International Journal of Mechanical Sciences 65 (2012) 147–156

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International Journal of Mechanical Sciences

0020-74

http://d

n Corr

E-m

Moham

Farhang

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

Numerical investigation of thermocapillary and buoyancy driven convectionof nanofluids in a floating zone

H. Aminfar a, M. Mohammadpourfard b,n, F. Mohseni a

a Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iranb Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz-53751-71379, Iran

a r t i c l e i n f o

Article history:

Received 3 January 2012

Received in revised form

23 August 2012

Accepted 23 September 2012Available online 3 October 2012

Keywords:

Nanofluid

Thermocapillary

Buoyancy

Floating zone

Mixture model

Marangoni number

03/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ijmecsci.2012.09.013

esponding author. Tel./fax: þ98 412 4327566

ail addresses: hh_aminfar@tabrizu.ac.ir (H. Am

madpour@azaruniv.edu (M. Mohammadpour

.Mohseni88@ms.tabrizu.ac.ir (F. Mohseni).

a b s t r a c t

In this paper, the results of adding nanoparticles (spherical shape Al2O3 with a diameter of 20 nm) to

silicone oil (Prf¼16.08) as a base fluid in a floating zone with aspect ratio equal to unity have been

reported. In the absence of gravity the thermocapillary convection, driven by surface tension gradient,

is dominant in the floating zone. In the range of Marangoni number considered here, the flow is steady

and axisymmetric. Numerical study was conducted on the effect of the increase in the volume fraction

of nanoparticles on various thermo–hydrodynamic parameters of the flow, with the use of two phase

mixture model and control volume technique. It has been concluded that adding nanoparticles and

increasing their volume fraction deteriorate thermocapillary convective characteristics. The effect of

gravity (by taking the buoyancy driven convection into account) has also been studied.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Newly, thermocapillary convection in fluid associated withgradients in surface tension has received a lot attention, for examplethermocapillary convection is important for mass and heat transportin binary fluids and crystal growth processes. The floating zone is acrucible-free method of growing higher quality crystals of semicon-ductors and is a typical model of investigating thermocapillaryconvection. So far there have been many experimental and numer-ical studies on thermocapillary convection in floating zone forvarious liquids with different Prandtl numbers [1–6]. Investigationsshow that for a low Marangoni number, the flow remains steadyand has an axisymmetrical structure. At the first critical Marangoninumber (Ma1

cr) , the transition from the axisymmetrical to thesteady three-dimensional state has been reported. The value of thefirst critical Marangoni numbers varies for different liquids anddifferent geometrical shapes [1–6]. Beyond the first critical Mar-angoni number instabilities will appear. Full description of theseinstabilities in thermocapillary convection can be found in [7,8].Shevtsova [9] have investigated the onset of instabilities in thermo-capillary convection numerically. They have also studied the effects

ll rights reserved.

.

infar),

fard),

of temperature-dependant viscosity on oscillatory thermocapillaryconvection in [10].

A new class of fluids, nanofluids, has been recently introduced.Nanofluids are uniformly dispersed suspensions of nanoparticles(e.g. carbon and metal oxides) in the size range of about10–50 nm in a base fluid (e.g. water and lubricants). One of thefascinating features of nanofluids is that they have anomaloushigh thermal conductivity at very low nanoparticles concentra-tion [11,12] and considerable enhancement of forced convectiveheat transfer [13–15].

In the main flow of a nanofluid several factors such as gravity,inertia, the phenomena of Brownian diffusion and thermophor-esis may coexist, which means that there exists slip velocitybetween the fluid and nanoparticles [16]. Single phase modelassumes that the fluid phase and nanoparticles move with thesame velocity, so it seems that two phase models are better toapply the nanofluid. Behzadmehr et al. [17], numerically studiedthe turbulent forced convection of a nanofluid in a circular tubeby using two phase mixture model. Their study indicated that, incomparison to experimental results the mixture model is moreprecise than the single phase model.

