Alexandria Engineering Journal (2016) 55, 1333–1339

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ORIGINAL ARTICLE

Heat transfer analysis of boundary layer flow overhyperbolic stretching cylinder

* Corresponding author. Tel.: +92 3335251806.

E-mail address: abidmajeed726@yahoo.com (A. Majeed).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

http://dx.doi.org/10.1016/j.aej.2016.04.0281110-0168 � 2016 Faculty of Engineering, Alexandria 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/).

Abid Majeed *, Tariq Javed, Irfan Mustafa

Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan

Received 23 October 2014; revised 15 March 2016; accepted 19 April 2016

Available online 9 May 2016

KEYWORDS

Boundary layer flow;

Hyperbolic stretching cylin-

der;

Heat transfer analysis;

Entropy generation

Abstract In the present study heat transfer and entropy generation analysis of boundary layer flow

of an incompressible viscous fluid over hyperbolic stretching cylinder are taken into account. The

governing nonlinear partial differential equations are normalized by using suitable transformations.

The numerical results are obtained for the partial differential equations by a finite difference scheme

known as Keller box method. The influence of emerging parameters namely curvature parameter

and Prandtl number on velocity and temperature profiles, skin friction coefficient and the Nusselt

number is presented through graphs. A comparison for the flat plate case is given and developed

code is validated. It is seen that curvature parameter has dominant effect on the flow and heat trans-

fer characteristics. The increment in the curvature of the hyperbolic stretching cylinder increases

both the momentum and thermal boundary layers. Also skin friction coefficient at the surface of

cylinder decreases but Nusselt number shows opposite results. Temperature distribution is decreas-

ing by increasing Prandtl number. Moreover, the effects of different physical parameters on entropy

generation number and Bejan number are shown graphically.� 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is anopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The boundary layer flow and heat transfer with stretching

boundaries received remarkable attention in modern industrialand engineering practice. The attributes of end product aregreatly reliant on stretching and rate of heat transfer at final

stage of processing. Due to this real-world importance, interestdeveloped among scientists and engineers to comprehend thisphenomenon. Common examples are the extrusion of metalsinto cooling liquids, food, plastic products, the reprocessing

of material in the molten state under high temperature. Duringthis phase of manufacturing process, the material passes into

elongation (stretching) and a cooling process. Such types ofprocesses are very handy in the production of plastic andmetallic made apparatus, such as cutting hardware tools, elec-tronic components in computers, rolling and annealing of cop-

per wires. In many engineering and industrial applications, thecooling of a solid surface is a primary tool for minimizing theboundary layer. Due to these useful and realistic impacts, the

problem of cooling of solid moving surfaces has turned out tobe an area of concern for scientists and engineers. Recentlyresearchers have shown deep interest in analyzing the charac-

teristics of flow and heat transfer over stretching surfaces. Inthis context, analysis of flow and heat transfer phenomenaover a stretching cylinder has its own importance in processes

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r, v

x*, uR Uw= Uo cosh(x*/L)

1334 A. Majeed et al.

such as fiber and wire drawing, hot rolling. Based on theseapplications, the boundary layer flow and heat transfer dueto a stretching cylinder was initially studied by Wang [1]. He

extended the work of Crane [2] to study the flow of heat trans-fer analysis of a viscous fluid due to stretching hollow cylinder.During last few years many researchers gave attention to ana-

lyze the fluid flow and heat transfer over the stretching cylinderand found the similarity solutions. Nazar et al. [3–5] producednumerical solution of laminar boundary layer flow, uniform

suction blowing and MHD effects over stretching cylinder inan ambient fluid. After that a lot of research problems havebeen done on the analysis of flow and heat transfer overstretching cylinder. Mukhopadhyay [6–9] considered chemical

solute transfer, mixed convection in porous medium, andMHD slip flow along a stretching cylinder. Effect of magneticfield over horizontal stretching cylinder in the presence of

