For Review Only · 2019. 6. 10. · For Review Only MHD ⁄ow of a kinetic postulate of liquids inaugurated ⁄uid under thermal radiation e⁄ects Azad Hussain a,1 Zainia Muneer

For Review Only

MHD flow of a kinetic postulate of liquids inaugurated fluidunder thermal radiation effects

Journal: Canadian Journal of Physics

Manuscript ID cjp-2018-0102.R1

Manuscript Type: Article

Date Submitted by theAuthor: 01-Apr-2018

Complete List of Authors: Hussain, Azad; University of GujratMuneer, Zainia; University of GujratMalik, M.Y.; QAUGhafoor, Saadia; university of gujrat

Keyword: Radiation effects, Porous medium, Stretching sheet, Heattransfer, MHD

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Canadian Journal of Physics

For Review Only

MHD �ow of a kinetic postulate of liquidsinaugurated �uid under thermal radiation e¤ects

Azad Hussaina,1 Zainia Muneera, M. Y. Malikb, Saadia GhafooraaDepartment of Mathematics, University of Gujrat, Gujrat 50700

bDepartment of Mathematics, Quaid-i-Azam University, Islamabad 44000,Pakistan

Abstract:The present study focuses on the non-Newtonian magneto-hydrodynamic �ow of a ki-

netic postulate of liquids inaugurated �uid past a porous plate in the appearance of thermalradiation e¤ects. Resemblance trans�gurations are used to metamorphose the governingequations for temperature and velocity into a system of ordinary di¤erential equations. Wethen solved these di¤erential equations subject to convenient boundary conditions by usingshooting method along with Runge-Kutta method. Heat transfer and characteristic �owresults are acquired for di¤erent compositions of physical parameters. These results are ex-tended graphically to demonstrate interesting attributes of physics of the problem. Nusseltnumber and skin friction coe¢ cients are also discussed via graphs and tables for di¤erentvalues of dimensionless parameters. Decline occurs in velocity pro�le due to escalating valuesof M. Temperature pro�le depicts growing behavior due to acceleration in the values of �and M. Nusselt number and skin friction curves represents rising behavior according to theirparameters.Keywords: Radiation e¤ects; Porous medium; Stretching sheet; Heat transfer; MHD.

1 Introduction

Materials which do not satisfy the property of Newtonian �uids are non-Newtonian �uids likedrilling muds, ketchup, tooth paste and apple sauce etc. Theoretical and experimental studiesof non-Newtonian �uids have obtained much concentration because of numerous industrialand technological applications. The inspection of �uid motion behavior of the non-Newtonian�uids becomes more complicated in contrast with Newtonian �uids because of the fact thatnon-Newtonian �uids do not show linear relation between shear stress and strain. Petroleumdrilling, extrusion of polymers, blasting of glasses and modelling of steel substances are thegrowing applications of non-Newtonian �uids. Rizwan Ul Haq et al. [1] studied numericallythe �ow of non-Newtonian nano�uid passed over a stretching sheet. Zeeshan et al. [2]analyzed non-Newtonian �uid �ow between two caoxial cylinders along a porous mediumwith variable viscosity and heat transfer. Domairry et al. [3] discussed the motion of curvedparticles using the �ow of a plane couette Newtonian �uid. Dogonchi et al. [4] observed heattransfer in a �ow of non-Newtonian �uid for cooling turbine disks applying DRA. Hussain etal. [5] investigated non-Newtonian nano�uid �ow through coaxial cylinders. Bhatti et al. [6]examined entropy generation using SLM as an experimental mechanism of optimization forthe �ow of non-Newtonian nano�uid using a permeable stretching surface.Eyring and Powell introduced a model in 1944 which today�s known with the name of

Eyring-Powell model. In literature there are numerous models of non-Newtonian �uids andEyring-Powell model is one of them. It is important than Cross model, Bingham model,Yasuda model and Power law model. The reason of its importance is that it is used to

1Corresponding author: E-mail: [email protected]

