non-Newtonian fluid in a C-Shaped cavity with magnetic field effect using finite difference lattice Boltzmann method

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Computers and Fluids

journalhomepage:www.elsevier.com/locate/compfluid

Numerical investigation of heat transfer in a power-law

non-Newtonian fluid in a C-Shaped cavity with magnetic field effect using finite difference lattice Boltzmann method

Saeed Aghakhani

^a

, Ahmad Hajatzadeh Pordanjani

^b

, Arash Karimipour

^a,∗

, Ali Abdollahi

^a

, Masoud Afrand

^a

aDepartment of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

bDepartment of Mechanical Engineering, Shahrekord University, Shahrekord, Iran

a rt i c l e i n f o

Article history:

Received 14 June 2018 Revised 14 September 2018 Accepted 17 September 2018 Available online 21 September 2018 Keywords:

Power-law non-newtonian fluid C-shaped cavity

Magnetic field

Finite Difference Lattice Boltzmann Method

a b s t r a c t

The present study aimed to investigate the natural convection heat transfer in a power-law, non- NewtonianfluidunderanappliedmagneticfieldinsideaC-shapedcavityusingtheFiniteDifferenceLat- ticeBoltzmannmethod(FDLBM).Temperaturedistributiononthewallontheleftsidewasnon-uniform andsinusoidalwiththecoldwallontherightside.Bothtopandbottomhorizontalwallsofthecavity wereinsulatedagainstheatandmasstransfer.TheBoussinesqapproximationwasusedduetonegligible densityvariations,makingthehydrodynamicfieldsensitivetothethermalfield.Furthermore,theD₂Q₉ latticearrangementwas usedforthedensityand energydistributionfunctions‏.Thisstudy investigates theeffectsoftheRayleighnumber,exponentialfunctionindex,aspectratio,andtheHartmannnumber ontheflowandtemperaturefields.Theresultsshowthattheheattransferrateincreaseswithincreasing Rayleighnumber.Moreover,itwasfoundthattheNusseltnumberdecreaseswithincreasingpower-law index(n)athigherRayleighnumbersandthatanincreaseintheHartmannnumberresultsinareduced heattransferrate.ThereductionintheheattransferratecausedbytheincreasedHartmannnumberin theshearthinningfluidswasmorethanthatinshearthickeningfluids.TheNusseltnumberdecreases withincreasingcavityaspectratiofortheNewtonianandshearthinningfluids,whereasforshearthick- eningfluids,theNusseltnumberinitiallyincreasedandthendecreased.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Heat transfer in closed cavities has many applications in dif- ferentindustries.Thesecavitiesareused incooling systems,elec- tricmachines,andmicroelectronicdevices.Consideringtheneces- sity of the optimal use of space, the use of closed cavities with differentprofiles, especiallyasymmetrical,isimportantindesign- ing these devices.In the closedcavities, the main mechanismof heat transfer is the natural convection. The heat transfer of a Newtonian fluidnaturalconvectioninaclosedcavitywithsimple conditions hasbeen extensivelyinvestigated [1–4]. Kefayatiet al.

[5]examinedaNewtonianfluidinarectangularcavity,whoseside wallswereat“Tc” and“Th” temperatures,usingLatticeBoltzmann method.Thetopandbottomwallswerealsoinsulated.Theyfound thattheheattransferrateincreaseswithincreasingRayleighnum-

∗ Corresponding author.

E-mail addresses: arash.karimipour@pmc.iaun.ac.ir , arash.karimipour@uniroma1.it (A. Karimipour).

ber. Y. He, C. Qi, Y. Hu e al. [6]also examined a similar geome- tryand came tosimilar conclusions asKefayati etal.Kalteh and Hasani [7]addressed the laminar natural convection heat trans- fer ina Newtonian fluid in an l-shaped cavity.The openingand end walls of the insulated cavity and the top walls were kept cold and the bottom walls were kept hot. They found that the increasedRayleighnumberincreases theheat transferrate. Guiet etal.[8]examinedthenaturalconvectionheattransferofaNew- tonian fluid in a closedcavity. In the middleof the bottomwall wasa dent with temperature T_h. The side wallswere alsoat Tc

temperature. They also found that the increased Rayleigh num- berincreasesthe heattransferrate. Ina numerical,finite-volume study, Mahmoodi [9] addressed an l-shaped cavity containing a Newtonian fluid. They examined the parameters of the Rayleigh numberandthedimensionsofthecavity.MahmoodiandHashemi [10]investigatedthenaturalconvectionheattransfer ofa Newto- nian fluid in a C-shaped cavity using the finite volume method.

The outer walls ofthe cavity were kept hot andthe inner walls were kept cold.The two ends ofthe cavity were insulated.They examined the Rayleigh parameter in the range from 103 to 106 https://doi.org/10.1016/j.compfluid.2018.09.012

0045-7930/© 2018 Elsevier Ltd. All rights reserved.

Nomenclature

AR aspectratioofenclosure B magneticfieldstrength C latticespeed

Cp specificheatatconstantpressure F externalforces

f densitydistributionfunctions

feq equilibriumdensitydistributionfunctions g internalenergydistributionfunctions

g_eq equilibriuminternalenergydistributionfunctions

g gravity

H enclosureheight Ha Hartmannnumber K theconsistencycoefficient L thicknessofenclosure n power-lawindex Nu nusseltnumber

P pressure

Pr prandtlnumber Ra Rayleighnumber T temperature

t time

x,y cartesiancoordinates u velocityinxdirection v velocityinydirection Greekletters

σ

^{theelectrical}conductivity

φ

^{relaxationtime}

τ

^shearstress

ζ

^{discreteparticlespeeds x latticespacing}

^{t timeincrement}

α

^thermaldiffusivity

ρ

^density

μ

^{dynamicviscosity}

ψ

^{streamfunctionvalue}

Subscripts

m average

C cold

H hot

x,y cartesiancoordinates

α

^{thenumberofthenode}

andtheaspectratiosfrom0.2to0.8.Theyfoundthattheincreased Rayleighnumberincreasestheheattransfer,andincreasedaspect ratiodecreased theheat transferrate. Other researchers reported theheattransferofnanofluids[11–17]besidesusinglattice Boltz- mannmethod[18–25].Giventhefactthatintheindustrythemag- neticfieldiscreatedby theelectriccurrent, manyresearchersin- vestigatedtheeffectofthe magneticfieldon theheattransfer in the cavities. Therefore, a lotof studies [26–34] were carried out onthecavitieswithelectricallyconductivefluidsinthevicinityof themagneticfieldwhichconductednumericalstudiesonthenat- uralconvectionheattransferofaNewtonianfluidwithaconstant appliedmagneticfield. Therightside wallofthe cavitywaskept cold,theleftsidewallwaskepthotandthetopandbottomwalls wereinsulated.Aconstant horizontalmagneticfieldwasalsoap- pliedto the cavity. They investigatedthe effects of Rayleigh and Hartmannumbersaswell asthevolumepercentofnanoparticles ontheheattransfer. Theresultsoftheir studies indicatedthat in theweak magnetic fields,the addition ofnanoparticles increases theheat transfer rate, whereas in strong magnetic fieldsthis re-

ducestheheattransferrate.SheikholeslamiandShehzad[35]car- riedoutanumericalstudyontheheattransferof[hydrated]cop- peroxidenanofluidnaturalconvectionusingthelatticeBoltzmann method. Theyassumed nanofluid asa Newtonian fluid. The cav- itywasunderconstant angularmagneticflux.Apartofthelower side waskept hot at T_h temperature andthe top wall wasinsu- lated.Thesidewallswerekeptcoldatalowtemperature.Theyin- vestigated theeffects ofRayleigh,Darcy,andHartmann numbers, as well as the volume percent of nanoparticleson heat transfer.

