The boundary layer flow and heat transfer of a nanofluid over a stretching sheet are numerically studied. Velocity slip is considered instead of no-slip condition at the boundary as is usually appears in the literature. The governing partial differential equations are transformed into ordinary ones using a similarity transformation, before being solved numerically. Numerical solutions of these equations are obtained using finite element method (FEM). The variations of velocity and temperature inside the boundary layer as well as the skin friction coefficient and the heat transfer rate at the surface for some values of the governing parameters, namely, the nanoparticle volume fraction and the slip parameter are presented graphically and discussed. Comparison with published results for the regular fluid is presented and it is found to be in excellent agreement.
1. Introduction
The fluid flow past a stretching plate was first investigated by Crane [1], where an exact analytical solution to the Navier-Stokes equations was reported. This problem was then extended to a permeable surface by P. S. Gupta and A. S. Gupta [2]. Grubka and Bobba [3] considered a more general case with power law surface temperature variation. They reported a series solution to the energy equation in terms of Kummer’s functions and presented several closed-form analytical solutions for specific conditions. Further, the unsteady flow past a stretching sheet was investigated by Andersson et al. [4], Ali and Mehmood [5], Ishak et al. [6–8], Hayat et al. [9], Sharma [10], and Sharma et al. [11] among others. The study of flow and heat transfer past a stretching sheet is important due to its many industrial applications such as in the polymer industry, where one deals with a stretching plastic sheet. The quality of the final product depends on the rate of heat transfer at the stretching surface (Sparrow and Abraham [12]).
Nanofluid is a new class of fluid with nanosized particles dispersed in a poor thermal conductivity base fluid, such as water and ethylene glycol, to increase its thermal conductivity. These particles, generally metal or metal oxide, increase conduction and convection coefficient, allowing for more heat transfer out of the coolant (Choi and Eastman [13]). It seems that the term nanofluid was first introduced by Choi and Eastman [13] and it was adopted by many researchers. The materials with sizes of nanometers possess unique physical and chemical properties. They can flow smoothly through microchannels without clogging them because they are small enough to behave similarly to liquid molecules (Khanafer et al. [14]). They are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop, and non-Newtonian behavior, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. A very interesting critical synthesis of the variants within the thermophysical properties of nanofluids has been recently presented by Khanafer and Vafai [15]. It has been shown that the experimental results for the effective thermal conductivity and viscosity reported by several authors are in disagreement. Thus, theoretical and experimental studies are essential to clarify the discrepancies in the results and in proper understanding of heat transfer enhancement characteristics of nanofluids. The study by Khanafer and Vafai [15] shows that it is not clear which analytical model should be used to describe the thermal conductivity of nanofluids. Additional theoretical and experimental research studies are required to clarify the mechanisms responsible for heat transfer enhancement in nanofluids. Correlations for effective thermal conductivity and viscosity are synthesized and developed in [15] in terms of pertinent physical parameters based on the reported experimental data. The broad range of current and future applications of nanofluids is discussed in the review article by Wong and Leon [16], which includes automotive, electronics, biomedical, and heat transfer applications besides other applications such as nanofluid detergent. In a recent review article by Saidur et al. [17], the authors also presented some applications of nanofluids in industrial, commercial, residential, and transportation sectors based on the available literatures. A critical review of the state-of-the-art of nanofluids research for heat transfer application was conducted by Wen et al. [18], Mahian et al. [19], and so forth. The enhanced thermal behavior of nanofluids could provide a basis for an enormous innovation for heat transfer intensification, which is of major importance to a number of industrial sectors including transportation, power generation, micromanufacturing, thermal therapy for cancer treatment, chemical and metallurgical sectors, and heating, cooling, ventilation, and air-conditioning. Nanofluids are also important for the production of nanostructured materials for the engineering of complete fluids, as well as for cleaning oil from surfaces due to their excellent wetting and spreading behavior (Ding et al. [20]). The capability of nanofluids to enhance thermal conductivity has attracted the interest of fluid dynamics community to conduct further studies. As a result, the research on nanofluids has progressed rapidly. The number of research articles and citations found in Web of Science and SCOPUS under “nanofluids” shows exponential increase, indicating the high interest in research activities in this topic. In the present paper, we consider the flow and heat transfer over a stretching sheet immersed in a nanofluid, with velocity slip effect at the boundary. The nanofluid model proposed by Tiwari and Das [21], which analyzes the behavior of nanofluids taking into account the solid volume fraction, is employed in the present paper.
