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A mixed convection flow and heat transfer of pseudo-plastic power law nanofluids past a stretching vertical plate
International Journal of Heat and Mass Transfer 105 (2017) 350–358
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A mixed convection flow and heat transfer of pseudo-plastic power law nanofluids past a stretching vertical plate Xinhui Si a,⇑, Haozhe Li a, Liancun Zheng a, Yanan Shen a, Xinxin Zhang b a b
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 1 June 2016 Received in revised form 22 September 2016 Accepted 28 September 2016
Keywords: Power law nanofluids Dual solutions Stability Bvp4c
a b s t r a c t A mixed convection flow and heat transfer of pseudo-plastic power law nanofluid past a stretching vertical plate is investigated. Three types of nano-particles, such as copper (Cu), aluminum oxide (Al2O3) and titanium oxide (TiO2), are considered. The generalized Fourier law proposed by Zheng for varying thermal conductivity of nanofluids, which is dependent on the power law of velocity gradient as well as the nanoparticles property, is taken into account. Dual solutions are obtained by Bvp4c with Matlab. The stability of the solution also is discussed by introducing the unsteady governing equations. Furthermore, a new interesting phenomenon is found: the local Nusselt number do not maintain the similar characteristics of Newtonian fluid near the point where the velocity ratio is equal to 0.5. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, many investigations have been made to explore the nanofluid technology in the field of enhanced heat transfer and new generation cooling technology, which overcomes the limitation of many conventional heat transfer fluids with low thermal conductivity, such as water, oil, and ethylene glycol mixture. Furthermore, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement, nanofluids are also very stable and have no additional problems [1], such as sedimentation, erosion, additional pressure drop and non-Newtonian behavior. Numerous methods have been taken to develop advanced heat transfer fluids with substantially higher conductivity by suspending nano(usually less than 100 nm)/micro or larger-sized particle materials in liquids [2]. The comprehensive references and the broad range of current and future applications on nanofluids can be found in the recent book [3] and in the review papers by Buongiorno [4],Wang and Mujumdar [5], Kakac and Pramuanjaroenkij [6], Wong and Leon [7]. Nanosfluid has been used to improve heat transfer for some convection problems due to its wide applications in electronics cooling, heat exchangers, and double pane windows. Furthermore, mixed convection is preferred and then many numerical studies about heat and mass transfer of nanofluids in enclosures have been ⇑ Corresponding author. E-mail address: [email protected] (X. Si). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.106 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
studied [8–11]. Some similarity solutions or analytical solutions also have been done. For example, Nield and Kuznetsov [12,13] studied the free convection of viscous and incompressible nanofluid past a vertical plate embedded in porous medium or not. Loganathan et al. [14] discussed unsteady natural convection flow of nanofluids past an infinite vertical plate with considering the radiation.Ahmad et al. [15] extended Blasius problem and Sakiadis problem to the case of nanofluids. Norfifah Bachok et al. [16] made another extension and extended the Blasius and Sakiadis problems in nanofluids by considering a uniform free streem parallel to a fixed or moving flat plate. Ahmad and Pop [17] investigated steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids. One of interesting problems, the existence of dual solutions, also have been proposed, which bring more insight on engineering applications. For example, Subhashini et al. [18–20] pointed out that the upper branch solutions are most physically relevant solution whereas the lower branch solutions seem to deprive physical significance or may have realistic meaning in different situations. Furthermore, the stability of the numerical solutions for the mixed convection also has been analyzed. Since Mahmood and Merkin [21] done the classical work and investigated the mixed convection on a vertical circular cylinder, Merkin [22], Merkin and Pop [23] also discussed the dual solutions occurring in mixed convection in a porous medium by perturbation method and analyzed the stability of the upper and lower branch of the solutions. Similarly, some interesting works also have been done [24,25].
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
In contrary, the number of studies on natural and mixed convection of nanofluids, where Non-Newtonian power law fluids is considered as the base fluid, is very small. Lin and Zheng et al. [26–28] studied the Marangoni convection of power law nanofluids, where they considered the thermal conductivity to be dependent on velocity or temperature gradient. Some other mass and heat transfer models about power law nanofluids [29–32] also have been proposed and solved numerically or analytically. The aim of this paper is to investigate the mixed convection mass and heat transfer of power law nanofluids past a stretching vertical plate. Here the CMC-Water (0.0–0.4%) is considered as a pseudo-plastic power law fluid [26–28] and regarded as the base fluid. Three types of nanoparticles are considered: copper (Cu), aluminum oxide (Al2O3) and titanium oxide (TiO2). According to the experimental studies, the thermophysical properties of CMC-Water(<0.4%) are similar to water [33,34]. Their related properties are given in Tables 1 and 2. The generalized Fourier law proposed by Zheng [26–28,35,36] for varying thermal conductivity of nanofluids is taken into account. The similar equations are solved numerically with Matlab. Effects of different parameters on velocity and temperature are discussed in detail. 2. Governing equations Consider the steady two-dimensional mixed convection flow and heat transfer of power law nanofluid over a vertical porous stretching plate. The base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. Here we assume that x-axis is along the vertical surface and y-axis is normal to the plate. u and v are the velocity components in the x and y directions, respectively. As shown in Fig. 1, the free stream velocity and the stretching velocity are assumed to be ue ¼ a and uw ¼ b. The temperature T w at the wall is a constant, T 1 is the temperature of the fluid far away from the plate. The suction/injection velocity along
n1 @u @T @T @T @ ðqcp Þnf u þv ¼ ; jnf @x @y @y @y @y where qnf ; knf ;
ð2:3Þ
lnf ; ðqcp Þnf ; bnf are the density of the nanofluid,
the thermal conductivity of the nanlfluid, the modified viscosity of the nanofluid, the heat capacitance of the nanofluid and the coefficient of the thermal expansion, respectively.
l ¼ lnf j @u jn1 is the @y
jn1 is the thermal difeffective viscosity of the nanofluid. K ¼ anf j @u @y j
fusivity proposed by Zheng et al. [26–28,35,36], where anf ¼ ðqcpnfÞ is nf
the modified thermal diffusivity of the nanofluid. The expression of above nanofluid parameters are given as follow:
qnf ¼ ð1 /Þqf þ /qs ; ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þs ; lf ; ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞs ; lnf ¼ 2:5 ð1 /Þ
jnf ðjs þ 2jf Þ 2/ðjf js Þ ¼ ; jf ðjs þ 2jf Þ þ /ðjf js Þ
ð2:4Þ
where qf ; qs are the reference density of the fluid fraction and the solid fraction, and kf ; ks are the thermal conductivity of the fluid fraction and solid fraction, respectively. / is the solid volume fraction parameter of the nanofluid, lf is the viscosity of the fluid fraction. Here the modified viscosity lnf of the nanofluid can be approximated as viscosity of the base fluid lf containing dilute
1
1 ðmf U 2n1 xn Þnþ1 , where U ¼ a þ b is the the plate is v w ¼ f w nþ1 composite velocity. Under these assumptions, the governing equations about the mixed convective flow and heat transfer of power-law nanofluid can be written as follow:
@u @ v þ ¼ 0; @x @y @u @u 1 @ þv ¼ u @x @y qnf @y
ð2:1Þ n1 ! @u @u ðqbÞnf þg lnf ðT T 1 Þ; @y @y qnf
ð2:2Þ
Fig. 1. The model of heat and mass transfer of mixed convection.
Table 1 Thermophysical properties of the base fluid and nanoparticles. Property
CMC-water (0.0–0.4%)
Copper (Cu)
Aluminum Oxide (Al2O3)
Titanium Oxide (TiO2)
cp (J/kg K)
4179 997.1 0.613 21
385 8933 400 1.67
765 3970 40 0.85
686.2 4250 8.9538 0.9
qðkg=m3 Þ j(W/mK) b 105 ð1=KÞ
Table 2 Dynamical properties of the carboxy methyl cellulose water.
