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Ain Shams Engineering Journal (2015) xxx, xxx–xxx

Ain Shams University

Ain Shams Engineering Journal

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MECHANICAL ENGINEERING

Entropy analysis of convective MHD flow of third

grade non-Newtonian fluid over a stretching sheet

* Corresponding author at: Shanghai Key Lab of Vehicle

Aerodynamics and Vehicle Thermal Management Systems, Tongji

University, 4800 Cao An Rd., Jiading, Shanghai 201804, China.

Tel.: +98 811 8257409; fax: +98 811 8257400.E-mail addresses: [email protected], mm_rashidi@yahoo.

com (M.M. Rashidi).

Peer review under responsibility of Ain Shams University.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.asej.2015.08.0122090-4479 � 2015 Faculty of Engineering, Ain Shams University. Production and hosting 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/).

Please cite this article in press as: Rashidi MM et al., Entropy analysis of convective MHD flow of third grade non-Newtonian fluid over a stretching sheet, AiEng J (2015), http://dx.doi.org/10.1016/j.asej.2015.08.012

M.M. Rashidi a,b,*, S. Bagheri c, E. Momoniat d, N. Freidoonimehr e

aShanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University,4800 Cao An Rd., Jiading, Shanghai 201804, ChinabENN-Tongji Clean Energy Institute of advanced studies, Shanghai, ChinacMechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, IrandDST/NRF Centre for Excellence in the Mathematical and Statistical Sciences, School of Computational and AppliedMathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South AfricaeYoung Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iran

Received 20 March 2015; revised 11 July 2015; accepted 19 August 2015

KEYWORDS

Entropy analysis;

Third grade fluid;

Non-Newtonian fluid;

Stretching sheet;

Magnetic field

Abstract The purpose of this article is to study and analyze the convective flow of a third grade

non-Newtonian fluid due to a linearly stretching sheet subject to a magnetic field. The dimensionless

entropy generation equation is obtained by solving the reduced momentum and energy equations.

The momentum and energy equations are reduced to a system of ordinary differential equations by

a similarity method. The optimal homotopy analysis method (OHAM) is used to solve the resulting

system of ordinary differential equations. The effects of the magnetic field, Biot number and Prandtl

number on the velocity component and temperature are studied. The results show that the thermal

boundary-layer thickness gets decreased with increasing the Prandtl number. In addition, Brownian

motion plays an important role to improve thermal conductivity of the fluid. The main purpose of

the paper is to study the effects of Reynolds number, dimensionless temperature difference, Brink-

man number, Hartmann number and other physical parameters on the entropy generation. These

results are analyzed and discussed.� 2015 Faculty of Engineering, Ain Shams University. Production and hosting 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/).

1. Introduction

The efficient use of energy and optimal consumption ofresources has motivated research into improving the efficiencyof industrial processes. Research on the improvement in heat

transfer as one of the most influential factors in energy con-sumption has been the main focus. Evaluation of entropy gen-eration and the use of special fluids like non-Newtonian fluids

are important and efficient methods for achieving optimal heattransfer.

n Shams

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http://dx.doi.org/10.1016/j.asej.2015.08.012


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Nomenclature

A constant

B magnetic induction vector (kg s�2 A�1)B0 constant magnetic flux density (kg s�2 A�1)Be Bejan numberBr Brinkman number

E1 fluid parameterE2 fluid parameterE3 fluid parameter

E4 Reynolds numberHa Hartmann numberk thermal conductivity (kg ms�3 K�1)

M Hartmann numberNG entropy generation number

P pressure (kg m�1 s�2)

Pr Prandtl number

Re Reynolds numberS0000 characteristic entropy generation rate (kg m�1 K�1 s�3)_S000gen rate of entropy generation (kg m�1 K�1 s�3)

T temperature (K)

T1 temperature at infinity (K)

Tw wall temperature (K)u, v components of velocity (m s�1)U1 free-stream velocity (m s�1)V velocity vector (m s�1)

vw uniform surface suction/blowingX dimensionless temperature differenceh dimensionless temperature

DT temperature differencel viscosity (kg m�1 s�1)a equivalent thermal diffusivity (m2 s�1)

q density of the fluid (kg m�3)m kinematic viscosity (m2 s�1)r electric conductivity