source/sink with suction/injection is studied by Elbashbeshy[10]. Fang and Yao [11], Vajravelu et al. [12], Munawaret al. [13,14], Wang [15], Lok et al. [16], Abbas et al. [17],

and Butt and Ali [18] have considered viscous swirling flow,axisymmetric MHD flow, unsteady vacillating flow, naturalconvection flow, axisymmetric mixed connection stagnation

point flow, radiation effects in porous medium and entropyanalysis of magnetohydrodynamics flow over stretching cylin-der, respectively. More recently, Majeed et al. [19] appliedChebyshev Spectral Newton Iterative Scheme (CSNIS) and

studied the heat transfer influence by velocity slip and pre-scribed heat flux over a stretching cylinder. Sheikholeslami[20] has calculated the effective thermal conductivity and vis-

cosity of the nanofluid by Koo–Kleinstreuer–Li (KKL) modeland provided the nanofluid heat transfer analysis over cylin-der. Javed et al. [21] discussed the combined effects of MHD

and radiation on natural convection flow over horizontalcylinder. Hajmohammadi et al. [22] considered flow and heattransfer of Cu-water and Ag-water nanofluid over permeable

surface with convective boundary conditions. Their main focuswas to discuss the effects of nano-particles volume fraction,and non-dimensional permeability parameter on skin frictionand convection heat transfer coefficient. Dhanai et al. [23]

investigated mixed convection nanofluid flow over inclinedcylinder. They utilized Buongiorno’s model of nanofluid andfound dual solutions with the velocity and thermal slip effects

in the presence of MHD.Aforementioned studies reveal that vast literature is present

on study of linearly stretching cylinder but to the best of our

knowledge, the problem over a cylinder with hyperbolicstretching velocity and entropy generation has not beenaddressed yet. The governing equations related to flow problemsolved numerically and the effects of involving parameters on

flow behavior are shown graphically. Entropy generation isthe quantification of thermodynamics irreversibility and itexists in all types of heat transfer phenomenon. Due to this fun-

damental importance the present work is optimized with theinclusion of entropy generation analysis. In this context thestudies [24–30] are quite useful to explore many aspects of

entropy generation which have not been taken into account yet.

2. Problem formulation

Consider the two-dimensional steady incompressible flow of aviscous fluid over a hyperbolic stretching circular cylinder of

fixed radius R (see Fig. 1). The basic equations that governthe flow and heat transfer phenomena will take the form as

@u

@x�þ @v

@r¼ 0; ð1Þ

u@u

@x�þ v @u

@r¼ m @

r@rr@u

@r

� �; ð2Þ

u@T

@x�þ v @T

@r¼ a @

r@rr@T

@r

� �ð3Þ

where u and v are be velocity components along x� and r direc-tions, T be the temperature and a ¼ k=qcp be the thermal dif-fusivity of the fluid. The relevant boundary conditions for thegoverning problem are as follows:

uðr; x�Þ ¼ Uðx�Þ; vðr; x�Þ ¼ 0; T ¼ Tw ¼ T1 þ AUðx�Þ at r ¼ Ruðr; x�Þ ! 0; T ¼ T1 as r ! 1

�;

ð4Þwhere A is a constant. Now introducing the following suitable

transformation

x ¼ x�

L; g ¼ r

2 � R22Rx�

Re1=2x� ; w ¼ m Re1=2x� fðg; xÞ;

hðg; xÞ ¼ T� T1Tw � T1 : ð5Þ

where stream function w is the non-dimensional functiondefined through usual relationship as

u ¼ @wr@r

; v ¼ � @wr@x

: ð6Þ

which satisfies the continuity Eq. (1). Upon using Eqs. (5) and(6) into Eqs. (2) and (3), we arrived at following transformed

equations

r2

R2fggg þ

2x

R Re1=2xfgg þ 1�

x

Rex

d Rexdx

� �f2g þ

x

2Rex

d Rexdx

ffgg

¼ x fg@fg@x

� fgg@f

@x

� �; ð7Þ

1

Pr

r2

R2hgg þ 2x

R Re1=2xfgg

!� xtanhxhfg þ

x

2 Rex

d Rexdx

fhg

¼ x fg@h@x

� hg @f@x

� �; ð8Þ

where Rex ¼ Uw xL=m (local Reynolds number) and Pr ¼ m=a(Prandtl number). Using the stretching velocityUw ¼ U0 cosh x in Eqs. (7) and (8), we get

Figure 1 Geometry of the flow.