The power law index KðNsn =m2 Þ
0.0
0.1
0.2
0.3
0.4
1.0 8.550 04
0.91 6.319 103
0.85 1.754 102
0.81 3.136 102
0.76 7.853 102
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
suspension of fine spherical particles and is given by Brinkman [37]. The effective thermal conductivity of the nanofluid is approximated by the Maxwell–Garnetts model, which restricts to spherical nanoparticles and does not account for other shapes of nanoparticles [38–46]. The corresponding boundary conditions are
v ¼ v w ðxÞ;
u ¼ uw ¼ b;
T ¼ T w ; y ¼ 0;
ð2:5Þ
u ! ue ¼ a; T ! T 1 ; y ¼ 1:
Introduce the stream function w such that u ¼ @w=@y and
v ¼ @w=@x, which are listed as follow: w ¼ ðmf U 2n1 xÞ where
mf ¼
U 2n nþ1 1 T T1 f ðgÞ; g ¼ ð Þ yx nþ1 ; h ¼ ; mf Tw T1
1
1 Pr
00 0
00
1 00 bf ðgÞf ðgÞ ¼ 0; nþ1
jnf 00 n1 0 0 1 h0 f ðgÞ ¼ 0; ðjf ðgÞj h Þ þ a nþ1 jf
ð2:7Þ
ð2:8Þ
bÞs where the coefficients are c ¼ 1 / þ / ððqqbÞ ; b ¼ 1 / þ / qqs ; f
f
a ¼ 1 / þ / ððqqccpp ÞÞfs , respectively. The Prandtl number Pr, the local Grashof number Grx , the local Reynold number Rex and the mixed convection parameter kðxÞ are given by
Grx ¼
g q2f bf ðT w T 1 Þx3
l2f
kðxÞ ¼
; Rex ¼
cp f lf qf Ux ; Pr ¼ ; lf kf
Grx gbf ðT w T 1 Þx ¼ ; Rex U2
ð2:9Þ
where kðxÞ is a function of x. Then the numerical solutions represent local similarity solutions since the parameters still depend on an independent variable x. It should be noted that the sign of k depends on the nature of the flow arising out from this situation. k > 0 corresponds to buoyancy assisting flow. In contrast, when k < 0, it corresponds to buoyancy opposing flow. Three special cases as k ¼ 0; n ¼ 1 are discussed by Norfifah Bachok et al. [16], Ahmad et al. [15] and Subhashini and Sumathi [19]. The corresponding boundary conditions become 0
0
f ð0Þ ¼ f w ; f ð0Þ ¼ e; hð0Þ ¼ 1; f ð1Þ ¼ 1 e; hð1Þ ¼ 0;
ð2:10Þ
where velocity ratio is e ¼ The shear stress and local Nusselt number on the surface are derived as follow: n1 @u @u n1 00 3n1 n 00 ¼ lnf U nþ1 Rex nþ1 f ð0Þ f ð0Þ; sw ¼ lnf @y @y b . U
Nux ¼
xqw ðxÞ jn1 ðT w T 1 Þ knf j @u @y
jy¼0 ¼
1
mf xn U n2
¼
1 aPr nþ1
jf jnf
y5 y1 ðn 1Þjy3 jn2 y03 y5 jy3 jn1
ð3:12Þ
:
The boundary conditions can be written to be
ð2:11Þ
1 !nþ1
ð3:13Þ The system of nonlinear differential equations can be solved by Bvp4c with Matlab. Here we assume that the Maximum residual is
lf qf .
ðjf jn1 f Þ þ hkðxÞc þ
y05
ð2:6Þ
Substituting above transformations into the governing Eqs. (2.1)–(2.3), the partial differential equations can be transformed into the following ordinary equations
ð1 /Þ2:5
y02 ¼ y3 ; 1 1 bay1 y3 Þð1 /Þ2:5 jy3 j1n ; y03 ¼ ðckðxÞy4 n nþ1 y04 ¼ y5 ;
y1 ð0Þ ¼ f w ; y2 ð0Þ ¼ e; y2 ð1Þ ¼ 1 e; y4 ð0Þ ¼ 1; y4 ð1Þ ¼ 0;
1
1 nþ1
y01 ¼ y2 ;
h0 ð0Þ h0 ð0Þ:
3. Numerical methods 3.1. Transformation of equations In order to solve the nonlinear differential Eqs. (2.7) and (2.8), we transfer this problem to a system of first-order equations. Here 0 00 we denote y1 ¼ f ; y2 ¼ f ; y3 ¼ f ; y4 ¼ h; y5 ¼ h0 , then
105 . In order to obtain the dual solutions, the number of the mesh, the infinity replaced by finite point and the initial guess values need to be adjusted according to the different physical parameters. Some solutions for special cases also are compared with previous ones to validate the accuracy and correctness, which is shown in Table 3. 4. Numerical solutions and discussion In this following section, we assume that the base fluid is 0.4% CMC-Water and will consider the velocity and temperature distribution influenced by different nano-particles and different physical parameters. In every figure, we will give illustration of velocity and temperature distribution corresponding to dual solutions for some fixed values. Table 4 gives some values of the dual solutions influenced by different physical parameters. And according to the Fig. 2, we can find that dual solutions exist for power law nanofluids and there is no solution as e < ek . Both types of nanofluids have the similar 00 trends as the base fluids are water or power law fluid for f ð0Þ. However, an interesting phenomenon also can be found for the power law nanofluids. Local Nusselt number decreases first and then increases again near the point e ¼ 0:5, which is different dramatically from the case as the base fluid is water. Furthermore, the 00 first solutions of f ð0Þ and h0 ð0Þ are stable and physically, while the second solutions are not [48,19].The stability of the dual solutions will be discussed in the next section. The influence of different nanoparticles on the velocity and temperature distribution can be found in Fig. 3 as other parameters are e = 0.5, k = 0.01, f w = 0.1, Pr = 6.2, / = 0.1 and n = 0.85. For the first solution, it is observed that the thickness of thermal boundary layer for Cu nanofluid is the more thinner than other two cases. The reason is that Cu has the highest thermal conductivity compared to TiO2 and Al2O3.The higher temperature gradients is caused from the reduced value of thermal diffusivity and, therefore, higher improvement in heat transfers. However, the difference between TiO2 and Al2O3 can be negligible. For the second solution, the velocity and temperature distribution shows a more obvious difference influenced by different nanoparticles. Fig. 4 gives the illustration of dual solutions of Cu-CMC nanofluid flow influenced by power law index as other parameters are k = 0.01,Pr = 6.2, / = 0.001, e = 0.5, f w = 0.1.For the first solution, it is observed that the velocity decreases and the thickness of velocity boundary layer increases with increasing power law index. The reason may be that the effective viscosity of the power law fluids decreases with the increasing power law number. The similar trends also happen on the second solution. Furthermore, the influence of the power law index on the second solution is more obvious.
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 Table 3 The comparison of Cu-water nanofluid with previous works as Pr ¼ 6:2. 00
f ð0Þ
/
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
n=1
0
0.5 0.2 0 1
0.397851 0.412369 0.332059 0.443762
0.171031 0.011421
0.3990 0.4124 0.3321 0.4438
0.1710 0.0114
0.3979 0.4124 0.3321 0.4438
0.1710 0.0114
0.1
0.5 0.2 0 1
0.467380 0.484476 0.390124 0.521295
0.200917 0.013433
0.4674 0.4844 0.3901 0.5218
0.2009 0.0134
0.2
0.5 0.2 0 1
0.484606 0.502304 0.404481 0.540537
0.208323 0.013928
0.4846 0.5023 0.4045 0.5405
0.2083 0.0139
Present results
Ishak et al. [47]
Bachok et al. [16]
Table 4 00 Dual solutions of f ð0Þ and h0 ð0Þ for Cu-CMC power law nanofluid as Pr ¼ 3; k ¼ 0:02; f w ¼ 0:1. 00
f ð0Þ
/¼0
h0 ð0Þ
/ ¼ 0:05
/¼0
/ ¼ 0:05
n
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
0.91
0.5 0.2 0 1
0.476333 0.446220 0.345915 0.48544
0.158882 0.01029
0.562692 0.511197 0.395003 0.54236
0.162790 0.00648
0.27960 0.47927 0.56913 0.91129
0.04073 0.00025
0.32487 0.48345 0.55368 0.80238
0.05626 0.00042
0.85
0.5 0.2 0 1
0.478113 0.448092 0.344111 0.48585
0.163068 0.00494
0.568494 0.515919 0.394770 0.54581
0.167186 0.00188
0.27994 0.47824 0.56144 0.87464
0.04428 0.00002
0.32767 0.48562 0.54991 0.75922
0.05955 0.00005
0.76
0.5 0.2 0 1
0.483640 0.453537 0.342869 0.49037
0.170801 0.000029
0.582823 0.526556 0.396378 0.55465
0.175234 0.001532
0.28228 0.47905 0.55165 0.81103
0.05064 0.00091
0.33508 0.49176 0.54641 0.68018
0.06542 0.00416
1.4
0.8 n=0.81, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.6998 w
2 3 k
1.2
0.6
1
0.4
n=0.91, Pr=6.2,λ=−0.01,φ=0.1,f =0.1, Cu,ε =−0.6325 w
k
n=1, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.7041 w
2 3 k
0.8
f’’(0)
−θ’(0)
0.2
0
0.6
0.4 −0.2
n=0.91, Pr=6.2,λ=−0.01,φ=0.1,f =0.1,Cu,ε =−0.6325 w
k
0.2 −0.4
w
−0.6 −0.8
n=0.81, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.6998 w
2 3 k
n=1, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.7041
−0.6
−0.4
−0.2
2 3 k
0
ε
0.2
0 0.4
0.6
0.8
1
−0.8
−0.6
−0.4
−0.2
0
ε
0.2
0.4
0.6
0.8
1
00
Fig. 2. Variations of f ð0Þ; h0 ð0Þ against e.
