Hindawi Publishing CorporationJournal of Computational Methods in PhysicsVolume 2013 Article ID 180516 11 pageshttpdxdoiorg1011552013180516

Research ArticleMHD-Conjugate Free Convection froman Isothermal Horizontal Circular Cylinder withJoule Heating and Heat Generation

NHM A Azim1 and M K Chowdhury2

1 School of Business Studies Southeast University House-64 Road-18 Block-B Banani Dhaka 1213 Bangladesh2Department of Mathematics Bangladesh University of Engineering and Technology Dhaka 1000 Bangladesh

Correspondence should be addressed to NHM A Azim nhmarif1gmailcom

Received 31 March 2013 Revised 27 July 2013 Accepted 15 August 2013

Academic Editor Ivan D Rukhlenko

Copyright copy 2013 NHM A Azim and M K Chowdhury This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The present work is devoted to the numerical study of laminar magnetohydrodynamic (MHD) conjugate natural convection flowfrom a horizontal circular cylinder taking into account Joule heating and internal heat generation The governing equations andthe associated boundary conditions for this analysis are made nondimensional forms using a set of dimensionless variables Thusthe nondimensional governing equations are solved numerically using finite difference method with Keller box scheme Numericaloutcomes are found for different values of the magnetic parameter conjugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter for the velocity and the temperature within the boundary layer as well as the skin frictioncoefficients and the rate of heat transfer along the surface It is found that the skin friction increases and heat transfer rate decreasesfor escalating value of Joule heating parameter and heat generation parameter Results are presented graphically with detaileddiscussion

1 Introduction

Two-dimensional laminar natural convection from a hori-zontal cylinder under various surface boundary conditionssuch as isothermal uniform heat flux and mixed boundaryconditions was investigated by several researchers employingthe different numerical techniques For example Merkin [1]analysed the free convection boundary layer on an isothermalhorizontal cylinder Kuehn and Goldstein [2] determinednumerical solution for the Navier-Stokes equations for lam-inar natural convection about a horizontal isothermal cir-cular cylinder They obtained the solutions of the Navier-Stokes and energy equations for natural-convection heattransfer from a horizontal isothermal cylinder and foundthat boundary layer conditions reached at the lower portionof the cylinder when the Rayleigh number becomes verylarge Wang et al [3] investigated numerical computation ofnatural convection flow about a horizontal cylinder using

splines They reported some new computations at very highRayleigh numbers which indicate the existence of attachedldquoseparationrdquo vortices in the downstream plume region

The combined effect of conduction and free convectionwhich is known as conjugate free convection has a significantimportance in many practical applications Gdalevich andFertman [4] investigated conjugate problems of natural con-vection Miyamoto et al [5] investigated the effects of axialheat conduction in a vertical flat plate on free convection heattransferThey discovered that axial heat conduction in the flatplate significantly affects the temperature distribution withinthe boundary layer Pozzi and Lupo [6] investigated the entirethermofluid-dynamic (TFD) field resulting from the couplingof natural convection along the surface of the flat plate andconduction inside a heated flat plate by two expansionsregular series and asymptotic expansions Kimura and Pop[7] analysed conjugate natural convection from a horizontalcircular cylinderThe results obtained by Kimura and Pop [7]

2 Journal of Computational Methods in Physics

showed that the ratio of thermal conductivities of the solidwall to that of fluid within the boundary layer substantiallyinfluenced the flow and heat transfer characteristics

On the other hand the electromagnetic fields are usedto control the heat transfer as in the convection flows andaerodynamic heating and heat is also produced by elec-tromagnetic fields such as MHD generators and pumps Asubstantial quantity of research has been done in the pres-ence of electromagnetic field of the flow and heat transfercharacteristics over a variety of geometries For exampleWilks [8] studiedMHD-free convection about a semi-infinitevertical plate in a strong cross field Hossain [9] studied theeffects of viscous dissipation and Joule heating on MHD-free convection flow with variable plate temperature Heconcluded that dissipative heat reduces the velocity fieldmore in the lower Prandtl number fluid than that of thehigher Prandtl number The author also revealed that thedissipative heat reduced temperature field faster in the higherPrandtl number than that in lower Prandtl number fluidAldoss et al [10] analysed MHD mixed convection from ahorizontal circular cylinder They found that the increase inthe magnetic field leads to a decrease in the flow field a risein the temperature values of the flow field and a decreasein the local wall shear stress and local Nusselt number Thecombined effect of viscous dissipation and Joule heating onMHD forced convection over a nonisothermal horizontalcylinder embedded in a fluid saturated porous medium wasstudied by El-Amin [11]

Many practical heat transfer applications involve theconversion of some form of mechanical electrical nuclearor chemical energy to thermal energy in the medium Suchmediums are said to involve internal heat generation Forexample a large amount of heat is generated in the fuelelements from atomic reactors as a result of atomic fissionthat serves as the heat source for the nuclear power plantsTheheat generated in the sun as a result of fusion of hydrogen intohelium makes the sun a large nuclear reactor that suppliesheat to the earth Possible heat generation effects maymodifytemperature distribution and therefore the particle deposi-tion rate The heat transfer in a laminar boundary layer flowof a viscous fluid over a linearly stretching continuous surfacewith viscous dissipationfrictional heating and internal heatgeneration was analysed by Vajravelu and Hadjinicolaou[12] They considered the volumetric rate of heat generation119902101584010158401015840[Wm3] as 119902101584010158401015840 = 119876

0(119879119891minus 119879infin) for 119879

119891ge 119879infin

and 119902101584010158401015840 = 0for 119879119891lt 119879infin where 119876

0is the heat generation constant The

effects of heat generationabsorption and the thermophoresison hydromagnetic flow with heat and mass transfer overa flat plate were investigated by Chamkha and Camille[13] Mendez and Trevino [14] studied the conjugate-naturalconvection heat transfer along a thin vertical plate withnonuniform internal heat generation Natural convectionflow from an isothermal horizontal circular cylinder in thepresence of heat generation was studied by Molla et al [15]They found that the skin-friction coefficient increases andthe Nusselt number decreases for increasing values of theheat generation parameters Mamun et al [16] investigatedMHD-conjugate heat transfer analysis for a vertical flat

plate in presence of viscous dissipation and heat generationRecently Azim et al [17] analysed viscous Joule heatingMHD-conjugate heat transfer for a vertical flat plate in thepresence of heat generation

In this study MHD-conjugate natural convection flowfrom a horizontal circular cylinder with Joule heating in thepresence of heat generation is considered To the best of ourknowledge this problem has not been considered beforeThe governing equations are transformed into a nondimen-sional form and the resulting nonlinear partial differentialequations are solved numerically using the implicit finitedifference method together with the Keller box technique[18 19]

2 Mathematical Analysis

Let us consider a steady laminar two-dimensional incom-pressible electrically conducting natural convection flowfrom an isothermal horizontal circular cylinder of radius 119886placed on a fluid of uniform temperature 119879

infin(see Figure 1)

The cylinder has a heated core region of temperature 119879119887

and the normal distance from the inner surface to the outersurface is 119887 with 119879

119887gt 119879infin A uniform magnetic field

having strength 1198610is acting normal at the cylinder surface

It is assumed that the fluid properties are constant andthe induced magnetic field is ignored In addition it isalso considered that the boundary layer thickness is verysmall compared with the external radius ldquo119886rdquo of the cylinderThe 119909-axis is taken along the circumference of the cylindermeasured from the lower stagnation point and the 119910-axis istaken normal to the surface Under the balance laws of massmomentum and energy and with the help of Boussinesqapproximation to the body force term in the momentumequation the equations governing this boundary layer nat-ural convection flow can be written as follows

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102+ 119892120573 (119879

119891minus 119879infin) sin(119909

119886) minus

1205901198610

2119906

120588

(2)

