Magnetohydrodynamic free convection of Al 2 O 3–water nanofluid considering thermophoresis and...

Computers & Fluids 94 (2014) 147–160

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Computers & Fluids

journal homepage: www.elsevier .com/ locate /compfluid

Magnetohydrodynamic free convection of Al2O3–water nanofluidconsidering Thermophoresis and Brownian motion effects

http://dx.doi.org/10.1016/j.compfluid.2014.01.0360045-7930/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: Department of Mechanical Engineering, BabolUniversity of Technology, Babol, Iran. Tel.: +98 9113968030.

E-mail address: [email protected] (M. Sheikholeslami).

M. Sheikholeslami a,b,⇑, M. Gorji-Bandpy a, D.D. Ganji a, P. Rana c, Soheil Soleimani d

a Department of Mechanical Engineering, Babol University of Technology, Babol, Iranb Department of Mechanical Engineering, University of Mazandaran, Babolsar, Iranc Department of Mathematics, Jaypee Institute of Information Technology, Noida, Indiad Department of Mechanical and Materials Engineering, Florida International University, Miami, FL 33199, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 October 2013Received in revised form 4 December 2013Accepted 28 January 2014Available online 10 February 2014

Keywords:MHDNanofluidCVFEMHeatlineThermophoresisBrownian

In this study MHD effect on natural convection heat transfer in an enclosure filled with nanofluid is inves-tigated. The transport equations used in the analysis took into account the effect of Brownian motion andthermophoresis parameters. The Navier Stokes equations in their vorticity-stream function form are usedto simulate the flow pattern, isotherms and concentration. The governing equations are solved via Con-trol Volume based Finite Element Method. The inner and outer circular walls are maintained at constanttemperatures while two other walls are thermally insulated. The heat transfer between cold and hotregions of the enclosure cannot be well understood by using isotherm patterns so heatline visualizationtechnique is used to find the direction and intensity of heat transfer in a domain. Effect of Hartmannnumber (Ha = 0, 30, 60 and 100), buoyancy ratio number (Nr = 0.1–4) and Lewis number (Le = 2, 4, 6and 8) on streamline, isotherm, isoconcentration and heatline are examined. Also a correlation for Nusseltnumber corresponding to active parameters is presented. The results indicate that Nusselt number is anincreasing function of buoyancy ratio number but it is a decreasing function of Lewis number and Hart-mann number. Also it can be concluded that as buoyancy ratio number increases the effects of otheractive parameters are more pronounced.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Heatline technique is an important method to visualize heattransport in enclosures. Isotherms are used to show the tempera-ture distribution in a domain, however, it is not easy to realize thedirection and intensity of heat transfer particularly in convectionproblems in which path of heat flux is not perpendicular to isothermdue to convection effect. Heatline is a useful tool for visualizationand analysis of not only direction but also intensity of heat transferin a domain. They provide corridors in where heat is transferredfrom hot to the cold regions by convection and/or conduction. Thistechnique was first proposed by Kimura and Bejan [1] to visualizeheat transport for convective heat transfer. Basak et al. [2] studiedthe natural convection in porous trapezoidal enclosures for uni-formly or non-uniformly heated bottom wall by presenting, stream-lines, isotherms and heatlines. They observed that heatlines areaffected with Darcy number. Oztop et al. [3] used heatline visualiza-

tion technique to understand heat transport path in an inclined non-uniformly heated enclosure filled with water based CuO nanofluid.Based on the heatline patterns, they observed three heat transfer re-gions. Numerical investigation of natural convection within poroussquare enclosures has been performed by Ramakrishna et al. [4].They illustrated from distribution of heatlines that significant heattransport occurs from hot bottom wall to the top portion side wallsat higher Darcy number. Sheikholeslami et al. [5] used heatline anal-ysis to simulate two phase simulation of nanofluid flow and heattransfer. Their results indicated that the average Nusselt numberdecreases as buoyancy ratio number increases until it reaches aminimum value and then starts increasing.

