Nonlinear radiative heat transfer in MHD three-dimensional flow of water based nanofluid over a non-linearly stretching sheet with convective boundary condition Academic research paper on "Mathematics"

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Nonlinear radiative heat transfer in MHD three-dimensional flow of water based nanofluid over a non-linearly stretching sheet with

convective boundary condition

qi B. Mahanthesha'b, B.J. Gireeshab'c'*, Rama Subba Reddy Gorlac

a Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India b Department of Mathematics, AIMS Institutes, Peenya-560058, Bangalore, India c Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, USA

Received 10 December 2015; received in revised form 12 February 2016; accepted 16 February 2016

Abstract

A theoretical investigation of the hydromagnetic three-dimensional boundary layer flow of nanofluid due to stretching sheet has been carried out in the presence of a non-linear thermal radiation, Soret and Dufour effects. Three different types of water-based nanofluids containing copper, aluminium oxide and titanium dioxide are taken into consideration. The governing boundary layer equations are transformed into a set of similarity equations using three dimensional non-linear type similarity transformations. The resultant equations are numerically solved by employing Runge-Kutta-Fehlberg fourth-fifth order method along with shooting scheme. Further, under some limiting case obtained results are compared with some previously published results and found in good agreement. The problem is governed eleven physical parameters such as magnetic parameter, radiation parameter, temperature ratio parameter, Prandtl, Schmidt, Soret, Dufour and Biot numbers, stretching ratio parameter, power index and nanoparticles volume fraction parameter. The effect of these parameters on various flow distributions is comprehensively discussed with the help of graphs and tables. It is found that, properties of the fluid can be changed by varying the concentration of nanoparticles and the nanoparticles enhance the thermal conductivity which results improvement in efficiency of heat transfer systems. © 2016 The Authors. Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Three-dimensional boundary layer flow; Heat and mass transfer; Nanofluid; Nonlinear thermal radiation; Soret and Dufour effect; Convective boundary condition

1. Introduction

The boundary layer flow analysis of an electrically conducting fluid due to a stretching sheet is of great interest because of their diverse engineering and industrial applications. MHD has immediate applications in designing of heat

Peer review under responsibility of Nigerian Mathematical Society. * Corresponding author at: Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, USA. E-mail addresses: bmanths@gmail.com (B. Mahanthesh), g.bijjanaljayanna@csuohio.edu (B.J. Gireesha), r.gorla@csuohio.edu (R.S.R. Gorla).

http://dx.doi.org/10.1016/j.jnnms.2016.02.003

0189-8965/© 2016 The Authors. Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access

article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/)._

Please cite this article in press as: Mahanthesh B, et al. Nonlinear radiative heat transfer in MHD three-dimensional flow of water based nanofluid over a non-linearly stretching sheet with convective boundary condition. Journal of the Nigerian Mathematical Society (2016), http://dx.doi.Org/10.1016/j.jnnms.2016.02.003

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2 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Nomenclature

a, b, c* Constants

B Magnetic field

Bo Constant

Bi Biot number

c Stretching ratio parameter

C Nanoparticle volume fraction (kg/m3 )

Cs Concentration susceptibility

Cw Concentration at the wall (kg/m3 )

Cœ Ambient nanofluid volume fraction (kg/m3 )

Cfx Skinfriction co-efficient along x direction

Cfy Skinfriction co-efficient along y direction

C p Specific heat coefficient (J/kg K)

Db Species diffusion coefficient

Df Dufour number

hc Heat transfer coefficient

k Thermal conductivity (W/mK)

km Mean absorption coefficient (m-1 )

kr Thermal diffusion coefficient

Le Lewis number

M Magnetic parameter

m w Mass flux at the sheet

Nu Local Nusselt number

n Power law index

Pr Prandtl number

qw Heat flux at the sheet

R Thermal radiation parameter

Rex, Re y Local Reynolds numbers

Sr Soret number

Sh Sherwood number

T Fluid temperature (K)

Tm Fluid mean temperature

Tc Surface temperature (K)

Tœ Ambient surface temperature (K)

u, v, w Velocity components along x, y and z directions

u w, vw Stretching velocities

x, y, z Coordinates (m)

M\, M2, M3, M4, M5 Expressions

Greek symbols

e Dimensionless temperature

0 p Temperature ratio parameter

Dimensionless nanoparticle volume fraction

<p Dimensionless concentration

p Density

anf Thermal diffusivity of the nanofluid

M Dynamic viscosity (kg m-1 s-1)

a Electrical conductivity

a * Stefan-Boltzmann constant (W m-2 K-4)

n Similarity variable

Tzx Surface shear stress in x direction

Tzy Surface shear stress in y direction

k Thermal conductivity

Cp Heat capacity of the nanofluid

v Kinematic viscosity

Superscript:

' Derivative with respect to n

Subscript:

f Fluid nf Nanofluid p Nanoparticles

exchangers, in space vehicle propulsion, in thermal protection, in magnetohydrodynamic (MHD) power generators, 1

MHD pumps, in polymer technology, in petroleum industry, in purification of crude oil and fluid droplets sprays. Its 2

relevance is also seen in the fields of stellar and planetary magnetospheres, aeronautics, chemical engineering and 3

electronics. In this view, many authors [1-6] have recently studied the MHD effects on flow problems with different 4

aspects. They found that, MHD effect have a significant role in thermal management applications. 5

