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

Abstract

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.

1. 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 single-step 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 Kakaç 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. [19–23], 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].

2. Problem Formulation

Consider a steady mixed convection flow of a nanofluid adjacent to a stretching vertical sheet with a linear velocity 𝑢_𝑤(𝑥)=𝑎𝑥, where 𝑎 is a constant and 𝑥 is the coordinate measured along the stretching sheet in the vertical direction. The flow takes place at 𝑦≥0, where 𝑦 is the coordinate measured normal to the stretching sheet. It is assumed that the temperature at the stretching surface is 𝑇_𝑤(𝑥)=𝑇_∞+𝑏𝑥, where 𝑏 is a constant and 𝑇_∞ 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 𝑥 and 𝑦 as (see Oztop and Abu-Nada [29]) 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢1𝜕𝑦=−𝜌_nf𝜕𝑝+𝜇𝜕𝑥𝜌^nf_nf𝜕²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²+𝜙𝜌_𝑠𝛽_𝑠+1−𝜙𝜌_𝑓𝛽_𝑓𝜌_nf𝑔𝑇−𝑇_∞,𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑣1𝜕𝑦=−𝜌_nf𝜕𝑝+𝜇𝜕𝑦𝜌^nf_nf𝜕²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦²,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝜕𝜕𝑦𝛼𝜕𝑥^nf𝜕𝑇+𝜕𝜕𝑥𝛼𝜕𝑦^nf𝜕𝑇,𝜕𝑦 (1)(2)(3)(4) subject to the boundary conditions 𝑢=𝑢_𝑤(𝑥),𝑣=0,𝑇=𝑇_𝑤(𝑥)at𝑦=0,𝑢⟶0,𝑇⟶𝑇_∞as𝑦⟶∞. (5) Here, 𝑢 and 𝑣 are the velocity components along the 𝑥 and 𝑦 axes, respectively, 𝑝 is the fluid pressure, 𝛽 is the thermal expansion coefficient (with subscripts 𝑓 and 𝑠 standing for fluid and solid, respectively), 𝑔 is the gravitational acceleration, 𝜌_nf is the effective density of the nanofluid,, 𝛼_nf is the effective thermal diffusivity of the nanofluid and 𝜇_nf is the dynamic viscosity of the nanofluid, which are given by (see Oztop and Abu-Nada [29]) 𝜌_nf=1−𝜙𝜌_𝑓+𝜙𝜌_𝑠,𝛼_nf=𝑘_nf𝜌𝐶_𝑝nf,𝜇_nf=𝜇_𝑓1−𝜙^2.5, (6) where 𝜙 is the nanoparticle fraction, 𝜌_𝑓 is the density of the fluid, 𝜌_𝑠 is the density of the solid, 𝜇_𝑓 is the dynamic viscosity of the fluid, 𝑘_nf is the effective thermal conductivity of the nanofluid, and (𝜌𝐶_𝑝)_nf is the heat capacity of the nanofluid, which are given by (see Oztop and Abu-Nada [29]) 𝑘_nf𝑘_𝑓=𝑘_𝑠+2𝑘_𝑓−2𝜙𝑘_𝑓−𝑘_𝑠𝑘_𝑠+2𝑘_𝑓+𝜙𝑘_𝑓−𝑘_𝑠,𝜌𝐶_𝑝nf=1−𝜙𝜌𝐶_𝑝𝑓+𝜙𝜌𝐶_𝑝𝑠. (7) The use of the effective thermal conductivity of the nanofluid 𝑘_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: 𝜓𝑥,𝑦=𝑎𝜈_𝑓^1/2𝑥𝑓𝜂,𝜃𝜂=𝑇−𝑇_𝑤𝑇_𝑤−𝑇_∞,𝑎𝜂=𝜈_𝑓^1/2𝑦, (8) where the stream function 𝜓 is defined in the usual way as 𝑢=𝜕𝜓/𝜕𝑦 and 𝑣=−𝜕𝜓/𝜕𝑥. Equation (1) is automatically satisfied, and (2) and (4), respectively, reduce to the following nonlinear ordinary differential equations: 11−𝜙^2.51−𝜙+𝜙𝜌_𝑠/𝜌_𝑓𝑓^′′′+𝑓𝑓^′′−𝑓^′2𝜙+𝜆1−𝜙𝜌_𝑓/𝜌_𝑠𝛽+𝜙𝛽^𝑠_𝑓+1−𝜙𝜙𝜌_𝑠/𝜌_𝑓+11−𝜙𝜃=0,𝑘Pr^nf/𝑘_𝑓1−𝜙+𝜙𝜌𝐶_𝑝/𝜌𝐶_𝑝𝑓𝜃^′′+𝑓𝜃^′−𝑓^′𝜃=0, (9) subject to the boundary conditions 𝑓(0)=0,𝑓^′(0)(0)𝑓=1,𝜃=1,_′𝜂⟶0,𝜃𝜂⟶0as𝜂⟶∞, (10) where primes denote differentiation with respect to 𝜂, Pr=𝜈_𝑓/𝛼_𝑓 is the Prandtl number, and 𝜆 is the buoyancy or mixed convection parameter, which is defined as 𝜆=Gr_𝑥Re²_𝑥=𝛽_𝑓𝑔𝑏𝑎², (11) with Gr_𝑥=𝑔𝛽_𝑓(𝑇_𝑤−𝑇_∞)𝑥³/𝜈²_𝑓 being the local Grashof number and Re_𝑥=𝑢_𝑤(𝑥)𝑥/𝜈_𝑓 being the local Reynolds number.

