Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation

MOLLIQ-04334; No of Pages 5 Journal of Molecular Liquids xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

3Q1

M.M. Rashidi a,b,⁎, N. Vishnu Ganesh c, A.K. Abdul Hakeem c, B. Ganga d a

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a r t i c l e

9 10 11 12 13

Article history: Received 22 April 2014 Received in revised form 8 June 2014 Accepted 29 June 2014 Available online xxxx

14 15 16 17 18 19 31 20

Keywords: Buoyancy effect MHD Nanofluid Shooting technique Thermal radiation Stretching sheet

Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Iran Mechanical Engineering Department, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, People's Republic of China Department of Mathematics, Sri Ramakrishna Mission Vidhyalaya College of Arts and Science, Coimbatore 641 020, India d Department of Mathematics, Providence College for Women, Coonoor 643 104, India b

i n f o

a b s t r a c t

43 44 45 46 47 48 49 50 51 52 53

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R

Nanofluids or so-called smart fluids research have attracted considerable attention in recent years owing to their significant applications in engineering and technology. The broad range of current and future applications involving nanofluids have been given in the publications [1–4]. Nanofluids enhance thermal conductivity of the base fluid enormously, which are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and nonNewtonian behaviour, due to the tiny size of nano elements and the low volume fraction of nano elements required for conductivity enhancement. These suspended nanoparticles can change the transport and thermal properties of the base fluid. In view of these applications the problems of heat transfer in nanofluids discussed in the recent publications [5–7]. The boundary-layer flow induced by a stretching surface has been the focus of extensive attempts during the past few decades in view of its broad range of applications in the polymer extrusion, in a melt spinning processes, aerodynamic extrusion of plastic sheets, glass fibre

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41 42

1. Introduction

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21 22 23 24 25 26 27 28 29 30

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In this paper, heat transfer of a steady, incompressible water based nanofluid flow over a stretching sheet in the presence of transverse magnetic field with thermal radiation and buoyancy effects are investigated. A similarity transformation is used to reduce the governing momentum and energy equations into non-linear ordinary differential equations. The resulting differential equations with the appropriate boundary conditions are solved by shooting iteration technique together with fourth-order Runge–Kutta integration scheme. Analytical solutions are presented in terms of hypergeometric function for the special case of horizontal stretching sheet. The effects of the physical parameters on the flow and heat transfer characteristics are presented through graphs and analysed for Cu-metal and Cu-oxide nanoparticles. The values of skin friction coefficient and the reduced Nusselt number are calculated and presented through tables. Furthermore, the limiting cases are obtained and are found to be in good agreement with the previously published results. © 2014 Published by Elsevier B.V.

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Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation

1

⁎ Corresponding author at: Mechanical Engineering Department, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, People's Republic of China. E-mail addresses: [email protected], [email protected] (M.M. Rashidi).

production, the cooling and drying of paper and textiles, water pipes, sewer pipes, irrigation channels, blood vessels etc. The boundary-layer flow of nanofluid over a stretching sheet is a current attractive topic among the researchers. Khan and Pop [8] analysed the development of steady boundary-layer flow, heat transfer and nanoparticle volume fraction over a linear stretching surface in a nanofluid. Kuznetsov and Nield [9] studied the classical problem of free convection boundarylayer flow of a viscous and incompressible fluid past a vertical flat plate in the case of nanofluids. Rashidi et al. [10] investigated the nano boundary-layer over a stretching surface using modified differential transform method. Recently the same author [11] analysed the nanofluid flow from a non-linearly stretching isothermal sheet with transpiration. Sheikholeslami et al. [12] studied the nanofluid flow in a rotating system with permeable sheet numerically. The problem of nanofluid flow in an asymmetric porous channel with expanding or contracting wall analysed by Hatami et al. [13]. The study of magnetic field effects has important applications in physics, chemistry and engineering. Many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid and that in the process of drawing, these strips are sometimes stretched. Mention may be made of drawing, annealing, and thinning of copper wires. In all these cases, the properties of the final product depend to a great extent on the rate of cooling by drawing such strips in an electrically conducting fluid subject to a magnetic field and the characteristic desired in the final product. The effects of magnetic field on nanofluid convection in different geometries are

http://dx.doi.org/10.1016/j.molliq.2014.06.037 0167-7322/© 2014 Published by Elsevier B.V.

