Results in Physics xxx (2017) xxx-xxx

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Results in Physics

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

Simultaneous effects of heat generation/absorption and thermal radiation in magnetohydrodynamics (MHD) flow of Maxwell nanofluid towards a stretched surface

Tasawar Hayata'b, Sajid Qayyuma'*, Sabir Ali Shehzadc, Ahmed Alsaedib

a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

b Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, P. O. Box 80207, Jeddah 21589, Saudi Arabia c Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal 57000, Pakistan

ARTICLE INFO

ABSTRACT

Article history:

Received 10 October 2016

Received in revised form 29 November 2016

Accepted 6 December 2016

Available online xxxx

Keywords:

Maxwell nanofluid

Magnetohydrodynamic (MHD)

Thermal radiation

Nonlinear convection

Flux conditions

Heat generation/absorption

Mathematical analysis of magnetohydrodynamic (MHD) three-dimensional nonlinear convective flow of Maxwell nanofluid towards a stretching surface is made in this article. Characteristics of heat transfer are examined under thermal radiation, heat generation/absorption and prescribed heat flux condition. Nanofluid model includes Brownian motion and thermophoresis. Dimensional nonlinear expressions of momentum, energy and concentration are converted into dimensionless systems by invoking suitable similarity variables. A well-known homotopic technique is implemented for dimensionless expressions. Impact of different quantities on velocities, temperature and concentration are scrutinized graphically and discussed in detail. The expressions of Nusselt and Sherwood numbers are calculated and addressed comprehensively. It is also seen that thermal radiation parameter enhances the temperature field and heat transfer rate.

© 2016 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/).

Introduction

The enhancement of heat transport characteristics through nanofluids over moving sheets is a hotspot topic of investigations nowadays due to its vast implications in industrial and technological processes like wire drawing, production of plastic films, melt spinning process, glass fiber technology and many more. The thermal engineers and scientists are working to recognize the unexpected thermal properties of nanoliquids. In fact, a nanoliquid is a combination of homogenous mixture of nanoparticles and ordinary base liquid. There exist various type common fluids such as organic liquids (refrigerants, ethylene, tri-ethylene glycol), water, lubricants, oils, bio liquids and polymeric solution. In the recent literature, it can be seen that the nanoliquids have great capability to improve thermal conductivity and thermal performance of ordinary liquids. The energy efficiency can also be enhanced by the utilization of nanoliquids. Heat transport enhancement effectiveness may depend on type of material, shape of particle and number of submerged nanoparticles. The nanoliquids can be used in the formation and structural process of MHD power generators, cooling of nuclear reactors, fiber production in textile, petroleum reser-

* Corresponding author. E-mail address: sajidqayyum94@gmail.com (S. Qayyum).

voirs, geothermal energy, vehicle transformer, safer surgery and cancer therapy processes. Choi [1] provided an experimental work on mechanism of various type nanoparticles and concluded that the nanoliquid is best suitable candidate for the betterment in heat transfer of ordinary liquids. Buongiorno [2] presented a detailed discussion on the effectiveness of nanoparticles in convective transport of ordinary fluids. Significance of CuO-water nanoparti-cles on surface of heat exchangers has been experimentally addressed by Pantzali et al. [3]. Hamad et al. [4] analyze the laminar flow of viscous liquid in a porous medium filled with nanopar-ticles. Combined impact of convection and viscous heating in nanofluid flow has been addressed by Pal and Mandal [5]. Some recent advancements in nanofluid flow can be seen in the attempts [6-24].

Convection over a heated/cooled moving sheet is one of fundamental aspect of the problems of heat and mass transport. The practical situation where free convection can take place along with external force is generally termed as ''mixed convection". The investigations on mechanism of mixed convection flows have gained special focus of recent researchers due to its practical and physical relevance in the applications of astrophysics, geophysics and modern technological devices. Turkyilmazoglu [25] computed and elaborated the analytical solution of mixed convective flow of second grade liquid with heat transport. Here flow generation is

http://dx.doi.org/10.1016/j.rinp.2016.12.009 2211-3797/® 2016 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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T. Hayat et al./Results in Physics xxx (2017) xxx-xxx

caused due to impermeable stretching boundary. Sheikholeslami et al. [26] addressed the impact of natural convective flow of CuO-water nanofluid filled in a cavity with sinusoidal walls. Mixed convective heat transport phenomenon of power law liquid due to moving conveyor is described by Sui et al. [27]. Mahantesh et al.

[28] studied the behavior of mixed convective chemically reactive flow of nanoliquid over stationary/moving surface. Abbasi et al.

[29] reported the thermal and solutal convection in 2D-flow of Maxwell fluid in presence of heat sink/source and nanoparticles.

The proper understanding of radiative heat transport mechanism is quite essential for the standard quality product in industrial processes. The role of radiative heat transfer is quite phenomenal in various engineering manufacturing processes like hypersonic fights, space vehicles, gas turbines, nuclear power plants, gas cooled nuclear reactors etc. Pal and Mandal [30] has explored the impacts of radiative convection in MHD flow of nanoliquid generated by a nonlinear boundary. Lin et al. [31] considered the Marangonic convection in laminar flow of copper-water nano-liquid driven by thermally exponential temperature. A numerical treatment for MHD flow of Ai2O3-water nanoparticles through thermal radiation is provided made by Sheikholeslami et al. [32]. Shehzad et al. [33] considered the effect of magnetic field and thermal radiation in laminar flow of 3D-Jeffrey nanofluid by a bidirectional moving sheet.

