slip ow of nano uid over a stretching vertical cylinder in...

Scientia Iranica B (2018) 25(4), 2098{2110

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Slip ow of nano uid over a stretching vertical cylinderin the presence of non-linear thermal radiation andnon-uniform heat source/sink

C.S. Sravanthi�

Department of Mathematics, The M.S. University of Baroda, Vadodara, Gujarat, India.

Received 24 July 2016; received in revised form 3 May 2017; accepted 18 September 2017

KEYWORDSNano uid;Stretching verticalcylinder;Non-linear thermalradiation;HAM.

Abstract. An analysis is performed to study axisymmetric mixed convective boundarylayer ow of a nano uid over a vertical stretching circular cylinder in the presence of non-linear radiative heat ux. The e�ects of non-uniform heat source/sink and slip ow arealso taken into consideration. Water as conventional base uid containing nanoparticlesof Copper (Cu) is used. By means of similarity transformations, the governing partialdi�erential equations are reduced to highly non-linear ordinary di�erential equations andthen solved analytically using Homotopy Analysis Method (HAM). A comparison is madewith the available results in the literature, which shows that our results are in very goodagreement with the known results. A parametric study of the physical parameters is madeand results are presented through graphs and tables. The results indicate that the thermalboundary layer is thicker for non-linear thermal radiation problem than for linear thermalradiation. It is also found out that heat transfer rate at the surface decreases with increasein both space- and time-dependent heat source/sink parameters.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Thermal conductivity rate of ordinary base uids,including water, ethylene glycol, and oil, is very low.However, nowadays, cooling of electronic devices is amajor industrial requirement. To achieve this, thenanoscale solid particles are submerged into host uids,which change thermophysical characteristics of these uids and dramatically enhance heat transfer rate.These uids are said to be nano uids. Choi [1] wasthe �rst who identi�ed this colloidal suspension. Thenano uids have applications in cooling of electronics,heat exchanger, nuclear reactor safety, hyperthermia,biomedicine, engine cooling, vehicle thermal manage-

*. E-mail address: srinivasan [email protected]

doi: 10.24200/sci.2017.4580

ment, and many others. Pop et al. [2] investigated themixed convective boundary layer ow of a nano uidover a vertical circular cylinder using homotopy anal-ysis method. A bulk of research articles on nano uidsand cylinder stretching ows is available in the litera-ture of which few can be seen in the [3-18].

Heat transfer, in uenced by thermal radiation,has applications in many technological processes, in-cluding nuclear power plants, gas turbines, and vari-ous propulsion devices for aircraft, missiles, satellites,and space vehicles. Many investigations [19-22] intothermal radiation e�ect by applying the linearizedform of Rosseland approximation, which is obtainedby assuming su�ciently small temperature di�erenceswithin the ow, have been carried out. However,linear radiation is not valid for high temperaturedi�erence and the dimensionless parameter used inthe linearized Rosseland approximation is the only

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C.S. Sravanthi/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2098{2110 2099

e�ective Prandtl number [23]. Recently, the nonlinearRosseland di�usion approximation has been adoptedby various researchers [24-33] for studying the nonlin-ear radiative heat transfer in various geometries. Thenonlinear Rosseland di�usion approximation is valid forsmall as well as large temperature di�erence of surfaceand ambient uid, unlike the linearized Rosselandapproximation, which is valid only for su�ciently smalltemperature di�erence.

The study of heat source/sink e�ects on heattransfer is very important, because these e�ects arecrucial in controlling the heat transfer. Hayat et al. [34]studied the non-uniform heat source on unsteady heatand mass transfer MHD ow of nano uids over astretching sheet. Manjulatha et al. [35] performed athermal analysis by conducting dusty uid ow in aporous medium over a stretching cylinder in the pres-ence of non-uniform source/sink using Runge KuttaFehlberg-45 Method. Subhas et al. [36] investigated thee�ect of non-uniform heat source on MHD heat transferin a liquid �lm over an unsteady stretching sheet. Ef-fect of non-uniform heat generation on unsteady MHDnon-Darcian ow over a vertical stretching surface withvariable properties was studied by Muthamilselvan etal. [37].

No-slip boundary condition (the assumption thata liquid adheres to a solid boundary) is used toobtain the solution to all the above-mentioned studies.The no-slip boundary condition is one of the centralassumptions of the Navier-Stokes theory. However,partial velocity slip (the non-adherence of the uid to asolid boundary) may occur on the stretching boundarywhen the uid is particulate, such as emulsions, suspen-sions, foams, and polymer solutions. Recently, manyresearchers [38-44] have investigated the ow problemstaking slip ow condition at the boundary into account.Polishing of arti�cial heart valves and internal cavitiesis an important technological application of the uidsthat exhibit boundary slip.

For some of the above-mentioned problems, nu-merical techniques have been developed to obtainthe accurate solution for years. But, due to somerestrictions [45], scientists have considered analyticalapproaches as an alternative. One of the most commonmethods in this �eld, which is widely applied in scienceand engineering, is perturbation technique. It must benoted that perturbation technique cannot be appliedto strongly nonlinear problems, as it strongly dependsupon small/large physical parameters. In order toavoid the dependency on small/large parameters, non-perturbation techniques such as Adomian decompo-sition method and variational iteration method havebeen developed. The major disadvantage of thesemethods is that they cannot ensure the convergenceof series solution. On the other hand, the HomotopyAnalysis Method (HAM) proposed by Liao [46] is a

general analytical approach to obtain series solutionsto strongly nonlinear equations, which can provide uswith a simple way to ensure the convergence of solutionseries. Recently, the HAM has been adopted by variousresearchers [47-50] to obtain solutions to various uid ow problems.

