Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786

Entropy analysis of magnetohydrodynamic flow and heat transfer dueto a stretching cylinder

Adnan Saeed Butt *, Asif Ali

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

A R T I C L E I N F O

Article history:

Received 15 April 2013

Received in revised form 14 August 2013

Accepted 18 August 2013

Available online 4 October 2013

Keywords:

Boundary layer flow

Stretching cylinder

Viscous dissipation

Entropy generation

A B S T R A C T

The present work investigates the entropy generation effects in magnetohydrodynamic flow and heat

transfer over a stretching cylinder. By using suitable similarity transformations, the governing partial

differential equations are reduced into non-linear ordinary differential equations which are then solved

numerically. Expressions for coefficient of skin friction, Nusselt number, entropy generation number are

obtained in dimensionless form. The effects of various parameters on these quantities are presented

through tables and graphs. A comparison of entropy generation due to heat transfer and entropy effects

due to fluid friction and magnetic field is made with the help of Bejan number. The influence of different

parameters involved in the problem on Bejan number is also shown with the aid of graphs.

� 2013 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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Journal of the Taiwan Institute of Chemical Engineers

jou r nal h o mep age: w ww.els evier . co m/lo c ate / j t i c e

1. Introduction

The concept of boundary layer theory was put forward byPrandtl [1] which brought a revolution in the study of flow andheat transfer phenomenon. Blasius [2] utilised this concept tostudy the steady two-dimensional flow past a stationary flatsurface. Sakiadis [3,4] discussed similar flow over a moving flatplate. Crane [5] studied the boundary layer flow over a stretchingsurface. Boundary layer flow over stretchable surfaces has vastapplications in manufacturing and industrial processes such asproduction of metallic and plastic sheets, paper production andglass blowing. Due to these applications, engineers and scientistshave examined this problem from different physical aspectsrelating to flow and heat transfer [6–15].

The study of viscous fluid flow and heat transfer outside ahollow stretching cylinder has importance in extrusion processes.Wang [16] investigated the viscous flow and heat transfer due touniformly stretching cylinder. He obtained exact similaritysolution of the problem and also compared asymptotic solutionsfor large Reynolds number to the numerical values. Ishak et al. [17]examined the magnetohydrodynamic flow and heat transfer due toa stretching cylinder and obtained numerical solutions with thehelp of keller box scheme. Burde [18] studied the axisymmetricmotion of a viscous fluid near a cylinder stretching in the axialdirection. Ishak and Nazar [19] presented the numerical solution offlow and heat transfer over a stretchable cylinder. Unsteady

* Corresponding author. Tel.: +92 051 03335422714.

E-mail address: adnansaeedbutt85@gmail.com (A.S. Butt).

1876-1070/$ – see front matter � 2013 Taiwan Institute of Chemical Engineers. Publis

http://dx.doi.org/10.1016/j.jtice.2013.08.018

viscous flow over an expanding stretching cylinder was examinedby Gang et al. [20]. Munawar et al. [21] analysed the unsteady flowof a viscous fluid over an oscillating stretching cylinder andobtained the numerical solution of the problem. Time dependentflow and heat transfer of a viscous fluid over a stretching cylinderwere examined by Munawar et al. [22].

In the modern age, one of the major concerns of scientists andengineers is to find methods which could control the wastage ofuseful energy. Especially in thermodynamical systems, energylosses can cause great disorder. This disorder in the system ismeasured in terms of energy. Bejan [23] proposed that thedisorders in flow and heat transfer processes can be examined interms of entropy generation. Based on his idea, many researchershave examined entropy generation effects in flow and heattransfer systems. San and Laven [24] analysed the entropygeneration due to heat and mass transfer through an isothermalchannel. Hijleh and Heilen [25] investigated the entropy genera-tion over an isothermal rotating cylinder. Yilbas [26] studied theentropy effects in concentric annulus with outer cylinder rotating.Makinde and Beg [27] discussed the entropy generation in areactive hydromagnetic channel. Entropy analysis of variableviscosity fluid flow between two concentric pipes with convectivecooling at the outer surface was made by Tshehla and Makinde[28]. Tayamol et al. [29] examined the entropy effects in fluid flowand heat transfer over a stretchable surface through a porousmedium. Butt and Ali [30] discussed the effects of magnetic fieldon entropy generation in flow and heat transfer phenomenon overa radially stretching sheet. Recently, Munawar et al. [31] studiedthe entropy production in viscous flow over an oscillatingstretching cylinder.

hed by Elsevier B.V. All rights reserved.

