume fraction, p is the pressure, v is the kinematic viscosity, ρ is the density of the nanofluid, α is the thermal diffusivity of the nanofluid, D B is the Brownian diffusion coefficient, D T is the thermophoretic diffusion coefficient and ∇ 2 is the Laplacean in the Cartesian coordinates x, y and z. Further, τ = (ρc) p /(ρc) f , where (ρc) f is the heat capacity of the nanofluid and (ρc) p is the effective heat capacity of the nanoparticle material. We assume that equations (1-6) are subject to the boundary conditions We look for a similarity solution of equations (1-6) of the following form: Substituting equation (8) into equations (2-6), we obtain the following ordinary differential equations: is the constant mass transfer parameter with s > 0 for suction and s < 0 for injection, respectively. It is worth also mentioning that λ = b/a is the stretching/shrinking parameter where λ > 0 corresponds to the stretching sheet and λ < 0 corresponds to the shrinking sheet. The dimensionless constants Pr, Le, Nb and Nt denote the Prandtl number, the Lewis number, the Brownian motion parameter and the thermophoresis parameter, respectively, which are defined as We notice that equations (9) and (10) have been studied by Wang 4 for an impermeable sheet (s = 0) and in a viscous fluid. The pressure p is given by Quantities of physical interest in this problem are the skin friction coefficient C f , the local Nusselt number Nu x and the Sherwood number Sh x which are defined as where τ w is the skin friction or shear stress at the surface of the plate, q w is the heat flux from the surface and q m is the mass flux of the nanoparticle volume fraction from the surface, and are given by where μ is the dynamic viscosity of the fluid and k is the thermal conductivity of the nanofluid. Using equations (8), (16) and (17), we get

Results and Discussions
The system of ordinary differential equations (9-12) with the boundary conditions (13) is solved numerically using the BVP4C function in Matlab. In this problem, solutions are obtained for the Prandtl number Pr = 6.8 (water-based nanofluid) and Lewis number Le = 2. The value of the suction parameter (s > 0) is chosen for 0.2 and the injection parameter (s < 0) is chosen for − 0.2. Meanwhile, the stretching sheet (λ > 0) takes the value λ = 0.5 and the shrinking sheet (λ < 0) takes the value λ = − 0.5. In order to verify the accuracy of the present method, comparison is made with Wang 4 in the absence of the nanofluid parameters. The results are shown in Tables 1 and 2, and they are found to be in good agreement and thus give us confidence on the accuracy of the method. Tables 3 and 4 show the numerical values for the reduced skin friction coefficient ″( ) f 0 , the reduced non-alignment function ′( ) g 0 , the reduced local Nusselt number θ − ′( ) 0 and the reduced nanoparticle volume fraction ϕ − ′( ) 0 with the variations of the Brownian motion parameter Nb, the thermophoresis parameter Nt and the Lewis number Le for the stretching sheet and shrinking sheet, respectively. It is observed that variations of Nb, Nt and Le have no effect on ″( ) f 0 and ′( ) g 0 . From the tables, it is found that the increase of the Brownian motion, thermophoresis and Lewis number parameters will reduce the local Nusselt number for both sheets, also for all suction/injection parameter s considered. Explanation of this phenomenon may be due to the enhancement of the collisions of the particles that results from the increasing of the Brownian motion and thermophoresis, which then increase the thermal boundary layer thickness, followed by the reduction of the local Nusselt number and the heat flux from the surface.  Further, by increasing the Lewis number, the thickness of the thermal diffusion layer is becoming larger than the mass diffusion layer, which leads to lower values of θ − ′( ) 0 . It should be noted that, increasing the Brownian motion, thermophoresis and Lewis number have different effects on the mass flux of the nanoparticle volume fraction. Generally, −φ′(0) is lower for the shrinking sheet when the fluid is injected.
On the other hand, the effects of the fluid suction and injection for the stretching and shrinking sheets are depicted on the Figs 2-9, respectively. It is seen in the Figs 2 and 6 that imposition of the fluid suction tends to increase ″( ) f 0 as well as the velocity profiles. Increment of ″( ) f 0 is not only because of the suction parameter but also due to the reduction of the value of λ. It is obvious that the interface shear stress is higher in the shrinking sheet. Moreover, we observed in the Figs 3 and 7 that the non-alignment function of the origin of the stagnation flow and the sheet can be reduced if the fluid is sucked into the surface of the sheet where it is more prominent in the shrinking sheet. It is worth mentioning that the non-alignment function is not affected by the variations in the Prandtl number. Further, Figs 4 and 8 show that the suction parameter also caused the reduction of the nanofluid temperature as the heat is transported into the interface of the sheet. Hence, as the thermal boundary thickness decreases, the heat flux from the surface is rising. The fluid suction also affected the nanoparticle volume fraction by reducing the profiles. This leads to the increasing of the interface mass transfer as shown in the

Conclusions
The present paper studied the steady laminar stagnation flow over a permeable stretching/shrinking sheet in the nanofluid using the Buongiorno's model. It is found that the second solutions exist in the shrinking region. Further, the non-alignment function of the stagnation flow and the sheet complicates the flow fields and can be increased using the fluid injection. Moreover, the skin friction at the surface of   the sheet is higher when the fluid is sucked into the surface. Different behavior is observed for the fluid injection. Generally, the Brownian motion, thermophoresis and Lewis number are the reducing factor of the heat transfer at the wall of both sheets. However, these parameters provide different effects for the rate of mass transfer. We mention to this end that the present paper can be extended by including the entropy effects into the governing equations by following, for example, the valuable books by Bejan 24,25 and the papers by Bejan 26 , Adboud and Saouli 27 , Makinde 28 , Butt and Ali 29,30 , Rashidi et al. 31 , etc. and also by considering the constructal law 32,33 .