No attempt has been made so far to analyzethe boundary layer ow of nano uid over a verticalstretching cylinder with non-linear thermal radiationand non-uniform heat source/sink under slip owboundary conditions. The present work aims to �ll thegap in the existing literature. In this article, we givea detailed discussion of the boundary layer slip owand heat transfer of nano uid over a stretching verticalcylinder in the presence of non-linear thermal radiationand non-uniform heat source/sink, analytically, viaHomotopy Analysis Method (HAM). The e�ects of dif-ferent emerging parameters are demonstrated throughgraphical representations and discussed at length.

2. Formulation of the problem

Let us consider the steady axisymmetric mixed con-vective boundary layer ow of a Cu-water nano uidover a vertical circular stretching cylinder with slipconditions. Flow analysis is carried out with non-linearthermal radiation and non-uniform heat source/sink.Cylindrical coordinates are chosen in such a way thatthe x-axis is along the axial direction of the cylinderwhile r-axis is normal to it (Figure 1). Stretchingvelocity of the cylinder is due to the application of twoforces to the cylinder, which are equal in magnitudebut opposite in direction, when origin is kept constant.Here, viscous dissipation is neglected. Under theseassumptions and using boundary layer approximations,the conservation laws are given as follows [2,24,34]:

@(ru)@x

+@(rw)@r

= 0; (1)

Figure 1. Physical model and coordinate system of theproblem.

2100 C.S. Sravanthi/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2098{2110

u@u@x

+ w@u@r

=�nfr

@@r

�r@u@r

�+'�s�s + (1� ')�f�f

�nfg(T � T1); (2)

u@T@x

+ w@T@r

= �nf1r@@r

�r@T@r

�� 1

(�cp)nf1r@(rqr)@r

+1

(�cp)nfq000: (3)

The corresponding boundary conditions are [38-40]:

u = U(x) +B1@u@r; w = 0;

T = Tw(x) +K1@T@r

at r = a;

u! 0; T ! T1 as r !1: (4)

Here, �nf is kinematic viscosity of the nano uid, �nf isdensity of nano uid, and �nf is thermal di�usivity ofthe nano uid, which are given by [3]:

�nf =�f

(1� ')2:5[(1� ')�f + '�s];

�nf = (1� ')�f + '�s;

�nf =knf

(�cp)nf; �nf =

�f(1� ')2:5 ;

(�cp)nf = (1� ')(�cp)f + '(�cp)s;

knfkf

=(ks � 2kf )� 2'(kf � ks)(ks + 2kf ) + '(kf � ks) ; (5)

where ' is the nanoparticle volume fraction (0 �' � 0:2 in which ' = 0 corresponds to the regularNewtonian uid. Also, ' beyond 0.2 is not physicallyrealizable due to accumulation) and (�cp)nf is the heatcapacity of the nano uid.

By using the Rosseland approximation [51], theradiative heat ux qr is given by:

qr = �4��3k�

@T 4

@y= �16��

3k� T3 @T@y

; (6)

where �� is the Stefan-Botzman constant and k� isabsorption coe�cient. It should be noted that by usingthe Rosseland approximation, the present analysis islimited to optically thick uids.

It is worth mentioning here that in the previousstudies on thermal radiation (see [19-22] and variousreferences therein), T 4 in Eq. (6) was expanded aboutthe ambient temperature T1. That means to simply

replace T 3 in Eq. (6) with T 31. Now, Eq. (3) can beexpressed as:

u@T@x

+ w@T@r

=��nf +

16��T 313k�(�cp)nf

�1r@@r

�r@T@r

�+

1(�cp)nf

q000: (7)

However, if we avoid the above-mentioned assumption,the radiative heat ux in Eq. (3) results in a highlynonlinear radiation expression, which is the subjectof the current study. Hence, the energy equation fornonlinear thermal radiation will be:

u@T@x

+ w@T@r

=1r@@r

��nf +

16��T 3

3k�(�cp)nf

�r@T@r

�+

1(�cp)nf

q000: (8)

The term q000 in the right-hand side of Eq. (3) is due tonon-uniform heat source, which is de�ned [34] as:

q000 =knfUx�f

[A�(Tw � T1)e�� +B�(T � T1)]; (9)

where A� and B� are the coe�cients of exponen-tially decaying space and temperature-dependent heatsource/sink, respectively. It is to be noted that the caseA� > 0; B� > 0 corresponds to internal heat source andA� < 0; B� < 0 corresponds to internal heat sink.

Consider the following similarity transforma-tions [39]:

� =r2 � a2

2a

sU�fx

; =pU�fxaf(�);

T � T1Tw � T1 = �(�); (10)

where is the stream function and the velocity com-ponents are de�ned as:

u =1r@ @r; w = �1

r@ @x

; (11)

which identically satisfy Eq. (1).Substituting Eqs. (5), (9), and (10) in Eqs. (2)

and (8), we obtain a system of dimensionless non-linearordinary di�erential equations:

1

(1� ')2:5h(1� ') + ' �s�f

i [(1 + 2 �)f 000 + 2 f 00]

+ ff 00�(f 0)2+(1�')+'

��s�f

��s�f

�(1�') + ' �s�f

�T � = 0;(12)


1Pr

1(1� ') + ' (�cp)s

(�cp)f"�knfkf

+Rd(1 + (�w � 1)�)3�0(1 + 2 �)�0

+knfkf

�A�e�� +B��

�#+ f�0 �Nf 0� = 0; (13)

subject to boundary conditions:

f 0 = 1 + S1f 00; f = 0; � = 1 + S2�0;

at � = 0;

f 0 ! 0; � ! 0 as � !1; (14)

where �T is the thermal buoyancy parameter (�T = 0corresponds to the case when there is no thermalbuoyancy e�ect, i.e., forced convection ow situation,while �T > 0 and �T < 0 correspond to assistingand opposing ows, respectively), is the curvatureparameter ( = 0 represents a at plate, while =1 represents vertical cylinder), Rd is the radiationparameter, and S1 and S2 are the velocity and thermalslip parameters, which are de�ned as:

�T =g�f (Tw � T1)L2

U20x

; =r

�fLU0a2 ;

Rd =16��T 313kfk�

; S1 = B1

sU0

�fL;

S2 = K1

sU0

�fL: (15)

The parameters of practical interest for the presentproblem are the skin-friction coe�cient and the localNusselt number, which are respectively given by thefollowing expressions [2,24]:

Cf =�w�fU2

0;

Nu =qwx

kf (Tw � T1); (16)

where �w is the shear stress at the surface of thecylinder and qw is the surface heat ux from the surfaceof the cylinder, which are given by:

�w = ��nf�@u@y

�r=a

;

qw = �knf�@T@y

�r=a

+ (qr)w: (17)

Using Eq. (15) in Eq. (16), the skin friction and localNusselt number in dimensionless form are:

pReCf = � f 00(0)�x)

(1� ')2:5 ;

NupRe

= ��knfkf

+Rd(1+(�w�1)�(0))3��0(0)�x: (18)

3. Series solution by means of HAM

For HAM solutions, the appropriate initial guesses canbe made as:

f0(�) =1

1 + S1[1� exp(��)];

�0(�) =1

1 + S2exp(��); (19)

and the auxiliary linear operators, Lf (f) and L�(�),which are the linear parts of the respective equations,are:

Lf (f) =d3fd�3 � df

d�;

L�(�) =d2�d�2 � �; (20)

satisfying the following properties:

Lf (C1 + C2 exp(�) + C3 exp(��)) = 0;

L�(C4 exp(�) + C5 exp(��)) = 0; (21)

where Ci, i = 1� 5, are constants.

3.1. Zeroth-order deformation problem

(1� p)Lf [f(�; p)� f0(�)] = p~fNf [f; �];

(1� p)L�[�(�; p)� �0(�)] = p~�N�[f; �]; (22)

where p 2 [0; 1] denotes the embedding parameter, ~indicates the non-zero auxiliary parameter, and:

Nf [f; �] =1

(1� ')2:5h(1� ') + ' �s�f

i�(1 + 2 �)

@3f(�; p)@�3 + 2

@2f(�; p)@�2

�+ f(�; p)

@2f(�; p)@�2 �

�@f(�; p)@�

�2

+(1� ') + '

��s�f

��s�f

�(1� ') + ' �s�f

�T �(�; p);


N�[f; �] =1

Pr1

(1� ') + ' (�cp)s(�cp)f"

knfkf

+Rd(1+(�w�1)�(�; p))3 @�(�; p)@�

(1+2 �)

!0+knfkf

(A�e�� +B��(�; p))#

+ f(�; p)@�(�; p)@�

�N @f(�; p)@�

�(�; p); (23)

subject to the following boundary conditions:

f(0; p)=0; f 0(0; p)=1+S1f 00(0; p); f 0(1; p)=0;

�(0; p) = 1 + S2�0(0; p); �(1; p) = 0: (24)

3.2. mth-order deformation problemsDi�erentiating the zeroth-order deformation prob-lems (24) for m times with respect to p, then dividingit by m! and setting p = 0, we get the following higher-order deformation problems:

Lf [fm(�)� �mfm�1(�)] = ~fRfm(�);

L�[�m(�)� �m�m�1(�)] = ~�R�m(�): (25)

And the boundary conditions are:

fm(0) = 0; f 0m(0) = S1f 00m(0); f 0m(1) = 0;

�m(0) = S2�0m(0); �m(1) = 0; (26)

where fm(�), �m(�), Rfm(�) and R�m(�) are calculatedby Eqs. (27) and (28), as shown in Box I, and:

�m =

(0 m � 11 m � 1

(29)

For p = 0 and p = 1, we can write:

f(�; 0) = f0(�); f(�; 1) = f(�);

�(�; 0) = �0(�); �(�; 1) = �(�); (30)

and with variation of p from 0 to 1, f(�; p), and�(�; p) vary from initial solutions f0(�) and �0(�) to�nal solutions f(�) and �(�), respectively. By Taylor'sseries, we have:

f(�; p) = f0(�) +1Xm=1

fm(�)pm;

fm(�) =1m!

@mf(�; p)@�m

��p=0

; (31)

�(�; p) = �0(�) +1Xm=1

�m(�)pm;

�m(�) =1m!

@m�(�; p)@�m

��p=0

: (32)

The value of the auxiliary parameter is chosen in such away that these two series (33) and (34) are convergentat p = 1.

Eq. (27) represents the system of non-homogeneous linear di�erential equations whose

fm(�) =1m!

@mf(�; p)@�m

; �m(�) =1m!

@m�(�; p)@�m

; (27)

Rfm(�) =1

(1� ')2:5h(1� ') + ' �s�f

i �(1 + 2 �)f 000m�1 + 2 f 00m�1

� +m�1Xn=0

fm�1�nf 00n �m�1Xn=0

f 0m�1�nf 0n

+(1� ') + '

��s�f

��s�f

�(1� ') + ' �s�f

�T �m�1;

R�m(�) =1

Pr1

(1� ') + ' (�cp)s(�cp)f

"�knfkf

+Rd(1 + (�w � 1)�m�1)3�0m�1(1 + 2 �)�0

+knfkf

�A�e�� +B��m�1

�#

+m�1Xn=0

fm�1�n�0n �Nm�1Xn=0

f 0m�1�n�n: (28)

Box I


general solutions are the sum of complementary andparticular solutions, which can be expressed as:

fm(�) = f�m(�) + Cm1 + Cm2 e� + Cm3 e

��;

�m(�) = ��m(�) + Cm4 e� + Cm5 e

��; (33)

where the constants Cmi (i = 1; 2; � � � 5) can be calcu-lated through the boundary conditions in Eq. (28).

