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Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids
International Journal of Heat and Mass Transfer 55 (2012) 1888–1895
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Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids Azizah Mohd Rohni a, Syakila Ahmad b,⇑, Ioan Pop c a
UUM College of Arts & Sciences, Physical Science Division, Building of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia c Faculty of Mathematics, University of Cluj, R-400082 Cluj, CP 253, Romania b
a r t i c l e
i n f o
Article history: Received 9 November 2010 Received in revised form 25 October 2011 Accepted 29 October 2011 Available online 16 December 2011 Keywords: Unsteady Shrinking sheet Suction Nanofluids Numerical results
a b s t r a c t The unsteady flow over a continuously shrinking surface with wall mass suction in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3 and TiO2 is numerically studied. Similarity equations are obtained through the application of similarity transformation techniques. The shooting method is used to solve the similarity equations for different values of the wall mass suction, the unsteadiness and the nanoparticle volume fraction parameters. The results of skin friction coefficient f 00 ð0Þ and heat transfer rate h0 (0) are presented in tables and graphs. It is found that dual solution exists for a certain range of wall mass suction s, volume fraction u and unsteadiness parameters A. The results of velocity and temperature profiles are also presented. It is seen that two values of boundary layer thickness g1 are obtained, which gives two different velocity and temperature profiles that satisfy the boundary conditions. It is also found that the nanoparticle volume fraction parameter u and types of nanofluid play an important role to significantly determine the flow behaviour. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In most forced convection boundary layer flow and heat transfer problem, the base fluid has a low thermal conductivity, which limits the heat transfer enhancement. However, the continuing miniaturization of electronic devices requires further heat transfer improvements from an energy saving viewpoint. An innovative technique, which uses a mixture of nanoparticles and the base fluid was first introduced by Choi [1] in order to develop advanced heat transfer fluids with substantially higher conductivities. The resulting mixture of the base fluid and nanoparticles having unique physical and chemical properties is referred to as a nanofluid. It is expected that the presence of the nanoparticles in the nanofluid increases the thermal conductivity and therefore substantially enhances the heat transfer characteristics of the nanofluid. Eastman et al. [2], Xie et al. [3], Jou and Tzeng [4] and Jana et al. [5] showed that higher thermal conductivity can be achieved in thermal systems utilising nanofluids. Hwang et al. [6] measured thermal conductivities of various nanofluids and showed that the thermal conductivity enhancement of nanofluids depended on the volume fraction of the suspended particles and the thermal conductivities of the particles and base fluids. Nanofluids can be defined as the dilution of nanometer-sized particles (smaller than 100 nm) in a fluid [7], and nanofluids can be produced by dispersed evenly ⇑ Corresponding author. Tel.: +60 4 653 4782; fax: +60 4 657 0910. E-mail addresses: [email protected] (A.M. Rohni), syakila.ahmad@ymail. com (S. Ahmad), [email protected] (I. Pop). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.11.042
nanoparticles in a base fluid, such as water, ethylene glycol and oil [8]. The nanofluids usually contain the nanoparticles such as metals, oxides, carbides, or carbon nanotubes, whereby these nanoparticles have unique chemical and physical properties [9]. Since the size of nanoparticles are in nanometer-sized, besides behaving similar as liquid molecules, they have the ability to flow smoothly through the microchannels easily [10], hence, nanofluids will enhance the thermal conductivity and convective heat transfer coefficient compared to the base fluid only [11]. Thus, nanofluids are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. According to Godson et al. [12], one of the main objectives of using nanofluids is to achieve the best thermal properties with the least possible (<1%) volume fraction of nanoparticles in the base fluid. There are many studies on the mechanism behind the enhanced heat transfer characteristics using nanofluids. The collection of papers on this topic is included in the book by Das et al. [7], and in the review papers by Kakaç and Pramuanjaroenkij [11], Maiga et al. [13], Buongiorno [14], Daungthongsuk and Wongwises [15], Trisaksri and Wongwises [16], and Wang and Mujumdar [8,17,18]. In this paper, the unsteady viscous flow over a continuously shrinking surface with wall mass suction in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3, and TiO2 is numerically studied. We use nanofluid equations model proposed by Tiwari and Das [19], as this model successfully applied in several papers [9,20–24]. Similarity equations are obtained through the application of a similarity transformation technique and the corresponding equations are solved numerically using
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Nomenclature Cf Cp (qCp)nf k knf Nu A p Rex T T1 s u, v uw
vw
x, y
skin friction coefficient specific heat at constant temperature heat capacitance of the nanofluid thermal conductivity thermal conductivity of the nanofluid Nusselt number unsteadiness parameter fluid pressure local Reynolds number fluid temperature fluid temperature of the ambient fluid wall mass transfer parameter velocity component along x and y directions velocity of the shrinking sheet velocity of the mass transfer Cartesian coordinates along the plate and normal to it, respectively
shooting method for different values of the wall mass suction, the unsteadiness and the nanoparticle volume fraction parameters in the based fluid of water with the Prandtl number of 6.2 [9]. Results show that multiple solutions exist for a certain range of wall mass suction and nanoparticle volume fraction parameters. For comparison purposes, the present results for a regular Newtonian fluid are computed, and they show excellent agreement with those obtained by Fang et al. [25]. In fact, to the present knowledge of the authors, no studies have been reported in the literature, which investigate the flow over a shrinking sheet in nanofluids. It is, however, important to point out that Miklavcˇicˇ and Wang [26] investigated the steady flow over a shrinking sheet, which is an exact solution of the Navier–Stokes equations. This new type of shrinking sheet flow is essentially a backward flow as discussed by Goldstein [27]. It was shown that wall mass suction is required to maintain the flow over a shrinking sheet. The flow induced by a shrinking sheet with constant velocity or power-law velocity distribution has been investigated recently by Fang et al. [28], Fang [29], and Fang and Zhang [30]. The shrinking sheet problem was also extended to other fluids by Hayat et al. [31] and Sajid et al. [32]. We mention also the recent paper by Zheng et al. [33] on the unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink. The work on the flow over a shrinking sheet was further generalized by Wang [34] to include a stagnation flow as the free stream. This problem may be encountered in a number of electronic cooling devices equipped with nanofluids. On the other hand, we mention that several papers have been recently published on steady boundary layer flow in nanofluids by Nield and Kuznetsov [35], Kuznetsov and Nield [36], Khan and Pop [37], and Ahmad and Pop [38]. However it is worth mentioning at this end that the study of nanofluids is still at its early stage, so that it seems it is very difficult to have a precise idea on the way the use of nanoparticles acts in convective heat transfer flows and complementary works are necessary to understand the heat transfer characteristics of nanofluids and identify new and unique applications for these fields [23].
2. Basic equations We consider the two-dimensional flow over a continuously unsteady shrinking sheet with mass transfer in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3 and
Greek letters effective thermal diffusivity of the nanofluid nanoparticle volume fraction similarity variable skin friction at the shrinking sheet effective dynamic viscosity of the nanofluid h dimensionless temperature q density qnf effective density of the nanofluid w stream function
anf u g sw lnf
Subscripts f fluid fraction s solid fraction nf nanofluid fraction Superscript 0 differentiation with respect to g
TiO2. We assume that the velocity of the shrinking sheet is uw(x, t) and the velocity of the mass transfer is vw(x, t), where x is the coordinate measured along the shrinking sheet and t is the time. It is also assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Assuming that the nanofluid is incompressible and laminar, and using the nanofluid model as proposed by Tiwari and Das [19], the governing equations of this problem, namely the two dimensional Navier–Stokes equations, are
@u @ v þ ¼0 @x @y
ð1Þ
@u @u @u 1 @p lnf þu þv ¼ þ @t @x @y qnf @x qnf
! @2u @2u þ @x2 @y2
@v @v @v 1 @p lnf þ þu þv ¼ @t @x @y qnf @y qnf
@2v @2v þ @x2 @y2
@T @T @T @2T @2T þu þv ¼ anf þ @t @x @y @x2 @y2
ð2Þ
! ð3Þ
! ð4Þ
where u and v are the velocity component in the x and y directions, respectively, p is the fluid pressure, T is the temperature of the nanofluid, lnf is the effective viscosity of the nanofluid and qnf is the effective density of the nanofluid and anf is the effective thermal diffusivity of the nanofluid, which are given by
anf ¼
knf ; ðqC p Þnf
qnf ¼ ð1 uÞqf þ uqs ; lnf ¼
lf ; ð1 uÞ2:5
ðqC p Þnf ¼ ð1 uÞðqC p Þf þ uðqC p Þs ; knf ðks þ 2kf Þ 2uðkf ks Þ ¼ kf ðks þ 2kf Þ þ uðkf ks Þ
ð5Þ
where u is the solid volume fraction of the nanofluid, qf is the density of the base fluid, qs is the density of the nanoparticle, lf is the dynamic viscosity of the base fluid, kf is the thermal conductivity of the base fluid and ks is the thermal conductivity of the solid nanoparticle. Density properties of the fluid and nanoparticles are given in Table 1 [21]. The viscosity lnf of the nanofluid can be approximated as viscosity of a base fluid lf containing dilute suspension of the spherical particles and is given by Brinkman [39]. Also, the
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3
Table 1 Thermophysical properties of fluid and nanoparticles [21].
