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Accepted Manuscript

Thermal stratification effects on MHD radiative flow of nanofluid over nonlinear stretching sheet with variable thickness

Yahaya Shagaiya Daniel, Zainal Abdul Aziz, Zuhaila Ismail, Faisal Salah

To appear in:

Journal of Computational Design and Engineering

S2288-4300(17)30124-0 http://dx.doi.Org/10.1016/j.jcde.2017.09.001 JCDE 104

Received Date: 3 July 2017

Revised Date: 8 September 2017

Accepted Date: 8 September 2017

Please cite this article as: Y. Shagaiya Daniel, Z. Abdul Aziz, Z. Ismail, F. Salah, Thermal stratification effects on MHD radiative flow of nanofluid over nonlinear stretching sheet with variable thickness, Journal of Computational Design and Engineering (2017), doi: http://dx.doi.org/10.1016/jocde.2017.09.001

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Thermal stratification effects on MHD radiative flow of nanofluid over nonlinear stretching sheet with variable thickness

a,bYahaya Shagaiya Daniel, a,bZainal Abdul Aziz*, a,bZuhaila Ismail, cFaisal Salah aDepartment of Mathematical science, Faculty of Sciences, Universiti Teknologi Mal

viaiaysi

81310 UTM Johor Bahru, Johor, Malaysia

bUTM Centre for Industrial and Applied Mathematics,

Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia ^Department of Mathematics, Faculty of Science, University of Kordofan, Elobied, 51111,Sudan.

Corresponding author:

Corresponding author: a,bZainal Abdul Aziz*

Mailing address: bUTM Centre for Industrial and Applied Mathematics, Institute Ibnu Sina for

Applied Mathe hor Bahru

Scientific and Industrial Research, 81310 UTM Johor Bahru, Johor, Malaysia

aDepartment of Mathematical science, Faculty of Sciences, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

E-mail: zainalaz@utm.my

Author: a,bYahaya Shagaiya Daniel, aDepartment of Mathematical science, Faculty of Sciences, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

bUTM Centre for Industrial and Applied Mathematics, Institute Ibnu Sina for Scientific and Industrial Research, 81310 UTM Johor Bahru, Johor, Malaysia

iil: shagaiya 12@gmail. com

thor: a,bZuhaila Ismail,

aDepartment of Mathematical science, Faculty of Sciences, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

bUTM Centre for Industrial and Applied Mathematics, Institute Ibnu Sina for Scientific and Industrial Research, 81310 UTM Johor Bahru, Johor, Malaysia

E-mail: zuhaila@utm.my Author: cFaisal Salah

Department of Mathematics, Faculty of Science, University of Kordofan, Elobied, 51111, Sudan. Email: faisal 19999@yahoo.com

Abstract

Sudan.

The combined effects of thermal stratification, applied electric and magnetic fields, thermal radiation, viscous dissipation and Joules heating are numerically studied on a boundary layer flow of electrical conducting nanofluid over a nonlinearly stretching sheet with variable thickness. The governing equations which are partial differential equations are converted to a couple of ordinary differential equations with suitable similarity transformation techniques and

are solved using implicit finite difference scheme. The electrical conducting nanofluid particle fraction on the boundary is passively rather than actively controlled. The effects of the emerging parameters on the electrical conducting nanofluid velocity, temperature, and nanoparticles concentration volume fraction with skin friction, heat transfer characteristics are examined with the aids of graphs and tabular form. It is observed that the variable thickness enhances the fluid velocity, temperature, and nanoparticle concentration volume fraction. The heat and mass transfer rate at the surface increases with thermal stratification resulting to a reduction in the fluid temperature. Electric field enhances the nanofluid velocity which resolved the sticking effects caused by a magnetic field which suppressed the profiles. Radiative heat transfer and viscous dissipation are sensitive to an increase in the fluid temperature and thicker thermal boundary layer thickness. Comparison with published results is examined and presented.

Keywords: MHD nanofluid; variable thickness; Thermal radiation; similarity solution; thermal stratification.