3.3. Convergence analysisIn HAM method, it is essential to ensure the con-vergence of the series solution. As pointed out byLiao [46], the convergence rate of approximation forthe HAM solution strongly depends on the value ofthe auxiliary parameter, ~. This auxiliary parameterprovides us with great freedom to adjust and controlthe convergence region of the series solution. Hence,in order to seek the permissible values of ~f and ~�,the functions of f 00(0) and �0(0) are plotted at the 20thorder of approximations. The values of ~f , ~�, and ~�are selected in such a way that curves are parallel tohorizontal axis, i.e., ~-axis [46]. Figures 2 and 3 clearlydepict the acceptable range for values of ~f (�0:8 �~f � �0:1) and ~�(�0:6 � ~� � �0:1). The currentcalculations are based on the value of ~f = ~� = �0:5.In order to ensure the convergence of solutions, Table 1is given. This table clearly shows that convergence isobtained at the 30th order of approximations.

Figure 2. ~ curve for f at 20th order of approximation.

Figure 3. ~ curve for � at 20th order of approximation.

Table 1. Convergence of HAM solutions for f 00(0) and�0(0).

Order ofapproximations

�f 00(0) ��0(0)

1 0.9385 0.98795 0.9806 1.443910 0.9906 1.771115 0.9926 1.938220 0.9925 2.016625 0.9921 2.048628 0.9919 2.055529 0.9919 2.056430 0.9919 2.0564

4. Results and discussion

To get clear physical insight into the present problem,computations have been carried out using the methoddescribed in the previous section for variations in thegoverning parameters and the results are graphicallyillustrated in Figures 4-14. In the present study,the following default parameter values are adoptedfor computations: ' = 0:1, �T = 0:1, A� = 0:01,B� = 0:01, S1 = 0:1, S2 = 0:2, = 1, �w = 1:5,N = 1, Pr = 6:2 (which physically corresponds to waterbased nano uid), and Rd = 1. Therefore, all graphsand tables correspond to these values unless otherwisestated.

In the present study, we have used Copper (Cu)

Figure 4. E�ect of ' on the velocity.

Figure 5. E�ect of on the velocity.


Figure 6. E�ect of �T on the velocity.

Figure 7. E�ect of S1 on the velocity.

Figure 8. E�ect of ' on the temperature for �w = 1:5.

Figure 9. E�ect of ' on the temperature for �w = 3.

Figure 10. E�ect of A� on the temperature.

Figure 11. E�ect of B� on the temperature.

Figure 12. E�ect of �w on the temperature.

Figure 13. E�ect of Rd on the temperature.


Figure 14. E�ect of S2 on the temperature.

nanoparticles with water as base uid and data relatedto the thermophysical properties of the uid andnanoparticles are listed in Table 2. For the veri�cationof accuracy of the applied HAM, a comparison ofthe present results corresponding to the values of[��0(0)] with the available published results of Grubkaet al. [52], Ali et al. [53], Abel et al. [54], and Yadavet al. [55] in the case of pure uid (' = 0) is madenumerically and presented in Table 3. The resultsare found in excellent agreement. Therefore, we arecon�dent that the present results are accurate.

The variation of dimensionless velocity, f 0(�), fordi�erent values of nanoparticle volume fraction, ', isshown in Figure 4 for Cu-water nano uid. It canbe observed that the dimensionless velocity increaseswith the nanoparticle volume fraction. This is due tothe fact that increase in ' leads to increase in energytransportation and, hence, uid ow. It is also worthnoting that velocity of the nano uid is higher than thatof the pure uid (' = 0).

The e�ect of curvature parameter, , on the veloc-

ity within the boundary layer of Cu-water nano uid ispresented in Figure 5. The velocity is found to increasewith increase in curvature parameter. This is due to thefact that increase in leads to shrinking of the cylinderand, hence, the space provided for the free stream of thenano uid increases. Thus, the nano uid velocity tendsto be free stream velocity. It is important to note in the�gure that for a at plate ( = 0), the dimensionlessvelocity is smaller than that for the vertical cylinder( = 1).

The behavior of thermal buoyancy parameter, �T ,on the momentum boundary layer can be visualizedfrom Figure 6. The e�ect of increasing values of thethermal buoyancy parameter, �T , is to increase thevelocity, f 0(�). Physically, �T > 0 means heatingof the uid or cooling of the surface (assisting ow),�T < 0 means cooling of the uid or heating of thesurface (opposing ow), and �T = 0 means absence ofconvection currents (forced convection ow). Hence,increase in the value of �T can lead to increase in thetemperature di�erence between uid and surface. Thisleads to enhancement of the velocity f 0(�).

Figure 7 demonstrates the e�ect of velocity slipparameter (S1) for vertical stretching cylinder on ve-locity. With increase in S1, the velocity is foundto decrease in the vicinity of the cylinder surface.This is because with increase in S1, the ow velocitynear the stretching wall will not be equal to thestretching velocity of the wall. Hence, pulling of thestretching wall can only be partly transmitted to the uid. Consequently, uid velocity decreases. S1 = 0corresponds to the case when there is no velocity slipe�ect.

Figures 8 and 9 present the in uence of nanoparti-

Table 2. Thermophysical properties of uid and nanoparticles [1].

Physicalproperties

Cp (J/kg K) � (kg/m3) K (W/m K) �X 10�5 (1/K)

Water 4179 997.1 0.613 21Cu 385 8933 400 1.67

Table 3. Comparison of numerical values of [��0(0)] in the present study with previous studies for special cases (' = 0,�T = 0, A� = 0, B� = 0, S1 = 0, S2 = 0, = 0, and �w = 1).

