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Results in Physics xxx (2016) xxx-xxx

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Results in Physics

journal homepage: www.journals.elsevier.com/results-in-physics

Transport and heat transfer of time dependent MHD slip flow of nanofluids in solar collectors with variable thermal conductivity and thermal radiation

Khadeeja Afzala, Asim Azizb'*

a School Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 45000, Pakistan b College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi 46070, Pakistan

ARTICLE INFO ABSTRACT

In this paper, the unsteady magnetohydrodynamic (MHD) boundary layer slip flow and heat transfer of nanofluid in a solar collector, modeled mathematically as a nonlinear stretching sheet is investigated numerically. The variable thermal conductivity is assumed as a function of temperature and the wallslip conditions are utilized at the boundary. The similarity transformation technique is used to reduce the governing boundary value problem to a system of nonlinear ordinary differential equations (ODEs) and then solved numerically. The numerical values obtained for the velocity and temperature depend on nanofluid volume concentration parameter, unsteadiness parameter, suction/injection parameter, thermal conductivity parameter, slip parameters, MHD parameter and thermal radiation parameter. The effects of various parameters on the flow and heat transfer characteristics are presented and discussed through graphs and tables.

© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/4XI/).

Article history:

Received 31 July 2016

Received in revised form 25 August 2016

Accepted 28 September 2016

Available online xxxx

Keywords: Nanofluid

Magnetohydrodynamic Partial slip Heat transfer Thermal radiation Variable thermal conductivity

Introduction

Solar energy is one of the cleanest renewable sources of available energy. Solar energy that reaches the earth is around 4 x 1015 MW and it is 200 times as large as the global utilization. Renewable forms of energy accounted for 2.1% of global energy consumption, up from 0.7% in 2001. Consequently, the utilization of solar energy and the technology of nanofluids attracted much more attention [1]. Nanofluid-based direct solar collectors are solar thermal collectors where nanoparticles in a liquid medium can scatter and absorb solar radiation and then converted into thermal energy in the focus of solar thermal concentrating systems. A comprehensive survey of convective transport in nanofluids was done by Buongiorno [2]. Experimental studies [3,4] show that the effective thermal conductivity of nanofluids increases under macro-scopically stationary conditions. Since then, a number of studies have been conducted on the thermal properties (mainly thermal conductivity) viscosity and convective heat transfer performance of nanofluids. The applications of nanofluids are, for example, cooling of electronic devices and vehicle, cancer therapy and surgery, drug delivery, detergency, oil and gas recovery, paper printing, tex-

* Corresponding author. E-mail address: aaziz@ceme.nust.edu.pk (A. Aziz).

tile manufacturing, nanoelectronics, nanophotonics, and nanomagnetics.

Choi [5] first coined the term nanofluid by introducing the nanoparticles in the base fluids and theoretically demonstrated the feasibility of the concept of nanofluids. Khan and Pop [6] first studied the boundary layer flow of a nanofluid past a stretching sheet and gave numerical results of the problem. Makinde and Aziz [7] presented the nanofluid flow induced by linearly stretching sheet. The magnetohydrodynamics flow of nanofluid over a vertical stretching permeable surface with suction/injection was presented by Kandasamy et al. [8]. The numerical study of MHD boundary layer flow of a Maxwell nanofluid past a stretching sheet was accomplished by Nadeem et al. [9]. The work is further extended by Bhattacharya and Layek [10] while studying MHD boundary layer flow of nanofluid over an exponentially stretching permeable sheet. An unsteady MHD flow, heat and mass transfer over a stretching sheet with a non-uniform heat source/sink is studied by Shankar and Yirga [11]. Nanofluid flow over an unsteady stretching surface in the presence of thermal radiation was studied by Das et al. [12]. Effects of thermal radiation on boundary layer flow of nanofluids over a permeable stretching flat plate was discussed by Motsumi [13]. Khan et al. [14] extended the previous work by including the thermal radiation and viscous dissipation in unsteady MHD boundary layer flow of nanofluids.