In this work, our aim is to have a three-dimensional flow of Maxwell fluid by a bidirectionally moving sheet. The literature survey shows that mostly the problems have been dealt with the consideration of two-dimensional flows. The three-dimensional problem is more realistic. Further, the present problem is coupled due to consideration of nonlinear thermal and concentration convection. The heat and mass species expressions are modeled in presence of motion of Brownian, thermophoresis and heat source/sink effects. The considered flow is thermally radiative through consideration of thermal radiation. The temperature and mass species flux conditions are taken for the analysis of present flow model. The governing mathematical expressions are first made non-dimensional and then treated by homotopic procedure [34-42] for the development of analytical solutions. The solutions acquired by HAM are preferred than the numerical solutions in perspective of the following points. (i) HAM gives the solutions within the domain of interest at each point while the numerical solutions hold just for discrete points in the domain. (ii) Algebraically developed approximate solutions require less effort and having a sensible measure of precision when compared to numerical solution which are more convenient for the scientist, an engineer or an applied mathematician. (iii) Although most of the scientific packages required some initial approximations for the solution are not generally convergent. In such conditions approximate solutions can offer better initial guess that can be readily advanced to the exact numerical solution in a limited iterations. Finally an approximate solution, if it is analytical, is most pleasing than the numerical solutions.

Modeling

Fig. 1. Physical sketch.

sis and Brownian motion are taken into account due to presence of nanoparticles. The relevant boundary layer equations are

du dv dw _ 0

dx + dy + ~dz ~ '

du du du d2u u^r- + v— + W— = V—T dx dy dz dz2

2 d_u 2 d2u dx2'

2 d2u _ +w2 _+2uv

dru dru dru --L 2 vw——+2uw——

dydz dxdz

dy2 dz2 dxdy

- u+Aw@H) + (T - Ti) + a2(T - Ti)2} +g{a3(C - Ci)+a4(C - C„)2},

dv dv dv d2v u^r- + v-—h w— = V—T dx dy dz dz2

2 d_v_ dx2'

2 d2 v

2 d2 v

—v+Aw^f Pi V dz

v2TTT+w2TTT + 2uv- „ dy2 dz2 dxdy

_ d2 v d2v - 2 vw—-+2uw—— dydz dxdz

We analyzed the steady three-dimensional nonlinear convec-tive flow of an incompressible Maxwell nanofluid induced by a permeable stretching surface at z = 0. The flow is considered due to stretched surface and the flow restricted in the domain z > 0 (see Fig. 1). The velocity components along the {x,y,z) directions are denoted by {u, v, w). The phenomena of heat and nanoparticle mass transfer have been studied in view of mixed convection. Strength of a uniform magnetic field B0 is applied in the z-direction. In addition the behavior of thermal radiation are taken into account with flux conditions. Characteristics of thermophore-

dT dT dT k

+ v— + w— = —i dx dy dz (pcp)

d2T dz2

(PcP)i dz (Pcp)i

dT dC dz Hz

-(T - T i),

dC dC dC d2C Dt (d2T u— + v— + w— = DB^T +

dz2 T dz2

sDT fdT dz

The relevant boundary conditions for the flow are

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u = uw(x) = ax, v = by, w = 0, at z = 0 and u

^ 0, v^ 0 as z^ oo.

The conditions for the prescribed heat flux (PHF) and prescribed concentration flux (PCF) are

PHF : -kf(g) = Tw at z=0 and T ! T„ when z-

PCF: -DbI | ) = Cw at z = 0 and C!C„ when z-

where m = (l/pf the kinematic viscosity, A the relaxation time, r the electrical conductivity, g the gravitational acceleration, a and a2 the linear and nonlinear thermal expansions coefficients, a3 and a4 the linear and nonlinear concentration expansion coefficients, Pf the fluid density, (cpf the fluid heat capacity, pp the particle density, (cp)p the particle heat capacity, kf the thermal conductivity, qr the radiative heat flux, s = (pcp)p/(pcpf the ratio between the effective nanoparticle material heat capacity and the base fluid heat capacity, DB the Brownian diffusion coefficient, DT the themophoretic diffusion coefficient, Q* the coefficient of heat generation/absorption, T and C the fluid temperature and concentration respectively, Tw and Cw the surface of temperature and concentration respectively, T„ the ambient fluid temperature and C„ the nanoparticle concentration far away from the surface.

The radiative heat flux qr through Rosseland approximation is

4ff* dT4 3k* ~dz

where r* and k* show the stefan-Boltzmann constant and the coefficient of mean absorption, respectively. We presume that difference of the temperature in the flow analysis is such that the term T4 may be expanded in a Taylor series. Hence expending T4 about T„ and omitting higher terms are obtain

T4u4T3T- 3T4.