Fig. 5 illustrates the effects of solid volume fraction on the velocity and temperature as other parameters are k = 0.001, f w = 0.1, Pr = 6.2, e = 0.5, / = 0.1, n = 0.85. For the first solution, the temperature decreases and the temperature gradient increases near the plate with the increasing solid volume fraction. This means that the increasing solid volume fraction results in the increase of heat transfer, which reduces to the thinning of the thermal boundary layers near the vertical walls. Meanwhile, the
velocity boundary layer thickness also decreases with the solid volume fraction. Furthermore, the second solution for velocity and temperature shows the similar trends compared with the first solution. Fig. 6 gives the profiles of suction parameter on the velocity and temperature as other parameter are k = 0.001, Pr = 6.2, e = 0.5, / = 0.1, n = 0.85. For the first solution, with the increasing suction parameter, much more fluid is carried away from the surface
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.6
1.5
1.4
Al O
2 3
TO
1.2
i 2
1
the first solution for f’(η)
1
Cu
Al O
0.8
2 3
the second solution for f’(η)
TO
i 2
0.6
0.5
Cu 0.4
the first solution for θ(η)
the second solution for θ(η)
0.2 0
0 −0.2 −0.4 0
2
4
η
6
8
−0.5
10
0
2
4
η
6
8
10
Fig. 3. Dual solutions for different power law nanofluids.
1.5
1.5 n=0.76 n=0.85 n=0.91
1
the second solution for f’(η)
1 the first solution for f’(η)
0.5
0.5 the second solution for θ(η)
the first solution for θ(η)
0
0 n=0.76 n=0.85 n=0.91
−0.5
0
2
4
η
6
8
10
−0.5
0
2
4
6 η
8
10
12
Fig. 4. Dual solutions for different n in Cu power law nanofluid.
1.5
1.5
the first solution for f’(η)
1
the second solution for f’(η)
1
0.5
0.5 the second solution for θ(η)
the first solution for θ(η)
0
0 φ=0.001
φ=0.001
φ=0.05
φ=0.05
φ=0.1 −0.5
0
2
4
6 η
8
10
φ=0.1 12
−0.5
0
5
Fig. 5. Dual solutions for different Cu volume fraction in power law nanofluid.
η
10
15
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.5
1.5
the first solution for f’(η) 1
1
the second solution for f’(η)
0.5
0.5
the second solution for θ(η)
the first solution for θ(η)
0
0 f =0.1
f =0.1
f =0.2
f =0.2
f =0.3
f =0.3
w
w
w
w
w
−0.5
0
2
4
η
6
8
10
−0.5
w
0
5
10
η
15
Fig. 6. Dual solutions for f w in Cu power law nanofluid.
which causes reduction in velocity gradient as it tries to maintain the same velocity over a small region near the surface. The velocity increases with the increasing value of permeability parameter. Moreover, the thermal boundary layer thickness decreases sharply with the increasing parameter f w . This also shows that the increasing suction velocity makes the temperature boundary layer thinner and the boundary layer becomes more stable. However, for the second solution, the opposite trends can be observed influenced by suction velocity. Fig. 7 illustrates the effects the velocity ratio on the velocity and temperature distribution as other parameters are k = 0.01, Pr = 6.2, e = 0.5, / = 0.001, n = 0.91, f w = 0.1. For the first solution, we can find that the temperature gradient and the velocity at the wall will increase with increasing e. However, as the distance is far away from the plate, the velocity will decrease with increasing e. For the second solution, the profile corresponding to temperature is put toward the right side with increasing parameter e. Fig. 8 exhibits the influence of mixed convection parameter on the velocity and temperature distribution as other parameters are Pr = 6.2, e = 0.5, / = 0.001, n = 0.91, f w = 0.1. For the first solution, it is observed that with the increasing mixed convection, velocity and temperature gradient will increase. Since the buoyancy forces
1.5
will prevent the movement of the fluid, fluids within the boundary layer get retarded acting as an adverse pressure gradient. The mixed convection parameter represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the thermal and velocity fields. For the second solution, the temperature profile is put toward right side with increasing mixed convection parameter k. The velocity has contrary trend corresponding to the case of first solution. 5. The stability of the numerical solution In order to analyze the stability of the numerical solution, the unsteady mixed convection problem is introduced. The governing equations are written as
@u @u @u 1 @ þu þv ¼ @t @x @y qnf @y
n1 ! @u @u ðqbÞnf þg lnf ðT T 1 Þ; @y @y qnf
ð5:14Þ ðqcp Þnf
@T @T @T @ ¼ þu þv @t @x @y @y
n1
@u jnf @y
!
@T ; @y
ð5:15Þ
1.5
1
ε=−0.50
1
the first solution for f’(η)
ε=−0.35 ε=−0.20
0.5
0.5 the first solution for θ(η)
the second solution for θ(η)
0
0 ε=−0.50 ε=−0.35
the second solution for f’(η)
ε=−0.20 −0.5
0
5
η
10
15
Fig. 7. Dual solutions for different
−0.5
0
5
e in Cu power law nanofluid.
10 η
15
20
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.5
1.6 1.4 1.2
1
the first solution for f’(η)
the second solution for f’(η)
1 0.8 0.6
0.5
the second solution for θ(η)
0.4
the first solution for θ(η)
0.2 0
0 λ=−0.01 λ=−0.03 λ=−0.05
−0.5
0
2
4
η
6
8
λ=−0.01
−0.2
λ=−0.03 λ=−0.05
−0.4
10
0
5
η
10
15
Fig. 8. Dual solutions for different k in Cu power law nanofluid.
Table 5 The smallest eigenvalue d0 for Cu-CMC power law nanofluid as Pr ¼ 6:2; k ¼ 0:01; f w ¼ 0:1. n = 0.91
n = 0.85
n = 0.76
/
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
0.1
0.5 0.2 0 0.25
0.198312 0.377343 0.454069 0.511108
0.10595 0.09631
0.197378 0.380363 0.453981 0.510199
0.10523 0.09461
0.196156 0.386269 0.471712 0.524424
0.10423 0.09381
0.05
0.5 0.2 0 0.25
0.193537 0.372135 0.448543 0.511968
0.10267 0.09904
0.188589 0.378842 0.462091 0.521913
0.10161 0.09652
0.190353 0.378402 0.456276 0.527553
0.10005 0.09600
where t represents time. The new dimensionless variables are introduced: 1
w ¼ ðmf U 2n1 xÞnþ1 f ðg; sÞ; hðg; sÞ ¼
g¼
U 2n
1 !nþ1
mf
1
yxnþ1 ;
T T1 ; Tw T1
s ¼ Utx1 ;
1 ð1 /Þ2:5 ð5:16Þ
then the partial differential equations can be obtained after one substitutes dimensionless transformation (5.16) into Eqs. (5.14) and (5.15): 1 ð1 /Þ
2:5
ðjf gg j
n1
tuting Eq. (5.19) into differential equations system (5.17) and (5.18) and linearizing, we can obtain:
1 bffgg bðf gs f g f gs s þ f s f gg sÞ ¼ 0; f gg Þg þ hkðxÞc þ nþ1 ð5:17Þ
½nðn 1ÞjF 00 jn2 F 000 g 00 þ njF 00 jn1 g 000
þ ckx þ dbg 0 þ
1 bðFg 00 þ gF 00 Þ ¼ 0; nþ1
ð5:20Þ
knf 1 ½ðn 1ÞjF 00 jn2 F 000 x0 þ ðn 1Þðn 2ÞjF 00 jn3 g 00 F 000 H0 kf Pr þ ðn 1Þg 000 jF 00 jn2 H0 þ jF 00 jn1 x00 þ ðn 1ÞjF 00 jn2 g 00 H00 1 aðF x0 þ H0 gÞ ¼ 0: þ dxa þ nþ1
ð5:21Þ
The corresponding boundary conditions are
1 Pr
jnf 1 hg f aðhs f g hs s þ f s hg sÞ ¼ 0: ðjf jn1 hg Þg þ a nþ1 jf gg
gð0Þ ¼ 0; g 0 ð0Þ ¼ 0; g 0 ð1Þ ¼ 0; ð5:18Þ
To investigate the stability of the steady flow solution, we express
f ðg; sÞ ¼ FðgÞ þ expðdsÞgðg; sÞ; hðg; sÞ ¼ HðgÞ þ expðdsÞxðg; sÞ;
ð5:19Þ
where FðgÞ; HðgÞ correspond to steady state solutions and d is the growth(or decay) rate of disturbances [22,23,25]. Here, functions g; x and their derivatives are assumed to be small compared to the steady solutions, respectively. Following Refs. [22,23,25], substi-
xð0Þ ¼ 0; xð1Þ ¼ 0:
ð5:22Þ
The homogeneous linear Eqs. (5.20) and (5.21) and the homogeneous boundary conditions (5.22) constitute an eigenvalue system problem with d as the eigenvalue. As a result, we need to get the smallest eigenvalue d0 . If d0 is negative, then there is an initial growth of disturbance and the flow is unstable. While if the value of d0 is positive, there is an initial decay and the solution is stable. In Table 5, the smallest eigenvalues for some values of physical parameters also have been given, which show the stability of the dual solutions.