2. Problem Formulation
Consider a two-dimensional flow over a flat sheet with heat transfer in a water based nanofluid containing Cu, CuO, Al_{2}O_{3}, or TiO_{2} nanoparticles. We assume that the sheet coincides with the plane y=0 and the flow is confined to y>0. Two equal and opposite forces are applied along the x-axis so that the sheet is stretched keeping the origin fixed. It is assumed that the sheet is stretched with velocity uw=cx, where c>0 is the stretching rate. It is also assumed that the base fluid (i.e., water) and the nanoparticles are in thermal equilibrium. The thermophysical properties of the water and nanoparticles are given in Table 1 (see Bachok et al. [22] and Turkyilmazoglu [23]). Assuming that the nanofluid is viscous and incompressible and using the nanofluid model proposed by Tiwari and Das [21], the governing boundary layer equations of mass, momentum, and thermal energy for nanofluids can be written as follows (see Tiwari and Das [21]):
(1)∂u∂x+∂v∂y=0,(2)ρnf(u∂u∂x+v∂u∂y)=μnf∂2u∂y2,(3)(ρcp)nf(u∂T∂x+v∂T∂y)=knf∂2T∂y2,
where y is the coordinate measured in the direction normal to the sheet, u and v are the velocity components along the x- and y-axes, respectively, T is the nanofluid temperature, ρnf is the effective density of the nanofluid, μnf is the effective dynamic viscosity of nanofluid, knf is the thermal conductivity, and (ρcp)nf is the heat capacity of the nanofluid, which are given by (see Vajravelu et al. [24] and Narayana and Sibanda [25])
(4)ρnf=(1-ϕ)ρf+ϕρs,μnf=μf(1-ϕ)2.5,(ρcp)nf=(1-ϕ)(ρcp)f+ϕ(ρcp)s,knfkf=(ks+2kf)-2ϕ(kf-ks)(ks+2kf)+ϕ(kf-ks),
where ϕ is the solid volume fraction of the nanofluid, ρf is the density of base fluid, ρs is the density of the nanoparticle, μf is the dynamic viscosity of the base fluid, (ρcp)f is the heat capacity of the base fluid, (ρcp)s is the heat capacity of the nanoparticle, kf is the thermal conductivity of the base fluid, and ks is the thermal conductivity of the solid nanoparticle.
Thermo-physical properties of water and nanoparticles.
ρ(kg m^{−3})
cp(J Kg^{−1} K^{−1})
k(W m^{−1} K^{−1})
Pure water (H_{2}O)
997.1
4179
0.6130
Copper (Cu)
8933
385.0
401.00
Copper oxide (CuO)
6320
531.8
76.500
Alumina (Al_{2}O_{3})
3970
765.0
40.000
Titanium oxide (TiO_{2})
4250
686.2
8.9538
Equations (1)–(3) are subjected to the following boundary conditions:
(5)u=uw+A∂u∂y,v=0,T=Twaty=0,u=0,T=T∞asy⟶∞,
where A is the velocity slip factor. It should be mentioned that such slip conditions were recently used in a series of papers [26–31]. It is worth mentioning at this end that fluid flow with slip is important in microelectromechanical systems (MEMS). The flow in these systems deviates significantly from the traditional no-slip flow because of the microscale dimensions of these devices. Rarefied gas flows with slip boundary conditions are often encountered in the microscale devices and low-pressure situations [32].
Following Ishak [33], we introduce the following similarity transformation:
(6)u=cxf′(η),v=-(cνf)1/2f(η),θ(η)=(T-T∞)Tw-T∞,η=ycνf,
where primes denote differentiation with respect to η. Using transformation (6), (1) is automatically satisfied, while (2) and (3), respectively, reduce to the following nonlinear ordinary differential equations:
(7)μnfμff′′′+((1-ϕ)+ϕρsρf)(ff′′-f′2)=0,1Prknfkfθ′′+((1-ϕ)+ϕ(ρcp)s(ρcp)f)fθ′=0
subject to the boundary conditions
(8)f(0)=0,f′(0)=1+γf′′(0),θ(0)=1,f′(η)⟶0,θ(η)⟶0asη⟶∞,
where γ=Ac/νf is the velocity slip parameter and Pr=μf/kf is the Prandtl number.