Subscripts

tot total1 infinity condition0 plate

2 M.M. Rashidi et al.

With the advancement of industry and improved engineer-ing expertise, entropy generation is seen as an appropriate

solution to improve efficiency in industrial processes. Bejanwas the first researcher to introduce this concept by meansof entropy generation minimization (EGM) [1]. EGM is also

known as second law analysis and thermodynamic optimiza-tion. In recent years much research has been done on this topicin different geometries. Analysis of the second law of thermo-

dynamics applied to an electrically conducting incompressiblenanofluid fluid flowing over a porous rotating disk with anexternally applied uniform vertical magnetic field has beenconsidered by Rashidi et al. [2]. Rashidi et al. [3] extended

the analysis from [2] by considering entropy generation inMHD and slip flow over a rotating porous disk [3]. Entropygenerated inside the boundary-layer of nanofluids over a flat

plate is studied analytically by Malvandi et al. [4]. Secondlaw analysis of a viscoelastic fluid over a stretching sheet influ-enced by a magnetic field with heat and mass transfer was

studied by Aıboud and Saouli [5]. Kummer’s functions wereused to analyze the model and show the influence of physicalparameters on entropy generation.

In this article the effects of a stretching sheet due to convec-

tive MHD flow of third grade non-Newtonian fluid were inves-tigated. In the same way a stretching sheet plays an importantrole in industry. It can be used in industrial applications such

as metal spinning, hot rolling and extrusion. The rates ofstretching and cooling have significant influence on the qualityand properties of the final product [6]. Chang et al. have stud-

ied the flow of a non-Newtonian fluid over a stretching sheetby determining explicit solutions [7]. Anderson et al. investi-gated the flow of a non-Newtonian power-law fluid over a

stretching sheet influenced by a magnetic field [8]. Chakrabartiand Gupta studied hydromagnetic flow and heat transfer dueto a stretching sheet [9]. Abel et al. studied magneto-hydrodynamic flow due to a stretching surface with heat and

mass transfer of a non-Newtonian fluid [10].

Please cite this article in press as: Rashidi MM et al., Entropy analysis of convective MEng J (2015), http://dx.doi.org/10.1016/j.asej.2015.08.012

Sarpakaya was the first researcher to study the effect ofmagnetic field on flows of non-Newtonian fluids [11]. Sid-

dappa and Subhas investigated the flow of a viscoelasticnon-Newtonian fluid due to a stretching plate [12]. Qasimet al. investigate the steady flow of a micropolar fluid over

a stretching surface with heat transfer in the presence of New-tonian heating [13]. Nadeem et al. consider the steady flow ofa Jeffrey fluid on a stretching sheet [14]. Prasad et al. investi-

gate MHD over a nonlinear stretching sheet of a power lawfluid [15]. Kishan and Reddy include the effects of suction/injection to the problem considered by Prasad et al. [15] toinvestigate a more complicated system of ordinary differential

equations [16]. Abel and Mahesha consider the effects of vari-able thermal conductivity and non-uniform heat source onheat transfer in a viscoelastic NHD fluid on a stretching sheet

[17]. Javeda et al. devote their paper to the study of an Eyr-ing–Powell fluid over a stretching sheet [18]. Mahmoud andMegahed investigate MHD flow of a power law fluid on an

unsteady stretching sheet [19]. Mukhopadhyay considers theflow of a Casson fluid on a nonlinearly stretching sheet [20].Prasad et al. extend the work of Javeda et al. [18] by consid-ering the flow of an Eyring–Powell fluid over a non-

isothermal stretching sheet [21]. Bhattacharyya extends thework of Mukhopadhyay [20] by considering the flow of aCasson fluid over a stretching sheet with thermal radiation

[22]. Boundary-layer flow and heat transfer of non-Newtonian fluids in porous media are investigated byChaoyang and Chuanjing [23]. In another studies, Nadeem

et al. [24–26] studied the oblique stagnation-point flow of vis-coelastic and Casson fluids numerically and analytically usingmidpoint integration scheme with Richardson’s extrapolation

and HAM. Further, Akbar et al. [27] discussed the twodimensional stagnation-point flow of carbon nanotubestoward a stretching sheet with water as the base fluid underthe influence of slip effects and convective boundary condi-

tion using a homogeneous model. Moreover, Noreen and

HD flow of third grade non-Newtonian fluid over a stretching sheet, Ain Shams


Convective MHD flow of third grade non-Newtonian fluid 3

Butt [28] investigated the peristaltic motion of a Cassonfluid of a non-Newtonian fluid accompanied in a horizontaltube.