X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
6. Conclusion A mixed convection flow and heat transfer of pseudo-plastic power law nanofluid past a stretching vertical plate is investigated. The generalized thermal conductivity proposed by Zheng is applied. The similar equations are solved by Matlab and the stability of the dual solutions is analyzed. Some conclusions can be drawn: Dual solutions are obtained for some values of the physical parameters. Compared both solutions,it is observed that the boundary layer thickness of the velocity for the first solution is thinner than the one for the second solution. This phenomenon also happens on the temperature boundary layer. Local Nusselt number shows the linearity near the point e ¼ 0:5 no longer for power law nanofluids. Cu has the highest thermal conductivity compared to TiO2 and Al2O3.The reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancement in heat transfers. Increasing power law index leads to the decreasing velocity and increasing temperature gradient. The second solution is unstable, and the first solution is stable.
Acknowledgments This work is supported by the National Natural Science Foundations of China (No. 11302024), the Fundamental Research Funds for the Central Universities (No. FRF-TP-15-036A3) and the foundation of the China Scholarship Council in 2014 (No. 154201406465041), Thanks Professor P. Lin giving us some helpful suggestion for the stability analysis. References [1] J.A. Estman, S.U.S. Choi, S. Li, W. Yu, L.J. Thomson, Anomalously increased effective thermal conductivities of ethylene glycol based nanofluid containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720. [2] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [3] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley, New Jersey, 2007. [4] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [5] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. [6] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [7] K.F.V. Wong, O.D. Leon, Applications of nanofluids: current and future, Adv. Mech. Eng. 2010 (2010), Article ID: 519659.. [8] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids 28 (2009) 630–640. [9] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [10] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Mass Transfer 29 (2008) 1326–1336. [11] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effect due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) 4506–4516. [12] D.A. Nield, A.V. Kuznetsov, The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792–5795. [13] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [14] P. Loganathan, P. Nirmal Chand, P. Ganesan, Radiation effects on an unsteady natural convective flow of a nanofluid past an infinite vertical plate, Nano 08 (2013) 1350001.
357
[15] S. Ahmad, A.M. Rohni, I. Pop, Blasius and Sakiadis problems in nanofluids, Acta Mech. 218 (2011) 195–204. [16] Norfifah Bachok, Anuar Ishak, I. Pop, Flow and heat transfer characteristics on a moving plate in a nanofluid, Int. J. Heat Mass Transfer 55 (2012) 642–648. [17] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Commun. Heat Mass Transfer 37 (2010) 987–991. [18] S.V. Subhashini, Nancy Samuel, I. Pop, Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface, Int. Commun. Heat Mass Transfer 38 (2011) 499–503. [19] S.V. Subhashini, R. Sumathi, I. Pop, Dual solutions in a double-diffusive convection near stagnation point region over a stretching vertical surface, Int. J. Heat Mass Transfer 55 (2012) 2524–2530. [20] S.V. Subhashini, R. Sumathi, Dual solutions of a mixed convection flow of nanofluids over a moving vertical plate, Int. J. Heat Mass Transfer 71 (2014) 117–124. [21] T. Mahmood, J.H. Merkin, Mixed convection on a vertical circular cylinder, J. Appl. Math. Phys. (ZAMP) 39 (1988) 186–203. [22] J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Eng. Math. 20 (1985) 171–179. [23] J.H. Merkin, I. Pop, Mixed convection boundary layer on a vertical cylinder embedded in a saturated porous medium, Acta Mech. 66 (1987) 251–262. [24] N. Afzal, A. Badaruddin, A.A. Elgarvi, Momentum and heat transport on a continuous flat surface moving in a parallel stream, Int. J. Heat Mass Transfer 36 (1993) 3399–3403. [25] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on selfsimilar boundary layer flow over moving surfaces, Int. J. Eng. Sci 44 (2006) 730–737. [26] Y.H. Lin, L.C. Zheng, X.X. Zhang, MHD Marangoni boundary layer flow and heat transfer of pseudo-plastic nanofluids over a porous medium with a modified model, Mech. Time Depend. Mater. 19 (2015) 519–536. [27] Y.H. Lin, L.C. Zheng, X.X. Zhang, Radiation effects on Marangoni convection flow and heat transfer in pseudo-plastic non-Newtonian nanofluids with variable thermal conductivity, Int. J. Heat Mass Transfer 77 (2014) 708–716. [28] Y.H. Lin, L.C. Zheng, Marangoni convection flow and heat transfer of power law nanofluids driven by temperature gradient with modified Fourier’s law, Int. J. Nonlinear Sci. Numer. 15 (6) (2014) 337–345. [29] N.A. Aini Mat, N.M. Arifin, R. Nazar, F. Ismail, N. Bachok, MHD mixed convection flow of a power law nanofluid over a vertical stretching sheet with radiation effect, AIP Conf. Proc. 1557 (604) (2013), http://dx.doi.org/ 10.1063/1.4824173. [30] M. Madhu, N. Kishan, Magnetohydrodynamic mixed convection stagnationpoint flow of a power-law non-newtonian nanofluid towards a stretching surface with radiation and heat source/sink, J. Fluid 2015 (2015) 14 page, Article ID 634186. [31] J.H. Kang, F.B. Zhou, W.C. Tan, T.Q. Xia, Thermal instability of a nonhomogeneous power-law nanofluid in a porous layer with horizontal throughflow, J. Non-Newton Fluid 213 (2014) 50–56. [32] W.A. Khan, R. Culham, Oluwole D. Makinde, Hydromagnetic blasius flow of power-law nanofluids over a convectively heated vertical plate, Can. J. Chem. Eng. 93 (2015) 1830–1837. [33] D.X. Jin, Y.H. Wu, J.T. Zou, Studies on heat transfer to pseudoplastic fluid in an agitated tank with helical ribbon impeller, Petro-Chem. Equip. 29 (2) (2000) 7– 9. [34] H. Zhang, A Study of the Boundary Layer on a Continuous Moving Surface in Power Law Fluids, University of Science and Technology Beijing, China, 2008. [35] Y. Zhang, L.C. Zheng, Analysis of MHD thermosolutal Marangoni convection with the heat generation and a first-order chemical reaction, Chem. Eng. Sci. 69 (2012) 449–455. [36] Y. Zhang, L.C. Zheng, Similarity solutions of Marangoni convection boundary layer flow with gravity and external pressure, Chin. J. Chem. Eng. 22 (4) (2014) 365–369. [37] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [38] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, Int. J. Heat Fluid Flow 29 (2008) 242–249. [39] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [40] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Therm. Eng. 27 (2007) 1327– 1337. [41] S. Palm, G. Roy, C.T. Nguyen, Heat transfer enhancement with the use of nanofluids in a radial flow cooling systems considering temperature dependent properties, Appl. Therm. Eng. 26 (2006) 2209–2218. [42] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [43] S. Ahmad, A.M. Rohni, I. Pop, Blasius and Sakiadis problems in nanofluids, Acta Mech. 218 (2011) 195–204. [44] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluid, Int. J. Heat Fluid Flow 21 (2000) 58–64.
358
X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
[45] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [46] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336.