2
Physical properties
Fluid phase (water)
Cu
Al2O3
TiO2
Cp (J/kg K) q (kg/m3) k (W/mK)
4179 997.1 0.613
385 8933 400
765 3970 40
686.2 4250 8.9538
t < 0 : u ¼ v ¼ 0;
T T1 hðgÞ ¼ ; Tw T1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c g¼ y mf ð1 atÞ
g 00 f 000 þ ff f 02 A f 0 þ f 00 ¼ 0 2 ð1 uÞ ½1 u þ uðqs =qf Þ 1
f " (0)
-5 -6 -7 -9
-8
f ðgÞ ! 0;
f ð0Þ ¼ 1;
hð0Þ ¼ 1
hðgÞ ! 0 as g ! 1
sw qw ; Nu ¼ qf u2w kf ðT w T 1 Þ
-5
-4
-3
-2
-1
0
• Ac = - 1.6551
-2 -4 -6
ð7Þ -8 -10 -12 -9
-8
-7
-6
-5
-4
-3
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-2
0
Fig. 2. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.1.
2
ð9Þ
Ac = - 1.7498
1
•
f " (0)
0
-1
Ac = - 8.3490 Ac = - 8.0117
-2
ð11Þ
ð12Þ
-1
A
ð8Þ
where A = a/c is the unsteadiness parameter. For the present situation, we assume a decelerating shrinking sheet with A 6 0 [25]. Quantities of interest in this problem are the skin friction coefficient Cf and the Nusselt number Nu, which are defined as
Cf ¼
-6
0
subject to the boundary conditions 0
-7
2.1 2.15 2.2 2.5 3.0
2
ð10Þ
f ð0Þ ¼ s;
= = = = =
A
knf =kf 1 g h i h00 þ f h0 mf 0 h A h þ h0 ¼ 0 Pr 1 u þ uðqC p Þ =ðqC p Þ 2 s f
0
s s s s s
-4
4
where s is the constant wall mass transfer parameter with s > 0 for suction and s < 0 for injection. Substituting (7) into Eqs. (2)–(4), we obtain the following ordinary differential equations 2:5
•
ð6Þ
where primes denote differentiation with respect to g and mf is the kinematic viscosity of the base fluid. Then, the wall mass transfer velocity becomes
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi mf c v w ð0; tÞ ¼ f ð0Þ ¼ s 1 at
Ac = - 1.6551
Fig. 1. Variation of f 00 ð0Þ with the parameter A for some values of s when u = 0.0.
where b, c and m are positive constants and a is a parameter showing the unsteadiness of the problem. It should be mentioned that the case m = 0 corresponds to a constant wall temperature and m = 1 to a linearly variation with x of the wall temperature, respectively. We look for a similarity solution of Eqs. (1)–(4) of the following form
cx u¼ f 0 ðgÞ; 1 at
•
Ac = - 8.3490
-2 -3
f " (0)
T ¼ T 1 for all x; y cx t P 0 : u ¼ uw ðx; tÞ ¼ ; v ¼ v w ðx; tÞ; 1 at m bx at y ¼ 0 T ¼ T w ðx; tÞ ¼ T 1 þ 1 at u ! 0; T ! T 1 as y ! 1
•
0 -1
effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnett’s model, which is found to be appropriate for studying heat transfer enhancement using nanofluids [13,20]. Very recently, Popa et al. [40] were making a comparison between the Maxwell model and the experimental data provided by Mintsa et al. [41] for thermal conductivity. Popa et al. [40] find that Maxwell’s model strongly overestimates the thermal conductivity of the nanofluid. Other several models for knf can be found in Kakaç and Pramunjaroenkij [11], Patel et al. [42] and Abu-Nada [43]. Eqs. (1)–(4) are subjected to the following initial and boundary conditions
Ac = - 3.8605
1
-3
-4 -9
•
•
-8
-7
-6
-5
-4
-3
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-2
-1
0
A Fig. 3. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Al2O3 when s = 2.2.
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2
4 2
1
Ac = - 3.2739
0 -2
Ac = - 8.3490
-6
•
-4 -5 -12
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-10
-8
-6 A
-4
-2
Cu
-12
Al 2O3
-14
TiO2
-16
• -10
Ac = - 1.7498
-8
-2 Ac = - 11.5101
Ac = - 3.2739
-4
•
-1
f " (0)
f " (0)
0
-3
•
•
-18
0
-12
-10
-8
-6
-4
-2
0
A
Fig. 4. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for TiO2 when s = 2.2.
Fig. 7. Variation of f 00 ð0Þ with unsteadiness parameter A for different nanofluids when u = 0.2 and s = 2.2.
5
14
0
-5
ϕ = 0.0 ϕ = 0.1
10
Ac = - 8.3490
8
-15
-20
-12
-10
-8
-6
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-4
-2
-θ'(0)
-10
6
ϕ = 0.2
4 2 0
0
A
-2
-12
Fig. 5. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2.
11.1
2
-6
-4
-2
0
•
10.9
-2
•
-4
-θ'(0)
Ac = - 8.0117
-8
10.7 10.6
Ac = - 11.5101
-12 -14
Cu
10.5
Al2O3
10.4 10.3
TiO2
-16
10.2 -8.5
-12
-10
Ac = - 8.3490
10.8
•
-6
-18
-8
(b)
11
0
-10
-10
A
4
f " (0)
m=0 m=1
12
• f " (0)
(a)
-8
-6
-4
-2
0
m=0 m=1
• -8
-7.5
-7
A
A Fig. 6. Variation of f 00 ð0Þ with unsteadiness parameter A for different nanofluids when u = 0.1 and s = 2.2.
Fig. 8. (a) Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for Cu–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2. (b) Enlargement of the area in circle in (a).
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14 m=0 m=1
13
7.4
12
• •
10
Cu Al 2O3 TiO2
7.2
Ac = - 8.3490
9
ϕ = 0.1
-θ'(0)
11
-θ'(0)
m=0
7.6
ϕ = 0.0
7
Ac = - 8.0117
8
6.8
•
7
•
• •
Ac = - 1.7498
6 5 -9
•
ϕ = 0.2
-8
-7
-6
-5
-4
-3
Ac = -3.7239
6.6
• Ac = -1.7498
-2
-1
6.4 -3.5
0
-3
-2.5
-2
Fig. 9. Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for Al2O3–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2.
-1.5
-1
-0.5
0
A
A
Fig. 11. Variation of h0 (0) with A when u = 0.2 for different nano fluid with m = 0 when s = 2.2 and Pr = 6.2.