While there are a significant number of publications on thebuoyancy driven convection [18–26], the thermocapillary con-vection in nanofluids is still hardly investigated. Unlike forcedconvection, experimental results show that in buoyancy drivenconvection, heat transfer coefficient decreases by increasing thenanoparticle volume fraction [18–21]. Putra et al. [18] conducted

Nomenclature

As aspect ratio (2H/D)Cp specific heat J= kg K

� �Nu Nusselt number (h H/k)k conductivity w= m K

� �Pr Prandtl number (mcp/k)vr radial velocity (m/s)vx axial velocity (m/s)v!

velocity vector (m/s)T temperature (K)P pressure (Pa)v!

pf slip velocity vector (m/s)v!

dr drift velocity vector (m/s)g gravitational acceleration (m/s2)dp nanoparticle diameter (m)D diameter of liquid bridge (m)H height of liquid bridge (m)h heat transfer coefficient (W/m2 K)Ma Marangoni number (9tT9DTHrcp/mk)Ra Rayleigh number (gbDTH3r2cp/mk)_Q Heat flow (W)

Th temperature at hot disc (K)

Tc temperature at cold disc (K)DT temperature difference between two discs Th�Tc(K)r radial directionx axial direction

Greek symbols

t surface tension (N/m)tT surface tension temperature gradient N= m K

� �ap particle volume fractionb thermal expansion coefficient (1/K)m dynamic viscosity (kg/ms)p density (kg/m3)

Subscripts

f pertaining to base fluideff pertaining to nanofluidm pertaining to mixturep pertaining to nanoparticles0 pertaining to reference condition

Fig. 1. Geometry of floating zone and coordinate system.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156148

the experiment for observation on the natural convective char-acteristics of water based Al2O3 nanofluids in a horizontalcylinder heated from one end and cooled from the other. Theirresults revealed that the presence of nanoparticles suspended inbase fluid and the increase in their concentration deteriorate thenatural convective heat transfer. Reduction of the natural con-vective heat transfer by increasing particles concentration char-acterized by reducing the Nusselt number for a given Rayleighnumber was experimentally observed. Other experiments alsoshowed similar result [19–21]. Numerical studies for buoyancydriven convection heat transfer of nanofluids have grown inrecent years, but in contrast to the experimental results manyof them showed that increasing the volume fraction of nanofluidscauses an increase in the Nusselt number [22–25]. Hwang et al.[26] showed that this discrepancy between the numerical and theexperimental results is due to models for effective viscosity, theyused two models for effective viscosity of water-Al2O3 nanofluids,Einstein’s model [27] and Pak and Cho’s model [28], and theore-tically showed that using Pak and Cho’s model degrades heattransfer coefficient relative to the base fluid due to the presence ofnanoparticles.

To the best knowledge of the authors, there is not such anexperimental or numerical observation on the thermocapillaryconvection in nanofluids. Thermocapillary convection is driven bysurface tension gradient due to temperature, so the effect ofnanoparticles on the surface tension gradient is important. Duringthe past few years some investigators reported the influence ofadding nanoparticles and nanotubes on surface tension [12,29–32].Das et al. [12] reported that in pool boiling the surface tensions ofthe water–Al2O3 nanofluids at small mass fraction of nanoparticlesare almost identical to that of water. Xue et al. [29] showed thatadding carbon nanotubes to water in boiling process increasessurface tension. Their results also indicated that although addingnanotubes changes the surface tension, the temperature gradient ofthe surface tension is almost unaffected. Su et al. [32] showed thatthe addition of carbon nanotubes has no effect on the surfacetension of carbon nanotubes–ammonia binary nanofluids.

In this paper thermocapillary and buoyancy driven convection ina floating zone for nanofluids have been investigated, using a two

phase mixture model. Marangoni number is well below Ma1cr and

flow field is steady and axisymmetric. Pak and Cho’s model [28] foreffective viscosity and Hamilton and Crosser model [33] for effectiveconductivity have been used in the buoyancy driven and thethermocapillary convection of oil silicone–Al2O3 nanofluids.

2. Governing equations and numerical method

The floating zone of radius R is suspended between twoparallel, concentric discs with different temperatures as shownschematically in Fig. 1. The temperature difference and heightbetween two discs are DT and H, respectively. Liquid surface hasbeen idealized to be adiabatic from the environmental gas andnon-deformable. The nanofluid has been assumed to have aconstant kinematical viscosity, thermal conductivity and specificheat. The fluid motion is driven by buoyancy and thermocapillaryforces which arise due to the temperature dependence of thedensity r and the surface tension t t¼ t0þtT T�T0ð ÞÞð . TheBoussinesq approximation has been used for the densityvariations.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156 149

The surface tension has been considered to be a linearlydecreasing function of the temperature. Regarding the previousinvestigations on the effect of nanoparticles and nanotubes onsurface tension [12,29,32], in this paper it has been assumed thatadding nanoparticles to the silicone oil does not affect thetemperature gradient of the surface tension. Silicone oil is aNewtonian fluid and regarding the low volume fraction ofnanoparticles used in this investigation, the nanofluid has beenassumed as a Newtonian fluid. Recently Chen and Xie [34]experimentally showed that, silicone oil based nanofluid withnanotubes behave in Newtonian manner in low volume fractions.

