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Results in Physics

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

On MHD nonlinear stretching flow of Powell-Eyring nanomaterial

Tasawar Hayata,b, Rai Sajjadc, Taseer Muhammad a'*, Ahmed Alsaedib, Rahmat Ellahic,d

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

b Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Department of Mathematics and Statistics, Faculty of Basic & Applied Sciences, International Islamic University, Islamabad 44000, Pakistan d Department of Mechanical Engineering, Bourns Hall A373, University of California, Riverside, CA 92521, USA

ARTICLE INFO ABSTRACT

This communication addresses the magnetohydrodynamic (MHD) flow of Powell-Eyring nanomaterial bounded by a nonlinear stretching sheet. Novel features regarding thermophoresis and Brownian motion are taken into consideration. Powell-Eyring fluid is electrically conducted subject to non-uniform applied magnetic field. Assumptions of small magnetic Reynolds number and boundary layer approximation are employed in the mathematical development. Zero nanoparticles mass flux condition at the sheet is selected. Adequate transformation yield nonlinear ordinary differential systems. The developed nonlinear systems have been computed through the homotopic approach. Effects of different pertinent parameters on velocity, temperature and concentration fields are studied and analyzed. Further numerical data of skin friction and heat transfer rate is also tabulated and interpreted.

© 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/).

Article history:

Received 24 November 2016

Received in revised form 18 December 2016

Accepted 24 December 2016

Available online xxxx

Keywords: Powell-Eyring fluid Magnetohydrodynamics Nanomaterial

Nonlinear stretching surface

Introduction

Alternative form of fluids that are composition of nanometer sized particles and convectional base liquids are termed as nanofluids. Nanoparticles utilized in the nanomaterials are basically made of metals (Ag, Cu, Al) or nonmetals (carbon nanotubes, graphite) and the base liquids include ethylene glycol, water or oil. Suspension of nanoparticles in the base liquids greatly varies the heat transfer characteristics and transport property. To obtain prominent thermal conductivity enhancement in the nanofluids, the studies have been processed both theoretically and experimentally. Applications of nanofluids in technology and engineering are nuclear reactor, vehicle cooling, vehicle thermal management, heat exchanger, cooling of electronic devices and many others. Moreover magneto nanofluids (MNFs) are helpful in removal of blockage in arteries, wound treatment, cancer therapy, hyperthermia and resonance visualization. Further the nanomaterials enhances the heat transfer rate of microchips in microelectronics, computers, fuel cells, transportation, biomedicine, food processing etc. The pioneer investigation regarding enhancement of thermal properties of base liquid through the suspension of nanoparticles was presented by Choi [1]. Later the development of mathematical relationship of nanofluid with Brownian diffusion and thermophoresis is presented by Buongiorno [2]. Turkyilmazoglu [3] derived the

* Corresponding author. E-mail address: taseer@math.qau.edu.pk (T. Muhammad).

exact analytical solutions for MHD slip flow of nanofluids by considering heat and mass transfer characteristics. Further relevant attempts on nanofluid flows can be quoted through the analysis [4-27] and various studies therein.

Non-Newtonian fluids are regarded very prominent for applications in chemical and petroleum industries, biological sciences and geophysics. The flow of non-Newtonian fluids due to stretching surface occurs in several industrial processes, for example, drawing of plastic films, polymer extrusion, oil recovery, food processing, paper production and numerous others. The well-known Navier-Stokes expressions are not appropriate to describe the flow behavior of non-Newtonian materials. However various constitutive relations of non-Newtonian materials are proposed in the literature due to their versatile nature. Such materials are categorized as differential, integral and rate types. The Powell-Eyring fluid model [28-33] is derived from kinetic theory of gases instead of empirical relation as in the case of the power-law model. Further it appropriately recovers Newtonian behavior at low and high shear rates. Ketchup, human blood, toothpaste, etc. are the examples of Powell-Eyring fluid.