Pr N Grubka et al.[52]

Ali et al.[53]

Abel et al.[54]

Yadav et al.[55]

Present study(by HAM)

0.72 1 0.8086 0.8086 | 0.8087 0.80871 1 1.0000 1.0000 1.0002 1.0000 1.00003 {1 0.0 | | {0.0039 0.03 0 1.1652 | | 1.1545 1.16423 1 1.9237 1.9237 1.9236 1.9237 1.92163 2 2.5097 | | 2.5159 2.509010 0 2.3080 | | 2.3000 2.3000


cle volume fraction, ', in both cases of linear (obtainedfrom the solution to Eq. (7)) and non-linear (obtainedfrom the solution to Eq. (8)) radiative heat uxes onthe temperature �(�) when �w = 1:5 and �w = 3:0,respectively. It is clear that as the nanoparticle volumefraction increases, temperature �(�) increases. Thisis due to the physical fact that thermal conductivityincreases with increase in copper volume fraction and,hence, thermal boundary layer thickness increases. Itcan also be observed that linear and non-linear resultsmatch better when �w approaches 1.

The impact of space-dependent heat source/sinkparameter, A�, on the dimensionless temperature isdepicted in Figure 10 for both linear and non-linearradiation cases. From Figure 10, it is observed thatincrease in space-dependent heat source/sink parame-ter leads to increase in the temperature in both linearand non-linear radiation cases. It is due to the factthat when A� > 0, thermal boundary layer generatesenergy, resulting in increasing temperature, and energyis absorbed when A� < 0, resulting in signi�canttemperature drop near the boundary layer. Similare�ect is observed in Figure 11 for the temperature-dependent heat source/sink parameter B� in bothlinear and non-linear radiative heat ux cases.

The e�ect of temperature ratio parameter, �w,on the dimensionless temperature has been plotted inFigure 12. It is observed that increase in �w leadsto intensi�cation of temperature. Increasing value of�w indicates larger wall temperature than the ambienttemperature. Hence, temperature increases with �w.We can �nd an overshoot near the surface of thecylinder for higher values of �w(> 5).

Figure 13 demonstrates the in uence of radiationparameter, Rd, on the dimensionless temperature, �(�),in both cases of linear and non-linear radiative heat uxes. Increase in Rd leads to increase in temperature.This is due to the fact that increase in radiation releasesthe heat energy from the ow, which in turn decreasesthe thickness of the thermal boundary layer. It can alsobe observed from the �gure that temperature becomeslarger with larger values of radiation parameter inthe case of non-linear radiation. However, this isnot the case for linear radiation. Thus, it can beconcluded that non-linear radiation has higher e�ecton the temperature than the linear radiation.

The impact of thermal slip parameter S2 on thedimensionless temperature is depicted in Figure 14 forboth linear and non-linear radiation cases. Increase inS2 results in decrease in temperature. The main reasonis that with increase in thermal slip parameter S2, lessheat is transferred to the uid from the stretching walland, thus, temperature decreases.

The density of nano uid increases with increase innanoparticle volume fraction. Due to this, skin friction(the drag force at the surface of the cylinder) also

Table 4. Numerical values of skin friction for variousvalues of S1, ', , and �T .

S1 �T ' (Re)12Cf

� 1�x

�0.0 0.1 0.1 1.0 1.7172

0.1 1.6270

0.0 1.6461

{0.1 1.6655

0.2 2.2833

0.0 1.1617

0.0 0.8891

increases with increasing nanoparticle volume fraction.This can be observed in Table 4. Moreover, it is notedthat skin friction is higher for the nano uid than forthe pure uid (' = 0). It can also be observed inTable 4 that the skin friction is higher for the verticalcylinder ( = 1) than for the at plate ( = 0); thisis due to large lateral surface area. It can be seenin Table 4 that with increase in thermal buoyancyparameter, �T , there is a decrease in skin friction. Thisagrees with the physical behavior that with increasein thermal buoyancy, velocity increases and, hence,there is decrease in momentum boundary layer and skinfriction. On the other hand, the skin friction decreaseswith slip velocity.

Table 5 provides the numerical values of Nusseltnumber (rate of heat transfer at the wall) for di�erentvalues of ', A�, B�, S2, , and �w with the other pa-rameters �xed for both linear and non-linear radiationcases. It shows that Nusselt number is an increasingfunction of ', i.e., the Nusselt number is higher fornano uid (' = 0:1) than for pure uid (' = 0).Increase in radiation parameter, Rd, leads to increase inNusselt number. This is due to the fact that for largevalues of radiation parameter, the thermal boundarylayer thickens. In the absence of thermal slip, S2, theNusselt number is found to be higher and it decreaseswith increasing thermal slip. The Nusselt number is adecreasing function of both space-dependent (A�) andtemperature-dependent (B�) heat source/sink param-eters. It is observed from the table that the Nusseltnumber is higher for a cylinder than for at plate. Thetable also depicts that the Nusselt number is lower forlinear radiation than for non-linear one.

5. Concluding remarks

The present work investigated the in uence of non-linear thermal radiation on mixed convective boundarylayer ow of Copper-Water nano uid over a verticalstretching circular cylinder. The e�ects of non-uniformheat source and slip ow were also incorporated.


Table 5. Numerical values of Nusselt number for various values of ', , A�, B�, Rd, and S2.