Magnetic field influence on natural convection has great impor-tance in many industrial applications such as crystal growth, metalcasting and liquid metal cooling blankets for fusion reactors.Rudraiah et al. [6] investigated numerically the effect of magneticfield on natural convection in a rectangular enclosure. They foundthat the magnetic field decreases the rate of heat transfer. Oztopet al. [7] studied the MHD natural convection in an enclosure fromtwo semi-circular heaters on the bottom wall. Their results showedthat flow strength decreases with increase of Hartmann number;

Nomenclature

Cp specific heat at constant pressureDB Brownian diffusion coefficientDT thermophoretic diffusion coefficient~g gravitational acceleration vectorHa Hartmann number ð¼ HBx

ffiffiffiffiffiffiffiffiffir=l

pÞ

k thermal conductivityL gap between inner and outer boundary of the enclosure

(=rout � rin)Le Lewis number ð¼ a=DBÞNb Brownian motion parameter ð¼ ðqcÞpDBð/h � /cÞ=

ðqcÞf aÞNt thermophoretic parameter ð¼ ðqcÞpDTðTh � TcÞ=

½ðqcÞf aTc�ÞNu Nusselt numberPr Prandtl number ð¼ l=qf aÞr non-dimensional radial distanceRa thermal Rayleigh number ð¼ ð1� /cÞqf0

gbL3ðTh � TcÞ=ðlaÞÞ

Nr buoyancy ratio number ð¼ ðqp � q0Þð/h � /cÞ=½ð1� /cÞqf0

bLðTh � TcÞ�ÞT fluid temperatureu, v velocity components in the x-direction and y-directionU, V dimensionless velocity components in the X-direction

and Y-direction

x, y space coordinates

Greek symbolsf angle measured from the insulated right planea thermal diffusivityu volume fractionr electrical conductivityl dynamic viscosityt kinematic viscosityx, X vorticity and dimensionless vorticityw and W stream function and dimensionless stream functionH dimensionless temperatureq fluid densityb thermal expansion coefficient

Subscriptsc coldh hotloc localave averagein innerout outer

148 M. Sheikholeslami et al. / Computers & Fluids 94 (2014) 147–160

means that magnetic field can be used as a control parameter forheat and fluid flow. MHD effect on natural convection heat transferin an inclined L-shape enclosure filled with nanofluid was studiedby Sheikholeslami et al. [8]. They found that enhancement in heattransfer has reverse relationship with Hartmann number and Ray-leigh number. Ellahi [9] studied the magnetohydrodynamic (MHD)flow of non-Newtonian nanofluid in a pipe. He observed that theMHD parameter decreases the fluid motion and the velocity profileis larger than that of temperature profile even in the presence ofvariable viscosities. Sheikholeslami et al. [10] analyzed the magne-tohydrodynamic nanofluid flow and heat transfer between twohorizontal plates in a rotating system. Mahmoudi et al. [11] stud-ied the entropy generation and enhancement of heat transfer innatural convection flow and heat transfer using Copper (Cu)–waternanofluid in the presence of a constant magnetic field. They ob-served that the entropy generation is decreased when the nanopar-ticles are present, while the magnetic field generally increases themagnitude of the entropy generation. Free convection heat transferin a concentric annulus between a cold square and heated ellipticcylinders in presence of magnetic field was investigated by Sheik-holeslami et al. [12]. They found that the enhancement in heattransfer increases as Hartmann number increases but it decreaseswith increase of Rayleigh number.

The above literature review reveal that the existence of mag-netic field has a noticeable effect on heat transfer reduction undernatural convection while in many engineering applications increas-ing heat transfer from solid surfaces is a goal. At this circumstance,the use of nanofluids with higher thermal conductivity can be con-sidered as a promising solution. The nanofluid can be applied toengineering problems, such as heat exchangers, cooling of elec-tronic equipments and chemical processes. Almost all of theresearchers assumed that nanofluids treated as the common purefluid and conventional equations of mass, momentum and energyare used and the only effect of nanofluid is its thermal conductivityand viscosity which are obtained from the theoretical models orexperimental data. These researchers assumed that nanoparticlesare in thermal equilibrium and there are not any slip velocities be-tween the nanoparticles and fluid molecules, thus they have a uni-

form mixture of nanoparticles. Khanafer et al. [13] conducted anumerical investigation on the heat transfer enhancement due toadding nanoparticles in a differentially heated enclosure. Theyfound that the suspended nanoparticles substantially increase theheat transfer rate at any given Grashof number. Rashidi et al. [14]considered the analysis of the second law of thermodynamics ap-plied to an electrically conducting incompressible nanofluid fluidflowing over a porous rotating disk. They concluded that using mag-netic rotating disk drives has important applications in heat trans-fer enhancement in renewable energy systems and industrialthermal management. Mixed convection and entropy generationof nanofluids flow in an open cavity heated from below with uni-form temperature was numerically investigated by Mehrez et al.[15]. They found that the heat transfer and the entropy generationincrease with the increase of Reynolds number, Richardson numberand volume fraction of nanoparticles, and vary with the aspect ratioof the cavity and nanoparticle types. Recently several authors inves-tigated about nanofluid flow and heat transfer [16–34].