Mahmoodi etal. [36] numerically studied the naturalconvection heat transfer of a Newtonian, alumina-waternanofluid using the lattice Boltzmannmethod. The cavity wasunder a constant and angularmagneticflux.Thebottomwallofthecavitywaskepthot andthesidewallswerekeptcold.Thetopwallwasalsoinsulated.

TheyexaminedtheeffectofchangesintheHartmannandRayleigh numbers,volumepercentofnanoparticles,andmagneticfield an- gle on the flowfield andthe heat transferrate. They found that the heat transferrate increasedwithan increase inthe Rayleigh numberanddecreasedwithanincreaseintheHartmannnumber.

In recent years, the study of naturalconvection heat transfer ofnon-Newtonianfluidsinaclosedcavityhasattractedmanyre- searchers.Thereasonistheuseofvarioustypesofnon-Newtonian fluidssuchasmoltenpolymermaterials,nanofluids,paints,adhe- sives, inks, organic materials, etc. in different industries such as food,nuclearreactors,oildrillingsystems,coolingsystems,etc.In non-Newtonianfluids, therelationship betweenshearstress vari- ations andthe applied shear rate is not linear and the duration of applying shear stress is very important for its rate. Hence, in non-Newtonianfluids,a coefficientsuch astheviscositydoesnot makesense todescribe theshearstress state.OzoeandChurchill [37]were amongthefirst toexaminethe heattransferofa non- Newtonian fluid inaclosed cavity.Theyused thepower-lawand the Ellis models to examine a shallow rectangular cavity heated fromthe lowerside andcooled fromthe upperside. Theyfound thatthecriticalRayleighnumberwouldincreasewithan increase in the power-law index to start a natural convection. Kim etal.

[38]examinedthenaturalconvectionheattransferinapower-law non-Newtonian fluid in a vertical cavity. The horizontal walls of the insulatingcavity andthe side wallswere assumed to be hot andcold.TheyfoundthatinaconstantRayleighnumber,thecon- vection power and the heat transfer rate of non-Newtonian flu- ids vary in comparison with Newtonian fluids; so that they are strengthened for a shear thinning fluid, and weaken for a shear thickeningfluid.Lamsaadietal.[39]conductedanumericalstudy on the power-law non-Newtonian fluid natural convection in a shallowcavityusingtheFiniteDifferenceMethod(FDM).Thecav- ityhadtallandinsulatedlateralwallspassing aconstantthermal flux.TheresultsshowedthatprovidedthePrandtlnumberandthe cavity aspect ratio are large enough, the heat transfer and flow power values donot depend on theseparameters. Therefore,for non-NewtonianfluidswithalargePrandtlnumberinshallowcav- ities, the parameters that influence the flow field and the heat transfer ratearethe power-lawindex (n)andthe Rayleigh num- ber. In two separate studies, Turan et al. [40,41] and Ternikand Rudolf[42]investigatedthenaturalconvectionofanon-Newtonian fluid ina rectangularandsquare cavitywithinsulatedhorizontal andfixed-temperaturevertical walls.The resultsshowedthat the flow field andthe heattransfer rateare affectedby the Rayleigh numberandthepower-lawindex(n).ButifthePrandtlnumberis large,fora specific Rayleighindexandnumber,the averageNus- seltnumber isnot affected by thePrandtl number.For a rectan- gular cavity with fixed temperatureboundary conditions on ver- tical walls, changes in the Nusselt number didnot increase uni- formlywithincreasing aspectratio.Vinogradovetal.[43] studied thenaturalconvectionofathinningfluidinsquareandrectangu- larcavitiesusingthepower-lawmathematicalmodel.Theyfounda

significantdifference betweenNewtonian andnon-Newtonianflu- ids regarding the heat transfer rate. Habibi et al. [44] examined thepower-lawnon-Newtonianfluidnaturalconvectioninasquare cavity withtwo concentric ducts,one of whichwas hot andthe otheronewascold.Theyexaminedtheinfluenceofpower-lawin- dexparameters(n),Rayleighnumber(Ra),Prandtlnumber(Pr)and aspectratioontheflowfield andtemperatureaswellastheheat transfer rate. The results of their research showedthat withthe increase inthepower-lawindexfrom0.6to1.4, theheattransfer ratedecreased.GuhaandPradhan[45]examinedtheheattransfer of anon-Newtonian fluidon a horizontalplate. Theyfound non- NewtonianfluidsofferbetterheattransferpropertiesthanNewto- nianpseudoplasticfluidsatthesamedevelopedPrandtlnumber.

Usingtheboundary-layertheory,Zhangetal.[46] conducteda numericalstudyontheheattransferandthesurfacefrictioncoef- ficientsofpower-lawnon-Newtonianfluidsattheboundarylayer.