' Rd A� B� S2Nu(Re)

�12� 1

�x

�Linear radiation

Non-linear radiation(�w = 1:1)

0.0 0.0 1.0 0.01 0.01 0.0 3.0203 3.40882.0 4.1351 4.7189

0.02 3.0158 3.40360.02 3.0139 3.4013

0.3 2.5085 2.5404

1.0 1.0 0.01 0.01 0.0 3.1862 3.63792.0 4.6339 5.4299

0.02 3.1817 3.63270.02 3.1798 3.6304

0.3 2.5120 2.7042

0.1 0.0 1.0 0.01 0.01 0.0 3.4704 3.83332.0 4.5134 5.0456

0.02 3.4634 3.82520.02 3.4602 3.8216

0.3 2.6571 2.8682

1.0 1.0 0.01 0.01 0.0 3.6785 4.11222.0 5.0903 5.8498

0.02 3.6714 4.10420.02 3.6682 4.1006

0.3 2.8172 3.0688

Di�erent from the linear radiation, the present prob-lem was governed by an additional temperature ratioparameter. The solutions were computed analyticallyby Homotopy Analysis Method (HAM). The majorconclusions are listed below:

1. Unlike previously reported results on thermal radi-ation e�ect, the current results are valid for bothsmall (�w � 1) and large temperature di�erences(�w > 1) between surface of the cylinder andenvironment. The results presented clearly indicatethat the temperature ratio parameter has signi�-cant e�ect on heat transfer characteristics;

2. Intensifying slip e�ect leads to reduction in velocityas well as temperature;

3. The non-linear thermal radiation e�ect is favorablefor the scienti�c and engineering processes involvinghigh thermal requirements when the temperatureis highly enhanced by increasing the values ofradiation parameter in the non-linear case;

4. By varying the nanoparticle volume fraction, the

ow and heat transfer characteristics could becontrolled;

5. Nusselt number decreases with rise in space- ortime-dependent heat source/sink parameters. Onthe other hand, the opposite trend is observed asradiation varies;

6. Both Nusselt number and skin-friction are higherfor nano uid than for pure uid. Also, they arehigher for vertical cylinder than for at plate.

Nomenclature

u;w Velocity components in the axial andradial directions (ms�1)

a Radius of the cylinder (m)x Axial coordinate of the cylinder (m)

U0 Reference velocity (ms�1)L Characteristic length (m)T1 Ambient temperature (K)


T0 Reference temperature (K)N Temperature exponentB1 Velocity slip parameterK1 Thermal slip parameter�s; �f Coe�cients of thermal expansion of

the solid and uid fractions (K�1)�s; �f Densities of the solid and uid fractions

(kg m�3)qr Radiative heat ux (Wm�2)knf Thermal conductivity of the nano uid

(Wm�1K�1)kf ; ks Thermal conductivity of the uid and

the solid fractions (Wm�1K�1)f Function related to the velocity �eld� Dimensionless temperature in the

nano uid,U Stretching velocity (ms�1), U = U0x=LTw(x) Prescribed surface temperature (K),

Tw(x) = T1 + T0(x=L)N

�x Dimensionless axial coordinate,�x = x=L

Re Local Reynolds number, Re = U0L=�fPr Prandtl number, Pr = �f (�cp)f=kf�w Temperature ratio parameter,

�w = Tw=T1Subscripts

f Base uids Nanoparticlesnf Nano uid

Superscript0 Primes denote di�erentiation with

respect to �

References

1. Choi, S.U.S. \Enhancing thermal conductivity of uidswith nanoparticles", ASME, USA, pp. 99-105, FED231/MD (1995).

2. Dinarvand, S., Abbassi, A., Hosseini, R., and Pop,I. \Homotopy analysis method for mixed convectiveboundary layer ow of a nano uid over a verticalcircular cylinder", Thermal Science, 19, pp. 549-561(2015).

3. Oztop, H.F. and Abu-Nada, E. \Numerical studyof natural convection in partially heated rectangularenclosures �lled with nano uids", Int. J. Heat FluidFlow, 29, pp. 1326-1336 (2008).

4. Hayat, T., Ijaz Khan, M., Farooq, M., Alsaedi, A., andImran Khan, M. \Thermally strati�ed stretching owwith Cattaneo-Christov heat ux", Int. J. Heat andMass Transfer, 106, pp. 289-294 (2017).

5. Hayat, T., Ijaz Khan, M., Farooq, M., Yasmeen T., andAlsaedi, A. \Water-carbon nano uid ow with variableheat ux by a thin needle", J. Molecular Liquids, 224,pp. 786-791 (2016).

6. Hayat, T., Tamoor, M., Khan, M.I., and Alsaedi, A.\Numerical simulation for nonlinear radiative ow byconvective cylinder", Results in Physics, 6, pp. 1031-1035 (2016).

7. Hayat, T., Khan, M.I., Waqas, M., and Alsaedi,A \Newtonian heating e�ect in nano uid ow by apermeable cylinder", Results in Physics, 7, pp. 256-262 (2017).

8. Hayat, T., Waqas, M., Ijaz Khan, M., and Alsaedi,A. \Analysis of thixotropic nanomaterial in a doublystrati�ed medium considering magnetic �eld e�ects",Int. J. Heat and Mass Transfer, 102, pp. 1123-1129(2016).

9. Hayat, T., Khan, M.I., Waqas, M., Yasmeen, T.,and Alsaedi, A. \Viscous dissipation e�ect in owof magneto nano uid with variable properties", J.Molecular Liquids, 222, pp. 47-54 (2016).

10. Alsaedi, A., Ijaz Khan, M., Farooq, M., Gull, N., andHayat, T. \Magnetohydrodynamic (MHD) strati�edbioconvective ow of nano uid due to gyrotactic mi-croorganisms", Advanced Powder Technology, 28, pp.288-298 (2017).

11. Hashim and Khan, M. \A revised model to analyzethe heat and mass transfer mechanisms in the ow ofCarreau nano uids", Int. J. Heat Mass Transf., 103,pp. 291-297 (2016).

12. Hayat, T., Qayyum, S., Waqas, M., and Alsaedi,A. \Thermally radiative stagnation point ow ofMaxwell nano uid due to unsteady convectively heatedstretched surface", .J Mol. Liq., 224, pp. 801-810(2016).

13. Malik, M.Y., Imad Khan, Hussain, A. and Salahud-din, T. \Mixed convection ow of MHD Eyring-Powell nano uid over a stretching sheet: A nu-merical study", AIP Advances, 5, 117118 (2016).DOI:10.1063/1.4935639.