There is no doubt that much attention in the past has been devoted to the flow caused by linear stretching velocity. However this consideration is not realistic in plastic industry. Hence some researchers studied the flow problem of nonlinear stretching surface. Gupta and Gupta [34] initially studied the flow by nonlinear stretching velocity. Heat transfer in flow of viscous fluid generated by nonlinear stretching velocity is analyzed by Vajravelu [35]. Cor-

http://dx.doi.org/10.1016/j.rinp.2016.12.039 2211-3797/® 2016 Published by Elsevier B.V.

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Nomenclature

u, v velocity components C concentration

1 dynamic viscosity C ambient fluid concentration

v kinematic viscosity B0 magnetic field strength

T temperature k thermal conductivity

T 11 ambient fluid temperature (Pc)f heat capacity of fluid

r electrical conductivity Db Brownian diffusion coefficient

a thermal diffusivity DT thermophoretic diffusion coefficient

(Pc)p effective heat capacity of nanoparticles uw surface velocity

Db Brownian diffusion coefficient Tw surface temperature

uw surface velocity a positive constant

a positive constant n power-law index

f similarity variable f ' dimensionless velocity

e dimensionless temperature / dimensionless concentration

K, K fluid parameters M magnetic parameter

Le Lewis number Pr Prandtl number

Nb Brownian motion parameter Nt thermophoresis parameter

sw wall shear stress Qw wall heat flux

sij extra stress tensor Cfx skin friction coefficient

Nux local Nusselt number Rex local Reynolds number

x, y coordinate axes

Pf density of base fluid

P, C material constants

tell [36] extended the work of [35] through prescribed surface temperature and constant surface temperature cases. Prasad et al. [37] investigated mixed convection flow over a nonlinear stretching surface. Mustafa et al. [38] analyzed axisymmetric nanoliquid flow past a nonlinear stretching surface. Magnetohydrodynamic (MHD) flow of second grade nanomaterial induced due to a nonlinear stretching sheet is studied by Hayat et al. [39].

This communication addresses the magnetohydrodynamic (MHD) flow of Powell-Eyring nanomaterial over a nonlinear stretching surface. Powell-Eyring fluid's constitutive relations are used in the problem formulation. Novel features regarding ther-mophoresis and Brownian motion are taken into consideration. Recently proposed condition for zero mass flux of nanoparticles at stretching sheet is taken into account. This condition describes that the nanofluid particle fraction on the boundary is passively rather than actively controlled, i.e., it is no longer assumed that one can control the value of the nanoparticle fraction at the wall but rather that the nanoparticle flux at the wall is zero. This change necessitates a rescaling of the parameters that are involved. The governing nonlinear systems have been computed through the homotopic approach [40-47]. Effectiveness of influential parameters on velocity, temperature and concentration fields have been studied in detail. Further numerical data of skin friction and heat transfer rate are explained.

Statement

Let us consider two dimensional (2D) magnetohydrodynamic flow of an incompressible Powell-Eyring nanomaterial. The flow is caused by a nonlinear stretching surface. Features of ther-mophoresis and Brownian motion are taken into consideration. The x- and y-axes are taken parallel and transverse to the stretching surface. The sheet at y = 0 is stretching along the x-direction with velocity uw{x) = axn where a and n are positive constants. Powell-Eyring fluid is electrically conducted subject to nonuniform magnetic field applied in the y-direction (see Fig. 1). Here the induced magnetic field is neglected for low magnetic Reynolds number [48-50]. Assumptions of low magnetic Reynolds number and boundary layer approximation are employed in the mathemat-

ical development. The extra stress tensor for Powell-Eyring fluid is [31]:

Tj =1 - + b sinh- ( r dx-

in which i stands for dynamic viscosity and b and C for material constants. Considering

sinh-1(i dUi) ~ 1 du-. 1(1 du-

C dxjj C dXj 6 V C dXj

1 duC dXj

The boundary layer expressions for two-dimensional (2D) mag-netohydrodynamic flow of Powell-Eyring nanofluid are [31,39]:

du dv .