Heat transfer analysis of boundary layer flow 1335

1þ 2g Kffiffiffiffiffiffiffiffiffiffiffiffifficosh x

p� �

fggg þ2Kffiffiffiffiffiffiffiffiffiffiffiffifficosh x

p fgg þ1

2ð1þ xtanhxÞffgg

� xtanhxf2g ¼ x fg@fg@x

� fgg@f

@x

� �; ð9Þ

1

Pr1þ 2g Kffiffiffiffiffiffiffiffiffiffiffiffiffi

cosh xp

� �hgg þ 2K

Prffiffiffiffiffiffiffiffiffiffiffiffifficosh x

p hg þ 12ð1þ xtanhxÞfhg

� xtanhxhfg ¼ x fg@h@x

� hg @f@x

� �ð10Þ

and boundary conditions take the new form as

fð0; xÞ ¼ 0; fgð0; xÞ ¼ 1; hð0; xÞ ¼ 1; fgð1; xÞ ¼ 0;hð1; xÞ ¼ 0: ð11Þwhere K represents curvature parameter.

If we consider x ¼ 0 and K ¼ 0; then Eq. (9) reduces to theSakiadis flow equation [31]

fggg þ1

2ffgg ¼ 0; ð12Þ

with boundary conditions

fð0Þ ¼ 0; fgð0Þ ¼ 1; fgð1Þ ¼ 0 ð13ÞThe quantities of physical significance are the skin friction

coefficient and Nusselt number which are defined as

Cf ¼ swqU2w

; Nu ¼ xqwkðTw � T1Þ ; ð14Þ

where sw be the wall shear stress and qw the constant heat fluxfrom the surface, which are defined as

sw ¼ l @u@r

� �r¼R

; qw ¼ �k@T

@r

� �r¼R

: ð15Þ

After using Eq. (15) into Eq. (14), the local skin frictioncoefficient and local Nusselt number take the form

Cf Re1=2x ¼ fggð0; xÞ; Nu Re�1=2x ¼ �hgð0; xÞ: ð16Þ

Table 1 Comparison with Rees and Pop [31].

x ¼ 0 Cf Re1=2x Nu Re�1=2xRef. [31] �0.4439 �0.3509Present �0.4439 �0.3509

3. Solution methodology

To solve nonlinear system of partial differential Eqs. (9), (10)subject to the set of boundary conditions (11), we employed

a very accurate and efficient implicit finite difference methodcommonly known as Keller box which is explained in detailin the book of Cebeci and Bradshaw [32]. In present problem,

the solution is obtained through following steps.

Step-1: Eqs. (9), (10) are written in terms of first order dif-

ferential equations. Then functions and derivatives arereplaced by central difference form in both coordinatedirections. This results in a set of nonlinear difference equa-tions for the number of unknown equal to the number of

difference equations.Step-2: obtained nonlinear difference equations are thenlinearized by using Newton’s quasi-linearization technique.

Step-3: and are solved using block tri-diagonal algorithm.

Now to initiate the process at x ¼ 0, it is first providedguess profile for all five dependent variables arising afterreduction to first order system and then obtained the solutionoutlined in steps 1-3. Iteration process is continued for all

values of x, until the difference in velocity and temperature

profiles between the consecutive iteration is less than 10�5.The accuracy of the employed numerical scheme has been

established in Table 1 by comparison with the known resultsobtained by Rees and Pop [31] for flat plate ðx ¼ 0Þ: This com-parison gives us confidence that the developed code is correctand has achieved the desired level of accuracy. Moreover, the

values of Cf Re1=2x and Nu Re

�1=2x are given in Table 2.