On the other hand, a couple of decades ago Choi [7] introduced the concept of the nanofluid very first. Nanofluid 6

is a dilute suspension of base fluid and nanoparticles. To enhance the thermal performance of the base fluid an 7

innovative technique of adding additives into the base fluid is applied. The nanofluids are potential heat transfer 8

fluids can be applied in many engineering and industrial processes due to enhanced thermo physical properties and 9

heat transfer performances. In heat transfer intensification, the nanofluids possess the some advantages; (i) large 10

surface area and therefore the heat transfer surface between fluid and particle is more, (ii) high dispersion stability, 11

(iii) Required no/low pumping power to achieve equivalent heat transfer intensification as compared with base fluid, 12

(iv) Adjustable fluid properties, by varying particle concentration, (v) Absorb thermal energy directly [8]. Masuda 13 et al. [9] observed the phenomenon that the enhancement of thermal conductivity is the main characteristic feature 14 of nanofluids. A comprehensive survey of nanofluids convective heat transport was presented by Buongiorno [10]. 15 Buongiorno proposed a nanofluid model; which includes the influence of Brownian motion and thermophoretic 16 diffusion due the inclusion of nanoparticles. Later, Tiwari and Das [11] presented another mathematical model to 17 study the thermal behaviours of nanofluid by considering the effective fluid properties. This model was extensively 18 used by many researchers [ 12-20]. 19

The convective heat transfer is significant in the processes such as thermal energy storage, gas turbines, nuclear 20

plants, etc. Aziz [21] investigated the Blasius flow of viscous fluid problem with the convective boundary condition. 21

He found that, thermal boundary layer thickness could be controlled by Biot number. Makinde [22] presented a 22

similarity solution for MHD mixed convection flow on a vertical plate with the convective boundary condition. They 23

found that, the temperature profile increases with an increase in Biot number. The heat and mass transfer past a 24

vertical plate in the presence of the magnetic field and convective boundary condition were studied by Makinde and 25

Aziz [23]. The same authors [24] have examined the boundary layer flow over a stretching sheet with a convective 26

heat exchange at the surface by utilizing nanofluids. Further, they concluded that, the convective boundary condition 27

is more realistic than prescribed surface temperature or prescribed heat flux. Hayat et al. [25] visualized the Soret and 28

Dufour effects in three-dimensional flow of Maxwell fluid with chemical reaction and convective boundary condition. 29

The flow and heat transfer of Eyring Powell fluid over a continuously moving surface in the presence of a free stream 30

velocity was investigated by Hayat et al. [26] and have used the convective boundary conditions in their problem 31

formulation. Recently, Gireesha and Mahanthesh [27] addressed the perturbation solution for viscoelastic fluid flow 32

and heat transfer with convective boundary condition in non-uniform channel. Besides, the radiative heat transfer 33

have wide occurrence in various applications, such as in nuclear power plants, gas turbines, propulsion devices for 34

space vehicles, missiles and aircraft etc. In view of these applications, many researchers [28-32] have considered the 35

4 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

B = B0(x + y)> |Z

A A A I

0, _ , Nanoparticles

(Macroscopic view)

= ¿/(.T + >0" ^ T ^ V,.. = + >0"

Fig. 1. Geometry of the flow problem.

1 influence of thermal radiation effect with different physical situations. To simplify the radiative heat flux the Rosseland

2 approximation has been employed. Further, they have assumed small temperature differences within the flow to make

3 out the linear radiative heat flux. But in recent years, many authors have an interest in the study of non-linear thermal

4 radiation effect (see [33-36]).

5 However, aforementioned studies are restricted to linearly stretching sheet. It is important to note that in many prac-

6 tical applications, the stretching of the sheet is not necessarily linear. It may be quadratic, exponential or non-linear.

7 Keeping this in view, the effects of various parameters on the two dimensional flow towards a nonlinearly stretching

8 sheet have been studied by many authors [37,30,38-40]. The 3D flow and heat transfer over a plane horizontal surface

9 stretched non-linearly was studied by Junaid et al. [41]. The same authors [42] have analyzed the three-dimensional

10 flow of nanofluid over an elastic sheet stretched non-linearly in two lateral directions.

11 The main aim of the present study is therefore to extend the work of [41] by taking the non-linear thermal radiation,

12 convective boundary condition, Soret and Dufour effects in a water based nanofluid. An efficient fourth-fifth order

13 Runge-Kutta-Fehlberg scheme is used to solve the boundary layer similarity equations. A parametric study is carried

14 out in order to understand the physics of the problem.

15 2. Mathematical formulation

16 Consider a three-dimensional steady boundary layer flow of an incompressible electrically conducting water based

17 nanofluid towards the convectively heated stretching sheet. The flow is induced due to stretching of the sheet in two

18 lateral x and y directions with velocity uw = a (x + y)n and vw = b (x + y)n correspondingly, where a, b and

19 n > 0 are constants. The fluid is water based nanofluid containing copper (Cu) or alumina (Al2O3) or Titanium

20 oxide (TiO2) nanoparticles. It is assumed that, the base fluid and nanoparticles are in thermal equilibrium and no slip

21 occurs between them. The temperature and concentration at the walls are Tw and Cw. Here Tw is characterized by a

22 temperature Tc and it is determined later, is the outcome of a convective heating process with heat transfer coefficient

23 hc. The ambient values of temperature and concentrations are denoted by T^ and C^. A variable kind of magnetic

24 field B = Bo (x + y) 2 applied in the z-direction as shown in Fig. 1, here Bo is a constant. The effect of Hall current,

25 viscous dissipation and Joule heating are neglected.