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categorize the non-Newtonian behavior. Malik et al. [7] examined the MHD Eyring-Powell�uid �ow over a stretching sheet. Nadeem et al. [8] observed endoscopic e¤ects on anEyring-Powell �uid �ow.The understanding of transport occurrence in porous materials have been basically origi-

nated by the research proceedings in chemical engineering industry and geophysical systems.The transport occurrence through porous media illustrates an consequential area of expedi-tious extension in the contemporary heat transfer research. The understanding of transportoccurrence in porous materials has been encouraged by a wide range of Agricultural, Environ-mental, Industrial and engineering applications. There are many examples of porous mediadomains in the chemical engineering and agriculture industry such as sand �lters, packed and�uidized beds, reservoirs and root zone etc. In recent years, there have been various studies[9� 16] on the �ows through porous media.In engineering processes, thermal radiations are possibly important when operating at

high temperatures. Thermal radiation e¤ects play an important role in managing the heattransfer processes. Heat transfer radiation is very mandatory in the design of numerouspropulsion gadgets for air craft missiles, dependable equipments, gas turbines, space vehicles,nuclear plants and satellites. The in�uences of thermal radiation are important in spacetechnology and the procedures that involves high temperatures. Ganji et al. [17] studiedconvective-radiation heat transfer in moving �ns with variable thermal conductivity andheat generation. Dogonchi et al. [18] observed heat transfer and buoyancy MHD nano�uid�ow across a stretching sheet in the existence of thermal radiation and joule heating impacts.Bhatti et al. [19] investigated thermal radiation and thermo-di¤usion e¤ects on Williamsonnano�uid across a porous stretching surface. Rizwan Ul Haq et al. [20] studied the e¤ectsof thermal radiations on MHD �ow of nano�uids moves over the stretching sheet. Tehseenet al. [21] investigated numerically entropy generation on MHD Carreau nano�uid withthermal radiation towards a shrinking sheet. Bhatti et al. [22] numerically investigatednonlinear thermal radiation entropy generation on MHD nano�uid using a porous shrinkingsheet. Nadeem et al. [23] analyzed the thermal radiation e¤ects on the Je¤rey �uid �owproceed across an exponentially stretching surface. Ganji et al. [24] investigated heat transferMHD nano�uid �ow connecting two parallel plates in the appearance of thermal radiations.Dogonchi et al. [25] examined heat transfer of a viscous incompressible �ow of MHD nano�uidbetween stretching or shrinking walls in the presence of thermal radiations. Ganji et al. [26]examined heat transfer and buoyancy �ow of MHD nano�uids considering Brownian motionover a lengthening sheet with thermal radiation e¤ects.Under the e¤ects of a transverse magnetic �eld the �ow and heat transfer of electro-

conductive �uids in circular pipes and channels occurs in magnetohydrodynamics pumps,accelerators, generators and �owmeters and have implementations in �ltration, puri�cationof crude oil, nuclear reactors, electrostatic �lters, thermal insulators, �uid droplets, geother-mal systems and others. Seventy years ago the interest in the external magnetic �eld e¤ectson physical heat processes appeared. During the late 1950s research in magnetohydrody-namics extended rapidly and have numerous applications. The growing attraction in thestudy of MHD appearance is also a¢ liated to the evolution of fusion reactors where plasmais restricted by a strong magnetic �eld. Rashidi et al. [27] investigated a successive lin-earization method for resolving stagnation point �ow under MHD e¤ects over a permeableshrinking sheet. Bhatti et al. [28] examined a numerical method which is occupied to solvethe stagnation-point �ow problem on a permeable stretching or shrinking sheet along a porousmedia with the e¤ects of MHD and heat transfer. Salahuddin et al. [29] observed MHD �owof Williamson �uid moves over a stretching sheet. Azeem Shahzad et al. [30] studied the

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power law model of viscous, incompressible, steady state MHD �ow over a vertically stretch-ing sheet. Bhatti and Rashidi et al. discussed entropy generation in detail on MHD �uid�ow using a proper channel as reported in Refs. [31 � 34]: Dogonchi and Ganji have donea satisfactory work on heat transfer and MHD nano�uid �ow using a suitable medium asdeclared in Refs. [35� 37]:Literature survey reveals that no analysis is accessible which discuss both MHD and

radiation e¤ects on the �ow of kinetic postulate of liquids inaugurated �uids. Therefore itis highly preferable to investigate the e¤ects of MHD and thermal radiations on the �owof kinetic postulate of liquids inaugurated �uids past a porous plate. Though the analyticsolution of the determining equations are uncompromising to depict the graphical solutionsof dimensionless parameters including in the problem, so the numerical calculation is carriedout through a familiar technique namely shooting method in order to achieve the desired levelof accuracy. The e¤ects on the �ow are explored through modelling of continuity, momentumand energy equations. The in�uences of di¤erent physical parameters on concerning pro�lesare discussed with the assistance of tables and graphs.