[47] A. Ishak, R. Nazar, I. Pop, Flow and heat transfer characteristics on a moving flat plate in a parallel stream with constant surface heat flux, Heat Mass Transfer 45 (2009) 563–567. [48] A. Postelnicu, I. Pop, Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge, Appl. Math. Comput. 217 (2011) 4359–4368.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A mixed convection flow and heat transfer of pseudo-plastic power law nanofluids past a stretching vertical plate Xinhui Si a,⇑, Haozhe Li a, Liancun Zheng a, Yanan Shen a, Xinxin Zhang b a b
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 1 June 2016 Received in revised form 22 September 2016 Accepted 28 September 2016
Keywords: Power law nanofluids Dual solutions Stability Bvp4c
a b s t r a c t A mixed convection flow and heat transfer of pseudo-plastic power law nanofluid past a stretching vertical plate is investigated. Three types of nano-particles, such as copper (Cu), aluminum oxide (Al2O3) and titanium oxide (TiO2), are considered. The generalized Fourier law proposed by Zheng for varying thermal conductivity of nanofluids, which is dependent on the power law of velocity gradient as well as the nanoparticles property, is taken into account. Dual solutions are obtained by Bvp4c with Matlab. The stability of the solution also is discussed by introducing the unsteady governing equations. Furthermore, a new interesting phenomenon is found: the local Nusselt number do not maintain the similar characteristics of Newtonian fluid near the point where the velocity ratio is equal to 0.5. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, many investigations have been made to explore the nanofluid technology in the field of enhanced heat transfer and new generation cooling technology, which overcomes the limitation of many conventional heat transfer fluids with low thermal conductivity, such as water, oil, and ethylene glycol mixture. Furthermore, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement, nanofluids are also very stable and have no additional problems [1], such as sedimentation, erosion, additional pressure drop and non-Newtonian behavior. Numerous methods have been taken to develop advanced heat transfer fluids with substantially higher conductivity by suspending nano(usually less than 100 nm)/micro or larger-sized particle materials in liquids [2]. The comprehensive references and the broad range of current and future applications on nanofluids can be found in the recent book [3] and in the review papers by Buongiorno [4],Wang and Mujumdar [5], Kakac and Pramuanjaroenkij [6], Wong and Leon [7]. Nanosfluid has been used to improve heat transfer for some convection problems due to its wide applications in electronics cooling, heat exchangers, and double pane windows. Furthermore, mixed convection is preferred and then many numerical studies about heat and mass transfer of nanofluids in enclosures have been ⇑ Corresponding author. E-mail address: [email protected] (X. Si). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.09.106 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
studied [8–11]. Some similarity solutions or analytical solutions also have been done. For example, Nield and Kuznetsov [12,13] studied the free convection of viscous and incompressible nanofluid past a vertical plate embedded in porous medium or not. Loganathan et al. [14] discussed unsteady natural convection flow of nanofluids past an infinite vertical plate with considering the radiation.Ahmad et al. [15] extended Blasius problem and Sakiadis problem to the case of nanofluids. Norfifah Bachok et al. [16] made another extension and extended the Blasius and Sakiadis problems in nanofluids by considering a uniform free streem parallel to a fixed or moving flat plate. Ahmad and Pop [17] investigated steady mixed convection boundary layer flow past a vertical flat plate embedded in a porous medium filled with nanofluids. One of interesting problems, the existence of dual solutions, also have been proposed, which bring more insight on engineering applications. For example, Subhashini et al. [18–20] pointed out that the upper branch solutions are most physically relevant solution whereas the lower branch solutions seem to deprive physical significance or may have realistic meaning in different situations. Furthermore, the stability of the numerical solutions for the mixed convection also has been analyzed. Since Mahmood and Merkin [21] done the classical work and investigated the mixed convection on a vertical circular cylinder, Merkin [22], Merkin and Pop [23] also discussed the dual solutions occurring in mixed convection in a porous medium by perturbation method and analyzed the stability of the upper and lower branch of the solutions. Similarly, some interesting works also have been done [24,25].
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In contrary, the number of studies on natural and mixed convection of nanofluids, where Non-Newtonian power law fluids is considered as the base fluid, is very small. Lin and Zheng et al. [26–28] studied the Marangoni convection of power law nanofluids, where they considered the thermal conductivity to be dependent on velocity or temperature gradient. Some other mass and heat transfer models about power law nanofluids [29–32] also have been proposed and solved numerically or analytically. The aim of this paper is to investigate the mixed convection mass and heat transfer of power law nanofluids past a stretching vertical plate. Here the CMC-Water (0.0–0.4%) is considered as a pseudo-plastic power law fluid [26–28] and regarded as the base fluid. Three types of nanoparticles are considered: copper (Cu), aluminum oxide (Al2O3) and titanium oxide (TiO2). According to the experimental studies, the thermophysical properties of CMC-Water(<0.4%) are similar to water [33,34]. Their related properties are given in Tables 1 and 2. The generalized Fourier law proposed by Zheng [26–28,35,36] for varying thermal conductivity of nanofluids is taken into account. The similar equations are solved numerically with Matlab. Effects of different parameters on velocity and temperature are discussed in detail. 2. Governing equations Consider the steady two-dimensional mixed convection flow and heat transfer of power law nanofluid over a vertical porous stretching plate. The base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. Here we assume that x-axis is along the vertical surface and y-axis is normal to the plate. u and v are the velocity components in the x and y directions, respectively. As shown in Fig. 1, the free stream velocity and the stretching velocity are assumed to be ue ¼ a and uw ¼ b. The temperature T w at the wall is a constant, T 1 is the temperature of the fluid far away from the plate. The suction/injection velocity along
n1 @u @T @T @T @ ðqcp Þnf u þv ¼ ; jnf @x @y @y @y @y where qnf ; knf ;
ð2:3Þ
lnf ; ðqcp Þnf ; bnf are the density of the nanofluid,
the thermal conductivity of the nanlfluid, the modified viscosity of the nanofluid, the heat capacitance of the nanofluid and the coefficient of the thermal expansion, respectively.
l ¼ lnf j @u jn1 is the @y
jn1 is the thermal difeffective viscosity of the nanofluid. K ¼ anf j @u @y j
fusivity proposed by Zheng et al. [26–28,35,36], where anf ¼ ðqcpnfÞ is nf
the modified thermal diffusivity of the nanofluid. The expression of above nanofluid parameters are given as follow:
qnf ¼ ð1 /Þqf þ /qs ; ðqcp Þnf ¼ ð1 /Þðqcp Þf þ /ðqcp Þs ; lf ; ðqbÞnf ¼ ð1 /ÞðqbÞf þ /ðqbÞs ; lnf ¼ 2:5 ð1 /Þ
jnf ðjs þ 2jf Þ 2/ðjf js Þ ¼ ; jf ðjs þ 2jf Þ þ /ðjf js Þ
ð2:4Þ
where qf ; qs are the reference density of the fluid fraction and the solid fraction, and kf ; ks are the thermal conductivity of the fluid fraction and solid fraction, respectively. / is the solid volume fraction parameter of the nanofluid, lf is the viscosity of the fluid fraction. Here the modified viscosity lnf of the nanofluid can be approximated as viscosity of the base fluid lf containing dilute
1
1 ðmf U 2n1 xn Þnþ1 , where U ¼ a þ b is the the plate is v w ¼ f w nþ1 composite velocity. Under these assumptions, the governing equations about the mixed convective flow and heat transfer of power-law nanofluid can be written as follow:
@u @ v þ ¼ 0; @x @y @u @u 1 @ þv ¼ u @x @y qnf @y
ð2:1Þ n1 ! @u @u ðqbÞnf þg lnf ðT T 1 Þ; @y @y qnf
ð2:2Þ
Fig. 1. The model of heat and mass transfer of mixed convection.
Table 1 Thermophysical properties of the base fluid and nanoparticles. Property
CMC-water (0.0–0.4%)
Copper (Cu)
Aluminum Oxide (Al2O3)
Titanium Oxide (TiO2)
cp (J/kg K)
4179 997.1 0.613 21
385 8933 400 1.67
765 3970 40 0.85
686.2 4250 8.9538 0.9
qðkg=m3 Þ j(W/mK) b 105 ð1=KÞ
Table 2 Dynamical properties of the carboxy methyl cellulose water.