26 Under these assumptions with usual boundary layer approximations, the conservation of mass, momentum,

27 temperature and concentration are given by;

du dv dw

28 — + — + - = 0, (2.1)

dx dy dz

du du du Mnf d2u OnfB2

29 u--+ v--+ w— = —---2----u, (2.2)

dx dy dz Pnf d z2 Pnf

dv dv dv Mnf d2v OnfB2

30 u--+ v--+ w— = —---tt----v, (2.3)

dv dv dv Mnf d2v OnfB2 u--+ v--+ w— = —---2---v, (2.3

d x d y d z pnf d z2 pnf

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx 5

Table 1

Comparison of the values of f ''(0) and g" (0) with that of Junaid et al. [42] with variation of ^.

n c Ф Junaid et al. [42] Present study

f '' (0) g'' (0) f '' (0) g" (0)

1 0 0 -1 0 -1 0

1 0.5 0 -1.224745 -0.612372 -1.22474 -0.61237

1 1 0 -1.414214 -1.414214 -1.41421 -1.41421

1 0.5 0.05 - - -1.35822 -0.67911

1 0.5 0.1 - - -1.43881 -0.71941

1 0.5 0.2 - - -1.49183 -0.74591

3 0 0 -1.624356 0 -1.62436 0

3 0.5 0 -1.989422 -0.994711 -1.98942 -0.99471

3 1 0 -2.297186 -2.297186 -2.29719 -2.29712

3 0.5 0.05 - - -2.20611 -1.10306

3 0.5 0.1 - - -2.33707 -1.16853

3 0.5 0.2 - - -2.4232 -1.2116

д T д T д T д2 T u— + V— + w — = anf

д x д C

д z д C

д Z 2

DBkT д2С

д С д С д2С u--+ V--+ w- = DB -TT +

д x д y д z д z2

(pCp)nf дz cs (Cp)nf дz2 '

DBkT д2T

Tm дz2 '

here и, v and w are the velocity components along the x, y and z-directions correspondingly, T and C are temperature and concentration of the nanofluid respectively, B —is applied magnetic field, DB—species diffusion coefficient, kT—thermal diffusion coefficient, Cs —concentration susceptibility, Tm fluid mean temperature and the effective fluid properties are defined as follows [18];

ßnf = (1 - Ф)

Pnf = (1 - ф) P f + (pPs,

(pCp)nf = (1 - ф) (pCp)f + Ф (pCp)

3 - 0 ф 1

°nf = Of

—S + 2 - ( ^ - Л Ф

Of 1 V Of

knf _ ks + 2kf - 2ф (kf - ks)

k f = ks + 2k f + Ф k f - ks

here ф—the volume fraction of nanoparticles, д—dynamic viscosity, p—density, k—thermal conductivity, Cp— specific heat, о —electrical conductivity, the subscript 'nf ' represents the nanofluid, ' f ' represents the base fluid and 's' represents the nanoparticles. It should be noted that, Eq. (2.5) is restricted to spherical shaped nanoparticles. The thermo physical properties of the base fluid and nanoparticles are taken from [18]. Subjected boundary conditions are given by;

u = u w, v = Vw, w = 0, -knf— = hc [Tc - T ], C = Cw, at z = 0

u ^ 0, v ^ 0, T ^ T,x, C ^ C^ as z ^ <x>. Now by following Rosseland approximation, the radiative heat flux qr is given by;

qr = -

4о * д T 4

3km д z

6 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

where a *—Stefan-Boltzmann constant and km —mean absorption coefficient. In this model, optically thick radiation is considered. The Eq. (2.8) is non-linear in T, now it can be linearise by assuming small temperature differences within the flow such that T4 can be expressed as a linear combination of the temperature, T4 can be expanded in Taylor's series about T^ as follows;

s T4 = T4 H 4T3 (T - T«) H бт! (T - T«)2 H

Now by neglecting higher order terms beyond the first degree in (T - T,), one can get

T4 = 4T3 T - 3T4.

(2.1О)

Using Eq. (2.10), the Eq. (2.8) takes the following form

д qr д z

16а * т3 д 2 T

д z 2'

Therefore, using (2.11) in the energy Eq. (2.4) we have;

(2.11)

д2 T DBkT д 2C

д T д T д T 16а * т3

u--H v--H w — = I am H--т-)— loi / ч о •

д x д y д z \m 3 (p Cp )nkm д z2 Cs(Cp)n д z2

(2.12)

The above equation corresponds to linear radiative heat transfer. But in this study we intend to investigate the non-linear thermal radiation effect, so by ignoring aforementioned assumption, Eq. (2.8) can be written as

4а* дT4 3„_

--=--T3 —.