14
7.5 m=0 m=1
13
m=1
ϕ = 0.0
7
12
•
10
Ac = - 8.3490
Ac = -3.7239
•
6
ϕ = 0.1
•
-θ'(0)
-θ'(0)
6.5
•
11
9
Ac = -1.7498
5.5
8
ϕ = 0.2
7
•
6
•
5 -12
•
Ac = - 3.7239
-8
-6 A
-4
Cu Al 2O3
TiO2
4.5
•
Ac = - 11.5101 -10
5
-2
4 -3.5
0
-3
-2.5
-2
-1.5
-1
-0.5
0
A
Fig. 10. Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for TiO2–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2.
Fig. 12. Variation of h0 (0) with A when u = 0.2 for different nano fluid with m = 1 when s = 2.2 and Pr = 6.2.
where sw is the skin friction at the shrinking sheet and qw is the heat flux from the shrinking sheet, which are given by
Table 2 Coordinates of turning points. Ac for u ¼ 0 (pure fluid) and different values of s when m ¼ 0 and m ¼ 1 with Pr ¼ 6:2.
sw ¼ lnf
@u @y y¼0
qw ¼ knf
@T @y y¼0
ð13Þ
Substituting (7) into (13) and using (12), we get
Re1=2 x Cf ¼
1 ð1 uÞ2:5
f 00 ð0Þ;
knf 0 Re1=2 Nu ¼ h ð0Þ x kf
ð14Þ
s
Turning points
2.10
(Ac ; f 00 ð0Þ) (Ac, h0 (0)) (Ac ; f 00 ð0Þ) (Ac, h0 (0)) (Ac ; f 00 ð0Þ) (Ac, h0 (0))
2.15 2.20
where Rex = uwx/mf is the local Reynolds number. 3. Results and discussion Numerical solutions of the governing ordinary differential Eqs. (9) and (10) with the boundary conditions (11) are obtained using the shooting method. The effects of the wall mass suction s ( > 0), the unsteadiness A and the solid volume fraction u parameters are analyzed for a viscous fluid (regular fluid u = 0), three different nanoparticles: Cu–water, Al2O3–water and TiO2–water as working
u=0 m=0
m=1
(1.6551, 0.1723) (1.6551, 12.0926) (3.8605,0.9159) (3.8605, 11.8377) (8.3490,3.0701) (8.3490, 11.0191)
(1.6551, 0.1723) (1.6551, 11.5257) (3.8605,0.9159) (3.8605, 11.2190) (8.3490,3.0701) (8.3490, 10.2887)
Table 3 Values of turning points Ac for f 00 ð0Þ at different nanoparticles with various values of u when s = 2.2, m = 0 and m = 1.
u 0 0.1 0.2
m=0
m=1
Cu
Al2O3
TiO2
Cu
Al2O3
TiO2
8.3490 <13.0000 <13.0000
8.3490 8.0117 1.7498
8.3490 11.5101 3.2739
8.3490 <13.0000 <13.0000
8.3490 8.0117 1.7498
8.3490 11.5101 3.2739
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fluids (u – 0) with Pr = 6.2 (water) and m = 0 (constant wall temperature) and m = 1 (linearly variation with x of the wall temperature) on the flow and heat transfer characteristics are analyzed. The values of u considered are 0.0, 0.1 and 0.2, while A takes negative values. The validation of the present results has been verified with the case of a regular fluid (u = 0), first studied by Fang et al. [25] in Fig. 1. The solution domain for Eqs. (9) and (10) with the boundary conditions (11) is illustrated in Figs. 1– 12. As in Fang et al. [25], we notice from these figures that there are regions of unique solution and regions of more than one solutions (dual solutions) with upper and lower branch solutions for each value of A(<0) under the same values of s and u. When A is equal to a certain critical value Ac (<0), there is only one solution, and when A is less than this critical value, there is no solution. Therefore, the solutions exist up to the critical values Ac beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations are not possible. With the increase of the wall mass suction parameter s, the solution domain expands with the critical values Ac moving to the left, as can be seen from Table 2. Meanwhile, Table 3 shows the influence of nanoparticle volume fraction parameter u on the solution domain for Cu, Al2O3 and TiO2, respectively. Both the wall skin friction or wall shear stress, f 00 ð0Þ and heat flux from the shrinking surface, h0 (0) also changes with the variation of s, A and u. Tables 4 and 5 present the upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.1 and 0.2, and various values of A with s = 2.1 and 2.2, Pr = 6.2, m = 0 and m = 1. The different values of m i.e m = 0 and m = 1 will only affect the heat transfer rate and not the skin friction coefficient as can be observed in Tables 2 to 4. It is seen from Tables 4 and 5, as well as from Figs. 1–12, that for certain values of s the upper branch solutions of both f 00 ð0Þ and h0 (0) generally decrease with the increase of the magnitude of A. We notice that at a certain value of A, f 00 ð0Þ becomes zero and continues decreasing to be negative. This implies that there is velocity overshoot near the shrinking sheet with a higher velocity in the fluid than the wall velocity. However, for a small value of s, f 00 ð0Þ can be positive for both solution branches. For the lower solution branch, generally f 00 ð0Þ decreases with the decrease of A. However, when A is close to the
Fig. 13. Velocity profiles for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2 and A = 1.
solution border point, it is possible for f 00 ð0Þ to slightly increase with decreasing A. Further, it should be mentioned that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and not physically realizable. The procedure for showing this has been described by Merkin [44], Weidman et al. [45] and Harris et al. [46], so that we will not repeat it here. Velocity f0 (g) and temperature h(g) profiles for Cu–water working fluid are shown in Figs. 13 and 14 for some values of u (0:0 6 u 6 0:2) when s = 2.2, A = 1, Pr = 6.2 and m = 0 (constant wall temperature or isothermal case), while the graphs of the velocity and temperature profiles for different nanofluids Cu, Al2O3 and TiO2 are plotted in Figs. 15 and 16 for u = 0.2, s = 3, m = 0 (constant wall temperature or isothermal case) and A = 1. Both the upper and lower branch solutions are presented here. It is seen from Figs. 13 and 14 that the boundary layer thickness of the velocity
Table 4 Upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.2 for various values of A with s = 2.1, Pr = 6.2, m = 0 and m = 1. A
0.0 0.2 0.4 0.6 1.0 3.0 5.0 9.0
m=0
m=1
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
2.5290 2.4621 2.3953 2.3283 2.1942 1.5212 0.8444 0.5173
0.5847 0.0481 0.4735 0.8408 1.4913 4.1448 6.4315 10.5898
6.8225 6.7672 6.7120 6.6567 6.5462 5.9927 5.4382 4.3268
6.6931 6.5915 6.5044 6.4211 6.2598 5.4883 4.7367 3.2556
2.5290 2.4621 2.3953 2.3283 2.1942 1.5212 0.8444 0.5173
0.5847 0.0481 0.4735 0.8408 1.4913 4.1448 6.4315 10.5898
6.4200 6.3588 6.2977 6.2365 6.1140 5.5001 4.8843 3.6483
6.0782 5.8680 5.7032 5.5513 5.2660 3.9769 2.7750 0.4744
Table 5 Upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.1 and 0.2 for various values of A with s = 2.2, m = 1 and Pr = 6.2. A
0.0 0.2 0.4 1.0 2.0 5.0 10.0 13.0
u = 0.1
u = 0.2
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
2.4795 2.4164 2.3532 2.1635 1.8464 0.8904 0.7148 1.6829
0.5564 0.0890 0.5140 1.5268 2.9187 6.4257 11.5264 14.3900
9.2724 9.2172 9.1620 8.9963 8.7198 7.8885 6.4987 5.6627
9.0621 8.9285 8.8204 8.5247 8.0621 6.7378 4.5995 3.3350
2.7182 2.6572 2.5962 2.4130 2.1071 1.1862 0.3564 1.2853
0.5455 0.2570 0.7533 1.9189 3.5073 7.4899 13.2696 16.5131
6.8117 6.7552 6.6987 6.5290 6.2459 5.3951 3.9727 3.1175
6.4891 6.2691 6.1044 5.6683 5.0058 3.1662 0.2701 1.4211
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Fig. 14. Temperature profiles for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2, m = 0 and A = 1.