The power law index KðNsn =m2 Þ
0.0
0.1
0.2
0.3
0.4
1.0 8.550 04
0.91 6.319 103
0.85 1.754 102
0.81 3.136 102
0.76 7.853 102
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
suspension of fine spherical particles and is given by Brinkman [37]. The effective thermal conductivity of the nanofluid is approximated by the Maxwell–Garnetts model, which restricts to spherical nanoparticles and does not account for other shapes of nanoparticles [38–46]. The corresponding boundary conditions are
v ¼ v w ðxÞ;
u ¼ uw ¼ b;
T ¼ T w ; y ¼ 0;
ð2:5Þ
u ! ue ¼ a; T ! T 1 ; y ¼ 1:
Introduce the stream function w such that u ¼ @w=@y and
v ¼ @w=@x, which are listed as follow: w ¼ ðmf U 2n1 xÞ where
mf ¼
U 2n nþ1 1 T T1 f ðgÞ; g ¼ ð Þ yx nþ1 ; h ¼ ; mf Tw T1
1
1 Pr
00 0
00
1 00 bf ðgÞf ðgÞ ¼ 0; nþ1
jnf 00 n1 0 0 1 h0 f ðgÞ ¼ 0; ðjf ðgÞj h Þ þ a nþ1 jf
ð2:7Þ
ð2:8Þ
bÞs where the coefficients are c ¼ 1 / þ / ððqqbÞ ; b ¼ 1 / þ / qqs ; f
f
a ¼ 1 / þ / ððqqccpp ÞÞfs , respectively. The Prandtl number Pr, the local Grashof number Grx , the local Reynold number Rex and the mixed convection parameter kðxÞ are given by
Grx ¼
g q2f bf ðT w T 1 Þx3
l2f
kðxÞ ¼
; Rex ¼
cp f lf qf Ux ; Pr ¼ ; lf kf
Grx gbf ðT w T 1 Þx ¼ ; Rex U2
ð2:9Þ
where kðxÞ is a function of x. Then the numerical solutions represent local similarity solutions since the parameters still depend on an independent variable x. It should be noted that the sign of k depends on the nature of the flow arising out from this situation. k > 0 corresponds to buoyancy assisting flow. In contrast, when k < 0, it corresponds to buoyancy opposing flow. Three special cases as k ¼ 0; n ¼ 1 are discussed by Norfifah Bachok et al. [16], Ahmad et al. [15] and Subhashini and Sumathi [19]. The corresponding boundary conditions become 0
0
f ð0Þ ¼ f w ; f ð0Þ ¼ e; hð0Þ ¼ 1; f ð1Þ ¼ 1 e; hð1Þ ¼ 0;
ð2:10Þ
where velocity ratio is e ¼ The shear stress and local Nusselt number on the surface are derived as follow: n1 @u @u n1 00 3n1 n 00 ¼ lnf U nþ1 Rex nþ1 f ð0Þ f ð0Þ; sw ¼ lnf @y @y b . U
Nux ¼
xqw ðxÞ jn1 ðT w T 1 Þ knf j @u @y
jy¼0 ¼
1
mf xn U n2
¼
1 aPr nþ1
jf jnf
y5 y1 ðn 1Þjy3 jn2 y03 y5 jy3 jn1
ð3:12Þ
:
The boundary conditions can be written to be
ð2:11Þ
1 !nþ1
ð3:13Þ The system of nonlinear differential equations can be solved by Bvp4c with Matlab. Here we assume that the Maximum residual is
lf qf .
ðjf jn1 f Þ þ hkðxÞc þ
y05
ð2:6Þ
Substituting above transformations into the governing Eqs. (2.1)–(2.3), the partial differential equations can be transformed into the following ordinary equations
ð1 /Þ2:5
y02 ¼ y3 ; 1 1 bay1 y3 Þð1 /Þ2:5 jy3 j1n ; y03 ¼ ðckðxÞy4 n nþ1 y04 ¼ y5 ;
y1 ð0Þ ¼ f w ; y2 ð0Þ ¼ e; y2 ð1Þ ¼ 1 e; y4 ð0Þ ¼ 1; y4 ð1Þ ¼ 0;
1
1 nþ1
y01 ¼ y2 ;
h0 ð0Þ h0 ð0Þ:
3. Numerical methods 3.1. Transformation of equations In order to solve the nonlinear differential Eqs. (2.7) and (2.8), we transfer this problem to a system of first-order equations. Here 0 00 we denote y1 ¼ f ; y2 ¼ f ; y3 ¼ f ; y4 ¼ h; y5 ¼ h0 , then
105 . In order to obtain the dual solutions, the number of the mesh, the infinity replaced by finite point and the initial guess values need to be adjusted according to the different physical parameters. Some solutions for special cases also are compared with previous ones to validate the accuracy and correctness, which is shown in Table 3. 4. Numerical solutions and discussion In this following section, we assume that the base fluid is 0.4% CMC-Water and will consider the velocity and temperature distribution influenced by different nano-particles and different physical parameters. In every figure, we will give illustration of velocity and temperature distribution corresponding to dual solutions for some fixed values. Table 4 gives some values of the dual solutions influenced by different physical parameters. And according to the Fig. 2, we can find that dual solutions exist for power law nanofluids and there is no solution as e < ek . Both types of nanofluids have the similar 00 trends as the base fluids are water or power law fluid for f ð0Þ. However, an interesting phenomenon also can be found for the power law nanofluids. Local Nusselt number decreases first and then increases again near the point e ¼ 0:5, which is different dramatically from the case as the base fluid is water. Furthermore, the 00 first solutions of f ð0Þ and h0 ð0Þ are stable and physically, while the second solutions are not [48,19].The stability of the dual solutions will be discussed in the next section. The influence of different nanoparticles on the velocity and temperature distribution can be found in Fig. 3 as other parameters are e = 0.5, k = 0.01, f w = 0.1, Pr = 6.2, / = 0.1 and n = 0.85. For the first solution, it is observed that the thickness of thermal boundary layer for Cu nanofluid is the more thinner than other two cases. The reason is that Cu has the highest thermal conductivity compared to TiO2 and Al2O3.The higher temperature gradients is caused from the reduced value of thermal diffusivity and, therefore, higher improvement in heat transfers. However, the difference between TiO2 and Al2O3 can be negligible. For the second solution, the velocity and temperature distribution shows a more obvious difference influenced by different nanoparticles. Fig. 4 gives the illustration of dual solutions of Cu-CMC nanofluid flow influenced by power law index as other parameters are k = 0.01,Pr = 6.2, / = 0.001, e = 0.5, f w = 0.1.For the first solution, it is observed that the velocity decreases and the thickness of velocity boundary layer increases with increasing power law index. The reason may be that the effective viscosity of the power law fluids decreases with the increasing power law number. The similar trends also happen on the second solution. Furthermore, the influence of the power law index on the second solution is more obvious.
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 Table 3 The comparison of Cu-water nanofluid with previous works as Pr ¼ 6:2. 00
f ð0Þ
/
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
n=1
0
0.5 0.2 0 1
0.397851 0.412369 0.332059 0.443762
0.171031 0.011421
0.3990 0.4124 0.3321 0.4438
0.1710 0.0114
0.3979 0.4124 0.3321 0.4438
0.1710 0.0114
0.1
0.5 0.2 0 1
0.467380 0.484476 0.390124 0.521295
0.200917 0.013433
0.4674 0.4844 0.3901 0.5218
0.2009 0.0134
0.2
0.5 0.2 0 1
0.484606 0.502304 0.404481 0.540537
0.208323 0.013928
0.4846 0.5023 0.4045 0.5405
0.2083 0.0139
Present results
Ishak et al. [47]
Bachok et al. [16]
Table 4 00 Dual solutions of f ð0Þ and h0 ð0Þ for Cu-CMC power law nanofluid as Pr ¼ 3; k ¼ 0:02; f w ¼ 0:1. 00
f ð0Þ
/¼0
h0 ð0Þ
/ ¼ 0:05
/¼0
/ ¼ 0:05
n
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
0.91
0.5 0.2 0 1
0.476333 0.446220 0.345915 0.48544
0.158882 0.01029
0.562692 0.511197 0.395003 0.54236
0.162790 0.00648
0.27960 0.47927 0.56913 0.91129
0.04073 0.00025
0.32487 0.48345 0.55368 0.80238
0.05626 0.00042
0.85
0.5 0.2 0 1
0.478113 0.448092 0.344111 0.48585
0.163068 0.00494
0.568494 0.515919 0.394770 0.54581
0.167186 0.00188
0.27994 0.47824 0.56144 0.87464
0.04428 0.00002
0.32767 0.48562 0.54991 0.75922
0.05955 0.00005
0.76
0.5 0.2 0 1
0.483640 0.453537 0.342869 0.49037
0.170801 0.000029
0.582823 0.526556 0.396378 0.55465
0.175234 0.001532
0.28228 0.47905 0.55165 0.81103
0.05064 0.00091
0.33508 0.49176 0.54641 0.68018
0.06542 0.00416
1.4
0.8 n=0.81, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.6998 w
2 3 k
1.2
0.6
1
0.4
n=0.91, Pr=6.2,λ=−0.01,φ=0.1,f =0.1, Cu,ε =−0.6325 w
k
n=1, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.7041 w
2 3 k
0.8
f’’(0)
−θ’(0)
0.2
0
0.6
0.4 −0.2
n=0.91, Pr=6.2,λ=−0.01,φ=0.1,f =0.1,Cu,ε =−0.6325 w
k
0.2 −0.4
w
−0.6 −0.8
n=0.81, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.6998 w
2 3 k
n=1, Pr=3,λ=−0.02,φ=0.1,f =0.2,Al O ,ε =−0.7041
−0.6
−0.4
−0.2
2 3 k
0
ε
0.2
0 0.4
0.6
0.8
1
−0.8
−0.6
−0.4
−0.2
0
ε
0.2
0.4
0.6
0.8
1
00
Fig. 2. Variations of f ð0Þ; h0 ð0Þ against e.