All the above studies assumed that the nanoparticle concentra-tion is uniform. It is believed that in natural convection of nanofl-uids, the nanoparticles could not accompany fluid molecules due tosome slip mechanisms such as Brownian motion and thermopho-resis, so the volume fraction of nanofluids may not be uniform any-more and there would be a variable concentration of nanoparticlesin a mixture. Nield and Kuznetsov [35] studied the natural convec-tion in a horizontal layer of a porous medium saturated by a nano-fluid. Their analysis revealed that for a typical nanofluid (with largeLewis number) the prime effect of the nanofluids is via a buoyancyeffect coupled with the conservation of nanoparticles, the contri-bution of nanoparticles to the thermal energy equation being a sec-ond-order effect. Khan and Pop [36] published a paper onboundary-layer flow of a nanofluid past a stretching sheet. Theyindicated that the reduced Nusselt number is a decreasing functionof each dimensionless number. The problem of double stratifica-tion on boundary layer flow and heat transfer induced due to ananofluid over a vertical plate was investigated by Ibrahim andMakinde [37]. The boundary layer flow and heat transfer over apermeable stretching sheet due to a nanofluid with the effects of

M. Sheikholeslami et al. / Computers & Fluids 94 (2014) 147–160 149

magnetic field, slip boundary condition and thermal radiation havebeen investigated by Ibrahim and Shankar [38]. Their results indi-cated that the local Nusselt number decreases with an increase inboth Brownian motion parameter and thermophoresis parameter.

Control Volume Finite Element Methods (CVFEM) is a schemethat uses the advantages of both finite volume and finite elementmethods for simulation of multi-physics problems in complexgeometries [39–45]. Sheikholeslami et al. [46] studied the problemof natural convection between a circular enclosure and a sinusoidalcylinder. They concluded that streamlines, isotherms, and thenumber, size and formation of the cells inside the enclosurestrongly depend on the Rayleigh number, values of amplitudeand the number of undulations of the enclosure.

The main aim of present study is to investigate magnetic field ef-fect on nanofluid flow and heat transfer in a semi-annulus enclosurevia Control Volume based Finite Element method. The combined ef-fects of thermophoresis and Brownian motion are considered to getthe gradient of nanoparticles’ volume fraction. In addition, heatlinevisualization technique is used to show where heat is transferredfrom hot to the cold regions by convection or conduction.

2. Geometry definition and boundary conditions

The schematic diagram and the mesh of the semi-annulusenclosure used in the present CVFEM program are shown inFig. 1. The inner and outer walls are maintained at constant tem-

Fig. 1. (a) Geometry and the boundary conditions; (b) the mesh of enclosure considerevolume.

peratures Th and Tc, respectively while the two other walls are ther-mally insulated. Also the boundary conditions of concentration aresimilar to temperature. It is also assumed that the uniform mag-

netic field (~B ¼ Bx~ex þ By~ey) of constant magnitude B ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2

x þ B2y

qis applied, where ~ex and ~ey are unit vectors in the Cartesian coordi-nate system. The orientation of the magnetic field form an angle kwith horizontal axis such that k ¼ Bx=By. The electric current J and

the electromagnetic force F are defined by J ¼ rð~V �~BÞ and

F ¼ rð~V �~BÞ �~B, respectively.

3. Mathematical modeling and numerical procedure

3.1. Problem formulation

The nanofluid’s density, q is

q ¼ /qp þ ð1� /Þqf ffi /qp þ ð1� /Þfqf0ð1� bðT � TcÞg ð1Þ

where qf is the base fluid’s density, Tc is a reference temperature, qf0is the base fluid’s density at the reference temperature, and b is thevolumetric coefficient of expansion. Taking the density of base fluidas that of the nanofluid, as adopted by Sheikholeslami et al. [5], thedensity q in Eq. (2), thus becomes

q ffi /qp þ ð1� /Þfq0ð1� bðT � TcÞg ð2Þ

q0 is the nanofluid’s density at the reference temperature.

d in this work; and (c) a sample triangular element and its corresponding control


The continuity, momentum under Boussinesq approximationand energy equations for the laminar and steady state natural con-vection in a two-dimensional enclosure can be written in dimen-sional form as follows:

@u@xþ @v@y¼ 0 ð3Þ

qf u@u@xþ v @u

@y

� �¼ � @P

@xþ l @2u

@x2 þ@2u@y2

!