2 Eyring-Powell �uid model:

The Cauchy stress tensor for an Eyring Powell �uid model which is given by [38] :

S = �rv + 1

�sinh�1

�1

Crv�; (1)

where � depicts shear viscosity, C and � are the material constant such that

sinh�1�1

Crv�� 1

Crv � 1

6

�1

Crv�3; j 1

Crv j<< 1: (2)

3 Mathematical modeling of the problem

In the present problem, we examined MHD �ow of an Eyring Powell �uid model past anin�nite penetrable plate.

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Fig. 1. Geometry of the problem.

The governing equations may be expressed in the form of

@�

@t+ div �v = 0; (3)

�dv

dt= div T + �b; (4)

�de

dt= T:L� div q+ �r: (5)

Where � indicates density, b denotes body force, e describes speci�c internal energy, qexpresses heat �ux vector, L denotes gradient of velocity and r represents radiant heating.The continuity equation for incompressible �uid will be of the form

div v = 0; (6)

v = �w0 = constant. (7)

Here w0 > 0 expresses suction at plate and w0 < 0 describes blowing. Using velocity �eldand continuity equation into eq. (6) ; we get

�w0dv

dz+ �

d2v

dz2+

1

�C

d2v

dz2� 1

2�C3

�dv

dz

�2d2v

dz2� �B20v =

@p

@y; (8)

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@p

@x= 0; (9)

@p

@z= 0: (10)

Eq. (8) can be written as

�w0dv

dz+ �

d2v

dz2+

1

�C

d2v

dz2� 1

2�C3

�dv

dz

�2d2v

dz2� �B20v = A1; (11)

where

@p

@y= constant = A1: (12)

Suitable boundary conditions for eq. (11) are

v (0) = 0; (13)

v (z)! V1 as z !1: (14)

Heat �ux q is expressed as

q = �A grad �: (15)

Radiation parameter is represented in the form

qr = �4��

3K�@T 4

@z: (16)

Using heat �ux and temperature �eld in eq. (5), we get

Ad2�

dz2+ �Cpw0

d�

dz+ �

�dv

dz

�2+

1

�C

�dv

dz

�2� 1

6�C3

�dv

dz

�4� 1

�Cp

@qr@z

= 0; (17)

where Cp represents the speci�c heat of �uid.The appropriate boundary condition for constant wall temperature may be demonstrated

as:

� (0) = 0; (18)

� (z) ! �1 as z !1:

The boundary condition for the insulated wall are given by

d�

dzj z!0 = 0;

� (1) ! �1: (19)

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3.1 Case 1: Constant wall temperature

Introducing the non-dimensional parameters

�v =v

V1; �z =

z

L; �� =

� � �1�0 � �1

: (20)

and

L =�U0�; =

��V 21w2

u2; u =w0V1;

� =V 21v

A (�0 � �1); E1 =

V1Cp (�0 � �1)

; (21)

E =1

�c�; � =

�

6�3C3� 1u2; Pr =

�CpA; � = E1 Pr; M =

�B20�1��2

; R =4��31K�K

(22)

energy and momentum equations takes the form

d2�v

d�z2+ E

d2�v

d�z2+

d�v

d�z� 3�

�d�v

d�z

�2d2�v

d�z2� AMv = A2; (23)

�1 +

4

3R

�d2��

d�z2+ Pr

d��

d�z+ �

�d�v

d�z

�2+ E�

�d�v

d�z

�2� ��

�d�v

d�z

�4= 0: (24)