The fluid we choose to study is a third-grade fluid nano-fluid. The flow behavior of this fluid cannot be classified bya Newtonian relationship. In addition, the fluid improves the

heat transfer rate. Examples of these fluids are different modelsof oil that are used for cooling and lubrication in industry,polymer solutions and molten polymers in chemical engineer-

ing and blood.In this paper we analyze entropy generation due to a stretch-

ing sheet with a non-Newtonian fluid in the presence of a mag-netic field. The momentum and energy equations that are used

to obtain the entropy generation are determined from a highlynonlinear system of differential equations [29]. We implementthe Optimal Homotopy Analysis Method (OHAM) to solve

the system of nonlinear governing equations. Liao introducedhomotopy as a general analytic method for solving nonlinearproblems [30–32]. Liao in particular implemented the HAM to

solve the resulting nonlinear ordinary differential equationwhenhe investigated the flow of second and third-order non-Newtonian power law fluids on a stretching sheet [33].

This paper is divided up as follows: in Section 2 the prob-lem statement and physical conditions are prescribed. In Sec-tion 3 the analytical method (OHAM) is explained and theresulting governing equations are solved. Entropy generation

is investigated in Section 4. Results and graphs are presentedin Section 5. Concluding remarks are made in Section 6.

2. Flow analysis and mathematical formulation

In this section we derive the system of nonlinear equationsmodeling the flow of a third grade fluid over a stretching sheet.

The flow is generated due to a plate that is stretching linearly inits own plane. An incompressible third grade nanofluid is situ-ated over the plate at y= 0 where the fluid occupies the space

y > 0. In the presence of a heat source or heat sink convectivecooling/heating is incorporated into our mathematical model.

The boundary-layer equations for this particular model are

given by the following:

@u

@xþ @v

@y¼ 0; ð1Þ

u@u

@xþ v

@u

@y¼ t

@2u

@y2þa1

qu

@3u

@y2@xþ @u

@x

@2u

@y2þ v

@3u

@y3þ3

@u

@x

@2u

@y@x

� �þ2a2

q@u

@y

@2u

@y@xþ6b

q@u

@y

� �2@2u

@y2�rB2

0

qu; ð2Þ

u@T

@xþ v

@T

@y¼ am

@2T

@y2: ð3Þ

Boundary conditions are assumed as follows:

u ¼ cx; v ¼ 0;�k@T

@y¼ HfðTf � TÞ; at y ¼ 0; ð4Þ

u ¼ 0; v ¼ 0;T ! T1; when y ! 1:

Cartesian coordinate axes are selected where y and x arenormal and along the flat plate respectively, u and v are the ele-

ments of the velocity in the x and y directions. T1, Tf are takenas ambient values of temperature and as the temperature dueto the convective cooling/heating process. Moreover q is the


density of the base fluid and t is the kinematic viscosity. c isconstant (where c P 0) and has ‘ time�1’ dimension. The non-linear equations are transformed to dimensionless form over

the normal distance from the plate g ¼ ffiffict

py, by using the sim-

ilarity transforms:

u ¼ cxf0ðgÞ; v ¼ � ffiffiffiffiffict

pfðgÞ; hðgÞ ¼ T� T1

Tf � T1; ð5Þ

The new governing equations are given by the following:

@3f

@g3� @f

@g

� �2

þ f@2f

@g2þ E1 2

@f

@g@3f

@g3� f

@4f

@g4

� �þ ð3E1 þ 2E2Þ @2f

@g2

� �2

þ 6E3E4

@2f

@g2

� �2@3f

@g3�M

@f

@g¼ 0;

ð6Þ@2h@g2

þ Prf@h@g

¼ 0: ð7Þ

The modified boundary conditions are given by thefollowing:

fð0Þ ¼ 0; f0ð0Þ ¼ 1; h0ð0Þ ¼ �cð1� hð0ÞÞ;f0ð1Þ ¼ 0; hð1Þ ¼ 0:

ð8Þ

We note that Pr, c, E4, (E1, E2, E3) as Prandtl number, the

Biot number, Reynold’s number and the fluid parameters,respectively and have the following scaling:

Pr ¼ tam; c ¼ Hf

ffiffitc

pk;E1 ¼ ca1

l ;E2 ¼ ca2l ;E3 ¼ cb3

l ;

E4 ¼ cx2

m ;M ¼ rB20

qc :ð9Þ

3. HAM solution

A well-known method for solving a system of nonlinear equa-tions is the homotopy analysis method (HAM). The analyticsolutions of the system of nonlinear ordinary differential equa-

tions are constructed in series form and the convergence of theobtained series solutions is carefully analyzed [33]. To solve theEqs. (6), (7) by using HAM we proceed as indicated below

[34,30].The first step is to choose the initial approximations as

follows:

F0ðgÞ ¼ 1� e�g; ð10Þh0ðgÞ ¼ c

1þ ce�g:

The next steps is chosen the auxiliary linear operators asfollows:

L1ðFÞ ¼ F000 � F0; ð11ÞL2ðhÞ ¼ h00 � h:

The nonlinear operators that are based on Eqs. (6)–(8) aredefined as follows:

N 1 ¼ @3bFðg;pÞ@g3 � @bFðg;pÞ

@g

� �2

þ bFðg;pÞ@2bFðg;pÞ@g2

þE1 2 @bFðg;pÞ@g

@3bFðg;pÞ@g3 � bFðg;pÞ@4bFðg;pÞ

@g4

� �þð3E1þ2E2Þ @2bFðg;pÞ

@g2

� �2

þ6E3E4@2bFðg;pÞ

@g2

� �2@3bFðg;pÞ

@g3 �M@bFðg;pÞ@g ;

N 2 ¼ @2 hðg;pÞ@g2 þPrð bFðg;pÞ @hðg;pÞ

@g :

ð12Þ



Table 1 Optimal values of convergence control parameters

versus different orders of approximation.

Order of approximation c f0 ch0

2 �0.575 �1.404

3 �0.571 �1.327

4 �0.514 �1.162

5 �0.672 �1.473

6 �0.688 �1.499


where p 2 ½0; 1� is the embedding parameter. Using the above

definitions, the so-called zero-order deformation equationsare constructed as [33]

ð1� pÞL1½ bFðg; pÞ � F0ðgÞ� ¼ p�hN 1½ bFðg; pÞ�; ð13Þð1� pÞL2½hðg; pÞ � h0ðgÞ� ¼ p�hN 2½ bFðg; pÞ; hðg; pÞ�:where HFðgÞ;HhðgÞ are the auxiliary functions and ⁄ is auxil-iary parameter. For p= 0 and p = 1:bFðg; 0Þ ¼ F0ðgÞ; bFðg; 1Þ ¼ FðgÞ; hðg; 0Þ ¼ h0ðgÞ;hðg; 1Þ ¼ hðgÞ: ð14Þ

So, the following series is generated:

bFðg; pÞ ¼ F0ðgÞ þX1m¼1

FmðgÞpm;

hðg; pÞ ¼ h0ðgÞ þX1m¼1

hmðgÞpm:ð15Þ

We note here that

FmðgÞ ¼ 1

m!

@m bFðg; pÞ@pm

��p¼0

; hmðgÞ ¼ 1

m!

@mhðg; pÞ@pm

��p¼0

: ð16Þ

The parameter ⁄ plays an important role in the convergence[34] so, if p is considered to p= 1 then ⁄ is selected such that

the series (17) are convergent. The Eq. (17) become:

bFðg; pÞ ¼ F0ðgÞ þX1m¼1

FmðgÞ; ð17Þ

hðg; pÞ ¼ h0ðgÞ þX1m¼1

hmðgÞ:

Eq. (15) is the zeroth-order equation and to obtain com-plete solutions of the series should be repeated several times

to achieve convergence so, mth-order deformation equationswith respect to p are defined as follows:

L1½ bFðg; pÞ � F0ðgÞ� ¼ �hR1;mðgÞ; ð18ÞL2½hðg; pÞ � h0ðgÞ� ¼ �hR2;mðgÞ:where for m 6 1, vm = 0 and for m> 1, vm = 1 and R1,m(g),R2,m(g) are given by