14. Hussain, A., Malik, M.Y., Salahuddin, T., Bilal, S.,and Awais, M. \Combined e�ects of viscous dissipationand Joule heating on MHD Sisko nano uid ow overstretching cylinder", Journal of Molecular Liquids,231, pp. 341-352 (2017).

15. Prasad, K.V., Vajravelu, K., Vaidya, H., and Santhi,S.R. \Axisymmetric ow of a nano uid past a verticalslender cylinder in the presence of a transverse mag-netic �eld", J. Nano uids, 5, pp. 101-109 (2016).

16. Prasad, K.V., Vajravelu, K., Vaidya, H., and Datti,P.S. \Axisymmetric ow over a vertical slender cylin-der in the presence of chemically reactive species",J. Appl. Comput. Math. (2015). DOI: 10.1007/s40819-015-0121-z

17. Prasad, K.V., Vajravelu, K., and Vaidya, H. \MHDCasson nano uid ow and heat transfer at a stretchingsheet with variable thickness", J. Nano uids, 5, pp.423-435 (2016).


18. Prasad, K.V., Vaidya, H., Vajravelu, K., Datti, V., andUmesh, V. \Axisymmetric mixed convective MHD owover a slender cylinder in the presence of chemicallyreaction", Int. J. App. Mech. and Eng., 21, pp. 121-141 (2016).

19. Pal, D., Mandal, G., and Vajravelu, K. \MHDconvection-dissipation heat transfer over a non-linearstretching and shrinking sheets in nano uids withthermal radiation", Int. J. Heat Mass Transf., 65, pp.481-490 (2013).

20. Zheng, L., Zhang, C., Zhang, X., and Zhang, J.\Flow and radiation heat transfer of a nano uid overa stretching sheet with velocity slip and temperaturejump in porous medium", J. Franklin Inst., 350, pp.990-1007 (2013).

21. Lin, Y., Zheng, L., and Zhang, X. \Radiation e�ectson Marangoni convection ow and heat transfer inpseudo-plastic non-Newtonian nano uids with variablethermal conductivity", Int. J. Heat Mass Transf., 77,pp. 708-716 (2014).

22. Zhang, C., Zheng, L., Zhang, X., and Chen, G. \MHD ow and radiation heat transfer of nano uids in porousmedia with variable surface heat ux and chemicalreaction", Appl. Math. Model., 39, pp. 165-181 (2015).

23. Magyari, E. and Pantokratoras, A. \Note on thee�ect of thermal radiation in the linearized Rosselandapproximation on the heat transfer characteristics ofvarious boundary layer ows", Int. Comm. in Heat andMass Transfer, 38, pp. 554-556 (2011).

24. Khan, M., Malik, R., and Hussain, M. \Nonlinearradiative heat transfer to stagnation-point ow of Sisko uid past a stretching cylinder", AIP Advances, 6,055315 (2016). DOI: 10.1063/1.4950946.

25. Hayat, T., Ijaz Khan, M., Waqas, M., Alsaedi, A.,and Farooq, M. \Numerical simulation for meltingheat transfer and radiation e�ects in stagnation point ow of carbon-water nano uid", Computer Methods inApplied Mechanics and Engineering, 315, pp. 1011-1024 (2017).

26. Hayat, T., Ijaz Khan, M., Farooq, M., Alsaedi, A.,and Yasmeen, T. \Impact of Marangoni convectionin the ow of carbon-water nano uid with thermalradiation", Int. J. of Heat and Mass Transfer, 106,pp. 810-815 (2017).

27. Yasmeen, T., Hayat, T., Ijaz Khan, M., Imtiaz, M.,and Alsaedi, A. \Ferro uid ow by a stretched surfacein the presence of magnetic dipole and homogeneous-heterogeneous reactions", Journal of Molecular Liq-uids, 223, pp. 1000-1005 (2016).

28. Hayat, T., Ijaz Khan, M., Imtiaz, M., and Alsaedi, A.\Similarity transformation approach for ferromagneticmixed convection ow in the presence of chemicallyreactive magnetic dipole", AIP Physics of Fluids, 28(2016). DOI: 10.1063/1.4964684.

29. Farooq, M., Ijaz Khan, M., Waqas, M., Hayat, T., andAlsaedi, A. \MHD stagnation point ow of viscoelasticnano uid with non-linear radiation e�ects", J. Molec-ular Liquids, 221, pp. 1097-1103 (2016).

30. Khan, M.I., Kiyani, M.Z., Malik, M.Y., Yasmeen,T., Khan, M.W.A., and Abbas, T. \Numerical in-vestigation of magnetohydrodynamic stagnation point ow with variable properties", Alexandria EngineeringJournal, 55, pp. 2367-2373 (2016).

31. Khan, M., Ali, H., and Hussain, M. \Magnetohydrody-namic ow of Carreau uid over a convectively heatedsurface in the presence of non-linear radiation", J.Mag. Mag. Mat., 412, pp. 63-68 (2016).

32. Hayat, T., Qayyum, S., Alsaedi, A., and Waqas,M. \Simultaneous in uences of mixed convection andnonlinear thermal radiation in stagnation point owof Oldroyd-B uid towards an unsteady convectivelyheated stretched surface", J. Mol. Liq., 224, pp. 811-817 (2016).