119906120597119879119891

120597119909+ V

120597119879119891

120597119910=

120581

120588119888119901

1205972119879119891

1205971199102+1205901198610

21199062

120588119888119901

+1198760

120588119888119901

(119879119891minus 119879infin)

(3)

The appropriate boundary conditions [7 20] of the system are

119906 = V = 0 119879119891= 119879 (119909 0)

120597119879119891

120597119910=

120581119904

119887120581119891

(119879119891minus 119879119887) on 119910 = 0 119909 gt 0

119906 997888rarr 0 119879119891997888rarr 119879infin

as 119910 997888rarr infin 119909 gt 0

(4)

To transform the governing equations (1)ndash(3) and the bound-ary conditions (4) from the (119909 119910) coordinates to the dimen-sionless coordinates (119909 119910) for the free convection dominated

Journal of Computational Methods in Physics 3

B0

B0

B0

B0

B0

B0

B0

T(x 0)

g Tinfin

Tf

yx

u

Tb

b

120601a

Figure 1 Physical model and coordinate system

regime the following nondimensional variables are intro-duced with Grashof number Gr = [1198921205731198863(119879

119887minus 119879infin)]]2

119909 =119909

119886 119910 =

119910

119886Gr14 119906 =

119906119886

]Gr minus12

V =V119886

]Grminus14 120579 =

119879119891minus 119879infin

119879119887minus 119879infin

(5)

where 120579 is the dimensionless temperature The nondimen-sional forms of the governing equations (1)ndash(3) are as follows

120597119906

120597119909+120597V

120597119910= 0 (6)

119906120597119906

120597119909+ V

120597V

120597119910+119872119906 =

1205972119906

1205971199102+ 120579 sin119909 (7)

119906120597120579

120597119909+ V

120597120579

120597119910=1

Pr1205972120579

1205971199102+ 1198691199062+ 119876120579 (8)

where 119872 = 12059011988621198610

2]120588Gr12 is the magnetic parameter 119869 =

120590]1198610

2Gr12120588119888119901(119879119887minus119879infin) is the Joule heating parameter119876 =

11987601198862120583119888119901Gr12 is the heat generation parameter and Pr =

120583119888119901120581 is the Prandtl numberThe boundary condition (4) can be written as in the

following dimensionless form

119906 = V = 0 120579 minus 1 = 120594120597120579

120597119910 on 119910 = 0 119909 gt 0

119906 997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(9)

where 120594 = (119887120581119891119886120581119904)Gr14 is the conjugate conduction para-

meter The present problem is governed by the magnitude ofmagnetic parameter119872 Joule heating parameter 119869 heat gen-eration parameter119876 and conjugate conduction parameter 120594The values of 120594 depend upon the ratios of 119887119886 and 120581

119891120581119904and

Grashof number Gr The ratios 119887119886 and 120581119891120581119904are less than

unity whereas Gr is large for free convection Therefore thevalue of 120594 is greater than zero Present analysis will refer tothe free convection problem without conduction for 120594 = 0

To solve (6)ndash(8) subject to the boundary condition (9)we assume the following transformations

120595 = 119909119891 (119909 119910) 120579 = 120579 (119909 119910) (10)

where 120579 is the dimensionless temperature and 120595 is the streamfunction usually defined as follows

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (11)

Substituting (11) into (6)ndash(9) new forms of the dimensionlessequations (7) and (8) are

1205973119891

1205971199103+ 119891

1205972119891

1205971199102minus (

120597119891

120597119910)

2

minus119872120597119891

120597119910+ 120579

sin119909119909

= 119909(120597119891

120597119910

1205972119891

120597119909120597119910minus1205972119891

1205971199102

120597119891

120597119909)

(12)

1

Pr1205972120579

1205971199102+ 119891

120597120579

120597119910+ 1198691199092(120597119891

120597119910)

2

+ 119876120579 = 119909(120597119891

120597119910

120597120579

120597119909minus120597120579

120597119910

120597119891

120597119909)

(13)

The corresponding boundary conditions as mentioned in (9)take the following form

119891 =120597119891

120597119910= 0 120579 minus 1 = 120594

120597120579

120597119910at 119910 = 0 119909 gt 0

120597119891

120597119910997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(14)

Equations (12) and (13) are solved numerically using implicitfinite difference method using Keller box scheme [18 19]based on the boundary conditions described in (14) Theshearing stress and the rate of heat transfer in terms of skinfriction coefficient and Nusselt number respectively can bewritten as [15]

119862119891=

120591119908

1205881198802infin

Nu =119886119902119908

120581 (119879119908minus 119879infin) (15)

where 120591119908= 120583(120597119906120597119910)

119910=0and 119902119908= minus120581(120597119879120597119910)

119910=0

Using the variables (5) and the boundary conditions into(14) we have

119862119891Gr14 = 11990911989110158401015840 (119909 0) NuGrminus14 = minus1205791015840 (119909 0) (16)

The results of the velocity profiles and temperature distribu-tions can be calculated by the following relations

119906 = 1198911015840(119909 119910) 120579 = 120579 (119909 119910) (17)

In (16) and (17) primes denote differentiation with respect to119910 only

3 Method of Solution

In the present studies we have employed implicit finitedifference method which was introduced by Keller [18]

4 Journal of Computational Methods in Physics

Table 1 Comparisons of the present numerical values with Merkin [1] and Molla et al [15] for different values of 119909 while with Pr = 10119872 = 00 120594 = 00 119869 = 00 and 119876 = 00

119909NuGrminus14 = minus1205791015840(119909 0) 119862

119891Gr14 = 11990911989110158401015840(119909 0)

Merkin [1] Molla et al [15] Present Merkin [1] Molla et al [15] Present00 04214 04214 04216 00000 00000 000001205876 04161 04161 04163 04151 04145 041391205873 04007 04005 04006 07558 07539 075281205872 03745 03740 03742 09579 09541 0952621205873 03364 03355 03356 09756 09696 0967851205876 02825 02812 02811 07822 07739 07718120587 01945 01917 01912 03391 03264 03239

00 05 10 15 20 25 30x

000

020

040

060

080

Skin

fric

tion

181 times 305

361 times 405451 times 505

(a)

000

010

020

030

040

Hea

t tra

nsfe

r

00 05 10 15 20 25 30x

181 times 305

361 times 405451 times 505

(b)

Figure 2 (a)The skin friction coefficient and (b) the rate of heat transfer against 119909 for different mesh configurations while Pr = 10119872 = 01120594 = 10 119869 = 001 and 119876 = 001

and elaborately described by Cebeci and Bradshaw [19]A brief discussion of the development of an algorithm ofimplicit finite difference method together with the Keller boxelimination scheme of the present analysis is given below

Equations (12) and (13) are written in terms of first-orderequations in 119910 which are then expressed in finite differenceform by approximating the functions and their derivatives interms of the central differences in both coordinate directionsDenoting the mesh points in the (119909 119910) plane by 119909

119894and 119910

119895

where 119894 = 1 2 3 181 and 119895 = 1 2 3 305 centraldifference approximations are made such that the equationsinvolving 119909 explicitly are centered at (119909

119894minus12 119910119895minus12

) and theremainder at (119909

119894 119910119895minus12

) where 119910119895minus12

= (119910119895+ 119910119895minus1)2 and so

forth This results in a set of nonlinear difference equationsfor the unknowns at 119909

119894in terms of their values at 119909

119894minus1 These

equations are then linearised by the Newtonrsquos method andare solved using a block-tridiagonal algorithm taken as theinitial iteration of the converged solution at 119909 = 119909