http://dx.doi.org/10.1016/j.rinp.2016.09.017 2211-3797/® 2016 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Noghrehabadi et al. [15] observed the effects of partial slip boundary conditions on the flow and heat transfer of nanofluids. Zheng et al. [16] analysed the effects of velocity slip with temperature jump on MHD flow and heat transfer of nanofluid over a porous shrinking sheet. Ellahi [17] presented the analytical solutions for the MHD flow of non-Newtonian nanofluids in a pipe with two different temperature dependent viscosity models. Reddy et al. [18] carried out an analysis to investigate the influence of variable thermal conductivity and partial velocity slip on hydromagnetic two-dimensional boundary layer flow of nanofluid over a stretching sheet with convective boundary condition. Noghrehabadi et al. [19] carried out a study on the effects of variable thermal conductivity and viscosity on the natural convective heat transfer of nanofluid over a vertical plate. A comprehensive literature survey on variable thermophysical properties of nanofluids is presented in [20-28].

Mathematical model

In this present paper, our main objective is to investigate the effects of nanoparticle volume concentration, variable thermal conductivity, partial slip, thermal radiation and applied transverse magnetic field on an unsteady boundary layer flow and heat transfer of two-dimensional, incompressible, electrically conducting nanofluid over a porous stretching sheet. The velocity and thermal slip conditions are taken in terms of shear stress and the stretching sheet is moving with the non-uniform velocity cx

U(x, t) =

1- at'

where c is the initial stretching rate and is the effective stretching rate (with at < 1). The surface of the plate admits the partial slip conditions in the presence of transverse magnetic field. The induced magnetic field is considered negligible as compared to applied magnetic field. The temperature of the plate is Tw and the flow far away from the plate is uniform and in the direction parallel to the plate. The temperature far away from the plate is T„. The geometry of the flow model is given in Fig. 1.

The governing equations under boundary layer approximation for the problem along with the slip boundary conditions are

du dv .

@x+dy=

du du du 1nf d2 u rnfB2(t)u --h u--h v-= —------

dt dx dy pnf dy2

, , (dl dT dT

Wf dt+ u dX + v dy.

.± K m dT) dy VKf (T) dy) dy

u(x, 0) = U„ + AHnfdu' v(x, 0) = V„ u ! 0, and T ! T1t as y n

TX 0) = TW+D1( |

Here, u and v are velocities in x and y directions respectively, t is the time, inf is the nanofluid dynamic viscosity, pnf is the nanofluid density, anf is the electrical conductivity of the nanofluid, T is the temperature of the nanofluid, Tw(x, t) = Tm + j2^ is the sheet surface (wall) temperature, (Cp)nf is the specific heat capacity of the nanofluid, Vw shows the mass transfer at the surface with Vw > 0 for injection and Vw < 0 for suction, A1 = A0V1 - at is the velocity slip factor and D1 = D0V1 - at is the thermal slip factor with A0 and D0 be the initial velocity and thermal slip respectively.

Following Reddy et al. [18], Arunachalam [29] and Maxwell [30], the nanofluid's physical parameters are given as:

1nf = (1 - /)2 5 ' Pnf = (1 - /)Pf + /Ps; (pCp)nf = 0 - /)(PCp)f + /(pCp)s TT

Kf (T ) = f1

(Ks + 2jf) - 2/(Kf - Ks)'

. (Ks + 2Kf ) + /(Kf - Ks) ,

(8) (9)

(10) (11)

Here / is nanoparticle volume fraction coefficient, i, p{ and (Cp)f are the dynamic viscosity, density and specific heat capacity of the base fluid, ps and (Cp)s are the density and specific heat of the nanoparticles, Knf is thermal conductivity of the nanofluid, Kf and Ks define the thermal conductivity of base fluid and nanoparticles, respectively. The electrical conductivity of the nanofluid and the radiative heat flux are

3 rt - 1

dy 3k* dy2' 1 '

Here, as is the nanoparticle electrical conductivity, af is the base fluid conductivity, k* is the mean absorption coefficient and r* is the Stefan Boltzmann constant.