3km д z 3km д z

16а *_ 3 д T

Now Eq. (2.4) has the following form

(2.13)

д T д T д T д u--H v--H w— = —

дx дy д z д z

16а *

т 3| дт1 h

DBkT д2C

3km (pCp)nf J dz \ Cs (Cp)nf 9z2

Now introduce three-dimensional non-linear type similarity transformations suggested by [42] as;

u = a (x + y)n f' (n), v = a (x + y)n g' (n),

n-l / n + 1 n — 1 , . ..

w = —javf (x + y) 2 I —— (f (n) + g(n)) +--Z— ^f (n) + g (n))

T - T« = (Tc - T«^(n), C - C« = (Cw - C«)y(n), n = /— (x + y)2 z

(2.14)

(2.15)

where primes denotes derivatives with respect to n. In view of Eq. (2.15), the continuity equation is automatically satisfied and the rest Eqs. (2.2), (2.3), (2.5), (2.12) and (2.14) will reduce to the following set of similarity equations;

n 1 ( )

Mi f ''' ( f + g) f '' - n(f ' + g') f ' - M5 Mf ' = О,

n 1 ( )

Mi g''' + — ( f + g) g'' - n(g' + f ') g' - M5Mg' = О,

, 4 \ .. í n + 1 .Dr ,,}

M4 ( 1 + 3 rJ в'' + PrM2¡ —+- ( f + + — <p"\= О,

Y\ff + R (1 + (в p -1) в )3\в'

+ PrM2

n + 1 . Df ..

—( f+g)*+M3 ^

1 ,, n + 1 ^ * +—

( f + g)*' + Sriï' = О.

(2.16) (2.17) (2.1S)

(2.19)

(2.20)

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Table 2

The numerical values of skin-friction distribution along x and y -direction for different values of n, M and c.

n M c Cfx Cfy

1 0 0.5 -2.606295 -1.303147

2 -3.592403 -1.796201

4 -4.361124 -2.180562

6 -5.013328 -2.506664

0.5 0.5 -2.884460 -1.442230

0.7 -3.037357 -2.126150

1.5 -3.584330 -5.376495

2 -3.887307 -7.774614

3 0 -4.233111 -2.116555

2 -4.908662 -2.454331

4 -5.499091 -2.749545

6 -6.030996 -3.015498

0.5 0.5 -4.412298 -2.206149

0.7 -4.675239 -3.272667

1.5 5.604945 -8.407418

2 -6.114631 -12.229262

Corresponding boundary conditions reduces to,

f = 1, g' = c, f = 0, g = 0, M46' = Bi (6 - 1), p = 1 at n = 0

f 0, g' 0, e 0, p -

where,

0, as n

(1 - 0)2'5 (l - 0 + 0pf)

m3 = A- 0+0

knf kf

M2 = 1 - 0 + 0

(P Cp)s (PCp) f

0 - 0 + 0f

( Of - 1) 0

(2.21)

Ol + 2

fa - 00J

and R, Pr, Df, 6p, Sc, Sr, Bi, c and M are correspondingly radiation parameter, Prandtl number, Dufour number, temperature ratio parameter, Schmidt number, Soret number, Biot number, stretching ratio parameter and magnetic parameter. In order to get similarity solution, here we take hc = c*/Vx1-n, where c* is constant. These parameters are defined by;

16a * Tœ

3kmk f V f

Db^t Tc - Tœ Tm vf Cw Cc

DBkT Cw - Cc

V fCsCpf Tc - Tc

a fBo2

(2.22)

It is worth to mention that; the linearly stretching sheet problem can be recovered for n = 1. In addition, when c = 0 the above equations reduce to two dimensional flow case. Further, the axisymmetric flow of nanofluid due to non-linear stretching sheet can be recovered for c = 1.

It is important to note that, when c = 0 = 0 and n = 1, the Eq. (2.16) and its corresponding boundary conditions takes following form;

f" ' + ff'' -( f' Y = 0, f (0) = 1, f (0) = 0,

f (c) = 0.

(2.23)

8 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Table 3

The numerical values Nusselt and Sherwood for different values of Sr, Df, R ,6p, p, Bi, M and c.

Sr Df R 0 p 0 Bi M c n = 1 Nu Sh n=3 Nu Sh

0 0.2 0.4 1.3 0.2 0.5 2 0.5 0.216326 0.943237 0.231693 1.374289

0.3 0.220071 0.906019 0.233179 1.339817

0.6 0.223454 0.869599 0.234051 1.309550

0.2 0 0.308510 0.892929 0.327023 1.322066

0.3 0.171941 0.932285 0.183489 1.366893

0.6 0.021067 0.979697 0.025852 1.421481

0 0.223108 0.917714 0.235214 1.351776

0.5 0.217800 0.918590 0.232072 1.350862

1 0.212663 0.919767 0.228662 1.350452

1.1 0.219903 0.918178 0.233346 1.350985

2 0.213570 0.919489 0.229653 1.351192

3 0.199832 0.922369 0.221866 1.351989

0 0.272889 0.913749 0.295065 1.381065

0.1 0.250259 0.905456 0.268202 1.346004

0.2 0.218854 0.918386 0.232729 1.350996

0.8 0.317610 0.902139 0.346744 1.332528

1.6 0.507121 0.871120 0.584488 1.294187

2.4 0.631193 0.850935 0.755919 1.266690

0 0.229551 1.038050 0.238349 1.438519

2 0.218854 0.918386 0.232729 1.350996

4 0.211617 0.836987 0.228070 1.279253

0.5 0.218854 0.918386 0.232729 1.350996

1.5 0.230718 1.245928 0.241344 1.796008

2.5 0.237256 1.511233 0.245875 2.154296

Table 4

The numerical values of Nusselt number and Sherwood number for different values of Bi when n = 3, M = 0.

