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Results in Physics xxx (2016) xxx-xxx

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Results in Physics

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Mixed convective flow of Maxwell nanofluid past a porous vertical stretched surface - An optimal solution

M. Ramzana'*, M. Bilalb, Jae Dong Chungc, U. Farooqd

a Department of Computer Science, Bahria University, Islamabad Campus, Islamabad 44000, Pakistan b Department of Mathematics, Faculty of Computing, Capital University of Science and Technology, Islamabad, Pakistan c Department of Mechanical Engineering, Sejong University, Seoul 143-747, Republic of Korea d Department of Mathematics, CIIT, Islamabad Campus, 44000, Pakistan

ARTICLE INFO

ABSTRACT

Article history:

Received 2 November 2016

Received in revised form 16 November 2016

Accepted 18 November 2016

Available online xxxx

Keywords: Mixed convection Maxwell nanofluid Soret and Dufour effects Optimal solution Porous medium

Present investigation is devoted to examine the mixed convective flow of Maxwell nanofluid with Soret and Dufour effects through a porous medium. Effects of variable temperature and concentration over a linearly permeable stretched surface are also taken into account. An optimal solution is obtained for the highly nonlinear set of differential equations using BVPh 2.0 Mathematica package. Graphs of different emerging pertinent parameters against velocity, temperature and concentration distributions are plotted and discussed accordingly. Numerically tabulated values of local Nusselt and Sherwood numbers are also part of this investigation. It is witnessed that concentration field is decreasing and increasing function of Brownian motion and thermophoretic parameters respectively. Further, opposite behavior of Soret number on temperature and concentration distributions is seen.

© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://

creativecommons.org/licenses/by/4.0/).

Introduction

The topic of heat transfer via porous media has been a hot subject due to its technological and engineering applications. Examples may include packed sphere beds, electro chemical processes, grain storage, insulation for buildings and lining of nuclear reactors, regeneration of heat exchangers, chemical catalytic reactors, and solar power collectors. Flagged investigations in this core area include numerous studies like Shehzad et al. [1] who examined 3D flow of Casson fluid through porous media. They carried out analysis in the presence of heat generation/absorption. Sheikholeslami et al. [2] debated flow of viscous nanofluid through a porous medium with four different nano materials and water as base fluid. Hayat et al. [3] explored influence of convective boundary conditions on magnetohydrodynamic (MHD) nanofluid flow through a porous medium over an exponentially stretching sheet using series solution technique. Makinde et al. [4] studied effects of unsteady magnetohydrodynamic, thermal radiation, chemical reaction, and thermophoresis on a vertical porous plate. They employed sixth order RK-technique accompanied by Nachtsheim and Swigert's shooting method. It was noticed that skin friction coefficient decreases and local Nusselt number increases against gradual

* Corresponding author. E-mail address: mramzan@bahria.edu.pk (M. Ramzan).

growing values of unsteady viscosity parameter. Extensive literature is also available pertaining flows through porous medium with most recent investigations referred at [5-7].

Recent studies have given a significant attention to non-Newtonian fluid flows which are produced by stretched surfaces. The non-Newtonian flows have wide range applications in engineering including aerodynamic emission of plastic films, thinning and annealing of copper wires and liquid film condensation process etc. [8]. Unlike viscous fluids, an obvious hurdle in mathematical modelling of these fluids is that a single constitutive equation cannot exhibit all characteristics of these fluid structures. That is why several non-Newtonian fluids models have been suggested by researchers in the literature. Maxwell fluid which is a class of viscoelastic fluid, can be quoted to represent the characteristics of fluid relaxation time. Here, shear-dependent viscosity's complicated effects are excluded and allows one to focus on the influence of elasticity of fluid on boundary layer characteristics. A pioneering work by Harris [9] arguing 2D flow of upper-convected Maxwell fluid encouraged follower researchers to investigate more avenues in this direction. Sadeghy et al. [10] proposed local similarity solutions by four dissimilar approaches with the findings that velocity decreases with an increase in local Deborah number. They considered Maxwell fluid flow over a moving flat plate known as Sakiadis flow. Kumari and Nath [11] discussed numerical solution of mixed convection stagnation point Maxwell fluid flow using finite differ-

http://dx.doi.org/10.1016/j.rinp.2016.11.036 2211-3797/® 2016 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY license (http://creativec0mm0ns.0rg/licenses/by/4.0/).