Method of solution

In order to solve the governing boundary value problem (2)-(6), we first employed the similarity transformation technique to transform the governing partial differential equations into a system of nonlinear coupled ordinary differential equation. The resulting ordinary differential equations are then solved numerically using MATLAB bvp4c code.

The following stream functions y) are introduced,

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which identically satisfies the continuity Eq. (2). The dimensionless stream function W(g) , 0(g) and the dimensionless similarity variable are

W(x ,y) =

\l Vf (1 - «tf

TW T on

Using Eqs. (7)-(15) the boundary value problem (2)-(6) transformed into

f"' + (1 - /)2

-M 1 +

Prj | 1 - / +

^ -/+/f {ff - f'2 - if+2 f' 'g)}

/ + / (f h' - f' h - A( h + gh')]

+ h" 1^1 + eh + j- PrNrj + eh'2 = 0 , (17)

f (0) = S, f (0) = 1 + \ 2 5 f '(0), h(0) = 1 + Xh' (0), (18) (1 - /) '

f (g)!0, and h(g)!0, as g (19)

where, Pr = is the Prandtl number, « = is thermal diffusivity

offlmd, Nr = f

is the thermal radiation parameter, A = « is

the unsteadiness parameter, M = is the magnetic parameter, S = -Vwy^-? is the suction/injection parameter, A = A0^if is the velocity slip parameter and X = is the thermal slip

parameter.

To get the numerical solutions of the system of ordinary differential Eqs. (16) and (17) along with conditions (18) and (19), we assume

y1 = f, y2 = f, y3 = f', y 4 = h, ys = h'. Using Eq. (20) Eqs. (16)-(19) become

y1 = y2;

y2 = y3 ;

y3 = -(1 - /)25

-M 1 +

(21) (22)

1 - / + {y1y3 - y2 - A(y2 + gya) }

y4 = y5 y5

f1 + KfNr + ey4)

Kf_ Knf

1 - / + / (qCl) {A(y4 + g yO

+y2y4 - y^} - e(y5)2

with the initial conditions

y1(0)=S, y2(0) = 1 +-

(1 - /)2 = 1 + Xy5(0), y5(0) = b.

5 y3(0), y3(0)=a, y4(0)

where a and b are unknown which are to be determined such that the boundary conditions y2(i) and y4 (1) are satisfied. To ensure the numerical accuracy, we have compared our results with Hayat [31] for several values of skin friction coefficient and Nusselt number, i.e., by assuming K = Pr = 1.0,A = Nr = 0.2, e = 0.0, X = 0.5, and / = 0.0. The comparison is shown in Table 1 and are found in excellent agreement. Thus, we are very much confident that the present results are accurate.

Results and discussion

In this section, the numerical results calculated for the velocity and temperature profiles are presented through graphs and tables. The computations are performed to study the effects of variation of unsteady parameter A, magnetic parameter M, velocity slip parameter K, nanoparticle volume fraction parameter /, suction and injection parameter S, thermal slip parameter X, thermal radiation Nr and variable thermal conductivity e on velocity and temperature profiles of Cu-water and Al2O3-water nanofluids. The behavior

of skin friction coefficient

and Nusselt number

0'(0) jj with the variation in physical parameters is also shown in Table 3.

The thermophysical properties of the base fluid water and the two different nanoparticles copper (Cu), Alumina (Al2O3) are given in Table 2.