R1;mðgÞ ¼ @3Fm�1ðgÞ@g3 �

Xm�1

n¼0

@FnðgÞ@g

@Fm�1�nðgÞ@g þ FnðgÞ @2Fm�1�nðgÞ

@g2

�þE1 2 @FnðgÞ

@g@3Fm�1�nðgÞ

@g3 � FnðgÞ @4Fm�1�nðgÞ@g4

� þð3E1 þ 2E2Þ

Xm�1

n¼0

@2FnðgÞ@g2

@2Fm�1�nðgÞ@g2

� þ6E3E4

Xn�1

m¼0

Xmj¼0

@2FjðgÞ@g2

@2Fm�jðgÞ@g2

!@3Fn�1�mðgÞ

@g3 �M @Fm�1ðgÞ@g ;

R2;mðgÞ ¼ @2hm�1ðgÞ@g2 � Pr

Xm�1

n¼0

FnðgÞ @hm�1�nðgÞ@g

� :

ð19Þwith the boundary conditions:

Fmð0Þ ¼ 0; Fmð1Þ ¼ 0; hmð1Þ ¼ 0; F0mð0Þ ¼ 1;

h0mð0Þ ¼ �cð1� hmð0ÞÞ: ð20Þ


Finally, for solving Eq. (18), the software MATHEMA-TICA is used by an appropriate developed program.

4. Optimal convergence control parameters

It must be remarked that the series solutions (18) contain thenonzero auxiliary parameters ⁄f and ⁄h, which determine the

convergence region and also rate of the homotopy series solu-tions. To obtain the optimal values of ⁄f and ⁄h, here the so-called average residual error defined by Refs. [35,36] were used

as follows:

e fm ¼ 1

kþ 1

Xkj¼0

N f

Xmi¼0

bFðgÞ !g¼jdg

24 352

dg; ð21Þ

ehm ¼ 1

kþ 1

Xkj¼0

N h

Xmi¼0

bFðgÞ;Xmi¼0

hðgÞ !

g¼jdg

24 352

dg; ð22Þ

Due to Liao [37]

etm ¼ e fm þ ehm; ð23Þwhere etm is the total squared residual error, dg = 0.5 and

k= 20. For example, a case has been considered whereE1 = E2 = E3 = 0.2, E4 = M = Pr = 1.0 and c = 0.1.Table 1 presents the optimal values of convergence control

parameters as well as the minimum values of total averagedsquared residual error versus different orders of approxima-tion. Table 2 displays the individual average squared residual

error at different orders of approximations using the optimalvalues from Table 1. In addition, Fig. 1 illustrates the maxi-mum average squared residual error at different orders ofapproximation. It can be obviously seen that the averaged

squared residual errors and total averaged squared residualerrors are getting smaller as the order of approximation isincreased. Hence, Optimal HAM gives us freedom to select

any set of local convergence control parameters to obtain con-vergent results.

5. Entropy generation

In this section we want to obtain entropy generation due toConvective MHD Flow of Third Grade Non-Newtonian Fluid

over a Stretching Sheet. From the work of Bejan [38,39] onentropy generation in the presence of a magnetic field withheat and mass transfer, the local volumetric rate of entropy

generation is obtained from

_S000gen ¼

k

T21

@T

@y

� �2

þ lT1

@u

@y

� �2

þ rB20

T1u2: ð23Þ



Table 2 Individual averaged squared residual errors using

optimal values of auxiliary parameters.

m Efm Eh

m

2 1.04 � 10�5 1.32 � 10�6

6 6.83 � 10�8 1.98 � 10�8

10 1.66 � 10�10 9.49 � 10�10

14 6.67 � 10�12 4.72 � 10�11

18 7.02 � 10�14 2.61 � 10�12

22 2.91 � 10�15 1.63 � 10�13

26 5.11 � 10�17 1.11 � 10�14

30 2.13 � 10�18 8.35 � 10�16

0.35

0.4


This equation includes the effective terms in entropy gener-ation that are defined as the first term in the local entropy gen-

eration due to heat transfer across a finite temperaturedifference; the next term is local entropy generation due to vis-cous dissipation, and the effect of the magnetic field is defined

by third term.

Figure 1 Maximum average squared residual error at different

orders of approximation.

η

F'(η)

0 2 4 6 80

0.2

0.4

0.6

0.8

1

E3=0.0E3=0.3E3=0.6E3=0.9

Figure 2 The effect of E3 on the velocity profile, E1 = E2 = 0.2,

E4 = 1.0,M = 0.5.