Fig. 16. Temperature profiles of different nanofluids for u = 0.2, s = 3, m = 0 and A = 1.
nanoparticles: Cu, Al2O3 and TiO2 is numerically studied in this paper. The ordinary differential equations governing the flow and heat transfer of the problem have been solved using shooting method. Qualitative (graphical) comparison has been made with the existing results in literature and it is found to be in good agreement. We also investigate the effects of the wall mass suction s(>0), the unsteadiness A(<0) and the solid volume fraction u parameters for the three different nanoparticles with Pr = 6.2 (water) and m = 0 (constant wall temperature) and m = 1 (linearly variation with x of the wall temperature) on the flow and heat transfer. We found that the flow and heat transfer are significantly influenced by these parameters. Acknowledgements
Fig. 15. Velocity profiles of different nanofluids for u = 0.2, s = 3 and A = 1.
boundary layer decreases, while boundary layer thickness of the temperature boundary layer increases with the volume fraction parameter u. It is also seen that the lower solution branch for velocity profiles as well as for temperature profiles exhibit a larger boundary layer thickness compared with the upper solution branches. But, the velocity boundary layer thickness is smaller for Cu compared with Al2O3 and TiO2, while the reverse is true for the thermal boundary layer, as can be seen from Figs. 15 and 16. Referring to Figs. 13 and 15, it is seen that the velocity profiles are influenced by both volume fraction parameter u and also type of nanoparticles. However, by looking at Figs. 14 and 16, the temperature profiles are sensitive to volume fraction parameter and not significantly sensitive to the type of nanoparticles. Thus, it would be interesting to compare these results with ones where other models for the effective viscosity of the nanofluid, lnf and for the effective thermal diffusivity of the nanofluid, anf are used [11,42,43]. 4. Conclusions The problem of unsteady shrinking surface with wall mass suction in a water based nanofluid containing different type of
The authors gratefully acknowledge the financial support received in the form of research grants: RU Grant (1001/PMATHS/ 811166) from Universiti Sains Malaysia and FRGS (203/PMATHS/ 6711234) from the Ministry of Higher Education, Malaysia. The authors also would like to thank the reviewers for their helpful comments and suggestions. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Division 231 (1995) 99–105. [2] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720. [3] H.Q. Xie, H. Lee, W. Youn, M. Choi, Nanofluids containing multiwalled carbon nanotubes and their enhanced thermal conductivities, J. Appl. Phys. 94 (2003) 4967–4971. [4] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Comm. Heat Mass Transfer 33 (2006) 727–736. [5] S. Jana, A. Salehi-Khojin, W.H. Zhong, Enhancement of fluid thermal conductivity by the addition of single and hybrid nano-additives, Thermochim. Acta 462 (2007) 45–55. [6] K.S. Hwang, J.H. Lee, S.P. Jang, Buoyancy-driven heat transfer of water-based Al nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (2007) 4003– 4010. [7] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley-Interscience, New Jersey, 2007. [8] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. [9] E. Abu-Nada, H.F. Oztop, Effect of inclination angle on natural convection in enclosures filled with Cu-water nanofluid, Int. J. Heat Fluid Flow 30 (2009) 669–678.
A.M. Rohni et al. / International Journal of Heat and Mass Transfer 55 (2012) 1888–1895 [10] 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. [11] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [12] L. Godson, B. Raja, D.M. Lal, S. Wongwises, Enhancement of heat transfer using nanofluids – An overview, Renew. Sustain. Energy Rev. 14 (2010) 629–641. [13] 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. [14] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [15] W. Duangthongsuk, S. Wongwises, Effect of thermophysical properties models on the predicting of the convective heat transfer coefficient for low concentration nanofluid, Int. Comm. Heat Mass Transfer 35 (2008) 1320–1326. [16] V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renew. Sustain. Energy Rev. 11 (2007) 512–523. [17] X.Q. Wang, A.S. Mujumdar, A review on nanofluids – Part I: Theoretical and numerical investigations, Brazilian J. Chem. Eng. 25 (2008) 613–630. [18] X.Q. Wang, A.S. Mujumdar, A review on nanofluids – Part II: experiments and applications, Brazilian J. Chem. Eng. 25 (2008) 631–648. [19] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [20] 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. [21] 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. [22] 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. [23] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copper–water nanofluids in a lid-driven enclosure, Comm. Nonlinear Sci. Numer. Simulat. 15 (2010) 1501–1510. [24] B. Ghasemi, S.M. Aminossadati, Periodic natural convection in a nanofluidfilled enclosure with oscillating heat flux, Int. J. Therm. Sci. 49 (2010) 1–9. [25] T.G. Fang, J. Zhang J. SS Yao, Viscous flow over an unteady shrinking sheet with mass transfer, Chin, Phys. Lett. 26 (29) pp. 1–473. [26] M. Miklavcˇicˇ, C.Y. Wang, Viscous flow due to a shrinking sheet, Quart. Appl. Math. 64 (2006) 283–290. [27] S. Goldstein, On backward boundary layers and flow in converging passage, J. Fluid Mech. 21 (1965) 33–45. [28] T. Fang, W. Liang, C.F. Lee, A new solution branch for the Blasius equation – a shrinking sheet problem, Comput. Math. Appl. 56 (2008) 3088–3095.
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[29] T. Fang, Boundary layer flow over a shrinking sheet with power-law velocity, Int. J. Heat Mass Transfer 51 (2008) 5838–5843. [30] T. Fang, J. Zhang, Thermal boundary layers over a shrinking sheet: an analytical solution, Acta Mech. 209 (2010) 325–343. [31] T. Hayat, Z. Abbas, M. Sajid, On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, ASME J. Appl. Mech. 74 (2007) 1165–1171. [32] M. Sajid, T. Hayat, T. Javed, MHD rotating flow of a viscous fluid over a shrinking surface, Nonlinear Dyn. 51 (2008) 259–265. [33] L. Zheng, L. Wang, X. Zhang, Analytic solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/ sink, Comm. Nonlinear Sci. Numer. Simulat. 16 (2011) 731–740. [34] C.Y. Wang, Stagnation flow towards a shrinking sheet, Int. J. Non Linear Mech. 43 (2008) 377–382. [35] 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. [36] 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. [37] W. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [38] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Comm. Heat Mass Transfer, in press (2010). [39] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [40] C.V. Popa, S. Fohanno, C.T. Nguyen, G. Polidori, On heat transfer in external natural convection flows using two nanofluids, Int. J. Therm. Sci. 49 (2010) 901–908. [41] H.A. Mintsa, G. Roy, C.T. Nguyen, D. Doucet, New temperature dependent thermal conductivity data for water-based nanofluids, Int. J. Therm. Sci. 48 (2009) 363–371. [42] H.E. Patel, T. Sundararajan, T. Pradeep, A. Dasgupta, N. Dasgupta, S.K. Das, A microconvection model for thermal conductivity of nanofluids, Pramana J. Phys. 65 (2005) 863–869. [43] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuOwater nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transfer 132 (2010) 052401-1–052401-9. [44] J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Engng. Math. 20 (1985) 171–179. [45] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on selfsimilar boundary layer flow over moving surface, Int. J. Eng. Sci. 44 (2006) 730–737. [46] S.D. Harris, D.B. Ingham, I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transport Porous Media 77 (2009) 267–285.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids Azizah Mohd Rohni a, Syakila Ahmad b,⇑, Ioan Pop c a
UUM College of Arts & Sciences, Physical Science Division, Building of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia c Faculty of Mathematics, University of Cluj, R-400082 Cluj, CP 253, Romania b
a r t i c l e
i n f o
Article history: Received 9 November 2010 Received in revised form 25 October 2011 Accepted 29 October 2011 Available online 16 December 2011 Keywords: Unsteady Shrinking sheet Suction Nanofluids Numerical results