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Nomenclature

a ,b, c dimensional constants Shx Sherwood number

C concentration of fluid Sr Soret number

cP specific heat T temperature of fluid

Cs concentration susceptibility Tm mean fluid temperature

Cw concentration on wall Tw wall temperature

C ambient concentration T 1 1 Ambient temperature

DB Brownian motion coefficient (u, v) velocity components

De mass diffusivity Uw(x) stretching velocity alongx -axis

Df dufour number V 0 stretching velocity alongy -axis

DT thermophoretic diffusion coeff. (x, y) coordinate axis

f ' dimensionless velocity am thermal diffusivity

g gravitational acceleration bT coefficient of thermal expansion

Grx Grashof number be coefficient of concentration expansion

jw mass flux b Deborah number

K thermal conductivity c porosity parameter

Kt thermal diffusion ratio P density of fluid

K permeability constant k mixed convection parameter

Le Lewis number fluid relaxation time

N Buoyancy ratio parameter V kinematic viscosity

Nb Brownian motion parameter w stream function

Nt thermophoresis parameter 0 dimensionless temperature

Nux Nusselt number g similarity variable

Pr Prandtl number / dimensionless concentration

Qw surface heat flux s ratio of effective heat capacity of nanoparticle and base

Rex Reynolds number fluid

S Suction parameter

ence method. Hayat et al. [12] found series solution of stagnation point magnetohydrodynamic over a stretching surface of an upper-convected Maxwell fluid. Motivated from above works, researchers have investigated two and three dimensional Maxwell fluid flows in numerous scenarios (see Shafique et al. [13] Awais et al. [14], Nadeem et al. [15], Qayyum et al.[16], and Abbasi et al. [17]).

Nanofluids are suspended ultra fine particles in base fluids (like water and organic liquids) with a size less than 100 nm. These nanoparticles consist of metals and their oxides, therefore, they have significantly higher thermal conductivity than base fluid. Recently, carbon nanomaterials with more diverse nature industrial applications including nanotubes [18,19], carbon nanoparti-cles [20,21], nanofibres [22], nanowires [23] and carbon nanorods [24] have been found in various nanostructures. A novel idea of "nanofluid" in heat transfer processes presented by Choi [25] has revolutionized the modern engineering and technological world. Nanofluids have numerous applications in metallurgical and chemical sectors, transportation, production of micro-sized products, thermal therapy to cure cancer, ventilation, and air-conditioning [26]. Following this coined work, Buongiorno [27] presented a more detailed study of nanofluids highlighting salient features of thermophoresis and Brownian motion. Using proposed model of Buongiorno, Kuznetsov and Nield [28] discussed nano-fluid flow past a vertical plate with convective boundary layer. Khan and Pop [29] conducted a comprehensive analysis of nanofluid flow over a stretched surface and discussed effects of thermophoresis and Brownian motion heat transfer using Keller-box numerical technique. Turkyilmazoglu [30] considering different nanoparticles like Ag, Cu, TiO2 and Ai2O3 examined flow of hydro-magnetic viscous fluid accompanied slip condition. Makinde et al. [31] discussed numerically magneto nanofluid neighboring stagnation point in the presence of buoyancy force and convective boundary conditions using RK-method of fourth order of shooting technique. Rashidi et al. [32] debated flow of MHD nanofluid over a permeable rotating disk with discussion of entropy generation