119894minus1 Now

to initiate the process at 119909 = 0 we first provide guess profilesfor all five variables (arising the reduction to the first-order

form) and use the Keller box method to solve the governingordinary differential equations Having obtained the lowerstagnation point solution it is possible to march step by stepalong the boundary layer For a given value of 119909 the iterativeprocedure is stopped when the difference in computing thevelocity and the temperature in the next iteration is less than10minus6 that is when |120575119891119894| le 10minus6 where the superscript denotes

the iteration number The computations were not performedusing a uniform grid in the 119910 direction but a nonuniformgrid was used and defined by 119910

119895= sinh((119895 minus 1)119901) with

119895 = 1 2 305 and 119901 = 100

4 Results and Discussion

The dimensionless governing equations (12) and (13) and theboundary conditions (14) contain a set of physical parame-ters Prandtl number Pr magnetic parameter 119872 conjugateconduction parameter 120594 Joule heating parameter 119869 and heatgeneration parameter119876The Prandtl numbers are consideredto be 163 144 10 and 0733 that correspond to Glycerin

Journal of Computational Methods in Physics 5

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

M = 01M = 03

M = 05M = 07

(a)

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)

Figure 3 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of magnetic parameter119872withPr = 10 120594 = 10 119869 = 001 and 119876 = 001

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

x

Skin

fric

tion

Cfx

00 05 10 15 20 25 30

(a)

M = 01M = 03

M = 05

M = 07

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 4 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119872 with Pr = 10 120594 = 10119869 = 001 and 119876 = 001

water steam and hydrogen respectively The remainingparameters are taken as follows magnetic parameter 119872 =

010ndash070 conjugate conduction parameter 120594 = 10ndash25Joule heating parameter 119869 = 001ndash10 and heat generationparameter 119876 = 001ndash012

A comparison of the local Nusselt number and the localskin friction factor obtained in the present work with 119872 =

00 120594 = 00 119869 = 00 119876 = 00 and Pr = 10 and obtainedby Merkin [1] and Molla et al [15] has been shown in Table 1and it has been observed that there is an excellent agreementamong these three results

A grid independent test has been shown by taking threedifferent grid configurations 181 times 305 361 times 405 and451 times 505 in Figures 2(a) and 2(b) as the skin frictioncoefficient and the rate of heat transfer while Pr = 10119872 = 01 120594 = 10 119869 = 001 and 119876 = 001 From thesefigures it can be concluded that the numerical solutionsare completely independent of the grid orientations In thepresent investigation 181 times 305 grid configuration has beenchosen for the numerical computation

Figures 3 5 7 9 and 11 illustrate the velocity andtemperature distributions at 119909 = 1205872 against 119910 the direction

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

2 Journal of Computational Methods in Physics

showed that the ratio of thermal conductivities of the solidwall to that of fluid within the boundary layer substantiallyinfluenced the flow and heat transfer characteristics

On the other hand the electromagnetic fields are usedto control the heat transfer as in the convection flows andaerodynamic heating and heat is also produced by elec-tromagnetic fields such as MHD generators and pumps Asubstantial quantity of research has been done in the pres-ence of electromagnetic field of the flow and heat transfercharacteristics over a variety of geometries For exampleWilks [8] studiedMHD-free convection about a semi-infinitevertical plate in a strong cross field Hossain [9] studied theeffects of viscous dissipation and Joule heating on MHD-free convection flow with variable plate temperature Heconcluded that dissipative heat reduces the velocity fieldmore in the lower Prandtl number fluid than that of thehigher Prandtl number The author also revealed that thedissipative heat reduced temperature field faster in the higherPrandtl number than that in lower Prandtl number fluidAldoss et al [10] analysed MHD mixed convection from ahorizontal circular cylinder They found that the increase inthe magnetic field leads to a decrease in the flow field a risein the temperature values of the flow field and a decreasein the local wall shear stress and local Nusselt number Thecombined effect of viscous dissipation and Joule heating onMHD forced convection over a nonisothermal horizontalcylinder embedded in a fluid saturated porous medium wasstudied by El-Amin [11]

Many practical heat transfer applications involve theconversion of some form of mechanical electrical nuclearor chemical energy to thermal energy in the medium Suchmediums are said to involve internal heat generation Forexample a large amount of heat is generated in the fuelelements from atomic reactors as a result of atomic fissionthat serves as the heat source for the nuclear power plantsTheheat generated in the sun as a result of fusion of hydrogen intohelium makes the sun a large nuclear reactor that suppliesheat to the earth Possible heat generation effects maymodifytemperature distribution and therefore the particle deposi-tion rate The heat transfer in a laminar boundary layer flowof a viscous fluid over a linearly stretching continuous surfacewith viscous dissipationfrictional heating and internal heatgeneration was analysed by Vajravelu and Hadjinicolaou[12] They considered the volumetric rate of heat generation119902101584010158401015840[Wm3] as 119902101584010158401015840 = 119876

0(119879119891minus 119879infin) for 119879

119891ge 119879infin

and 119902101584010158401015840 = 0for 119879119891lt 119879infin where 119876

0is the heat generation constant The

effects of heat generationabsorption and the thermophoresison hydromagnetic flow with heat and mass transfer overa flat plate were investigated by Chamkha and Camille[13] Mendez and Trevino [14] studied the conjugate-naturalconvection heat transfer along a thin vertical plate withnonuniform internal heat generation Natural convectionflow from an isothermal horizontal circular cylinder in thepresence of heat generation was studied by Molla et al [15]They found that the skin-friction coefficient increases andthe Nusselt number decreases for increasing values of theheat generation parameters Mamun et al [16] investigatedMHD-conjugate heat transfer analysis for a vertical flat

plate in presence of viscous dissipation and heat generationRecently Azim et al [17] analysed viscous Joule heatingMHD-conjugate heat transfer for a vertical flat plate in thepresence of heat generation

In this study MHD-conjugate natural convection flowfrom a horizontal circular cylinder with Joule heating in thepresence of heat generation is considered To the best of ourknowledge this problem has not been considered beforeThe governing equations are transformed into a nondimen-sional form and the resulting nonlinear partial differentialequations are solved numerically using the implicit finitedifference method together with the Keller box technique[18 19]

2 Mathematical Analysis

Let us consider a steady laminar two-dimensional incom-pressible electrically conducting natural convection flowfrom an isothermal horizontal circular cylinder of radius 119886placed on a fluid of uniform temperature 119879

infin(see Figure 1)

The cylinder has a heated core region of temperature 119879119887

and the normal distance from the inner surface to the outersurface is 119887 with 119879

119887gt 119879infin A uniform magnetic field

having strength 1198610is acting normal at the cylinder surface

It is assumed that the fluid properties are constant andthe induced magnetic field is ignored In addition it isalso considered that the boundary layer thickness is verysmall compared with the external radius ldquo119886rdquo of the cylinderThe 119909-axis is taken along the circumference of the cylindermeasured from the lower stagnation point and the 119910-axis istaken normal to the surface Under the balance laws of massmomentum and energy and with the help of Boussinesqapproximation to the body force term in the momentumequation the equations governing this boundary layer nat-ural convection flow can be written as follows

120597119906

120597119909+120597V

120597119910= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102+ 119892120573 (119879

119891minus 119879infin) sin(119909

119886) minus

1205901198610

2119906

120588

(2)

119906120597119879119891

120597119909+ V

120597119879119891

120597119910=

120581

120588119888119901

1205972119879119891

1205971199102+1205901198610

21199062

120588119888119901

+1198760

120588119888119901

(119879119891minus 119879infin)

(3)

The appropriate boundary conditions [7 20] of the system are

119906 = V = 0 119879119891= 119879 (119909 0)

120597119879119891

120597119910=

120581119904

119887120581119891

(119879119891minus 119879119887) on 119910 = 0 119909 gt 0

119906 997888rarr 0 119879119891997888rarr 119879infin

as 119910 997888rarr infin 119909 gt 0

(4)

