Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles

APT 1384

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Contents lists available at ScienceDirect

Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

2

Original Research Paper

6 4 7 5

Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles

8

N. Sandeep a, A. Malvandi b,⇑

9 10 12 11 13 1 2 5 8 16 17 18 19 20 21 22 23 24 25 26 27

a b

Department of Mathematics, VIT University, Vellore 632014, India Department of Mechanical Engineering, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran

a r t i c l e

i n f o

Article history: Received 15 June 2016 Received in revised form 10 August 2016 Accepted 29 August 2016 Available online xxxx Keywords: Graphene nanoparticles MHD Liquid film flow Non-uniform heat source/sink Non-Newtonian nanofluid

a b s t r a c t Recent days, graphene is emanating as one of the most encouraging nanomaterials due to its continuous electrical conducting behaviour even at zero carrier concentrations. Heat transfer in non-Newtonian fluids plays a major role in technology and in nature due to its stress relaxation, shear thinning and thickening properties. With this incentive, we investigate the flow and heat transfer characteristics of electrically conducting liquid film flow of water based non-Newtonian nanofluids dispensed with graphene nanoparticles. For this investigation, we proposed a mathematical model for the flow of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface in the presence of transverse magnetic field and non-uniform heat source/sink. Numerical results are carried out by employing Runge-Kutta-Felhberg integration scheme. The influence of pertinent parameters on reduced Nusselt number, friction factor, flow and heat transfer is discussed with the assistance of graphs. Embedding the graphene nanoparticles effectively enhances the thermal conductivity of Jeffrey nanofluid when compared with the Oldroyd-B and Maxwell nanofluids. Deborah number in terms of relaxation time plays a major role in convective heat transfer. Ó 2016 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan.

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46

1. Introduction

47

Nowadays, heat transfer of liquid film flow plays a major role in understanding the heat exchangers design, paper production, wire coating, cooling process, etc. Heat transfer of non-Newtonian flows has wide range of applications in design of industrial equipment, polymer sheet manufacturing, food stuffs, etc. To increase the thermal conductivity of the base fluids, nanometer sized high thermal conductivity nano-materials are suspended in base fluids and named it as nanofluid. Recent days, nanofluids are playing important role in cooling and heating applications. Graphene is one of the most encouraging nanomaterials due to its continuous electrical conducting behaviour even at zero carrier concentration. The electron movement in graphene is very high when compared with the other existing materials. External magnetic fields are capable to set the thermal and physical properties of magnetic-nanofluids and regulate the flow and heat transfer characteristics. This is very helpful in metallurgical and controlling process. Heat transfer process is non-uniform in the nature. It is depends on the temperature and the existed space differences. The space and temperature

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

⇑ Corresponding author. E-mail address: [email protected] (A. Malvandi).

dependent heat source/sink helps to regulate the heat transfer performance of the existed phenomena. In view of all the above mentioned applications, the study of flow and convective heat transfer in electrically conducting liquid film flow of water based non-Newtonian nanofluids dispensed with graphene nanoparticles in the presence of non-uniform heat source/sink have a great importance in industry as well as in engineering applications. But Navier-Stokes equations are not enough to elaborate the rheological properties of the fluids. The boundary layer behaviour of MHD flow over a stretching surface was initially discussed by Sakiadas [1]. Wang [2] was the first person who discussed about the thin film flow over a stretching sheet. Further, Andersson et al. [3,4] developed the Wang’s work by considering the power-law fluid under variable physical properties. Later on Wang and Pop [5] proposed a homotopy analysis method to analyze the heat transfer characteristics of liquid film flow of power-law fluid. In all the above existed nanofluid flow models, nanoparticles are passively involved in enhancing the thermal conductivity of the flow. In 2006, Buongiorno [6] proposed a mathematical model for active participation of the nanoparticles to enhance the effective thermal conductivity of the nanofluid. Further, the researchers [7–9] discussed the heat transfer characteristics of the liquid film flow of Newtonian and non-Newtonian flows over a stretching sheet by considering the various physical proper-

http://dx.doi.org/10.1016/j.apt.2016.08.023 0921-8831/Ó 2016 Published by Elsevier B.V. on behalf of The Society of Powder Technology Japan.