Theyfoundthat thedistributionofthermalboundarylayerisde- pendent not only on the velocity but alsoon the power-lawin- dex, aswell ason the Prandtl number. Nelson et al. [47]exam- ined the three-dimensional natural convection heat transfer in a container surrounded by a cavity. They filled the container with apower-lawnon-Newtonianfluidandfilleditssurroundings with air.Theresultsoftheirstudyincludedthestreamlinesandisother- mallinesindifferentRayleighnumbersforeachpower-lawNewto- nianandnon-Newtonianfluids. DrawingonthelatticeBoltzmann method‏,intwoseparatestudies,Kefayati[48,49]investigatedheat transfer by natural and mixed convection in a power-law non- Newtonian fluid withan appliedmagnetic field, ina square cav- ity.Thehorizontalwallsofthiscavitywereinsulated,andtheleft sidewallswerekeptcoldataconstanttemperatureandtheright sidewallsweresinusoidallykepthot.Hefoundthattheheattrans- fer rate increases withthe increase in the Rayleigh number and decreaseintheHartmann number.Nusseltnumberalsoincreases withthedecreaseinthepower-lawindex.Hisresultsalsoshowed thatthelatticeBoltzmannmethodwassuitableforanalyzingthese issues. In two separate studies, Raisi [50,51] investigated the be- haviorofnon-Newtonianfluidsandexponentialfunctionmodelin thesquare cavities.He examinedtheeffectoftwo barriersinside a cavity filled with a power-law non-Newtonian fluid. The bot- tom wall of thiscavitywas underconstant heat flux. Hisresults showedthattheNusseltnumberincreaseswiththeincreaseinthe Rayleigh number and the conductivity coefficient of barriers. He alsofoundthat theNusseltnumberdecreaseswithanincreasein theexponential functionindexinallRayleighnumbers.Increased lengthofbarriersinsmallRayleighnumbersincreasestheNusselt number, butathigher Rayleighnumbers, increasedlength ofthe barriersreducesNusseltnumber.Inhisotherwork,acavityfilled withanon-Newtonianfluid,andwarmedupfromthebottomwall, wasexamined. Hisresultsinthisstudyalso showedthatNusselt numberdecreasedwiththeincreaseinthesourcelengthforshear thinningfluids.Duetotheimportanceofnon-Newtonianfluidsin theindustry,thehydrodynamicandthermalbehaviorofthistype offluidswasexamined inthe presentstudy.Thenaturalconvec- tion of a power-law, viscid, incompressible, non-Newtonian fluid wasnumericallysimulatedinthisstudyunderthemagneticfluxin a continuousandlaminarstatusin aC-shapedcavity.The Lattice Boltzmann methodwas used tosolve the equations,and Boussi- nesqapproximation wasusedtoapproximatedensity.Finally,the effects ofparameterssuch astheRayleighandHartmannumbers aswell as thepower-law indexandthe aspectratio onthe flow field,temperaturefield,andtheheattransferrate.

2. Problemstatementandgoverningequations

The desired geometry, in accordance with Fig. 1, is a two- dimensional C-shaped cavity. The cavity has been filled with a

Fig. 1. The desired geometry.

non-Newtonian power-law fluid. The left wall of the cavity was kept at aT(^y)=Tc+(^TH−Tc)^sin(

π

^y/H) ^warm temperature. The topandbottomwallsofthe cavitywerealso insulated.Theright were keptat acold temperatureof T_c. Aconstant magneticfield ofmagnitudeB₀wasappliedtothecavityparalleltothehorizon- talaxis.Itwasassumedthattherewasanon-slipconditiononall walls.

In this research, a laminar and consistent flow wasassumed.

Apartfromthefluiddensity,whichchangeswithtemperatureand ismodeledusingtheBoussinesqapproximation,andthefluidvis- cosity,which is a function ofthe shear rate, the remaining fluid properties were assumed to be constant. The Prandtl number is not constant as the viscosity of a non-Newtonian fluid depends on the shearrate andhas a wide range ofvariations. Numerical studies use the nominal Prandtl as obtained from Eq.(8). Given that,usually,non-Newtonianfluidsusedintheindustryhavelarge Prandtl numbers,the Prandtl numberwasassumed to be 100 in thepresentstudy[52].

Thedimensionlessgoverningequationswere usedforthetwo- dimensional laminar flow of the non-Newtonian, incompressible fluidinsidethecavity,andtheBoussinesqapproximationwasused totakeintoaccounttheimpactofthethermalfieldonthehydro- dynamicfield. Theseequations,irrespectiveoftheheat generated by the magneticfield andthe effects ofradiation, are asfollows [44,48,53]:

du dx+

∂

^v

∂

^{y=0 (1)}

u

∂

^u

∂

^x+v

∂

^v

∂

^y=−

ρ

¹

∂

^p

∂

^x+

ρ

¹

∂ τ

^xx

∂

^{x +}

∂ τ

^xy

∂

^y

−

σ

^B2

ρ

^{u (2)}

u

∂

^v

∂

^x+v

∂

^v

∂

^y=−

1

ρ ∂

^p

∂

^y+

1

ρ ∂ τ

^xx

∂

^{x +}

∂ τ

^xy

∂

^y

+g

β (

^T−Tc

)

⁽³⁾

u

∂

^T

∂

^x+v

∂

^T

∂

^y

=

α ∂

^2T

∂

^{x2 +}

∂

^2T

∂

^y2

(4)

Foranon-Newtonianfluidthatfollowstheexponentialfunction model,theshearstresstensorisexpressedbyEq.(5):

τ

ij=2

μ

^aDij=

μ

^a

∂

^ui

∂

^xj

+

∂

^vj

∂

^yi

(5)

D_ijisthesheartensorrateinthetwo-dimensionalCartesianco- ordinates,and

μ

^{a istheapparent viscosityofthe}non-Newtonian fluid,which isobtainedinthe two-dimensionalCartesian coordi-

Fig. 2. The D 2Q 9model.

natesofEq.(6):

μ

^a=K

2

∂

^u

∂

^x

2

+

∂

^v

∂

^y

2

+

∂

^v

∂

^xj

+

∂

^u

∂

^yi

2

⁽

ⁿ⁻¹

⁾

2

(6)

In Eq.(6), K and n are the constants of exponential function model.Kisthecoefficientandnistheexponentialfunctionindex.

Forpseudo-plasticfluidsn>1andforthinningfluidsn<1,while

Fig. 3. mirror reflection boundary condition.

n=1representsNewtonianfluids.Theboundaryconditionsofthe problemareasfollows:

•Bottomwall:

0≤x≤100

y=0 u=v=0

∂

^T

∂

^{y =0}

•Topwall:

0≤x≤100

y=1 u=v=0

∂

^T

∂

^{y =0}

Fig. 4. Comparison of the streamline for various Rayleigh numbers and power-law indexes at the Ha = 20 and AR = 0.4.

Table 1

Nusselt number of mean for Ra = 10 ⁵, Ha = 20 , and AR = 0 . 4 , three exponential function indices n = 0.7, n = 1, n = 1.5.

n = 0 . 7

80 ×80 100 ×100 120 ×120 140 ×140 160 ×160 180 ×180

Nu m 2.608 2.717 2.768 2.836 2.838 2.838

max 0.846 0.852 0.857 0.856 0.856 0.856

n = 1

Nu m 2.596 2.687 2.701 2.706 2.709 2.709

max 0.737 0.746 0.752 0.754 0.754 0.754

n = 1 . 5

Nu m 2.302 2.321 2.339 2.341 2.341 2.341

max 0.465 0.475 0.477 0.478 0.478 0.478

•leftwall:

0≤y≤100

x=0 u=v=0 T

(

^y

)

^=Tc+

(

^TH−Tc

)

^sin

( π

^y/H

)

•Rightwall:

0≤y≤100

x=1 u=v=0 T=TC &

∂

^T

∂

^y=0

(7) 3. Non-dimensionalequations

To have dimensionless equations of (1)-(4), the following pa- rameters were used with the boundary conditions of equations (7). Eq.8 indicates the definitionof dimensionlessRayleigh (Ra), Prandtl(Pr),andHartmann(Ha)numbers:

X=x l, X=y

l,U= u

α

l

Ra^0.5

,V= v

α

l

Ra^0.5

P= p

ρ α

l

2

Ra ,

θ

= T−Tc

T_H−Tc

Pr=

μ

^a

ρα

^{, Ra=}

ρ

^g

β

^l3

(

^Th−T_c)

μ

a

α

^{, Ha=B0l}

σ

μ

a

(8)

Withthesedimensionlessvariables,thedimensionlesscontinu- ity,momentum,andenergyequationsarewrittenasfollows:

∂

^U

∂

^Y+

∂

^V

∂

^{Y=0 (9)}

U

∂

^U

∂

^Y+V

∂

^U

∂

^{Y =−}

∂

^P

∂

^X+√PrRa

∂ τ

XX

∂

^{X +}

∂ τ

XY

∂

^Y

−PrHa²

√Ra U (10)

U

∂

^V

∂

^X+V

∂

^V

∂

^Y=−

1

ρ ∂

^P

∂

^Y+

√Pr Ra

∂ τ

^XX

∂

^{X +}

∂ τ

^XY

∂

^Y

+Pr

θ

⁽¹¹⁾

U

∂θ

∂

^X+V

∂θ

∂

^Y

= 1

√Ra

∂

²

θ

∂

^{X2 +}

∂

²

θ

∂

^Y2

(12)

τ

^IJ=2

μ

^∗aDIJ=

μ

^∗a

∂

^Ui

∂

^Xj

+

∂

^Vj

∂

^Yi

(13)

μ

^∗a=K

2

∂

^U

∂

^X

2

+

∂

^V

∂

^Y

2

+

∂

^V

∂

^X+

∂

^U

∂

^Y

2

⁽

ⁿ⁻¹

⁾

2

(14)

Boundary conditions arechanged using thedimensionless pa- rametersasfollows:

•Bottomwall:

0≤X≤100

Y=0 U=V=0

∂θ

∂

^{Y =0}

•Topwall:

0≤X≤100

Y=1 U=V=0

∂θ

∂

^Y=0

•leftwall:

0≤Y≤100

X=0 U=V=0

θ (

^Y

)

^=sin

( π

^Y

)

•Rightwall:

0≤Y ≤100

X=1 U=V=0

θ

⁼¹

∂θ

∂

^Y=0

(15)

4. Numericalmethod

Inrecentyears,latticeBoltzmannmethodhasbecomeapower- fultechniqueinsimulatingcomputational fluiddynamics.In fact, thismethodwasbasedontheLatticeGasAutomata(LGA)method.

However, Lattice Boltzmann equations can be extracted in a fi- nite difference form of the Boltzmann transfer equation. The is- suediscussedintheLatticeBoltzmannequationistheparticledis- tribution function. In the above study, a new method called the finite element Boltzmann method was used to solve the equa- tions. A brief description of this method has already been pro- vided.Infact,oneofthemostimportantissuesinthelatticeBoltz- mannmethodistheabilitytosimulatenon-Newtonianfluidsdue to changes in thelocal viscosity witha velocity gradient. There- fore,since the relaxationtime is not constant, itcannot be used tosolvemanyproblems.Recently,thefiniteelement latticeBoltz- mannmethodhasbeen ofinterestto researchersforthe simula- tion of non-Newtonianfluids [54–58]. Fu et al.[59–61] reported thenewequationsfortheequilibriumdistributionfunctioninstead ofMaxwellequations.Infact,theyconsideredtheequilibriumdis- tributionfunctionvariableandtherelaxationtimeconstant.There- fore, this method has removed the limitation of the Boltzmann network method. Additionally, in this method, the pressure is a variableparameterthat isnot directlyobtainedfromthe density, unlikeordinaryLBM.Infact,theyhaveusedfinitedifferencelattice Boltzmannmethodtosolvethe problem.Finally,thismethodhas thecapacitytoextractshearstressfromthelatticeBoltzmannbase equation in theflow fields.In addition,in thismethod, different boundaryconditionscanbeappliedwithouttheproblemsandlim- itationsthatexistintheordinarylatticeBoltzmannmethod.More workscanbeaddressedinthisway[64–77].Thedistributionfunc- tion f represents the probability of the presence of a particle at velocity

ξ

^{atlocation r,attime t,basedontheLatticeBoltzmann}

equation:

∂

^fα

∂

^{t +}

ξ

α.

∇

^fα=− 1

εφ

f_α−f^eq_α

(16)

Where,

ϕ

^,

ξ

α, and f^eq_α represent the relaxation time, the micro- scopicvelocity distribution, thedensitydistribution function,and the equilibrium distribution function, respectively. Also,

ε

^{is a}

Table 2

Results of the present study with those of [62] and [63] .

n = 0 . 7 n = 0 . 8 n = 0 . 9 n = 1 . 0 n = 1 . 1 n = 1 . 2 n = 1 . 3 n = 1 . 5 Ra = 10 ⁵

Present work 1.31 1.18 1.09 1 0.92 0.89 0.84 0.81

Khezzar et al. [62] 1.3 1.18 1.09 1 0.93 0.89 0.86 0.82

Percentage error (%) 0.7 0.0 0.0 0.0 1.1 0.0 2.3 1.2

Kefayati [63] 1.32 1.19 1.1 1 0.92 0.88 0.85 0.81

Percentage error (%) 0.7 0.8 0.9 0.0 0.0 1.1 1.2 0.0

Ra = 10 ⁴

Present work 1.04 1.03 1.02 1.0 0.98 0.97 0.94 0.93

Khezzar et al. [62] 1.03 1.02 1.01 1.0 0.99 0.97 0.95 0.93

Percentage error (%) 0.9 0.9 1.0 0.0 1.0 0.0 1.0 0.0

Kefayati [63] 1.04 1.03 1.02 1.0 0.98 0.96 0.94 0.92

Percentage error (%) 0.0 0.0 0.0 0.0 0.0 1.0 0.0 1.8

smallparameterthatisconsideredequaltothetimestep(

ε

=^t).

Therelaxationtime isconsideredas

ϕ

=1; theequation distribu- tionsectioncanbewrittenasEq.17:

∂

^fα

∂

^{t +}

ξ

α.