HIGHLIGHTS

1. MHD nanofluid due to stretching sheet with variable thickness with passively controlled.

2. A similarity transformation is used then solved by Keller box method.

3. Electric and magnetic fields are taken into account in velocity and energy analysis.

4. Combined effect of thermal stratification is examined.

5. The comparison is also made with the existing published data.

Nomenclature

b positive constant [s " 1 ]

uniform transverse magnetic field applied magnetic field skin friction coefficient

t [m2s~1]

specific heat constant ambient concentration DB Brownian diffusion coefficient [m 2 s" -1] thermophoresis diffusion coefficient uniform electric field factor electrical field parameter applied electric field Eckert number

dimensionless stream function heat transfer coefficient J Joule current

thermal conductivity kf , kp thermal conductivity of the base fluid and nanoparticle [ Wm" 1K~ 1] L e Lewis number

magnetic field parameter N buoyancy ratio parameter

Brownian motion parameter N t thermophoresis parameter Nu local Nusselt number P r Prandtl number

C7m wall mass flux [//g s_ 1 m _ 2 ] qr radiative heat flux [ VKm _ 2 ] wall heat flux radiation parameter R ex local Reynolds number

thermal stratification parameter local Sherwood number temperature of the fluid reference temperature constant temperature at the wall ambient temperature velocity component and direction component V velocity fluid [ms_ 1 ] wall mass transfer

Greek symbols

a wall thickness parameter a^- the base fluid thermal diffusivity [m 2 s_ 1] Steffan-Boltzmann constant electrical conductivity //n_f volumetric volume expansion coefficient of the nanofluid ?7 dimensionless similarity variable X dynamic viscosity of the fluid [P aS] v kinematic viscosity of the fluid [m 2 s_1] p,pn/ density [//g m _ 3 ] particle density density of the fluid heat capacity of the fluid effective heat capacity of a nanoparticle

xp stream function

a electrical conductivity

p concentration of the fluid

p w nanoparticle volume fraction at the surface

p co nanoparticle volume fraction at large values of

0 dimensionless temperature

< dimensionless concentration

t ratio between the effective heat transfer capacity and the heat capacity of the fluid

rw surface shear stress

S chemical reaction parameter

Subscripts

condition at the free stream W condition at the wall/surface

acity of the flui

1. Introduction

More recently, a new class of fluids known as nanofluids has drawn attentions of researchers in diverse areas of science and engineering technology as result of wide coverage of industrial applications of these fluids. This new innovation aims at enhancing the thermal conductivities and the convective heat transfer of fluids through suspensions of ultrafine nanoparticles in the base fluids[1]. Nanofluid is a mixture of an ultrafine nanoparticle of diameter less than 100nm dispersed in the conventional basic fluid namely water, toluene, ethylene, and oil. Some common metallic nanoparticles are copper, silver, silicon, aluminum, and titanium which tends to enhances the thermal conductivities and hence convective heat transfer rate of such fluids, which increases the energy transport strength and enactment [2-7]. Considering variable thickness due to flow, it has gained consideration due to widely advances recently in the area of engineering enhancement in the fields of mechanical, civil, architectural etc [8-10]. This is rooted in the innovative work of Fang et al. [11] against variable thickness using pure fluid. The surface medium of variable thickness have influential values and significant noticed in industrial and engineering processes. It aims at reducing the heaviness of supplementary component and

enhance the operation of devices. Consequently, this drew the attention of various researchers [12-17] for flow behavior against stretching sheet involving variable thickness.

The study of nanofluids with effects of magnetic fields has enormous applications in the fields of metallurgy and engineering advancement [18-30]. Such significant are derived in stretching of plastic sheets, polymer industry and metallurgy by hydromagnetic. In the area of metallurgical processes which involves cooling of continuous strips/filaments through drawing them from nanofluid [31]. Drawing these strips involves stretched at some point in time and also annealing and thinning of copper wires [32]. In these process, the desired properties of final product strongly depend by the virtue on the level of cooling from side to side drawing such strips in an electrically conducting fluid with impacts of magnetic fields [33]. Nanofluids are primarily aimed at cooling devices in the computer (cooling of microchips) and electronics devices (microfluidic) applications. Investigation showed that nanofluids with the influence of magnetic field, by varying the electromagnetic field, it absorbs energy and gives a controllable hyperthermia, which acts as super-para-magnetic fluid see the works of [34].