4. Result and discussion

An inspection of the governing partial differential equations

indicates the presence of two emerging parameters: (a) the cur-vature parameter K, (b) and the Prandtl number Pr.

The effects of these parameters are discussed in the forth-coming figures. In Figs. 2 and 3, effects of curvature parameter

K on velocity and temperature profile are plotted. It isobserved that curvature has significant effects on velocityand temperature profiles. In Fig. 2 the trend for velocity profile

is rapidly decreasing in the region (0 < g < 0:42) and thenincreasing after g ¼ 0:42 for increasing values of curvatureparameter K and consequently as a result the boundary layer

thickness increases with the increase in curvature of the cylin-der. This behavior of the fluid velocity is due to the fact thatcurvature K and radius of cylinder have reciprocal relation-

ship. In other words increase in K tends to decrease in radiusof cylinder. In this way, due to lesser surface area of the cylin-der, the increase in velocity gradient at the surface increasesand consequently enhances the shear stress per unit area.

Fig. 2 also depicts that an increase in the curvature of the cylin-der leads to augment in boundary layer thickness, as comparedto the flat plate case (K ¼ 0). In Fig. 3 the temperature profilesdecrease near the surface of the cylinder as K increases, andafterward rise significantly and the thermal boundary-layerthickness increases. This variation of the temperature profile

is due to the reason that as the curvature increases, the radiusof cylinder reduces the surface area which is intact with thefluid also decreases. It is also important to mention that heatis transferred into the fluid in modes: conduction at the cylin-

der surface and convection for the region g > 0. Now, as thearea of surface of the cylinder decreases, a slender reductionin the temperature profile occurs close to the surface of the

cylinder, since a smaller amount of heat energy is transferredfrom the surface to the fluid through conduction phenomena.On the other hand, the thermal boundary-layer thickness

increases, because of the heat transport in the fluid due toenhanced convection process all around the cylinder, whichis evident from Fig. 3. Fig. 4 is plotted to observe the conse-

quence of curvature parameter K on the coefficient of skin fric-

tion Cf Re1=2x . It is noticed that as K increases, the wall skin

friction coefficient Cf Re1=2x decreases. This is due to the reason

that, for larger curvature of the cylinder, the velocity gradient

at the surface becomes increased rapidly as compared to the

Table 2 Numerical values of Cf Re1=2x and Nu Re

�1=2x for

various values of K when Pr ¼ 0:7 at x = 0.5.K CfRe

1=2x Nu Re

�1=2x

0 �0.55387 0.410120.5 �0.71911 0.600431 �0.88994 0.788941.5 �1.05690 0.96676

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

η

f η (

η, x

)

K = 1.5, 1, 0.5, 0.0

Figure 2 Effects on velocity profile due to different values of K

at x ¼ 3.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

η

θ (η

, x )

K = 1.5, 1, 0.5, 0

Figure 3 Effects on temperature profile due to different values of

K while Pr ¼ 0:7 and x ¼ 3.

0 0.5 1 1.5 2 2.5 3−2

−1.5

−1

−0.5

0

x

Cf R

e1/2 x

K=0, 0.5, 1, 1.5

Figure 4 Variations in Skin-friction coefficient for different

values of K against x.

0 0.5 1 1.5 2 2.5 3

0.5

1

1.5

x

Nu

Re−

1/2

x

K = 0, 0.5, 1, 1.5

Figure 5 Variations in Nusselt number for different values of K

against x while Pr ¼ 0:7.

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

η

f η (

η, x

)

x = 0.5, 1.0, 1.5, 3

Figure 6 Velocity profile for different values of x at K ¼ 0:5.