dX + dy =

du du u dx + v dy =

'du\ 2 d2 u

Pf pC J dy2 2pfPC3 \9yJ dy

rB2(x) ---i

dT dT d2T (pc)

usr + VTT = + 7-x IT

dx dy dy2 (pc)f \ Tt

piDl (dl)2 + Db( dT dC

dC dC „ d2C\ Dt

Utt + Vtt = Db h^ dx dy \ dy2 T i

d2T dy2

Here u and v show the velocity components along the horizontal and vertical directions, B(x) = B0xnr1 represents the non-uniform magnetic field, v^ = p^ stands for kinematic viscosity, Pf for density of base liquid, r for electrical conductivity, T for temperature, C for concentration, (pc)p for effective heat capacity of nanoparticles, a = k/(pc)f for thermal diffusivity of fluid, k for thermal conductivity, (pc)f for heat capacity of fluid, DT for thermophoretic

128 129

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Fig. 1. Geometry of the problem.

diffusion coefficient, DB for Brownian diffusivity and T1 for ambient fluid temperature. The associated boundary conditions are [8,39]:

u = uw(x) = axn, v = 0, T = Tw, Db @y + ^ @y = 0 at y = 0, (7)

u ! 0, T ! T1, C ! Cl as y -

where a shows the rate of the stretching surface, Tw the temperature of the stretching surface, n the power-law index and C„ the ambient fluid concentration. Introducing the suitable transformations

u = axnf(f), v = -(«)1/2x^(f + n^rf ),1

f = p^)1^, 0(O=£b, /(0=^/

Expression (3) is vanishes identically while (4)-(8) yield

^ k kf f- f2

(1 + K T + ff" -2

n + 1 =0;

0''' + Prf 0 + N60'/' + Nt 0'^ = 0,

/'' + LePrf /' + N 0'' = 0, Nb

f = 0, f = 1, 0 = 1, Nb/' + Nt0' = 0 at f = f ! 0, 0 ! 0, / ! 0 as f !1

(10) (11) (12) (13)

Here K and A stand for fluid parameters, Pr for Prandtl number, M for magnetic parameter, Nt for thermophoresis parameter, Nb for Brownian motion parameter and Le for Lewis number. These parameters are defined by

K - A - uW M2 — rB» Pr — m

K = ibC , A = 2mxC-2 ' M = Pfa' Pr = a'

fj (pc)pDT (Tw-Ti) M (fiC)pDBC,

Nt = -VT-- , Nb =

1 Le_a

(qc)f V , Le Db :

(qc)f vTi

Skin friction coefficient and local Nusselt number are

Cfx Nux = xqw

' k(Tw - Ti) '

in which wall shear stress (sw) and heat flux (qw) are given by

1 @u 1

bC*, @y ebC*3 W ) y=0

qw = -'< f)

In dimensionless variables

Re1/2Cfx = (1 + K)f''(0) -1 (^KAf 3(0)), > Re-1/2Nux = -y/'n+L0' (0), >

where Rex = uwx/v stands for local Reynolds number.

Homotopic solutions

The appropriate initial approximations f 0 , 00 , /0) in homotopic solutions are defined as

f0(f) = 1 - exp(-f), 00(f) = exp(-f), /(f)

= -Nb exp(-f),

and auxiliary linear operators (£j , L0 , L^) are

d3f df d20 d2 / Lf = df3 - df L = df2 - 0 L/ = df2 -

The above auxiliary linear operators satisfy the following characteristics:

Lf [E + E2 exp(f)+E3 exp(-f)] = 0 , L0 [E4exp(f) +E5exp(-f)] = 0, L/ [£6exp(f)+£7exp(-f)= 0 ,

where £* (r = 1 - 7) elucidate the arbitrary constants. Deformation problems at zeroth-order are

(1 - p*)L/ f (f ,p) -f,(f)] = phNf f (f ,p)], (21)

(1 - p)Lh [0(f,p) - 00(f)] = phNhf(f ,p), 0(f ,p) , /(f ,p)], (22)