For simplicity we remove the bars in the eqs. (23-24) and get

d2v

dz2+ E

d2v

dz2+

dv

dz� 3�

�dv

dz

�2d2v

dz2� AMv = A2; (25)

�1 +

4

3R

�d2�

dz2+ Pr

d�

dz+ �

�dv

dz

�2+ E�

�dv

dz

�2� ��

�dv

dz

�4= 0: (26)

A1 and A2 are constant. The non-dimensionlised boundary conditions are

v (0) = 0; (27)

v ! 1 as z !1;

and

� (0) = 1; (28)

� ! 0 as z !1: (29)

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3.2 Case 2: Insulated plate

We de�ne non- dimensional forms below

�� =� � �1�b � �1

; (30)

where �b represents the bulk temperature and the remaining parameters are same in theeq. (21) and (22). Also the equations of motion are remain same as the eqs. (25� 26). Nowwe de�ne the Eckert number here that is

E� =V 21

cp (�b � �1): (31)

Now, following the eqs. (24) and (25) ; the appropriate boundary conditions are

d�

dzj z!0 = 0; (32)

� ! 0 as z !1:

Nusselt number and skin friction of �ow �eld are expressed as

Nuy =yqw

A(Tu � T1)and Cf =

�w12�V 2

: (33)

In above equation Nusselt number is represented as Nuy and skin friction is denoted withCf . Here, �u indicates wall shear stress and qw represents wall heat �ux that are explainedas

�w =

��dv

dz+

1

�C

dv

dz� 1

6�C3(dv

dz)3�z!0

and qw = �A�@T

@z

�z!0

; (34)

after applying suitable similarity transformation in the above relation, we get interestingmonumental quantities in the form

1

2Cf Re = v

0(0) +Mv0(0)� �(v0(0))3; Nuy = ��0(0) (35)

Here, Re represents Reynold number.

4 Numerical Solution

Resemblance trans�gurations are used to get ordinary di¤erential equations from the �ow aris-ing governing equations which are highly nonlinear in nature. Shooting technique [14� 18]along with Runge Kutta method is used to �nd the solution of these equations. First of allwe covert energy and momentum equations in �rst order form i. e,

v0 =� v0

1 + E � 3�(v0)2 ; (36)

�00 =1�

1 + 43R�(� Pr �0 � �(v0)2 � E�(v0)2 + ��(v0)4): (37)

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Now, we clarify new variables that are applied to reduce the higher order ordinary di¤er-ential equations into �rst order i.e,

v = y1; v0 = y2; v

00 = y02; � = y3; �0 = y4; �

00 = y04: (38)

After putting the new variables, we obtain the new system of ordinary di¤erential equa-tions i.e.,

y01 = y2; y03 = y4; (39)

y02 =� v0

1 + E � 3�(v0)2 ; (40)

y04 =1�

1 + 43R�(� Pr �0 � �(v0)2 � E�(v0)2 + ��(v0)4): (41)

along with the boundary conditions

y1(0) = 0; y1(1) = 1; y3(0) = 1; y3(1) = 0: (42)

