Appl. Math. Mech. -Engl. Ed., 33(7), 923–930 (2012) DOI 10.1007/s10483-012-1595-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2012
Applied Mathematics and Mechanics (English Edition)
Similarity solutions to viscous flow and heat transfer of nanofluid over nonlinearly stretching sheet∗ M. A. A. HAMAD1,2 , M. FERDOWS3 (1. Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt; 2. School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia; 3. Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh)
Abstract The boundary-layer flow and heat transfer in a viscous fluid containing metallic nanoparticles over a nonlinear stretching sheet are analyzed. The stretching velocity is assumed to vary as a power function of the distance from the origin. The governing partial differential equation and auxiliary conditions are reduced to coupled nonlinear ordinary differential equations with the appropriate corresponding auxiliary conditions. The resulting nonlinear ordinary differential equations (ODEs) are solved numerically. The effects of various relevant parameters, namely, the Eckert number Ec, the solid volume fraction of the nanoparticles φ, and the nonlinear stretching parameter n are discussed. The comparison with published results is also presented. Different types of nanoparticles are studied. It is shown that the behavior of the fluid flow changes with the change of the nanoparticles type. Key words nanofluid, nonlinearly stretching sheet, similarity solution, nonlinear ordinary equation, partial differential equation, viscous flow Chinese Library Classification O357 2010 Mathematics Subject Classification
1
76Dxx
Introduction
It is known that nanofluids can tremendously enhance the heat transfer characteristics of the original (base) fluid. Heat transfer is an important process in physics and engineering, and therefore improvements in heat transfer characteristics will improve the efficiency of many processes. A nanofluid is a fluid in which nanometer-sized particles are suspended in a conventional heat transfer fluid to improve the heat transfer characteristics. Thus, nanofluids have many applications in industry such as coolants, lubricants, heat exchangers, and micro-channel heat sinks. Therefore, numerous methods have been taken to improve the thermal conductivity of these fluids by suspending nano/micro sized particle materials in liquids. The reported breakthrough in substantially increasing the thermal conductivity of fluids by adding very small amounts of suspended metallic or metallic oxide nanoparticles (Cu, CuO, Al2 O3 ) to the fluid[1–2] , or alternatively using nanotube suspensions[3–4] conflicts with the classical theories[5–13] , of estimating the effective thermal conductivity of suspensions. There have been published several recent numerical studies on the modeling of natural convection heat transfer in nanofluids, Congedo et al.[14] , Ghasemi and Aminossadati[15] , Ho et al.[16–17] , and several others. These studies used ∗ Received Jun. 6, 2011 / Revised Dec. 30, 2011 Corresponding author M. A. A. HAMAD, E-mail: m
[email protected]
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traditional finite difference and finite volume techniques with the tremendous call on computational resources that these techniques necessitate. Hamad et al.[18] used the application of a one-parameter group to present similarity reductions for problems of magnetic field effects on free-convection flow of a nanofluid past a semi-infinite vertical flat plate following a nanofluid. Hamad and Pop[19] investigated theoretically the unsteady magnetohydrodynamic (MHD) flow of a nanofluid past an oscillatory moving vertical permeable semi-infinite flat plate with constant heat source in a rotating frame of reference. Hamad[20] found the analytical solutions of convective flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the presence of a magnetic field. Hamad and Ferdows[21] investigated heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing. The comprehensive references on nanofluid can be found in the recent book by Das et al.