Please cite this article in press as: N. Sandeep, A. Malvandi, Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.08.023

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Nomenclature u, v uw(x) cp B0 A ; B S Cfx f k M Nux Pr Rex T T1

velocity components along x and y directions (m s
1) stretching velocity of the sheet (m s
1) specific heat capacitance applied magnetic field strength non-uniform heat source/sink parameters unsteadiness parameter skin friction coefficient dimensionless velocity thermal conductivity (W m
1 K
1) magnetic field parameter local Nusselt number Prandtl number local reynolds number temperature of the fluid (K) temperature of the fluid in the free stream (K)

Greek symbols / solid volume fraction of the nanoparticles k dimensionless film thickness c1 relaxation time

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

ties. The non-Newtonian fluid properties for modeling the engineering problems were studied by Chhabra [10]. To enhance the cyclic performance of lithium ion batteries Wu et al. [11] used graphene doped Co3O4 nanoparticles and found the enhanced reversible life of the battery. Further, the researchers [12–14] used graphene based composites for energy storage in tribological and electronics applications. Deborah and Erickson number predictions for non-Newtonian flow models were studied by Kelin et al. [15]. Liquid film flow over a graphene-mica slit pore was experimentally investigated by Severin et al. [16]. The thermal properties of graphene and its applications were experimentally studied by the researchers [17–19]. The flow and heat transfer characteristics of chemically reacting couple stress, Power-Law and Maxwell fluids are respectively studied by the researchers [20–22]. Flow boiling characteristics of water-graphene nanofluid and graphene thin film flow with chemical deposition was illustrated by the authors [23,24]. Flow and heat transfer characteristics of Jeffrey, Maxwell and Oldroyd-B fluids over a stretching/shrinking sheet was individually studied by the researchers [25–28]. The heat transfer in magnetohydrodynamic thin film flow of a second grade fluid past a vertical belt was numerically studied by Gul et al. [29]. Applications and properties of graphene doped materials was experimentally studied by Rao et al. [30]. Effect of variable thermal conductivity on Maxwell fluid flow over a stretching sheet in porous medium was numerically elaborated by Singh and Agarwal [31]. Unsteady thin film flow of Newtonian and non-Newtonian magnetic-flows over a stretching sheet by considering the viscous dissipation, internal heat generation, thermal radiation and variable thermal conductivity effects was studied by the researchers [32–35]. Recently, the researchers [36–38] analyzed the heat and mass transfer in Newtonian and non-Newtonian flows past uniform/non-uniform thickness stretching sheet by considering the various physical effects and presented dual solutions for each case. Adegbie et al. [39] studied the heat and mass transfer in Maxwell nanofluid flow over a melting surface with variable thermophysical properties. Further, the researchers [40–42] continued their research on analyzing the heat and mass transfer characteristics of MHD flows by considering the Oldroyd-B fluid. Recently, Malvandi et al. [43–45] studied the film boiling of magnetic nanofluid over a vertical plate by considering the thermophoresis and Brownian motion effects.

c2 c3 d1 d2 d3 n f

r h

q l t e1 to e5

ratio of the relaxation and retardation times retardation time Deborah number with respect to relaxation time Deborah number with respect to ratio of relaxation and retardation time Deborah number with respect to retardation time stream function similarity variable electrical conductivity (m X m
1) dimensionless temperature fluid density (kg m
3) dynamic viscosity (Pa S) kinematic viscosity (m2 s
1) nanofluid constants

Subscripts f base fluid nf nanofluid s solid nanoparticle

Raju et al. [46] developed a mathematical model for the flow of thee different types of non-Newtonian fluids over a cone. Very recently, Yadav et al. [47–50] studied the free convective heat transfer in rotating nanofluid layer by considering the rotating porous medium with variable physical aspects. Effect of magnetic field on the free convection of onset of nanofluid was numerically investigated by Yadv et al. [51]. Further, the researchers [52–55] studied the natural convection of magnetohydrodynamic flow in vertically orientated Hele-Shaw cell suspending with the nanoparticles. Nanoparticle migration effect on heat transfer enhancement of nanofluid over a vertical cylinder was numerically studied by Malvandi et al. [56–58]. The researchers [59–62] studied the effect of various physical parameters on MHD flow of nanofluid by considering the uniform and non-uniform thickness stretching surfaces, upper paraboloid of revolution and film flow over a stretching surface. In all the above investigations, authors discussed the heat or heat and mass transfer characteristics of either Newtonian or non-Newtonian flows over a steady or unsteady stretching sheet with one or more physical aspects. But to the author’s best knowledge no studies has been reported yet on analyzing the flow and heat transfer characteristics of electrically conducting liquid film flow of water based non-Newtonian nanofluids dispensed with graphene nanoparticles. For this investigation, we proposed a combined model for the flow of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface in the presence of transverse magnetic field and non-uniform heat source/sink. Numerical results are carried out by enforcing Runge-Kutta-Felhberg integration scheme. Obtained results are discussed with the help of graphs.