5 Graphical Results and Discussions

The boundary value problem given by eqs. (25) and (26) subject to boundary conditionsin eqs. (27)-(29) are trans�gured into initial value problem in order to use Runge-Kuttamethod. In the domain of graphical render, e¤ects of pertinent parameters on velocity andtemperature pro�les are discussed brie�y. Fig. 1 describes the geometry of the problem.Fig. 2 displays the e¤ects of M on velocity distribution. The velocity distribution goesdown due to increment in the values of M . Fig. 3 indicates the e¤ects of on temperaturedistribution. Temperature distribution is growing down by accelerating where � = 0:7; Pr= 7.4, � = 0:5; M = 0:2; R = 0:1. Fig. 4 points out the corollary of � on temperature pro�le.Temperature pro�le indicates decreasing behavior owing to escalating � = 0:01; 0:9; 1:9; 2:9where the values of other parameters are Pr = 7:4; � = 1:1; M = 1:1; R = 0:9; = 0:01.Temperature pro�le for di¤erent values of � = 0:1; 1:1; 2:1; 3:1 is represented in Fig.5. Thetemperature pro�le goes up because of increasing � for �xed values of Pr = 7:1; � = 0:5;M = 0:3; R = 0:5; = 0:05. Fig. 6 renders the in�uence of M on temperature pro�le whenthe values of �; Pr; �; and R presumed to be 1:0; 7:4; 2:9; 0:1 and 0:1 respectively. It is tobe noted that Temperature pro�le depicts increasing behavior due to escalating values of M .Fig. 7cleri�es the temperature pro�le for various values of Pr = 7:1; 9:1; 11:1; 13:1. As thevalues of Pr are escalating, the temperature �eld illustrates the declination in annular gapwhere the values of other parameters are = 0:1; M = 0:01; � = 1:1; � = 1:0 and R = 1:5.Fig. 8. delineates the outcomes of R on temperature distribution for the values R = 0:1;0:4; 1:0; 2:9. The temperature pro�le goes down owing to increment in R where = 0:01;M = 1:0; Pr = 7:1; � = 1:5 and � = 2:1. Fig. 9 construes the consequences of M and onskin friction coe¢ cient. The skin friction coe¢ cient curve accelerates owing to accretion inM . Fig. 10 indicates the in�uences of Pr and on Nusselt number. The curve of Nusseltnumber is going up due to increment in Pr. Figs. 11-13 explicate stream lines for di¤erentvalues of . Stream lines are going far from the origin due to escalating values of : Figs.

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14-16 represent the three dimensional graphs of v(z) for di¤erent values of . The threedimensional graphs shows that the curve is going to be bend because of accelerating . Table1 describes the values of d�

dzat the wall for perceptible parameters. In third and fourth case,

values represent the rate of change in temperature is displaying increasing behavior due toincrease in the values of � and M . Rate of change in temperature is sighted declination dueto boost in the values of , �, Pr and R. Table 2 exhibits the in�uence of di¤erent parameterson skin friction coe¢ cient. The values indicates that skin friction is accelerating with theincrement in the values of but shows declination due to increment in the values of � andM:Table 3 reveals the signi�cance of �, Pr and R on Nusselt number. The values of Nusseltnumber are growing down due to variation in � and R but depicts increasing behavior in caseof Pr.

Fig. 2. In�uence of M on velocitydistribution.

Fig. 3. E¢ cacy of on temperature�eld.

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For Review OnlyFig. 4. E¤ects of � on temperature

distribution.

Fig. 5. Impact of � on temperaturepro�le.

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For Review OnlyFig. 6. E¢ cacy of M on temperature

�eld.

Fig. 7. In�uence of Pr ontemperature distribution.

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For Review OnlyFig. 8. Impact of R on temperature

pro�le.

Fig. 9. In�uences of M and on skinfriction.

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For Review OnlyFig. 10. E¤ects of and Pr on

Nusselt number.

Fig. 11. Stream lines for = 0:1.

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For Review OnlyFig. 12. Stream lines for = 0:2.

Fig. 13. Stream lines for = 0:3.

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For Review OnlyFig. 14. Three dimensional graph of

v(z) for = 0:1:

Fig. 15. Three dimensional graph ofv(z) for = 0:2:

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For Review OnlyFig. 16. Three dimensional graph of

v(z) for = 0:3:

Table 1. Values of d�dzat the wall.

� 0:01 0:9 1:9 2:9R = 0:9 = 0:01Pr = 7.4 �0:491 �0:715 �0:966 �1:218� = 1:1M = 1:1

0:01 0:02 0:03 0:04R = 0:1� = 0:5Pr = 7.4 �0:109 �0:151 �0:199 �0:251M = 0:2� = 0:7

� 0:1 1:1 2:1 3:1R = 0:5 = 0:05Pr = 7.1 �0:236 �0:182 �0:128 �0:075M = 0:3� = 0:5

M 0:01 0:5 0:9 1:3R = 0:1 = 0:1Pr = 7.4 �0:424 �0:318 �0:231 �0:145� = 2:9� = 1:0

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Pr 7:1 9:1 11:1 13:1R = 1:5 = 0:1� = 1:1 �0:233 �0:289 �0:349 �0:411M = 0:01� = 1:0

R 0:1 0:4 1:0 2:9 = 0:01� = 1:5Pr = 7.1 �0:012 �0:019 �0:048 �0:076M = 1:0� = 2:1