[22] and in the review papers by Trisaksri and Wongwises[23] , Wang and Mujumdar[24] , and Kakac and Pramuanjaroenkij[25]. Heat, mass, and momentum transfer in the laminar boundary layer flow on a stretching sheet are important due to its applications to polymer technology and metallurgy. Gupta and Gupta[26] stated that the stretching of the sheet may not necessarily be linear. Vajravelu[27] studied flow and heat transfer in a viscous fluid over a nonlinear stretching sheet without viscous dissipation, but the heat transfer in this flow was analyzed only in the case when the sheet was held at a constant temperature. Raptis and Perdikis[28] investigated the steady twodimensional flow of an incompressible viscous and electrically conducting fluid over a nonlinearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field. Bataller[29] presented a numerical analysis in connection with the boundary 1 layer flow induced in a quiescent fluid by a stretching sheet with velocity ux (x) = x 3 along with [30] heat transfer. Prasad and Vajravelu examined the hydromagnetic laminar boundary layer flow and heat transfer in a power law fluid over a non-isothermal stretching sheet. There have also been several recent studies, Ziabakhsh et al.[31] employed the homotopy analysis method (HAM) to compute an approximation to the solution for the problem of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. Akyildiz and Siginer[32] presented an analytical solutions for the velocity and temperature fields in a viscous fluid flowing over a nonlinearly stretching sheet by the Galerkin Legendre spectral method. Prasad et al.[33] presented a numerical solution for the steady two-dimensional mixed convection MHD flow of an electrically conducting viscous fluid over a vertical stretching sheet in its own plane. The stretching velocity and the transverse magnetic field were assumed to vary as a power function of the distance from the origin. Afzal[34] studied the laminar boundary layer flow over a nonlinearly stretching two-dimensional sheet, or axisymmetric plane or the body of revolution arising from nonlinear power law stretching velocity. In the present study, the similarity solution of two-dimensional flow and heat transfer of an incompressible viscous nanofluid past non-linear stretching surface is presented. The resulting ordinary differential equations are then solved numerically. The aim is to investigate the influence of various nanofluid parameter (the solid volume fraction of the nanoparticles φ) and the effect of nonlinear stretching parameter n on flow and heat-transfer characteristics.
2
Formulation of problem
Consider the steady laminar two-dimensional flow of an incompressible viscous fluid coinciding with the plane y = 0, with the flow being confined to y > 0. Two equal and opposite forces are introduced along the x-axis so that the wall is stretched whilst keeping the position of the origin fixed (see Ref. [35]). The temperature at the stretching surface is deemed to have a constant value Tw while the ambient temperature has a constant value T∞ . It is further assumed that the regular fluid and the suspended nanoparticles are in the thermal equilibrium
Similarity solutions to viscous flow and heat transfer of nanofluid
925
and no slip occurs between them. The thermo physical properties of the nanofluid are given in Ref. [36]. Under the above assumptions, the boundary layer equations governing the flow and temperature in the presence of heat source or heat sink are (using the boundary layer approximations and taking into account viscous dissipation (last term in the energy equation)) ∂u ∂v + = 0, ∂x ∂y
(1)
u
∂u μnf ∂ 2 u ∂u +v = , ∂x ∂y ρnf ∂y 2
(2)
u
∂2T ∂T μnf ∂u 2 ∂T +v = αnf 2 + , ∂x ∂y ∂y (ρcp )nf ∂y
(3)
where x and y are the coordinates along and perpendicular to the sheet, u and v are the velocity components in the x- and y-directions, respectively, T is the local temperature of the fluid. Further, ρnf is the effective density, μnf is the effective dynamic viscosity, (ρcp )nf is the heat capacitance, αnf is the effective thermal diffusivity, and κnf is the effective thermal conductivity of the nanofluid, which are defined as (see Refs. [36–37]) ⎧ μf ⎪ ρnf = (1 − φ)ρf + φρs , μnf = , ⎪ ⎪ ⎪ (1 − φ)2.5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ κnf (ρcp )nf = (1 − φ)(ρcp )f + φ(ρcp )s , αnf = , (4) (ρc p )nf ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ κ + 2k − 2φ(κ − κ ) ⎪ ⎪ s f f s ⎪ , ⎩ κnf = κf κs + 2κf + 2φ(κf − κs ) where φ is the solid volume fraction of the nanoparticles. The appropriate boundary conditions for the problem are given by ⎧ n 2n ⎨ u = uw (x) = cx , v = 0, T = Tw (x) = T∞ + bx at y = 0, ⎩
(5) u → 0,
T → T∞ as y → ∞,
where b and c are positive constants, and n is the nonlinear stretching parameter. By introducing the following non-dimensional variables (see Ref. [35]): ⎧ η = y c(n + 1)/(2νf ) x(n−1)/2 , u = cxn F (η), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ n−1 (n + 1)cνf (n−1)/2 x ηF (η) , F (η) + v = − ⎪ ⎪ 2 n+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ θ(η) = (T − T∞ )/(Tw − T∞ ).