In Figs. 2 and 3 the influence of unsteady parameter A on velocity and temperature profiles for Pr = 6.2, Nr = 0.1, S = 0.1, M = 1.0, K = 1.0, X = 1.0, e = 1.0 and / = 0.1 is presented. The graphs are plotted for both Cu-water and Al2O3-water nanoflu-ids. It is observed that the velocity decreases with the increase in unsteadiness parameter A. This in turn decreases the thickness of the momentum boundary layer. Moreover, the velocity attains it's maximum value near the surface but gradually decreases to zero at the free stream far away from the plate satisfying the boundary conditions, thus supporting the validity of the obtained numerical results. In Fig. 3 the temperature profiles 0(g) decrease with increase in unsteadiness parameter A for a given distance from the plate. This will enhance the rate of heat transfer and decreases the thickness of thermal boundary layer. The crossover point (g ~ 2.4) is also observed for temperature profile in Fig. 3, i.e., the temperature decreases with the increasing values of A before crossing over point whereas it increases slightly after this. We may explain this phenomenon near the boundary as increase in unsteadiness parameter accelerates the nanoparticles causing more collisions near the surface and makes the wall temperature higher than the ambient temperature and due to this high wall temperature nanoparticles move to cooler area causing an increase in the temperature profile. The comparison of curves in Figs. 2 and 3 shows the impact of A on temperature profiles is more pronounced than that on the velocity profiles. Moreover, it is observed from Table 3 that, as unsteadiness parameter A increases, both skin friction coefficient and Nusselt number increase for both Cu-water and Al2O3 water based nanofluids.

The influence of magnetic parameter M on velocity and temperature profiles is shown in Figs. 4 and 5 respectively. Here, Pr = 6.2, Nr = 0.1, S = 0.1, A = 0.2, K = 1.0, X = 1.0, e = 1.0 and / = 0.1. It is observed that the increase in M causes decrease in the velocity of nanofluid. This is due to the fact that the applied transverse magnetic field produces a body force, called the Lorentz

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Table 1

Comparison of values of -f(0) and -h(0) with Hayat [31].

M S -f'(0) T. Hayat -f"(0) Present -h (0) T. Hayat -h (0) Present

0.25 1.0 0.601575 0.601571 0.715271 0.715316

1.0 0.2 0.575633 0.575634 0.501501 0.501691

1.0 0.5 0.602285 0.602285 0.571204 0.57133

Table 2

Thermophysical properties of the base fluid and nanoparticles.

Physical Base fluid Nanoparticles Nanoparticles

properties Water Cu Al2Û3

Cp (J/kgK) 4179 385 765

q(kg/m3) 997.1 8933 3970

k(W/mK) 0.613 400 40

r(n.m)-1 0.05 5.96 x 107 10-12

' 0.25 0.2 0.15 0.1 0.05 0

Fig. 2. Velocity profiles for variation in unsteadiness parameter.

Fig. 3. Temperature profiles for variation in unsteadiness parameter.

301 force, which resists the motion of the fluid. Since the magnetic

302 parameter is inversely proportional to the density by M = ГВ-,

303 hence increase in M causes decrease in density and consequently

304 the temperature of the fluid rises. Table 3, concluded that the skin

305 friction coefficient increases with increase in M due to the influ-

306 ence of the Lorentz force. Whereas, decrease in Nusselt number

307 is observed due to the decrease in heat transfer rate.