Please cite this article as: M.M. Rashidi, et al., Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.06.037

54 55 56 57 58 59 60 61 62 63 64 65 66 67 Q2 68 69 70 71 72 73 74 75 76 77 78 79

95 96 97 98

104 105 106 107 108 109 110 111 112 113 114 115

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93 94

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91 92

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89 90

"

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117 118

R

∂u ∂v þ ¼ 0; ∂x ∂y

2

ð1Þ

#

∂u ∂v 1 ∂ u 2 u þv ¼ μ þ g ðρβÞnf ðT−T ∞ Þ−σ B0 u ; ∂x ∂y ρnf nf ∂y2

C

ð2Þ

knf ∂2 T ∂T ∂T 1 ∂qr − ; þv ¼ ∂x ∂y ðρCpÞnf ∂y2 ðρCpÞnf ∂y

U

u

ð3Þ

124

where u, v are the velocity components along x and y-axes respectively, g is the acceleration due to gravity, σ is the electric conductivity, B0 is the constant magnetic field strength and qr is the radiative heat flux.

t1:1 t1:2

Table 1 Thermo-physical properties of water and nano particles [29].

t1:3 t1:4 t1:5 t1:6

Pure water Copper (Cu) Copper oxide(CuO)

3

4

131 132

ð4Þ

Therefore Eq. (3) is simplified to u

129 130

134

knf ∂2 T ∂T ∂T 16σ  T 3∞ ∂2 T þ  ; þv ¼ 2 3knf ðρCpÞnf ∂y2 ∂x ∂y ðρCpÞnf ∂y

ð5Þ 136

where ρnf is the effective density of the nanofluid, μnf is the effective dynamic viscosity of the nanofluid, (ρCp)nf is the heat capacitance, βnf 137 thermal expansion coefficient of the nanofluid and knf is the thermal 138 conductivity of the nanofluid are given as in Vishnu Ganesh et al. [25] 139 2:5

μ nf ¼ μ f =ð1−φÞ ; ρ ¼ ð1−ϕÞρ f þ ϕρs ;  nf      ¼ ð1−ϕÞ ρC p f þ ϕ ρC p s ; ðρβÞnf ¼ ð1−ϕÞðρβÞ f þ ϕðρβÞs ρC p nf    ks þ 2k f −2ϕ k f −ks   knf =k f ¼  ks þ 2k f þ 2ϕ k f −ks : 141

The boundary conditions of Eqs. (1)–(3) are

u ¼ uw ðxÞ ¼ a x u→0

v¼0

T ¼ T w at y ¼ 0 T→T ∞ as y→∞;

ð6Þ

where μf is the dynamic viscosity of the basic fluid, ρf is the density of the pure fluid, βf is the thermal expansion of base fluid, (ρCp)f is the specific heat parameter of the base fluid, kf is the thermal conductivity of the base fluid, ρs is the density of the nanoparticles, βs is the thermal expansion of the nanoparticles, (ρCp)s is the specific heat parameter of the nanoparticles, ks is the thermal conductivity of the nanoparticles and a is a constant. By using similarity transformations η¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi a=v f y;

0

u ¼ a x f ðηÞ;

143 144 145 146 147 148 149

pffiffiffiffiffiffiffiffi v ¼ − a v f f ðηÞ;

θðηÞ ¼ ðT−T ∞ Þ=ðT w −T ∞ Þ: 151

N

120 121

123

4

T ≅4T ∞ T−3T ∞ :

F

Consider the steady laminar two-dimensional, radiative flow of an incompressible viscous nanofluid along a semi-infinite vertical stretching sheet. We also consider the influence of a constant magnetic field of strength B0 which is applied normally to the sheet. The x-axis is taken along the stretching surface and the y-axis is normal to the flow direction. The temperature at the stretching surface takes the constant value Tw, while the ambient value, attained as y tends to infinity, takes the constant value T∞. The fluid is a water based nanofluid containing two different types of nanoparticles: Copper (Cu) and Copper oxide (CuO). It is assumed that the base fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermophysical properties of the nanofluid are considered as given in Table 1. Under the above assumptions, the governing equations for this problem can be written as