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(1 - p)L/[/(f, p)-/c(f)] = PhN/ff p), f(f, p), /(f, p)], (23) L9[hm (f)-Xd em-1(f)] = hRR f

f (0, p) = 0, f'(0, p) = 1, 0(0, p) = 1, Nb/'(0, p)+Nt h ' (0, p) = 0,

f'(1, p) = 0, e(i, p) = 0, /(1, p) = 0,

Nf f (f,p)]-<-+K ) f+f f - (^K f f

2n n +1

Ne f (f, p), f(f, p), /(f, p)]

d2'd df2

n @fde .. df d/ .. (df

+Pr(f df+Nbdf df+M df

N / [f (f, p), f(f, p), / (f, p)]-f + LePrf + N f •

@/f Nt @2fe

@f Nb @f2

Here Nf, Nh and N/ are nonlinear operators and hf, hh and h/ the nonzero auxiliary parameters. For p = 0 and p = 1 one obtains

f(f,0)-f0(f), f(f, 1)—f(f),

h(f,0)-hc(f), h(f, 1) - e(f),

/(f,0) — /0(f), /(f, 1)-/(f).

When p changes from 0 to 1 then f (f,p), h(f,p) and /(f,p) display alteration from primary approximations f0(f), h0(f) and /0(f) to desired ultimate solutions f (f), h(f) and /(f). The following expressions are derived via Taylor's series expansion:

f(f, p)—f0(f) + Efm(f)pm, fm(f) —

f(f,p)—he(f) + £hm(f)pm, hm(f) —

1 dmf (f, p)

m! dp'

1 dmf(f, p)

m! dpd

/(f, p) — /0(f) + ¿/m(f)pm, /m(f) — m

m 1 ! @

The convergence regarding Eqs. (31)-(33) is strongly based upon the suitable choices of hf, hh and h/. Choosing suitable values of hf, hh and h/ so that Eqs. (31)-(33) converge at p = 1 then

f (f)—f <,(f) + £f m(f), m—1

h(f) — hc(f)^ hm(f), m—1

/(f) — /0(f) + ^/m(f).

The mth-order deformation problems are presented as follows:

Lf |fm (f)-Vmfm-1(f)] — hRf (f), (37)

L/[/m (f)-Vm/m-1(f)] — hR f (f),

fm (0)—fm (0) —hm (0)—0, > Nb/m (0) + Nt h'm (0) —0, I fm(l) — hm(l) — /m(l) — 0 J

Rf (f)—(1 + k)/™-, +z fm-1-kfd - ( ^ ) KA^/-1-kYTk-fi

2 M KA^-kf ^ / k—0 l—0

^ff -(-+1) Mfd-1, (41)

/if-1 if-1 , Rif(f)—hf-1 + Pr If™-1-khk+NbY,hf-1-k/k+n^hf-1-kh'k

R f (f) — /f-1 + LeP^Xfm-1-k/k + ^ hf-1

0, if 6 1,

1, if > 1,

In terms of special solutions (ff, 0f,/d ), the general solutions fm, h™, /m) of the Eqs. (37)-(39) are defined by the following expressions:

fm (f) —fm (f) + E + E2 exp(f) + E exp(-f),

hm (f) — hf (f)+E4exp(f)+E5exp(-f), /m (f) — /f (f) + E6 exp(f) + E7 exp(-f),

in which (r — 1 - 7) through the boundary conditions (40) are given by

p2 — p4 — E6 — 0, E3 —

df f (f)

E — - - ff (0),

E5 — -hf(0), E7 —

d/m (f)

,Nt ( dhimii)

f—0 + Nbl p5 + df

Convergence analysis

Here the homotopic solutions (34) - (36)contain the nonzero auxiliary parameters hf, hh and h/. Such auxiliary parameters play a significant role to control and adjust the region of convergence. To get the suitable values of auxiliary parameters, the h-curves are sketched at 25th order of deformations. Fig. 2 displays that the convergence zone lies within the ranges -1.8 6 hf 6 -0.1, -1.75 6 hh 6 -0.15 and -1.7 6 h/ 6 -0.2. The residual errors for velocity, temperature and concentration distributions are calculated through the following expressions:

Dm —

f hf )] 2df, f ha)] 2df,

Ad — I [Rm (f, h/)\ df.

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K = A = M= 0.1,Pr= n= 1.2, A{ = 0.2, Ni, = 0.3, Le = 1.0 0.0010

0.0005

A^, 0.0000

-1.5 -1.0 -0.5

Fig. 2. The h-curves for f (f), 6(f) and /(f).