∇

Xf_α=0 (17)

Eq.(17)iswrittenusingtheLax-Wendroff method.Thecollision pointsoftheequationarewritteninisolatedformasfollows:

∂

^fα

∂

^{t =−}

1

εφ

f_α

(

^X,t

)

^−feqα

(

^X,t

)

(18)

Thedistributionsectionisusedastheappropriateconditionfor solvingthecollisionpointandEq.(3)canbewrittenasEq.(19)us- ingtheEulermethodandchoosing

ε

=fand

ϕ

=1:

f_α

(

^x,t+

^t

)

^−fα

(

^X,t

)

^{t =−}

1

εφ

f_α

(

^X,t

)

^−feqα

(

^X,t

)

(19) Therefore:

f_α

(

^X,t+

^t

)

=f^eq_α

(

^X,t

)

(20) For thedensity distributionfunction, Chapman-Enskog expan- sionwasusedintheformofEq.(21):

f_α=f^eq_α +

ε

^f(_α¹⁾+

ε

^2f(α²⁾+O

ε

³

(21) Where:

8 α=0

f_α⁽ⁿ⁾= 8 α=0

f_α⁽ⁿ⁾

ξ

α^x= 8 α=0

f⁽_αⁿ⁾

ξ

α^y=0, n≥1 (22)

8 α=0

f^eq_α =

ρ

=constant (23)

8 α=0

f^eq_α

ξ

α^x=

ρ

^{u (24)}

8 α=0

f^eq_α

ξ

α^y=

ρ v

(25)

8 α=0

f^eq_α

ξ

_α²x=

ρ

^u2+p−

τ

^{xx (26)}

8 α=0

f^eq_α

ξ

_α²y=

ρ v

²+p−

τ

^{yy (27)}

8 α=0

f^eq_α

ξ

α^x

ξ

α^y=

ρ v

u−

τ

^{xy (28)}

Navier-Stokes equations can be extracted fromthe Boltzmann equationusingthementionedequations:

∂ ( ρ

^u

)

∂

^{x +}

∂ ( ρ

^v

)

∂

^{y =0 (29)}

∂ρ

^u2

∂

^{x +}

∂ ρ

^vu

∂

^{y =−}

∂

^p

∂

^x+

∂ τ

xx

∂

^{x +}

∂ τ

xy

∂

^y

+O

( ε )

⁽³⁰⁾

∂ρ

^v2

∂

^{y +}

∂ ρ

^vu

∂

^{x =−}

∂

^p

∂

^y+

∂ τ

^yy

∂

^{y +}

∂ τ

^xy

∂

^x

+O

( ε )

⁽³¹⁾

Inthe above equations,

ρ

^{isthe fluiddensitywhich hasbeen}

assumedtobeconstant.Thephysicalvalues(v,u),p,and

τ

^xx,

τ

^yy,

τ

^{xy are,}respectively, thevelocity inthedirectionofXandY, the pressureandfluidtensions.

If f^eq_α is suitable using Eqs. (22) to (28), the solution of Eq.(16)isactuallythesolutionoftheNavier-Stokesequationswith the error of O(

ε

^{). As for the first suggestion forthe variable feq}_α

formicro-flows whichdonotfollowclassicapproaches,a second- orderpolynomialseries

ξ

αwasassumed.Therefore,f^eq_α is the equi- librium distribution function withthe Maxwell-Boltzmann distri- butionfunction,whichisobtainedfromquadraticorcubiccut-off expansionwithUand

ξ

α,whichiscompletelydifferent.

f^eq_α =A_α+

( ξ

α

)

XAX_α+

( ξ

α

)

YAY_α+

( ξ

α

)

²XBXX_α+

( ξ

α

)

²YBYY_α +

( ξ

α

)

X

( ξ

α

)

YBXY_α (32)

ξ

α,which isthe distribution ofmicroscopic velocity, dependson the selected model.The modelused to discretize the Boltzmann equationonameshisgenerallyrepresentedasDnQm,wherende- notesthedimensionandm denotesthenumberofpathsallowed forparticlemotion.

TheD₂Q₉modelisthemostcommontwo-dimensionalmethod.

Fig.2showsthistypeofgrid.

ξ

α isexpressedastheEq.(22),whereCis thevirtualparticle velocityonthegrid,whichmustbedeterminedaccordingtoother parameters[59]

ξ

α=

⎧ ⎨

⎩

0

α

⁼⁰

c

(

^Cos

γ

α,Sin

γ

α

)

^,

γ

α=^(α−1₂^)π,

α

^=1,2,3,4

c√

2

(

^Cos

γ

α,Sin

γ

α

)

,

γ

α=^(α−₂^5)π +^π₄,

α

=5,6,7,8 (33) Otherparametersintheequationforthedistributionofmicro- scopicvelocityareasfollows:

A0=

ρ

−2

ρ

c² −

ρ |

^u

|

²

c² +

τ

XX+

τ

YY

c² , A1=A2=0 Ax1=

ρ

^U

2c²,AX2=0 AY₁=

ρ

^V

2c²,AY₂=0

Fig. 5. Comparison of the isotherms for various Rayleigh numbers and power-law indexes at the Ha = 20 and AR = 0.4.

BXX1= 1 2c⁴

P+

ρ

^U2−

τ

XX

, BXX2=0

BYY1= 1 2c⁴

P+

ρ

^V2−

τ

^YY

, BYY2=0 BXY₂= 1

4c²

( ρ

^VU−

τ

XY

)

, BXY₁=0, I,J=X,Y (34) ThegivencoefficientsinEq.(35)aredefinedasfollows:

A1=A3=A5=A7 A2=A4=A6=A8

AX1=AX3=AX5=AX7 AX2=AX4=AX6=AX8

AY₁=AY₃=AY₅=AY₇ AY₂=AY₄=AY₆=AY₈ BXX1=BXX3=BXX5=BXX7 BXX2=BXX4=BXX6=BXX8

BYY1=BYY3=BYY5=BYY7 BYY2=BYY4=BYY6=BYY8

BXY₂=BXY₄=BXY₆=BXY₈

τ

IJ= Pr

μ

α K√

Ra

∂

^U

∂

^Y+

∂

^V

∂

^X

(35) The statementrelatedtothe magneticfield andthebuoyancy force inthe momentum equationof (36)isadded todistribution section:

f_α=A_α+

( ξ

α

)

XAX_α+

( ξ

α

)

YAY_α A₀=A₁=A₂=0

Fig. 6. Comparison of the average Nusselt number (NU m) for various power-law indexes and Rayleigh.

AX1= PrHa²

√Ra

2c² , AX2=0 AY1=Pr

θ

2c², AY2=0 (36)

TheBoltzmannenergyequationisshowninEq.(37)regardless oftheviscosityloss:

∂

^gα

∂

^{t +}

ξ

α.