87 88

128



Here σ* is the Stefan–Boltzmann constant and k nf is the mean absorption coefficient of the nanofluid. Further we assume that the temperature difference within the flow is such that T4 may be expanded in a Taylor series. Hence, expanding T4 about T∞ and neglecting higher order terms we get

O

102 103

86

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2. Mathematical formulation

84 Q3 85

  ∂T 4   qr ¼ − 4σ =3k nf : ∂y

P

101

82 83

Using Rosseland approximation for radiation (see Abdul Hakeem 125 et al. [26]) we have 126

T

99 100

studied in [14–23]. Hamad [24] have done an analytical work on water based nanofluid flow over a stretching sheet with magnetic field effect. Recently, Vishnu Ganesh et al. [25] studied the natural convection MHD flow of water based nanofluid over a stretching sheet numerically. Radiative heat transfer flow is very important in manufacturing industries for the design of reliable equipment, nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles. In view of this Abdul Hakeem et al. [26] investigated the thermal radiation effects on Walter's liquid B fluid over a stretching sheet. Hady et al. [27] investigated the radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretched sheet numerically. Very recently Vishnu Ganesh et al. [28] investigated the radiation effects on a nanofluid flow over a horizontal stretching sheet for some metal nanofluids in the presence of magnetic field both numerically and analytically. Keeping this in mind, we investigated the MHD convective flow of water based nanofluid over a vertical stretching sheet with thermal radiation effect. The governing nonlinear partial differential equations along with the boundary conditions are first transformed into a dimensionless form and then the equations are solved numerically by the fourth-order Runge–Kutta method with the shooting technique.

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M.M. Rashidi et al. / Journal of Molecular Liquids xxx (2014) xxx–xxx

E

2

The fundamental equations of the boundary-layer (Eq. (2)) and (Eq. (5)) are transformed to ordinary differential ones that are locally 152 valid as follows: 153 h     ‴ 2:5 ″ 02 0 ð1−φÞ þ φ ρs =ρ f f f − f −Mnf þ λθ ð1−φÞ f þ ð1−φÞ  i þφ ðρβÞs =ðρβÞ f ¼ 0; ð7Þ

″

θ þ

ρ (kg/m3)

Cp (J/kg K)

k (W/m K)

β × 105 (K−1)

997.1 8933 6320

4179 385 531.8

0.613 401 76.5

21 1.67 1.80

   .   ρC p 3N Prk f 1−φ þ φ ρC p s

f

ð3N þ 4Þknf

155 156 0

f θ ¼ 0:

ð8Þ 158

The corresponding boundary conditions (Eq. 6) turn into f ¼0 0 f →0

0

f ¼1 θ→0

θ¼1 as

at

η¼0 η→∞;

ð9Þ

Please cite this article as: M.M. Rashidi, et al., Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.06.037

M.M. Rashidi et al. / Journal of Molecular Liquids xxx (2014) xxx–xxx

161 162 163

where λ = gβf(Tw − T∞)/a u is the buoyancy parameter, Mn = σB20/ρf a is the magnetic parameter, Pr = vf/αf is the Prandtl number and N = knfknf⁎/ 4σT3∞ is the radiation parameter. Another important characteristics of the present investigation are the skin friction coefficient and the reduced Nusselt number which are defined as follows: Cf ¼ −

  ∂u 1 −1=2 0 ¼− Rex f ð0Þ; ρ f u2w ∂y y¼0 ð1−φÞ2:5 μ nf

problem. In this method we have to choose a suitable finite value of 188 η → ∞ say n∞. We set following first order system: 189 0

y1 ¼ y2 ; 0 y2 ¼ y3 ;        0 2:5 2 ð1−φÞ þ φ ρs =ρ f y3 ¼ −ð1−φÞ y1 y3 −y2 −Mny2 þ λy4 ð1−φÞ þ φ ðρβ Þs =ðρβ Þ f ;

0

y5 ¼ −

165

Nu ¼

y1 ð0Þ ¼ 0;

One can note that if λ = 0, the problem under consideration reduces to the boundary-layer flow of nanofluid along a horizontal stretching sheet considered by Vishnu Ganesh et al. [28]. The analytical solution of Eq. (7) with the boundary conditions (Eq. 9) for λ = 0 is −αη given by f ðηÞ ¼ 1−eα , Where