To transform the governing equations (1)ndash(3) and the bound-ary conditions (4) from the (119909 119910) coordinates to the dimen-sionless coordinates (119909 119910) for the free convection dominated

Journal of Computational Methods in Physics 3

B0

B0

B0

B0

B0

B0

B0

T(x 0)

g Tinfin

Tf

yx

u

Tb

b

120601a

Figure 1 Physical model and coordinate system

regime the following nondimensional variables are intro-duced with Grashof number Gr = [1198921205731198863(119879

119887minus 119879infin)]]2

119909 =119909

119886 119910 =

119910

119886Gr14 119906 =

119906119886

]Gr minus12

V =V119886

]Grminus14 120579 =

119879119891minus 119879infin

119879119887minus 119879infin

(5)

where 120579 is the dimensionless temperature The nondimen-sional forms of the governing equations (1)ndash(3) are as follows

120597119906

120597119909+120597V

120597119910= 0 (6)

119906120597119906

120597119909+ V

120597V

120597119910+119872119906 =

1205972119906

1205971199102+ 120579 sin119909 (7)

119906120597120579

120597119909+ V

120597120579

120597119910=1

Pr1205972120579

1205971199102+ 1198691199062+ 119876120579 (8)

where 119872 = 12059011988621198610

2]120588Gr12 is the magnetic parameter 119869 =

120590]1198610

2Gr12120588119888119901(119879119887minus119879infin) is the Joule heating parameter119876 =

11987601198862120583119888119901Gr12 is the heat generation parameter and Pr =

120583119888119901120581 is the Prandtl numberThe boundary condition (4) can be written as in the

following dimensionless form

119906 = V = 0 120579 minus 1 = 120594120597120579

120597119910 on 119910 = 0 119909 gt 0

119906 997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(9)

where 120594 = (119887120581119891119886120581119904)Gr14 is the conjugate conduction para-

meter The present problem is governed by the magnitude ofmagnetic parameter119872 Joule heating parameter 119869 heat gen-eration parameter119876 and conjugate conduction parameter 120594The values of 120594 depend upon the ratios of 119887119886 and 120581

119891120581119904and

Grashof number Gr The ratios 119887119886 and 120581119891120581119904are less than

unity whereas Gr is large for free convection Therefore thevalue of 120594 is greater than zero Present analysis will refer tothe free convection problem without conduction for 120594 = 0

To solve (6)ndash(8) subject to the boundary condition (9)we assume the following transformations

120595 = 119909119891 (119909 119910) 120579 = 120579 (119909 119910) (10)

where 120579 is the dimensionless temperature and 120595 is the streamfunction usually defined as follows

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (11)

Substituting (11) into (6)ndash(9) new forms of the dimensionlessequations (7) and (8) are

1205973119891

1205971199103+ 119891

1205972119891

1205971199102minus (

120597119891

120597119910)

2

minus119872120597119891

120597119910+ 120579

sin119909119909

= 119909(120597119891

120597119910

1205972119891

120597119909120597119910minus1205972119891

1205971199102

120597119891

120597119909)

(12)

1

Pr1205972120579

1205971199102+ 119891

120597120579

120597119910+ 1198691199092(120597119891

120597119910)

2

+ 119876120579 = 119909(120597119891

120597119910

120597120579

120597119909minus120597120579

120597119910

120597119891

120597119909)

(13)

The corresponding boundary conditions as mentioned in (9)take the following form

119891 =120597119891

120597119910= 0 120579 minus 1 = 120594

120597120579

120597119910at 119910 = 0 119909 gt 0

120597119891

120597119910997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(14)

Equations (12) and (13) are solved numerically using implicitfinite difference method using Keller box scheme [18 19]based on the boundary conditions described in (14) Theshearing stress and the rate of heat transfer in terms of skinfriction coefficient and Nusselt number respectively can bewritten as [15]

119862119891=

120591119908

1205881198802infin

Nu =119886119902119908

120581 (119879119908minus 119879infin) (15)

where 120591119908= 120583(120597119906120597119910)

119910=0and 119902119908= minus120581(120597119879120597119910)

119910=0

Using the variables (5) and the boundary conditions into(14) we have

119862119891Gr14 = 11990911989110158401015840 (119909 0) NuGrminus14 = minus1205791015840 (119909 0) (16)

The results of the velocity profiles and temperature distribu-tions can be calculated by the following relations

119906 = 1198911015840(119909 119910) 120579 = 120579 (119909 119910) (17)

In (16) and (17) primes denote differentiation with respect to119910 only

3 Method of Solution

In the present studies we have employed implicit finitedifference method which was introduced by Keller [18]

4 Journal of Computational Methods in Physics

Table 1 Comparisons of the present numerical values with Merkin [1] and Molla et al [15] for different values of 119909 while with Pr = 10119872 = 00 120594 = 00 119869 = 00 and 119876 = 00

119909NuGrminus14 = minus1205791015840(119909 0) 119862

119891Gr14 = 11990911989110158401015840(119909 0)

Merkin [1] Molla et al [15] Present Merkin [1] Molla et al [15] Present00 04214 04214 04216 00000 00000 000001205876 04161 04161 04163 04151 04145 041391205873 04007 04005 04006 07558 07539 075281205872 03745 03740 03742 09579 09541 0952621205873 03364 03355 03356 09756 09696 0967851205876 02825 02812 02811 07822 07739 07718120587 01945 01917 01912 03391 03264 03239

00 05 10 15 20 25 30x

000

020

040

060

080

Skin

fric

tion

181 times 305

361 times 405451 times 505

(a)

000

010

020

030

040

Hea

t tra

nsfe

r

00 05 10 15 20 25 30x

181 times 305

361 times 405451 times 505

(b)

Figure 2 (a)The skin friction coefficient and (b) the rate of heat transfer against 119909 for different mesh configurations while Pr = 10119872 = 01120594 = 10 119869 = 001 and 119876 = 001

and elaborately described by Cebeci and Bradshaw [19]A brief discussion of the development of an algorithm ofimplicit finite difference method together with the Keller boxelimination scheme of the present analysis is given below

Equations (12) and (13) are written in terms of first-orderequations in 119910 which are then expressed in finite differenceform by approximating the functions and their derivatives interms of the central differences in both coordinate directionsDenoting the mesh points in the (119909 119910) plane by 119909

119894and 119910

119895

where 119894 = 1 2 3 181 and 119895 = 1 2 3 305 centraldifference approximations are made such that the equationsinvolving 119909 explicitly are centered at (119909

119894minus12 119910119895minus12

) and theremainder at (119909

119894 119910119895minus12

) where 119910119895minus12

= (119910119895+ 119910119895minus1)2 and so

forth This results in a set of nonlinear difference equationsfor the unknowns at 119909

119894in terms of their values at 119909

119894minus1 These

equations are then linearised by the Newtonrsquos method andare solved using a block-tridiagonal algorithm taken as theinitial iteration of the converged solution at 119909 = 119909

119894minus1 Now

to initiate the process at 119909 = 0 we first provide guess profilesfor all five variables (arising the reduction to the first-order

form) and use the Keller box method to solve the governingordinary differential equations Having obtained the lowerstagnation point solution it is possible to march step by stepalong the boundary layer For a given value of 119909 the iterativeprocedure is stopped when the difference in computing thevelocity and the temperature in the next iteration is less than10minus6 that is when |120575119891119894| le 10minus6 where the superscript denotes

the iteration number The computations were not performedusing a uniform grid in the 119910 direction but a nonuniformgrid was used and defined by 119910