Considering these assumptions and using two phase mixturemodel the dimensional conservation equations for steady statecondition when viscous dissipation is neglected are as follows:

Continuity equation:

r: rm,0 v!

m

� �¼ 0 ð1Þ

Momentum equation:

r rm,0 v!

m v!

m

� �¼�rpþr mm r v

!m

� �þr aP rP v

!dr,p v!

dr,p

� ��rm,0 T�T0ð Þbm g

!ð2Þ

Energy equation:

r½ aPrPcp,p v!

pþ 1�aPð Þrf cp,f v!

f ÞT� ¼r kmrTð Þ:�

ð3Þ

Volume fraction:

r aPrP v!

m

� �¼�r aPrP v

!dr,p

� �ð4Þ

where

v!

m ¼aprP v!

pþ 1�aPð Þrf v!

f

rm,0

ð5Þ

v!

dr,p ¼ v!

p� v!

m ð6Þ

are the mass-averaged velocity and drift velocity respectivelyand ap is the volume fraction of nanoparticles that is a quantitybetween 0 and 1.

Two phase mixture model can be used for two phase flowswhere the phases are strongly coupled and move at differentvelocities. The model solves the continuity, the energy and themomentum equation for the mixture. The momentum equationfor the mixture is the sum of the momentum equations for theindividual phases. The volume fraction equation is the continuityequation for the particles phase. Clearly the volume fractionequation and the other equations of the flow are coupled.

The slip velocity is defined as the velocity of a secondary phase(p) relative to the velocity of the primary phase (f):

v!

pf ¼ v!

p� v!

f ð7Þ

The drift velocity is related to the slip velocity

v!

dr,p ¼ v!

pf�aprP

rm,0

v!

f� v!

p

� �ð8Þ

The slip velocity is determined by Eq. (9) suggested byManninen et al. [35] while Eq. (10) by Schiller and Naumann[36] is used to calculate the drag function fdrag.

v!

pf ¼rpd2

p

18 mf f drag

rp�rf

rp

a!

ð9Þ

f drag ¼1þ0:15 Re0:687

p Repr1000

0:0183 Rep Rep41000

(ð10Þ

where Rep ¼ vm dprm,0=mm and the acceleration ( a

!) in Eq. (9) is:

a!¼ g!� v!

mr� �

v!

m ð11Þ

The mixture physical properties in the above equations are:Mixture density:

rm,0 ¼ aPrpþ 1�aPð Þrf ð12Þ

Mixture dynamic viscosity (Pak and Cho’s model) [23]:

mm ¼ 1þ39:11apþ533:9a2p

� �mf ð13Þ

Mixture thermal conductivity (Hamilton and Crosser model [33]for spherical nanoparticles):

km ¼kpþ2kf�2ap kf�kp

� �kpþ2kf þap kf�kp

� �" #

kf : ð14Þ

Mixture thermal expansion coefficient [32]:

bm ¼1

1þ 1�aPð Þrf =aPrP

bp

bf

þ1

1þaPrP= 1�aPð Þrf

" #bf ð15Þ

Zeng et al. [6] showed, in a case quite similar to the oneconsidered here (silicone oil floating zone with As¼1) whenMarangoni number is less than 1.98�104 (Ma1

cr), flow field issteady and axisymmetric. The aforementioned non-linear andcoupled partial differential governing equations for the axisym-metric condition subjected to the following boundary conditions:

At x ¼ 0 (cold disc):

vm,r ¼ 0; vm,x ¼ 0; T ¼ Tc: ð16Þ

At x¼H (hot disc):

vm,r ¼ 0; vm,x ¼ 0; T ¼ Th ð17Þ

At r¼ 0 (axial axis):

vm,r ¼ 0;@vm,x

@r¼ 0;