33. Mustafa, M., Mushtaq, A., Hayat, T., and Ahmad,B. \Nonlinear radiation heat transfer e�ects in thenatural convective boundary layer ow of nano uidpast a vertical plate: A numerical study", PLOS ONE,9(9): e103946 (2014).https://doi.org/10.1371/journal.pone.0103946

34. Hayat, T., Qayyum, S., Alsaedi, A., and Asghar,S. \Radiation e�ects on the mixed convection owinduced by an inclined stretching cylinder withnon-uniform heat source/sink", PLOS ONE, 12(4):e0175584 (2017).https://doi.org/10.1371/journal.pone.0175584

35. Manjunatha, P.T., Gireesha, B.J., and Prasanna Ku-mar, B.C. \Thermal analysis of conducting dusty uid ow in a porous medium over a stretching cylinderin the presence of non-uniform source/sink", Int. J.Mech. Mat. Eng. (2014). DOI: 10.1186/s40712-014-0013-8

36. Subhas Abela, M., Jagadish, T., and MahanteshNandeppanavar, M. \E�ect of non-uniform heat sourceon MHD heat transfer in a liquid �lm over an unsteadystretching sheet", Int. J. Non-Linear Mechanics, 44,pp. 990-998 (2009).

37. Muthtamilselvan, M., Prakash, D., and Doh, D.H. \Ef-fect of non-uniform heat generation on unsteady MHDnon-Darcian ow over a vertical stretching surface withvariable properties", J. Applied Fluid Mechanics, 7,pp. 425-434 (2014).

38. Hayat, T., Farooq, M., and Alsaedi, A. \Stagnationpoint ow of carbon nanotubes over stretching cylinderwith slip conditions", Open Phys., 13, pp. 188-197(2015).

39. Hayat, T., Qayyum, A., and Alsaedi, A. \E�ectsof heat and mass transfer in ow along a verticalstretching cylinder with slip conditions", Eur. Phys. J.Plus, 129(63) (2014). DOI:10.1140/epjp/i2014-14063-9.

40. Mukhopadhyay, S. \Chemically reactive solute transferin a boundary layer slip ow along a stretching cylin-der", Front Chem. Sci. Eng., 5, pp. 385-391(2011).

41. Sravanthi, C.S. \Homotopy analysis solution of MHDslip ow past an exponentially stretching inclined sheet


with Soret-Dofour e�ects", J. the Nigerian Mathemat-ical Society, 35, pp. 208-226 (2016).

42. Masood Khan and Ali, H. \E�ects of multiple slipon ow of magneto-carreau uid along wedge withchemically reactive species", Neural Comput Applic,(2016). DOI: 10.1007/s00521-016-2825-2016

43. Salahuddin, T., Malik, M.Y., Arif Hussain, Awais,M., and Bilal, S. \Mixed convection boundary layer ow of Williamson uid with slip conditions over astretching cylinder by using Keller box method", Int.J. Nonlinear Sciences and Numerical Simulation, 18,pp. 9-17 (2017).

44. Bilal, S., Khalil Ur Rehman, Hamayun Jamil, Malik,M.Y. and Salahuddin, T. \Dissipative slip ow alongheat and mass transfer over a vertically rotatingcone by way of chemical reaction with Dufour andSoret e�ects", AIP Advances, 6, 125125 (2016). DOI:http://dx.doi.org/10.1063/1.4973307.

45. Liao, S.J., Beyond Perturbation: Introduction to Ho-motopy Analysis Method, Boca Raton: Chapman &Hall/CRC Press (2003).

46. Liao, S.J., Homotopy Analysis Method in NonlinearDi�erential Equations, Beijing and Heidelberg: HigherEducation Press & Springer (2012).

47. Hayat, T., Khan, M.I., Waqas, M., Alsaedi, A.,and Yasmeen, T. \Di�usion of chemically reactivespecies in third grade uid ow over an exponentiallystretching sheet considering magnetic �eld e�ects",Chinese Journal of Chemical Engineering, 25, pp. 257-263 (2017).

48. Hayat, T., Khan, M.I., Alsaedi, A., and Khan, M.I.\Homogeneous-heterogeneous reactions and meltingheat transfer e�ects in the MHD ow by a stretchingsurface with variable thickness", J. Mol. Liq., 223, pp.960-968 (2016).

49. Hayat, T., Khan, M.I., Farooq, M., Gull, N., andAlsaedi, A. \Unsteady three-dimensional mixed con-vection ow with variable viscosity and thermal con-ductivity", J. Mol. Liq., 223, pp. 297-310 (2016).

50. Hayat, T., Khan, M.I., Farooq, M., Alsaedi, A., andKhan, M.I. \Thermally strati�ed stretching ow withCattaneo-Christov heat ux", Int. J. Heat and MassTransfer, 106, pp. 289-294 (2017).

51. Rosseland, S., Astrophysik und AtomtheoretischeGrundlagen, Springer, Berlin, pp. 41-44 (1931).

52. Grubka, L.J. and Bobba, K.M. \Heat transfer char-acteristics of a continuous stretching surface withvariable temperature", J. Heat Transfer, 107, pp. 248-250 (1985).

53. Ali, F.M., Nazar, R., Ari�n, N.M., and Pop, I. \E�ectof Hall current on MHD mixed convection boundarylayer ow over a stretched vertical at plate", Mecca-nica, 46, pp. 1103-1112 (2011).

54. Abel, M.S., Tawade, J.N., and Nandeppanavar, M.M.\MHD ow and heat transfer for the upper-convectedMaxwell uid over a stretching sheet", Meccanica, 47,pp. 385-393 (2012).

55. Yadav, R.S. and Sharma, P.R. \E�ects of porousmedium on MHD uid ow along a stretching cylin-der", Annals of Pure and Applied Mathematics, 6, pp.104-113 (2014).

Biography

C.S. Sravanthi has been working as an AssistantProfessor of Mathematics in the Department of Mathe-matics at the Maharaja Sayajirao University of Barodasince 2013. In 2008, she obtained MSc degree from S.V. University, Tirupati, India. In 2015, she obtainedPhD degree from Mahila University, India. Her mainworks in mathematics are in the �eld of uid mechanics.Since 2008, she has given lectures on mathematics tograduate and post-graduate students of science andengineering in various parts of India. She has givenmany presentations of papers in international andnational conferences. She has published more than 10papers in the �elds of mathematics and uid mechanicsin reputed international journals.

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