In different flows of practical relevance in nature as well as in many engineering devices, the environment is thermally stratified [35]. Stratification is a formation or deposition of layers which occur as results of temperature difference or variations of densities or due to the presence of different fluids. The discharge of hot fluid into enclosed regions more often leads in a stable thermal stratification containing lighter fluid overlying denser fluid [36]. These thermal stratification influence of convective heat transfer due to stretching sheet is of paramount importance in polymer extrusion processes, the object after passing through a die, enters the fluid for cooling of at certain range of temperature [23]. The level at which these objects are cooled a vital bearing in mind on the desired properties of the finished product. These applications involve heat rejection into the environment namely lakes, seas, and rivers; thermal energy storage systems like solar ponds; and heat transfer through thermal sources namely condensers of power plants [37-39].

Related works pointed out that thermal stratification effects in electrical magnetohydrodynamic (MHD) boundary layer stretched flow of revised nanofluid model [40] containing nanoparticles with water base fluid due to variable thickness is not investigated yet. Consequently, our main goal here is four folds. Firstly to examine thermal stratification effects through the heat in the

magnetohydrodynamic (MHD) flow. Secondly to analyze electric field impact. Thirdly to address thermal radiation and Joule heating in view of Heat transfer phenomenon. Fourth to present formulation in the company of Brownian motion and thermophoresis. In manufacturing processes, the raw material passes through the die for the extrusion in a liquefied state under high temperature, with densities gradient leads to thermal stratification. Moving surface into a cooling medium is a mathematical tool for the process of heat treatment in the fields of engineering technology noticed in mechanical, civil, architectural involved variable thickness. The governing mathematical system which is partial differential equations is converted to a system of nonlinear coupled of ordinary differential equations by similarity transformation t echniq ues. The resulted non-dimensional nonlinear convective effect are solved using implicit finite difference. Behaviors of various pertinent parameters on the velocity, temperature, and nanoparticle concentration are examined. Skin friction coefficient and Nusselt number are compared and analyzed. A comparative assessment of the present and previous data is made with judgment.

2. Mathematical formulation 2.1 Flow formulation

Consider a steady two-dimensional flow of magnetohydrodynamic (MHD) nanofluid due to a nonlinearly stretching sheet with variable thickness. The velocity of the stretching sheet is

denoted as LV ( x) = tto ( x + b ) n> and the surface is taken at y = A x (x + b ) "as the nonlinear stretching surface variable, in which for n = 1 the stretching sheet is of the same thickness. The boundary layer equations of the fluid flow are consist of the continuity equation, the momentum equation, energy equation and concentration equation. The laminar incompressible flow of viscous nanofluid in the presence of applied magnetic field and magnetic field are taken into consideration. The flow is due to stretching of a sheet from a slot through two equal and opposite force and thermally radiative. The magnetic and electric fields obey the Ohm's law define where is the Joule current, is the electrical conductivity and represent the fluid velocity. The magnetic field of strength and electric field of strength is

applied normal to the flow field. The induced magnetic field is so small, so the induced magnetic field and Hall current impacts are insignificant. We choose the Cartesian coordinate system such

that u and v are the velocity components of the fluid in the x and y-direction see Fig. 1. The combined effects of thermal radiation, viscous dissipation, magnetic and electrical fields are incorporated. The nanofluid flow due to a nonlinear stretching sheet with variable thickness is considered. The investigation of the nanofluid which involves particles and liquid. The fluid properties are indicated with subscript as (pcp) and that of particles properties as (pcp) . The

magnetohydrodynamic (MHD) boundary layer flow equation of an incompressible nanofluid are given as:

Variable sheet thickness

du dv dx dy

x — dire

Fig. 1 Physical configuration of the geometry

direction momentum equation

du du 1 dP (d2u d2u\ a

" te + » 5T ~ ^ + + a^J V£(*)SW Wu)

y — direction momentum equation

dv dv 1 dP (d2v d2v\ a

The uniqueness for this, the electrical conducting nanofluid in the presence of applied magnetics and electric fields which are incorporated in the governing equation of the nanofluid flow field. The electrically conducting implies the nanofluid possess electricity force and it may make the fluid heat transfer impact changeable, fluid due to nonlinear stretching sheet with variable thickness with defined variables for magnetic field factor is

the electrical conductivity, is the electrical field factor, are the

kinematic viscosity of the fluid and the fluid density. Follow with the boundary conditions for the velocity component [41]:

y = A1(x + b) 2 , u = Uw(x) = UQ(x + b)n, v

y -> oo: U -> 0 (4)

where is the stretching sheet constant, is the power law index, and is the wall notation. To obtain similarity solution of equations (1)-(3) the nondimensionalized variables are presented as

[42-44]:

( * \1/2 fn+1 A1/2

ip = [—^vUQ(x + b)n) F(0, K = y {-}—Uo(x + b)11'1) ,<P(V) = (.<P~ <Poo)/q> 71 1/2 /

= - (^^vU0(x + zo71"1) (f(0 + ' = (T~ Too)/(Tw - W,

u = U0(x + b)nF(O> (5)

Using an order magnitude analysis of the direction momentum equation (normal to the stretching sheet) applying the normal boundary layer equation:

U » V

du du dv dv

dy ^ dx' dx' dy ^

After the boundary layer equation [35, 45-47], substitute equations (5) and (6) into (1)-(3), we obtained the transformed ordinary differential equation as:

F"XO + FiOF'XO-^j(FXO)2+M(E1-F'(0) = 0 (7)

Here F(<Q is the dimensionless velocity, M = 2aB^/p fU0(n + 1 ) is the magnetic field parameter, E1=E0/B0U 0(x + b)n is the electric parameter. The transformed boundary conditions are presented as:

Here a = A! 2. Taking F (0 = f (Ç — a) = f (t]). The

/'"(!/) + /0?)f 0?) - (/'0?))2 + M(Et - /'Cl/)) = 0 (9)

^ = 0: ftf) = F'(£) = 1,

f -> oo: F\0 = 0 (8)

ie above equation (7) and (8)

becomes:

/(0) = a—— , /'( 0) = 1, /'( oo) = 0 (10)

where a is the wall thickness parameter. The skin friction coefficient is defined in terms of shear stress and density:

T (11)

where tw is the surface shear stress expressed in terms of i dynamic viscosity of the fluid stretching surface defined as

(1Z-) y=A1(x+b) 2

For the local skin-friction coefficient is presented in non-dimensional form as

ReJ2cfx =

■/"( 0)

Here is the local Reynolds number.

2.2 Heat convection formulation

The energy field for temperature can be expressed in terms of thermal radiation, thermal

stratification, viscous dissipation and the Joule heating as [48-51]:

dy pcp \dx2

dT dT k (d2T d2T\ ( /d(p dT d<pdT\ DT\/dTy /07Y~n u ^ + ^ = — [ ^ + ^ j + T - + - - j + - j + j j j

1 /dqr\ u ¡du\2 a

[-T- + TT +-(uB(x) — E(x))

Cpc)f\dyj pcp \dyj pcp

Where T, <p is the fluid temperature and concentration, the ambient values of temperature and nanoparticle fraction attained to a constant value of and . The radiative heat flux via Rosseland approximation can be written as [52, 53];

4<r* dT4

We assumed less temperature gradient within the viscous fluid flow in such a way that can be

ture gra ction of

expressed as a linear function of temperature. By expanding T4 using Taylor's series approach about a free stream temperature . Neglecting higher order terms in the above equation which resulted to an approximated:

4TlT - 3T¿

Using equation (15)-(16) into equation (14), we have dT

k (dzT dzT

_ ó*T\ J /d(pdT d<pdT\ DT /dTy /Ô7V

" " " n dy2 ) [ B \dx dx dy dy) Tœ \dx) \dy)

dy pcp \dx2 dy2

16d2T [L (du

+ 3k*pcpdy2 + pcp (ôy) + E(x))