@T

@r¼ 0 ð18Þ

At r¼D/2 (free surface):

vm,r ¼ 0; m @vm,x

@r¼ t; @T

@r¼ 0 ð19Þ

Some important dimensionless variables can be drawn fromprevious governing equations including the Rayleigh number ofnanofluid Raeff and the Marangoni number of nanofluid Maeff:

Raef f ¼gbmDTH3r2

mcp,m

mmkm; Maef f ¼

tT9DTHrmcp,m

mmkmð20Þ

where

cp,m ¼ aPcp,pþ 1�aPð Þcp,f ð21Þ

is mixture specific heat.The mentioned coupled non-linear differential equations were

discretized with the control volume technique. For the convectiveand diffusive terms a second order upwind method was usedwhile the SIMPLE procedure was utilized for the velocity–pressure coupling. A structured non-uniform grid has been usedto discretize the computational domain. It is finer near the freesurface and the discs. Several different grid distributions havebeen examined to ensure that the calculated results are gridindependent. The used grid for the present calculations consistedof 100�100 nodes. As shown in Table 1, increasing the gridnumbers does not significantly change the heat transfer coeffi-cient for the hot disc.

In order to validate the model and the numerical procedure,comparisons with the previously published numerical and experi-mental works have been done. Numerical results of Li and Hu [37]for thermocapillary convection in microgravity condition in afloating zone with As¼2 and Ma¼50 for silicon (Pr¼0.01) have

Table 1Grid independent test. (Maf¼10,000, As¼1 and aP¼0.04).

Node number r� x heff at hot disc (W/m2K)

r direction

80�100 86.47668446

100�100 86.29466782

100�80 86.22373495

x direction

100�80 86.52597192

100�100 86.29466782

100�120 86.19460792

Fig. 2. Comparison of dimensionless axial velocity along free surface for floatin

Fig. 3. Comparison of Nusselt number versus Rayleigh number for water-Al2O3 4% nanofl

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156150

been used for validating numerical procedure in the floating zone.Fig. 2 indicates comparisons of the present results for axialdimensionless velocity along the free surface of the floating zonewith the results of Li and Hu [37]. Very good agreement betweenthe results is seen.

Also experimental work of Putra [18] for natural convection ofnanofluid (water and 4 vol.% Al2O3) in a horizontal cylinder withL/D¼0.5 has been used for validating the numerical procedure,mixture model and models for effective viscosity and conductiv-ity, where L and D are the length and diameter of the cylinderrespectively. Fig. 3 indicates comparison of the present results foreffective Nusselt number for various effective Rayleigh numbers

g zone As¼2 (Ma¼50, Pr¼0.01) with the corresponding numerical result.

uid in a horizontal cylinder (L/D¼0.5) with the corresponding experimental result.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156 151

with the experimental results. It should be noted that, for thenumerical simulation of natural convection of nanofluid in thehorizontal cylinder, a 3D model has been used and the assump-tion of spherical nanoparticles with the diameter of 15 nm hasbeen utilized. Other numerical procedures and models are thesame as mentioned earlier. Good agreement between the experi-mental and numerical results is seen.

3. Results and discussion

The results are presented for thermocapillary convection, withzero gravity condition, and also for thermocapillary and buoyancydriven convection of silicone oil–Al2O3 nanofluids in a floatingzone with As¼1. The nanoparticles are assumed to be sphericalwith a diameter of 20 nm. The bulk temperature variation is lowenough for the reliability of Boussinesq approximation in the

Table 2Physical and geometrical conditions.

rf ¼818 kg=m3

cp,f ¼1966 J=kg K

kf ¼0.1 w= m K

bf ¼ 0:001341=K

mf ¼ 8:18� 10�4 kg= m s

tT¼�8.39�10�5 N= m K

20:1� 10�6 N= m

¼rP¼3950 kg/m3

cp,p¼775 J/kg K

kP¼42.34 w/m.K

bp¼5.4�10�6 1= K

Tc ¼ 300 K

H¼5 mm

g¼9.81 m/s2

Fig. 4. Temperature contour lines in the absence of gravity for Maf¼10,000 and As¼

range of applied Marangoni and Rayleigh numbers. Properties ofthe studied nanofluids and some geometrical conditions arepresented in Table 2.

