Mixed Convection Flow Adjacent to a Stretching Vertical Sheet in a Nanofluid

The characteristics of fluid flow and heat transfer over a stretching vertical sheet immersed in a nanofluid are investigated numerically in this paper. Three different types of nanoparticles, namely, copper Cu, alumina Al2O3, and titania TiO2, are considered, using water as the base fluid. It is found that nanofluid with titania nanoparticles has better enhancement on the heat transfer rate compared to copper and alumina nanoparticles. For a particular nanoparticle, increasing the nanoparticle fraction is to reduce the skin friction coefficient and the heat transfer rate at the surface.


Introduction
Nanofluids are defined as dilute suspensions of particles with at least one critical dimension smaller than around 100 nm and are also known as two-phase fluid [1].Nanofluids are produced by dispersing nanometer-scale solid particles into base liquids with low thermal conductivity such as water, ethylene glycol (EG), and oils [2].Nanofluids can be Newtonian or non-Newtonian fluids.The term "nanofluids" was first introduced by Choi [3].There are mainly two techniques used to produce nanofluids which are the singlestep and the two-step methods (see Akoh et al. [4] and Eastman et al. [5]).Both of these methods have advantages and disadvantages as discussed by Wang and Mujumdar [2].The materials with sizes of nanometers possess unique physical and chemical properties [6].They can flow smoothly through microchannels without clogging them because they are small enough to behave similar to liquid molecules [7].This fact has attracted many researchers such as Abu-Nada and Oztop [6], Tiwari and Das [8], El Bécaye Maïga et al. [9], Polidori et al. [10], Talebi et al. [11], Akbari et al. [12], and Shahi et al. [13] to investigate the heat transfer characteristics of nanofluids in cavities, and they found that the presence of the nanoparticles in the fluids increases appreciably the effective thermal conductivity of the fluid and consequently enhances the heat transfer characteristics.The recent book by Das et al. [14] and the more recent review paper by Kakac ¸and Pramuanjaroenkij [15] present an excellent collection of the work done on nanofluids.
Motivated by the above investigations, the present paper considers the problem of mixed convection boundary layer flow adjacent to a stretching vertical sheet immersed in a nanofluid, using the nanofluid model proposed by Tiwari and Das [8], which was also used by several authors (cf.Abu-Nada and Oztop [6], Talebi et al. [11], Abu-Nada [16], Ahmad et al. [17], Rohni et al. [18], Bachok et al. [19][20][21][22][23], Yacob et al. [24,25], and Muthtamilselvan et al. [26]).As mentioned by El Bécaye Maïga et al. [9], the study of boundary layer flow is important because the reduction of the thermal boundary layer thickness due to the presence of particles and their random motion within the base fluid may have important contributions to such heat transfer improvement as well.It should be pointed out that there is only a very little work done on boundary layer flow of nanofluids, as can be seen in the recent papers by Nield and Kuznetsov [27] and Kuznetsov and Nield [28].
It is worth mentioning that nanofluids, with nanosized particles, appear to have a very high thermal conductivity and may be able to meet the rising demand as an efficient heat transfer agent.Scientists and engineers have started showing interest in the study of heat transfer characteristics of these nanofluids.However, a clear picture about the heat transfer through these nanofluids is yet to emerge.Taking into account the rising demands of modern technology, including chemical production, power stations, and microelectronics, there is a need to develop new types of fluids that will be more effective in terms of heat exchange performance [26].

Problem Formulation
Consider a steady mixed convection flow of a nanofluid adjacent to a stretching vertical sheet with a linear velocity   () = , where  is a constant and  is the coordinate measured along the stretching sheet in the vertical direction.The flow takes place at  ≥ 0, where  is the coordinate measured normal to the stretching sheet.It is assumed that the temperature at the stretching surface is   () =  ∞ + , where  is a constant and  ∞ is the constant temperature of the ambient fluid.
The basic equations of continuity, momentum, and energy for nanofluids can be obtained, written in Cartesian coordinates  and  as (see Oztop and Abu-Nada [29]) subject to the boundary conditions Here,  and V are the velocity components along the  and  axes, respectively,  is the fluid pressure,  is the thermal expansion coefficient (with subscripts  and  standing for fluid and solid, respectively),  is the gravitational acceleration,  nf is the effective density of the nanofluid,,  nf is the effective thermal diffusivity of the nanofluid and  nf is the dynamic viscosity of the nanofluid, which are given by (see Oztop and Abu-Nada [29]) where  is the nanoparticle fraction,   is the density of the fluid,   is the density of the solid,   is the dynamic viscosity of the fluid,  nf is the effective thermal conductivity of the nanofluid, and (  ) nf is the heat capacity of the nanofluid, which are given by (see Oztop and Abu-Nada [29]) The use of the effective thermal conductivity of the nanofluid  nf is approximated by the Maxwell-Garnetts model and is restricted to spherical nanoparticles, where it does not account for other shapes of nanoparticles.This model is found to be appropriate for studying heat transfer enhancement using nanofluids (Abu-Nada [16]).
We look for a similarity solution of ( 1)-( 4) with the boundary conditions ( 5) by introducing the following transformation: where the stream function  is defined in the usual way as  = / and V = −/.Equation ( 1) is automatically satisfied, and ( 2) and ( 4), respectively, reduce to the following nonlinear ordinary differential equations: subject to the boundary conditions where primes denote differentiation with respect to , Pr = ]  /  is the Prandtl number, and  is the buoyancy or mixed convection parameter, which is defined as with Gr  =   (  −  ∞ ) 3 /] 2  being the local Grashof number and Re  =   ()/]  being the local Reynolds number.
The pressure p can now be determined from (3) and is given by The physical quantities of interest are the skin friction coefficient   and the Nusselt number Nu, which are defined as where   is the surface shear stress and   is the surface heat flux, which are defined as Using variables in (8), we obtain