Thus using Eq. (10) in Eq. (9), we get 16r*T3 dT

Qr = -

3k* dz ' Now Eqs. (4) and (11) yield

kf ( d2T

@T @T @T

u Tx + v dy + w dz = (pop)

xDb{ sTz dC

sDT (dT

1 16ff*T 1 d2T

dzJ (p°p)f (T - T i).

3k* dz2

(Pcp)f

Making use the following transformations

g = zpm, T = Ti + pm Tf 0(g), C = Ci + pa %/(g),

u = axf '(g), v = ayg'(g), w = -Vavf (g)+g(g)),

incompressibility condition is identically satisfied and Eqs. and (12) are reduce to

r + (1 + H2a)(f + g)f'' - if')2 + «(2(f + g)f'f'' - ( + g)2f

- H2f + At (1 + b1<?)0 + AtN*(1 + b2/)/ = 0,

g'" + (1 + H2aa)(f + g)g" - (g')2 + a(2(f + g)g'g" - (f + g)2g"') - H2g' = 0,

(13) (2)-(8)

1 R h' ' + Pr(f+g)h' + Pr Nbh/' + Pr Nt (h )2 + Pr yh = 0,

(16) (17)

/'' + Scf + g)/' + ^ 8'' - 0,

f (g)-0, f (g)-1, g(g)-0, g' (g)-d, 8' (g)--1, /' (g)--1 at g = 0,

f (g) ! 0,g(g) ! 0,8(g) ! 0/(g) ! 0 as g (18)

where a the Deborah number in term of relaxation time, Ha the magnetic parameter, kT the mixed convection parameter, b1 the nonlinear convection parameter due temperature, b2 the nonlinear convection parameter due concentration, Pr the Prandtl number, N* the ratio of concentration to thermal buoyancy forces, R the radiation parameter, y the heat generation/absorption parameter, Sc the Schmidt number, Nb the Brownian motion parameter, Nt the thermophoresis parameter, d the ratio parameter, Gr the Grashof number in term of temperature and Gr* the Grashof number in term of concentration. The involved quantities are defined as:

a_ ka H _ rB0 , Grx b _a2^/mJa(Tw/kf) „ _a4pmAl(Cw/0B)

a - ka, Ha - f, at -Rej, P1 - aa1 , b2 - a ,

N* = = «3(Cw/DB), r = ^, Pr = BL, Nb =

(pcp)pDB(Cw/DB)

~ «1 (Tw/kf ) _(pcp)pDT (Tw/kf ) Q '(Tw/kf )

(Pcp)f f

(pcp)f mTi

(p°p)fû

Sc=if, d=a,

Dß> a '

The local Nusselt Nux and Sherwood Shx numbers are defined for the given problem in the forms

kf (T - Ti)'

Db(C - Ci) '

where qw the surface heat flux and jw the concentration flux which can be expressed as

Qw = -kf(dT

= -kf 1 +

(Qr )w

~3kf¥

dTh Jw-»b(@CL «2')

In dimensionless form the local Nusselt Nux and Sherwood Shx numbers are given by

Re-1/2mx = (1 + 3^ h^), Re-1/2shx = /(0), (22)

in which Rex = uwx denotes the local Reynolds number. Series solutions

Here we take initial approximations (f0,g0, h0, /0) and auxiliary linear operators (Lf , Lg, Lh, L/ for the homotopy analysis method are taken as follows:

f0(g) = (1 -exp(-g)), g0(g) = d(1 -exp(-g)), 00(g) = exp(-g) and /0(g) = exp(-g),

Lff )= 0 - f, Lg (g)= 3gg - f, L0(0)

= 3g2- e and l/(/) = dg2 -

T. Hayat et al./Results in Physics xxx (2017) xxx-xxx

with the related properties

Lf [Ai + A2 exp(-g) + A3 exp(g)] = 0, (25)

£g [A4 + A5 exp(-g) + A6 exp(g)] = 0, (26)

lh[A7 exp(-g) + A8 exp(g)] = 0, (27)

l/[A9 exp(-g) + A10 exp(g)] = 0, (28)

where Ai (i = 1 - 10) denotes the arbitrary constants.

Zeroth-order problems

The zeroth-order deformation problems are

(1 -p)lf [/(g;p) -f 0(g)] = pHf hfnf [/(g;p),g(g;p),h(g;p), /(g;p)],

(1 - p)lg [g(g ;p) - g,(g)] = pHghgng f(g;p), g(g ;p)],

(1 - p)L [0(g; p) - 60(g)] = pHs%N s f (g; p), g (g; p), f (g; p), /(g; p)J ,

(1 -p)L/ |/(g;p)-/0(g)] = jpH/h/N/ f (g;p),g(g;p),6(g;p),/(g;p)J,

f (0; p) = 0, f'(0; p) = 1, f'(i; p) = 0, g (0; p) = 0, g'(0; p) = d, g'(i; p) = 0, f'(0;p) = -1, f(i; p) = 0, /'(0; p) = -1, /(1; p) = 0,

N/ [.f (g; p), g(g;p), 0(g; p),/(g; p)J _a3/ (g;p)