In the absence of gravity, thermocapillary effect is the onlysource of convection. Marangoni number of base fluid (Maf) wasassumed to be 10,000. As the reason is explained later Fig. 4shows that by adding Al2O3 nanoparticles to the silicone oil in thefloating zone and increasing the volume fraction of nanoparticles,contours of static temperature become a little more uniform.It should be mentioned that in Fig. 4 the right boundary is the freesurface, the left boundary is the axis and the up and the downboundaries are the hot disc and the cold disc respectively.

In a floating zone under the influence of thermocapillaryconvection, since the thermocapillary convection is driven fromthe free surface, axial velocity of fluid in that surface is animportant parameter. Increase or decrease in the free surfaceaxial velocity can greatly influence the hydrodynamic of the fluidin the floating zone. Fig. 5 shows that, by increasing the volumefraction of Al2O3 nanoparticles the free surface axial velocitydecreases. It is expected that the decrease in the axial velocityof the free surface causes the decrease in the velocity magnitudeof the other points in the floating zone consequently. Axialvelocity of the fluid in the free surface due to the temperaturegradient of the surface tension is from the hot disk to thecold disk.

Fig. 6 shows the effect of increasing the volume fraction ofAl2O3 nanoparticles on the heat transfer coefficient (h¼ 4 _Q =pD2

Th�Tcð Þ where _Q ¼�kmpD2=4@T=@x) and the Nusselt number.Nusselt number and heat transfer coefficient ratio of the nanofluidto the base fluid decreases by increasing the volume fraction. Theseinteresting results clearly indicate that, thermocapillary convectivecharacteristics are deteriorated by increasing the volume fraction ofnanoparticles. It’s worth mentioning that, the reduction in theNusselt number ratio is more than the reduction in the heat transfercoefficient ratio and the difference between these ratios increases as

1 (a) aP¼0 (b) aP¼0.01 (c) aP¼0.02 (d) aP¼0.03 (e) aP¼0.04 and (f) aP¼0.05.

Fig. 5. Axial velocity along free surface in the absence of gravity for Maf¼10,000 and As¼1.

Fig. 6. Nanofluid heat transfer coefficient and Nusselt number ratio versus volume fraction in the absence of gravity for Maf¼10,000 and As¼1.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156152

the volume fraction decreases, since thermal conductivity of thenanofluid increases by the volume fraction increment. It is alsointeresting to know that adding 5 vol.% Al2O3 nanoparticles to thesilicone oil base fluid can cause significant decrease (about 40%) inthe ratio of the nanofluid to the base fluid Nusselt number.

In the absence of gravity, when the thermocapillary convectionis dominant, hydrodynamic and heat transfer in the floating zoneis a function of Marangoni number. Adding Al2O3 nanoparticles to

the silicone oil and increasing their volume fraction change theeffective properties of the fluid, so the Marangoni number of thenanofluid varies with that of the base fluid. The effect of increas-ing the volume fraction on the ratio of the nanofluid Marangoninumber to the base fluid Marangoni number is presented in Fig. 7.Increasing the volume fraction decreases the Marangoni numberof the nanofluid. The reason is that, by increasing the volumefraction of nanoparticles, the nanofluid viscosity, the conductivity

Fig. 7. Ratio of nanofluid Marangoni number to base fluid Marangoni number versus volume fraction in the absence of gravity for Maf¼10,000 and As¼1.

Fig. 8. Effective Nusselt number versus effective Marangoni number for silicone oil and silicone oil–Al2O3 5% in the absence of gravity for Maf¼10,000 and As¼1.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156 153

and density increase and the nanofluid specific heat decreases, Eqs.(12–15, 21), but the increase in the nanofluid viscosity and con-ductivity and the decrease in the nanofluid specific heatare more significant than the increase in the nanofluid density, sothe effective Marangoni number (Marangoni number of the nano-fluid) decreases. The decrease in the Marangoni number of thenanofluid by increasing the volume fraction explains the deteriora-tion of thermocapillary convective characteristics and changes instatic temperature contours as indicated in the previous results.

Fig. 8 indicates the variation of Nusselt number versus Maran-goni number for silicone oil and for silicone oil nanofluid (5 vol.%Al2O3). As expected, by increasing the effective Marangoni number,Nusselt number increases in both fluids. Even in the same effective

Marangoni number, effective Nusselt number of the silicone oilnanofluid is slightly smaller than the pure silicone oil.