a b s t r a c t The unsteady flow over a continuously shrinking surface with wall mass suction in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3 and TiO2 is numerically studied. Similarity equations are obtained through the application of similarity transformation techniques. The shooting method is used to solve the similarity equations for different values of the wall mass suction, the unsteadiness and the nanoparticle volume fraction parameters. The results of skin friction coefficient f 00 ð0Þ and heat transfer rate h0 (0) are presented in tables and graphs. It is found that dual solution exists for a certain range of wall mass suction s, volume fraction u and unsteadiness parameters A. The results of velocity and temperature profiles are also presented. It is seen that two values of boundary layer thickness g1 are obtained, which gives two different velocity and temperature profiles that satisfy the boundary conditions. It is also found that the nanoparticle volume fraction parameter u and types of nanofluid play an important role to significantly determine the flow behaviour. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In most forced convection boundary layer flow and heat transfer problem, the base fluid has a low thermal conductivity, which limits the heat transfer enhancement. However, the continuing miniaturization of electronic devices requires further heat transfer improvements from an energy saving viewpoint. An innovative technique, which uses a mixture of nanoparticles and the base fluid was first introduced by Choi [1] in order to develop advanced heat transfer fluids with substantially higher conductivities. The resulting mixture of the base fluid and nanoparticles having unique physical and chemical properties is referred to as a nanofluid. It is expected that the presence of the nanoparticles in the nanofluid increases the thermal conductivity and therefore substantially enhances the heat transfer characteristics of the nanofluid. Eastman et al. [2], Xie et al. [3], Jou and Tzeng [4] and Jana et al. [5] showed that higher thermal conductivity can be achieved in thermal systems utilising nanofluids. Hwang et al. [6] measured thermal conductivities of various nanofluids and showed that the thermal conductivity enhancement of nanofluids depended on the volume fraction of the suspended particles and the thermal conductivities of the particles and base fluids. Nanofluids can be defined as the dilution of nanometer-sized particles (smaller than 100 nm) in a fluid [7], and nanofluids can be produced by dispersed evenly ⇑ Corresponding author. Tel.: +60 4 653 4782; fax: +60 4 657 0910. E-mail addresses: [email protected] (A.M. Rohni), syakila.ahmad@ymail. com (S. Ahmad), [email protected] (I. Pop). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.11.042
nanoparticles in a base fluid, such as water, ethylene glycol and oil [8]. The nanofluids usually contain the nanoparticles such as metals, oxides, carbides, or carbon nanotubes, whereby these nanoparticles have unique chemical and physical properties [9]. Since the size of nanoparticles are in nanometer-sized, besides behaving similar as liquid molecules, they have the ability to flow smoothly through the microchannels easily [10], hence, nanofluids will enhance the thermal conductivity and convective heat transfer coefficient compared to the base fluid only [11]. Thus, nanofluids are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. According to Godson et al. [12], one of the main objectives of using nanofluids is to achieve the best thermal properties with the least possible (<1%) volume fraction of nanoparticles in the base fluid. There are many studies on the mechanism behind the enhanced heat transfer characteristics using nanofluids. The collection of papers on this topic is included in the book by Das et al. [7], and in the review papers by Kakaç and Pramuanjaroenkij [11], Maiga et al. [13], Buongiorno [14], Daungthongsuk and Wongwises [15], Trisaksri and Wongwises [16], and Wang and Mujumdar [8,17,18]. In this paper, the unsteady viscous flow over a continuously shrinking surface with wall mass suction in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3, and TiO2 is numerically studied. We use nanofluid equations model proposed by Tiwari and Das [19], as this model successfully applied in several papers [9,20–24]. Similarity equations are obtained through the application of a similarity transformation technique and the corresponding equations are solved numerically using
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Nomenclature Cf Cp (qCp)nf k knf Nu A p Rex T T1 s u, v uw
vw
x, y
skin friction coefficient specific heat at constant temperature heat capacitance of the nanofluid thermal conductivity thermal conductivity of the nanofluid Nusselt number unsteadiness parameter fluid pressure local Reynolds number fluid temperature fluid temperature of the ambient fluid wall mass transfer parameter velocity component along x and y directions velocity of the shrinking sheet velocity of the mass transfer Cartesian coordinates along the plate and normal to it, respectively
shooting method for different values of the wall mass suction, the unsteadiness and the nanoparticle volume fraction parameters in the based fluid of water with the Prandtl number of 6.2 [9]. Results show that multiple solutions exist for a certain range of wall mass suction and nanoparticle volume fraction parameters. For comparison purposes, the present results for a regular Newtonian fluid are computed, and they show excellent agreement with those obtained by Fang et al. [25]. In fact, to the present knowledge of the authors, no studies have been reported in the literature, which investigate the flow over a shrinking sheet in nanofluids. It is, however, important to point out that Miklavcˇicˇ and Wang [26] investigated the steady flow over a shrinking sheet, which is an exact solution of the Navier–Stokes equations. This new type of shrinking sheet flow is essentially a backward flow as discussed by Goldstein [27]. It was shown that wall mass suction is required to maintain the flow over a shrinking sheet. The flow induced by a shrinking sheet with constant velocity or power-law velocity distribution has been investigated recently by Fang et al. [28], Fang [29], and Fang and Zhang [30]. The shrinking sheet problem was also extended to other fluids by Hayat et al. [31] and Sajid et al. [32]. We mention also the recent paper by Zheng et al. [33] on the unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink. The work on the flow over a shrinking sheet was further generalized by Wang [34] to include a stagnation flow as the free stream. This problem may be encountered in a number of electronic cooling devices equipped with nanofluids. On the other hand, we mention that several papers have been recently published on steady boundary layer flow in nanofluids by Nield and Kuznetsov [35], Kuznetsov and Nield [36], Khan and Pop [37], and Ahmad and Pop [38]. However it is worth mentioning at this end that the study of nanofluids is still at its early stage, so that it seems it is very difficult to have a precise idea on the way the use of nanoparticles acts in convective heat transfer flows and complementary works are necessary to understand the heat transfer characteristics of nanofluids and identify new and unique applications for these fields [23].
2. Basic equations We consider the two-dimensional flow over a continuously unsteady shrinking sheet with mass transfer in a water based nanofluid containing different type of nanoparticles: Cu, Al2O3 and
Greek letters effective thermal diffusivity of the nanofluid nanoparticle volume fraction similarity variable skin friction at the shrinking sheet effective dynamic viscosity of the nanofluid h dimensionless temperature q density qnf effective density of the nanofluid w stream function
anf u g sw lnf
Subscripts f fluid fraction s solid fraction nf nanofluid fraction Superscript 0 differentiation with respect to g
TiO2. We assume that the velocity of the shrinking sheet is uw(x, t) and the velocity of the mass transfer is vw(x, t), where x is the coordinate measured along the shrinking sheet and t is the time. It is also assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Assuming that the nanofluid is incompressible and laminar, and using the nanofluid model as proposed by Tiwari and Das [19], the governing equations of this problem, namely the two dimensional Navier–Stokes equations, are
@u @ v þ ¼0 @x @y
ð1Þ
@u @u @u 1 @p lnf þu þv ¼ þ @t @x @y qnf @x qnf
! @2u @2u þ @x2 @y2
@v @v @v 1 @p lnf þ þu þv ¼ @t @x @y qnf @y qnf
@2v @2v þ @x2 @y2
@T @T @T @2T @2T þu þv ¼ anf þ @t @x @y @x2 @y2
ð2Þ
! ð3Þ
! ð4Þ
where u and v are the velocity component in the x and y directions, respectively, p is the fluid pressure, T is the temperature of the nanofluid, lnf is the effective viscosity of the nanofluid and qnf is the effective density of the nanofluid and anf is the effective thermal diffusivity of the nanofluid, which are given by
anf ¼
knf ; ðqC p Þnf
qnf ¼ ð1 uÞqf þ uqs ; lnf ¼
lf ; ð1 uÞ2:5
ðqC p Þnf ¼ ð1 uÞðqC p Þf þ uðqC p Þs ; knf ðks þ 2kf Þ 2uðkf ks Þ ¼ kf ðks þ 2kf Þ þ uðkf ks Þ
ð5Þ
where u is the solid volume fraction of the nanofluid, qf is the density of the base fluid, qs is the density of the nanoparticle, lf is the dynamic viscosity of the base fluid, kf is the thermal conductivity of the base fluid and ks is the thermal conductivity of the solid nanoparticle. Density properties of the fluid and nanoparticles are given in Table 1 [21]. The viscosity lnf of the nanofluid can be approximated as viscosity of a base fluid lf containing dilute suspension of the spherical particles and is given by Brinkman [39]. Also, the
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3
Table 1 Thermophysical properties of fluid and nanoparticles [21].