119895= sinh((119895 minus 1)119901) with

119895 = 1 2 305 and 119901 = 100

4 Results and Discussion

The dimensionless governing equations (12) and (13) and theboundary conditions (14) contain a set of physical parame-ters Prandtl number Pr magnetic parameter 119872 conjugateconduction parameter 120594 Joule heating parameter 119869 and heatgeneration parameter119876The Prandtl numbers are consideredto be 163 144 10 and 0733 that correspond to Glycerin

Journal of Computational Methods in Physics 5

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

M = 01M = 03

M = 05M = 07

(a)

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)

Figure 3 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of magnetic parameter119872withPr = 10 120594 = 10 119869 = 001 and 119876 = 001

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

x

Skin

fric

tion

Cfx

00 05 10 15 20 25 30

(a)

M = 01M = 03

M = 05

M = 07

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 4 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119872 with Pr = 10 120594 = 10119869 = 001 and 119876 = 001

water steam and hydrogen respectively The remainingparameters are taken as follows magnetic parameter 119872 =

010ndash070 conjugate conduction parameter 120594 = 10ndash25Joule heating parameter 119869 = 001ndash10 and heat generationparameter 119876 = 001ndash012

A comparison of the local Nusselt number and the localskin friction factor obtained in the present work with 119872 =

00 120594 = 00 119869 = 00 119876 = 00 and Pr = 10 and obtainedby Merkin [1] and Molla et al [15] has been shown in Table 1and it has been observed that there is an excellent agreementamong these three results

A grid independent test has been shown by taking threedifferent grid configurations 181 times 305 361 times 405 and451 times 505 in Figures 2(a) and 2(b) as the skin frictioncoefficient and the rate of heat transfer while Pr = 10119872 = 01 120594 = 10 119869 = 001 and 119876 = 001 From thesefigures it can be concluded that the numerical solutionsare completely independent of the grid orientations In thepresent investigation 181 times 305 grid configuration has beenchosen for the numerical computation

Figures 3 5 7 9 and 11 illustrate the velocity andtemperature distributions at 119909 = 1205872 against 119910 the direction

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Journal of Computational Methods in Physics 3

B0

B0

B0

B0

B0

B0

B0

T(x 0)

g Tinfin

Tf

yx

u

Tb

b

120601a

Figure 1 Physical model and coordinate system

regime the following nondimensional variables are intro-duced with Grashof number Gr = [1198921205731198863(119879

119887minus 119879infin)]]2

119909 =119909

119886 119910 =

119910

119886Gr14 119906 =

119906119886

]Gr minus12

V =V119886

]Grminus14 120579 =

119879119891minus 119879infin

119879119887minus 119879infin

(5)

where 120579 is the dimensionless temperature The nondimen-sional forms of the governing equations (1)ndash(3) are as follows

120597119906

120597119909+120597V

120597119910= 0 (6)

119906120597119906

120597119909+ V

120597V

120597119910+119872119906 =

1205972119906

1205971199102+ 120579 sin119909 (7)

119906120597120579

120597119909+ V

120597120579

120597119910=1

Pr1205972120579

1205971199102+ 1198691199062+ 119876120579 (8)

where 119872 = 12059011988621198610

2]120588Gr12 is the magnetic parameter 119869 =

120590]1198610

2Gr12120588119888119901(119879119887minus119879infin) is the Joule heating parameter119876 =

11987601198862120583119888119901Gr12 is the heat generation parameter and Pr =

120583119888119901120581 is the Prandtl numberThe boundary condition (4) can be written as in the

following dimensionless form

119906 = V = 0 120579 minus 1 = 120594120597120579

120597119910 on 119910 = 0 119909 gt 0

119906 997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(9)

where 120594 = (119887120581119891119886120581119904)Gr14 is the conjugate conduction para-

meter The present problem is governed by the magnitude ofmagnetic parameter119872 Joule heating parameter 119869 heat gen-eration parameter119876 and conjugate conduction parameter 120594The values of 120594 depend upon the ratios of 119887119886 and 120581

119891120581119904and

Grashof number Gr The ratios 119887119886 and 120581119891120581119904are less than

unity whereas Gr is large for free convection Therefore thevalue of 120594 is greater than zero Present analysis will refer tothe free convection problem without conduction for 120594 = 0

To solve (6)ndash(8) subject to the boundary condition (9)we assume the following transformations

120595 = 119909119891 (119909 119910) 120579 = 120579 (119909 119910) (10)

where 120579 is the dimensionless temperature and 120595 is the streamfunction usually defined as follows

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (11)

Substituting (11) into (6)ndash(9) new forms of the dimensionlessequations (7) and (8) are

1205973119891

1205971199103+ 119891

1205972119891

1205971199102minus (

120597119891

120597119910)

2

minus119872120597119891

120597119910+ 120579

sin119909119909

= 119909(120597119891

120597119910

1205972119891

120597119909120597119910minus1205972119891

1205971199102

120597119891

120597119909)

(12)

1

Pr1205972120579

1205971199102+ 119891

120597120579

120597119910+ 1198691199092(120597119891

120597119910)

2

+ 119876120579 = 119909(120597119891

120597119910

120597120579

120597119909minus120597120579

120597119910

120597119891

120597119909)

(13)

The corresponding boundary conditions as mentioned in (9)take the following form

119891 =120597119891

120597119910= 0 120579 minus 1 = 120594

120597120579

120597119910at 119910 = 0 119909 gt 0

120597119891

120597119910997888rarr 0 120579 997888rarr 0 as 119910 997888rarr infin 119909 gt 0

(14)

Equations (12) and (13) are solved numerically using implicitfinite difference method using Keller box scheme [18 19]based on the boundary conditions described in (14) Theshearing stress and the rate of heat transfer in terms of skinfriction coefficient and Nusselt number respectively can bewritten as [15]

119862119891=

120591119908

1205881198802infin

Nu =119886119902119908

120581 (119879119908minus 119879infin) (15)

where 120591119908= 120583(120597119906120597119910)

119910=0and 119902119908= minus120581(120597119879120597119910)

119910=0

Using the variables (5) and the boundary conditions into(14) we have

119862119891Gr14 = 11990911989110158401015840 (119909 0) NuGrminus14 = minus1205791015840 (119909 0) (16)

The results of the velocity profiles and temperature distribu-tions can be calculated by the following relations

119906 = 1198911015840(119909 119910) 120579 = 120579 (119909 119910) (17)

In (16) and (17) primes denote differentiation with respect to119910 only

3 Method of Solution

In the present studies we have employed implicit finitedifference method which was introduced by Keller [18]

4 Journal of Computational Methods in Physics

Table 1 Comparisons of the present numerical values with Merkin [1] and Molla et al [15] for different values of 119909 while with Pr = 10119872 = 00 120594 = 00 119869 = 00 and 119876 = 00

119909NuGrminus14 = minus1205791015840(119909 0) 119862

119891Gr14 = 11990911989110158401015840(119909 0)

Merkin [1] Molla et al [15] Present Merkin [1] Molla et al [15] Present00 04214 04214 04216 00000 00000 000001205876 04161 04161 04163 04151 04145 041391205873 04007 04005 04006 07558 07539 075281205872 03745 03740 03742 09579 09541 0952621205873 03364 03355 03356 09756 09696 0967851205876 02825 02812 02811 07822 07739 07718120587 01945 01917 01912 03391 03264 03239

00 05 10 15 20 25 30x

000

020

040

060

080

Skin

fric

tion

181 times 305

361 times 405451 times 505

(a)

000

010

020

030

040

Hea

t tra

nsfe

r

00 05 10 15 20 25 30x

181 times 305

361 times 405451 times 505

(b)

Figure 2 (a)The skin friction coefficient and (b) the rate of heat transfer against 119909 for different mesh configurations while Pr = 10119872 = 01120594 = 10 119869 = 001 and 119876 = 001

and elaborately described by Cebeci and Bradshaw [19]A brief discussion of the development of an algorithm ofimplicit finite difference method together with the Keller boxelimination scheme of the present analysis is given below