and explored that such study is really beneficial in energy conversion for mechanical systems of space vehicles with nuclear propulsion and energy generators. Mustafa et al. [33] studied nanofluid flow near a stagnation point over an exponentially stretched surface. They found the solution of the problem using Homotopy Analysis method (HAM) and MATLAB's built in bvp4c software to calculate numerical solution and found that thermophoretic impact strengthens with growth in nanoparticle volume fraction. Sheikholeslami and Ganji [34] examined Cu-water nanofluid flow between parallel plates. They used Maxwell-Garnetts and Brink-man models were to find effects of viscosity and thermal conductivity. Kuznetsov and Nield [35] reviewed flow of nanofluid through a vertical plate with convective boundary conditions and disclosed that control of nano particle fraction is passive rather than active. Afterwards, researchers have extensively investigated about two and three dimensional nanofluid structures [36-45].

To address the aforesaid subjects, need was felt to model a mathematical problem that encircle all issues and solve them with an appropriate method. Due to obvious restrictions in numerical methods [46], analytical techniques are considered as a replacement by the researchers. Amongst these, perturbation technique is most common and extensively practiced method to address a variety of engineering and science problems [47]. This tool is highly dependent on small/large parameters which is considered as a major disadvantage of this method and restricts it to handle highly nonlinear problems. To elude this constraint, non-perturbation methods like Adomian decomposition method [48] and variational iteration technique [49] were introduced. But these methods cannot guarantee series solutions' convergence. However, Liao's proposed homotopy analysis method (HAM) [50] has answered this question. This technique gives solution to highly nonlinear equations with an ample freedom to guarantee convergence of the problem. Additionally, unlike numerical methods, HAM can be applied to the problems having boundary conditions with far field characteristics. Further to HAM, Liao's newly proposed Optimal HAM [51] is a strong tool that guarantee the conver-

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gence of series solution. His idea of averaged squared residual error has led to an optimal convergence which triggered the convergence of series solution.

We here discussed the effects of Soret/Dufour and mixed convection on the flow of Maxwell nanofluid in the presence of variable temperature and concentration conditions. The proposed highly nonlinear problem is solved by using BVPh 2.0 Mathematica package [52,53] to find an optimal solution. Numerous graphs are drawn to highlight the impact of various emerging parameters against involved distributions. Numerically calculated values of local Nusselt and Sherwood numbers are shown in the form of table and are well deliberated.

Problem formulation

We assume two dimensional Maxwell nanofluid flow past a vertical stretched surface (with velocity uw(x)) with variable temperature Tw(x), variable concentration Cw(x), uniform ambient temperature Tm, and uniform ambient concentration in a porous medium. We also consider amalgamated effects of Soret and Dufour with mixed convection. The buoyancy effects and density variation are also considered. Boussinesq approximation is taken for both temperature and concentration profiles (see Fig. 1). The governing equations representing the proposed model are [54]:

du dv .

dx+dy = 0

du du d2 u , 2 @ u „ _

Utt + v^- = V — - kA u2 —y + v2 —y + 2uv dx dy dy2 \ dx2 dy2 dxdy

+ g[ßi(T - Ti) + bc(C - Ci

dT dT d2T DeKT d2C

u dx + v dy - a™

dy2 + ~CsCp

■+ s

D @c dT Di (dT. B dy dy + Ti V dy

d2C Dt d2T

@c d£_DeKi d2! D — —

u dx + v dy - Tm dy2 + db dy2 + Ti dy2 ' with appropriate boundary conditions

u - uw(x) - ax, v - -V0, T - Tw(x) - Ti + bx, C - Cw(x) - Ci + cx at y - 0,

U ! 0, dUU ! 0, T ! T1, C ! Ci as y -

Here, velocity components u and v are along x and y-axes respectively. Also, Db, T , C,g, Dj, am, bj, k1, and s = (pd)p/(pd)f are the Brownian motion coefficient, fluid temperature, nano particle concentration, gravitational acceleration, thermophoretic diffusion coefficient, thermal diffusivity, coefficient of thermal expansion, relaxation time, and the ratio of effective heat capacity of the nanoparticle to the fluid respectively. Further, (a > 0) and (c > 0) are positive constants. However, b > 0 denotes heated plate (Tw > Tm) and for a cooled surface (Tw < Tm) respective constant is b < 0. Using the following transformations [54]