∇

^gα=− 1

εφ

g_α−g^eq_α

g_α

(

^x,t+

^t

)

^−gα

(

^x,t

)

^{t =−}

1

εφ

g

(

^x,t

)

^−geqα

(

^x,t

)

(37)

g_α is the energy distribution function. It is like the momentum equationandisdiscretizedalongthevelocity.Theequilibriumand temperaturedistributionfunctionsaredefinedbyEq.(38):

g^eq_α =A_α+

( ξ

α

)

XAX_α+

( ξ

α

)

YAY_α A0=

θ

^{, A1=A2=0}

AX1=

U

θ

⁻√¹ Ra

∂ ∂θ

^X

2c² , AX2=0 AY1=

V

θ

^−√1

Ra

∂θ ∂

^Y

2c² , AY2=0 (38)

Oneoftheboundaryconditionsusedinthecavityisthebound- aryconditionofthemirrorreflection.Thisboundaryconditionex- presses thesame conditionofnon-slip inthe wall.To apply this boundary condition, the flow distribution functions must be re- flected in the directionof their motionafter their collision with thewall.Fig. 3showshowthe mirrorreflectionboundarycondi- tionisapplied.Thedistributionfunctionsmarkedwiththedashed lineareunknownfunctions.

Therefore, the unknown functionsin Fig. 3 are calculated ac- cordingto thedefinitionofthemirrorreflectionboundarycondi- tion,whichcanbegeneralizedforotherboundaries.

f2=f4

f₅=f₇

f6=f8 (39)

Theboundaryconditionfortheconstantandsinusoidaltemper- atureisalsogiveninEqs.(40).

Hotwall Coldwall

g1=

(

^sin

( π ) (

^w1+w3

) )

^{−g3 g3=}

(

^Tc

(

^w1+w3

) )

^−g1 g₅=

(

^sin

( π ) (

^w5+w₇

) )

^−g7 g₇=

(

^Tc

(

^w5+w₇

) )

^−g5

g8=

(

^sin

( π ) (

^w8+w6

) )

−g6 g6=

(

^Tc

(

^w8+w6

) )

−g8 (40) In order to havea measure to determine the amount ofheat transferoflocalandaverageNusseltnumbersontherightandleft sidewallsiscalculated,respectively,usingtheEqs.(41)and(42): Nu=2

−

∂θ

∂

^X

X=0,X=1

(41)

Nu_Ave= 1

0

NudY</ce (42)

5. Gridindependenceandvalidation

To investigate the independence of the solution process from the cavity mesh size, according to Table 1, for Ra=10⁵, Ha= 20,andAR=0.4,three exponentialfunctionindicesn=0.7,n=1, n=1.5wereevaluated.Table1showstheNusseltnumberofmean

Num on the side wallsand the maximum stream function ^max for 6 grids. Due to the small difference between the results of 140×140, 160×160, and 180×180 grids in Table 1, 160×160 meshhasbeenconsideredtobepropertothecalculations.

The results ofthisresearch are compared withseveralearlier studiestoverifytheaccuracyofthewrittenprogram.Thenatural convectionofapower-lawnon-Newtonianfluidinasquare-shaped cavitywithfixed-temperatureverticalwallsandinsulatedhorizon- tal walls which were numerically investigated in Refs. [62] and [63]. In Table 2, the results of the present study are compared withtheresultspresentedinvariousreferences,whichareingood agreementwitheachother.

6. Resultsanddiscussion 6.1. TheeffectofRayleighnumber

In this section, by keeping the parameters of Ha=20 and AR=0.4constant,theeffectofchangingtheRayleighnumberand exponential function indexon theflow andtemperature fieldsis investigated. Fig. 3 shows the flow field for the three Rayleigh numbers 10³,10⁴,10⁵ andthree exponential indices 1, 0.7, 1.4. In all ofthe diagrams,a clockwise-spinningvortexis formed inthe cavity.Inallexponentialfunctionindices,itisobservedthatwith increasing Rayleighnumberduetoincreasedbuoyancy, thenatu- ralconvectioninthecavityisstrengthenedandthedensityofthe streamlines along the wallsincreases.Therefore, thevelocities of thevorticesincreaseandthemaximumvalueoftheflowfunction atthecenterofthevortexincreases.AtN=0.7,duetothelowap- parentviscosityofthefluid,theflowpenetrationintothenarrow sideofthecavityincreaseswithanincrease intheRayleighnum- ber,andtheflowvelocityincreasesinthatpart.But,atN=1.4,due to the high apparent viscosity of the non-Newtonian fluid, fluid penetration does not change much with increasing the Rayleigh number.AthighRayleighnumbers,itisobservedthatwithanin- creasein theexponential function index, thebuoyancy decreases due to the increased apparent viscosity. Therefore,with reduced buoyancy,naturalconvectionbecomesweakerandtheflowveloc- itydecreases.Inthe lowerRayleighnumbers, thedecreaseinthe velocityislesssensible.Itisalsoobservedinallthestatesthatthe flowpenetration within thenarrowrightside isreducedwithan increaseintheindexoftheexponentialfunction.Thisismoresig- nificant athigherRayleigh numbers, indicating a decreasein the flowrateathigherRayleighnumbers.

Fig.5plotsthetemperaturefieldagainst Rayleighnumberand the exponential index forHa=20, AR=0.4. It is evident that, at lower Rayleigh numbers, the isothermal lines are regular and in the form ofa hot temperaturewall profile (a sinusoidal profile).

Thisindicatesthatthedominantmechanismofheattransferisdi- rectedatlowRayleighnumbers.WiththeincreaseintheRayleigh numberduetoincreaseinthebuoyancy,thedominantheattrans- fermechanismischangedtonaturalconvection.AstheRileynum- berincreases,thedensityofthelinesincreasesalongthehotwall, andisothermallinesbecomemoreirregular.Theheattransferrate increaseswithanincreaseinthedensityofisothermallines.With an increaseinthe exponentialindex, duetotheincreasedappar- entviscosityofthefluid,thefluidconvectionstrengthisweakened andtheflowvelocitydecreases.Therefore,withanincreaseinthe indexof theexponential function, the contributionof conductive heattransferisincreasedandthecontributionofthenaturalcon- vection heattransferisreduced.Thisreducestheheattransferat highRayleighnumbers.

In Fig. 6, the average Nusselt number is plotted on a hot wall of the cavity for different Rayleigh numbers considered for threetypesofNewtonian,non-Newtonianshearthinningandnon- Newtonian shear thickening fluids. An increase in the Rayleigh

Fig. 7. Comparison of the streamlines for various Hartmann numbers and power-law indexes at AR = 0.4 and Ra = 10 ⁵.

number due to the higher flow velocity promotes buoyancy and increasestheheattransfer rate.However,atlower Rayleighnum- bers, conductionisthe dominantheattransfer mechanisminthe cavity.Therefore,duetothesmallerscaleofheattransferby con- ductioncomparedwithconvection,loweraverageNusseltnumbers

are observed compared with the higher Rayleigh numbers. With anincreaseintheRayleighnumber,andconsequentlytheincrease intheheattransfer,theaverageNusseltnumberalsoincreases.In theshearthinningfluids,thenaturalconvectionheattransfercon- tributionishigherthanthatoftheconductiveheattransfer,com-

Fig. 8. Comparison of isotherms for various Hartmann numbers and power-law indexes at AR = 0.4 and Ra = 10 ⁵.