131

2. Mathematical formulation

161

Consider an unsteady, electrically conducting liquid film flow of magnetohydrodynamic Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface. The elastic sheet starts from a narrow slit, which is located at the origin of a coordinate system (x, y). Here xaxis is measured along the stretching surface with stretched velocity uw(x, t) = bx/(1
at), where b, a constants and y-axis is normal to it. It is assumed that T0 and Tr are the slit and reference temper-

162

atures and T s ðx; tÞ ¼ T 0 þ T r ðbx =2v f Þð1
atÞ
1:5 is the wall tem2

Please cite this article in press as: N. Sandeep, A. Malvandi, Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.08.023

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perature of the fluid, where v f is the kinematic viscosity of the base fluid. An external magnetic field BðtÞ ¼ B0 ð1
atÞ
0:5 is applied normal to the stretching sheet as shown in Fig. 1. Induced magnetic field is neglected in this study. Nanoparticles used in the preparation of nanofluids are non-magnetic. With the above assumptions, the governing conservation equations in unsteady state in terms of similarity variable n can be expressed as (see Refs. [22,46,62])

177 179

@2n @2n
¼ 0; @x@y @y@x

180



qnf

ð1Þ

@2 n @n @ 2 n @2 n þ @y
@n þ 1 @t@y @x@y @x @y2


 2 @n @y


  @n

 2 @ 3 n @3 n @n þ @n
2 @y @x @y3 @x2 @y

@3 n @x @x@y2



9 > =

¼ h3
4  i
> lnf @ n @ n @n @ 4 n @n @ 4 n @2 n @3 n @2 n @3 n @n ; þ c3 @t@y
rnf B2 ðtÞ @y ; 3 þ @y @x@y3
@x @y4
@x@y @y3 þ @y2 @y2 @x ð1þc Þ @y3

c

2

ð2Þ

182 183

  @T @n @T @n @T @2T þ
¼ knf 2 ðqcp Þnf @t @y @x @x @y @y þ

 kf u w   0 A ðT s
T 0 Þf þ B ðT
T 0 Þ ; xv f ð3Þ

185 186

with the boundary conditions

187

u ¼ uw ; 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203

@u @y

¼ 0;

v ¼ 0; @T @y

)

T ¼ T s at y ¼ 0

¼ 0; at y ¼ h;

v¼

dh dt

as y ¼ hðtÞ

ð4Þ

where n is the stream function, u and v are the velocity components along x and y directions respectively, qnf is the density of the nanofluid, lnf is dynamic viscosity of the nanofluid, rnf is the electrical conductivity of nanofluid, BðtÞ is the applied magnetic field, T is the fluid temperature, knf is thermal conductivity of the nanofluid, ðcp Þnf represent the heat capacitance of the nanofluid, A and B are the space and temperature dependent heat source/sink parameters, c1 ; c2 and c3 represents the relaxation time, ratio of the relaxation to retardation times and retardation time respectively. Eq. (2) deals with different nano-film flow models based on the following conditions (Ref. [46]) (i) Jeffrey nanofluid: c1 = 0, c2 – 0, c3 – 0, (ii) Maxwell nanofluid: c1 – 0, c2 = 0, c3 = 0, (iii) Oldroyd-B nanofluid: c1 – 0, c2 = 0, c3 – 0.