Bi Ordinary fluid Cu-water nanofluid AI2O3-water nanofluid TÍO2-water nanofluid

Nu Sh Nu Sh Nu Sh Nu Sh

0.2 0.35622 1.53719 0.60354 1.44261 0.59940 1.55300 0.55478 1.54676

0.4 0.66765 1.50393 1.10288 1.41299 1.09983 1.52405 1.02047 1.51745

0.6 0.94109 1.47493 1.52094 1.38831 1.52213 1.49977 1.41523 1.49275

2 2.17004 1.34714 3.19811 1.29065 3.24915 1.40209 3.04595 1.39244

5 3.18815 1.24502 4.39618 1.22239 4.51683 1.33231 4.25752 1.31998

10 3.73794 1.19158 4.99580 1.18878 5.16179 1.29755 4.87738 1.28371

50 4.29720 1.13868 5.58547 1.15613 5.80232 1.26358 5.49436 1.24822

100 4.37573 1.13138 5.66727 1.15613 5.89163 1.25889 5.58045 1.24332

500 4.44001 1.12543 5.73412 1.14797 5.96468 1.25507 5.65977 1.23882

1000 4.44813 1.12468 5.74256 1.14751 5.97391 1.25458 5.66780 1.23836

10000 4.45546 •1.12401 5.75018 1.14709 5.98224 1.25415 5.66861 1.23832

100000 4.45620 1.12394 5.75094 1.14705 5.98308 1.25410 5.66869 1.23831

1 000 000 4.45627 1.12393 5.75102 1.14704 5.98316 1.25410 5.66869 1.23831

1.00E+07 4.45627 ^1.12393 5.75103 1.14704 5.98317 1.25410 5.66869 1.23831

The exact solution of the above system is given by;

f (n) = 1 - e-n.

(2.24)

3 The most important physical quantities for the problem in engineering point of view are the skin-friction coeffi-

4 cients, local Nusselt number and local Sherwood number, which are defined by the following relations:

P fu W

(x + y) qw kf (Tf — ) '

(x + y ) m w Db (Cw — Cœ)

(2.25)

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx 9

Table 5

The numerical values of Nusselt number and Sherwood number for different values of Bi when n = 1, M = 0.

Bi Ordinary fluid Cu-water nanofluid AI2O3 -water nanofluid TiO2-water nanofluid

Nu Sh Nu Sh Nu Sh Nu Sh

X.2 X.34845 1.1X4X1 X.58557 1.X4X53 X.58195 1.11552 X.53917 1.1113X

X.4 X.638X8 1.X7351 1.X3893 1.X14X6 1.X375X 1.X8959 X.96419 1.X8497

X.6 X.88X86 1.X4818 1.39766 X.99327 1.4X124 1.X69X5 1.3X542 1.X64XX

2 1.84545 X.94999 2.66384 X.921X3 2.71XX3 X.99658 2.54592 X.9893X

5 2.5X85X X.8852X 3.42576 X.87859 3.51675 X.95323 3.3188X X.94418

1X 2.82696 X.855X3 3.77218 X.85958 3.8881X X.93366 3.67599 X.92375

5X 3.12963 X.827X2 4.X9459 X.842X8 4.236X1 X.91558 4.X11X9 X.9X486

1XX 3.17X75 X.82326 4.138X7 X.83974 4.283X8 X.91315 4.X5645 X.9X233

5XX 3.2X419 X.82X22 4.17338 X.83974 4.32135 X.91118 4.X9333 X.9XX27

1XXX 3.2X419 X.81984 4.17783 X.83974 4.32617 X.91X93 4.X9798 X.9XXX1

1XXXX 3.2122X X.81949 4.18184 X.83738 4.33X52 X.91X71 4.1X217 X.89978

1XXXXX 3.21258 X.81946 4.18229 X.83735 4.33X96 X.91X69 4.1X259 X.89976

1 XXX XXX 3.21262 X.81946 4.18229 X.83735 4.331XX X.91X68 4.1X263 X.89976

1.XXE+X7 3.21262 X.81946 4.18229 X.83735 4.331XX X.91X68 4.1X263 X.89975

Fig. 2. Axial velocity for various values of stretching ratio parameter c.

where Tzx and Tzy are shear stresses at the wall along x and y directions, qw —heat flux at the wall and mw —mass flux at the wall. They are defined as below;

d u dw

Tzx = Yz + HX)^

_ dT '

qw ~-knf[ Tz)z=,

Í dv dw Tzy = f Tz + ~dJjz=o , d C

m w = - Db 1 — .

d z /z=0

(2.26)

Using similarity variables and above relations, the Eq. (2.25) takes the following form;

Re°x5 Cfx =

-TT f (X)

Rexy5c1-5Cfy = 1 g'' (X)

Re-0-5Nu = - kknf 6' (X), Re-X5Sh = -cp' (X),

(2.27)

10 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 3. Transverse velocity for various values of stretching ratio parameter c.

1.0-1-■-1-■-1-■-1-■-L

Fig. 4. Temperature and concentration for various values of stretching ratio parameter c.