When y = A ! (x + b ) ^ : T = Tw = T0 + c (x + b)n,

Here a f = k /pcp,jj.,a *, p Pf , and pp is the fluid thermal diffusivity, the kinematic viscosity, the Steffan-Boltzmann constant, the density, the fluid density and particles density respectively. We also have B 0,DB,DT , x = (pc)p /(pc)f which represents the magnetic field factor, the Brownian diffusion coefficient, the thermophoresis diffusion coefficient, the ratio between the effective heat transfer capacity of the ultrafine nanoparticle material and the heat capacity of the fluid. The boundary conditions are defined as:

y^œ: T ~^Tœ=T0 + d(x + b)n, (18)

Here T0 is the reference temperature, c > 0 & d > 0 are constants. The transformed energy equation after applying boundary layer approximation is presented as:

(l+| m)B-

I 2 n t

+ Pr ifO' - -—-f'O + Nbcp'O' + Nt6 + Ec(f")2 + MEc(f - Ej2

jsj') = 0 (19)

Boundary conditions

0(0) = 1 - st, 0(oo) = 0 (20)

Here is the Prandtl number, is the Brownian motion

parameter, is the thermophoresis parameter,

U wIcp (Tw — T œ) is the Eckert number, s t = c/d is the thermal stratification parameter, and Rd = /k* k is the radiation parameter. These parameters affect the flow behavior, the rate

of heat and mass transfer. The physical importance of Brownian motion which is about the diffusion equation produces an approximation of the time generation of the density function corresponding to the position of the particles moving under the Brownian motion.

Nusselt number is defined as:

Nux = (x + b)ilT:iqw/k{Tw - T) (21)

Where q w is the wall heat flux given as: 16dT

qw = -\k +

3k* Jdy

y=A1(x+b) 2

By using above equations we get the dimensionless form as:

/ 4 \ 1 + n

CRexrlf2 =-(l + -Rd) ¡—0X0) (23)

2.3 Volume fraction Concentration formulation

The mass diffusion analysis for the boundary layer flow of the nanoparticle fraction p with energy field for temperature T is presented as:

dcp dcp (d2(p d2p\ DT/d2T d<

After applying the boundary layer approximation to equation (27), it reduces to:

dp dp (d2p\ DT(d2T\

uTy+"Ty = D\w)+rXw) (25)

Subjected to boundary conditions:

When y = A ! (x + b ) ^ : DB^ + ^ = 0

7 1V J B dy Tocdy

y —> 00: (p ^ (poo (26)

Applying the similarity variables defined in equation (6) into equation (28) the dimensionless concentration resulted as:

= 0 (27)

Boundary conditions:

Nb<p'( 0) + Nte'( 0) = 0, 0(oo) = o (28)

Where Le = v/DB, is the Lewis number, Nb = (p c) pDBp(p c) fV denotes the Brownian motion parameter, represent the thermophoresis parameter.

The Sherwood number is defined as:

/• -(1~n)

(x + b) 2 qm

Snx =---

DB(pco

Where is the wall mass flux, is presented as:

9m — B gy

y^iXx+b) 2

The applying the above equations we have:

/ 1/. 1 + n

Shx ReJ2 = - —- 0'(O) (31)

3. Results and discussion

The set of nonlinear highly ordinary differential equations (9), (19), (27) with the respective boundary conditions (10), (20) and (28) are solved numerically using Keller box method [54], for the velocity, temperature and concentration fields. The computation is repeated until some convergence criterion is satisfied up to the desired accuracy of a 10-5 level. Comparison with the existing results published by Fang et al. [11] shows a perfect agreement, as presented in Tables 1 and 2. Table 3 displayed the variation of the skin friction coefficient, and Table 4 displayed local Nusselt number in relation to magnetic field , thermal radiation , Eckert number , Lewis number , Brownian motion , thermophoresis , Prandtl number , power law index , electric field Ex, and thermal stratification s t parameters. In Table 3, the skin friction coefficient ases by increasing while it decreases for higher values and . Generated

erical values of the local Nusselt number for different involving parameters are presented in Table 4. It is noted that the local Nusselt number decreases for higher values of and , however it increases for higher values of and .