1336 A. Majeed et al.

flat plate (K ¼ 0). Fig. 5 is plotted to observe the consequenceof curvature parameter K on the Nusselt number Nu Re�1=2x . Itis noticed that as K increases the Nusselt number Nu Re�1=2xalso increases. In Fig. 6 and 7, it is observed that both velocityand temperature boundary layer thickness decrease by increas-ing the value of x. The effects of different values of Prandtl

number Pr on temperature profile have been discussed inFig. 8. It is observed that temperature is rapidly decreasingfor increasing Pr and thermal boundary layer thickness

decreases. It is obvious from the graph that fluids with lowPrandtl number slow down the cooling process and fluids withhigh Prandtl number speed up the cooling process. The behav-

ior of temperature profile shows that Prandtl number can beused to control the cooling process. This could be managedby using high Prandtl number fluids such as oils and lubri-

cants. Fig. 9 shows that Nusselt number Nu Re�1=2x increasesalong the surface of cylinder and with the increase of Pr. Thisis because of fact that by increasing the value of Pr thermalboundary layer decreases and heat transfer rate enhances. This

finding also testify the results shown in Fig. 8.

5. Entropy generation analysis

Using the boundary layer approximation, the local volumetricrate of entropy generation EG for a Newtonian fluid over ahyperbolic stretching cylinder is defined as follows:

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

η

θ (η

, x )

x = 0.5, 1, 1.5, 3

Figure 7 Temperature profile for different values of x at K ¼ 0:5and Pr ¼ 0:7.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

η

θ (η

, x )

Pr = 0.025, 0.7, 2, 3.6, 5.5

Figure 8 Temperature profile for different value of Pr at x ¼ 3.

Heat transfer analysis of boundary layer flow 1337

EG ¼ kT21

@T

@r

� �2|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

Entropy effects due to heat transfer

þ lT1

@u

@r

� �2|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

Entropy effects due to fluid friction

ð17Þ

After using results of Eq. (5) and Eq. (6) in Eq. (17), follow-ing form is obtained

EG ¼ kr2ðTw � T1Þ2U0 cosh xðhgðg; xÞÞ2

LR2T21xm

þ r2lðU0 cosh xÞ3ðfggðg; xÞÞ2

LR2T1xmð18Þ

Above equation can be written as

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

x

Nu

Re−

1/2

x

Pr = 8, 6, 4, 2

Figure 9 Variation in Nusselt number for different values of Pr

against x.

N�E ¼EGE0

¼ ð1þ 2KgÞReL cosh xx

� ðhgðg; xÞÞ2 þBrðcosh xÞ2ðfggðg; xÞÞ2

X�

" #ð19Þ

where N�E is the entropy generation number which is the ratioof local volumetric rate of entropy generation EG and charac-

teristic entropy generation rate E0 ¼ kðTw � T1Þ2=L2T21,ReL ¼ U0L=m is the Reynolds number, Br ¼ lU20=kðTw � T1Þis the Brinkman number, and X� ¼ ðTw � T1Þ=T1 is thedimensionless temperature difference and the ratio BrX��1 isgroup parameter. The Bejan number Be serves as a substituteof entropy generation parameter and it represents the ratiobetween the entropy generation due to heat transfer and the

total entropy generation due to heat transfer and fluid friction.It is defined by

Be ¼ NEHTNEHT þNEFF ð20Þ

Here

NEHT ¼ ð1þ 2KgÞReL cosh xx

ðhgðg; xÞÞ2 ð21Þ

NEFF ¼ ð1þ 2KgÞReL cosh3x

xBrX��1ðfggðg; xÞÞ2 ð22Þ

Bejan number Be can also be written as Be ¼ 1=1þ U�,where U� ¼ NEFF=NEHT is called irreversibility ratio.