Physical quantities of interest are the skin friction coefficient Cf and the local Nusselt number Nu, which are defined as
(9)Cf=τwρfuw2,Nu=qwkf(Tw-T∞),
where τw is the surface shear stress and qw is the surface heat flux, which are given by
(10)τw=μnf(∂u∂y)y=0,qw=-knf(∂T∂y)y=0.
Using the similarity variables (6), we obtain
(11)Rex1/2Cf=1(1-ϕ)2.5f′′(0),Rex-1/2Nu=-knfkfθ′(0),
where Rex=uwx/νf is the local Reynolds number.
It should be mentioned that for a regular fluid (ϕ=0) and without slip (γ=0), (7) reduces to
(12)f′′′+ff′′-f′2=0,(13)1Prθ′′+fθ′=0
along with the boundary conditions
(14)f(0)=0,f′(0)=1,θ(0)=1,f′(η)⟶0,θ(η)⟶0asη⟶∞.
Equation (12) subject to the associated boundary conditions (14) admits the closed-form analytical solution
(15)f(η)=1-e-η
which has been first reported by Crane [1], while the solution for the thermal field in terms of Kummer's functions is given by Grubka and Bobba [3] as
(16)θ(η)=e-PrηM(Pr,Pr+1,-Pre-η)M(Pr,Pr+1,-Pr).
Thus,
(17)f′′(0)=-1,θ′(0)=-Pr+Pr2Pr+1M(Pr+1,Pr+2,-Pr)M(Pr,Pr+1,-Pr),
where M(a,b,z) denotes the Kummer’s function (see Abramowitz and Stegun [34]).
3. Method of Solution
The set of ordinary differential equations (7) and (8) are highly nonlinear and cannot be solved analytically. Therefore, the finite element method [35–38] is implemented to solve this system numerically. For computational purposes, η∞ has been fixed as 8. The dimensionless spatial coordinate is discretized by uniform elements. We did experiment with different step sizes (h=0.5,0.1,0.05,0.01,0.005, and 0.004) for γ=ϕ=0.1 and Pr=6.2 as shown in Table 2 and observed that a very slight change occurs for step size h<0.01, but the computational time increases too much. Thus, for the computational purpose, h=0.01 is taken for presentation of the results.
The values of |f′′(0)| and -θ′(0) for CuO-water nanofluid with γ=ϕ=0.1 and Pr = 6.2.
Step size h
|f′′(0)|
-θ′(0)
0.5
0.758528
1.230633
0.1
0.896314
1.396718
0.05
0.916250
1.402676
0.01
0.931318
1.403155
0.005
0.932160
1.402452
0.004
0.931852
1.402065
The Gauss quadrature formula has been used to calculate the integrals. Owing to the nonlinearity of the system of equations, an iterative scheme has been used to solve it. An initial guess is taken at each node point. The system of equations is then linearized by incorporating the functions, which are assumed to be known values of the functions f and θ. After applying the given boundary conditions, the remaining system of equations has been solved using Gauss-elimination method. This gives us new values of unknowns. This process continues till the absolute differences of two successive iterate values of unknowns are less than the accuracy of 0.0001.
3.1. Code Verification
In order to verify the accuracy of the applied numerical scheme, comparisons of the present results corresponding to the values of heat transfer coefficient for γ=0 and ϕ=0 are made with the available results of Grubka and Bobba [3], Chen [39], Mukhopadhyay et al. [40], and Ishak et al. [41, 42] as well as the series solution given by (17), as presented in Table 3. The results are found in an excellent agreement and thus give confidence that the numerical results in our case are accurate.
Values of -θ′(0) for several values of Prandtl number Pr with γ=0.
ϕ
Pr
Reference [3]
Reference [39]
Reference [40]
References [41, 42]
Present results
Numerical
Analytical equation (17)
0
0.72
0.4631
0.46315
0.466965907
0.4631445610
1
0.5820
0.58199
0.5820
0.5820
0.582652158
0.5819767068
3
1.1652
1.16523
1.1652
1.1652
1.165101358
1.165245952
10
2.3080
2.30796
2.3080
2.307564415
2.308003944
0.1
6.2
1.474324348
0.2
1.245900088
4. Results and Discussion
Numerical solutions to the system of ordinary differential equations (7) and (8) were obtained using finite element method (FEM). A study has been made on nondimensional parameters and results are presented in Figures 1–10. For numerical computations, the following default parameter values have been prescribed: γ=ϕ=0.1 and Pr=6.2.