T

The analytical solution of Eq. (8) with the boundary conditions (Eq. 9) absence of buoyancy parameter is obtained in terms of hyper geometric function as

!1 3N ð3N þ 4ÞCα 2 þ 3N −e−αη 3N   BM ; ; C −3Nη ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2 C B !C θðηÞ ¼ e ð3Nþ4ÞCα2 B B C; 3N ð3N þ 4ÞCα 2 þ 3N −3N @ A M ; ; ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2

R

E

C

0

N C O

knf    C¼ k f Pr 1−ϕ þ ϕ ðρCpÞs =ðρCpÞ f

R

where

The dimensionless wall temperature gradient θ ' (0) is obtained as

181

!  2 −3N 9N

þ θ ð0Þ ¼ ð3N þ 4ÞCα ð3N þ 4ÞCα ð3N þ 4ÞCα 2 þ 3N !1 0 2 2 3N þ ð3N þ 4ÞCα ð6N þ 8ÞCα þ 3N −3N ; ; BM C ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2 ð3N þ 4ÞCα 2 C B C: ! B B C 3N ð3N þ 4ÞCα 2 þ 3N −3N @ A ; ; M 2 2 2 ð3N þ 4ÞCα ð3N þ 4ÞCα ð3N þ 4ÞCα 

U

0

y4 ð0Þ ¼ 1:

ð13Þ

To solve Eq. (12) with Eq. (13) as an initial value problem we must need the values for y3(0) i.e. f″(0) and y5(0) i.e. θ′(0) but no such values are given. The initial guess values for f″(0) and θ′(0) are chosen and the fourth order Runge–Kutta integration scheme is applied to obtain the solution. Then we compare the calculated values of f(η) and θ(η) at η∞ with the given boundary conditions f(η∞) = 0 and θ(η∞) = 0, and adjust the values of f″(0) and θ′(0) using shooting iteration technique to give better approximation for the solution. The process is repeated until we get the results correct up to the desired accuracy of 10−9 level, which fulfils the convergence criterion.

192

4. Results and discussion

202

The transformed momentum and energy Eqs. (7) and (8) subjected to the boundary conditions Eq. (9) were numerically solved by using fourth-order Runge–Kutta method along with shooting technique. The Prandtl number is fixed as 6.2 which is for the base fluid water. The effect of magnetic parameter on the wall normal velocity profile f(η) and the temperature profile θ(η) of Cu–water is shown in Fig. 1. It is observed that the velocity profile decreases with the magnetic parameter (Mn) and the temperature profile increases with the increasing of magnetic parameter. This is due to the fact that with the increasing Mn, the Lorentz force associated with the magnetic field increases and it produces more resistance to the flow and the nanoparticles. But the magnetic field enhances the temperature at all points leading to increase in thermal boundary-layer thickness. Fig. 2 exhibits the effect of buoyancy parameter on velocity and temperature profiles of Cu–water. It is clear that the velocity profile of the nanofluid rises with the rising values of buoyancy parameter (λ). Since the momentum boundary-layer thickness increases with increasing values of buoyancy parameter enabling more flow. The rising values of buoyancy parameter results in thinning of the thermal boundarylayer and hence produces an increase in the heat transfer rate.

203 204

E

176

180

and

D

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r    α¼ ð1−ϕÞ2:5 Mn þ 1−ϕ þ ϕ ρs =ρ f

177 178

y2 ð0Þ ¼ 1

R O

173 174

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P

171 172

y1 y5 ;

O

respectively. Here Rex = uwx/νf is the local Reynolds number. 3. Analytical solutions for the special case

169 170

f

with the boundary conditions

ð11Þ 168

s

ð3N þ 4Þknf

ð12Þ

! !     3  x 16σ T ∞ ∂T 1=2 3N þ 4 0 ¼ −Re θ ð0Þ; knf þ x 3knf k f ðT w −T ∞ Þ 3N ∂y y¼0

183

1

Cu-Water

0.8

f(η) & θ(η)

166

0

y4 ¼y5 ;      3N Prk f 1−φ þ φ ρC p = ρC p 

ð10Þ

F

160

3

0.6

0.4

Mn= 2, 4, 6 Mn= 2, 4, 6

0.2

4. Numerical method for solution 0 184 185 186 187

The nonlinear coupled differential Eqs. (7) and (8) along with the boundary conditions (Eq. 9) form a two point boundary value problem and is solved using shooting technique together with fourth order Runge–Kutta integration scheme by converting it into an initial value

0

1

2

3

4

5

η Fig. 1. Effect of Mn on velocity and temperature profiles of Cu–water with ϕ = 0.12, λ = 1, N = 2 and Pr = 6.2.