(6)
Using (6), (1)–(3) lead to the following non-dimensional ordinary differential equations (ODEs): 2n F 2 = 0, F + (1 − ϕ)2.5 (1 − ϕ + ϕ(ρs /ρf )) F F − (7) n+1 1 knf Ec 4n F θ = 0, θ + F 2 + (1 − ϕ + ϕ(ρcp )s /(ρcp )f ) F θ − 2.5 P r kf (1 − ϕ) n+1
(8)
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and the corresponding boundary conditions (5) become ⎧ ⎨ F = 0, F = 1, θ = 1 at η = 0, ⎩ where P r =
ν α
(9)
F → 0,
θ → 0 as η → ∞,
is the Prandtl number, and Ec is the Eckert number, which is defined as Ec =
u2w . cp (Tw − T∞ )
(10)
The skin friction coefficient Cf and the local Nusselt number N ux are defined as ∂T μnf ∂u xκnf , N ux = − . Cf = 2 ρf uw ∂y y=0 κf (Tw − T∞ ) ∂y y=0
(11)
Using (6) and (11), we get 1
Rex2 Cf = where Rex =
3
xuw νf
n + 1 12 1 F (0), (1 − ϕ)2.5 2
−1
Rex 2 N ux = −
κnf n + 1 12 θ (0), κf 2
(12)
is the local Reynolds number based on the stretching velocity uw .
Results and discussion
The transformed system of coupled nonlinear ordinary differential equations (7) and (8) with the boundary conditions (9) have been solved numerically for some values of the governing parameters ϕ, P r, Ec, and n using the shooting method with Mathematic package. In order to test the accuracy of our results, we compare our results with those in Ref. [35] when neglecting the effects of φ. We notice that the comparison shows an excellent agreement, as presented in Tables 1 and 2. Table 3 shows the values of F (0) and θ (0) for various ϕ when Ec = 0 and 0.1 with n = 10 and P r = 10 for two different nanofluids (Cu and Ag) with the same base fluid (water). In order to highlight the important features of the flow and the heat transfer characteristics, the numerical values are plotted in Figs. 1–6. These figures show the velocity profiles (see Figs. 1 and 4), the temperature profiles (see Figs. 2–3), the variation of reduced skin friction at the wall (see Fig. 5), and the variation of the reduced Sherwood number (see Fig. 6) at the wall for different values of the physical parameters. Table 1 n
Present results
Cortell[35]
0.0 0.2 0.5 1.0 3.0 10.0 20.0
0.636 9 0.765 9 0.889 7 1.004 3 1.148 1 1.234 2 1.257 4
0.627 6 0.766 8 0.889 5 1.000 0 1.148 6 1.234 9 1.257 4
Table 2 n 0.75 1.50 7.00 10.0
Comparison of results for −F (0) when φ = 0
Comparison of results for −θ (0) when P r = 5
Ec = 0
Ec = 0.1
Present results
Cortell[35]
Present results
Cortell[35]
3.124 6 3.567 2 4.184 8 4.256 0
3.125 0 3.567 7 4.185 4 4.256 0
3.015 6 3.456 6 4.065 9 4.135 4
3.017 0 3.455 7 4.065 7 4.135 3
Similarity solutions to viscous flow and heat transfer of nanofluid Table 3
Effect of solid volume fraction on F (0) and θ (0) when P r = 10.0 and n = 10.0 −θ (0)
−F (0)
ϕ
0.05 0.10 0.15 0.20
927
Ec = 0
Ec = 0.1
Cu
Ag
Cu
Ag
Cu
Ag
1.400 49 1.477 69 1.517 94 1.528 80
1.436 46 1.537 12 1.593 00 1.613 99
5.621 89 5.172 37 4.772 57 4.413 06
5.577 54 5.091 04 4.659 91 4.273 71
5.368 53 4.884 17 4.455 81 4.072 41
5.312 81 4.782 34 4.315 13 3.898 74