Skin friction coefficient against ϕ for different values of γ for CuO-water nanofluid.
Heat transfer coefficient against ϕ for different values of γ for CuO-water nanofluid.
Skin friction coefficient against ϕ for different water based nanofluids.
Heat transfer coefficient against ϕ for different water based nanofluids.
Velocity variation with η for different values of γ for CuO-water nanofluid.
Temperature variation with η for different values of γ for CuO-water nanofluid.
Velocity variation with η for different water based nanofluids.
Temperature variation with η for different water based nanofluids.
Velocity variation with η for different values of ϕ for CuO-water nanofluid.
Temperature variation with η for different values of ϕ for CuO-water nanofluid.
Variations of the reduced skin friction coefficient f′′(0)/(1-ϕ)2.5 and the reduced local Nusselt number -(knf/kf)θ′(0) as a function of nanoparticle fraction parameters ϕ under different values of the slip parameter γ are shown in Figures 1 and 2. It is seen in Figure 1 that the skin friction coefficient increases (in absolute sense) with ϕ, whereas it decreases with velocity slip parameter. It means that in the no-slip velocity condition (γ=0) the highest surface shear stress occurs for pure fluid (ϕ=0) as well as nanofluid. These observations (for pure fluid) show good agreement with the results of Mukhopadhyay and Gorla [43].
Figure 2 shows the heat transfer rate with ϕ for different values of velocity slip parameter γ. Heat transfer rate increases with ϕ but decreases with velocity slip parameter. Hence, to achieve a high heat transfer rate, less slip on the fluid-solid interface is desired. Also, the heat transfer rate is higher for a nanofluid (ϕ≠0) compared to a regular fluid (ϕ=0). Figures 3 and 4, respectively, present the variations of the skin friction coefficient and the local Nusselt number with ϕ for different types of nanofluids: Cu-water, TiO_{2}-water, CuO-water, and Al_{2}O_{3}-water. It is seen that the local Nusselt number, which represents the heat transfer rate at the surface, is the highest for Al_{2}O_{3}-water nanofluid.
The velocity and temperature profiles for different values of the physical parameters are given in Figures 5–10. Figures 5 and 6 show the velocity and temperature distributions with different values of the velocity slip parameter. It is clear from Figure 5 that the velocity decreases in the presence of slip at the fluid-solid interface and decreases monotonically to zero far away from the solid surface. In case of the no-slip condition, the fluid velocity adjacent to the solid surface is equal to the velocity of the stretching sheet; then f′(0)=1, which is clearly satisfied in this figure. Figure 6 indicates that an increase in the velocity slip parameter tends to increase the temperature of the fluid and consequently decrease the temperature gradient, which represents the rate of heat transfer. This observation is in agreement with the results presented in Figure 2.
Figures 7 and 8 show the velocity and temperature distributions in the boundary layer for nanofluid with different nanoparticles. It is observed that both velocity and temperature profiles are influenced by the types of the nanoparticle. It is also seen that the velocity boundary layer thickness is smaller for Cu compared to CuO, TiO_{2}, and Al_{2}O_{3} nanoparticles.
Finally, Figures 9 and 10, respectively, depict the effects of nanoparticle volume fraction ϕ on the velocity and temperature distributions for CuO-water nanofluid. The effect of ϕ on the flow field is less significant than the thermal field. With increasing the concentration of the nanoparticle, the velocity decreases throughout the boundary layer region. It is due to the fact that nanoparticles produce friction in fluid, which retards the flow. Nanoparticles also serve to increase the thermal conductivity of the base fluid; as a result, heat is transferred from the sheet to the fluid with faster rate and consequently warms the thermal boundary layer region. Therefore, as the concentration of the nanoparticles in the base fluid increases, the temperature in the thermal boundary layer also increases, which is clearly visible in Figure 10.
5. Conclusions
In this paper, the problem of two-dimensional flow of a viscous and incompressible water based nanofluid over a stretching flat sheet with velocity slip conditions was numerically studied. The governing partial differential equations for mass, momentum, and energy were transformed into ordinary differential equations using a similarity transformation. These equations were solved numerically using finite element method. We found that velocity slip parameter reduces the rate of heat transfer. The results also indicate that with the increase of nanoparticle volume fraction, the skin friction coefficient as well as the heat transfer rate increases.
Acknowledgments
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions. This work was supported by research Grants 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).
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