Fig. 5 illustrates the effects of solid volume fraction on the velocity and temperature as other parameters are k = 0.001, f w = 0.1, Pr = 6.2, e = 0.5, / = 0.1, n = 0.85. For the first solution, the temperature decreases and the temperature gradient increases near the plate with the increasing solid volume fraction. This means that the increasing solid volume fraction results in the increase of heat transfer, which reduces to the thinning of the thermal boundary layers near the vertical walls. Meanwhile, the
velocity boundary layer thickness also decreases with the solid volume fraction. Furthermore, the second solution for velocity and temperature shows the similar trends compared with the first solution. Fig. 6 gives the profiles of suction parameter on the velocity and temperature as other parameter are k = 0.001, Pr = 6.2, e = 0.5, / = 0.1, n = 0.85. For the first solution, with the increasing suction parameter, much more fluid is carried away from the surface
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.6
1.5
1.4
Al O
2 3
TO
1.2
i 2
1
the first solution for f’(η)
1
Cu
Al O
0.8
2 3
the second solution for f’(η)
TO
i 2
0.6
0.5
Cu 0.4
the first solution for θ(η)
the second solution for θ(η)
0.2 0
0 −0.2 −0.4 0
2
4
η
6
8
−0.5
10
0
2
4
η
6
8
10
Fig. 3. Dual solutions for different power law nanofluids.
1.5
1.5 n=0.76 n=0.85 n=0.91
1
the second solution for f’(η)
1 the first solution for f’(η)
0.5
0.5 the second solution for θ(η)
the first solution for θ(η)
0
0 n=0.76 n=0.85 n=0.91
−0.5
0
2
4
η
6
8
10
−0.5
0
2
4
6 η
8
10
12
Fig. 4. Dual solutions for different n in Cu power law nanofluid.
1.5
1.5
the first solution for f’(η)
1
the second solution for f’(η)
1
0.5
0.5 the second solution for θ(η)
the first solution for θ(η)
0
0 φ=0.001
φ=0.001
φ=0.05
φ=0.05
φ=0.1 −0.5
0
2
4
6 η
8
10
φ=0.1 12
−0.5
0
5
Fig. 5. Dual solutions for different Cu volume fraction in power law nanofluid.
η
10
15
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.5
1.5
the first solution for f’(η) 1
1
the second solution for f’(η)
0.5
0.5
the second solution for θ(η)
the first solution for θ(η)
0
0 f =0.1
f =0.1
f =0.2
f =0.2
f =0.3
f =0.3
w
w
w
w
w
−0.5
0
2
4
η
6
8
10
−0.5
w
0
5
10
η
15
Fig. 6. Dual solutions for f w in Cu power law nanofluid.
which causes reduction in velocity gradient as it tries to maintain the same velocity over a small region near the surface. The velocity increases with the increasing value of permeability parameter. Moreover, the thermal boundary layer thickness decreases sharply with the increasing parameter f w . This also shows that the increasing suction velocity makes the temperature boundary layer thinner and the boundary layer becomes more stable. However, for the second solution, the opposite trends can be observed influenced by suction velocity. Fig. 7 illustrates the effects the velocity ratio on the velocity and temperature distribution as other parameters are k = 0.01, Pr = 6.2, e = 0.5, / = 0.001, n = 0.91, f w = 0.1. For the first solution, we can find that the temperature gradient and the velocity at the wall will increase with increasing e. However, as the distance is far away from the plate, the velocity will decrease with increasing e. For the second solution, the profile corresponding to temperature is put toward the right side with increasing parameter e. Fig. 8 exhibits the influence of mixed convection parameter on the velocity and temperature distribution as other parameters are Pr = 6.2, e = 0.5, / = 0.001, n = 0.91, f w = 0.1. For the first solution, it is observed that with the increasing mixed convection, velocity and temperature gradient will increase. Since the buoyancy forces
1.5
will prevent the movement of the fluid, fluids within the boundary layer get retarded acting as an adverse pressure gradient. The mixed convection parameter represents a measure of the effect of the buoyancy in comparison with that of the inertia of the external forced or free stream flow on the thermal and velocity fields. For the second solution, the temperature profile is put toward right side with increasing mixed convection parameter k. The velocity has contrary trend corresponding to the case of first solution. 5. The stability of the numerical solution In order to analyze the stability of the numerical solution, the unsteady mixed convection problem is introduced. The governing equations are written as
@u @u @u 1 @ þu þv ¼ @t @x @y qnf @y
n1 ! @u @u ðqbÞnf þg lnf ðT T 1 Þ; @y @y qnf
ð5:14Þ ðqcp Þnf
@T @T @T @ ¼ þu þv @t @x @y @y
n1
@u jnf @y
!
@T ; @y
ð5:15Þ
1.5
1
ε=−0.50
1
the first solution for f’(η)
ε=−0.35 ε=−0.20
0.5
0.5 the first solution for θ(η)
the second solution for θ(η)
0
0 ε=−0.50 ε=−0.35
the second solution for f’(η)
ε=−0.20 −0.5
0
5
η
10
15
Fig. 7. Dual solutions for different
−0.5
0
5
e in Cu power law nanofluid.
10 η
15
20
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X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358 1.5
1.6 1.4 1.2
1
the first solution for f’(η)
the second solution for f’(η)
1 0.8 0.6
0.5
the second solution for θ(η)
0.4
the first solution for θ(η)
0.2 0
0 λ=−0.01 λ=−0.03 λ=−0.05
−0.5
0
2
4
η
6
8
λ=−0.01
−0.2
λ=−0.03 λ=−0.05
−0.4
10
0
5
η
10
15
Fig. 8. Dual solutions for different k in Cu power law nanofluid.
Table 5 The smallest eigenvalue d0 for Cu-CMC power law nanofluid as Pr ¼ 6:2; k ¼ 0:01; f w ¼ 0:1. n = 0.91
n = 0.85
n = 0.76
/
e
1st solution
2st solution
1st solution
2st solution
1st solution
2st solution
0.1
0.5 0.2 0 0.25
0.198312 0.377343 0.454069 0.511108
0.10595 0.09631
0.197378 0.380363 0.453981 0.510199
0.10523 0.09461
0.196156 0.386269 0.471712 0.524424
0.10423 0.09381
0.05
0.5 0.2 0 0.25
0.193537 0.372135 0.448543 0.511968
0.10267 0.09904
0.188589 0.378842 0.462091 0.521913
0.10161 0.09652
0.190353 0.378402 0.456276 0.527553
0.10005 0.09600
where t represents time. The new dimensionless variables are introduced: 1
w ¼ ðmf U 2n1 xÞnþ1 f ðg; sÞ; hðg; sÞ ¼
g¼
U 2n
1 !nþ1
mf
1
yxnþ1 ;
T T1 ; Tw T1
s ¼ Utx1 ;
1 ð1 /Þ2:5 ð5:16Þ
then the partial differential equations can be obtained after one substitutes dimensionless transformation (5.16) into Eqs. (5.14) and (5.15): 1 ð1 /Þ
2:5
ðjf gg j
n1
tuting Eq. (5.19) into differential equations system (5.17) and (5.18) and linearizing, we can obtain:
1 bffgg bðf gs f g f gs s þ f s f gg sÞ ¼ 0; f gg Þg þ hkðxÞc þ nþ1 ð5:17Þ
½nðn 1ÞjF 00 jn2 F 000 g 00 þ njF 00 jn1 g 000
þ ckx þ dbg 0 þ
1 bðFg 00 þ gF 00 Þ ¼ 0; nþ1
ð5:20Þ
knf 1 ½ðn 1ÞjF 00 jn2 F 000 x0 þ ðn 1Þðn 2ÞjF 00 jn3 g 00 F 000 H0 kf Pr þ ðn 1Þg 000 jF 00 jn2 H0 þ jF 00 jn1 x00 þ ðn 1ÞjF 00 jn2 g 00 H00 1 aðF x0 þ H0 gÞ ¼ 0: þ dxa þ nþ1
ð5:21Þ
The corresponding boundary conditions are
1 Pr
jnf 1 hg f aðhs f g hs s þ f s hg sÞ ¼ 0: ðjf jn1 hg Þg þ a nþ1 jf gg
gð0Þ ¼ 0; g 0 ð0Þ ¼ 0; g 0 ð1Þ ¼ 0; ð5:18Þ
To investigate the stability of the steady flow solution, we express
f ðg; sÞ ¼ FðgÞ þ expðdsÞgðg; sÞ; hðg; sÞ ¼ HðgÞ þ expðdsÞxðg; sÞ;
ð5:19Þ
where FðgÞ; HðgÞ correspond to steady state solutions and d is the growth(or decay) rate of disturbances [22,23,25]. Here, functions g; x and their derivatives are assumed to be small compared to the steady solutions, respectively. Following Refs. [22,23,25], substi-
xð0Þ ¼ 0; xð1Þ ¼ 0:
ð5:22Þ
The homogeneous linear Eqs. (5.20) and (5.21) and the homogeneous boundary conditions (5.22) constitute an eigenvalue system problem with d as the eigenvalue. As a result, we need to get the smallest eigenvalue d0 . If d0 is negative, then there is an initial growth of disturbance and the flow is unstable. While if the value of d0 is positive, there is an initial decay and the solution is stable. In Table 5, the smallest eigenvalues for some values of physical parameters also have been given, which show the stability of the dual solutions.