To obtain the dimensionless number for the entropy gener-ation rate, the Eqs. (23), (6) are used to obtain Eq. (24) that isdefined as

_S000gen ¼

kDT2c

T21t

h0ðgÞ2 þ lx2c3

T1tf 00ðgÞ2 þ rB2

0c2x2

T1f 0ðgÞ2: ð24Þ

Then by using the dimensionless numbers

Re ¼ c l2=m;Br ¼ lu2w=kDT;X ¼ DT=T1;Ha ¼ B0l

ffiffiffirl

r; ð25Þ

where Re, Br, X are defined as the Reynolds number, Brink-

man number and dimensionless temperature difference num-bers, and entropy generation is given by

NG ¼ Reh02ðgÞ þ ReBr

Xf 002ðgÞ þHa2

Br

Xf 0ðgÞ2; ð26Þ

η

θ(η)

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

E3=0.0E3=0.3E3=0.6E3=0.9

Figure 3 The effect of E3 on the temperature distribution with

constant parameter when E1 = E2 = 0.2, E4 = 1.0, Pr = 1.0,

c = 0.1, M= 0.5.

η

F'(η)

0 2 4 6 8

0.2

0.4

0.6

0.8

1 M=0.0M=0.3M=0.6M=0.9

Figure 4 The effect of M on the velocity profile when

E1 = E2 = E3 = 0.2, E4 = 1.0.



η

θ(η)

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

M=0.0M=0.3M=0.6M=0.9

Figure 5 The effect of M on the temperature distribution with

constant parameter for E1 = E2 = E3 = 0.2, E4 = 1.0, Pr = 1.0,

c= 0.3.

η

θ( η)

0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Pr=0.71Pr=1.0Pr=1.5Pr=2.0

Figure 6 The effect of Pr on the temperature distribution with

constant parameter for E1 = E2 = E3 = 0.2, E4 = 1.0, c = 0.3,

M= 0.5.

η

θ( η)

0 2 4 6 80

0.2

0.4

0.6 γ=0.3γ=0.6γ=0.9γ=1.2

Figure 7 The effect of c on the temperature distribution with

constant parameter when E1 = E2 = E3 = 0.2, E4 = 1.0,

pr= 1.0, M= 0.5.

η

NG

0 1 2 3 4 50

5

10

15

20

25

M=0.0M=0.3M=0.6M=0.9

Figure 8 The effect of the M on the entropy generation number

with constant parameter for E1 = E2 = E3 = 0.2, E4 = 1.0,

pr= 1.0, c= 0.1, Br = 5.0, X = 1.0, Re = 5.0.


where NG is defined as

NG ¼_S000gen

_S0000

; ð27Þ

where _S000gen is the local volumetric entropy generation rate and

_S000 characteristic entropy generation rate given by _S0000 ¼

kðDTÞ2=l2T21.

6. Results and discussion

In this section we discuss the effects of the different parameterson the velocity profile, temperature distribution and entropy


generation function. Firstly, the effect of E3 on the velocityand temperature distribution is illustrated in Figs. 2 and 3.From Fig. 2, it is clear that the velocity profile and conse-

quently momentum boundary-layer are increasing functionsof E3. This physically implies that E3 is a shear thinningparameter and causes the fluid to be less viscous for increasing

values of E3. In addition, Fig. 4 displays that the temperatureof the fluid is reduced by increasing the value of cE3.

Figs. 4 and 5 illustrate the effect of magnetic parameter on

the velocity profile and temperature distribution. A drag-likeforce that called Lorentz force is created by the influence ofthe vertical magnetic field on the electrically conducting fluid.This force has the tendency to slow down the flow over the



η

NG

0 1 2 3 4 50

5

10

15

20γ=0.3γ=0.6γ=0.9γ=1.2

Figure 9 The effect of the c on the entropy generation number

when with constant parameter when E1 = E2 = E3 = 0.2,

E4 = 1.0, Pr = 1.0, c= 0.1, Br = 5.0, X = 1.0, Re = 5.0.

η

NG

0 1 2 3 4 50

5

10

15

20

25

E3=0.0E3=0.3E3=0.6E3=0.9

Figure 10 The effect of the E3 on the entropy generation number

with constant parameter for E1 = E2 = 0.2, E4 = 1.0, Pr= 1.0,

c = 0.1, M= 0.5, Br = 5.0, X = 1.0, Re = 5.0.