Equations (12) and (13) are written in terms of first-orderequations in 119910 which are then expressed in finite differenceform by approximating the functions and their derivatives interms of the central differences in both coordinate directionsDenoting the mesh points in the (119909 119910) plane by 119909

119894and 119910

119895

where 119894 = 1 2 3 181 and 119895 = 1 2 3 305 centraldifference approximations are made such that the equationsinvolving 119909 explicitly are centered at (119909

119894minus12 119910119895minus12

) and theremainder at (119909

119894 119910119895minus12

) where 119910119895minus12

= (119910119895+ 119910119895minus1)2 and so

forth This results in a set of nonlinear difference equationsfor the unknowns at 119909

119894in terms of their values at 119909

119894minus1 These

equations are then linearised by the Newtonrsquos method andare solved using a block-tridiagonal algorithm taken as theinitial iteration of the converged solution at 119909 = 119909

119894minus1 Now

to initiate the process at 119909 = 0 we first provide guess profilesfor all five variables (arising the reduction to the first-order

form) and use the Keller box method to solve the governingordinary differential equations Having obtained the lowerstagnation point solution it is possible to march step by stepalong the boundary layer For a given value of 119909 the iterativeprocedure is stopped when the difference in computing thevelocity and the temperature in the next iteration is less than10minus6 that is when |120575119891119894| le 10minus6 where the superscript denotes

the iteration number The computations were not performedusing a uniform grid in the 119910 direction but a nonuniformgrid was used and defined by 119910

119895= sinh((119895 minus 1)119901) with

119895 = 1 2 305 and 119901 = 100

4 Results and Discussion

The dimensionless governing equations (12) and (13) and theboundary conditions (14) contain a set of physical parame-ters Prandtl number Pr magnetic parameter 119872 conjugateconduction parameter 120594 Joule heating parameter 119869 and heatgeneration parameter119876The Prandtl numbers are consideredto be 163 144 10 and 0733 that correspond to Glycerin

Journal of Computational Methods in Physics 5

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

M = 01M = 03

M = 05M = 07

(a)

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)

Figure 3 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of magnetic parameter119872withPr = 10 120594 = 10 119869 = 001 and 119876 = 001

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

x

Skin

fric

tion

Cfx

00 05 10 15 20 25 30

(a)

M = 01M = 03

M = 05

M = 07

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 4 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119872 with Pr = 10 120594 = 10119869 = 001 and 119876 = 001

water steam and hydrogen respectively The remainingparameters are taken as follows magnetic parameter 119872 =

010ndash070 conjugate conduction parameter 120594 = 10ndash25Joule heating parameter 119869 = 001ndash10 and heat generationparameter 119876 = 001ndash012

A comparison of the local Nusselt number and the localskin friction factor obtained in the present work with 119872 =

00 120594 = 00 119869 = 00 119876 = 00 and Pr = 10 and obtainedby Merkin [1] and Molla et al [15] has been shown in Table 1and it has been observed that there is an excellent agreementamong these three results

A grid independent test has been shown by taking threedifferent grid configurations 181 times 305 361 times 405 and451 times 505 in Figures 2(a) and 2(b) as the skin frictioncoefficient and the rate of heat transfer while Pr = 10119872 = 01 120594 = 10 119869 = 001 and 119876 = 001 From thesefigures it can be concluded that the numerical solutionsare completely independent of the grid orientations In thepresent investigation 181 times 305 grid configuration has beenchosen for the numerical computation

Figures 3 5 7 9 and 11 illustrate the velocity andtemperature distributions at 119909 = 1205872 against 119910 the direction

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

4 Journal of Computational Methods in Physics

Table 1 Comparisons of the present numerical values with Merkin [1] and Molla et al [15] for different values of 119909 while with Pr = 10119872 = 00 120594 = 00 119869 = 00 and 119876 = 00

119909NuGrminus14 = minus1205791015840(119909 0) 119862

119891Gr14 = 11990911989110158401015840(119909 0)

Merkin [1] Molla et al [15] Present Merkin [1] Molla et al [15] Present00 04214 04214 04216 00000 00000 000001205876 04161 04161 04163 04151 04145 041391205873 04007 04005 04006 07558 07539 075281205872 03745 03740 03742 09579 09541 0952621205873 03364 03355 03356 09756 09696 0967851205876 02825 02812 02811 07822 07739 07718120587 01945 01917 01912 03391 03264 03239

00 05 10 15 20 25 30x

000

020

040

060

080

Skin

fric

tion

181 times 305

361 times 405451 times 505

(a)

000

010

020

030

040

Hea

t tra

nsfe

r

00 05 10 15 20 25 30x

181 times 305

361 times 405451 times 505

(b)

Figure 2 (a)The skin friction coefficient and (b) the rate of heat transfer against 119909 for different mesh configurations while Pr = 10119872 = 01120594 = 10 119869 = 001 and 119876 = 001

and elaborately described by Cebeci and Bradshaw [19]A brief discussion of the development of an algorithm ofimplicit finite difference method together with the Keller boxelimination scheme of the present analysis is given below

Equations (12) and (13) are written in terms of first-orderequations in 119910 which are then expressed in finite differenceform by approximating the functions and their derivatives interms of the central differences in both coordinate directionsDenoting the mesh points in the (119909 119910) plane by 119909

119894and 119910

119895

where 119894 = 1 2 3 181 and 119895 = 1 2 3 305 centraldifference approximations are made such that the equationsinvolving 119909 explicitly are centered at (119909

119894minus12 119910119895minus12

) and theremainder at (119909

119894 119910119895minus12

) where 119910119895minus12

= (119910119895+ 119910119895minus1)2 and so

forth This results in a set of nonlinear difference equationsfor the unknowns at 119909

119894in terms of their values at 119909

119894minus1 These

equations are then linearised by the Newtonrsquos method andare solved using a block-tridiagonal algorithm taken as theinitial iteration of the converged solution at 119909 = 119909

119894minus1 Now

to initiate the process at 119909 = 0 we first provide guess profilesfor all five variables (arising the reduction to the first-order

form) and use the Keller box method to solve the governingordinary differential equations Having obtained the lowerstagnation point solution it is possible to march step by stepalong the boundary layer For a given value of 119909 the iterativeprocedure is stopped when the difference in computing thevelocity and the temperature in the next iteration is less than10minus6 that is when |120575119891119894| le 10minus6 where the superscript denotes

the iteration number The computations were not performedusing a uniform grid in the 119910 direction but a nonuniformgrid was used and defined by 119910

119895= sinh((119895 minus 1)119901) with

119895 = 1 2 305 and 119901 = 100

4 Results and Discussion

The dimensionless governing equations (12) and (13) and theboundary conditions (14) contain a set of physical parame-ters Prandtl number Pr magnetic parameter 119872 conjugateconduction parameter 120594 Joule heating parameter 119869 and heatgeneration parameter119876The Prandtl numbers are consideredto be 163 144 10 and 0733 that correspond to Glycerin

Journal of Computational Methods in Physics 5

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

M = 01M = 03

M = 05M = 07

(a)

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)

Figure 3 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of magnetic parameter119872withPr = 10 120594 = 10 119869 = 001 and 119876 = 001

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

x

Skin

fric

tion

Cfx

00 05 10 15 20 25 30

(a)

M = 01M = 03

M = 05

M = 07

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 4 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119872 with Pr = 10 120594 = 10119869 = 001 and 119876 = 001

water steam and hydrogen respectively The remainingparameters are taken as follows magnetic parameter 119872 =

010ndash070 conjugate conduction parameter 120594 = 10ndash25Joule heating parameter 119869 = 001ndash10 and heat generationparameter 119876 = 001ndash012

