Scholarly article on topic 'MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness'

MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness Academic research paper on "Nano-technology"

CC BY-NC-ND
0
0
Share paper
Academic journal
Results in Physics
OECD Field of science
Keywords
{MHD / "Variable thicked surface" / "Powell-Eyring nanofluid" / "Zero mass flux conditions"}

Abstract of research paper on Nano-technology, author of scientific article — T. Hayat, Ikram Ullah, A. Alsaedi, M. Farooq

Abstract This research explores the magnetohydrodynamic (MHD) boundary layer flow of Powell-Eyring nanofluid past a non-linear stretching sheet of variable thickness. An electrically conducting fluid is considered under the characteristics of magnetic field applied transverse to the sheet. The mathematical expressions are accomplished via boundary layer access, Brownian motion and thermophoresis phenomena. The flow analysis is subjected to a recently established conditions requiring zero nanoparticles mass flux. Adequate transformations are implemented for the reduction of partial differential systems to the ordinary differential systems. Series solutions for the governing nonlinear flow of momentum, temperature and nanoparticles concentration have been executed. Physical interpretation of numerous parameters is assigned by graphical illustrations and tabular values. Moreover the numerical data of drag coefficient and local heat transfer rate are executed and discussed. It is investigated that higher wall thickness parameter results in the reduction of velocity distribution. Effects of thermophoresis parameter on temperature and concentration profiles are qualitatively similar. Both the temperature and concentration profiles are enhanced for higher values of thermophoresis parameter.

Academic research paper on topic "MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness"

ELSEVIER

Contents lists available at ScienceDirect

Results in Physics

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

MHD flow of Powell-Eyring nanofluid over a non-linear stretching sheet with variable thickness

T. Hayata,b, Ikram Ullaha'*, A. Alsaedib, M. Farooqc

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 80203, Jeddah 21589, Saudi Arabia c Department of Mathematics, Ripha International University, Islamabad 44000, Pakistan

ARTICLE INFO ABSTRACT

This research explores the magnetohydrodynamic (MHD) boundary layer flow of Powell-Eyring nanofluid past a non-linear stretching sheet of variable thickness. An electrically conducting fluid is considered under the characteristics of magnetic field applied transverse to the sheet. The mathematical expressions are accomplished via boundary layer access, Brownian motion and thermophoresis phenomena. The flow analysis is subjected to a recently established conditions requiring zero nanoparticles mass flux. Adequate transformations are implemented for the reduction of partial differential systems to the ordinary differential systems. Series solutions for the governing nonlinear flow of momentum, temperature and nanoparticles concentration have been executed. Physical interpretation of numerous parameters is assigned by graphical illustrations and tabular values. Moreover the numerical data of drag coefficient and local heat transfer rate are executed and discussed. It is investigated that higher wall thickness parameter results in the reduction of velocity distribution. Effects of thermophoresis parameter on temperature and concentration profiles are qualitatively similar. Both the temperature and concentration profiles are enhanced for higher values of thermophoresis parameter.

© 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/).

CrossMark

Article history:

Received 28 October 2016

Received in revised form 26 November 2016

Accepted 6 December 2016

Available online 21 December 2016

Keywords: MHD

Variable thicked surface Powell-Eyring nanofluid Zero mass flux conditions

Introduction

Nanofluids are the materials that consist of small quantities of nanometer-size particles, known as nanoparticles. Typically the nanoparticles are made of oxides like alumina, titania and copper oxide, carbides and metals including copper and gold. Diamond and Carbon nanotubes have also been utilized in nanofluids. These particles have ability to increase thermophysical properties of the base liquids. The base fluids include oil, ethylene glycol, water, biofluids, polymer solutions and some lubricants. Further magnetic nanofluid is a single material having both the magnetic and liquid properties. Magnetic field interaction with nanofluids is useful to deal with the situations such as cooling of nuclear reactors via liquid sodium. The magneto nanofluids have remarkable involvement in nonlinear optical materials, optical switches, optical modulators, tunable optical fiber filters, magnetic resonance imaging, optical grating, blockage removal in the arteries, drug delivery and hyperthermia etc. Choi [1] was the first one who introduced this colloidal suspension. Buongiorno [2] studied the convective transport phenomena in nanofluid. He constructed a mathematical

* Corresponding author. E-mail address: ikramu020@yahoo.com (I. Ullah).

model to study the nanofluid flow comprising Brownian motion and thermophoretic dispersion of nanoparticles. The two-dimensional stretched flow of nanofluid is conducted by Khan and Pop [3]. Makinde and Aziz [4] extended this analysis by considering convective boundary conditions. They demonstrated that convective heating significantly affects the thermal boundary layer. Having such facts in mind, many engineers and scientists are busy in the investigations of flows of nanofluid via various aspects. Few representative studies in this direction can be seen in the attempts (see [5-10].

The analysis of magnetohydrodynamic flow has received the attention of researchers due to its widespread engineering and industrial applications. Such applications include the design of cooling systems by adding liquid metals, accelerators, MHD generators, nuclear reactors, energy storage, flow meters and pumps. Having such concept in mind the researchers explored the behavior of MHD flows in various physical aspects. Heat transfer characteristics in MHD flow of nanofluid are investigated by Sheikholeslami et al. [11]. Rashidi et al. [12] explored the magnetohydrodynamic (MHD) flow of nanofluid over a stretching sheet. The boundary layer stagnation point flow of Jeffrey fluid towards a stretching sheet with convective boundary condition is addressed by Hayat et al. [13]. Zhang et al. [14] studied the radiative MHD

http://dx.doi.org/10.1016/j.rinp.2016.12.008 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/).

flow of nanofluid in a porous medium with variable surface heat flux and chemical reaction effects. Hayat et al. [15] examined the magnetic field effect in three-dimensional flow of second grade nanofluid past a stretching surface. Further the idea of ferromag-netism is remarkable in advance technology and industry. Ferroliq-uid is a magnetic colloidal suspension, with typical dimension less then 15 nm submerge in liquid transporter. The frequently utilized attractive material is single domain particles of iron, magnetite, hematite or cobalt ferrite and the base liquids such as kerosene or water. A ferrofluid reacts as a liquid that is influenced by an externally utilized magnetic field can be used to direct and control the flow of ferrofluids, due to which it is relevant in various fields such as mechanical designing, electronic packing, aerospace, bioengineering and thermal engineering. Motivated by all the aforementioned facts, various engineers and scientists are engaged in the discussion of flows of ferrofluids via various aspects. Stretched flow of ferrofluid flow with magnetic dipole is firstly examined by Andersson and Valnes [16]. Consequences of magnetic dipole on ferrofluid flow by a stretching sheet is inspected by Zeeshan et al. [17]. Yasmeen et al. [18] studied the ferrofluid flow in the existence of magnetic dipole and homogeneous-heterogamous reactions. Non-linear stretched MHD flow of second grade nanoliq-uid is investigated by Hayat et al. [19].

Flow analysis over stationary or moving surface has key interest of the investigators for its vast applications in engineering. Few such applications may include consideration of liquid films, polymer extrusion, metal working process, crystal growth process, drawing of filaments, the manufacturing of rubber and plastic sheets, artificial fibers and continuous cooling of fiber. Sakiadis [20] analyzed the flow of viscous fluid over continuos surface with uniform velocity. Cortell [21] studied the nonlinear stretched flow in the presence of viscous dissipation and thermal radiation. Mukhopadhyay et al. [22] examined the solutal stratification effect in boundary layer flow of viscous fluid over a permeable stretching sheet. Sheikholeslami and Ganji [23] disclosed the behavior of nanofluid in stretching flow through rotating frame. Three-dimensional flow caused by an exponentially stretching sheet is discussed by Mustafa et al. [24]. Khan et al.[25] examined the MHD stagnation point flow of variable viscosity nanofluid past a stretching sheet. However it is noticed that the variable thicked stretching sheet seems more realistic then the flat stretching surface. In particular the sheets with variable thickness are useful in civil, mechanical, marine and aeronautical design and structures. The utilization of variable thickness improves the use of material and reduced the weight of structural elements. In spite of such importance very little attention is yet paid to the flow by variable thicked surface. Fang et al. [26] firstly attempted the boundary layer flow induced by a stretching sheet with variable thickness. Flow of nanofluid over a non-linear stretching surface with velocity of variable thickness is tackled by Elbashbeshy et al. [27]. Khader et al. [28] analyzed the slip phenomenon in boundary layer flow induced by a stretching sheet of variable thickness. MHD and heat generation effects flow of nanofluid caused by stretching surface of variable thickness are explored by Abdel-Wahed et al. [29]. The characteristics of variable thickness in the flow of Maxwell fluid over a stretching sheet is presented by Hayat et al. [30].

The main purpose of present attempt is to analyze the feature of nanoparticles in MHD flow of Powell-Eyring fluid over a stretching sheet with variable thickness. The influences of Brownian motion and thermophoresis are taken into consideration. More realistic boundary conditions are imposed at the boundary [31]. The relevant mathematical formulation is established under boundary layer approach. Homotopic technique [32-40] is implemented for the convergent series solutions. Plots of emerging physical parameters are presented and analyzed.

Mathematical analysis

Powell-Eyring nanofluid over an impermeable nonlinear heated stretching sheet with variable thickness is considered. An incompressible fluid is selected electrically conducting. A non-uniform magnetic field B(x) = B0(x + hf"^ is imposed transverse to the stretching sheet. Magnetic Reynolds number is chosen small. Induced magnetic and electric fields are not accounted. Brownian and thermophoresis in the nanofluid model are considered. Newly developed condition for mass flux is imposed. The fluid configuration is such that the x-axis is presumed along the sheet while y-axis is transverse to it. The stretching surface has the nonlinear velocity Uw = U0(x + h)n where U0 is the reference velocity and h the dimensional constant. Further it is assumed that sheet at

y = A(x + by" is not flat (where n is the velocity power index and A is assumed very small so that the sheet retain adequately thin). We also noticed that for n = 1 the problem reduces to a flat sheet. The governing expressions for the considered flow are given by:

du dv _ ...

+ dy = 0 (1)

du du , 1 ,d2u

u dx+t dy = (v+psd dy2

fdu\ 2 d2u rB2(x) pbd> dy2 2pbd3VdyJ dy2 ~u'

dT dT d2T u7¿ + D7¡y = v-nfj-â + s

dT dC DT

1dy dy + T1

dC dC n d2C Dt (d2T dx dy dy2 Ti \dy2

u = Uw(x) = Uo(x + b)n, t = 0, T = Tw Db dC + Dt dT = oat y = A(x + b)1?,

dy Ti dy

C ! Ci as y

Here u and t are the corresponding velocity components parallel to x- and y-directions respectively, i the dynamic viscosity, v = i/p designates the kinematic viscosity, p the fluid density, d and b are the material liquid parameters of Powell-Eyring model,

kthe thermal conductivity, anf (= jjc^ the thermal diffusivity of liquid, s the ratio of the heat capacity of fluid of the nanoparticles material to the effective heat capacity of the base fluid, DB indicates the Brownian diffusion coefficient, DT represents the ther-mophoretic diffusion, T the temperature of the fluid, C the nanoparticles concentration, Tw and Tm are the sheet and ambient fluid temperatures and Cœ the ambient fluid nanoparticles concentration. Transformations are expressed as follows [19,30]:

u = Uo(x + b)nF'(g),

U(g) =

vUo(x + b)n

F(g)+ gF'(g)

n - 1 n + 1

1\ Uo(x + b)n

©(g) =

TT 1 1 1

Tw Trx

Incompressibility condition is satisfied identically and Eqs. (2)-(6) take the following forms

u ! 0, t ! 0,

©" + Pr(F©' + Nb©U' + N ©) = 0,

U'' + PrLeFU' + (N- )©" = 0,

F(a) = K^) ' F (a) = 1' ®(a) = 1'

U' (a) + ©' (a) = 0, F' (l) ! 0, ©(l) ! 0' U(l) ! 0

(1 + N)F'''+ FF'' -( F'2 - N( ^ ) kF F'''-( MF' = 0, N = JL, k = -U^ (x + b)

\3n-1 .,r2

(8) (9) (10)

In the above expressions primes designate differentiation with

respect to g where a = Ay (^±1) U* is the wall thickness parameter

and g = a = AyJ(^J1) U7 shows the plate surface. Upon using F(g) =f (g - a) =f (f) , ©(g) = 0(g - a) = 0(f), U(g) = /(g - a) = /(f) Eqs. (8)-(12) yield (see Fig. 1)

(1 + N)f''' + f'f'' - (J+L)f'2 - N(n^)kT2r

M2f' = 0,

h'' + Prf h + Nbh'/' + Nt hH ) = 0,

/'' + PrLef /' + (*) h'' = 0,

f (0) = a^), f' (0) = 1, h(0) = 1, /' (0) + (I) h' (0) = 0 f (i) = 0, h(i) = 0, /(i) = 0,

where N and k are the fluid parameters, M represents the magnetic parameter, Nb the Brownian motion parameter, Pr the Prandtl number, Nt the thermophoresis parameter, Le presents Lewis number and prime indicates differentiation with respect to f. The non-dimensional parameters are

u-»0, T-» TaC C».

Variable sheet thickness

Uw M = (/„(* + i-)n, r = rw, ^ + = 0-

Fig. 1. Physical flow model.

dßß'

4d2 v '

Pr — — N — T 1

Pr" Of, Nb = T v ,

Nt = t

Dt(Tw - Ti) vT„

, Le = D ■ (17)

Surface drag coefficient and surface heat transfer are expressed as follows:

(x+b)qw k(Tw - Ti),

Tw —

1\ du 1 /duY

1nf + ßd) dy - 6ßdHdyj

y=A(X+b) 2

(12) qw=jD

y=A(x+b) 2

In dimensionless form we have [37,38]:

CfRef = P2nTT)((1 + N)f'(0) (f'''(0))3

NuxRe-1/2 = -y^h' (0) ,

where Rex designates the local Reynolds number presented by

Uw(x + b)/v.

Series expressions and convergence

The initial guesses, operators and deformation problem are given below

f(f) = a

1 - n 1 + n

1 - e-f , h0(f)=e-f , /0(0 = -N)e-f ,

Lf = f''' - f , Lh = h'' - h, L/ = /'' - / ,

Lf [CÎ + C2ef + C3e-f = 0 , Lh [C4ef + C5e-f

= 0 , L/ [C6ef + C7 ef = 0 ,

(21) (22)

in which C (i = 1 - 7) represents the arbitrary constants. The problem for zeroth and mth order are

(1 - p)Lf f (f ,p) -f0(f)] = phfNf f (f ,p))], (24)

(1 - p)L [f(f ,p) - 00(C)] = phhNhf (f ,p) , 0(f ,p), /(f ,p)] , (25) (1 - p)L/ [/(f,p) - Mi)] = ph/N/[f(f,p), 0(f,p), /(f,p)], (26) f (0 , p) = a(1=a), f (0 , p) = 1, f (1 , p)= 0 , f(0 , p) = 1,l

('N^^ f i

(i ,p) = 0, /'(0 ,p)+ Nt)e'(0 ,p) = 0 , /(i ,p) = 0 ,

Nf (f;p^ = c + N) dfj+/ f - (n2+r) (@0

- Nk(n + 1/2M -

dfV df3 Vn + V df

N h [h(f, p) , / (f, p) , f (f; Ä]= f + Prf dh+PrNb f+PrN

N/[/ (f, P), 9(f, p), f (f;p)] = f + LePrf + (N) f ■ (30)

/i=a2,M=a3 = a,N=a5,n=a5

Lf[fm(0 - Xjm-M = hf Rf c6[em(C) - Xmem-i(C)} = hRmf ),

L*[/m(Q-Xm /m-l(f)} = h/ R m®,

fm(0) =fm(0) = fm(l) = 0, 9m(0) = 9m(l) = 0,1

/m(0)^t) em(0)=o, /m(i)=o, f

m-1 / 2n \m-1 Rf(f = (1 + N)fm-i(f) + Yfm-1-kfk - —1)Yj'm-1-fk

k=0 v + ' k=0

m-1 k / 2 \ - Nk(n + 1 /2)Yf"m-1-kTfkJ? - ZTTM2fm-1 (f), k=0 l=0 v'+ V

m-1 m-1

R m(f) = ffm-1(f) + PrYfm-1-kd'k + PTNb£ /m-1-A k=0 k=0

+ hm-1-A, (36)

m-1 /N \

R m(f)=/m-1 (f)+LePrYjm-1-k/'k + UT c^o, (37)

k=0 VW

1, m > 1:

The general solutions for the governing equations are expressed as follows:

fm(f)=f m(f) + C1 + C2ef + C3 e-f, 9m(f) = e*m(f) + C4ef + C5e-f,

Fig. 2. The /-curves for f (f), 9(f) and /(C).

X = 0.2, M = 0.3 = ce, N = 0.5, Nb=0A,Nt = 0.2, Pr = 1.0 = Le, n = 0.5

Fig. 3. The /-curves for f (f), 9(f) and /(f).

0, m 6 1,

/m(0 = /m(0+Qel + C7 e-f, (41)

in which fm(f), 8*m(f) and /*m(f) denote the special solutions. The constants Q (i = 1 - 7) are calculated through the imposed boundary conditions (34):

C2 = C4 = C6 = o,C3 = fpl n,CI = -C3 -fm(0), 1

lf=° I (42)

C; = -em(0)C7 = /m(0) + (t) K(0) + em(0)). J

Here p designates the embedding parameter and hf, hh and h/ the auxiliary parameters. To show convergence of obtained series solutions (36)-(38), it is necessary to display the h-curves. The region of the graph parallel to h-axis shows the interval of convergence. Hence h-curves are interpreted at 20th order of deformations (see Figs. 2 and 3). It is noted that the ranges of acceptable values of (hf, hh and %) are -1.4 6 hf 6 -0.3, -1.6 6 hh 6 -0.4 and -1.6 6 h/ 6 -0.2. Table 1 presents the convergence of the resultant series solutions through numerical data. It is demonstrated that 35th order of deformations are adequate for the convergence analysis.

Table 1

HAM solutions convergence when n = 0:9, M = 0:3 = a, N = 0:5, Nb = 0:1, Nt = 0:3 = k

and Le = 1.0 = Pr.

Order of deformations -f '(0) -99 (0) /' (0)

1 0.9709 0.9100 1.8200

5 0.8948 0.7340 1.4679

10 0.8624 0.6865 1.3730

15 0.8537 0.6865 1.3730

25 0.8510 0.6928 1.3855

30 0.8501 0.6967 1.3934

35 0.8497 0.6980 1.3960

40 0.8497 0.6980 1.3960

Discussion

Figs. 4-14 are prepared to explore the characteristics of non-dimensional velocity f (n), temperature 8(n) and nanoparticles concentration /(n) for distinct values of emerging parameters like thickness parameter a, fluid parameters k and N, Prandtl number Pr, magnetic parameter M, thermophoresis parameter Nt, Lewis

Fig. 4. Effect of a on f ({).

Fig. 5. Effect of k on f ({).

Fig. 6. Effect of N on f ({).

Effect of a on 0(f).

Fig. 7. Effect of a on S({).

Effect of M on 0(f)

Fig. 8. Effect of M on S({).

number Le and Brownian motion parameter Nb. The behavior of wall thickness parameter a on the velocity profile f (n) is displayed in Fig. 4. Here it is clearly shown that increasing values of wall thickness parameter a results diminution in the fluid velocity f (n). In fact larger values of wall thickness parameter correspond to small deformation due to stretching of wall and hence the velocity profile decreases. Characteristics of fluid parameter k on the velocity field f (n) is presented in Fig. 5. Here the fluid velocity f (n) and momentum boundary layer thickness reduce for larger values of k. From Fig. 6 we have seen that the velocity profile f (n) enhances for larger values of N. Further the momentum boundary layer thickness also enhances. Fig. 7 exhibits the aspect of wall thickness parameter a on 0(n). It is revealed that an enhancement in a reduce the temperature field 0(n) and thickness of thermal boundary layer. The behavior of magnetic parameter on 0(n)is portrayed in Fig. 8. Temperature profile 0(n) and thermal boundary layer thickness show increasing behavior for higher values of M. An increment in M produces higher Lorentz force

Fig. 9. Effect of Nb on S({).

Fig. 10. Effect of Nt on S({).

Fig. 11. Effect of Pr on S({).

Fig. 12. Effect of Nt on /({).

Fig. 13. Effect of Nb on /({).

(resistive force) which has the characteristic to covert some thermal energy into heat energy. Therefore temperature profile increases. Variations in temperature vs. n for distinct values of Brownian motion Nb and thermophoresis parameters Nt are examined in Figs. 9 and 10. Here clearly the temperature 0(n) and associated thickness of thermal boundary layer are higher for the enhanced values of Nt and Nb. Also we noticed that the variation in temperature due to thermophoresis parameter Nt is higher than the variation due Brownian motion parameter Nb. As the Brownian motion parameter increases, the collision between the particle enhances which produces more heat and consequently the temperature profile increases. Fig. 11 indicates the impact of temperature profile 0(n) for diverse values of Prandtl number Pr. Here it is analyzed that temperature distribution decreases for higher Pr. Physically Prandtl number manipulates the momentum and thermal boundary layers thickness. Higher Prandtl number implies a thinner thermal boundary layer thickness which keep the temperature distribution uniform across the boundary layer. Hydromagnetic boundary layer dominant the thermal boundary layer. Low Prandtl

Table 3

Numerical values of surface heat transfer rate parameters M, n, a, N, Pr, Le, Nt, k and Nb.

Nu.Re 1/2 ] for several values of the

n = 0.5, or = 0.3, /1 = 0.2, M =0.3, Pr= 1.0, Nt= 0.2, JVi=0.1

Effect of Le on 0(f)

Fig. 14. Effect of Le on /({).

Table 2

Numerical data of Cf Re1/2 for distinct values of involved parameters n, k, M, N and a when Nb = 0.1, Nc = 0.2, Pr = 1.0, Le = 1.0.

M n k a N -cf Re1'2

0.0 0.5 0.2 0.3 0.5 0.6684

0.4 0.6999

0.6 0.9012

0.3 0.4 0.2 0.3 0.5 0.6849

0:6 0.7840

0.8 0.8737

0.3 0.5 0.0 0.3 0.5 0.7206

0:3 0.7443

0:6 0.7745

0.3 0.5 0.2 0.0 0.5 0.7361

0:2 0.7359

0:4 0.7358

0.3 0.5 0.2 0.3 0.0 0.9397

0:3 0.8043

0:5 0.7359

number fluids have higher thermal conductivities so that heat can diffuse faster than for higher Pr fluid. Fig. 12 depicts the characteristics of thermophoresis parameter Nt on the nanoparticles concentration /(£). It is concluded that higher values of ther-mophoresis parameter results in enhancement of nanoparticles concentration. It illustrates that for larger thermophoresis parameter more particles are pushed far from the hot surface which results in the enhancement of concentration profile /(g). Fig. 13 indicates the impact of Brownian motion parameter Nb on nanoparticles concentration profile /(g). Larger values of Nb give rise to the nanoparticles concentration /(g) and related boundary layer thickness. Fig. 14 is presented to characterize the behavior of Le on nanoparticles concentration field /(£). From this Fig. we see that an increment in Le leads to lower nanoparticles concentration field. Lewis number has inverse relation with the coefficient of Brownian diffusion. The coefficient of Brownian diffusion becomes smaller parallel to the large values of Le which is responsible in the reduction of nanoparticles concentration and the corresponding thickness of boundary layer. The numerical data of -/"(0), -0'(0) and /' (0) at distinct order of approximation are analyzed in Table 1 when n = 0.5 = N, a = 0.3 = M, k = 0.2 = Nt, Pr= 1.0 = Le and

M n N Pr k Le Nt Nb a -v^' (0)

0.0 0.5 0.5 1.0 0.2 1.0 0.2 0.1 0.3 0.6161

0:3 0.5919

0:7 0.5596

0:3 0.0 0:5 1.0 0.5 1.0 0.2 1.0 0.2 0.1 0.3 0.6139 0.6045 0.6181

0:3 0.5 0.0 0:3 0:6 1.0 0.2 1.0 0.2 0.1 0.3 0.5752 0.5946 0.6088

0:3 0.5 0.5 0.7 1.0 1.3 0.2 1.0 0.2 0.1 0.3 0.5207 0.6045 0.7092

0:3 0.5 0.5 1.0 0.0 0:4 0:7 1.0 0.2 0.1 0.3 0.6056 0.6034 0.6014

0:3 0.5 0.5 1.0 0.2 0.7 1.0 1.3 0.2 0.1 0.3 0.6084 0.6045 0.6017

0:3 0.5 0.5 1.0 0.2 1.0 0.0 0:2 0:4 0.1 0.3 0.6217 0.6045 0.5888

0:3 0.5 0.5 1.0 0.2 1.0 0.2 0.2 0:5 0:7 0.3 0.6046 0.6136 0.6219

0:3 0.5 0.5 1.0 0.2 1.0 0.2 0.1 0.0 0:4 0:8 0.5468 0.6240 0.7045

Nb = 0.1. It is concluded that the convergence of series solution is achieved at 35th-order of approximations. Tables 2 and 3 are prepared for variations of surface drag coefficient and local Nusselt number (heat transfer rate) for diverse values of the involved parameters. It is evident from Table 2 that the surface drag coefficient enhances when n, k and M increase while it reduces with the increment of a and N. Table 3 demonstrates that local Nusselt number shows the increasing behavior for higher values of Pr, Nb and a whereas it decreases when M, k, Le and Nt are increased.

Closing remarks

We inspected the feature of MHD stretched flow of Powell-Eyring nanofluid over a surface with variable thickness. Main outcome is explicitly stated below.

• An increment in wall thickness parameter a results in decrease of velocity /'(n) and temperature 0(n).

• Enhancement in magnetic field shows an increment of temperature.

• Enhancement in the fluid temperature is observed for higher thermophoresis parameter.

• An increment in Brownian motion parameter Nb corresponds to temperature enhancement while it reduces the nanoparticles concentration field.

• Surface drag coefficient increases for larger values of velocity power index, magnetic parameter and fluid parameter.

References

[1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, ASME, USA (1995) 99-105, FED 231/MD.

[2] Buongiorno J. Convective transport in nanofluids. J. Heat Transfer 2006;128:240-50.

[3] Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transfer 2010;53:2477-83.

Makinde OD, Aziz A Boundary layer flow of a nanofluid past a stretching sheet [23 with a convective boundary condition. Int. J. Therm. Sci. 2011;50:1326-32. Turkyilmazoglu M. Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids. Chem. Eng. Sci. 2012;84:182-7. [24

Makinde OD, Khan WA, Khan ZH. Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/ shrinking sheet. Int. J. Heat Mass Transfer 2013;62:526-33. [25

Zeeshan A, Baig M, Ellahi R, Hayat T. Flow of viscous nanofluid between the concentric cylinders. J. Comput. Theor. Nanosci. 2014;11:646-54. Sheikholeslami M, Abelman S, Ganji DD. Numerical simulation of MHD [26 nanofluid flow and heat transfer considering viscous dissipation. Int. J. Heat Mass Transfer 2014;79:212-22. [27

Hayat T, Farooq M, Alsaedi A. Thermal radiation effect in MHD flow of Powell-Eyring nanofluid induced by a stretching cylinder. Open Phys. 2015;13:188-97. [28

Abbasi FM, Shehzad SA, Hayat T, Ahmad B. Doubly stratified mixed convection flow of Maxwell nanofluid with heat generation/absorption. J. Magn. Magn. Mater. 2016;404:159-65. [29

Sheikholeslami M, Gorji-Bandpy M, Ganji DD. Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid. Powder Technol. 2014;254:82-93. [30

Rashidi MM, Ganesh NV, Hakeem AKA, Ganga B. Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation. J. Mol. Liq. 2014;198:234-8. [31

Hayat T, Asad S, Mustafa M, Alsaedi A. MHD stagnation point flow of Jeffrey fluid over a convectively heated stretching sheet. Comput. Fluids 2015;108:179-85. [32

Zhang C, Zheng L, Zhang X, Chen G. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical [33 reaction. Appl. Math. Model. 2015;39:165-81.

Hayat T, Ullah I, Muhammad T, Alsaedi A. Magnetohydrodynamic (MHD) [34

three-dimensional flow of second grade nanofluid by a convectively heated

exponentially stretching surface. J. Mol. Liq. 2016;220:1004-12.

Andersson HI, Valnes OA. Flow of a heated ferrofluid over a stretching sheet in [35

the presence of a magnetic dipole. Acta Mech. 1998;128:39-47.

Zeeshan A, Majeed A, Ellahi R. Effect of magnetic dipole on viscous ferro-fluid

past a stretching surface with thermal radiation. J. Mol. Liq. 2016;215:549-54. [36

Yasmeen T, Hayat T, Khan MI, Imtiaz M, Alsaedi A. Ferrofluid flow by a

stretched surface in the presence of magnetic dipole and homogeneous-

heterogeneous reactions. J. Mol. Liq. 2016;233:1000-5. [37

Hayat T, Aziz A, Muhammad T, Alsaedi A, Mustafa M. On

magnetohydrodynamic flow of second grade nanofluid over a convectively

heated nonlinear stretching surface. Adv. Powder Technol. 2016;27: [38

1992-2004.