2
Physical properties
Fluid phase (water)
Cu
Al2O3
TiO2
Cp (J/kg K) q (kg/m3) k (W/mK)
4179 997.1 0.613
385 8933 400
765 3970 40
686.2 4250 8.9538
t < 0 : u ¼ v ¼ 0;
T T1 hðgÞ ¼ ; Tw T1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c g¼ y mf ð1 atÞ
g 00 f 000 þ ff f 02 A f 0 þ f 00 ¼ 0 2 ð1 uÞ ½1 u þ uðqs =qf Þ 1
f " (0)
-5 -6 -7 -9
-8
f ðgÞ ! 0;
f ð0Þ ¼ 1;
hð0Þ ¼ 1
hðgÞ ! 0 as g ! 1
sw qw ; Nu ¼ qf u2w kf ðT w T 1 Þ
-5
-4
-3
-2
-1
0
• Ac = - 1.6551
-2 -4 -6
ð7Þ -8 -10 -12 -9
-8
-7
-6
-5
-4
-3
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-2
0
Fig. 2. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.1.
2
ð9Þ
Ac = - 1.7498
1
•
f " (0)
0
-1
Ac = - 8.3490 Ac = - 8.0117
-2
ð11Þ
ð12Þ
-1
A
ð8Þ
where A = a/c is the unsteadiness parameter. For the present situation, we assume a decelerating shrinking sheet with A 6 0 [25]. Quantities of interest in this problem are the skin friction coefficient Cf and the Nusselt number Nu, which are defined as
Cf ¼
-6
0
subject to the boundary conditions 0
-7
2.1 2.15 2.2 2.5 3.0
2
ð10Þ
f ð0Þ ¼ s;
= = = = =
A
knf =kf 1 g h i h00 þ f h0 mf 0 h A h þ h0 ¼ 0 Pr 1 u þ uðqC p Þ =ðqC p Þ 2 s f
0
s s s s s
-4
4
where s is the constant wall mass transfer parameter with s > 0 for suction and s < 0 for injection. Substituting (7) into Eqs. (2)–(4), we obtain the following ordinary differential equations 2:5
•
ð6Þ
where primes denote differentiation with respect to g and mf is the kinematic viscosity of the base fluid. Then, the wall mass transfer velocity becomes
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi mf c v w ð0; tÞ ¼ f ð0Þ ¼ s 1 at
Ac = - 1.6551
Fig. 1. Variation of f 00 ð0Þ with the parameter A for some values of s when u = 0.0.
where b, c and m are positive constants and a is a parameter showing the unsteadiness of the problem. It should be mentioned that the case m = 0 corresponds to a constant wall temperature and m = 1 to a linearly variation with x of the wall temperature, respectively. We look for a similarity solution of Eqs. (1)–(4) of the following form
cx u¼ f 0 ðgÞ; 1 at
•
Ac = - 8.3490
-2 -3
f " (0)
T ¼ T 1 for all x; y cx t P 0 : u ¼ uw ðx; tÞ ¼ ; v ¼ v w ðx; tÞ; 1 at m bx at y ¼ 0 T ¼ T w ðx; tÞ ¼ T 1 þ 1 at u ! 0; T ! T 1 as y ! 1
•
0 -1
effective thermal conductivity of the nanofluid knf is approximated by the Maxwell–Garnett’s model, which is found to be appropriate for studying heat transfer enhancement using nanofluids [13,20]. Very recently, Popa et al. [40] were making a comparison between the Maxwell model and the experimental data provided by Mintsa et al. [41] for thermal conductivity. Popa et al. [40] find that Maxwell’s model strongly overestimates the thermal conductivity of the nanofluid. Other several models for knf can be found in Kakaç and Pramunjaroenkij [11], Patel et al. [42] and Abu-Nada [43]. Eqs. (1)–(4) are subjected to the following initial and boundary conditions
Ac = - 3.8605
1
-3
-4 -9
•
•
-8
-7
-6
-5
-4
-3
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-2
-1
0
A Fig. 3. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Al2O3 when s = 2.2.
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2
4 2
1
Ac = - 3.2739
0 -2
Ac = - 8.3490
-6
•
-4 -5 -12
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-10
-8
-6 A
-4
-2
Cu
-12
Al 2O3
-14
TiO2
-16
• -10
Ac = - 1.7498
-8
-2 Ac = - 11.5101
Ac = - 3.2739
-4
•
-1
f " (0)
f " (0)
0
-3
•
•
-18
0
-12
-10
-8
-6
-4
-2
0
A
Fig. 4. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for TiO2 when s = 2.2.
Fig. 7. Variation of f 00 ð0Þ with unsteadiness parameter A for different nanofluids when u = 0.2 and s = 2.2.
5
14
0
-5
ϕ = 0.0 ϕ = 0.1
10
Ac = - 8.3490
8
-15
-20
-12
-10
-8
-6
ϕ
= 0.0
ϕ
= 0.1
ϕ
= 0.2
-4
-2
-θ'(0)
-10
6
ϕ = 0.2
4 2 0
0
A
-2
-12
Fig. 5. Variation of f 00 ð0Þ with A for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2.
11.1
2
-6
-4
-2
0
•
10.9
-2
•
-4
-θ'(0)
Ac = - 8.0117
-8
10.7 10.6
Ac = - 11.5101
-12 -14
Cu
10.5
Al2O3
10.4 10.3
TiO2
-16
10.2 -8.5
-12
-10
Ac = - 8.3490
10.8
•
-6
-18
-8
(b)
11
0
-10
-10
A
4
f " (0)
m=0 m=1
12
• f " (0)
(a)
-8
-6
-4
-2
0
m=0 m=1
• -8
-7.5
-7
A
A Fig. 6. Variation of f 00 ð0Þ with unsteadiness parameter A for different nanofluids when u = 0.1 and s = 2.2.
Fig. 8. (a) Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for Cu–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2. (b) Enlargement of the area in circle in (a).
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14 m=0 m=1
13
7.4
12
• •
10
Cu Al 2O3 TiO2
7.2
Ac = - 8.3490
9
ϕ = 0.1
-θ'(0)
11
-θ'(0)
m=0
7.6
ϕ = 0.0
7
Ac = - 8.0117
8
6.8
•
7
•
• •
Ac = - 1.7498
6 5 -9
•
ϕ = 0.2
-8
-7
-6
-5
-4
-3
Ac = -3.7239
6.6
• Ac = -1.7498
-2
-1
6.4 -3.5
0
-3
-2.5
-2
Fig. 9. Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for Al2O3–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2.
-1.5
-1
-0.5
0
A
A
Fig. 11. Variation of h0 (0) with A when u = 0.2 for different nano fluid with m = 0 when s = 2.2 and Pr = 6.2.
14
7.5 m=0 m=1
13
m=1
ϕ = 0.0
7
12
•
10
Ac = - 8.3490
Ac = -3.7239
•
6
ϕ = 0.1
•
-θ'(0)
-θ'(0)
6.5
•
11
9
Ac = -1.7498
5.5
8
ϕ = 0.2
7
•
6
•
5 -12
•
Ac = - 3.7239
-8
-6 A
-4
Cu Al 2O3
TiO2
4.5
•
Ac = - 11.5101 -10
5
-2
4 -3.5
0
-3
-2.5
-2
-1.5
-1
-0.5
0
A
Fig. 10. Variation of h0 (0) with A for some values of u(0 6 u 6 0.2) for TiO2–water working nano fluid with m = 0 and m = 1 when s = 2.2 and Pr = 6.2.