-+(1 + H2a) f(g;p)+g(g;p)

,a2f(g;p) 1 ag2

■,(f, -, ~\\@f(g;p)@2f(g;p) (f -N -, a3f(g;p)

2(f(g;p)+g(g;p)) @g2 ;- (f(g;p)+g(g;p)) ag3

ag ag2

af (g;p)

ag a ag

+1t( 1 + b1f(g; p)) f(g; p)+kTN^ 1 + b2/(g;p))/(g;p),

Ng f (g;p),g (g;p)] =

a3g(g;~) + (1 + Hjja) (f (g;p) + f (g;p)^a2M

ag (g;p)

f (g; p)+g (g;p)

2(f (g;p)+g (g;p) 2 a3 g (g; p)#

ag (g; p) a2 g (g;p)

2 ag (g; p) a ag '

ag2 (35)

N s f (g; p), g (g; p), f(g; p),/(g; p)]

= ( 1 + 4 R) «+P^/(g;p) + g (g; ft)a h(g; p)

+ Pr Nb as(g;p)a/(g;p) + Pr a s(g;p)

+ Pr yf (g; p),

N / f (g; p), g (g; p), f(g; p),/(g; p)]

a2/(g; p)

" ag2 +(g; p)+g (g;p)

N a2 g(g; p) N ag2 '

a/(g; p) ag

Here p e [0,1] represents the embedding quantities, hf, hg, hh and h/ the non-zero auxiliary parameters and Nf, Ng, Nh and N/ the nonlinear operators.

mth-order deformation problems

The mth-order deformation problems are

Lf fm(g) - Vjm-1(g)] = hfRm(g)>

Lg[gm(g) - Vmgm-1(g^ = hg(g)>

Lh[0m(g)- Vmhm-1(g)] = hh<(g),

L/[/m(g) - Vm/m-1 (g)] = h/Rm (g)

fm(0) = 0, fm(0) = 0 and fm(g)! 0 when g

gm(0) = 0, gm(0) = 0 and g'm(n) ! 0 when g

hm(0) = 0 and hm(g) ! 0 when g

/m(0) = 0 and /m(g) ! 0 when g

k=0 m-1 k

Rm(g) = /m-1 (g) + (1 + [/m-1-kfk - gm-1-kfk']

2(f m-1-k + gm-1-k

-Yj'm-1-kfk + aEE k=0 k=0 l=0

_ / fm-1-f k-l + gm-1-kgk-l '

V +2fm-1-kgk-l

-H2jm-1(g)+ kT (1 + b!0m-1 (g))Sm-1 (g)

+kTN*(1 + ft/m-1 (g))/m-1 (g),

Rm (g) = gm-1 (g) + (1 + H2 a)E [/m-1-kgk' - gm-1-kgk]

2fm-1-k + gm-1-k)g'k-lg'l

k=0 m-1

-Yj£m-1-kgk + k=0 k=0 -

( fm-1-kf k-l + gm-1-kgk-l

+2fm-1-kgk-l

( 4 \ m-i

rm(g) = 1 + 3^ sm_1 (g) + Pr£(Sm_1_^k + Sm_1_kgk)

m- 1 m- 1

+ Sm-1-k/k + Sm-1-kSk + PrCSm-1 (g),

Rm(g) = /m-1(g)+(/m-1-k/k+/m-1-kgk) k=0

+Nt sm-1(g),

0, m 6 1,

1, m > 1,

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/ (g; 0)= ¡0 (g), g (g; 0)= g0(g), 0(g; 0) = 00(g), /(g; 0) = /0(g) when p = 0,

/(g; 1)= f (g), /(g; 1)= g(g), b(g; 1) = 0(g), /(g; 1) = /(g) when p = 1. (48)

It is analyzed that when p increases from 0 to 1 then /(g;p), /(g;p), 0(g;p) and /(g;p)vary from the initial solutions f0(g),g0(g),00(g) and /0(g) to the final solutions f(g),g(g),0(g) and /(g) respectively. We can write the solutions in view of Taylor's series as follows:

/(g; p)= f0(g)^Efm(g)pm with fm(g) =

1 ôm/(g;p)

m! @pm

g (g; p)= g0(g) + X]gm(g)pm With gm (g) =

b(g;p)= 00(g) ^0m(g)pm with 0m(g) =

1 ômg(g; p)

m! @ p~m

1 @m 0(g; p)

m! @ p~m

/(g;p)- /0(g) + (g)Pm with /m(g) - m@mP)

m-1 ' P p-0

The series of f, g, 8 and / through Taylor's series are selected convergent for p~ - 1 and thus (see Fig. 3)

f (g)= f 0(g)^Ef m(g),

g(g)= g0(g) + ^ gm (g), m=1

0(g) = 00(g) + £ 0m(g),

/(g) = /0(g) + J]/m(g),

The general solutions (fm, gm, 0m, /m) of Eqs. (29)-(32) in view of special solutions (fm,gm, 0m, /m) are

fm(g) = fm(g) + A1 + A2 exp( g) + A3exp(g)

gm (g) = gm (g) + A4 + A5 exp(-g) +A6 exp(g),

0m(g) = 0m (g) + A7 exp(-g) +Ag exp(g),

/m(g) = /m(g) + A9 exp(-g) +A10exp(g).