þ rB2ðv sin k cos k� u sin2 kÞ ð4Þ

qf u@v@xþ v @v

@y

� �¼ � @P

@yþ l

@2v@x2 þ

@2v@y2

!

� ð/� /cÞ qp � qf0

� �g þ ð1� /cÞqf0

ðT � TcÞg

þ rB2ðu sin k cos k� v cos2 kÞ ð5Þ

u@T@xþ v @T

@y¼ a

@2T@x2 þ

@2T@y2

!þðqcÞpðqcÞf

DB@/@x

:@T@xþ @/@y

:@T@y

� ��

þ DT=TcÞ ð@T@xÞ

2

þ @T@y

� 2( ) #

ð6Þ

u@/@xþ v @/

@y¼ DB

@2/@x2 þ

@2/@y2

( )þ DT

Tc

� @2T@x2 þ

@2T@y2

( )ð7Þ

Boundary conditions are

T ¼ Th; / ¼ /h on the inner circular boundaryT ¼ Tc; / ¼ /c on the outer circular boundary@T=@n ¼ @/=@n ¼ 0:0 on two other insulation boundariesw ¼ 0 all the solid boundaries

ð8Þ

The stream function and vorticity are defined as follows:

u ¼ @w@y

; v ¼ � @w@x

; x ¼ @v@x� @u@y

ð9Þ

The stream function satisfies the continuity Eq. (3). The vorticityequation is obtained by eliminating the pressure between the twomomentum equations, i.e. by taking y-derivative of Eq. (4) and sub-tracting from it the x-derivative of Eq. (5). Also the following non-dimensional variables should be introduced:

X ¼ xL; Y ¼ y

L; X ¼ xL2

a; W ¼ w

a; H ¼ T � Tc

Th � Tc;

U ¼ /� /c

/h � /c;U ¼ uL

a;V ¼ vL

að10Þ

where in Eq. (10) L = rout � rin = rin.By using these dimensionless parameters the equations

become:

@W@Y

@X@X�@W@X

@X@Y

� ¼Pr

@2X

@X2þ@2X

@Y2

!þPrRa

@H@X�Nr

@H@X

�

þHa2Pr �dVdY

tankþ�

dUdY

tan2 kþdUdX

tan k�dVdX

ð11Þ

@W@Y

@H@X� @W@X

@H@Y¼ @2H

@X2 þ@2H

@Y2

!þ Nb

@U@X

@H@Xþ @U@Y

@H@Y

�

þ Nt@H@X

� 2

þ @H@Y

� 2 !

ð12Þ

@W@Y

@U@X� @W@X

@U@Y¼ 1

Le@2U

@X2 þ@2U

@Y2

!þ Nt

Nb Le@2H

@X2 þ@2H

@Y2

!ð13Þ

@2W

@X2 þ@2W

@Y2 ¼ �X ð14Þ

where thermal Rayleigh number, the buoyancy ratio number, Pra-ndtl number, the Brownian motion parameter, the thermophoreticparameter, Lewis number and Hartmann number of nanofluid aredefined as: Ra ¼ ð1� /cÞqf0

gbL3ðTh � TcÞ=ðlaÞ; Nr ¼ ðqp � q0Þð/h � /cÞ=½ð1� /cÞqf0

bLðTh � TcÞ�; Pr ¼ l=qf a; Nb ¼ ðqcÞpDBð/h�/cÞ= ððqcÞf aÞ; Nt ¼ ðqcÞpDTðTh � TcÞ= ½ðqcÞf aTc�; Le ¼ a=DB

and Ha ¼ LBx

ffiffiffiffiffiffiffiffiffir=l

p, respectively.

Eq. (11) has been obtained using small temperature gradient ina dilute suspension of nanoparticles. The boundary conditions asshown in Fig. 1 are:

H ¼ 1:0; U ¼ 1:0 on the inner circular boundaryH ¼ 0:0; U ¼ 0:0 on the outer circular boundary@H=@n ¼ @U=@n ¼ 0:0 on two other insulation boundariesW ¼ 0:0 on all solid boundaries

ð15Þ

The values of vorticity on the boundary of the enclosure can beobtained using the stream function formulation and the knownvelocity conditions during the iterative solution procedure.