Table 2. Values of skin friction coe¢ cient with respect to ; � and M: � M 1

2Cf Re

0:1 3:2 0:1 0:1072970:12 0:1075090:14 0:1077560:1 0:107297

3:3 0:1071903:4 0:1070843:2 0:107297

0:2 0:1172730:3 0:127251

Table 3. Values of Nusselt number with respect to Pr; � and R:� Pr R ��0(0)0:1 7:1 0:5 0:2363350:2 0:2309440:3 0:225554

7:1 0:2363357:2 0:2387527:3 0:2411787:1 0:236335

0:6 0:2241840:7 0:213929

6 Concluding Remarks

An investigation has been contented to catechize the nature of kinetic theory of liquidsoriginated �uids. Conservation laws are used to generate mathematical modelling of theproblem. Numerical solution is obtained with the assistance of shooting technique along

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with Runge-kutta method. To explore the �ow problem, consequences of di¤erent parametersupon temperature and velocity �elds are explained. Graphs and tables are used to visualizethe behavior of di¤erent parameters. Some impressive points of present investigation are asfollows:

1. Velocity pro�le shows decline due to increment in the values of M .

2. Temperature pro�le shows declination because of increment in �, ,R and Pr but in-creases due to acceleration in the values of � and M .

3. Skin friction curve is growing up because of boost in the values of pertinent parameters.

4. Nusselt number curve rises due to acceleration in the value of Pr.

Nomenclatureq heat �ux, [kgm2s�3]

p �uid pressure, [kgm�1s�2]

Nu Nusselt number, [�]Pr Fluid Prandtl number, [�]Re Reynolds number, [�]T Temperature, [K]

� similarity transformations in terms of z and t

� �uid dynamic viscosity, [kgms�1]

� transformed �uid temperature

� �uid density, [kgm�3]

v �uid kinematic viscosity, [ms�2]

t time

Cp speci�c heat, [J=K]

b body force, [N=m3]

qr radiation parameter

M dimensionless magnetic parameter

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17. A. S. Dogonchi and D. D. Ganji, � Convection-radiation heat transfer study of moving�n with Temperature-dependent thermal conductivity, heat transfer coe¢ cient and heatgeneration�. Applied Thermal Engineering, vol. 103, 705-712, 2016.

18. A. S. Dogonchi and D. D. Ganji, � E¤ects of Cattaneo-Christov heat �ux on buoy-ancy MHD nano�uid �ow and heat transfer over a stretching sheet in the presenceof Joule heating and thermal radiation impacts�. Indian Journal of physics, DOI :10.1007/s12648-017-1156-2.

19. M. M. Bhatti and M. M. Rashidi, �E¤ects of Thermo-di¤usion and Thermal radia-tion on Williamson nano�uid over a porous shrinking/stretching sheet�. Journal ofMolecular liquids, vol. 221, 567-73, 2016.

20. Rizwan Ul Haq, S. Nadeem, Z. H. khan and Noreen Sher Akbar , � Thermal radiationand slip e¤ects on MHD stagnation point �ow of nano�uid over a stretching sheet�.Physica E: low-Dimensional systems and Nanostructures, vol. (65), 17-23, 2015.

21. M. M. Bhatti, T. Abbas, M. M. Rashidi and M. El-Sayed Ali, � Numerical simulationof Entropy Generation with Thermal radiation on MHD carreau nano�uid towards ashrinking sheet�. Entropy, vol. 18(6), 200, 2016.

22. M. M. Bhatti, T. Abbas and M. M. Rashidi, � Numerical study of entropy generationwith nonlinear thermal radiation on magnetohydrodynamics non-Newtonian nano�uidthrough a porous shrinking sheet�. Journal of Magnetics, vol. 21(3), 468-475, 2016.

23. S. Nadeem, S. Zaheer and Tiegang Fang, � E¤ects of Thermal radiation on the bound-ary layer �ow of a Je¤rey �uid over an exponentially stretching surface�. NumericalAlgorithm, vol. (57), 187-205, 2011.