Figures 1 and 2 are the graphical representations of the velocity F (η) and the temperature θ(η) for various values of the Cu nanoparticles volume fraction for three values of nonlinear stretching parameter n (0.0, 0.5, and 10.0) when Ec = 0.1 and P r = 6.8 (water). It can be seen from Fig. 1 that the momentum boundary layer thickness decreases with the increase in φ. Then, the existence of nanoparticles leads to more thinning of the boundary layer. While, from Fig. 2, the thermal boundary layer thickness increases with the increase in the nanoparticle volume φ. Also the thermal boundary layer for Cu-water is greater than for pure water (φ = 0). This is because copper has high thermal conductivity and the addition of it increases the thermal conductivity for the fluid. Therefore, the thickness of the thermal boundary layer increases. Furthermore, it can be observed that both momentum and thermal boundary layers decrease as the nonlinear stretching parameter increases.
Fig. 1
Velocity profiles for various φ when n = 0.0, 0.5, and 10.0 with Ec = 0.1
Fig. 2
Temperature profiles for various φ when n = 0.0, 0.5, and 10.0 with Ec = 0.1
Figure 3 shows the effect of Eckert number Ec for Cu-water on the temperature when n = 0.5. It can be seen that the thermal boundary layer increases when Ec increases. Also we note that at each value of Ec the thickness of the thermal boundary layer in Cu-water is greater than that in pure water. Figure 4 shows the behavior of the velocity and the temperature for different types of nanofluids when φ = 0.1, n = 0.5, P r = 6.8, and Ec = 0.1. It can be seen that both momentum and thermal boundary layer thicknesses change with change in the type of nanoparticles. This means the addition of nanoparticles in regular fluid plays an important role. Figures 5 and 6 show the variation in shear stress and heat transfer rates versus the nonlinear stretching parameter n for Cu-water. Figures 5 shows the effect of the nanoparticles volume fraction φ on the shear stress, while Fig. 6 depicts the effect φ on the heat transfer rates for two values of Eckert number Ec (0.0 and 0.2). It can be seen from Fig. 5 that the shear stress increases with the increase in φ. It can also be seen that the change in the shear stress increases with the increase in n. Also from Fig. 6, it is noted that the heat transfer rates increases with
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the increase in φ and the change in the heat transfer rates increases with the increase in n. Also, it can be observed that the heat transfer rates for Ec = 0 is higher than that for Ec = 0.
Fig. 3
Temperature profiles for various Ec when φ = 0.0 (regular fluid) and φ = 0.1 with n = 0.5
Fig. 4
Velocity and temperature profiles for different types of nanofluids when φ = 0.1, n = 0.5, Pr = 6.8, and Ec = 0.1
Fig. 5
Variation of skin friction with n for various ϕ when Pr = 6.8
Fig. 6
Variation of local Nusselt number with n when Ec = 0.0 and 0.2 with Pr = 6.8
4
Conclusions
The problem of two-dimensional laminar free convection flow of a nanofluid over a nonlinear stretching surface has been studied. A similarity solution is presented and the numerical solutions are analyzed and discussed. This solution depends on the nanoparticle volume fraction φ, nonlinear stretching parameter n, Eckert number Ec, and the Prandtl number P r. The base fluid is water with the Prandtl number of P r = 6.8. We have investigated the way in which the velocity and temperature profiles as well as the surface skin friction and the surface heat flux depend on these parameters. It is shown that the inclusion of nanoparticles into the base fluid of this problem is capable to change the flow pattern for the problem under consideration.
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