The local Nusselt number on the hot circular wall can be ex-pressed as:

Nuloc ¼ �@H@n

ð16Þ

where n is the direction normal to the inner cylinder surface. Theaverage number on the hot circular wall is evaluated as:

Nuave ¼1

0:5p

Z 0:5p

0NulocðfÞdf ð17Þ

The heatlines are adequate tools for visualization and analysisof 2D convection heat transfer, through an extension of the heatflux line concept to include the advection terms. Heat function(H) are defined in terms of the energy equation as

@H@Y¼ UH� @H

@Xð18aÞ

� @H@X¼ VH� @H

@Yð18bÞ

3.2. Numerical procedure

Triangular elements are considered as the building block of thediscretization using CVEM. The values of variables are approxi-mated with linear interpolation within the elements. A control vol-ume is created by joining the center of each element in the supportto the mid-points of the element sides that pass through the cen-tral node i, which creates a close polygonal control volume (seeFig. 1(b)).

To illustrate the solution procedure using the CVFEM, one canconsider the general form of advection–diffusion equation for nodei in integral form:

�Z

VQ dV �

ZA

kr/ � ndAþZ

Aðv � nÞ/dA ¼ 0 ð19Þ

or point form

�r � ðkr/Þ þ r � ðv/Þ � Q ¼ 0 ð20Þ

Table 1Comparison of the average Nusselt number Nuave for different grid resolution at Ra = 105, Nr = 4, Nt = Nb = 0.5, Le = 8, Ha = 100 and Pr = 10.

Mesh size 41 � 121 51 � 151 61 � 181 71 � 211 81 � 241 91 � 271 101 � 301 111 � 311

Nuave 3.371411 2.599407 1.850726 1.845539 1.840354 1.84026 1.840814 1.839254

Table 3Average Nusselt number versus at different Grashof number under various strengthsof the magnetic field at Pr = 0.733.

Ha Gr = 2 � 104 Gr = 2 � 105

Present Rudraiah et al. [6] Present Rudraiah et al. [6]

0 2.5665 2.5188 5.093205 4.919810 2.26626 2.2234 4.9047 4.805350 1.09954 1.0856 2.67911 2.8442

100 1.02218 1.011 1.46048 1.4317

Table 2Comparison of the present results with previous works for different Rayleigh numberswhen Pr = 0.7.

Ra Present Khanafer et al. [13] Davis [47]

103 1.1432 1.118 1.118104 2.2749 2.245 2.243105 4.5199 4.522 4.519


which can be represented by the system of CVFEM discrete equa-tions as:

½ai þ Qci þ Bci�/i ¼Xni

j¼1

ai;j/Si;jþ Q Bi

þ BBið21Þ

In above, the a‘s are the coefficients, the index (i, j) indicates the jthnode in the support of node i, the index Si,j provides the node num-ber of the jth node in the support, the B‘s account for boundary con-ditions, and the Q‘s for source terms. For the selected triangularelement which is shown in Fig. 1(c) this approximation withoutconsidering the source term leads to

�ðak1 þ au

1Þ/i þ ðak2 þ au

2Þ/Si;3þ ðak

2 þ au2Þ/Si;4

¼ 0 ð22Þ

Using upwinding the advective coefficients, identified with thesuperscripts ( )u, are given by

au1 ¼max½qf 1;0� þmax½qf 2;0�

au2 ¼max½�qf 1;0�

au3 ¼max½�qf 2;0�

ð23Þ

Fig. 2. Comparison of isotherms between the present work and experimental study of (a)when (a) Ro/Ri = 2.6, Ra = 4.7 � 104 and (b) Ro/Ri = 2.36, Ra = 0.9 � 105.

And the diffusion coefficients, identified with the superscripts( )k, are given by

ak1 ¼ �kf 1N1xD y!f 1 þ kf 1N1yD x!f 1 � kf 2N1xD y!f 2 þ kf 2N1yD x!f 2



ð24Þ

In Eq. (23) the volume flow across face 1 and 2 in the directionof the outward normal, is

qf 1 ¼ v � nAjf 1 ¼ v f 1x D y!f 1 � v f 1

y D y!f 1

qf 2 ¼ v � nAjf 2 ¼ v f 2x D y!f 2 � v f 2

y D y!f 2

ð25Þ

The value of the diffusivity at the mid-point of face 1 can be ob-tained as

kf 1 ¼ ½N1k1 þ N2k2 þ N3k3�f 1 ¼5

12k1 þ

512

k2 þ2

12k3 ð26Þ

and at the mid-point of face 2

kf 2 ¼ ½N1k1 þ N2k2 þ N3k3�f 2 ¼5

12k1 þ

212

k2 þ5

12k3 ð27Þ

The velocity components at the mid-point of face 1 are:

v f 1x ¼ 5

12 vx1 þ 512 vx2 þ 2

12 vx3

v f 1y ¼ 5

12 vy1þ 5

12 vy2þ 2

12 vy3

ð28Þ

And on face 2:

v f 2x ¼ 5

12 vx1 þ 212 vx2 þ 5

12 vx3

v f 2y ¼ 5

12 vy1þ 2

12 vy2þ 5

12 vy3

ð29Þ

These values can be used to update the ith support coefficientsthrough the following equation:

ai ¼ ai þ ak1

ai;3 ¼ ai;3 þ ak2

ai;4 ¼ ai;4 þ ak3

ð30Þ

In Eq. (24), moving counter-clockwise around node i, the signeddistances are:

Kuehn and Goldstein [48]; and Laboni and Guj [49] for viscous flow (/ = 0, Pr = 0.71)

Ha 0= Ha 30= Ha 100=Is

othe

rms

Stre

amlin

es

max min0.033, 32.47Ψ Ψ= = − max min0.0, 12.71Ψ Ψ= = − max min0, 2.223Ψ Ψ= = −

Isoc

once

ntra

tion

Hea

tline

s

Fig. 3. Comparison of the isotherms, streamlines, isoconcentration and heatline contours for different values of thermal Hartmann number whenNr = 0.1, Le = 8, Nt = Nb = 0.5, Ra = 105 and Pr = 10.


Ha 0= Ha 30= Ha 100=

Isot

herm

s St

ream

lines

max min12.09, 42.68Ψ Ψ= = − max min6.636, 21.64Ψ Ψ= = − max min0.0, 6.115Ψ Ψ= = −

Isoc

once

ntra

tion

Hea

tline

s

Fig. 4. Comparison of the isotherms, streamlines, isoconcentration and heatline contours for different values of thermal Hartmann number whenNr = 4, Le = 8, Nt = Nb = 0.5, Ra = 105 and Pr = 10.


D x!f 1 ¼ x33 �

x26 �

x16 ; D x!f 2 ¼ � x2

3 þx36 þ

x16

D y!f 1 ¼ y33 �

y26 �

y16 ; D y!f 2 ¼ � y2

3 þy36 þ

y16

ð31Þ

the derivatives of the shape functions are:

N1x ¼ @N1@x ¼

ðy2�y3Þ2Vele ; N1y ¼ @N1

@y ¼ðx3�x2Þ

2Vele

N2x ¼ @N2@x ¼


@y ¼ðx1�x3Þ

2Vele

N3x ¼ @N2@x ¼


@y ¼ðx2�x1Þ

2Vele

ð32Þ

Ha 0= Ha 30= Ha 100=

Isot

herm

s St

ream

lines

max min14.5, 63.33Ψ Ψ= = − max min11.35, 25.83Ψ Ψ= = − max min=0.0, 9.203Ψ Ψ = −

Isoc

once

ntra

tion

Hea

tline

s

Fig. 5. Comparison of the isotherms, streamlines, isoconcentration and heatline contours for different values of thermal Hartmann number whenNr = 4, Le = 2, Nt = Nb = 0.5, Ra = 105 and Pr = 10.


and the volume of the element is

Vele ¼ ðx2y3 � x3y2Þ þ x1ðy2 � y3Þ þ y1ðx3 � x2Þ2

ð33Þ

The obtained algebraic equations from the discretization proce-dure using CVFEM are solved by the Gauss–Seidel Method.

3.3. Implementation of source terms and boundary conditions

The boundary conditions for the present problem can be en-forced using BBi

and BCias follows:

Insulated boundary : BBi¼ 0 and BCi

¼ 0 ð34Þ

Ha 0= Ha 30= Ha 100=

Isot

herm

sSt

ream

lines

max min0.038, 33.24Ψ Ψ= = − max min0, 13.07Ψ Ψ= = − max min=0.0, 2.32Ψ Ψ = −

Isoc

once

ntra

tion

Hea

tline

s

Fig. 6. Comparison of the isotherms, streamlines, isoconcentration and heatline contours for different values of thermal Hartmann number whenNr = 0.1, Le = 2, Nt = Nb = 0.5, Ra = 105 and Pr = 10.