24. A. S. Dogonchi, K. Divsalar and D. D. Ganji,� Flow and heat transfer of MHD nano�uidbetween parallel plates in the presence of thermal radiation�. Comput. Methods. Appl.Mech. Engrg, vol. 310, 58-76, 2016.

25. A. S. Dogonchi and D. D. Ganji, � Investigation of MHD nano�uid �ow and heat transferin a stretching/shrinking convergent/divergent channel considering thermal radiation�.Journal of Molecular Liquids, vol. 220, 592-603, 2016.

26. A. S. Dogonchi, K. Divsalar and D. D. Ganji, � Thermal radiation e¤ect on thenano�uid buoyancy �ow and heat transfer over a stretching sheet considering Brownianmotion�. Journal of Molecular Liquids, vol. 223, 521-527, 2016.

27. M. M. Bhatti, M. Ali Abbas and M. M. Rashidi, � A robust numerical method forsolving stagnation point �ow over a permeable shrinking sheet under the in�uence ofMHD�. Applied Mathematics and Computation, vol. 316, 381-389, 2018.

28. M. M. Bhatti, T. Abbas and M. M. Rashidi, �A numerical simulation of MHDstagnation-point �ow over a permeable stretching/shrinking sheet in a porous me-dia with heat transfer�. Iranian Journal of Science and Technology Transactions A:Science, vol. 41, 799-785, 2017.

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29. T. Salahudddin, M. Y. Malik, Arif Hussain, S. Bilal and M. Awais, � MHD �ow ofCattanneo-Christov heat �ux model for Williamson �uid over a stretching sheet withvariable thickness: using numerical approach�. Journal of Magnetism and magneticmaterials, vol. 401, 991-997, 2016.

30. A. Shahzad and Ramzan Ali, � MHD �ow of a non-Newtonian power law �uid over avertical stretching sheet with the convective boundary condition� . Walailak Journalof Science and Technology, vol. 10, 43-56, 2013.

31. M. M. Bhatti and M. M. Rashidi, �Numerical simulation of entropy generation onMHDnano�uid towards a stagnation point �ow over a stretching surface�. InternationalJournal of Applied and Computational Mathematics, vol. 3, 2275-2289, 2017.

32. M. M. Bhatti andM. M. Rashidi, �Entropy generation with nonlinear thermal radiationin MHD boundary layer �ow over a permeable shrinking/stretching sheet : Numericalsolution�. Journal of Nano�uids, vol. 5, 543-548(6), 2016.

33. M. M. Bhatti, M. M. Rashidi and I. Pop, � Entropy generation with nonlinear heat andmass transfer on MHD boundary layer over a moving surface using SLM�. NonlinearEngineering, vol. 6(1), 43-52, 2017.

34. M. M. Bhatti, S. R. Mishra, T. Abbas and M. M. Rashidi, �A mathematical modelof MHD nano�uid �ow having gyrotactic microorganisms with thermal radiation andchemical reaction e¤ects�. Neural Computing and Applications, 1-13, 2016.

35. A. S. Dogonchi, M. Alizadeh and D. D. Ganji, � Investigation of MHD Go-waternano�uid �ow and heat transfer in a porous channel in the presence of thermal ra-diation e¤ects�. Advanced Powder Technology, vol. 28(7), 1815-1825, 2017.

36. A. S. Dogonchi and D. D. Ganji, �Impact of Cattaneo-Christov heat �ux on MHDnano�uid �ow and heat transfer between parallel plates considering thermal radiatione¤ect�. Journal of the Taiwan Institute of Chemical Engineers, vol. 80, 52-63, 2017.

37. A. S. Dogonchi and D. D. Ganji, �Analytical investigation of convective heat transferof a logitudinal �n with temperature-dependent conductivity, heat transfer coe¢ cientand heat generation�. International Journal of Physical sciences, vol. 9(21), 466-474,2014.

38. R. E. Powell and H. Eyring, �Mechanisms for the relaxation theory of viscosity�. Na-ture, vol. 154, 427-428, 1944.

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For Review Only · 2019. 6. 10. · For Review Only MHD ⁄ow of a kinetic postulate of liquids inaugurated ⁄uid under thermal radiation e⁄ects Azad Hussain a,1 Zainia Muneer