1 here Rex = uw(X+y) and Rey = Vw(X+y) are the local Reynolds number along the x and y directions respectively.

2 3. Numerical method and validation

3 Nonlinear ordinary differential equations (2.16)-(2.20) subjected to the boundary conditions (2.21) are integrated

4 by means of shooting method coupled with fourth-fifth order Runge-Kutta-Fehlberg scheme. The Eqs. (2.16)-(2.20)

5 represent a two-point boundary value problem is solved by converting it into an initial-value problem. Here both f

6 and g are of order three and both 6 and y are of order two. By the principle superposition, first we reduced them into

7 a set of ten first order simultaneous differential equations. Later, the shooting scheme is employed to guess missing

8 initial conditions by an iterative process until the boundary conditions are satisfied. The shooting technique is based on

9 Maple functioning 'shoot' algorithm. A detailed explanation of the Shooting method on maple implementation can be

10 found in Meade et al. [43]. Then the resultant system of the initial value problem is solved by Runge-Kutta-Fehlberg

11 scheme.

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

0 1 2^3 4 5

Fig. 5. Axial and transverse velocity for various values of nanoparticle volume fraction p.

The Runge-Kutta-Fehlberg method is a standard method, sometimes abbreviated RKF45 and is typically employed to obtain moderate accuracy solutions for non-stiff initial value problems. The method uses a pair of Runge-Kutta methods, one used to propagate the solution, and the other for error control. Both the absolute and relative error tolerances apply for this method. The solution of the related initial value problem provides the solution to the original boundary value problem. In this method, it is most important to choose the suitable finite values of n^. Following the standard practice in the boundary layer analysis, the asymptotic boundary conditions at n^ are replaced by n6. In the numerical computation, the step size is taken as An = 0.001 with the convergence criteria 10-6. The formula of RKF-45 method is given below;

ym+1 = ym + h

ym+1 = ym+h ko = f (xm, ym) ,

(_ h _ hko

k1 = f I Xm + 4, ym + -43

25 1408 2197

-ko +--k2 +--k3

216 0 2565 2 4109 3

16 6656 28561,

-ko +--k2 +--k3

135 0 12825 2 56430 3

—k4 +--k5

50 4 55 5

k2 = f \Xm + - h, ym + — k0 + — k1 h

, _ 12 _

k3 = f [Xm + — h, ym +

k4 = f I Xm + h, ym +

1932 7200 7296 . 7

k0 — -zrzz k1 + -rrrz k2 h

439 3860

216k0 — 8k1 + "513"k2

2197 845

/_ h _ ( 8 ^ 3544 1859 11 \ 7

k5 = f xm +—, ym +--k0 + 2k1--ko +--k3--£4 h

5 J ym + 2, ym + ^ 27 0 + 1 2565 2 + 4104 3 40 )

here (3.1) and (3.2) are fourth and fifth order Runge-Kutta respectively.

In order to check the accuracy of the employed numerical method, a comparison of f" (0) and g" (0) is made with that of Junaid et al. [42] for ordinary fluid. The comparative results are presented in Table 1. We observed from this table that, comparison results are found to be an excellent agreement and this confirms that the numerical method adopted in the present work gives accurate results.

12 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 6. Temperature and concentration for various values of nanoparticle volume fraction p.

Fig. 7. Axial and transverse velocity for various values of power law index n.

1 4. Results and discussion

2 The problem of MHD three-dimensional boundary layer flow and heat transfer of a nanofluid past a convectively

3 heated non-linear stretching sheet was numerically investigated. The non-linear thermal radiation, Soret and Dufour

4 effects are present. The nanoparticle volume fraction 0 is chosen in the range of 0 < 0 < 0.2. It is worth to mention

5 that, ordinary fluid situation is recovered when 0 = 0. A parametric study is conducted, in order to analyze the

6 Q2 influence of pertinent parameters on different flow fields. The results of parametric study are presented through the

7 graphs 2-22 and Tables 2-5.

8 Figs. 2 and 3 plotted to show the velocities (along the x and y directions) respectively, for various values of stretch-

9 ing ratio parameter. From these figures it is observed that an increase in stretching ratio parameter leads to decline the

10 velocity along x-direction, whereas an opposite trend can be observed for velocity along x-direction. Physically, the

11 large values of c (=b/a) lead to either increase in b or decrease in a. As a result, the velocity along the y direction

12 increased and velocity along x direction decreased correspondingly. The variations of stretching sheet ratio parameter

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx 13

\ Green :6(л)

\ Blue : <p(n )

Bi = R = 0.5, Df = Sr = ф = 0.2

Le = 2, Pr = 6.2, с = 0.6

e = 1.5, M = 0.5

n = 1,2, 3,4, 5

--1-1-1-1--, —'"["""—1-

0 1 2 ~ 3 4 5

Fig. 8. Temperature and concentration for various values of power law index n.

on the temperature and concentration profiles were illustrated in Fig. 4. The intensity of the cooler fluid from the am- 1

bient to hotter fluid near the surface is increased as the stretching ratio parameter increases. Consequently, the thermal 2

boundary layer becomes thin and this is responsible for the decrease in temperature as well as concentration profile. 3

The similar outcome is observed by Junaid et al. [42]. 4

Fig. 5 depicts the behaviour of velocities (along the x and y directions) for the influence of the nanoparticle volume 5

fraction parameter. It is evident from this figure that, the velocity distribution is decreased rapidly with an increase in 6