Sakiadis BC. Boundary layer behavior on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axisymmetric flow. AIChE J. 1961;7:26. [39

Cortell R. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys. Lett. A 2008;372:631-6. Mukhopadhyay S, Bhattacharyya K, Layek GC. Mass transfer over an [40 exponentially stretching porous sheet embedded in a stratified medium. Chem. Eng. Commun. 2014;201:272-86.

Sheikholeslami M, Ganji DD. Numerical investigation for two phase modeling of nanofluid in a rotating system with permeable sheet. J. Mol. Liq. 2014;194:13-9.

Mustafa M, Mushtaq A, Hayat T, Alsaedi A. Radiation effects in three-dimensional flow over a bi-directional exponentially stretching sheet. J. Taiwan Inst. Che. Eng. 2015;47:43-9.

Khan WA, Makinde OD, Khan ZH. Non-aligned MHD stagnation point flow of variable viscosity nanofluids past a stretching sheet with radiative heat. Int. J. Heat Mass Transfer 2016;96:525-34.

Fang T, Zhang J, Zhong Y. Boundary layer flow over a stretching sheet with variable thickness. Appl. Math. Comput. 2012;218:7241-52. Elbashbeshy EMA, Emam TG, Abde-wahed MS. Flow and heat transfer over a moving surface with non-linear velocity and variable thickness in a nanofluid in the presence of thermal radiation. Can. J. Phys. 2013;91:699-708. Khader MM, Megahed AM. Numerical solution for boundary layer flow due to a nonlinearly stretching sheet with variable thickness and slip velocity. Eur. Phys. J. Plus 2013;128:100.

Abdel-Wahed MS, Elbashbeshy EMA, Emam TG. Flow and heat transfer over a moving surface with non linear velocity and variable thickness in a nanofluids in the presence of Brownian motion. Appl. Math. Comput. 2015;254:49-62. Hayat T, Farooq M, Alsaedi A, Al-Solamy F. Impact of Cattaneo-Christov heat flux in the flow over a stretching sheet with variable thickness. AIP Adv. 2015;5:087159.

Kuznetsov AV, Nield DA. Natural convective boundary-layer flow of a nanofluid past a vertical plate: a revised model. Int. J. Therm. Sci. 2014;77:126-9.

Liao SJ. Homotopy Analysis Method in Nonlinear Differential Equations. Heidelberg: Springer & Higher Education Press; 2012. Turkyilmazoglu M. Solution of the Thomas-Fermi equation with a convergent approach. Commun. Nonliner Sci. Numer. Simul. 2012;17:4097-103. Abbasbandy S, Hashemi MS, Hashim I. On convergence of homotopy analysis method and its application to fractional integro-differential equations. Quaestiones Mathematicae 2013;36:93-105.

Ramzan M, Farooq M, Alhuthali MS, Malaikah HM, Cui W, Hayat T. Three dimensional flow of an Oldroyd-B fluid with Newtonian heating. Int. J. Numer. Methods Heat Fluid flow 2015;25:68-85.

Hayat T, Ullah I, Muhammad T, Alsaedi A, Shehzad SA. Three-dimensional flow of Powell-Eyring nanofluid with heat and mass flux boundary conditions. Chin. Phys. B 2016;25:074701.

Hayat T, Zubair M, Ayub M, Waqas M, Alsaedi A. Stagnation point flow towards nonlinear stretching surface with, Cattaneo-Christov heat flux. Eur. Phys. J. Plus 2016;131:355.

Hayat T, Waqas M, Ijaz Khan M, Alsaedi A. Impacts of constructive and destructive chemical reactions in magnetohydrodynamic (MHD) flow of Jeffrey liquid due to nonlinear radially stretched surface. J. Mol. Liq. 2017;225:302-10.

Subhashini SV, Sumathi R, Pop I. Dual solutions in a thermal diffusive flow over a stretching sheet with variable thickness. Int. Commun. Heat Mass Transfer 2013;48:61-6.

Hayat T, Bashir G, Waqas M, Alsaedi A. MHD flow of Jeffrey liquid due to a nonlinear radially stretched sheet in presence of Newtonian heating. Res. Phys. 2016;6:817-23.