Fixed Value Boundary : BBi¼ /value � 1016 and BCi

¼ 1016 ð35Þ

where /value is the prescribed value on the boundary. The volumesource terms can be applied to Eq. (21) as:

Xelements

j¼1

ZVj

Q dV � Q iVi ð36Þ

or after linearizing the as source term

QiVi ¼ �QCi/i þ QBi

ð37Þ

4. Grid testing and code validation

To guarantee the grid-independency of the present solution amesh testing procedure was conducted. Different mesh

Nr 0.1,Ha 100= = Nr 4,Ha 100= =

Ha 0,Le 2= = Ha 100,Le 8= =

Nr 0.1,Le 2= = Nr 4,Le 8= =

Fig. 7. Effects of Hartmann number, buoyancy ratio number and Lewis number on Local Nusselt number at Nt = Nb = 0.5, Ra = 105 and Pr = 10.



combinations were explored for the case of Ra = 105, -Nr = 4, Nt = Nb = 0.5, Le = 8, Ha = 100 and Pr = 10 as shown in Table1. The present code was tested for grid independence by calculat-ing the average Nusselt number on the inner circular wall. In har-mony with this, it was found that a grid size of 81 � 241 ensures agrid-independent solution. The convergence criterion for the ter-mination of all computations is:

maxgridjCsþ1 � Csj 6 10�7 ð38Þ

where s is the iteration number and C stands for the independentvariables (X, W, H, U). The present FORTRAN code is validated bycomparing the obtained results for Pr = 0.7 with other works re-ported in the literature [13,47]. As shown in Table 2, they arein a very good agreement. Another test for validation of the cur-rent code was performed for the case of natural convection in asquare enclosure in the presence of magnetic field. In this testcase, the average Nusselt number using different Gr and Ha num-ber have been compared with those obtained by Rudraiah et al.[6] as shown in Table 3. Furthermore, the present numerical solu-tion is validated by comparing the results of present code againstthe results of experimental study of Kuehn and Goldstein [48],

Ha 0=

Ha 60=

Fig. 8. Effects of Hartmann number, buoyancy ratio number and Lewis num

Laboni and Guj [49] at the different Rayleigh numbers as shownin Fig. 2.

5. Results and discussion

MHD effect on natural convection heat transfer in an enclosurefilled with nanofluid is investigated numerically using CVFEM. Ef-fects of Hartmann number (Ha = 0, 30, 60 and 100), buoyancy rationumber (Nr = 0.1–4) and Lewis number (Le = 2, 4, 6 and 8) on flowand heat transfer characteristics are examined. Brownian motionparameter of nanofluids (Nb = 0.5), thermophoretic parameter ofnanofluids (Nt = 0.5), thermal Rayleigh number (Ra = 105) and Pra-ndtl number (Pr = 10) are fixed.

Effects of Hartmann number, Lewis number and buoyancy rationumber on isotherms, streamlines, isoconcentration and heatlinecontours are shown in Figs. 3–6. When the magnetic field is im-posed on the enclosure, the velocity field suppressed owing tothe retarding effect of the Lorenz force. So intensity of convectionweakens significantly. The braking effect of the magnetic field isobserved from the maximum stream function value. The core ofvortex is shift downward vertically as the Hartmann number

Ha 30=

Ha 100=

ber on average Nusselt number at Nt = Nb = 0.5, Ra = 105 and Pr = 10.

Table 4Constant coefficient for using Eq. (39).

aij i = 1 i = 2 i = 3 i = 4 i = 5 i = 6

j = 1 3.069651 �0.16499 0.578896 0.012949 �0.01115 �0.02536j = 2 4.906119 �7.59508 0.585453 3.681993 �0.01115 �0.28078j = 3 0.819671 �0.56997 0.947823 0.094258 �0.00422 0.021492


increases. Also imposing magnetic field leads to omit the thermalplume over the inner wall. At high Hartmann number, the conduc-tion heat transfer mechanism is more pronounced. For this reasonthe isotherms are parallel to each other. Increasing Hartmannnumber causes the concentration boundary layer thickness nearinner wall to increase.

The heat flow within the enclosure is displayed using the heatfunction obtained from conductive heat fluxes (oH/oX, oH/oY) aswell as convective heat fluxes (VH, UH). Heatlines emanate fromhot regimes and end on cold regimes illustrating the path of heatflow. As seen heatline has two regions rotate in different direction.The lower one is greater than another which means that more heattransfer occurs in this region. As Hartmann number increases heat-lines become weaker because of reduction of heat transfer rate byapplying magnetic field. The domination of conduction heat trans-fer in high Hartmann number can be observed from the heatlinepatterns since no passive area exists.