Figs. 10–16 are drawn to discuss the influence of Prandtl

number Pr, curvature parameter K, group parameter BrX��1

and Reynolds number ReL on entropy generation numberN�E and Bejan number Be. In Figs. 10 and 11, the overall effectof Prandtl number is the enhancement in N�E and Be. This isbecause the temperature gradient increases with the larger val-ues of Pr. It is also noted that when Pr < 1 a small variation invalues of Be at surface of the cylinder is observed and it

increases for larger g and attains maximum value i.e.,Be ! 1. On the other hand when Pr > 1, large values of Beat surface of the cylinder are reported. These effects start

decreasing and Be ! 0 far away from the surface. Figs. 12and 13 are plotted to see the effects of K on N�E and Be. Bothfigures depict that entropy generation is more dominant forstretching cylinder case K > 0 in comparison with flat plat caseK ¼ 0. In Fig. 13, Be decrease at the surface stretching cylinder(g ¼ 0) and reverse behavior is observed far away from the

0 1 2 3 4 50

5

10

15

20

25

η

NE*

Pr = 0.1, 0.3, 0.7, 3, 5, 7

Figure 10 Effects of Pr on N�E whenK ¼ 0:5; ReL ¼ 2; BrX��1 ¼ 1 at x ¼ 0:2.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

η

Be

Pr = 0.1, 0.3, 0.7, 3, 5, 7

Figure 11 Effects of Pr on Be when

K ¼ 0:5; ReL ¼ 2; BrX��1 ¼ 1 at x ¼ 0:2.

0 1 2 3 4 50

10

20

30

40

η

NE*

K = 0, 0.3, 0.7, 1

Figure 12 Effects of K on N�E whenReL ¼ 2; BrX��1 ¼ 1; Pr ¼ 7 at x ¼ 0:2.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

η

Be

K = 0, 0.3, 0.7, 1

Figure 13 Effects of K on Be when Pr ¼ 7; BrX��1 ¼ 1 atx ¼ 0:2.

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

η

NE* BrΩ −1 = 0.1, 0.3, 0.7, 1

Figure 14 Effects of BrX��1 on N�E whenPr ¼ 7; ReL ¼ 2; K ¼ 0:5 at x ¼ 0:2.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

η

Be

BrΩ* −1 = 0.1, 0.3, 0.7, 1

Figure 15 Effects of BrX��1 on Be whenK ¼ 0:5; ReL ¼ 2; Pr ¼ 7 at x ¼ 0:2.

0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

η

NE*

ReL = 1, 2, 3, 5

Figure 16 Effects of ReL on N�E when

K ¼ 0:5; Pr ¼ 7; BrX��1 ¼ 1 at x ¼ 0:2.

1338 A. Majeed et al.

surface. The variations of group parameter BrX��1 on N�E andBe are depicted in Figs. 14 and 15. The enhancement in BrX��1

results in augmentation of N�E and these effects are opposite onBe. The values of Be are decreasing with increase in BrX��1

because the fluid friction dominates when BrX��1 increasesand this results in decrement of Be. The effects of ReL onentropy generation number are expressed in Fig. 16. This fig-ure reveals that the entropy generation number has gained

an increasing trend with increase in ReL.

6. Concluding remarks

A study is performed to investigate the flow and heat transfercharacteristics over a nonlinear stretching cylinder. The non-linear stretching velocity is considered as hyperbolic function.

An analysis of entropy generation is also presented and resultsare computed numerically with Keller box method. Fromabove study it is observed that:

� Increase in the values of curvature parameter K leads toincrease in velocity and temperature distribution.

� skin-friction coefficient reduces and Nusselt numberenhances with an increase in curvature parameter K.

� The temperature reduces and Nusselt number enhances byincreasing Prandtl number Pr.

� Increasing trend of curvature parameter, Prandtl numberand group parameter results in enhancement of entropygeneration.

Heat transfer analysis of boundary layer flow 1339

Acknowledgment

The authors are grateful to reviewers for their constructive

guideline which definitely improve this version of manuscript.

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Heat transfer analysis of boundary layer flow over hyperbolic stretching cylinder1 Introduction2 Problem formulation3 Solution methodology4 Result and discussion5 Entropy generation analysis6 Concluding remarksAcknowledgmentReferences