increase numeric

Table 1: Comparison of — f''(0 ) when a = 0 . 2 5,& Et = M = 0 for varying values of m

n Ref [11]—f' ( 0 ) Present result

10.00 1.1433 1.143332

9.00 1.1404 1.140404

7.00 1.1323 1.132297

5.00 1.1186 1.118601

3.00 1.0905 1.090503

1.00 1.0000 1.000008

0.50 0.9338 0.933832

0.00 0.7843 0.784284

-1/3 0.5000 0.500001

-0.5 0.0833 0.083322

-0.51 0.0385 0.038525

-0.55 -0.1976 -0.197610

-0.60 -0.8503 -0.850207

-0.61 -1.2244 -1.224426

Table 2: Comparison of f ' '( 0 ) when a = 0 . 5 , & Et = M = 0 for varying values of m

n Ref[11] —f' ( 0 ) Present result

10.00 1.0603 ► 1.060342

9.00 1.0589 1.058933

7.00 1.0550 1.055061

5.00 1.0486 1.048627

3.00 1.0359 1.035883

2.00 1.0234 1.023420

1.00 1.0000 1.000008

0.50 0.9799 0.979950

0.00 0.9576 0.957646

-1/3 1.0000 1.000000

-0.5 1.1667 1.166661

-0.51 1.1859 1.185899

-0.55 1.2807 1.280724

-0.60 1.4522 1.452177

-0.70 2.0967 2.096621

Table 3: Numerical values for Skin friction for different values of M , E!t n,& a when

M n Ei a -/"( 0)

0.1 2.0 0.1 0.25 1.079498

0.5 1.177340

1.0 1.299003

1.5 1.412475

0.1 3.0 1.109261

5.0 1.137842

7.0 1.151717

2.0 0.2 1.055921

0.3 1.033923

0.4 1.012958

0.1. 0.5 1.043214

0.75 1.008310

1.0 0.974764

1.25 0.942551

4: Numerical values for local Nusselt number for different values of

M Rd Ec Le Nb Nt n E1 st Pr -fl'(O)

0.1 0.2 0.1 1.5 0.1 0.1 2.0 0.1 0.1 0.72 0.721557

0.5 0.725550

1.0 0.715291

1.5 0.704856

0.1 0.4 0.638225

0.6 0.574814

0.8 0.524756

1.0 0.484152

0.2 0.2 0.700506

0.4 0.658399

0.6 0.616286

0.8 0.574166

0.1 1.7 0.721933

2.0 0.722432

2.5 0.723135

3.0 0.723719

10 0.2 0.727572

0.3 0.727572

0.4 0.727572

0.1 0.2 0.739844

0.3 0.752317

0.4 0.764990

0.1 3.0 0.753037

4.0 0.767716

5.0 0.777263

2.0 0.2 0.748429

0.3 0.763959

0.4 0.776391

0.1 0.2 0.707060

0.3 0.686655

0.4 0.666356

0.1 1.0 0.895679

3.0 1.706871

5.0 2.248331

7.0 2.674167

Figs. 2-4 displayed the dimensionless velocity fields for various values of magnetic field parameter electric field parameter and nonlinear stretching sheet parameter . Variation in velocity with an increase in magnetic field parameter with effects of variable thickness can be seen from Fig. 2. It is noticed that an increase in M, the velocity profiles reduced close to the wall and suddenly increase near the stretching sheet surface as result of electrical force. The Lorentz force which acts as a retarding force tends to enhance the frictional resistance opposing the nanofluid movement in the hydrodynamic boundary layer thickness. In the case of presence

of variable thickness, the velocity and momentum boundary layer is higher. The effects of electric field parameter on the velocity profiles, is presented in Fig. 3 with effects of variable thickness. Increasing in the values E-l accelerate the nanofluid flow more significantly near the stretching sheet surface with thicker hydrodynamic boundary layer thickness. This is as result of Lorentz force that is arising due to an electrical force acting as accelerating force which tends to reduce the frictional resistance leading to shifting the stream away from the nonlinear stretching sheet. The velocity increase as the momentum boundary layer becomes thicker in the presence of variable thickness parameter to that of absence. Fig. 4 portrays the influence of nonlinear stretching sheet parameter in the presence of electric field parameter ( ) and absence

( ). Higher values of resulted in an increase in the velocity profiles and thicker

momentum boundary layer thickness. In the case of presence of electric field, the flow shifts away from the stretching surface at initial stage with an increase in the velocity field. The velocity gradient reduced as the nonlinear stretching sheet rises.

The variation of temperature field for various values of Prandtl number thermal radiation parameter Eckert number and thermal stratification parameter is investigated in the Figs 5-8. From Fig. 5, we examined for larger Pr > 1 momentum boundary layer thickness is greater than thermal boundary layer thickness with thicker thermal boundary layer thickness with a presence of variable thickness. Since Prandtl number is the momentum diffusivity to thermal diffusivity. The fluid temperature decreases close to the nonlinear

stretching sheet surface significantly for an increase in the values of Prandtl number. The rate of heat transfer at the surface increase with increases in values of Pr. Radiation impacts on the temperature field is depicted in Fig. 6. An increase in values of Rd enhances the heat flux from the nonlinear stretching sheet which resulted in an increase in the fluid's temperature. Hence the temperature field and thermal boundary layer increase with an increase in Rd. In the absence of variable thickness, the thermal boundary layer thickness is lower compared to presence. The temperature gradient reduced for higher values of thermal radiation. Fig. 7 illustrate that temperature is an increasing function of the Eckert number Ec . The fluid temperature and thermal boundary layer thickness increase significantly in the presence of variable thickness. Eckert number is the ratio of kinetic energy to enthalpy. For higher values of kinetic energy rises with consequently enhances the nanofluid temperature. The rate of heat transfer at the surface reduced for much quantity of viscous dissipation. Fig. 8 is sketched to analyze the variation in temperature field for varying thermal stratification parameter . It is worth notice that fluid temperature is a decreasing function of thermal stratification with a rise in variable thickness parameter. The effective convective potential that coexists between the nonlinear stretching sheet and the ambient nanofluid decreased with an increase in . In view of this, the fluid temperature and thermal boundary layer thickness reduced for higher thermal stratification. The rate of heat transfer at the surface increases for an increase in the amount of thermal stratification.

Figs. 9-11 are plotted to examine the behavior of Brownian motion parameter Nb, thermophoresis parameter Nt, and Lewis number Le on the dimensionless nanoparticles concentration field ф (j ) . Fig. 9 it is obvious that the nanoparticle concentration and its solutal boundary layer thickness is smaller for larger Brownian motion parameter with thicker concentration boundary layer as resulted of variable thickness. The nanoparticle concentration gradient at the boundary wall is controlled passively at the surface by the expression of and temperature gradient. The effects of thermophoresis parameter on the concentration profile ф(jj)is demonstrated in Fig. 10. It is noticed that the nanoparticle concentration increases with an increase in thermophoresis parameter and higher with variable thickness. This is as result of thermophoresis force developed by the rate of mass transfer at the surface creates a smooth flow far from the nonlinear stretching sheet surface. In view of this more heated fluid shift away from the surface and hence with much amount of thermophoresis,

the fast flow from the nonlinear stretching sheet due to the presence of thermophoresis force resulting to enhancement in the nanoparticle concentration boundary layer thickness. The influence of Lewis number Le on nanoparticle concentration is revealed in Fig. 11. It is worth noticing that the nanoparticle concentration reduced significantly with higher values of Lewis number but thicker with variable thickness. This reduction in nanoparticle concentration and solutal boundary layer thickness is as results of the change in Brownian diffusion coefficient. It noted that higher values Lewis number associated to weaker Brownian diffusion coefficient. The concentration gradient increases with higher values of Lewis number.

0.9 0.8 0.7 0.6 S 0.5

0.4 0.3 0.2 0.1 0

a =0.0 a =1.0

M=0.0, 0.5, 1.0, 1.5

Pr=6.2, Rd = Ec=0.2, E =0.01, s =0.1, Nb=Nt=0.5, Le=10, n=4.0.

3456789 10

Fig.2 Influence of M on the velocity profile f' (7 )

ence o

0.9 0.8 0.7 0.6 3 0.5

0.4 0.3 0.2 0.1

a =0.0 a =1.0

E =0.0, 0.1, 0.2, 0.3

Pr=6.2, Rd=Ec=0.2, M=0.1, s =0.1, Nb=Nt=0.5, Le=i0, n=4.0.

0 1 2 3 4 5 6 7

Fig.3 Influence of Et on the velocity profile f' (r)

0.9 0.8 0.7 0.6

0.3 0.2 0.1

0 12 3 4

6 7 8 9 10

Fig.4 Influence of on the velocity profile

0.9 0.8 0.7 0.6 3 0.5

0.4 0.3 0.2 0.1 0

-a =0.0

a =1.0

Pr=0.72, 3.0, 6.2, 10.0

Rd=Ec=0.2, E =M= 1.0, st=0.1, Nb=Nt=0.5, Le=10, n=4.0.

0 0.5 1 1.5 2 2.5 3 3.5

Fig.5 Influence of Pr on the temperature pri

rofile 0 (r

0.9 0.8 0.7 0.6 0.5 0.4

a=0.0 a =1.0

Pr=6.2, Ec=0.2, E =M= 1.0, s =0.1,

' '1 ' t '

Nb=Nt=0.5, Le=10, n=4.0.

Fig.6 Influence of Rd on the temperature profile 0 (j)

0123456789 10

Fig.8 Influence of s t on the temperature profile 0( 7 )

0.1 0 -0.1 -0.2 -0.3 2 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

Fig.9 Influence of Nb on the concentration profile 0 ( 7 )

-0.1 -0.2 -0.3 -0.4 0.5 6 0.7 -0.8

:■ l L L L L "

\ a =0.0 _

a =1.0

Nt=0.1, 0.2, 0.3, 0.4

Pr=6.2, Rd=Ec=0.2, E =M=0.1, '

^=0.1, Nb=0.5, Le=10, n=4.0

■ r r r r r

Fig.10 Influence of N t on the concentration profile 0 (7 )

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

a=0.0 a =1.0

Pr=6.2, Rd=Ec=0.2, E =M=0.1,

' ' 1 '

s =0.1, Nb=Nt=0.5, n=4.0

Fig.11 Influence of L e on the concentration profile <p ( t])

4. Conclusion

ind Joule hi

The impact of thermal radiation and Joule heating due to the magnetohydrodynamic (MHD) flow of nanofluid against a nonlinearly stretching sheet with variable thickness are examined in the presence of thermal stratification. Thermal stratification and passively controlled nanoparticle concentration on the boundary condition in the presence of applied magnetic and electric fields makes this investigation novel one. The main conclusion of this study is stated as follows:

High velocity is obtained for higher values of electric field and nonlinear stretching

sheet.

The temperature reduces with increasing values of the thermal stratification parameter and Prandtl number.

Magnetic field effects on electrical conducting nanofluid suppressed the flow at the initial stage and after some distance enhances due to electrical force near the nonlinear stretching sheet surface which leads to enhancement of the skin friction coefficient.

iv. The radiative heat transfer and viscous dissipation in the presence of variable thickness plays a dominant role in the temperature of the nanofluid.

v. Lewis number and thermophoresis of nanoparticles reduces the nanoparticle concentration volume fraction for higher values but enhanced the mass transfer rate at the controlled surface.

vi. Brownian movement and thermophoresis nanoparticle deposition in the presence of variable thickness and electric field reaches a significant role on the nanoparticle concentration volume fraction due to random movement of ultrafine nanoparticle suspended in the base fluid with passively controlled boundary at the wall.

Acknowledgments

her Educi

The authors would like to acknowledge Ministry of Higher Education and Research Management

Centre, UTM for the financial support through GUP with vote number 11H90, Flagship grants

with vote numbers 03G50 and 03G53 for this

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Conflict of interest

The authors declare no conflict of interest.

Boundary Layer

Fig. 1 Physical configuration of the geometry