Invoking Eq. (33) the values of A,- (i = 1 - 10) are

A = @f m(g)

A4 = -

@gm (g)

-fm(0), A2=¡M

@0m(g)

fm , @gm(g)

^=0 - gm(8), A5 =-mr

@/m(g)

A8 0; A9

A10 0.

a = 0.1, Ha = 0.5, Ar = 0.1, A = 0.1, ft = 01, A™ = 0.5, J? = 0.1, Pr = 1.0, iV4 = 0.7, JV, = 0.2, y = 0.1, Sc = 1.0, <5 = 0.2

Fig. 2. h-curves forf''(0) and g''(0).

0.1, Ha = 0.5, Aj. = 0.1, ft = 0.1, ft = 0.1, = 05, « = 0.1, Pr = 1.0, JV4 = 0.7, AT, = 0.2, y = 0.1, Sc = 1.0, tf = 0.2

0"(O)■

Fig. 3. h-curves for 0'(0) and /''(0).

Convergence analysis

A homotopy analysis method gives us opportunity and a simple approach to adjusting and control the convergence region of the homotopic solutions. For such analysis the auxiliary parameters hf, hg, h8 and h/ are appeared which plays a significant role for the convergence analysis of the homotopic solution. Therefore we have portrayed the h-curves at 21th- order of approximations (see Figs. 1 and 2). It is seen that the acceptable values of these auxiliary parameters are in the ranges -1.7 6 hf 6 -0.6,

1.9 6 hg 6

0.45, -1.3 6 h8 6 -0.4 and -1.25 6 6 -0.2. Further Figs. 4-7 show the plots for the residual errors regarding f, g, 8 and /.

Analysis

This subsection aims to investigate behavior of different physical quantities on the velocities, temperature and concentration distributions. Variations in velocities fields f (g) and g'(g) for various values of Deborah number a, magnetic parameter Ha, mixed convection parameter aT , nonlinear convection parameter b1 (in view of temperature), concentration buoyancy parameter N* and ratio parameter d are addressed in Figs. 8-13. Behavior of Deborah number a on the velocities f (g) and g'(g) is captured in Fig. 8. The velocity fields and related boundary layer thickness decrease for larger Deborah number a. This is due to fact that the increase of Deborah number relaxation time increases as a result both the velocities f (g) and g'(g) and thickness of momentum boundary layer decreases. Fig. 9 exhibit the velocities along x- and y-

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components for various values of magnetic parameter Ha. It is observed that velocities fields and related boundary layer thickness decreases when Ha increase. Here an increase in Lorentz force produces more resistance to the fluid motion. Therefore velocities decreases. Velocity fields in Fig. 10 are plotted for different values of mixed convection parameter kT. When mixed convection parameter enhances, the velocity field along the x-direction increases while the velocity field along the y-direction decreases. Since mixed convection parameter leads to higher buoyancy forces which results in the enhancement of x-component of velocity. Fig. 10 demonstrates the characteristics of b1 on f (g) and g'(g). Here we see that f (g) and g (g) have opposite behavior for larger nonlinear convection parameter. Characteristics of N* i.e., ratio of concentration to thermal buoyancy forces on the x- and y-components of velocity is captured in Fig. 11. It is examined that x-component of the velocity increases while y-component decreases for larger values of N*. Since N* is the ratio of concentration and thermal buoyancy forces so with the increment of N* the corresponds to higher concentration buoyancy force which results x-component of the velocity f (g) enhanced. Influence of ratio parameter d on the velocities f (g) and g'(g) are delineated in Fig. 13. Increasing d implies to an increase in the rate of stretching along y-direction due to which velocity field g'(g) increasing whereas f (g) indicates the decreasing behavior.

Influences of dimensionless Deborah number a, magnetic parameter Ha, mixed convection parameter kT, radiation parameter R, Prandtl number Pr, Brownian motion parameter Nb, ther-mophoresis parameter Nt and heat generation/absorption parameter c on the temperature distributions 0(g) are demonstrated in Figs. 14-21. Fig. 14 disclosed the behavior of Deborah number a on the temperature distribution. Temperature and thermal boundary layer are increasing function of Deborah number a. Fig. 15 elucidates that larger values of magnetic parameter Ha correspond to higher temperature and thickness of layer. Physically, Lorentz force enhances due to higher magnetic parameter Ha which creates resistance to the fluid motion and consequently some valuable energy is transform into heat. This phenomenon introduce the temperature field enhances. Effect of mixed convection parameter kT on the temperature curves is observed in Fig. 16. It is found that the temperature field and associated layer thickness decreases for larger kT. The curves of temperature field 0(g) for various values of R are shown in Fig. 17. Here temperature and thickness of thermal boundary layer are reduced for larger radiation parameter. Physically, in the radiation process more heating to the working fluid which results the temperature enhances. Fig. 18 represents the effect of Prandtl number Pr on the temperature profile. Tempera-

a=XT=ß1=ß1=R=y=0.l, Pr=Sc=l .0, N,=0=0.2, Ha=0.5, A"=0.5, JV6=0.7

i 0.00 ¿im

Fig. 5. Residual error for hg.