Fig. 9. Local Nusselt number distribution on the right wall at Ra = 10 ⁵and AR = 0.4, for Hartmann numbers of Ha = 0 to 40 and different power-law indexes (a) N = 0.7 (b) N = 1.0 (c) N = 1.4.

pared withthe shear thickening fluids. It can be seen that with theincrease inthe power-lawindexandthe increaseinthe con- tributionofconductiveheattransfercomparedwiththeconvection heattransfer,theaverageNusseltnumberonthehotwallalsode- creases.

6.2. TheeffectofHartmannnumber

In this section, the temperature and flow fields are com- pared for three Newtonian, non-Newtonian shear thinning and non-Newtonian shear thickening fluids by maintaining Ra=105, AR=0.4,andchangingtheHartmannnumberinthe0to40range.

Figs.7and8respectivelyshowtheflowfieldandthetemperature field. Withrespect tothestreamlines(Fig.7), withan increasein theexponentialindexforafixedHartmannumber,thedensityof thelinesonthesideofthewallsdecreases;andwiththeincrease intheapparentviscosityofthefluid,thevelocity offluidconvec- tionalsodecreases.Withtheincreaseinthepower-lawindex,the maximum value of the flow function that is located in the cen- ter ofthe cavity isalso reduced, leading to the reduction ofthe vortex velocity inthe cavity.With theincrease in thepower-law indexandincrease inthefluid apparentviscosity, theflowpene-

trationinthenarrowpartoftherightsideofthecavityincreases leadingto the increase in thevelocity in thispart. Onthe other hand,themagnetic field actsasa barrierandcreates adrag-like forceagainst theflow.Ina fixedexponentialfunction index,with theincreaseintheHartmannnumber,theLorentzforce increases andthepoweroftheflowvorticesdecreases.Thiscanbe seenby minimizingthemaximumvalueoftheflowfunction.AstheHart- mannumberincreases,thevortexvelocitydecreases.Ontheother hand,asthe field is parallel withthe horizontalaxis ofthe cav- ity,itcanbeseenthatwiththeincreaseintheHartmannnumber, theflowoffluidmovesfurthertowardtheendsofthewallstothe rightsideofthecavityandpenetratesmoreintothenarrowerside ofthecavity.

Fig.8showsthetemperaturefieldfortheabovevalues.Itisev- identthatwiththeincreaseinthepower-lawindex,theapparent viscosityofthefluiddecreases andnaturalconvectionheattrans- ferisweakened.Itisalsoseeninthefigurethatthedensityofthe linesalongthecold andhot wallsisreduced.Therefore,withthe increaseinthepower-law,theheattransferratefromthecoldand hot walls of the cavity decreases. With an increase in the Hart- mannnumber,thenaturalconvectionheattransferofthefluidin- sidethe cavitydecreasesduetothe increaseofthe Lorentzforce

Fig. 10. velocity distributions of u at H = 0.5 at Ra = 10 ⁵ and AR = 0.4, for differ- ent power-law indexes: two lines first (N = 0.7), two lines seconds (N = 1) and two lines thirds (N = 1.4) and Hartmann numbers dashed lines (Ha = 10) and solid lines (Ha = 40).

andthereductionofthefluidconvectionpower.Therefore,thein- fluenceoftheisothermal linesinsidethecavitydecreasesleading tothedecreasedheattransfer rate.Thus, thehighestrateofcon- densation occurred along the walls and the highest rateof heat transferin the shear thinning fluid occurredin the absence ofa magnetic field. It is also seen at higher Hartmann numbers that theisothermallinesbecomesimilartothetemperatureprofilesap- pliedtothehotwall,indicating thattheheattransfermechanism ismoreconductiveinthiscase.

Fig. 9 showsthe local Nusseltnumber on the hot cavitywall foraRayleighnumberof10⁵ andtheaspectratioof0.4fordiffer- entHartmann numbers andthreedifferent power-lawindices.In allthreegraphsit canbeseen thatwithan increaseintheHart- mannnumberanddecreaseinthenaturalconvectionpowerofthe

Fig. 12. Comparison of the average Nusselt number for various power-law indexes and Hartmann numbers.

fluid inthe cavity, thelocal Nusseltnumber onthe hot wall de- creases.Thisdecreaseintheshearthinningfluidishigherdueto the higher fluid convection at lower Hartmann numbers. Due to theshapeofthesinusoidaltemperatureprofileonthehotwall,it isobservedthat theNusseltnumberisnegativeintheupperand lowerpartsofthewall;thatisheatistransferredfromthefluidto thewall.ThemaximumlocalNusseltnumberoccursinthemiddle ofthehotwallthathasthemaximumtemperature.Acomparison between the three graphs revealsthat the local Nusselt number decreaseswithincreasingthepower-lawindexanddecreasingthe fluidconvectionpower.

Fig. 10 shows the profile of horizontal velocity component at H=0.5 for Rayleigh 10⁵, aspect ratio 0.4, and three different power-lawindicesintwoweak andstrongmagneticfields.It can beseenthatwiththeincreaseinthepower-lawindexandincrease intheapparentviscosityofthefluid,thevelocityofthefluidalso

Fig. 11. Velocity profiles at H = 0.5 at Ra = 10 ⁵and AR = 0.4, for Hartmann numbers (a) N = 0.7 (b) N = 1.4.

Fig. 13. Comparison of streamlines for various aspect ratio and power-law indexes at Ha = 20 and Ra = 10 ⁵.

decreases andlower maximumvaluesare observedfortheveloc- ity. It can also be seen that withthe increase in the Hartmann numberandLorentz force, thefluid convectionandfluid velocity insidethecavityalsodecrease.Thisdecreaseintheflowratedue totheincreaseintheHartmannnumberintheshearthinningflu- ids is greater than the shearthickening fluids. Fig. 11showsthe profileofverticalvelocitycomponentatH=0.5forRayleighnum- ber 10⁵, aspect ratio 0.4, anddifferent magnetic fields for shear thickeningandshearthinningfluids.

Fig.11alsoshowsthatwiththeincreaseintheHartmannnum- ber, the velocity of the vortices inside the cavity decreases and less velocity occurs inthe cavity. It is also observed by compar- ing two graphs that, ina shear thinning fluid, more velocity oc- cursinsidethecavityduetoitslowerviscositycomparedwiththe shearthickeningfluid.Themaximumvelocityintheshearthinning fluid isobserved inthe vicinityof walls,which canalso increase theheattransferratefromthewalls.Ontheotherhand,itcanbe seenthatwiththeincreaseintheHartmannnumberintheshear thinningfluid,thevelocityislessthanthatoftheshearthickening fluid,duetoitslowerviscosityintheshearthinningfluid.There- fore,thistype offluid ismostaffectedbythemagneticfield;and the fluid velocity is much reducedat higher Hartmann numbers andstrongmagneticfields.