Results and Discussion
The nonlinear ordinary differential equations ( 9) subject to the boundary conditions (10) were solved numerically by means of an implicit finite-difference scheme known as the Keller-box method as described in the book by Cebeci and Bradshaw [32].We have considered three different types of nanoparticles, namely, copper (Cu), alumina (Al 2 O 3 ), and titania (TiO 2 ) with water as the base fluid.Following Abu-Nada [16], we considered the range of nanoparticle fraction as 0 ≤  ≤ 0.2.The Prandtl number of the base fluid (water) is kept constant at 7. It is worth mentioning that the present study reduces to the classical viscous fluid when  = 0. Therefore, in order to validate the present numerical method used, we have compared our results with those obtained by Grubka and Bobba [30] and Ishak et al. [31] for different values of Pr when  = 0 as shown in Table 1.The comparisons are found to be in a very good agreement.The thermophysical properties of fluid and nanoparticles are listed in Table 2.
Figure 1 shows the temperature distribution for the fluid with nanoparticle Al 2 O 3 when  = 1.It can be seen that the thermal boundary layer thickness increases as the nanoparticle fraction  is increased and consequently decreases the surface temperature gradient.This observation is consistent with the surface temperature gradient −  (0) illustrated in Figure 2. The velocity distribution for Cu nanoparticles when  = 1 is presented in Figure 3.It can be seen that the velocity boundary layer thickness decreases with , which in turn increases the velocity gradient at the surface.
Figure 4 displays the variation of the skin friction coefficient   (0) for different types of nanoparticles when  = 0.1, while the respective surface temperature gradient that represents the heat transfer rate at the surface is presented in Figure 5.We note that the values of   (0) obtained are quite similar for TiO 2 and Al 2 O 3 .It is in agreement with the velocity gradient displayed in Figure 6.Further, from Table 2 and Figure 4, we observe that the nanoparticle with higher density (i.e., Cu) has the lowest skin friction coefficient.The thermal boundary layer thickness as shown in Figure 7 decreases with a decrease in thermal conductivity  (or low thermal diffusivity  nf ), which in turn gives rise to the Nusselt number −  (0) as illustrated in Figure 5. Furthermore, the values of   (0) and −  (0) increase as the buoyancy parameter  is increased.Note that the entire values of −  (0) are always positive; that is, the heat is transferred from the hot sheet to the cold fluid.From Figures 1, 3, 6, and  7, it is observed that the velocity and temperature profiles satisfy the far field boundary conditions (10) asymptotically, thus supporting the validity of the numerical results obtained.
The variations of   (0) and −  (0) with  for different types of nanoparticles when  = 2 are depicted in Figures 8 and 9, respectively.It is clear that both of them decrease with increasing values of .We notice that the effect of nanoparticle fraction on the skin friction coefficient is more pronounced for Cu.A decrease in thermal conductivity is to enhance the heat transfer rate at the surface as shown in Figures 5 and 9. Conversely, the opposite behaviors are observed for the effect of nanoparticle fraction .The thermal conductivity ratio with  for different types of nanoparticles is presented in Figure 10.As expected, the thermal conductivity ratio increases with , which follows from the relationship given by (7).However, it decreases with decreasing value of thermal conductivity of the nanoparticles.

Conclusions
The problem of steady two-dimensional laminar flow adjacent to a stretching vertical sheet immersed in a nanofluid was studied numerically.The governing partial differential equations were transformed into a system of nonlinear ordinary differential equations using a similarity transformation, before being solved numerically by the Keller-box method.It was found that nanofluid with TiO 2 nanoparticles which has lower thermal conductivity has better heat transfer capability    compared to Al 2 O 3 -water and Cu-water nanofluids.For a particular nanoparticle, increasing nanoparticle fraction is to reduce the skin friction coefficient and the heat transfer rate at the surface.

Figure 8 :
Figure 8: Variation of   (0) with  for different types of nanoparticles when Pr = 7 and  = 2.

Table 2 :
Thermophysical properties of fluid and nanoparticles.