a=\T=ß1=ß2=R=y=0.1, Pr=Sc=1.0, N¡=6=0.2, H„=0.5, JV*=0.5, Nb=0.7

Fig. 6. Residual error for hh.

a=\T=ßi=ß2=R=y=0.1, Pr=Sc=l .0, JV,=á=0.2, #„=0.5, N'=0.5, Nb=0.7

Am 0.00-

a=\T=ß1=ß2=R=y=0.l, Pr=Sc=l .0, N,=6=0.2, //„=0.5, N'=0.5, Nb=0.7

Am 0.00-

Fig. 4. Residual error for hf.

Fig. 7. Residual error for h/

ture distribution has increasing effects near the surface and it has decreasing behavior far away from the surface. Physically, higher Pr corresponds to lower thermal diffusivity which results in the reduction of temperature. It can be seen from Fig. 19 that temperature and thickness of associated boundary layer are enhanced when the values of Brownian motion parameter Nb increase. Fig. 20 is sketched to see the characteristics of thermophoresis quantity Nt on the temperature distribution. It is

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Htt = 0.5, AT = 0.1, j8i = 0.1, j32 = 0.1, N* = 0.5, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, y = 0.1, Sc = 1.0, 6 = 0.2

12 3 4

Fig. 8. Impact of a on f (g) and g' (g).

02 = 0.1, N„ = 0.7, 6 = 0.2

Fig. 11. Impact of b1 on f (g) and g'(g).

f'OAg'fa) 1.0

a = 0.1, XT = 0.1, = 0.1, /?2 = 0.1, /V* = 0.5, R - 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, 7 = 0.1, Sc = 1.0, Ô = 0.2

Ha = 0.0, 0.5, 1.0, 1.5

> -'-'jjj^

12 3 4

Fig. 9. Impact of Ha on f (g) and g (g).

fmg'w 1.0

12 3 4

Fig. 12. Impact of N* on f' (g) and g' (g).

f'W.g'W 1.0

or = 0.1, Htt = 0.5, j8i = 0.1, y32 = 0.1, iV* = 0.5, /i =0.1, Pr = 1.0, 7Vé = 0.7, = 0.2, y = 0.1, Sc = 1.0, 6 = 0.2

12 3 4

Fig. 10. Impact of AT on f (g) and g' (g).

a = 0.1, Ha - 0.5, AT = 0.1, fii - 0.1, fc = 0.1, N* = 0.5, R = 0.1, Pr = 1.0, Nb = 0.7, Nt =02, y = 0.1, Sc = 1.0

S = 0.1, 03, 0.6, 0.9

/'01) g'in)

12 3 4

Fig. 13. Impact of d on f (g) and g (g).

499 evident that temperature and thermal boundary layer thickness

500 are enhanced via larger Nt. Behavior of heat generation/absorption

501 parameter y on the temperature field are elucidated in Fig. 21. It is

502 concluded that the temperature and thickness of thermal boundary

503 layer increase when heat generation parameter (y > 0) increases

504 while reverse situation is observed for heat absorption situation

505 (y < 0). Obviously in heat generation parameter y > 0 process

506 more heat is produced which result in the enhancement of temper-

507 ature field.

Variations in dimensionless concentration distributions /(g) for 508

different values of Deborah number a, mixed convection parame- 509 ter aT, Brownian motion parameter Nb, thermophoresis parameter 510

Nt and Schmidt number Sc are addressed in the Figs. 22-26. Fig. 22 511

is prepared to see the behavior of Deborah number a on concentra- 512

tion field. It is found that the concentration curves and associated 513

layer thickness enhances for larger a. Fig. 23 elucidates that the 514

higher values of mixed convection parameter aT correspond to 515 low concentration and thickness of layer. Influence of Brownian 516

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Htt = 0.5, XT = 0.1, Pi = 0.1, Pz = 0.1, iV* = 0.5, /f = 0.1, Pr = 1.0, Nb = 0.7, AJ = 0.2, y = 0.1, Se = 1.0, S = 0.2

a = 0.0, 0.5, 1.0, 1.5

Fig. 14. Impact of a on S(g).

0.1, = 0.7, 0.2

1 2 3 4 5

Fig. 17. Impact of R on S(g).

a = 0.1, Xt = 0.1, Pi - 0.1, PÏ - 0.1, N* = 0.5, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, 7 = 0.1, Se = 1.0, S = 0.2

Ha = 0.0, 0.5, 1.0, 1.5

Fig. 15. Impact of Ha on S(g).

a = 0.1, Ha = 0.5, XT = 0.1, Pi = 0.1, 182 = 0.1, N* = 0.5, R = 0.1, Nb = 0.7, Nt = 0.2, y = 0.1, Se = 1.0, ô = 0.2

Pr = 0.1, 03, 0.6, 0.9

Fig. 18. Impact of Pr on S(g).

517 motion parameter Nb on the concentration field is drawn in Fig. 24.

518 From Fig. 24, we scrutinized that an increase in Nb creates a reduc-

519 tion in the concentration field. Fig. 25 indicates that both concen-

520 tration profile and boundary layer thickness increases by

521 increasing values of thermophoresis parameter Nt. The thermal

522 conductivity of the fluid enhances in presence of nanoparticles.

523 For higher values of Nt gives rise to the thermal conductivity of

524 the fluid. Such higher thermal conductivity shows the larger con-

525 centration. Analysis of Schmidt number Sc on the concentration

field is presented in Fig. 26. Physically Schmidt number Sc is the 526 momentum to mass diffusivities ratio. Thus with the increase of 527

Schmidt number Sc the mass diffusivity decreases which is respon- 528

sible in the reduction of concentration field. 529

Table 1 is set up for the convergence of the homotopic solutions. 530

It is observed that 17th orders of approximations are sufficient for 531

the x- component of velocity, 15th orders of approximations are 532

enough for y- component of velocity while 20th orders of approx- 533

imations are sufficient for temperature and concentration fields for 534

01 = o.i,

, Pr = 1.0, i, S = 02

T. Hayat et al./Results in Physics xxx (2017) xxx-xxx

•Pin)

1 2 3 4 5

Fig. 20. Impact of Nt on 0(g).

a = 0.1, Ha = 0.1, 0i = 0.1, 02 = 0.1, V* =0.5, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 02, y = 0.1, Sc = 1.0, ô = 0.2

At = 0.0, 0.5, 1.0, 1.5

2 3 4 5

Fig. 23. Impact of AT on /(g).

.1, 01 = 0.1, 0.1, Pr = 1.0, ; 1.0, S = 02

0.1, 02

2 3 4 5

Fig. 21. Impact of y on 0(g).

a = 0.1, Ha = 0.5, AT = 0.1, 0! = 0.1, 02 = 0.1, N* = 0.5, R = 0.1, Pr = 1.0, Nt = 02, y = 0.1, Sc = 1.0, ô = 0.2

Nb = 0.1, 03, 0.6, 0.9

Fig. 24. Impact of Nb on /(g).

535 a convergent of the series solutions. Influences of different param-

536 eter on Nusselt number are shown in Table 2. The values of Nusselt

537 number increases by increasing mixed convection parameter aT,

538 nonlinear convection parameter due temperature b1 and radiation

539 parameter R while it presents decreasing behavior for larger Deb-

540 orah number a, magnetic parameter Ha, Prandtl number Pr, Brow-

541 nian motion parameter Nb, thermophoresis parameter Nt and heat

542 generation/absorption parameter y. Table 3 is constructed to

543 examine the behavior of various parameters on Sherwood number.

It is recognized that Sherwood number enhances mixed convection 544

parameter aT , nonlinear convection parameter due concentration 545

b2, Brownian motion parameter Nb, Schmidt number Sc and ratio 546

parameter dwhile it reduces with increasing values of Deborah 547

number a, magnetic parameter Ha and thermophoresis parameter 548

Nt. Table 4 presents the comparison of f (0) and g (0) for the var- 549

ious values of dwith previous published works. It is observed that 550

present results are in good agreement with the previous solutions 551

in limiting sense. 552

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10 T. Hayat et al./Results in Physics xxx (2017) xxx-xxx

Ф № Concluding remarks

Three-dimensional nonlinear convective flow of Maxwell nano- 554

fluid over a permeable stretching surface with the effects heat gen- 555

eration/absorption and thermal radiation are addressed. Major 556

outcomes are: 557

• Velocity fields f (g) and g' (g) decays for higher values of Debo- 558 rah number a. 559

• Dual behavior of the temperature profile due to Prandtl number 560 Pr. 561

• Temperature and concentration curves are increasing function 562 of the thermophoresis parameter Nt. 563

• The velocities f (g) and g'(g) have opposite behavior for larger 564 kT, b1, N* and d. 565

• An increase in Schmidt number Sc gives a reduction in the con- 566 centration and enhancement of Sherwood number. 567

Table 1

Convergence analysis of the series solutions for different order of approximations when a = 0:1,Ha = 0:5, kT = 0:1, ft = 0:1, = 0:1, N* = 0:5, R = 0:1, Pr = 1:0,Nb = 0:7, Nt = 0:2, c = 0:1, Sc = 1:0 and d = 0:2.

Order of approximation -f '(0) -g '(0) -S' '(0) /' '(0)

1 1.11034 0.18153 ^ 0.41667 0.15714

5 0.89578 0.18776 0.79444 0.20651

10 0.76605 0.19249 1.0643 0.27393

15 0.72258 0.19381 1.1842 0.32838

17 0.71851 0.19381 1.2022 0.33993

20 0.71851 0.19381 1.2091 0.34738

25 0.71851 0.19381 1.2091 0.34738

30 0.71851 0.19381 1.2091 0.34738

35 0.71851 0.19381 1.2091 0.34738

Table 2

Local Nusselt number Re-1/2Nux for various physical quantities when = 0-1, N* = 0.5, Sc = 1.2 and d = 0.2.