308 Effect of variation of nanoparticle volume fraction / on temper-

309 ature and velocity profiles is illustrated in Figs. 6 and 7 for

Pr = 6.2, Nr = 0.1, S = 0.1, A = 0.2, M = 1.5, Л = 1.5, X = 1.5 and 310

e = 1.0. It is observed from Fig. 6 that an increase in / causes 311

decrease in velocity profiles for Cu water based nanofluids. 312

Whereas for Al2O3 water based nanofluids the behavior is slight 313

different i.e near the surface (at about g = 0.5) the velocity profiles 314

show the cross-over point, i.e., the velocity profiles decrease inside 315

the boundary layer, while they increase outside the boundary layer 316

(for details see, Hamad et al. [32]). In Fig. 7 the increase in / causes 317

an increase in temperature due to the fact that when the volume of 318

nanoparticles increases, the thermal conductivity also increases 319

which leads to the increase in thickness of thermal boundary layer. 320

From Table 3, the reduction in skin friction coefficient and Nusselt 321

number is observed with the increase in volume friction 322

parameter. 323

In Fig. 8 the effects of velocity slip parameter Л on the velocity 324

profile are presented. Here Pr = 6.2, Nr = 0.1, S = 0.1, A = 325

0.2,M = 1.0, X = 1.0, e = 1.0 and / = 0.1. The comparison of 326

curves shows that the fluid velocity within the boundary layer 327

decreases with an increase in the velocity slip at the boundary. 328

The increase in magnitude of the slip parameter allows more fluid 329

to slip past the plate and accordingly the flow velocity through the 330

boundary layer will decrease because pulling of the stretching can 331

be only partly transmitted to the fluid. The temperature profiles in 332

Fig. 9 show the increase in temperature and thermal boundary 333

layer thickness with an increase in velocity slip at the boundary. 334

It is observed from Table 3 that increase in Л leads to decrease in 335

skin friction coefficient both for Cu water and Al2O3 water based 336

nanofluids. As expected, the slip is to reduce the friction at the 337

solid fluid interface and thus reduces the skin friction coefficient. 338

Figs. 10-13 demonstrate the effects of variation of suction/ 339

injection parameter on temperature and velocity profiles, respec- 340

tively. Since applying suction (S > 0) leads to draw the amount of 341

fluid particles into the wall that's why increase in S causes decrease 342

in velocity of the nanofluids as shown in Fig. 10. Opposite behavior 343

is noted for injection (S < 0) in Fig. 12. The physical explanation for 344

such behavior is that; since more fluid is injected, the heated fluid 345

is pushed farther from the wall where due to less influence of the 346

viscosity, the flow is accelerated. It is also observed from Figs. 11- 347

13 that increasing S for the case of suction (S > 0) decreases the 348

temperature within boundary layer whereas increasing S for the 349

case of injection (S < 0) enhances the temperature profiles. That 350

is, the imposition of suction on the surface causes reduction in 351

the thermal boundary layer thickness and the injection causes an 352

increase in the thermal boundary layer thickness. It can be seen 353

from Table 3 that the skin friction coefficient and Nusselt number 354

show the gradual reduction with the increase in injection parame- 355

ter and the gradual raise with the increase in suction parameter. 356

Thermal conduction is the spontaneous transfer of thermal 357

energy from the region of high temperature to the region of low 358

temperature. Fig. 14 for Pr = 6.2, Nr = 0.1, S = 0.1,A = 0.2, M = 359

1.0, Л = 1.0, X = 1.0 and / = 0.1, show the effects of an increase 360

in the thermal conductivity parameter e on temperature profiles. 361

The increase in e results in, increase of fluid temperature across 362

the boundary layer. This in-turn increase the thickness of thermal 363

boundary layer. This is because Kf > Knf when e > 0 and hence an 364

increase in e results in increase of thermal conductivity, thereby 365

raising the temperature. Furthermore, the Nusselt number 366

Cu - water

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Table 3

Skin friction coefficient -f '(0) and Nusselt number -h (0) for Pr = 6.2, S = 0.1, X = 1.0.

Nr A M A e / -f''(0) Cu-water -f''(0) Al2Os - water -h (0) Cu-water -h (0) A,

0.0 0.63024 0.58571 0.66548 0.68687

0.1 0.2 1.0 1.0 1.0 0.1 0.64264 0.59908 0.71080 0.72807

0.5 0.65922 0.61703 0.77246 0.77893

0.5 0.60769 0.56619 0.73195 0.74576

0.1 0.2 1.0 1.0 1.0 0.1 0.64264 0.59908 0.71080 0.72807

1.5 0.66891 0.62470 0.69392 0.71323

0.5 0.97460 0.8838 0.75868 0.77303

0.1 0.2 1.0 1.0 1.0 0.1 0.64264 0.59908 0.71080 0.72807

1.5 0.48219 0.45604 0.56982 0.57870

0.0 0.45685 0.45685 0.44311 0.44311

0.1 0.2 1.5 1.5 1.0 0.1 0.49826 0.47230 0.54032 0.55071

0.2 0.53131 0.48870 0.64404 0.66905

0.1 0.5 0.71080 0.72807

0.5 0.2 1.0 1.0 1.0 0.1 0.56592 0.58454

1.0 1.5 0.47110 0.48884

0.0 0.77869 0.79130

0.1 0.2 1.0 1.0 0.5 0.1 0.74420 0.75933

1.0 0.71080 0.72807

Fig. 4. Velocity profiles for variation in magnetic parameter.