A comparison of the local Nusselt number and the localskin friction factor obtained in the present work with 119872 =

00 120594 = 00 119869 = 00 119876 = 00 and Pr = 10 and obtainedby Merkin [1] and Molla et al [15] has been shown in Table 1and it has been observed that there is an excellent agreementamong these three results

A grid independent test has been shown by taking threedifferent grid configurations 181 times 305 361 times 405 and451 times 505 in Figures 2(a) and 2(b) as the skin frictioncoefficient and the rate of heat transfer while Pr = 10119872 = 01 120594 = 10 119869 = 001 and 119876 = 001 From thesefigures it can be concluded that the numerical solutionsare completely independent of the grid orientations In thepresent investigation 181 times 305 grid configuration has beenchosen for the numerical computation

Figures 3 5 7 9 and 11 illustrate the velocity andtemperature distributions at 119909 = 1205872 against 119910 the direction

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Journal of Computational Methods in Physics 5

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

M = 01M = 03

M = 05M = 07

(a)

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)

Figure 3 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of magnetic parameter119872withPr = 10 120594 = 10 119869 = 001 and 119876 = 001

M = 01M = 03

M = 05

M = 07

000

020

040

060

080

x

Skin

fric

tion

Cfx

00 05 10 15 20 25 30

(a)

M = 01M = 03

M = 05

M = 07

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 4 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119872 with Pr = 10 120594 = 10119869 = 001 and 119876 = 001

water steam and hydrogen respectively The remainingparameters are taken as follows magnetic parameter 119872 =

010ndash070 conjugate conduction parameter 120594 = 10ndash25Joule heating parameter 119869 = 001ndash10 and heat generationparameter 119876 = 001ndash012

A comparison of the local Nusselt number and the localskin friction factor obtained in the present work with 119872 =

00 120594 = 00 119869 = 00 119876 = 00 and Pr = 10 and obtainedby Merkin [1] and Molla et al [15] has been shown in Table 1and it has been observed that there is an excellent agreementamong these three results

A grid independent test has been shown by taking threedifferent grid configurations 181 times 305 361 times 405 and451 times 505 in Figures 2(a) and 2(b) as the skin frictioncoefficient and the rate of heat transfer while Pr = 10119872 = 01 120594 = 10 119869 = 001 and 119876 = 001 From thesefigures it can be concluded that the numerical solutionsare completely independent of the grid orientations In thepresent investigation 181 times 305 grid configuration has beenchosen for the numerical computation

Figures 3 5 7 9 and 11 illustrate the velocity andtemperature distributions at 119909 = 1205872 against 119910 the direction

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

6 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

120594 = 10120594 = 15

120594 = 20120594 = 25

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

120594 = 10120594 = 15

120594 = 20120594 = 25

(b)

Figure 5 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

120594 = 10120594 = 15

120594 = 20120594 = 25

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 6 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of conjugate conductionparameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001

along the normal to the surface of the cylinder and Figures 46 8 10 and 12 depict the skin friction coefficients and heattransfer rates against 119909 at 119910 = 0 (along the surface of thecylinder) for different values of the magnetic parameter con-jugate conduction parameter Prandtl number Joule heatingparameter and heat generation parameter respectively

Themagnetic parameter is the ratio of the magnetic forceto the inertia force Hence the magnetic force is importantwhen it is the order of one and the flow is considered ashydromagnetic flow The flow is hydrodynamic for 119872 ≪

1 For small value of 119872 the motion is hardly affected by

the magnetic field and for large value of 119872 the motion islargely controlled by themagnetic fieldThe increasing valuesof the magnetic parameters increase magnetic-field strengthwhich is acting normal to the cylinder surface that reducesfluid motion as observed in Figure 3(a) As a result the skinfriction at the surface to the cylinder is decreased which isshown from Figure 4(a) Heat is produced due to the interac-tion between magnetic field and fluid motion consequentlytemperature within the thermal boundary layer increases forincreasing value of the magnetic parameters as revealed fromFigure 3(b) Increasing thermal energy within the boundary

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Journal of Computational Methods in Physics 7

00 15 30 45 60 75 90y

000

008

016

024

032Ve

loci

tyf

998400

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

(b)

Figure 7 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

030

Hea

t tra

nsfe

r

Pr = 0733Pr = 1000

Pr = 1440Pr = 1630

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 8 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of Pr with119872 = 01 120594 = 10119869 = 001 and 119876 = 001

layer reduces the temperature difference between core regionand boundary layer region which ultimately decreases theheat transfer rate as illustrated in Figure 4(b)

The velocity and temperature are illustrated in Figure 5and the variation of the local skin friction coefficient andlocal rate of heat transfer are depicted in Figure 6 for differentvalues of conjugate conduction parameter 120594 with Pr = 10119872 = 01 119869 = 001 and 119876 = 001 Increasing value of conju-gate conduction parameter120594 resists conduction from the core

region to the boundary layer and consequently deceleratesconvection within the boundary layer as a result velocity andtemperature decrease for increasing values of the conjugateconduction parameters at a particular value of 119910 whichare presented in Figures 5(a) and 5(b) respectively As thevelocity decreases the skin friction at the surface decreasesfor increasing value of conjugate conduction parameter 120594as observed in Figure 6(a) Since increasing value of theconjugate conduction parameters resists conduction from

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

8 Journal of Computational Methods in Physics

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

J = 001J = 040

J = 070J = 100

(a)

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

J = 001J = 040

J = 070J = 100

(b)

Figure 9 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119869 with Pr = 10119872 = 01120594 = 10 and 119876 = 001

Skin

fric

tion

Cfx

00

015

030

045

060

075

090

J = 001J = 040

J = 070J = 100

x

00 05 10 15 20 25 30

(a)

J = 001

J = 010J = 020J = 030

000

005

010

015

020

025

Hea

t tra

nsfe

r

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 10 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of 119869with Pr = 10119872 = 01120594 = 10 and 119876 = 001

the core region to the boundary layer as mentioned earlierit of course resists thermal energy transfer which is observedfrom Figure 6(b)

The increasing value of Prandtl number increases vis-cosity and decreases the thermal action of the fluid There-fore the velocity and temperature of fluid are expectedto decrease with the increasing Prandtl number which areobserved in Figures 7(a) and 7(b) respectively Decreasingvelocity of the fluid leads to decrease skin friction andthe decreasing temperature within in the boundary layer

increases the temperature difference between core (inner)region and boundary layer region which eventually increasesheat transfer rate from the core region to the boundary layerregion as depicted in Figures 8(a) and 8(b) respectively

The effects of the Joule heating parameters on the velocityand temperature are presented in Figure 9 and on the skinfriction and rate of heat transfer they are illustrated inFigure 10 respectively with Pr = 10 119872 = 01 120594 =

10 and 119876 = 001 The Joule heating parameter havingthe magnetic-field strength increases the temperature and

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Journal of Computational Methods in Physics 9

00 15 30 45 60 75 90y

000

008

016

024

032

Velo

city

f998400

Q = 001Q = 005

Q = 008Q = 012

(a)

Q = 001Q = 005

Q = 008Q = 012

000

020

040

060

080

Tem

pera

ture

120579

00 15 30 45 60 75 90y

(b)Figure 11 (a) Variation of velocity profiles and (b) variation of temperature distributions against 119910 for varying of 119876 with Pr = 10119872 = 01120594 = 10 and 119869 = 001

Skin

fric

tion

Cfx

000

020

040

060

080

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

(a)

000

005

010

015

020

025

Hea

t tra

nsfe

r

Q = 001Q = 002

Q = 003

Q = 004

x

00 05 10 15 20 25 30

Nu x

(b)