w - xvavf (g), 8(g)--, i

Tw Tcx

Satisfaction of Eq. (1) is obvious and Eqs. (2)-(5) come to the form

f" + ff" - f02 + ß(2ff'f" - f2f") - yf' + k(8 + N/) - 0,

8" + f 8 - 8f + Df /'' + Nb8/' + Nt82 - 0,

(11) (12)

/" + PrLe(f /' - /f) + SrLed" + 8' = 0,

f (0)=S, f(0) = 1, 8(0) = 1, /(0) = 1,

f (l) ! 0, f"(l) ! 0, 8(l) ! 0, /(l) ! 0,

with Nt,Le = am/DB,Nb,Df,Pr = v/am , k,b(p 0),N, y, and Sr are thermophoresis parameter, Lewis number, Brownian motion parameter, Dufour number, Prandtl number, dimensionless mixed convection parameter, Deborah number, dimensionless concentration buoyancy parameter, dimensionless porosity parameter, and Soret number respectively. Defining these parameters

2 _ gßTb _ gßT (Tw-Ti)x3/V2 _ Grx N_ ßc (Cw-Ci) k a2 uWx2/v2 Re2 , ßT(Tw-Ti) ,

y_ß_a 2 D _ DeKT (Cw-Ci)

y - aK, ß - Uk1, Uf - CsCp(Tw-Ti)v ,

S DeKT(Tw-Ti) Nb _ (.pd)pDB(Cw-Ci) Nt _ (pd)„DT(Tw-Ti)

T /V it r \ , Nb n , Nt n •

Tm am (Cw - Ci )

(qd)fV

(pd)fTiV

Here, Rex = uwx/v, Grx = gbT(Tw - T1)x3/o2 are the local Reynolds and Grashof numbers. Moreover, k > 0 , k < 0, and k = 0 depict supporting flow (heated plate), opposing flow (cooled plate) and forced convection flow. Moreover, Ncan take positive values (N > 0) and negative values (N < 0) with N = 0 (in the absence of mass transfer). The local Nusselt and Sherwood numbers are symbolized by:

Fig. 1. Schematic flow problem.

Nu — xqw Sh — xjw NUx - k(Tw - Ti), Shx - Db(Cw - Ci) '

where qw and jw are represented by

qw --k{S)y-0 Jw --DB{Dy-0-

In non-dimensional, local Nusselt, and Sherwood numbers are presented as

Re-1/2NUx - -8(0), Re-1/2Shx - -/'(0).

Series solution development

Here, we intend to interpret the convergence of the series solutions by renowned Optimal Homotopy analysis method (OHAM) [50,52]. The initial estimates and the respective operators are required for the homotopic solutions. For the present flow, these are depicted as follows:

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f>(g) = S + 1 - exp(-g), 00(g) = exp(-g) , /0(g) = exp(-g),

L d3f df „ h „ ,

Following the foot steps given in [50]. The general solutions of Eqs. (7)-(10) are given by

fm(g) -fm(g) + Cl + C2exp(g) + C3exp(-g), (19)

hm(g) - hm(g) + C4exp(g) + C5exp(-g) , (20)

/m(g) - /m(g) + Ceexp(g) + C7exp(-g), (21)

where f'm(g), h„(g) and /m(g) represent the special solutions and

C2 - C4 - C6 - 0 , Ci - -C3 -fm(0),

,@f m(g)

C5 = -hm(0), C7 = -/m (0) ,

with Q (i = 1 - 7) are the arbitrary constants.

Optimal solution

As suggested by Liao [51], averaged squared residual errors that can result in excellent approximations of optimal convergence control parameters are assumed to be:

e5 = — V

m k+1 u

Nfijy (g) , E?(g) , £/(g)

i=0 i=0

1 k +1S

n* Ë? (g), E?(g) , £/(g) , i=0 i=0

fm k +

N/ £/ (g), £%) , £/(g)

i=0 i=0

g=jdg.

g=jdg.

g=jdg.

•dg ,

•dg ,

•dg (25)

where k is an integer. The overall squared residual error ecm is given

ft = ef

e + e+ f '

with dg = 0.5, and k = 20. Mathematica BVPh 2.0 package is used to minimize these errors. At 3rd order of approximation, the values of optimal convergent control parameters are hf = -0.75293 , hh = -0.90738 and h^ = -0.933951 with total averaged squared error em = 0.000141212. At 3rd order of approximation with S = 0.5 , y = k = Pr = Le = N = 1, Sr = 0.2 , Nt = Df = b = 0.1, and Nb = 0.8, the values of averaged squared residual errors are given in Table 1. It can be observed that increasing values of higher order of approximations results in decrease in averaged squared residual errors. Fig. 2 is portrays the propensity of average squared residual error Co = (hf = hh = h^) versus an optimal value of all three auxiliary control parameters hf , hh and h^ at 2nd, 4th and 6th iterations using Mathematica package BVPh 2.0. It can be perceived that increasing values of order of iterations give rise to optimal convergence control parameters to a -0.67 converging value.

Table 1

Averaged squared residual errors for varied order of approximations.

m ff fh f/ cm

2 5:79 x 10-4 1:48 x 10-4 7:14 x 10-4

6 1:42 x 10-4 4:05 x 10-6 7:05 x 10-6

10 2:10 x 10-5 8:18 x 10-7 3:70 x 10-6

16 1:41 X 10-5 6:30 x 10-8 3:09 x 10-6

20 1:07 x 10-5 1:42 x 10-12 2:82 x 10-6

26 1:06 X 10-5 5:26 x 10-14 2:66 x 10-6

30 1:01 X 10-5 1:24 x 10-16 1:33 x 10-6

10' Iff Iff' Iff1 Iff' Iff' Iff' Iff'

\ Total a ver axai squared residua/ error \ ai different orders of iteration — • — • — ■ — 2nd order — — — — 4 th order - 6th order

. —• " x/

.«-•"' / / .___•—" / /

\\ /■ /

Fig. 2. Minimum averaged squared residual errors for 2nd, 4th, and 6th order of approximations.

Results and discussion

The purpose of this section is to deliberate the significant characteristics of promising parameters on velocity, temperature, and nanoparticle concentration profiles. Fig. 3 portrays the impact of suction parameter S on the velocity profile. It is witnessed that velocity field is diminishing function of S. Impact of Deborah number b on the velocity distribution is given in Fig. 4. It is noticed that velocity profile is a waning function of Deborah number. The effect of porosity parameter y on velocity field is depicted in Fig. 5. It is witnessed that velocity distribution is dwindling function of porosity parameter. Physically, an increase in resistance against the fluid flow is observed by increasing thickness of porous medium which results in decrease in fluid velocity. Fig. 6 shows the assisting flow (k > 0) which speed up the fluid's flow for positive gravitational

ß = Df = Nt = 0.1, Sr = 0.2, y = 2.0, X = N = Pr = Le= 1.0, Nb = 0.8,

2 3 4 5

Fig. 3. Effect of S on f'(g).

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S = Df = Nt = 0.1, Sr = 0.2, y = 2.0, A = N = Pr = Le = 1.0, Nb = 0.8,

Fig. 4. Effect of b on f (g).

ß = Df = Nt = 0.1, Sr = 0.2, y = 2.0, S = 0.5, N = Pr = Le = 1.0, Nb = 0.8,

12 3 4

Fig. 7. Effect of k < 0 on f' (g).

ß = Df = Nt = 0.1, Sr = 0.2, 5 = 0.5, X = N = Pr = Le=1.0, Nb = 0.8,

y = 0.0, 0.5,1.2,2.0

Fig. 5. Effect of c on f (g).

ß = Df = Nt = 0.1, Sr = 0.2, y = 2.0, 5 = 0.5, N = X = Le = 1.0, Nb = 0.8,

Pr = 0.7, 1.0,1.4,2.0

2 3 4 5

Fig. 8. Influence of Pr on 0(g).

f'fo) m

337 force and hence results in an increase in fluid's velocity. On the

338 other hand, Fig. 7 depicts the opposing flow (k < 0) which resists

339 the fluid's flow. In Figs. 8 and 9, we observe the effect of Pr on tem-

340 perature profile 0(g) and nanoparticle concentration profile /(g).

341 Increasing values of Prandtl number cause an attenuation in both

342 temperature and nanoparticle concentration distributions. This is

343 because of the fact that a feebler thermal diffusivity is witnessed

344 for higher Prandtl number. Figs. 10 and 11 exhibit the effect of

345 the Dufour number Df on 0(g) and /(g). It is found that tempera-

346 ture and concentration profiles increase and decrease respectively

versus increasing values of Dufour number. Higher values of 347

Dufour number lower temperature and ultimately larger tempera- 348

ture distribution is observed. On the contrary, an opposite behavior 349

is witnessed in case of concentration field. Figs. 12 and 13 illustrate 350

that the Soret number Sr decreases temperature profile while there 351

is an increase in concentration profile and boundary layer thick- 352

ness. Higher temperature difference and a lower concentration dif- 353

ference are observed because of increasing values of the Soret 354

number. This variation in the temperature and concentration dif- 355

ferences is liable for the decrease in the temperature and an 356

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357 increase in the concentration. It is also noticed that the Dufour and

358 Soret numbers have fairly contrary effects for temperature and

359 nanoparticle concentration fields. The consequences of Brownian

360 motion parameter Nb on temperature and concentration distribu-

361 tion are depicted in Figs. 14 and 15. It is examined that tempera-

362 ture profile is larger for higher values of Brownian motion

363 parameter. An increase in Brownian motion parameter Nb amplify

364 the random motion of the fluid particles which produces more heat

365 and reduces the concentration of the fluid. Figs. 16 and 17 demon-

366 strate the influence of thermophoresis parameter Nt on tempera-

367 ture and nanoparticle concentration fields. It is perceived that

368 with an increase in thermophoresis parameter both the tempera-

ture profile and thermal boundary layer thickness also increase. 369

It is also shown that this enhancement in thermophoresis param- 370

eter pushes the nanoparticles away from the hot surface which 371

results in an increase in volume fraction distribution. 372

A comparison in the limiting case is presented in Table 2, where 373

a very good agreement is observed for the Nusselt number when 374 different values of suction/injection parameter and Prandtl number 375

are considered. 376

Table 3a(a) and 3b(b) show the values of the local Nusselt 377

number NuxRe-1/2 and the local Sherwood number ShRe-1/2. The 378

magnitude of the local Nusselt number increases for S, k,N, Pr 379

and Sr. However, it decreases for values of Nb, b, Df, y, Le, and Nt. 380

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I = D, = Nt = 0.1, Sr = 0.2, y = 2.0, S =0.5, N = Pr = Le = 1.0, A = 1.0,

Fig. 16. Influence of Nt on 6(g).

ß = Df = Nt = 0.1, Sr = 0.2, y = 2.0, S = 0.5, N = Pr = Le= 1.0, A = 1.0,

Nt = 0.0, 0.2, 0.5, 0.9

1 2 3 4 5 6

Fig. 17. Influence of Nt on /(g).

Table 2

Comparison of -6 (0) for some values of Pr and S when

ß = y = k = Df = Nb = Nt = / = 0.

S Pr Ishak et al. [55] Hayat et al. [56] Present

-1.5 0.72 0.4570 0.4570273 0.4570271

1 0.5000 0.5000000 0.5000000

10 0.6542 0.6451648 0.6451645

0 0.72 0.8086 0.8086314 0.8086313

1 1.0000 1.0000000 1.0000000

3 1.9237 1.92359132 1.9359130

1.0 3.7207 3.7215968 3.7215958

1.5 0.72 1.4944 1.4943687 1.4943680

1 2.0000 2.0000621 2.0000620

10 16.0842 16.096248 16.096232

381 The magnitude of local Sherwood number decreases for increasing

382 values of b, y, Sr and Nt whereas it increases for large values of

383 Df, S, N, Pr, k, Nb, and Le.

384 Conclusions

385 It is of great interest in this exploration to examine effects of

386 mixed convection, Soret and Dufour past a permeable medium of

387 Maxwell nanofluid flow. Effects of variable temperature and con-

388 centration over a linearly porous stretched surface are also taken

389 into account. An optimal solution is obtained for the highly nonlin-

390 ear set of differential equations using BVPh 2.0 Mathematica pack-

Table 3a

Local Nusselt number NuxRe~1/2 and the local Sherwood number ShRe~1/2 against values of y, k,N, S, b and Pr when Df = 0.1,Le = 1, Sr = 0.2,Nb = 0.8 and Nt = 0.1 are fixed.

S ß y k N Pr -6 (0) -/' (0)

0.0 0.1 2.0 1.0 1.0 1.0 0.71104 0.89301

0.3 0.79696 1.01679

0.5 0.85983 1.10661

0.9 0.99873 1.30359

0.5 0.0 0.86690 1.11816

0.2 0.85263 1.09559

0.4 0.83795 1.07326

0.1 0.5 0.91493 1.19556

1.0 0.89498 1.16286

1.5 0.87680 1.13337

2.0 0.5 0.81772 1.03883

0.8 0.84445 1.08202

1.2 0.87379 1.12906

1.0 -0.2 0.80842 1.02316

-0.1 0.81357 1.03149

0.5 0.84071 1.07596

1.0 0.7 0.77431 0.84566

1.2 0.88622 1.27852

1.5 0.89045 1.53416

Table 3b

Local Nusselt number NuxRe~1/2

and the local Sherwood number ShRe,

against

values of Sr, Nb, Df,Le, and Nt when S = 0.5, ß = 0.1, y = 2.0, k = N = Pr = 1.0 are fixed.

Df Le Sr Nb Nt -6' (0) -/' (0)

0.0 f 1.0 0.2 0.8 0.1 0.92430 1.08556

0.2 0.79244 1.12890

0.4 0.64798 1.17570

0.1 0.7 0.93603 0.82755

1.2 0.81957 1.28184

1.5 0.76852 1.53285

1.0 0.0 0.82745 1.21843

0.1 0.84341 1.16386

0.4 0.89416 0.98468

0.2 0.95038 0.77806

0.3 1.07399 0.89003

0.5 0.97908 1.02149

1.0 0.78921 1.14098

0.8 0.0 0.87116 1.17100

0.3 0.83699 0.98765

0.5 0.81388 0.88051

age. Consideration of the problem along with its proposed solution is unique and has been not discussed in the literature before. The significant findings of the present study are listed below:

• Nanoparticle concentration distribution is a decreasing and increasing function of Nb and Nt.

• Velocity distribution reduces with an increase in values of S.

• Q and / decrease with growing values of Pr.

• The impact of Df on Q and / are opposite.

• Local Nusselt and Sherwood numbers are larger for increasing values of S and k.

Competing interests

The authors have not any competing interests in the manuscript.

Acknowledgement

This research is supported by Korea Institute of Energy Technology Evaluation and Planning (KETEP) granted financial resource

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from the Ministry of Trade, Industry & Energy of Korea (No. 20132010101780).

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