204

For real life applications, the nanoparticle concentration is very small. With the help of Taylor’s series, the nanofluid constants are defined as follows (Ref. [33]) lnf lf

1 ¼ ð1
2:5/Þ ;

knf kf

¼ 1 þ 3ðk
1Þ/ ; kþ2

qnf qf

¼ 1
/ þ /r; rnf rf

d ¼ ðqcpp Þs ;

f

ðqcp Þf

3ðr
1Þ/ ¼ ðrþ2Þ
ð r
1Þ/ þ 1 ðqc Þ

where r ¼ qqs ;

ðqcp Þnf

f

f

f¼

0:5

b v f ð1
atÞ

y;

210

@ f þd3 ð1þd2 Þ @f3






2
3  @2 f @4 f @ f f @4 f
f þS 2 þ þ 4 @f4 @f2 @f4 @f3

2

@ f e2 k f @f 2
S




2 @f þf @ f @f 2 @f2

213 214 215 216 217 218 219

222

where n is the stream function such that u = ny and v =
nx, f is the similarity variable, b > 0 is the dimensionless film thickness (see Ref. [22]), f and h are dimensionless velocity and temperature. With the help of Eqs. (5) and (6), the Eqs. (2) and (3) transformed as 3

212

ð6Þ

T ¼ T 0 þ T r ðbx =2v f Þð1
atÞ
1:5 hðgÞ;

e1

211

220


 v b 0:5 n ¼ b ð1
f atÞ xf ðgÞ;

2



207

ð5Þ

> > > > > ;

where / is the volume fraction of the nano particles. qf, qs are the densities of the base fluid and solid nanoparticles, kf, kf are the thermal conductivities of the base fluid and solid nanoparticles, rf, rs are the electrical conductivities of the base fluid and solid nanoparticles, (cp)f, (cp)s are the specific heat capacity of the base fluid and solid nano particles at constant pressure. The subscripts f and s refer to fluid and solid particles respectively. To get inside analysis of the problem, we use following similarity transformation.




206

208

9 ¼ ð1
/Þ þ /d > > > > > =

r ¼ rrfs ;

k ¼ kks ;

205

223 224 225 226 227

9 > > > =

228

ð7Þ

230


2
3  > 2@ f @f @ 2 f >
@f
e3 M @f
d f
2f ¼ 0; > ; 1 @f @f @f2 @f @f3

231

  @f @h 3 1 @h @f e4 2
Pre5 k 2 h
f þ Sh þ Sf þ A þ B h ¼ 0; @f @f 2 2 @f @f @f @2h

ð8Þ With the reduced boundary conditions 0

f ð0Þ ¼ 0;

f ð0Þ ¼ 1;

f ð1Þ ¼ S=2;

00

234

235

000

f ð0Þ ¼ 0;

f ð1Þ ¼ 0;

hð0Þ ¼ 1;

ð9Þ

h0 ð1Þ ¼ 0;

where primes denotes the differentiation with respect to f, d1, d2 and d3 are the Deborah numbers in terms of relaxation time, ratio of the relaxation to retardation times and retardation time, respectively. Pr is the Prandtl number, M is magnetic field parameter, S is unsteadiness parameter, k is the dimensionless film thickness, A is the space dependent heat source/sink, B is the temperature dependent heat source/sink and e1, e2, e3, e4 and e5 are the nanofluid constants, which are given by

Pr ¼

ðlcp Þf

e1 ¼

lnf lf

kf

;

;

M¼

rf B20 ; bqf

233

S ¼ ab ;

d1 ¼ c1 b;

d3 ¼ c3 b;

e2 ¼ qqnff ; e3 ¼ rrnff ; e4 ¼ kknff ; e5 ¼

ðqcp Þnf ðqcp Þf

;

9 = k ¼ b2 ; >

237 238 239 240 241 242 243 244 245

246

> ; ð10Þ

248

The physical quantities of engineering interest, the local Nusselt number Nux and friction factor Cfx are given by

249 250

251

Re
0:5 Nux x

Fig. 1. Physical model of the problem.

1 knf 0 ¼
h ð0Þ; b kf

C fx Re0:5 x

1 1 00 f ð0Þ; ¼ b ð1
2:5/Þ

ð11Þ

where Rex ¼ uvw x is the local Reynolds number. f

Please cite this article in press as: N. Sandeep, A. Malvandi, Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.08.023

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3. Numerical solution

256

The set of nonlinear ordinary differential equations (7) and (8) with the reduced boundary conditions of Eq. (9) are solved numerically by employing Runge-Kutta-Felhberg integration scheme. We reduce the system of Eqs. (7) and (8) as initial valued problem by assuming

257 258 259 260

261

263 264

f ¼ f 1;

@f ¼ f 2; @f

h ¼ f 6;

@h ¼ f 7; @f

@2f 2

@f @2h @f2

¼ f 3;

@3f 3

@f

¼ f 4;