Fig. 12. Variation of h0 (0) with A when u = 0.2 for different nano fluid with m = 1 when s = 2.2 and Pr = 6.2.
where sw is the skin friction at the shrinking sheet and qw is the heat flux from the shrinking sheet, which are given by
Table 2 Coordinates of turning points. Ac for u ¼ 0 (pure fluid) and different values of s when m ¼ 0 and m ¼ 1 with Pr ¼ 6:2.
sw ¼ lnf
@u @y y¼0
qw ¼ knf
@T @y y¼0
ð13Þ
Substituting (7) into (13) and using (12), we get
Re1=2 x Cf ¼
1 ð1 uÞ2:5
f 00 ð0Þ;
knf 0 Re1=2 Nu ¼ h ð0Þ x kf
ð14Þ
s
Turning points
2.10
(Ac ; f 00 ð0Þ) (Ac, h0 (0)) (Ac ; f 00 ð0Þ) (Ac, h0 (0)) (Ac ; f 00 ð0Þ) (Ac, h0 (0))
2.15 2.20
where Rex = uwx/mf is the local Reynolds number. 3. Results and discussion Numerical solutions of the governing ordinary differential Eqs. (9) and (10) with the boundary conditions (11) are obtained using the shooting method. The effects of the wall mass suction s ( > 0), the unsteadiness A and the solid volume fraction u parameters are analyzed for a viscous fluid (regular fluid u = 0), three different nanoparticles: Cu–water, Al2O3–water and TiO2–water as working
u=0 m=0
m=1
(1.6551, 0.1723) (1.6551, 12.0926) (3.8605,0.9159) (3.8605, 11.8377) (8.3490,3.0701) (8.3490, 11.0191)
(1.6551, 0.1723) (1.6551, 11.5257) (3.8605,0.9159) (3.8605, 11.2190) (8.3490,3.0701) (8.3490, 10.2887)
Table 3 Values of turning points Ac for f 00 ð0Þ at different nanoparticles with various values of u when s = 2.2, m = 0 and m = 1.
u 0 0.1 0.2
m=0
m=1
Cu
Al2O3
TiO2
Cu
Al2O3
TiO2
8.3490 <13.0000 <13.0000
8.3490 8.0117 1.7498
8.3490 11.5101 3.2739
8.3490 <13.0000 <13.0000
8.3490 8.0117 1.7498
8.3490 11.5101 3.2739
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fluids (u – 0) with Pr = 6.2 (water) and m = 0 (constant wall temperature) and m = 1 (linearly variation with x of the wall temperature) on the flow and heat transfer characteristics are analyzed. The values of u considered are 0.0, 0.1 and 0.2, while A takes negative values. The validation of the present results has been verified with the case of a regular fluid (u = 0), first studied by Fang et al. [25] in Fig. 1. The solution domain for Eqs. (9) and (10) with the boundary conditions (11) is illustrated in Figs. 1– 12. As in Fang et al. [25], we notice from these figures that there are regions of unique solution and regions of more than one solutions (dual solutions) with upper and lower branch solutions for each value of A(<0) under the same values of s and u. When A is equal to a certain critical value Ac (<0), there is only one solution, and when A is less than this critical value, there is no solution. Therefore, the solutions exist up to the critical values Ac beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations are not possible. With the increase of the wall mass suction parameter s, the solution domain expands with the critical values Ac moving to the left, as can be seen from Table 2. Meanwhile, Table 3 shows the influence of nanoparticle volume fraction parameter u on the solution domain for Cu, Al2O3 and TiO2, respectively. Both the wall skin friction or wall shear stress, f 00 ð0Þ and heat flux from the shrinking surface, h0 (0) also changes with the variation of s, A and u. Tables 4 and 5 present the upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.1 and 0.2, and various values of A with s = 2.1 and 2.2, Pr = 6.2, m = 0 and m = 1. The different values of m i.e m = 0 and m = 1 will only affect the heat transfer rate and not the skin friction coefficient as can be observed in Tables 2 to 4. It is seen from Tables 4 and 5, as well as from Figs. 1–12, that for certain values of s the upper branch solutions of both f 00 ð0Þ and h0 (0) generally decrease with the increase of the magnitude of A. We notice that at a certain value of A, f 00 ð0Þ becomes zero and continues decreasing to be negative. This implies that there is velocity overshoot near the shrinking sheet with a higher velocity in the fluid than the wall velocity. However, for a small value of s, f 00 ð0Þ can be positive for both solution branches. For the lower solution branch, generally f 00 ð0Þ decreases with the decrease of A. However, when A is close to the
Fig. 13. Velocity profiles for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2 and A = 1.
solution border point, it is possible for f 00 ð0Þ to slightly increase with decreasing A. Further, it should be mentioned that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and not physically realizable. The procedure for showing this has been described by Merkin [44], Weidman et al. [45] and Harris et al. [46], so that we will not repeat it here. Velocity f0 (g) and temperature h(g) profiles for Cu–water working fluid are shown in Figs. 13 and 14 for some values of u (0:0 6 u 6 0:2) when s = 2.2, A = 1, Pr = 6.2 and m = 0 (constant wall temperature or isothermal case), while the graphs of the velocity and temperature profiles for different nanofluids Cu, Al2O3 and TiO2 are plotted in Figs. 15 and 16 for u = 0.2, s = 3, m = 0 (constant wall temperature or isothermal case) and A = 1. Both the upper and lower branch solutions are presented here. It is seen from Figs. 13 and 14 that the boundary layer thickness of the velocity
Table 4 Upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.2 for various values of A with s = 2.1, Pr = 6.2, m = 0 and m = 1. A
0.0 0.2 0.4 0.6 1.0 3.0 5.0 9.0
m=0
m=1
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
2.5290 2.4621 2.3953 2.3283 2.1942 1.5212 0.8444 0.5173
0.5847 0.0481 0.4735 0.8408 1.4913 4.1448 6.4315 10.5898
6.8225 6.7672 6.7120 6.6567 6.5462 5.9927 5.4382 4.3268
6.6931 6.5915 6.5044 6.4211 6.2598 5.4883 4.7367 3.2556
2.5290 2.4621 2.3953 2.3283 2.1942 1.5212 0.8444 0.5173
0.5847 0.0481 0.4735 0.8408 1.4913 4.1448 6.4315 10.5898
6.4200 6.3588 6.2977 6.2365 6.1140 5.5001 4.8843 3.6483
6.0782 5.8680 5.7032 5.5513 5.2660 3.9769 2.7750 0.4744
Table 5 Upper and lower branch solutions of f 00 ð0Þ and h0 (0) for Cu–water nanofluid when u = 0.1 and 0.2 for various values of A with s = 2.2, m = 1 and Pr = 6.2. A
0.0 0.2 0.4 1.0 2.0 5.0 10.0 13.0
u = 0.1
u = 0.2
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
fu00 ð0Þ
fl00 ð0Þ
h0u ð0Þ
h0l ð0Þ
2.4795 2.4164 2.3532 2.1635 1.8464 0.8904 0.7148 1.6829
0.5564 0.0890 0.5140 1.5268 2.9187 6.4257 11.5264 14.3900
9.2724 9.2172 9.1620 8.9963 8.7198 7.8885 6.4987 5.6627
9.0621 8.9285 8.8204 8.5247 8.0621 6.7378 4.5995 3.3350
2.7182 2.6572 2.5962 2.4130 2.1071 1.1862 0.3564 1.2853
0.5455 0.2570 0.7533 1.9189 3.5073 7.4899 13.2696 16.5131
6.8117 6.7552 6.6987 6.5290 6.2459 5.3951 3.9727 3.1175
6.4891 6.2691 6.1044 5.6683 5.0058 3.1662 0.2701 1.4211
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Fig. 14. Temperature profiles for some values of u (0.0 6 u 6 0.2) for Cu when s = 2.2, m = 0 and A = 1.