Please cite this article as: M.M. Rashidi, et al., Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.06.037

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M.M. Rashidi et al. / Journal of Molecular Liquids xxx (2014) xxx–xxx

1

a

Cu-Water

Cu-Water 1 0.8

0.6

λ= 1,2, 3

0.4

f(η) & θ(η)

f(η) & θ(η)

0.8

λ= 1,2, 3

φ= 0.03, 0.06, 0.09

0.6 For smaller Magnetic field (Mn=0.2)

0.4

φ= 0.03, 0.06, 0.09

0.2 0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

η

0

0

0.5

1

244 245 246 247 248 249 250

2.5

3

3.5

4

0.4

O P

0.6

Cu-Water

φ= 0.03, 0.06, 0.09

For higher Magnetic field (Mn=2)

φ= 0.03, 0.06, 0.09

D

f(η) & θ(η)

0.8

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η

T

E

0.2

1

0.8

0.6

Fig. 4. Effect of ϕ on temperature profile of Cu–water with Mn = 0. 2, λ = 1, N = 2 and Pr = 6.2.

presented in Table 3. This table reveals that the skin friction coefficient increases with the increasing values of Mn, and N and decreases with ϕ and λ. The reduced Nusselt number increases with λ and N and also decreases with ϕ and Mn. The skin friction coefficient values of Cu nanofluid are greater than CuO nanofluid. The reduced Nusselt number values of Cu are lesser than CuO nanofluid.

251 252

5. Conclusion

257

1

Cu-Water

0.8

0.4 N= 2, 3, 4

0.6 CuO, Cu 0.4 CuO, Cu 0.2

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η Fig. 3. Effect of N on temperature profile of Cu–water with Mn = 2, ϕ = 0.12, λ = 1, N = 2 and Pr = 6.2.

253 254 255 256

The steady two dimensional, viscous incompressible water based 258 MHD nanofluid flow over a vertical stretching sheet with thermal 259

f(η) & θ(η)

242 243

C

240 241

E

238 239

R

236 237

R

234 235

O

232 233

C

230 231

N

228 229

U

226 227

θ(η)

224 225

1

R O

b The effect of radiation parameter (N) on the temperature profile of the nanofluid demonstrated in Fig. 3. The rising values of radiation parameter lead to decrease the temperature of the nanofluid. The presence of radiation parameter leads to thinning of the thermal boundary-layer. Fig. 4a and b show the effect of nanoparticle volume fraction on the velocity and temperature profiles of the nanofluid. It can be seen that the increasing values of nanoparticle volume fraction parameter decrease the velocity profile in the presence of smaller magnetic field (Fig. 4a) and increase the velocity profile in the presence of higher magnetic field (Fig. 4b). The presence of the magnetic field always leads to thickening of the thermal boundary-layer. The effects of Cu metal and Cu-oxide nanoparticles on velocity and temperature profiles are shown in Fig. 5. It is found that the velocity profile of the CuO water is higher than the Cu–water and the Cu–water rises the temperature than CuO water. The effect of magnetic parameter and the nanoparticle volume fraction on the skin friction coefficient is shown in Fig. 6. It is noted that the skin friction increases with the increasing values of Mn. The increasing values of nanoparticle volume fraction increases the skin friction coefficient for certain values of Mn and decreases the skin friction for large values of Mn. In order to verify the present numerical solution we have compared our results with those of Wang [30] for rated heat transfer in horizontal stretching sheet case in the absence of magnetic field, thermal radiation and nanoparticle volume fraction and an excellent agreement is obtained. Table 1 shows the thermo–physical properties of water and nano particles. In Table 2 we have compared the results for the reduced Nusselt number. The values of local skin friction coefficient and the reduced Nusselt number for Cu and CuO nanofluids are calculated and

2

η

Fig. 2. Effect of λ on velocity and temperature profiles of Cu–water with Mn = 2, ϕ = 0.12, N = 2 and Pr = 6.2.

223

1.5

F

0

0

0

0.5

1

1.5

2

2.5

3

3.5

4

η Fig. 5. The velocity and temperature profiles for Cu and CuO nanofluids. ϕ = 0.15, Mn = 2, λ = 2, N = −2 and Pr = 6.2.

Please cite this article as: M.M. Rashidi, et al., Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.06.037

M.M. Rashidi et al. / Journal of Molecular Liquids xxx (2014) xxx–xxx

2.2

Table.3 Values of − f ″ (0) and −θ ′ (0) for various values of governing parameters.

Cu-Water 2

− f ″ (0) φ= 0.03, 0.06, 0.09

Mn

φ= 0.03, 0.06, 0.09

1.4

ϕ

1.2 1

λ

0.8 0

1

2

3

4

N

5

Mn

260

278

Acknowledgements

279 280

The authors wish to express his sincere thanks to the honourable referees and the editor for their valuable comments and suggestions to improve the quality of the paper. One of the authors (N.V.G.) gratefully acknowledges the financial support of Rajiv Gandhi National Fellowship (RGNF), UGC, New Delhi, India. (F1-17.1/2012-13/RGNF2012-13-SC-TAM-16936).

273 274 275 276

281 282 283 284 Q4

C

E

R

271 272

R

269 270

N C O

267 268

U

265 266

T

277

263 264

1.12493100 1.37601235 1.59574518 1.38939862 1.37601235 1.35302258 1.91209861 1.62626070 1.37601235 1.34973397 1.37601235 1.38724832

0.92890862 0.88544087 0.84801483 0.93553568 0.88544087 0.81376547 0.71908156 0.82990241 0.88544087 0.71647710 0.88544087 0.96827471

0.94800696 0.90160918 0.86211196 0.94977810 0.90160918 0.83237271 0.75296042 0.84810822 0.90160918 0.73058138 0.90160918 0.98527512

t3:5 t3:6 t3:7 t3:8 t3:9 t3:10 t3:11 t3:12 t3:13 t3:14 t3:15 t3:16

[1] S.K. Das, S.U.S. Choi, W. Yu, T. Pradeep, Nanofluids: science technology, Wiley, New Jersey, 2007. [2] S. Kakac, A. Pramuanjaroenkij, Int. J. Heat Mass Transfer 52 (2009) 3187–3196. [3] X.-Q. Wang, A.S. Mujumdar, Braz. J. Chem. Eng. 25 (2008) 613–630. [4] X,.-Q. Wang, A.S. Mujumdar, Braz. J. Chem. Eng. 25 (2008) 631–648. [5] M. Sheikholeslami, M. Gorji-Bandpy, S. Soleimani, Int. Commun. Heat Mass Transfer 47 (2013) 73–81. [6] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, J. Mol. Liq. 194 (2014) 179–187. [7] M. Sheikholeslami, D.D. Ganji, Powder Technol. 253 (2014) 789–796. [8] W.A. Khan, I. Pop, Int. J. Heat Mass Transfer 53 (2010) 2477–2483. [9] A.V. Kuznetsov, D.A. Nield, Int. J. Therm. Sci. 49 (2010) 243–247. [10] M.M. Rashidi, E. Erfani, Int. J. Numer. Methods Heat Fluid Flow 21 (7) (2011) 864–883. [11] M.M. Rashidi, N. Freidoonimehr, A. Hosseini, O. Anwar Bég, T.-K. Hung, Meccanica (2013), http://dx.doi.org/10.1007/s11012-013-9805-9. [12] M. Sheikholeslami, D.D. Ganji, J. Mol. Liq. 194 (2014) 13–19. [13] M. Hatami, M. Sheikholeslami, D.D. Ganji, J. Mol. Liq. 195 (2014) 230–239. [14] S.M. Aminossadati, A. Raisi, B. Ghasemi, Int. J. Non-Linear Mech. 46 (10) (2011) 1373–1382. [15] B. Ghasemi, S.M. Aminossadati, A. Raisi, Int. J. Therm. Sci. 50 (9) (2011) 1748–1756. [16] A.H. Mahmoudi, I. Pop, M. Shahi, Int. J. Therm. Sci. 59 (2012) 126–140. [17] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Energy 60 (2013) 501–510. [18] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, P. Rana, Soheil Soleimani, Comput. Fluids 94 (2014) 147–160. [19] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Soheil Soleimani, J. Mol. Liq. 193 (2014) 174–184. [20] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, J. Taiwan Inst. Chem. Eng. (2014), http://dx.doi.org/10.1016/j.jtice.2014.03.010. [21] M. Sheikholeslami, M. Gorji-Bandpy, R. Ellahi, Mohsan Hassan, Soheil Soleimani, J. Magn. Magn. Mater. 349 (2014) 188–200. [22] M. Sheikholeslami, M. Gorji-Bandpy, Powder Technol. 256 (2014) 490–498. [23] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, Powder Technol. 254 (2014) 82–93. [24] M.A.A. Hamad, Int. Commun. Heat Mass Transfer 38 (2011) 487–492. [25] N. Vishnu Ganesh, B. Ganga, A.K. Abdul Hakeem, J. Egypt. Math. Soc. (2013), http:// dx.doi.org/10.1016/j.joems.2013.08.003. [26] A.K. Abdul Hakeem, N. Vishnu Ganesh, B. Ganga, J. King Saud Univ. Eng. Sci. (2013), http://dx.doi.org/10.1016/j.jksues.2013.05.006. [27] F.M. Hady, F.S. Ibrahim, S.M. Abdel-Gaied, M.R. Eid, Nanoscale Res. Lett. 7 (2012) 229–242. [28] N. Vishnu Ganesh, A.K. Abdul Hakeem, R. Jayaprakash, B. Ganga, J. Nanofluid. 3 (2014) 154–161. [29] A.V. Domkundwar, V.M. Domdundwar, Heat and Mass transfer data book, Dhanpa rai and co (p) Ltd, Educational and Technical Publishers, Delhi, 2004. [30] C.Y. Wang, J. Appl. Math. Mech. (ZAMM) 69 (1989) 418–420.