Fig. 5. Temperature profiles for variation in magnetic parameter.

367 decreases with the increase in e because large amount of heat

368 transfer causes decrease in temperature gradient. The effect of

369 variation of radiation parameter Nr on temperature profile is delin-

370 eated in Fig. 15 for Pr = 6.2, S = 0.1, A = 0.2, A = 1.0, X =

371 1 .0; e = 1 .0 and / = 0.1. Thermal radiation parameter is the ratio

372 of thermal radiation to the conductive radiation. Raising Nr depicts

Fig. 6. Velocity profiles for variation in volume fraction parameter.

Fig. 7. Temperature profiles for variation in volume fraction parameter.

the dominance of thermal heat transfer over conductive heat trans- 373

fer. Consequently, large amount of heat is transferred into the sys- 374

tem, raising the temperature of the boundary layer. Increase in Nr 375

leads to the decrease in Nusselt number as observed from Table 3. 376

This behavior is true for both the Cu and Al2O3 water based 377

nanofluids. 378

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■E 0.3

A = 0.5, 1.0, 1.5

Cu - water - Al203 - water

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fig. 11. Temperature profiles for variation in suction S > 0.

Fig. 8. Velocity profiles for variation in velocity slip parameter.

Fig. 9. Temperature profiles for variation in velocity slip parameter.

Fig. 12. Velocity profiles for variation in injection S < 0.

S = 0.1, 0.2, 0.3

0 1 2 3

6 7 8 9

Fig. 10. Velocity profiles for variation in suction S > 0.

Fig. 13. Temperature profiles for variation in injection S < 0.

379 Fig. 16 presented the reduction in the thickness of thermal

380 boundary layer due to the increase in thermal slip parameter X.

381 The increase in thermal slip parameter causes the less transfer of

382 heat from sheet to the fluid which leads to decrease in boundary

383 layer temperature. The temperature gradient at the surface also

384 decreases due to the increase in thermal slip, and is observed

385 through variation in Nusselt number given in Table 3. This also rep-

386 resent less heat transfer rate at the surface.

Concluding remarks

In this paper, we studied the MHD slip flow and heat transfer of 388

nanofluids with variable thermal conductivity and thermal radia- 389

tion in a solar collectors modeled as a over a porous stretching sur- 390

face. We have summarized our results based on the key 391

parameters such as, unsteadiness parameter, MHD parameter, 392

nanoparticle volume concentration parameter, thermal conductiv- 393

ity parameter, velocity and thermal slip parameters, suction/injec- 394

Cu - water

AL03 - water

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Fig. 14. Temperature profiles for variation in thermal conductivity parameter.

Fig. 15. Temperature profiles for variation in thermal radiation parameter.

Fig. 16. Temperature profiles for different values of Thermal slip parameter X.

395 tion parameters and thermal radiation parameter. We can con-

396 clude the following results from our investigation:

397 1. The increase in suction (S > 0) and unsteadiness parameter A

398 decreases the velocity and temperature distribution within

399 the boundary layer. This will enhance the heat transfer rate

400 and reduce the thickness of momentum boundary layer.

401 2. The increase in magnetic parameter M, nanoparticle volume

402 concentration parameter / and velocity slip parameter A

Physics xxx (2016) xxx-xxx 7

decreases the velocity and increases the temperature distribu- 403

tion within the boundary layer. 404

3. The Cu — water based nanofluids are better thermal conductor 405 than Al2O3 — water based nanofluids for solar collectors. 406

4. The increase in thermal conductivity parameter e and thermal 407 radiation parameter Nr raised the fluid temperature across the 408 boundary layer. This will result in an increase in the thickness 409 of thermal boundary layer. 410

The present simplified model can be generalized to include the 411

effects of variable viscosity, variable porosity, multidimensional 412

MHD slip flow and heat transfer of non-Newtonian regular and 413

nanofluids. Clearly there is an opportunity for experimental work 414

on these systems. 415

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