@4f @f4

¼ f 5;

¼ f 8;

ð12Þ

h i 8 9 2 e1 > >
ð1þd f þ d 3 ðf 3 þ 2Sf4 Þ
> > 4 Þ 2 > > < =  ð1 þ d Þ 2 3 2 2 ; f5 ¼  ðf 1 f 3
Sðf 2 þ f 3 f=2Þ
f 1 > e1 ðSf=4
f 1 Þ
 4 5 þ e3 Mf2 > > > e k > > 2 2 : ;
d f f
2f f f 1

1 4

1 2 3

ð13Þ

266 267

@2h @f2

¼

1



e4

  3 1 Pre5 k 2f 2 f 6
f 1 f 7 þ Sf6 þ Sff 7
A f 2
B f 6 ; 2 2 ð14Þ

269 270

271

with boundary conditions as

f 1 ¼ f 2 ¼ 0; 273

f 1 ¼ S=2;

f 3 ¼ 0;

f 3 ¼ 0;

f 4 ¼ a;

f 6 ¼ 1;

at f ¼ 0;

f 7 ¼ 0 as f ¼ 1;

ð15Þ

276

We guess the values undefined initial conditions. Eqs. (12)–(14) are integrated by using Runge-Kutta fourth order method with the successive iterative step length is 0.01.

277

4. Results and discussion

278

The effect of various physical parameters namely, volume fraction of nanoparticles, unsteadiness parameter, film thickness parameter, space and temperature dependent heat source/sink parameters, etc., on velocity and temperature fields along with reduced Nusselt number and friction factor for three different non-Newtonian nanofluids. For computation work, we consider the pertinent parameters as S ¼ A ¼ B ¼ 0:5; k ¼ 0:3; M ¼ 1; / ¼ 0:1; d1 ¼ d2 ¼ d3 ¼ 1:2. Apart from the changed parameters as shown in the plots, these are conserved as invariable. The thermophysical properties of water and graphene nanoparticles are depicted in Table 1. Figs. 2 and 3 elucidate the effect of nanoparticle volume fraction on velocity and temperature fields of Maxwell, Jeffrey and Oldroyd-B nanofluids. It is evident that rising values of / enhances the temperature as well as the velocity field. It is interesting to mention that the momentum boundary layer of Oldroyd-B nanofluid and thermal boundary layer of Jeffrey nanofluid are highly influenced by the variation in volume fraction of nanoparticles. Rise in the momentum boundary layer of Oldroyd-B nanofluid may happen due to an increase in the relaxation and retardation times. Physically, increasing the Deborah number for relaxation and retardation times improves the shear thinning property of

274 275

279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299

Fig. 2. The effect of / on velocity field.

Table 1 Thermophysical properties of graphene and water (see Refs. [17,19]). Thermo physical properties 3

q (kg/m ) cp (J/kg K) k (W/m K) r (S/m)

Water

Graphene

997 4076 0.605 0.005

2250 2100 2500 1  107

Fig. 3. The effect of / on temperature field.

the fluid. But, for Jeffrey fluid relaxation time is zero, this causes to thickening the fluid and encourages the thermal conductivity and hence the temperature field. This leads to the further conclusion that graphene nanoparticles effectively enhance the thermal conductivity of Jeffrey nanofluid when compared with the Oldroyd-B and Maxwell nanofluids. Figs. 4 and 5 depict the effect of transverse magnetic field on velocity and temperature fields of Maxwell, Jeffrey and OldroydB nanofluids. It is observed a rise in the temperature field and depreciation in the velocity field for increasing values of magnetic field parameter. This is due to the well known fact that the increasing values of magnetic field parameter boosts the resistive type force, called a Lorentz force. This force declines the momentum boundary layer and enhances the thermal boundary layer thickness. It is also observed that Jeffrey nanofluid is highly affected by the Lorentz’s force when compared with other two nanofluids. This proves that the fluids with shear thinning property are effectively opposing the resistive forces. We have noticed a similar type of results as discussed above for rising values of unsteadiness parameter, which is displayed in Figs. 6 and 7. Interestingly, thermal and momentum boundary layers of Oldroyd-B nanofluid are highly influenced by the increasing values of unsteadiness parameter (we have seen a similar behaviour for Jeffrey nanofluid in above case). Physically, unsteadiness causes to develop the buoyancy forces acting on the flow. These

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1 M=0.5,1,1.5

0.95

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.9

0.9

0.8

0.85

S=0.5,1,1.5

0.7

( )

I f ( )

0.8 0.75

0.6

0.7 0.65

0.5

0.6

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.55 0.5

0

0.1

0.2

0.3

0.4

0.4 0.5

0.6

0.7

0.8

0.9

0.3

1

0

Fig. 4. The effect of M on velocity field.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

1

Fig. 7. The effect of S on temperature field.