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786 781

The present paper examines the entropy effects in magnetohy-drodynamic flow and heat transfer over a stretching cylinder. Theinfluence of various parameters on velocity, temperature, entropygeneration number and Bejan number are presented and discussedwith the help of graphs.

2. Mathematical formulation

Consider a steady laminar boundary layer flow of an electricallyconducting viscous fluid due to stretching of a hollow cylinder ofradius a in the axial direction. The x-axis is taken along the axis ofcylinder while r-axis is along the radial direction. A uniformmagnetic field of strength Bo is applied in the radial direction. Theeffects of induced magnetic field are negligible as magneticReynolds number is assumed to be small. The surface of stretchingcylinder is held at constant temperature Tw and the ambienttemperature is considered to be T1. The effects of viscousdissipation are also present. Further, it is assumed that Tw > T1.The schematic diagram of the problem is shown in Fig. 1. Then, theboundary layer equations governing the flow and heat transferphenomenon are

@ðruÞ@xþ @ðrvÞ

@r¼ 0; (1)

u@u

@xþ v

@u

@r¼ n

r

@@r

r@u

@r

� �� sB0

2u

r; (2)

u@T

@xþ v

@T

@r¼ a

r

@@r

r@T

@r

� �þ n

c p

@u

@r

� �2

; (3)

subject to the boundary conditions:

u ¼ uw ¼ 2cx; v ¼ 0; T ¼ Tw; at r ¼ a;u ! 0; T ! T1; as r ! 1: (4)

where x and r are the axial and radial directions, u and v are thevelocity components in x and r directions, s is the electricalconductivity, r is the density of fluid, n is the kinematic viscosity, ais the thermal diffusivity, cp is the specific heat at constantpressure, T is the temperature of the fluid, Tw is the temperature atthe surface of stretching cylinder, T1 is the temperature of theambient fluid and c > 0 is a positive constant.

In order to transform the Eqs. (1)–(4) in dimensionless form,following similarity transformations are taken

h ¼ r

a

� �2

; u ¼ 2cx f 0ðhÞ; v ¼ � caffiffiffiffihp f ðhÞ; u ¼ T � T1

Tw � T1; (5)

Substituting (5) into Eqs. (2)–(4), the equations take the form

h f 000 þ f 00 þ Reð f f 00 � f 02Þ � M f 0 ¼ 0; (6)

hu00 þ u0 þ Pr Re fu0 þ Pr Ec h f 002 ¼ 0; (7)

Fig. 1. Schematic diagram of the problem.

with corresponding boundary conditions

f ð1Þ ¼ 0; f 0ð1Þ ¼ 1; uð1Þ ¼ 1;f 0ð1Þ ! 0; uð1Þ ! 0:

(8)

Here Re ¼ ca2=2n is the Reynolds number, M ¼ sB2oa2=4nr is the

magnetic parameter, Pr ¼ mc p=k is the Prandtl number and Ec ¼u2

w=c pðTw � T1Þ is the Eckert number.The physical quantities that are of interest for the engineers are

the skin friction coefficient and the local Nusselt number which aredefined as:

C f ¼tw

ru2w=2

; Nu ¼ aqw

kðTw � T1Þ; (9)

where tw and qw are the shear stress and the heat transfer ratefrom the surface of the cylinder respectively and are given as:

tw ¼ m@u

@r

� �r¼a

; qw ¼ k@T

@r

� �r¼a

(10)

By using (5), we get

C fRe x

a

� �¼ f 00ð1Þ; Nu ¼ �2u0ð1Þ: (11)

3. Entropy generation

The entropy generation rate SG for a viscous fluid in thepresence of magnetic field is defined by

SG ¼k

T12

@T

@r

� �2" #

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}entropy contribution

due toheat transfer

þ mT1

@u

@r

� �2

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}entropy contribution

due tofluid friction

þ sB02

T1u2

|fflfflfflffl{zfflfflfflffl}entropy contribution

due tomagnetic field

:

(12)

In terms of non-dimensional variables, the entropy generationhas the form

Ns ¼ SG

SGo

¼ 1

Xhu02 þ 1

X

Br

Vh f 0 0

2 þ 1

X

Br

VM f 0

2; (13)

where SGo ¼ 4kðTw � T1Þ2=T12L2 is the characteristic entropy

generation rate, V ¼ Tw � T1=T1 is the dimensionless tempera-ture difference and X ¼ ða=LÞ2 is the dimensionless parameter.Here Ns is the ratio of the entropy generation rate SG tocharacteristic entropy generation rate SGo and is known as entropygeneration number. Eq. (13) can be written in the form:

Ns ¼ NH þ N f þ Nm ¼ NH þ NF ; (14)

where NH ¼ ð1=XÞhu02

is the entropy generation due to heattransfer, N f ¼ ð1=XÞðBr=VÞh f 00

2is the entropy generation due to

fluid friction and Nm ¼ ð1=XÞðBr=VÞM f 02

is the local entropygeneration due to magnetic field. Here NF ¼ N f þ Nm.

Another parameter known as Bejan number is defined asfollows:

Be ¼ NH

NH þ NF: (15)

From Eq. (15), it is quite obvious that the value of Bejan numberlies between 0 and 1. When Bejan number is greater than 0.5, theentropy effects due to heat transfer dominate. On the other handthe value of Bejan number is less than 0.5, the entropy effects dueto fluid friction and magnetic field are significant. The contributionof entropy due to heat transfer is equal to that of fluid friction andmagnetic field, when Be ¼ 0:5.

Table 1Comparison of our values of skin friction f 00ð1Þ with those of Wang [16] in the

absence of magnetic field i.e., M ¼ 0:0.

R f 00ð1Þ [16] f 00ð1Þ Present

0.1 �0.48181 �0.48181

0.2 �0.61748 �0.61748

0.5 �0.88220 �0.88220

1.0 �1.17776 �1.17775

2.0 �1.59590 �1.59589

5.0 �2.41745 �2.41743

10.0 �3.34445 �3.34446

Table 2Comparison of our values of Nusselt number u0ð1Þ with those of Wang [16] when

M ¼ 0:0; Ec ¼ 0:0.

R Pr u0ð1Þ [16] u0ð1Þ Present

1.0 0.7 �0.5880 �0.5880

2.0 �1.065 �1.064

7.0 �2.059 �2.059

20.0 �3.521 �3.521

10.0 0.7 �1.568 �1.568

2.0 �3.035 �3.035

7.0 �6.160 �6.159

20.0 �10.77 �10.77

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786782

4. Results and discussions

The non-linear differential equations (6)–(7) subject to theboundary conditions (8) are solved numerically using the shootingtechnique with fourth-order Runge Kutta scheme. In order to solvethe equations, the semi-infinite domain 1 � h < 1 is changed by afinite domain 1 � h < h1, where h1 is chosen such that thenumerical solution approximates close to the boundary conditions.

Fig. 3. Effects of magnetic parameter M on temperatu

Fig. 2. Effects of magnetic parameter M on

The numerical procedure is carried out by guessing out the valuesof f 00ð1Þ and u0ð1Þ which are then improved until the end boundaryconditions are satisfied. The accuracy goal is kept equal to 10�6 asthe convergence criteria. In Tables 1 and 2, a comparison of ourresults with those reported by Wang [16] is made in the absence ofmagnetic field and viscous dissipation. It is observed there is a goodagreement between the values of the both studies which shows thevalidity of our numerical scheme.

re profiles uðhÞ when Re ¼ 5:0; Pr ¼ 0:7; Ec ¼ 0:2.

velocity profiles f 0ðhÞ when Re ¼ 5:0.

Fig. 4. Effects of Prandtl number Pr on temperature profiles uðhÞ when M ¼ 1:0; Re ¼ 5:0; Ec ¼ 0:2.

Fig. 5. Effects of Eckerd number Ec on temperature profiles uðhÞ when M ¼ 1:0; Re ¼ 5:0; Pr ¼ 0:7.

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786 783

In order to explore the physical aspects of the problem, theeffects of various parameters on velocity and temperature profilesare presented in Figs. 2–5. Fig. 2 shows the variation in velocityprofile f 0ðhÞ against h for different values of magnetic parameterM. It is noticed that with an increase in magnetic parameter, thetransport rate is reduced due to Lorentz force which producesresistance to flow. In Fig. 3, it is noticed that due to resistance to theflow, the temperature uðhÞ increases with increase in magneticparameter M. Fig. 4 depicts the effects on Prandtl number Pr onuðhÞ. It is observed that the thermal boundary layer thicknessdecreases with increase in Pr which is in agreement with thephysical fact. On the other hand, it can be seen in Fig. 5 that anincrease in Eckerd number Ec causes the temperature of the fluid toincrease.

Figs. 6–9 are presented to show the influence of physicalperimeters involved in the considered problem on entropygeneration number Ns. Fig. 6 illustrates that entropy generationnumber Ns increases with increase in magnetic parameter M nearthe surface of the stretching cylinder. Also, it is observed that as thedistance from the surface increases, the effects of entropy start todepreciate and far away from the stretching cylinder, these effectsare negligible. This is quite true as the Lorentz force which opposesthe fluid motion results in enhancement of entropy effect near the

stretching cylinder. Fig. 7 shows the effects of Reynolds number Re

on entropy generation number. As the value of Reynolds numberincreases, the velocity and the temperature gradients increases (asdepicted in Tables 1 and 2) which cause the entropy effects toenhance near the surface of the stretching cylinder. Far away fromthe surface, these effects diminish. The effects of group parameterBr=V on Ns are increasing as illustrated in Fig. 8. The influence ofthe dimensionless parameter X on entropy generation number areshown in Fig. 9. It is observed that the entropy generation ratedecreases with increase in X.

In order to explore whether heat transfer entropy effectsdominate over entropy due to fluid friction and magnetic field orvice versa, Figs. 10–12 are plotted. Fig. 10 illustrates that near thesurface of the stretching cylinder entropy effects due to fluidfriction and magnetic field become dominant over the heat transferentropy effects with increase in magnetic parameter M. However,with increase in distance from the cylinder surface, entropy effectsdue to heat transfer become strong and in the far away region,these effects are fully dominant. In Fig. 11, it is observed that withan increase in Reynolds number Re, the entropy effects due to fluidfriction and magnetic field lessen a bit near the stretching cylindersurface. However, as the distance increases, heat transfer entropyeffects become prominent and in the far away region, these effects

Fig. 6. Effects of magnetic parameter M on entropy generation number Ns when Re ¼ 5:0; Pr ¼ 0:7; Br=V ¼ 1:0; X ¼ 0:5.

Fig. 7. Effects of Reynolds number Re on entropy generation number Ns when M ¼ 1:0; Pr ¼ 0:7; Br=V ¼ 1:0; X ¼ 0:5.

Fig. 8. Effects of group parameter Br=V on entropy generation number Ns when M ¼ 1:0; Re ¼ 5:0; Pr ¼ 0:7; X ¼ 0:5.

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786784

Fig. 9. Effects of dimensionless parameter X on entropy generation number Ns when M ¼ 1:0; Re ¼ 5:0; Pr ¼ 0:7; Br=V ¼ 1:0.

Fig. 10. Effects of magnetic parameter M on Bejan number Be when Re ¼ 5:0; Pr ¼ 0:7; Br=V ¼ 1:0; X ¼ 0:5.

Fig. 11. Effects of Reynolds number Re on Bejan number Be when M ¼ 1:0; Pr ¼ 0:7; Br=V ¼ 1:0; X ¼ 0:5.

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786 785

Fig. 12. Effects of group parameter Br=V on Bejan number Be when M ¼ 1:0; Re ¼ 5:0; Pr ¼ 0:7; X ¼ 0:5.

A.S. Butt, A. Ali / Journal of the Taiwan Institute of Chemical Engineers 45 (2014) 780–786786

are fully dominant. Similar effects are observed in case of groupparameter Br=V a shown in Fig. 12.

5. Conclusions

In the present article, magnetohydrodynamic flow and heattransfer over a stretching cylinder is considered and entropyeffects are examined. Similarity transformations are used to reducethe equations which are then solved numerically. The followingconclusions are drawn from the investigation:

� The magnetic field causes resistance to the flow which results indecrease in velocity and increase in temperature.� The thermal boundary layer thickness decreases with Prandtl

number Pr and increases with Eckerd number Ec.� The entropy generation number Ns increases with magnetic

parameter M, Reynolds number Re and group parameter Br=Vand decreases with dimensionless parameter X.� The entropy effects due to fluid friction and magnetic field are

prominent near the stretching cylinder surface and in the faraway region, heat transfer entropy effects are dominant.� The fluid friction and magnetic field entropy effects become

more dominant near the stretching surface of cylinder withincrease in M. However, a very slight decrease in these effects areobserved with increase in Re and Br=V.

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