Figure 12 (a) Variation of skin friction coefficients and (b) variation of rate of heat transfer against 119909 for varying of119876with Pr = 10119872 = 01120594 = 10 and 119869 = 001

eventually the fluidmotion is accelerated as plotted in Figures9(b) and 9(a) respectively The increased temperature forincreasing Joule heating parameter increases the temperaturewithin the boundary layer which results in the heat transferrate decrease as illustrated in Figure 10(b) The variation oflocal skin friction coefficient increases for the increasing 119869 asdepicted in Figure 10(a) This is an expected behavior as fluidmotion increases for increasing Joule heating parameter

Figure 11 illustrates the effect of the heat generationparameters on the fluid velocity and temperature profilesrespectively It is clear that as the heat generation parameter

increases both the fluid velocity and temperature of the fluidincrease Figure 12 depicts the variation of the heat generationparameters on the skin friction coefficient and the heattransfer rate with Pr = 10119872 = 01 120594 = 10 and 119869 = 001It is observed that the local Nusselt number decreases butthe skin friction coefficient increases with increasing heatgeneration parameter

5 Conclusion

MHD-conjugate natural convection flow from horizontalcylinder taking into account Joule heating in the presence of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

10 Journal of Computational Methods in Physics

heat generation is studied Numerical results were obtainedfor the physical parameters and discussed From the resultsit is established that the velocity of the fluidwithin the bound-ary layer decreases with increasing magnetic parameterconjugate conduction parameter and Prandtl number whileit increases for increasing Joule heating parameter and heatgeneration parameter The temperature within the boundarylayer increases for increasing magnetic parameter Jouleheating parameter and heat generation parameter whereas itdecreases for increasing conjugate conduction parameter andPrandtl numberMoreover the skin friction along the surfaceof the cylinder decreases for increasing magnetic parameterconjugate conduction parameter and Prandtl number andit increases for increasing Joule heating parameter and heatgeneration parameter Furthermore the rate of heat transferalong the surface increases for increasing Prandtl numberwhile it decreases for remaining parameters

Nomenclature

119886 Outer radius of the cylinder119887 Thickness of the cylinder1198610 Applied magnetic field

119862119891119909 Skin friction coefficient

119888119901 Specific heat

119891 Dimensionless stream function119892 Acceleration due to gravity119869 Joule heating parameter119872 Magnetic parameterNu119909 Local Nusselt number

Pr Prandtl number119876 Heat generation parameter119879119891 Temperature at the boundary layer region

119879119904 Temperature of the solid of the cylinder

119879119887 Temperature of the inner cylinder

119879infin Temperature of the ambient fluid

119906 V Velocity components119906 V Dimensionless velocity components119880infin Dimensionless free stream velocity

119909 119910 Cartesian coordinates120573 Coefficient of thermal expansion120594 Conjugate conduction parameter120595 Dimensionless stream function120588 Density of the fluid] Kinematic viscosity120583 Viscosity of the fluid120579 Dimensionless temperature120590 Electrical conductivity120581119891 Thermal conductivity of the ambient fluid

120581119904 Thermal conductivity of the ambient solid

References

[1] J HMerkin ldquoFree convection boundary-layer on an isothermalhorizontal cylinderrdquo in Proceedings of the ASME-AIChE HeatTransfer Conference pp 1ndash4 St Louis Miss USA 1976

[2] T H Kuehn and R J Goldstein ldquoNumerical solution to theNavier-Stokes equations for laminar natural convection about

a horizontal isothermal circular cylinderrdquo International Journalof Heat and Mass Transfer vol 23 no 7 pp 971ndash979 1980

[3] P Wang R Kahawita and T H Nguyen ldquoNumerical computa-tion of the natural convection flow about a horizontal cylinderusing splinesrdquoNumerical Heat Transfer A vol 17 no 2 pp 191ndash215 1990

[4] L B Gdalevich and V E Fertman ldquoConjugate problems ofnatural convectionrdquo Inzhenerno-Fizicheskii Zhurnal vol 33 pp539ndash547 1977

[5] M Miyamoto J Sumikawa T Akiyoshi and T NakamuraldquoEffects of axial heat conduction in a vertical flat plate on freeconvection heat transferrdquo International Journal ofHeat andMassTransfer vol 23 no 11 pp 1545ndash1553 1980

[6] A Pozzi and M Lupo ldquoThe coupling of conduction withlaminar natural convection along a flat platerdquo InternationalJournal of Heat and Mass Transfer vol 31 no 9 pp 1807ndash18141988

[7] S Kimura and I Pop ldquoConjugate natural convection from ahorizontal circular cylinderrdquoNumerical Heat Transfer A vol 25no 3 pp 347ndash361 1994

[8] G Wilks ldquoMagnetohydrodynamic free convection about asemi-infinite vertical plate in a strong cross fieldrdquo Zeitschrift furAngewandte Mathematik und Physik vol 27 no 5 pp 621ndash6311976

[9] M A Hossain ldquoViscous and Joule heating effects onMHD-freeconvection flow with variable plate temperaturerdquo InternationalJournal of Heat andMass Transfer vol 35 no 12 pp 3485ndash34871992

[10] T K Aldoss Y D Ali and M A Al-Nimr ldquoMHD mixedconvection from a horizontal circular cylinderrdquoNumerical HeatTransfer A vol 30 no 4 pp 379ndash396 1996

[11] M F El-Amin ldquoCombined effect of viscous dissipation andJoule heating onMHD forced convection over a non-isothermalhorizontal cylinder embedded in a fluid saturated porousmediumrdquo Journal of Magnetism and Magnetic Materials vol263 no 3 pp 337ndash343 2003

[12] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation andinternal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[13] A J Chamkha and I Camille ldquoEffects of heat genera-tionabsorption and thermophoresis on hydromagnetic flowwith heat and mass transfer over a flat surfacerdquo InternationalJournal of Numerical Methods for Heat and Fluid Flow vol 10no 4 pp 432ndash448 2000

[14] F Mendez and C Trevino ldquoThe conjugate conduction-naturalconvection heat transfer along a thin vertical plate with non-uniform internal heat generationrdquo International Journal of Heatand Mass Transfer vol 43 no 15 pp 2739ndash2748 2000

[15] M M Molla M A Hossain and M C Paul ldquoNatural con-vection flow from an isothermal horizontal circular cylinder inpresence of heat generationrdquo International Journal of Engineer-ing Science vol 44 no 13-14 pp 949ndash958 2006

[16] A A Mamun Z R Chowdhury M A Azim and M MMolla ldquoMHD-conjugate heat transfer analysis for a vertical flatplate in presence of viscous dissipation and heat generationrdquoInternational Communications in Heat and Mass Transfer vol35 no 10 pp 1275ndash1280 2008

[17] M A Azim A A Mamun and M M Rahman ldquoViscous Jouleheating MHD-conjugate heat transfer for a vertical flat plate inthe presence of heat generationrdquo International Communicationsin Heat and Mass Transfer vol 37 no 6 pp 666ndash674 2010

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Journal of Computational Methods in Physics 11

[18] H B Keller ldquoNumerical methods in the boundary layer theoryrdquoAnnual Review of Fluid Mechanics vol 10 pp 417ndash433 1978

[19] T Cebeci and P Bradshaw Physical and Computational Aspectsof Convective Heat Transfer Springer NewYork NYUSA 1984

[20] I Pop and D B Ingham Convective Heat Transfer PergamonOxford UK 1st edition 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

FluidsJournal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

ISRN Astronomy and Astrophysics

Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Condensed Matter PhysicsAdvances in

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

Physics Research International

Volume 2013

ISRN High Energy Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Advances in

Astronomy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

GravityJournal of

ISRN Condensed Matter Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2013

ISRN Thermodynamics

Volume 2013Hindawi Publishing Corporationhttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

High Energy PhysicsAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

PhotonicsJournal of

ISRN Optics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ThermodynamicsJournal of