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

Authors: Nor Azizah Yacob; Anuar Ishak; Roslinda Nazar; Ioan Pop
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 velocityuw(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 Tw(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 coordinatesx and y as (see Oztop and Abu-Nada [29])
(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=-1ρnf∂p∂x+μnfρnf(∂2u∂x2+∂2u∂y2)+ϕρsβs+(1-ϕ)ρfβfρnfg(T-T∞),(3)u∂v∂x+v∂v∂y=-1ρnf∂p∂y+μnfρnf(∂2v∂x2+∂2v∂y2),(4)u∂T∂x+v∂T∂y=∂∂x(αnf∂T∂x)+∂∂y(αnf∂T∂y),
subject to the boundary conditions
(5)u=uw(x),v=0,T=Tw(x)aty=0,u⟶0,T⟶T∞asy⟶∞.
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, ρnf is the effective density of the nanofluid,, αnf is the effective thermal diffusivity of the nanofluid and μnf is the dynamic viscosity of the nanofluid, which are given by (see Oztop and Abu-Nada [29])
(6)ρnf=(1-ϕ)ρf+ϕρs,αnf=knf(ρCp)nf,μnf=μf(1-ϕ)2.5,
where ϕ is the nanoparticle fraction, ρf is the density of the fluid, ρs is the density of the solid, μf is the dynamic viscosity of the fluid, knf is the effective thermal conductivity of the nanofluid, and (ρCp)nf is the heat capacity of the nanofluid, which are given by (see Oztop and Abu-Nada [29])
(7)knfkf=ks+2kf-2ϕ(kf-ks)ks+2kf+ϕ(kf-ks),(ρCp)nf=(1-ϕ)(ρCp)f+ϕ(ρCp)s.
The use of the effective thermal conductivity of the nanofluid knf 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:
(8)ψ(x,y)=[aνf]1/2xf(η),θ(η)=T-TwTw-T∞,η=[aνf]1/2y,
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:
(9)1(1-ϕ)2.5[(1-ϕ)+ϕ(ρs/ρf)]f′′′+ff′′-f′2+λ[ϕ(1-ϕ)(ρf/ρs)+ϕβsβf+1-ϕϕ(ρs/ρf)+(1-ϕ)]θ=0,1Prknf/kf(1-ϕ)+ϕ(ρCp)/(ρCp)fθ′′+fθ′-f′θ=0,
subject to the boundary conditions
(10)f(0)=0,f′(0)=1,θ(0)=1,f′(η)⟶0,θ(η)⟶0asη⟶∞,
where primes denote differentiation with respect to η, Pr=νf/αf is the Prandtl number, and λ is the buoyancy or mixed convection parameter, which is defined as
(11)λ=GrxRex2(=βfgba2),
with Grx=gβf(Tw-T∞)x3/νf2 being the local Grashof number and Rex=uw(x)x/νf being the local Reynolds number.The pressurep can now be determined from (3) and is given by
(12)1ρnfp=-12v2+μnfρnf∂v∂y+constant.The physical quantities of interest are the skin friction coefficientCf and the Nusselt number Nu, which are defined as
(13)Cf=τwρfuw2,Nu=xqwkf(Tw-T∞),
where τw is the surface shear stress and qw is the surface heat flux, which are defined as
(14)τw=μnf(∂u∂y)y=0,qw=-knf(∂T∂y)y=0.
Using variables in (8), we obtain
(15)Rex1/2Cf=(μnfμf)f′′(0),Rex-1/2Nu=-(knfkf)θ′(0).


## 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 gradientθ′(0) for λ=0, ϕ=0, and ϕ=0.1.
ϕ
=
0
ϕ
=
0.1
Pr
Grubka and Bobba [30]
Ishak et al. [31]
Present results
Present results
Cu
Al2O3
TiO2
0.72
-
0.8086
—
-
0.8086
-
0.6146
-
0.6590
-
0.6724
1
-
1.0000
-
1.0000
-
1.0000
-
0.7765
-
0.8228
-
0.8392
3
-
1.9237
-
1.9237
-
1.9237
-
1.5703
-
1.6175
-
1.6479
7
—
-
3.0722
-
3.0723
-
2.5593
-
2.6057
-
2.6535
10
-
3.7207
-
3.7207
-
3.7207
-
3.1171
-
3.1634
-
3.2211Table 2
Thermophysical properties of fluid and nanoparticles.
Physical properties
Fluid phase (water)
Cu
A
l
2
O
3
Ti
O
2
C
p (J/kg K)
4179
385
765
686.2
ρ (KG/m3)
997.1
8933
3970
4250
k (W/m K)
0.613
400
40
8.9538
α
×
1
0
7 (m2/s)
1.47
1163.1
131.7
30.7
β
×
1
0
-
5 (1/K)
21
1.67
0.85
0.9Figure1 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 profilesθ(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 profilesf′(η) for different values of ϕ when Pr=7 and λ=1 using Cu nanoparticle.Figure4 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 αnf), 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 off′′(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 profilesf′(η) for different types of nanoparticles when Pr=7, ϕ=0.1, and λ=1.Figure 7
Temperature profilesθ(η) for different types of nanoparticles when Pr=7, ϕ=0.1, and λ=1.The variations off′′(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 off′′(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 withϕ 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.



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