It should be mentioned that negative Nr values (opposing buoy-ancy forces) showed more complex and interesting flow patterns,such as multi-cells, which is worthy for presentation and discus-sion. So in this study results for positive Nr (aiding buoyancyforces), where the temperature and species induced buoyancyforces aides each other, are not considered. For Nr = 0, the speciesinduced buoyancy force has no effect on flow; the flow is solely dri-ven by the thermal buoyancy force. However, the effect of speciesinduced buoyancy increases as Nr value increases, and reaches acertain value, where the effect of thermally induced buoyancy be-

1.69Nr =

* 0.4Ha =

Fig. 9. Variation of Nuave for v

comes negligible in comparison to the solutal one. For small Nr va-lue, the flow is mainly driven by the thermal buoyancy force. WhenNr increases a reverse thermal plume appears at f = 90�. This phe-nomena is due to existing one counter clockwise eddy at thisregion. By increasing Hartmann number, the two main eddiesmerged into one counterclockwise eddy. The isoconcentrationsare more distorted with increase of solutal forces. As Nr increasesthe upper region of healine counters divided into two smaller oneand this new region disappear with increase of Hartmann number.

The mass flow is given by wmax � dsv , where the solutal bound-ary layer thickness is given by ds � (RaLeNr)�1/4 and v � (RaLeNr)1/2

so wmax � (RaLeNr)1/4. The solutal boundary layer, ds, becomes thin-ner by increasing of Le. Heatlines are found to be more distorted asLe augments.

Effects of Hartmann number, buoyancy ratio number and Lewisnumber on Local Nusselt number are shown in Fig. 7. As seen LocalNusselt number increases as buoyancy ratio number increases butit decreases with increase of Hartmann number and Lewis number.Generally as f increases local Nusselt number decreases due toincrement of thermal boundary layer thickness. In absence of mag-netic field Nuloc has one local maximum near the bottom wall. Theoccurrence of maxima for Nuloc is due to dense heatlines based onconductive heat transport occurring at this portion. This point dis-appears at high Hartmann number. Local Nusselt number profilehas minimum point at f = 90� because of existing of thermal plumeat this region. Also this figure shows that effect of Lewis number onNuloc is negligible at low buoyancy ratio number.

The corresponding polynomial representation of such model forNusselt number is as follows:

Nu ¼ a13 þ a23Y1 þ a33Y2 þ a43Y21 þ a53Y2

2 þ a63Y1Y2

Y1 ¼ a11 þ a21Leþ a31Nr þ a41Le2 þ a51Nr2 þ a61Le Nr

Y2 ¼ a12 þ a22Ha� þ a32Nr þ a42Ha�2 þ a52Nr2 þ a62Ha�Nr

ð39Þ

5Le =

5

arious input parameters.


where aij can be found in Table 4 for example a21 equals to(�0.16499). Effects of Hartmann number, buoyancy ratio numberand Lewis number on average Nusselt number are shown in Figs.8 and 9. Presence of magnetic field leads to disappear the thermalplume over inner wall and makes the isotherms parallel to eachother due to domination of conduction mode of heat transfer.Therefore average Nusselt number decreases with increase of Hart-mann number. As Lewis number increases, thermal boundary layerthickness increases and in turn Nusselt number decreases. Effect ofbuoyancy ratio number on Nuave is in contrast with Ha and Le. Alsoit can be found that effect of Hartmann number and Lewis numberare more pronounced at higher values of buoyancy ratio number.

6. Conclusions

Magnetic field effect on nanofluid flow and heat transfer is stud-ied numerically using Control Volume based Finite Element Meth-od. The model used for the nanofluid incorporates the effects ofBrownian motion and thermophoresis. The effects of Hartmannnumber, buoyancy ratio number and Lewis number on the flowand heat transfer characteristics have been examined. Results arepresented in the form of streamline, isotherm, isoconcentrationand heatline plots. The results show that Nusselt number has directrelationship with buoyancy ratio number while it has reverse rela-tionship with Hartmann number and Lewis number. Existing of nopassive area in heatline contours at high Hartmann number indi-cates the domination of conduction heat transfer. Also it can beseen that effect of Hartmann number and Lewis number on Nusseltnumber are more obvious with increase of solutal forces.

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