-0.0005

-0.0010

Fig. 4. hh-curve for the residual error Am

To get the suitable range for h, the h-curves for the residual errors of velocity, temperature and concentration distributions are plotted in Figs. 3-5. It is observed that the correct results up to fifth decimal place are obtained when we choose the values of hfrom this range. HAM solutions convergence via Table 1 is satisfactorily achieved by considering 20th orders of approximation.

Discussion

This portion has been arranged to explore the impacts of several effective parameters including fluid parameter K, magnetic parameter M, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, Prandtl number Pr and power-law index n on velocity f (f), temperature 0(f) and concentration /(f) distributions. Fig. 6 shows the impact of fluid parameter K on velocity distribution f (f). Both velocity field and momentum boundary layer thickness are increased for larger K. Behavior of M on velocity distribution f (f)is presented in Fig. 7. Here both velocity and momentum boundary layer thickness decay for M. Fig. 8 shows influence of power-law index n for velocity f (f). By increasing n, both the velocity and momentum boundary layer thickness have been reduced. Here n = 1 corresponds to linear stretching surface case and n — 1 for nonlinear stretching surface. The impacts of fluid parameter K, magnetic parameter M, thermophoresis parameter

a£ 0.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 ñf

Fig. 3. hf-curve for the residual error Afm.

0.10 0.05

A i 0.00

-IS -1.0 -0.5 0.0

Fig. 5. h/-curve for the residual error Am.

Table 1

Homotopic solutions convergence when K = A = M = 0.1, Pr = n = 1.2 , Nt = 0.2 , Nb = 0.3 and Le = 1.0.

Order of approximations -f '(0) -h (0) /' (0)

1 0.98485 0.70000 0.46667

5 0.98476 0.64522 0.43015

10 0.98475 0.64301 0.42867

15 0.98475 0.64292 0.42862

20 0.98475 0.64294 0.42863

25 0.98475 0.64294 0.42863

30 0.98475 0.64294 0.42863

35 0.98475 0.64294 0.42863

Nt, Prandtl number Pr and power-law index n for temperature 0(f) have been displayed in the Figs. 9-13 respectively. It is observed that by increasing magnetic parameter M, thermophore-sis parameter Nt and power-law index n, both the temperature distribution and thermal boundary layer thickness are increased whereas opposite behavior is seen for fluid parameter K and Prandtl number Pr. It is a valuable fact to mention here that the properties of liquid metals are characterized by small values of Prandtl number (Pr < 1), which have larger thermal conductivity but smaller viscosity, whereas higher values of Prandtl number (Pr > 1) associate with high-viscosity oils. Particularly Prandtl number Pr = 0.72 ,1.0 and 6.2 are associated to air, electrolyte

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Fig. 8. Plots of f (f) for n.

379 solution such as salt water and water respectively. Moreover it is

380 also observed that Nt portrays the strength of thermophoresis

381 effects. Higher Nt leads to more strength to thermophoresis. The

Fig. 11. Plots of S(C) for Nt.

variations in concentration field /(f)for various values of fluid 382

parameter K, magnetic parameter M, thermophoresis parameter 383

Nt, Brownian motion parameter Nb, Lewis number Le, Prandtl num- 384

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K=A = M = 0.1, n = 1.2, N, = 0.2, Nb = 0.3, Le = 1.0.

M= 0.0,0.5,0.8,1.0

SiawMi^...

K=A = 0.1, Pr = n = 1.2, N, = 0.2, Nb = 0.3, Le = 1.0.

Fig. 12. Plots of 0(C) for Pr.

Fig. 15. Plots of /(f) for M.

K=A = M = 0.1, Pr = 1.2, N, = 0.2, Nb = 0.3, Le = 1.0.

n - 0.0,0.2,0.6,1.8

Fig. 13. Plots of 0(0 for n.

Nt = 0.1,0.4,0.7,1.0

I=A = M = 0.1, Pr = n = 1.2, Nb = 0.3, Le = 1.0.

Fig. 16. Plots of /(f) for Nt.

K = A = M = 0.1, Pr = n = 1.2, N, = 0.2, U = 1.0.

Fig. 14. Plots of /(f) for K.

Fig. 17. Plots of /(f) for Nb.