Buoyancy driven and thermocapillary convection both exist infloating zone in the presence of gravity. Gravity is assumed to bein negative x direction. As mentioned, the thermocapillary con-vection causes a downward flow near the free surface whilebuoyancy driven convection acts in the opposite direction sinceupper disk is hot and lower disk is cold. Fig. 9 shows the effect ofadding Al2O3 nanoparticles and increasing the volume fraction ofthe nanoparticles to the silicone oil in the floating zone on thecontours of static temperature in the presence of the thermo-capillary and buoyancy driven convection when Maf¼10,000 andRaf¼32,038. Like in the previous case, increasing the volume

Fig. 9. Temperature contour lines for Maf¼10,000, Raf¼32,038 and As¼1 (a) aP¼0 (b) aP¼0.01 (c) aP¼0.02 (d) aP¼0.03 (e) aP¼0.04 (f) aP¼0.05.

Fig. 10. Axial velocity along free surface for Maf¼10,000, Raf¼32,038 and As¼1.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156154

fraction slightly changes the temperature contours, and makethem a little more uniform. As the reason is explained later,changes become even smaller in comparison to the zero gravitycondition.

Fig. 10 shows that, in the presence of the thermocapillary andbuoyancy driven convection, like in the previous case, theincrease in the volume fraction of Al2O3 nanoparticles decreasesthe free surface velocity.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156 155

Fig. 11 shows that, in the presence of gravity, the heat transfercoefficient ratio of the nanofluid to the base fluid decreases byincreasing the volume fraction. It also indicates that, in thepresence of thermocapillary and buoyancy driven convectionreduction in the heat transfer coefficient ratio is less than thereduction in the heat transfer coefficient ratio in the presence ofthe thermocapillary convection alone. As mentioned, in thepresent model for the floating zone, the thermocapillary andbuoyancy driven convection act in opposite directions, so the factthat by increasing the volume fraction, reduction in the heattransfer coefficient ratio in the presence of gravity is less than this

Fig. 11. Nanofluid heat transfer coefficient ratio versus vo

Fig. 12. Nanofluid Nusselt number versus volume fr

reduction in the absence of gravity, indicates that the buoyancydriven convective heat transfer coefficient is deteriorated byincreasing the volume fraction. This is in agreement with theprevious investigations for buoyancy driven convection for dif-ferent geometries.

Fig. 12 shows the variation of the effective Nusselt numberversus the volume fraction in the presence of thermocapillary andbuoyancy driven convection and also shows comparison betweenthese results and the results in the presence of the thermocapil-lary convection alone. As expected, in the presence of thethermocapillary and buoyancy driven convection the Nusselt

lume fraction for Maf¼10,000, Raf¼32,038 and As¼1.

action for Maf¼10,000. Raf¼32,038 and As¼1.

H. Aminfar et al. / International Journal of Mechanical Sciences 65 (2012) 147–156156

number is smaller. By increasing the volume fraction, the effectiveNusselt number decreases in both cases, but difference betweenthe Nusselt numbers also decreases by increasing the volumefraction. Again, since the thermocapillary and buoyancy drivenconvection act in opposite directions, this result indicates that thebuoyancy driven Nusselt number is also deteriorated by increas-ing the volume fraction of nanoparticles. So both the thermo-capillary and buoyancy driven convective characteristics aredeteriorated by increasing the volume fraction of the nanoparti-cles in the floating zone.

4. Conclusion

This article describes a numerical investigation of thermoca-pillary and buoyancy driven convection of silicone oil (Prf¼16.08)based nanofluid with Al2O3 nanoparticles in a floating zonewith As¼ 1. Two phase mixture model and control volumetechnique have been used to study the flow. The effect ofincreasing the volume fraction on the temperature contour lines,free surface axial velocity, heat transfer coefficient and Nusseltnumber has been studied. The results indicated that, in theabsence of gravity, when thermocapillary effect is the only sourceof convection, increasing the volume fraction of nanoparticlescauses deterioration in nanofluid Marangoni number (effectiveMarangoni number), nanofluid heat transfer coefficient and nano-fluid Nusselt number. In the presence of gravity, increasing thevolume fraction of nanoparticles decreases both the thermocapil-lary and buoyancy driven nanofluid heat transfer coefficient andas a result the nanofluid Nusselt number.

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