η

NG

0 1 2 3 4 50

5

10

15

20

25

30

35

40

Re=1.0Re=2.0Re=3.0Re=5.0Re=10.0

Figure 11 The effect of the Re on the entropy generation

number with constant parameter when E1 = E2 = E3 = 0.2,

E4 = 1.0, Pr= 1.0, c= 0.1, M= 0.5.

η

NG

0 1 2 3 4 50

10

20

30

40Br=1.0Br=2.0Br=3.0Br=5.0Br=10.0

Figure 12 The effect of the Br on the entropy generation number

with constant parameter for E1 = E2 = E3 = 0.2, E4 = 1.0,

Pr= 1.0, c= 0.1, M= 0.5, Ha = 1.0, X = 1.0, Re = 5.0.


vertical surface. According to the above explanation, the veloc-ity boundary-layer thickness gets depressed and the tempera-

ture distribution increases slightly with the increase in themagnetic parameter. It is important to mention that the largeresistances on the fluid particles, which cause heat to be gener-

ated in the fluid apply as the magnitude of the magnetic fieldincreases.

The effect of the Prandtl number on the temperature distri-bution is shown in Fig. 6. The thermal boundary-layer thick-

ness decreases when the magnitude of the Prandtl number isincreased. It physically means that the flow with large Prandtlnumber prevents spreading of the heat in the fluid.

The effect of the Biot number on the dimensionlesstemperature profile is demonstrated in Fig. 7. The thermal


boundary-layer thickness increases as the Biot number

increases in magnitude. The effects of different flow physicalparameters on the entropy generation number are plotted inFigs. 8–13. The results show that increasing Biot number,

Reynolds number, Brinkman number and Hartmann numbercause an increase in the entropy generation number. Alterna-tively, decreasing the magnitude of the above parameters

achieves the main goal of second law of thermodynamics,the minimizing of entropy generation. In addition, Fig. 8displays that by increasing the magnitude of the magnetic

parameter the entropy generation function increases up to acertain distance from the sheet (about g = 1). After this point,the behavior mentioned above is reversed. Roughly the samebehavior can be observed from Fig. 10 for different value



η

NG

0 1 2 3 4 50

50

100

150

Ha=1.0Ha=2.0Ha=3.0Ha=4.0Ha=5.0

Figure 13 The effect of the Ha on the entropy generation

number with constant parameter when E1 = E2 = E3 = 0.2,

E4 = 1.0, Pr = 1.0, c = 0.1,M = 0.5, Br = 5.0, X = 1.0,

Re = 5.0.


of E3. This parameter changes the entropy generation param-eter which decreases for increasing value of E3 up to a certain

distance and then reverses.

7. Conclusion

In this paper, after solving the governing equations analyti-cally by means of the optimal Homotopy analysis method(OHAM), entropy generation was studied in detail. The effect

of various parameters on the velocity and temperature, such asthe magnetic parameter and Prandtl number are investigated.In addition, the dependence of the entropy generation numberon the magnetic parameter, Prandtl number, Reynolds num-

ber, Hartmann number and the dimensionless temperature dif-ference is investigated.

From the obtained results we can note the following:

1. Brownian motion plays an important role to improve ther-mal conductivity of the fluid.

2. The thermal boundary-layer thickness gets decreased withincreasing magnitude of the Prandtl number.

3. Effect of the magnetic field on the fluid flow and tempera-

ture distribution is important.4. By increasing the magnitude of the magnetic parameter, the

entropy generation function increases up to a certain dis-tance from the sheet.

Conflict of Interests

The authors declare that they have no conflicts of interests.

Acknowledgments

EM thanks the National Research Foundation of South Africafor their support.


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1982.

Mohammad Mehdi Rashidi is a professor of

Mechanical Engineering at the Bu-Ali Sina

University, Hamedan, Iran. He has published two

books: Advanced Engineering Mathematics

with Applied Examples of MATHEMATICA

Software (2007) (320 pages) (in Persian), and

Mathematical Modelling of Nonlinear Flows

of Micropolar Fluids (Germany, Lambert

Academic Press, 2011). He has published over

195 (125 of them are indexed in Scopus)

journal articles and 43 conference papers.

Professor Rashidi is a reviewer of several journals (over 160 refereed

journals) and the editor of Fifty International Journals.




















































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