P

Numerical 0.91135769 1.89540340

radiation effect is investigated numerically using shooting technique. The presence of magnetic field leads to decrease the nanofluid velocity and increase the nanofluid temperature. The rising of buoyancy parameter increases the velocity and decreases the temperature of the nanofluid. The velocity decreases with the rising of nanoparticle volume fraction in the presence of smaller magnetic field and the nanoparticle fraction has an opposite effect on the velocity profile in the presence of higher magnetic field. The presence of nanoparticles in the base fluid increases the temperature profile. The velocity of the CuO water is higher than the Cu–water and the Cu–water rises the temperature than CuO water. The skin friction coefficient increases with the increasing values of magnetic parameter and the radiation parameter and decreases with buoyancy parameter and nanoparticle volume fraction parameter. The reduced Nusselt number increases with buoyancy and radiation parameters and decreases with nanoparticle volume fraction and magnetic parameters. The skin friction coefficient values of Cu nanofluid are greater than CuO nanofluid. The reduced Nusselt number values of Cu are lesser than CuO nanofluid.

261 262

1.20278694 1.44214325 1.65402315 1.44726274 1.44214325 1.42985157 1.99005641 1.69504351 1.44214325 1.41599846 1.44214325 1.45336332

O

Present results the absence of N, λ and ϕ = 0 Analytical 0.91136 1.89540

t3:4

D

0.91136 1.89540

CuO

E

2 7

t2:3 t2:4 t2:5 t2:6

References

Table 2 Comparison of results for the reduced Nusselt number −θ′(0). Wang [30]

Cu

Note: While studying the effect of individual parameters the following values are assumed ϕ = 0.12, M = 2, λ = 1, N = 2 and Pr = 6.2.

Fig. 6. Variation of skin friction coefficient with λ = 1, N = 2 and Pr = 6.2.

Pr

1 2 3 0.10 0.12 0.15 −1 0 1 1 2 3

t3:3

CuO

F

1.6

− θ ′ (0)

t3:1 t3:2

Cu

R O

Re1/2 Cf x

1.8

t2:1 t2:2

5

Please cite this article as: M.M. Rashidi, et al., Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.06.037

t3:17 t3:18

285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 Q5 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327