The pressure 𝑝 can now be determined from (3) and is given by 1𝜌_nf1𝑝=−2𝑣²+𝜇_nf𝜌_nf𝜕𝑣𝜕𝑦+constant. (12)

The physical quantities of interest are the skin friction coefficient 𝐶_𝑓 and the Nusselt number Nu, which are defined as 𝐶_𝑓=𝜏_𝑤𝜌_𝑓𝑢²_𝑤,Nu=𝑥𝑞_𝑤𝑘_𝑓𝑇_𝑤−𝑇_∞, (13) where 𝜏_𝑤 is the surface shear stress and 𝑞_𝑤 is the surface heat flux, which are defined as 𝜏_𝑤=𝜇_nf𝜕𝑢𝜕𝑦_𝑦=0,𝑞_𝑤=−𝑘_nf𝜕𝑇𝜕𝑦_𝑦=0. (14) Using variables in (8), we obtain Re_𝑥^1/2𝐶_𝑓=𝜇_nf𝜇_𝑓𝑓^′′(0),Re_𝑥^−1/2𝑘Nu=−^nf𝑘_𝑓𝜃^′(0). (15)

3. 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 (Al₂O₃), and titania (TiO₂) with water as the base fluid. Following Abu-Nada [16], we considered the range of nanoparticle fraction as 0≤𝜙≤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 𝜙=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 𝜙=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.

Table 1 

Surface temperature gradient 𝜃^′(0) for 𝜆=0, 𝜙=0, and 𝜙=0.1.

Table 2 

Thermophysical properties of fluid and nanoparticles.

Figure 1 shows the temperature distribution for the fluid with nanoparticle Al₂O₃ when 𝜆=1. It can be seen that the thermal boundary layer thickness increases as the nanoparticle fraction 𝜙 is increased and consequently decreases the surface temperature gradient. This observation is consistent with the surface temperature gradient −𝜃^′(0) illustrated in Figure 2. The velocity distribution for Cu nanoparticles when 𝜆=1 is presented in Figure 3. It can be seen that the velocity boundary layer thickness decreases with 𝜙, which in turn increases the velocity gradient at the surface.

Figure 1 

Temperature profiles 𝜃(𝜂) for different values of 𝜙 when Pr=7 and 𝜆=1 using Al2O3 nanoparticle.

Figure 2 

Variation of −𝜃′(0) with 𝜆 for different values of 𝜙 when Pr=7 using Al2O3 nanoparticle.

Figure 3 

Velocity profiles 𝑓′(𝜂) for different values of 𝜙 when Pr=7 and 𝜆=1 using Cu nanoparticle.

Figure 4 displays the variation of the skin friction coefficient 𝑓^′′(0) for different types of nanoparticles when 𝜙=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 𝑓^′′(0) obtained are quite similar for TiO₂ and Al₂O₃. 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 𝑘 (or low thermal diffusivity 𝛼_nf), which in turn gives rise to the Nusselt number −𝜃^′(0) as illustrated in Figure 5. Furthermore, the values of 𝑓^′′(0) and −𝜃^′(0) increase as the buoyancy parameter 𝜆 is increased. Note that the entire values of −𝜃^′(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.

Figure 4 

Variation of 𝑓′′(0) with 𝜆 for different types of nanoparticles when Pr=7 and 𝜙=0.1.

Figure 5 

Variation of −𝜃′(0) with 𝜆 for different types of nanoparticles when Pr=7 and 𝜙=0.1.

Figure 6 

Velocity profiles 𝑓′(𝜂) for different types of nanoparticles when Pr=7, 𝜙=0.1, and 𝜆=1.

Figure 7 

Temperature profiles 𝜃(𝜂) for different types of nanoparticles when Pr=7, 𝜙=0.1, and 𝜆=1.

The variations of 𝑓^′′(0) and −𝜃^′(0) with 𝜙 for different types of nanoparticles when 𝜆=2 are depicted in Figures 8 and 9, respectively. It is clear that both of them decrease with increasing values of 𝜙. 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 𝜙. The thermal conductivity ratio with 𝜙 for different types of nanoparticles is presented in Figure 10. As expected, the thermal conductivity ratio increases with 𝜙, which follows from the relationship given by (7). However, it decreases with decreasing value of thermal conductivity of the nanoparticles.

Figure 8 

Variation of 𝑓′′(0) with 𝜙 for different types of nanoparticles when Pr=7 and 𝜆=2.

Figure 9 

Variation of −𝜃′(0) with 𝜙 for different types of nanoparticles when Pr=7 and 𝜆=2.

Figure 10 

Variation of thermal conductivity ratio with 𝜙 for different types of nanoparticles when Pr=7 and 𝜆=1.

4. 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 TiO₂ nanoparticles which has lower thermal conductivity has better heat transfer capability compared to Al₂O₃-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.

Acknowledgments

The financial supports received from the Ministry of Higher Education, Malaysia (Project code: FRGS/1/2012/SG04/UKM/01/1), and the Universiti Kebangsaan Malaysia (Project code: DIP-2012-31) are gratefully acknowledged.