Fig. 12 showsthe average Nusseltnumber on thehot wall of the cavity, for Rayleigh number 10⁵, the aspect ratioof 0.4, and

thedifferent magneticfields inthe threedifferentpower-law in- dices.Inlow Hartman numbers,withthedecrease inthe power- lawindex due to the decrease in the apparent viscosity, Nusselt number increases moderately due to the increase in the natural convectionheat transfer rateby removing the apparent viscosity.

Withthe increase in the Hartmann numberin all power-law in- dices, the average Nusseltnumber decreases. This is also due to theincreaseintheLorentz forceand, consequently,thereduction ofbuoyancyandnaturalconvectionheattransfer ratein thecav- ity.Itcanbesaidthat withan increaseintheHartmannnumber, the heat transfermechanism changes fromnatural convection to conductionandthemain heattransferisconductedinthecavity.

Withlower viscosity,thefluidismoresensitivetothechangesin the magnitudeof the magnetic field. With greater Lorentz force, thevelocityisloweratlowpower-lawindexandapparent viscos- ity.ItisthenseenthattheslopeoftheaverageNusseltnumberfor ashearthinningfluidismuchhigherthanthatoftheshearthick- eningfluid;asatHa=40,thevelocityofnaturalconvectioninthe shearthinningfluidislessthanthatoftheshearthickeningfluid andtheheattransferrateislowerinthehotwall.

6.3.Theeffectofchangesintheaspectratio

Inthis section, theeffect ofchanging the aspect ratiofor the Rayleigh number 10⁵ and the Hartman number 20 on the flow

Fig. 14. Comparison of isotherms for various aspect ratio and power-law indexes at Ha = 20 and Ra = 10 ⁵.

andtemperaturefield isexamined.At eachstep, bychangingthe power-law index, this parameter is simultaneously examined. In Fig.13,theflowfieldandinFig.14,thetemperaturefieldareplot- ted forthe parameters mentioned. It can be seen that with the increase in the aspect ratio and the shrinking space for the de- velopmentof flowin the cavity,the densityofthe linesand the velocityinthecavitydecrease.Itisalsoobservedthatwiththein- creaseinthelevelofaspectratio,themaximumvalueoftheflow functionisalsoreduced, indicatinga reductioninthestrength of theclockwisevortexgeneratedinthecavity.Withtheincrease in thepower-lawindexandtheapparentviscosity,adecreaseinthe densityofthelinesandconsequentlyinthevelocityinthecavity wasobserved.Withasimultaneousincrease inthepower-lawin- dexandtheaspectratioduetothenarrowingofthespacetocre- ateavortexandincreaseintheapparentviscosityofthefluid,the maximumamount of flow field reduces enormously anda weak vortexisproduced.

In Fig.14, itcan be seen that withthe increase inthe aspect ratioandthecoldtemperatureapproaching thehot wall,thenat- uralconvection inthe cavityis reducedand theconductive heat transferrateincreases.Therefore,thedensityofisothermallinesis reducedalongthehotwallandtheheattransferrateincreases.At higherpower-law indices, dueto higherapparent viscosity, with narrowing thespace forthe fluid flow, the naturalconvection of fluid is reducedand the heat transfer mechanism movesfurther

towardstheconduction;sothatintheshearthickeningfluidsand aspectratioof0.6,itisobservedthatthemainmechanismisthe conductiveheattransfer,andtheshapeofisothermallinesissim- ilartothehotwalltemperatureprofile.

InFig.15,thelocalNusseltnumberisplottedonahotwallof the cavityforthree aspect ratios, power-lawindexand fixed pa- rametersofHa=20andRa=10⁵.Inallofthepower-law indices, thenaturalconvectioninthecavitydecreasesandtheconductive heat transferincreases withan increase inthe aspect ratio.Con- sequently, the local Nusselt number on the wall is reduced due tothehighernaturalconvectionheattransferratecomparedwith conductiveheat transferrate.Withtheincrease inthepower-law indexandtheapparentfluidviscosity,thenaturalconvectionheat transferofthefluidandthelocalNusseltnumberdecrease.

The average Nusseltnumber is plotted in Fig. 16, on the hot wall of the cavity for three aspect ratios of 0.2, 0.4 and 0.6, and three Newtonian, non-Newtonian shear thinning, and non- Newtonianshearthickeningfluids.Itcanbeseenthat,atdifferent aspect ratios,theaverage Nusseltnumberisthe largestforshear thinning,Newtonianandshearthickeningfluidsinorderofmagni- tude.Thisisalsoduetoastrongernaturalconvectioninthefluids withlowerapparentviscosity.Byincreasingtheaspectratiointhe Newtonianandsearthinningfluids,theNusseltnumberdecreases moderatelyduetothereductioninthenaturalheattransfer.How- ever, inshear thickening fluids, dueto the weak naturalconvec-

Fig. 15. Local Nusselt number distribution on the right wall at Ra = 10 ⁵and Ha = 20, for various aspect ratio and different power-law indexes (a) N = 0.7 (b) N = 1.0 (c) N = 1.4.

tionheattransfermechanismcausedbythehighapparentviscos- ity,asthecoldfixed-temperaturewallapproachesthehotwallof thecavity,theconductiveheattransfermechanismincreaseslead- ing totheincreasedheat transferrateinthefirstplace.However, withafurtherincreaseinthecavityaspectratio,duetotheshrink- ingspaceoftheflowandthereductionofthestrengthofthefluid vortices,the naturalconvectionheattransferisalmost eliminated andtheaverageNusseltnumberisalsoreduced.

7. Conclusion

In this study, natural convection heat transfer in a two- dimensional C-shaped cavitywith an applied magnetic field was investigated for a power-law non-Newtonian fluid using the fi- niteelementlatticeBoltzmannmethod.Thetopandbottomwalls werethermallyinsulated,whereastheleftwallhadanon-uniform temperature distribution. The effect of Rayleigh number, Hart- mannumber, aspect ratioandexponential function indexon the

streamlinesdistribution,isothermallinesdistribution,localNusselt numberandaverageNusseltnumberwereinvestigated.Theresults areasfollows:

1. Anincrease in theRayleigh numberpromotes naturalcon- vection inside the cavity, increasing the heat transfer rate.

Theimprovednaturalconvection causedby theincrease in theRayleigh number wasmore significant for n<1(shear thinningfluids).

2. At higher Rayleigh numbers, the amount of Nusselt num- berdecreasesbyincreasingthepower-lawindex(N).Infact, theaverageNusseltnumberincreasesinshear-thinningflu- ids (n<1) compared with Newtonian fluids (n=1) while decreasingforshear thickening fluids(n-1)compared with Newtonian fluids. Changes inthe average Nusseltnumbers atn<1aremostaffectedbyvariationsofn.

3. Byincreasing theHartmann number,the fluidvelocity and flow power decrease. Therefore, an increased Hartmann