1

1

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.9

0.95 =0.1,0.2,0.3

0.9

0.8

I f ( )

( )

0.85 0.7

0.6

0.8

0.75

M=0.5,1,1.5

0.7

0.5

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.65 0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 5. The effect of M on temperature field.

0.95 0.9

I f ( )

0.85 0.8 S=0.5,1,1.5

0.7 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.65 0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 6. The effect of S on velocity field.

325 326 327

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 8. The effect of k on velocity field.

1

0.75

0.6

forces decline the velocity and encourage the temperature field. It is evident to conclude that the shear thickening fluids are highly influenced by the buoyancy forces caused by the unsteadiness.

Figs. 8 and 9 illustrate the effect of film thickness parameter on velocity and temperature profiles of Maxwell, Jeffrey and Oldroyd-B nanofluid. We observed a fall in velocity and temperature fields for increasing values of film thickness parameter. Generally, rising the film thickness does not allow the fluid to flow. This may be the reason for depreciating the velocity and temperature fields. We also observed a sudden fall in velocity field of Jeffrey fluid for increasing values of film thickness. This may happen due to rise in retardation time. Figs. 10 and 11 show the effect of space and temperature dependent heat source/sink parameters on temperature fields of all three nanofluids. This is due to the well known fact that the positive values of space and temperature dependent heat source/sink parameters acts like heat generators and hence boosts the thermal boundary layer thickness. It is also observed that the temperature field of Jeffrey nanofluid is effectively enhances by adding the additional heat to the flow. This concludes that the fluid particle interaction is high in Jeffrey nanofluid when compared with the other two nanofluids. Figs. 12 and 13 depict the effect of nanoparticle volume fraction on skin friction coefficient and local Nusselt number of Maxwell, Jeffrey and Oldroyd-B nanofluids. It is observed a rise in friction factor and a fall in heat transfer rate for increasing values of magnetic field parameter. It is also observed that friction factor is high

Please cite this article in press as: N. Sandeep, A. Malvandi, Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles, Advanced Powder Technology (2016), http://dx.doi.org/10.1016/j.apt.2016.08.023

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1

-0.3 -0.35

=0.1,0.2,0.3

0.9

skin friction coefficient

0.8

( )

0.7 0.6 0.5

-0.45 -0.5 -0.55 -0.6 -0.65 -0.7

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.4 0.3

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

-0.4

-0.75 -0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.15 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.95 0.9

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

Local Nusselt number

1.1

0.85 0.8

( )

0.2

Fig. 12. The effect of / on skin friction coefficient.

Fig. 9. The effect of k on temperature field.

0.75 0.7 0.65 0.6

0.1

A* =0.5,1,1.5

1.05

1

0.95

0.55 0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.9

1

0

Fig. 10. The effect of A⁄ on temperature field.

0.5

0.6

0.7

0.8

0.9

1

0.9

1

-0.4

skin friction coefficient

0.8

( )

0.4

-0.3

0.85

0.75 0.7 0.65 0.6

B * =0.5,1,1.5

0.55 0.5

0.3

-0.2

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.9

0.2

Fig. 13. The effect of / on local Nusselt number.

1 0.95

0.1

-0.5 -0.6 -0.7 -0.8

Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

-0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

M Fig. 11. The effect of B⁄ on temperature field. 352 353 354 355

in Oldroyd-B nanofluid when compared with other two. This may happen due to the fact that Oldroyd-B nanofluid is shear thickening fluid. Adding the nanoparticle concentrations leads to enhance the thickening of the fluid and hence improves the wall friction.

Fig. 14. The effect of M on skin friction coefficient.