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Le = 0.9,1.1,1.3,1.5

K = A = M = 0.1, Fr = n = 1.2, N, = 0.2, Nt = 0.3.

Fig. 18. Plots of /(0 for Le.

Pr=0.7,1.0,1.3,1.6

K = A = M = 0.1, n = 1.2, N, = 0.2, iV4 = 0.3, Le = 1.0.

Fig. 19. Plots of /(0 for Pr.

» = 0.0,0.2,0.6,1.8

K = A = M = 0.1, ft = 1.2, N, = 0.2, iVj = 0.3, Le = 1.0,

Table 2

Numerical data of skin friction coefficient for K, A, M and n.

-Rei/2C,

0.0 0.1 0.1 1.2 1.0832

0.1 1.1324

0.2 1.1802

0.1 0.0 0.2 0.4 0.1 1.2 1.1361 1.1288 1.1214

0.1 0.1 0.0 0.1 0.2 1.2 1.1276 1.1324 1.1468

0.1 0.1 0.1 0.8 1.0 1.2 0.9629 1.0511 1.1324

Table 3

Numerical data of local Nusselt number for K, M, Nt, Nb, Le, Pr and n when A = 0.1.

K M Nt Nb Le Pr n -Re-1/2 Nux

0.0 0.1 0.2 0.3 1.0 1.2 1.2 0.6638

0.1 0.6742

0.2 0.6836

0.1 0.0 0.1 0.2 0.2 0.3 1.0 1.2 1.2 0.6753 0.6743 0.6715

0.1 0.1 0.1 0.2 ^ 0.3 0.3 1.0 1.2 1.2 0.6848 0.6743 0.6639

0.1 0.1 0.2 0.1 0.2 0.3 1.0 1.2 1.2 0.6743 0.6743 0.6743

0.1 0.1 0.2 0.3 0.0 0.5 1.0 1.2 1.2 0.6951 0.6824 0.6743

0.1 0.1 0.2 0.3 1.0 0.8 1.0 1.2 1.2 0.5177 0.6001 0.6743

0.1 0.1 0.2 0.3 1.0 1.2 0.8 1.0 1.2 0.6172 0.6463 0.6743

Fig. 20. Plots of /(0 for n.

ber Pr and power-law index n are displayed in the Figs. 14-20. 385

Concentration field through these sketches enhances for larger 386

magnetic parameter M, thermophoresis parameter Nt and power- 387

law index n whereas reverse trend is observed for fluid parameter 388

K, Brownian motion parameter Nb, Lewis number Le and Prandtl 389

number Pr. Table 2 depicts the numerical data of skin friction coef- 390

ficient for several effective parameters K, A, M and n. Skin friction 391

coefficient is higher for larger K, M and n while the reverse behav- 392

ior is noticed through n. Table 3 is presented to analyze the numer- 393

ical data of local Nusselt numbers via different parameters. Here 394

local Nusselt number increases for larger fluid parameter K, Prandtl 395

number Pr and power-law index n whereas opposite result holds 396

for magnetic parameter M, thermophoresis parameter Nt and 397

Lewis number Le. There is no significant change of Nb on local Nus- 398

selt number. 399

Conclusions 400

Magnetohydrodynamic (MHD) flow of Powell-Eyring nanoma- 401

terial bounded by a nonlinear stretching surface is investigated. 402

Main observations of presented analysis are: 403

• Larger values of fluid parameter K depict increasing behavior for 404

velocity field while opposite behavior holds for temperature 405

and concentration fields. 406

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• Impact of magnetic parameter M on temperature and concentration fields is quite opposite to that of velocity field.

• Temperature and concentration fields through Prandtl number Pr are qualitatively similar.

• An increase in Lewis number Le causes a decay in the concentration field and related concentration layer thickness.

• Behaviors of Brownian motion Nb and thermophoresis Nt parameters on concentration field are different.

• Skin friction coefficient is higher for larger values of K, M and n while the reverse trend is noticed through A.

• Heat transfer rate at the surface (local Nusselt number) is lower when the larger values of magnetic M and thermophoresis Nt parameters are taken into account.

• The present results lead to the hydrodynamic flow situation when M = 0.

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