ф. Since, as increase in nanoparticle volume fraction intensifies the friction force within the fluid, consequently the 7

velocity profile as well as its corresponding boundary layer thickness decreases notably. Further, it is observed that 8

the velocity distributions are much higher in the case of regular fluid (ф = 0) than that of nanofluid (ф = 0); this was 9

the similar outcome observed by Gireesha et al. [14]. The response of temperature and concentration profile for the 10

variation of the nanoparticle volume fraction parameter is shown in Fig. 6. This figure shows that, the temperature and 11

concentration distributions in the boundary layer are increasing functions of ф. As displayed in Fig. 7; for large values 12

of power-law index n, the velocities and their corresponding boundary layer thickness diminishes. It is observed from 13

Fig. 8 that, the temperature plus concentration distributions decreases with increase in n. 14

Figs. 9 and 10 shows the behaviour of axial velocity, transverse velocity, temperature and concentration profiles 15

for different values of magnetic parameter. It is well known that the magnetic parameter represents the importance of 16

magnetic field on the flow. Clearly the presence of a magnetic field retards the flow. That is, both axial and transverse 17

velocity profile decreases by increasing magnetic parameter. The presence of the Lorentzian hydromagnetic drag 18

obstructs the boundary layer flow strongly and serves as a potent control mechanism. Further, it is seen from Fig. 10 19

that, both the thermal and solute boundary layer thickness is greatly enhanced with strong magnetic field. 20

Fig. 11 describes the effect of thermal radiation parameter on the dimensionless temperature profile. This figure is 21

plotted to compare both linear and non-linear thermal radiation effects on thermal boundary layer. In this figure, the 22

solid lines correspond to linear thermal radiation and dotted lines correspond to non-linear thermal radiation and they 23

are obtained by solving the Eqs. (2.18) and (2.19) correspondingly. It is seen from this figure that, the temperature in 24

the boundary layer increases with an increase in radiation parameter. The central reason behind this outcome is that, by 25

strengthening the radiation parameter, the Rosseland radiative absorptive km decreases. Consequently, the divergence 26

of radiative heat flux 9qr/9y increases, which in turn the rate of radiative heat transfer into the fluid increases. The 27

higher radiative heat transfer to the fluid is responsible for the increase in thermal boundary layer growth. Further, 28

the temperature profile is higher for non-linear thermal radiation as compared with linear thermal radiation; which 29

was the similar result observed by Pantokratoras and Fang [33] and Mushtaq et al. [36]. Therefore, we can conclude 30

that, the non-linear thermal radiation is more suitable for heating processes like nuclear power plants, gas turbines and 31

thermal energy storages. 32

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 9. Axial and transverse velocity for various values of magnetic parameter M.

Fig. 10. Temperature and concentration for various values of magnetic parameter M.

The effect of temperature ratio parameter 6p (=Tc/ Tto) on both temperature and concentration profile is illustrated in the Fig. 12. When 6p is larger then both temperature and concentration curves are higher, so that the temperature and concentration in the boundary layer are higher too. This is due to the fact that, the stretching sheet temperature (Tc) increases with temperature ratio parameter, accordingly the thermal boundary layer thickness augmented. The Figs. 13 and 14 are aimed to shed a light on the influence of the Biot number (Bi) on temperature and concentration profiles in the boundary layer. It is interesting to note that, the thermal boundary layer thickens on the right hand side of the sheet by increasing Biot number. Physically speaking, the thermal resistance of the sheet reduced and convective heat transfer to the fluid on the right side of the sheet is augmented for larger values of Biot number. Additionally, the influence of Bi on fluid temperature can be observed in three ways, namely Bi < 1, Bi > 1 and Bi —> to; and they correspond to uniform, non-uniform and constant wall temperature respectively. It is also observed that, as Bi tends to a very large number, then temperature approaches to unity. Mathematically, the boundary condition 6' = — Bi(1 — 6) can be written as (6 = 1 + 6'/Bi). This implies the temperature 6 approaches to 1 as Bi —> to. Further, it is noticed from Fig. 14 that, the concentration distribution and its corresponding boundary layer thickness increases with Biot number.

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 11. Temperature and concentration for various values of radiation parameter R.

Fig. 12. Temperature and concentration for various values of temperature ratio parameter dp.

The influence of Dufour number on temperature and concentration profiles is shown in Fig. 15. It is observed that 1

for larger values of Dufour number, the temperature as well as concentration distribution in the boundary layer is also 2

larger. This is because; by increasing Dufour number implies that the fluid heats strongly. As a result, the temperature 3

profile increases significantly. On the other hand, the Soret number describes the impact of temperature gradients 4

inducing imperative mass diffusion effects. Fig. 16 illustrates that an increase in Soret number shows dual behaviour 5

by temperature field. Also, the concentration distribution and its associated boundary layer thickness increases for the 6

larger Soret number. This outcome is agreed very well with the results obtained by Hayat et al. [25]. 7

An expected outcome has been observed for the influence of Prandtl number on temperature and concentration 8

profile. That is the thermal and solute boundary thickness decreases for large values of Prandtl number as shown 9

in Fig. 17. It can be noticed from the Fig. 18 that, the temperature distribution in the boundary layer increases for 10

larger Lewis number. In contrast to this, for larger Lewis number the concentration profile and its associated boundary 11

layer thickness decreased rapidly. From Figs. 2 to 18 we observed an interesting fact that, the velocity along x and 12

y directions, temperature and concentration distributions are higher for linear stretching sheet than that of non-linear 13

stretching sheet. This outcome is consistent with the results obtained by Junaid et al. [42]. 14

16 B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 13. Temperature for various values of Biot number Bi.

Fig. 14. Concentration for various values of radiation parameter R.