Figs. 14 and 15 illustrate the influence of magnetic field parameter on skin friction coefficient and local Nusselt number of Maxwell, Jeffrey and Oldroyd-B nanofluids. It is clear that increasing

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1.2

-0.3

1.15

-0.4

skin friction coefficient

Local Nusselt Number

N. Sandeep, A. Malvandi / Advanced Powder Technology xxx (2016) xxx–xxx

1.1

1.05

1 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.95

0.9

-0.5

-0.6

-0.7

-0.8

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0

0.1

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Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.5

0.6

0.7

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1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.9

1

M Fig. 18. The effect of k on skin friction coefficient.

Fig. 15. The effect of M on local Nusselt number.

-0.2

1.6 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

1.4

Local Nusselt number

skin friction coefficient

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

1.2

1

0.8 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4

S

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 16. The effect of S on skin friction coefficient. Fig. 19. The effect of k on local Nusselt number.

1.35 Table 2 Comparison of values
h0 (0) for different values of S when M = 0 and Pr = 1 (for Newtonian case).

Local Nusselt number

1.3 1.25

1.15 1.1 Solid : Jeffrey nanofluid Dashed : Maxwell nanofluid Dotted : Oldroyd-B nanofluid

1.05 1 0.95

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S Fig. 17. The effect of S on local Nusselt number. 359 360 361

S

/=0 Ref. [62]

Present results

1.0 1.2 1.4 1.6 1.8

2.67722 1.99959 1.44775 0.95669 0.48453

2.677221 1.999590 1.447753 0.956692 0.484534

1.2

values of magnetic field parameter depreciate the both skin friction coefficient and local Nusselt number. In particular, the effect of Lorentz’s is high on Jeffrey fluid. This may happen due to the shear

thinning property of Jeffrey fluid. Figs. 16 and 17 depict the effect of unsteadiness parameter on skin friction coefficient and local Nusselt number of Maxwell, Jeffrey and Oldroyd-B nanofluids. We observed a rise in heat transfer rate as well as the friction factor for increasing values of unsteadiness parameter. When compared with Maxwell and Jeffrey nanofluids, we noticed a hike in heat transfer rate of Oldroyd-B nanofluid. This is due to the fact that shear thickening fluids have higher thermal conductivity when compared with shear thinning fluids. We observed a similar type of result in heat transfer rate for rising values of film thickness

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Table 3 Comparison of values f00 (0) for different values of S when A ¼ B ¼ k ¼ M ¼ 0 (for Maxwell fluid case). S

0.8 1.2

/=0 Ref. [22]

Present results


1.261479
1.377850


1.2614790
1.3778501

379

parameter as displayed in Fig. 18. But reverse trend has been observed for skin coefficient, which is depicted in Fig. 19. Till now no numerical investigations reported on the flow and heat transfer of thin film flows by considering the graphene nanoparticles. So, for validating the present results we assumed that the volume fraction of nanoparticles / = 0 and compared the obtained results with published results for special cases and found a favourable agreement. This is displayed in Tables 2 and 3.

380

5. Conclusion

381

Due to the numerous applications in design of industrial equipment and cooling process, in this study we presented a mathematical model for analyzing the flow and heat transfer in liquid film flow of non-Newtonian nanofluids in the presence of variable heat source/sink and transverse magnetic field. We developed a mathematical model for the flow of Jeffrey, Maxwell and Oldroyd-B nanofluids past a stretching surface. Numerical results are carried out by employing Runge-Kutta-Felhberg integration scheme. The influence of pertinent parameters on reduced Nusselt number, friction factor, flow and heat transfer is discussed with the assistance of graphs. We strongly believe that this study gives a good scope to the researchers to identify the high thermal conductivity nonNewtonian fluids which give good heat transfer enhancement by suspending the various high thermal conductivity nanoparticles. The numerical findings of the present study are as follows:

372 373 374 375 376 377 378

382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410

411 412 413 414 415 416 417 418 419 420 421 422 423 424

 Deborah number with relaxation and retardation times plays a major role in deciding the shear thinning and thickening behaviour of the non-Newtonian nanofluid.  Suspension of graphene nanoparticles effectively enhances the thermal conductivity of Jeffrey nanofluid.  Deborah numbers with relaxation and retardation times encourage the heat transfer rate.  Increasing the nanoparticle concentration effectively enhances the friction factor of Jeffrey, Maxwell and Oldroyd-B nanofluids.  Space and temperature dependent heat source/sink parameters regulate the thermal boundary layers of all three nanofluids.  Lorentz force effectively controls the momentum boundary layer of Jeffrey nanofluid.  This study is very useful for designing the heat exchanger equipments.

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