Parameters (fixed values) Parameters Re-1=2NUx

Ha = 0.5, kT = 0.1, b = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, c = 0.1 a 0.1 0.24527

0.2 0.23811

0.5 0.21909

a = 0.1, kT = 0.1, ft = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, c = 0.1 Ha 0.1 0.26372

0.2 0.26133

0.5 0.24527

a = 0.1, Ha = 0.5, ft = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, c = 0.1 kT 0.1 0.24527

0.2 0.29151

0.5 0.35111

a = 0.1, Ha = 0.5, kT = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, c = 0.1 b1 0.1 0.24527

0.2 0.25165

0.5 0.26591

a = 0.1, Ha = 0.5, kT = 0.1, b1 = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2, c = 0.1 R 0.1 0.24527

0.2 0.27512

0.5 0.35502

a = 0.1, Ha = 0.5, kT = 0.1, b1 = 0.1, R = 0.1, Nb = 0.7, Nt = 0.2, c = 0.1 Pr 0.8 0.25737

1.0 0.24527

1.5 0.23234

a = 0.1, Ha = 0.5, kT = 0.1, b1 = 0.1, R = 0.1, Pr = 1.0, Nt = 0.2, c = 0.1 Nb 0.2 0.39776

0.5 0.30215

0.7 0.24527

a = 0.1, Ha = 0.5, kT = 0.1, b1 = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, c = 0.1 Nt 0.2 0.24527

0.4 0.18920

0.6 0.13818

a = 0.1, Ha = 0.5, kT = 0.1, ft = 0.1, R = 0.1, Pr = 1.0, Nb = 0.7, Nt = 0.2 c 0.1 0.24527

0.2 0.17656

0.5 0.10374

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Table 3

Local Sherwood number Re^'/2Sux for various physical parameters when b1 = 0.1, N* = 0.5, R = 0.1, Pr = 1.2 and c = 0.1

Parameters (fixed values) Parameters Re-1 =2 Sux

Ha = 0.5, kT = 0.1, b2 = 0.1, Nb = 0.7, Nt = 0.2, Sc = 1.0, d = 0.2 a 0.1 0.2 0.5 0.60092 0.59591 0.57917

a = 0.1, kT = 0.1, b2 = 0.1, Nb 0.7, Nt = 0.2, Sc = 1.0, d = 0.2 Ha 0.1 0.2 0.5 0.62424 0.61927 0.60092

a = 0.1, Ha = 0.5, b2 = 0.1, Nb = 0.7, Nt = 0.2, Sc = 1.0, d = 0.2 kT 0.1 0.2 0.5 0.60092 0.62185 0.65335

a = 0.1, Ha = 0.5, kT = 0.1, Nb = 0.7, Nt = 0.2, Sc = 1.0, d = 0.2 b2 0.1 0.2 0.5 0.60092 0.61086 0.61136

a = 0.1, Ha = 0.5, kT = 0.1, b2 = 0.1, Nt = 0.2, Sc = 1.0, d = 0.2 Nb 0.2 0.5 0.7 0.44642 0.56533 0.60092

a = 0.1, Ha = 0.5, kT = 0.1, b2 = 0.1, Nb = 0.7, Sc = 1.0, d = 0.2 Nt 0.2 0.4 0.6 0.60092 0.56473 0.54129

a = 0.1, Ha = 0.5, kT = 0.1, b2 = 0.1, Nb = 0.7, Nt = 0.2, d = 0.2 Sc 0.8 1.0 1.5 0.52858 0.60092 0.75701

a = 0.1, Ha = 0.5, kT = 0.1, b2 = 0.1, Nb = 0.7, Nt = 0.2, Sc = 0.2 d 0.2 0.3 0.5 0.60092 0.61782 0.65431

Table 4

Comparison of /"(0) and g"(0) for the various values of d by HAM, HPM and exact solutions with Ariel [45].

d Ariel [45] Ariel [45] Present results

-f '(0) -g '(0) -f '(0) -g '(0) -f '(0) -g '(0)

0.0 1.00000 0.00000 1.000000 0.000000 1.000000 0.000000

0.1 1.02025 0.06684 1.020259 0.066847 1.020260 0.066847

0.2 1.03949 0.14873 1.039495 0.148736 1.039495 0.148737

0.3 1.05795 0.24335 1.057954 0.243359 1.057955 0.243360

0.4 1.07578 0.34920 1.075788 0.349208 1.075789 0.349209

0.5 1.03909 0.46520 1.093095 0.465204 1.093095 0.465205

0.6 1.10994 0.59052 1.109946 0.590528 1.109947 0.590529

0.7 1.12639 0.72453 1.126397 0.724531 1.126398 0.724532

0.8 1.14248 0.86668 1.421488 0.866682 1.142490 0.866677

0.9 1.15825 1.01653 1.158253 1.016538 1.158254 1.016539

1.0 1.17372 1.17372 1.173720 1.173720 1.173722 1.173722

• Qualitative behavior of local Nusselt and local Sherwood numbers are similar when a,Ha, kT and Nt increases.

Uncited references [43,44].

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