Figs. 19-22 illustrates the characteristics of skin friction co-efficient along x and y directions, Nusselt number and Sherwood number for different nanoparticles like Cu, TiO2 and Al2O3. It is observed that, the skin friction coefficient is higher for Al2O3-water nanofluid than that of TiO2-water nanofluid and Cu-water nanofluid in order. Since the density of Cu-water nanofluid is higher than Al2O3 and TiO2-water nanofluids, the friction factors for Cu-water nanofluid is found to be lower. The Nusselt and Sherwood number will be decreased by strengthening the magnetic parameter. Physically, the thermal and solute boundary layer thickens for larger values of magnetic parameter; thus, the rate of heat and mass transfer retards significantly. Further, the Nusselt and Sherwood number is higher for TiO2-water nanofluid than that of Al2O3-water nanofluid and Cu-water nanofluid in order.

Table 2 shows the values of C fx and C fy for different values of the magnetic parameter, power law index and stretching ration parameter. We obtained the negative values for the skin friction factor of the flow. The negative value of friction drag means that the sheet exerts a drag force on the fluid. Impact of magnetic parameter, index and stretching ratio parameter on skin friction co-efficient due the flow is similar in a qualitative manner. That is, both C fx and Cfy are decreasing functions of M, n and c.

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx 17

Fig. 15. Temperature and concentration for various values of Dufour number Df.

Fig. 16. Temperature and concentration for various values of Soret number Sr.

Table 3 presents the numerical values of Nusselt and Sherwood number for different values of Sr, Df, R, 1

6p, 0, Bi, M and c. Here, the Nusselt number increases for large values of Sr, Bi, c and n whereas it decreases 2

with an increase in Df, R, 6p, 0 and M. The Sherwood number is an increasing function of Df, R, 6p, c and n 3

while it is a decreasing function of Sr, Bi, M and 0. In addition, the Nusselt and Sherwood number are higher in the 4

case of a non-linear stretching sheet than that of linear stretching sheet. 5

Tables 4 and 5 presents the variation in Nusselt number and Sherwood number due to the influence of the Biot 6

number for linear and non-linear stretching cases correspondingly. It is observed that, the Nusselt and Sherwood 7

number significantly increases as the Biot number vary from 0.2 to 100, then after a slight variation is observed in 8

both Nusselt and Sherwood number (as Bi —> to). This is due to the fact that, the Biot number increases the thermal 9

boundary layer thickness as we previously observed in Figs. 13 and 14, which in turn cause for higher Nusselt number. 10

The obtained results of the convective boundary condition are in accordance with Makinde and Aziz [24]. 11

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 17. Temperature and concentration for various values of Prandtl number Pr.

Fig. 18. Temperature and concentration for various values of Lewis number Le.

1 5. Concluding remarks

2 The fully developed heat and mass transfer of water based nanofluid with non-linear thermal radiation have been

3 studied numerically. Both linear and non-linear stretching sheet case were also analysed on the flow fields. The

4 effects of various physical parameters on different flow fields were examined. In light of the present study following

5 concluding remarks can be drawn;

6 • The properties of the fluid could be controlled by varying the nanoparticle volume fraction.

7 • Different flow characteristics are affected significantly by the types of nanoparticles.

8 • The influence of power-law index n is qualitatively same on momentum, thermal and solute boundary layer.

9 • Magnetic field obstructs the momentum boundary layer flow strongly and serves as a potent control mechanism.

10 • The Dufour and Soret number shows the opposite behaviour on thermal field.

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 19. The skin-friction distribution along x-direction for different nanoparticles versus of nanoparticle volume fraction p.

Fig. 20. The skin-friction distribution along y-direction for different nanoparticles versus of nanoparticle volume fraction p.

• The non-linear thermal radiation has high impact on flow fields as compared with linear thermal radiation. 1

• By varying the Biot number, the temperature and concentration distributions could be controlled easily. 2

• The Nusselt and Sherwood number at the stretching surface are higher in case of a non-linear stretching sheet than 3 that of linear stretching sheet. 4

For many industrial and technological applications in liquid-based systems involving stretchable equipments, the 5

present results may be accommodating. 6

Acknowledgments 7

The authors are grateful to the reviewers for their constructive remarks. Further, one of the authors (B. J. Gireesha) 8 is thankful to the University Grants Commission, India, for the financial support under the scheme of Raman Q3 9

Fellowship-2014 for Post-Doctoral Research for Indian Scholars in USA. 10

B. Mahanthesh et al. / Journal of the Nigerian Mathematical Society xx (xxxx) xxx-xxx

Fig. 21. The Nusselt number for different nanoparticles versus of nanoparticle volume fraction p.

Fig. 22. The Sherwood number for different nanoparticles versus of nanoparticle volume fraction p.

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