X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
6. Conclusion A mixed convection flow and heat transfer of pseudo-plastic power law nanofluid past a stretching vertical plate is investigated. The generalized thermal conductivity proposed by Zheng is applied. The similar equations are solved by Matlab and the stability of the dual solutions is analyzed. Some conclusions can be drawn: Dual solutions are obtained for some values of the physical parameters. Compared both solutions,it is observed that the boundary layer thickness of the velocity for the first solution is thinner than the one for the second solution. This phenomenon also happens on the temperature boundary layer. Local Nusselt number shows the linearity near the point e ¼ 0:5 no longer for power law nanofluids. Cu has the highest thermal conductivity compared to TiO2 and Al2O3.The reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancement in heat transfers. Increasing power law index leads to the decreasing velocity and increasing temperature gradient. The second solution is unstable, and the first solution is stable.
Acknowledgments This work is supported by the National Natural Science Foundations of China (No. 11302024), the Fundamental Research Funds for the Central Universities (No. FRF-TP-15-036A3) and the foundation of the China Scholarship Council in 2014 (No. 154201406465041), Thanks Professor P. Lin giving us some helpful suggestion for the stability analysis. References [1] J.A. Estman, S.U.S. Choi, S. Li, W. Yu, L.J. Thomson, Anomalously increased effective thermal conductivities of ethylene glycol based nanofluid containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720. [2] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [3] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley, New Jersey, 2007. [4] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [5] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. [6] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [7] K.F.V. Wong, O.D. Leon, Applications of nanofluids: current and future, Adv. Mech. Eng. 2010 (2010), Article ID: 519659.. [8] S.M. Aminossadati, B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, Eur. J. Mech. B/Fluids 28 (2009) 630–640. [9] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [10] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Mass Transfer 29 (2008) 1326–1336. [11] C.J. Ho, M.W. Chen, Z.W. Li, Numerical simulation of natural convection of nanofluid in a square enclosure: effect due to uncertainties of viscosity and thermal conductivity, Int. J. Heat Mass Transfer 51 (2008) 4506–4516. [12] D.A. Nield, A.V. Kuznetsov, The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 52 (2009) 5792–5795. [13] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 49 (2010) 243–247. [14] P. Loganathan, P. Nirmal Chand, P. Ganesan, Radiation effects on an unsteady natural convective flow of a nanofluid past an infinite vertical plate, Nano 08 (2013) 1350001.
357
[15] S. Ahmad, A.M. Rohni, I. Pop, Blasius and Sakiadis problems in nanofluids, Acta Mech. 218 (2011) 195–204. [16] Norfifah Bachok, Anuar Ishak, I. Pop, Flow and heat transfer characteristics on a moving plate in a nanofluid, Int. J. Heat Mass Transfer 55 (2012) 642–648. [17] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Commun. Heat Mass Transfer 37 (2010) 987–991. [18] S.V. Subhashini, Nancy Samuel, I. Pop, Effects of buoyancy assisting and opposing flows on mixed convection boundary layer flow over a permeable vertical surface, Int. Commun. Heat Mass Transfer 38 (2011) 499–503. [19] S.V. Subhashini, R. Sumathi, I. Pop, Dual solutions in a double-diffusive convection near stagnation point region over a stretching vertical surface, Int. J. Heat Mass Transfer 55 (2012) 2524–2530. [20] S.V. Subhashini, R. Sumathi, Dual solutions of a mixed convection flow of nanofluids over a moving vertical plate, Int. J. Heat Mass Transfer 71 (2014) 117–124. [21] T. Mahmood, J.H. Merkin, Mixed convection on a vertical circular cylinder, J. Appl. Math. Phys. (ZAMP) 39 (1988) 186–203. [22] J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Eng. Math. 20 (1985) 171–179. [23] J.H. Merkin, I. Pop, Mixed convection boundary layer on a vertical cylinder embedded in a saturated porous medium, Acta Mech. 66 (1987) 251–262. [24] N. Afzal, A. Badaruddin, A.A. Elgarvi, Momentum and heat transport on a continuous flat surface moving in a parallel stream, Int. J. Heat Mass Transfer 36 (1993) 3399–3403. [25] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on selfsimilar boundary layer flow over moving surfaces, Int. J. Eng. Sci 44 (2006) 730–737. [26] Y.H. Lin, L.C. Zheng, X.X. Zhang, MHD Marangoni boundary layer flow and heat transfer of pseudo-plastic nanofluids over a porous medium with a modified model, Mech. Time Depend. Mater. 19 (2015) 519–536. [27] Y.H. Lin, L.C. Zheng, X.X. Zhang, Radiation effects on Marangoni convection flow and heat transfer in pseudo-plastic non-Newtonian nanofluids with variable thermal conductivity, Int. J. Heat Mass Transfer 77 (2014) 708–716. [28] Y.H. Lin, L.C. Zheng, Marangoni convection flow and heat transfer of power law nanofluids driven by temperature gradient with modified Fourier’s law, Int. J. Nonlinear Sci. Numer. 15 (6) (2014) 337–345. [29] N.A. Aini Mat, N.M. Arifin, R. Nazar, F. Ismail, N. Bachok, MHD mixed convection flow of a power law nanofluid over a vertical stretching sheet with radiation effect, AIP Conf. Proc. 1557 (604) (2013), http://dx.doi.org/ 10.1063/1.4824173. [30] M. Madhu, N. Kishan, Magnetohydrodynamic mixed convection stagnationpoint flow of a power-law non-newtonian nanofluid towards a stretching surface with radiation and heat source/sink, J. Fluid 2015 (2015) 14 page, Article ID 634186. [31] J.H. Kang, F.B. Zhou, W.C. Tan, T.Q. Xia, Thermal instability of a nonhomogeneous power-law nanofluid in a porous layer with horizontal throughflow, J. Non-Newton Fluid 213 (2014) 50–56. [32] W.A. Khan, R. Culham, Oluwole D. Makinde, Hydromagnetic blasius flow of power-law nanofluids over a convectively heated vertical plate, Can. J. Chem. Eng. 93 (2015) 1830–1837. [33] D.X. Jin, Y.H. Wu, J.T. Zou, Studies on heat transfer to pseudoplastic fluid in an agitated tank with helical ribbon impeller, Petro-Chem. Equip. 29 (2) (2000) 7– 9. [34] H. Zhang, A Study of the Boundary Layer on a Continuous Moving Surface in Power Law Fluids, University of Science and Technology Beijing, China, 2008. [35] Y. Zhang, L.C. Zheng, Analysis of MHD thermosolutal Marangoni convection with the heat generation and a first-order chemical reaction, Chem. Eng. Sci. 69 (2012) 449–455. [36] Y. Zhang, L.C. Zheng, Similarity solutions of Marangoni convection boundary layer flow with gravity and external pressure, Chin. J. Chem. Eng. 22 (4) (2014) 365–369. [37] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [38] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, Int. J. Heat Fluid Flow 29 (2008) 242–249. [39] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transfer 46 (2003) 3639–3653. [40] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Therm. Eng. 27 (2007) 1327– 1337. [41] S. Palm, G. Roy, C.T. Nguyen, Heat transfer enhancement with the use of nanofluids in a radial flow cooling systems considering temperature dependent properties, Appl. Therm. Eng. 26 (2006) 2209–2218. [42] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [43] S. Ahmad, A.M. Rohni, I. Pop, Blasius and Sakiadis problems in nanofluids, Acta Mech. 218 (2011) 195–204. [44] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluid, Int. J. Heat Fluid Flow 21 (2000) 58–64.
358
X. Si et al. / International Journal of Heat and Mass Transfer 105 (2017) 350–358
[45] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [46] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336.
[47] A. Ishak, R. Nazar, I. Pop, Flow and heat transfer characteristics on a moving flat plate in a parallel stream with constant surface heat flux, Heat Mass Transfer 45 (2009) 563–567. [48] A. Postelnicu, I. Pop, Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge, Appl. Math. Comput. 217 (2011) 4359–4368.