Fig. 16. Temperature profiles of different nanofluids for u = 0.2, s = 3, m = 0 and A = 1.
nanoparticles: Cu, Al2O3 and TiO2 is numerically studied in this paper. The ordinary differential equations governing the flow and heat transfer of the problem have been solved using shooting method. Qualitative (graphical) comparison has been made with the existing results in literature and it is found to be in good agreement. We also investigate the effects of the wall mass suction s(>0), the unsteadiness A(<0) and the solid volume fraction u parameters for the three different nanoparticles with Pr = 6.2 (water) and m = 0 (constant wall temperature) and m = 1 (linearly variation with x of the wall temperature) on the flow and heat transfer. We found that the flow and heat transfer are significantly influenced by these parameters. Acknowledgements
Fig. 15. Velocity profiles of different nanofluids for u = 0.2, s = 3 and A = 1.
boundary layer decreases, while boundary layer thickness of the temperature boundary layer increases with the volume fraction parameter u. It is also seen that the lower solution branch for velocity profiles as well as for temperature profiles exhibit a larger boundary layer thickness compared with the upper solution branches. But, the velocity boundary layer thickness is smaller for Cu compared with Al2O3 and TiO2, while the reverse is true for the thermal boundary layer, as can be seen from Figs. 15 and 16. Referring to Figs. 13 and 15, it is seen that the velocity profiles are influenced by both volume fraction parameter u and also type of nanoparticles. However, by looking at Figs. 14 and 16, the temperature profiles are sensitive to volume fraction parameter and not significantly sensitive to the type of nanoparticles. Thus, it would be interesting to compare these results with ones where other models for the effective viscosity of the nanofluid, lnf and for the effective thermal diffusivity of the nanofluid, anf are used [11,42,43]. 4. Conclusions The problem of unsteady shrinking surface with wall mass suction in a water based nanofluid containing different type of
The authors gratefully acknowledge the financial support received in the form of research grants: RU Grant (1001/PMATHS/ 811166) from Universiti Sains Malaysia and FRGS (203/PMATHS/ 6711234) from the Ministry of Higher Education, Malaysia. The authors also would like to thank the reviewers for their helpful comments and suggestions. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME Fluids Eng. Division 231 (1995) 99–105. [2] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Appl. Phys. Lett. 78 (2001) 718–720. [3] H.Q. Xie, H. Lee, W. Youn, M. Choi, Nanofluids containing multiwalled carbon nanotubes and their enhanced thermal conductivities, J. Appl. Phys. 94 (2003) 4967–4971. [4] R.Y. Jou, S.C. Tzeng, Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, Int. Comm. Heat Mass Transfer 33 (2006) 727–736. [5] S. Jana, A. Salehi-Khojin, W.H. Zhong, Enhancement of fluid thermal conductivity by the addition of single and hybrid nano-additives, Thermochim. Acta 462 (2007) 45–55. [6] K.S. Hwang, J.H. Lee, S.P. Jang, Buoyancy-driven heat transfer of water-based Al nanofluids in a rectangular cavity, Int. J. Heat Mass Transfer 50 (2007) 4003– 4010. [7] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: Science and Technology, Wiley-Interscience, New Jersey, 2007. [8] X.Q. Wang, A.S. Mujumdar, Heat transfer characteristics of nanofluids: a review, Int. J. Therm. Sci. 46 (2007) 1–19. [9] E. Abu-Nada, H.F. Oztop, Effect of inclination angle on natural convection in enclosures filled with Cu-water nanofluid, Int. J. Heat Fluid Flow 30 (2009) 669–678.
A.M. Rohni et al. / International Journal of Heat and Mass Transfer 55 (2012) 1888–1895 [10] 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. [11] S. Kakaç, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [12] L. Godson, B. Raja, D.M. Lal, S. Wongwises, Enhancement of heat transfer using nanofluids – An overview, Renew. Sustain. Energy Rev. 14 (2010) 629–641. [13] 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. [14] J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transfer 128 (2006) 240–250. [15] W. Duangthongsuk, S. Wongwises, Effect of thermophysical properties models on the predicting of the convective heat transfer coefficient for low concentration nanofluid, Int. Comm. Heat Mass Transfer 35 (2008) 1320–1326. [16] V. Trisaksri, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renew. Sustain. Energy Rev. 11 (2007) 512–523. [17] X.Q. Wang, A.S. Mujumdar, A review on nanofluids – Part I: Theoretical and numerical investigations, Brazilian J. Chem. Eng. 25 (2008) 613–630. [18] X.Q. Wang, A.S. Mujumdar, A review on nanofluids – Part II: experiments and applications, Brazilian J. Chem. Eng. 25 (2008) 631–648. [19] R.K. Tiwari, M.K. Das, Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002–2018. [20] 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. [21] 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. [22] 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. [23] M. Muthtamilselvan, P. Kandaswamy, J. Lee, Heat transfer enhancement of copper–water nanofluids in a lid-driven enclosure, Comm. Nonlinear Sci. Numer. Simulat. 15 (2010) 1501–1510. [24] B. Ghasemi, S.M. Aminossadati, Periodic natural convection in a nanofluidfilled enclosure with oscillating heat flux, Int. J. Therm. Sci. 49 (2010) 1–9. [25] T.G. Fang, J. Zhang J. SS Yao, Viscous flow over an unteady shrinking sheet with mass transfer, Chin, Phys. Lett. 26 (29) pp. 1–473. [26] M. Miklavcˇicˇ, C.Y. Wang, Viscous flow due to a shrinking sheet, Quart. Appl. Math. 64 (2006) 283–290. [27] S. Goldstein, On backward boundary layers and flow in converging passage, J. Fluid Mech. 21 (1965) 33–45. [28] T. Fang, W. Liang, C.F. Lee, A new solution branch for the Blasius equation – a shrinking sheet problem, Comput. Math. Appl. 56 (2008) 3088–3095.
1895
[29] T. Fang, Boundary layer flow over a shrinking sheet with power-law velocity, Int. J. Heat Mass Transfer 51 (2008) 5838–5843. [30] T. Fang, J. Zhang, Thermal boundary layers over a shrinking sheet: an analytical solution, Acta Mech. 209 (2010) 325–343. [31] T. Hayat, Z. Abbas, M. Sajid, On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet, ASME J. Appl. Mech. 74 (2007) 1165–1171. [32] M. Sajid, T. Hayat, T. Javed, MHD rotating flow of a viscous fluid over a shrinking surface, Nonlinear Dyn. 51 (2008) 259–265. [33] L. Zheng, L. Wang, X. Zhang, Analytic solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/ sink, Comm. Nonlinear Sci. Numer. Simulat. 16 (2011) 731–740. [34] C.Y. Wang, Stagnation flow towards a shrinking sheet, Int. J. Non Linear Mech. 43 (2008) 377–382. [35] 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. [36] 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. [37] W. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [38] S. Ahmad, I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, Int. Comm. Heat Mass Transfer, in press (2010). [39] H.C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20 (1952) 571–581. [40] C.V. Popa, S. Fohanno, C.T. Nguyen, G. Polidori, On heat transfer in external natural convection flows using two nanofluids, Int. J. Therm. Sci. 49 (2010) 901–908. [41] H.A. Mintsa, G. Roy, C.T. Nguyen, D. Doucet, New temperature dependent thermal conductivity data for water-based nanofluids, Int. J. Therm. Sci. 48 (2009) 363–371. [42] H.E. Patel, T. Sundararajan, T. Pradeep, A. Dasgupta, N. Dasgupta, S.K. Das, A microconvection model for thermal conductivity of nanofluids, Pramana J. Phys. 65 (2005) 863–869. [43] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuOwater nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transfer 132 (2010) 052401-1–052401-9. [44] J.H. Merkin, On dual solutions occurring in mixed convection in a porous medium, J. Engng. Math. 20 (1985) 171–179. [45] P.D. Weidman, D.G. Kubitschek, A.M.J. Davis, The effect of transpiration on selfsimilar boundary layer flow over moving surface, Int. J. Eng. Sci. 44 (2006) 730–737. [46] S.D. Harris, D.B. Ingham, I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transport Porous Media 77 (2009) 267–285.