natural convection heat transfer enhancement in a ...€¦ · ied natural convection in a vertical...

Salam Hadi Hussain1Mechanical Engineering Department,

College of Engineering,

Babylon University,

Babylon Province 00964, Iraq

e-mail: [email protected]

Ahmed Kadhim HusseinMechanical Engineering Department,

College of Engineering,

Babylon University,

Babylon Province, Iraq

e-mail: [email protected]

Natural Convection HeatTransfer Enhancementin a Differentially HeatedParallelogrammic EnclosureFilled With Copper-WaterNanofluidLaminar natural convection of a nanofluid consists of water and copper in a differentiallyheated parallelogrammic enclosure has been studied numerically using the finite volumemethod (FVM). Governing equations are solved over a wide range of Rayleigh numbers(104�Ra� 106), skew angles (�60 deg�U�þ60 deg), aspect ratios (0.5�AR� 4),and solid volume fractions (0�u� 0.2). Effects of all these parameters on flow and ther-mal fields are presented in form of streamline, isotherm contours and average Nusseltnumber. It is shown that the heat transfer rate increases remarkably by the addition ofcopper-water nanofluid and the shape of the convection vortices is sensitive to the skewangle variation. [DOI: 10.1115/1.4027448]

Keywords: Nanofluid, natural convection, aspect ratio, parallelogrammic enclosure

1 Introduction

Thermal fluids are very important for heat transfer in manyindustrial applications. Low thermal conductivity of conventionalheat transfer fluids such as water, oil, and ethylene glycol mixtureis a serious limitation in improving the performance of theseindustrial applications. To overcome this disadvantage, there is astrong motivation to develop advanced heat transfer fluids withsubstantially higher thermal conductivity. One way is to add smallsolid particles which is called nanofluids [1]. Nanofluid is a sus-pension of solid–liquid mixture nanoparticles (1–100 nm diame-ter) in a traditional liquids. The nanofluid is stable, introduce verysmall pressure drop and it can pass through nano channels [2,3].Nanofluids are utilized in numerous biological, and scientificapplications, such as blood clogging, cell transport in arteries andveins, drug and gene delivery and manufacturing of nanocompo-sites because the presence of nanoparticles in fluids increases theireffective thermal conductivities [4]. In the literature, variousresearch works have been focused mainly on natural convectionin enclosure filled with nanofluid. Keblinski et al. [5] studiedmechanisms of heat flow in suspensions of nanosized particles.Khanafer et al. [6] studied buoyancy-driven heat transfer enhance-ment in a rectangular enclosure containing nanofluids. Jou andTzeng [7] simulated numerically natural convection of copper-water nanofluids in a rectangular enclosure. Ho and Lin [8] stud-ied natural convection in a vertical square enclosure filled withAl2O3-water nanofluid.They observed a substantial heat transferreduction except the cases of the particle mass fraction not higherthan 1 wt.% Hwang et al. [9] investigated the natural convectionin a rectangular cavity heated from below. They concluded thatthe ratio of heat transfer coefficient of nanofluids to that of basefluid was decreased as size of nanoparticles increased. The works

of Refs. [10,11] conducted a literature review on heat transfercharacteristics of nanofluids. While Daungthongsuk and Wongw-ises [12] performed a review of convective heat transfer for nano-fluids. Ho et al. [13] studied the effects due to uncertainties ineffective dynamic viscosity and thermal conductivity of nanofluidon natural convection in a square enclosure with alumina–waternanofluid. Santra et al. [2] investigated the natural convection in asquare cavity filled with copper-water nanofluid. Oztop and Abu-Nada [14] simulated natural convection in a partially heated rec-tangular enclosure using nanofluids. It was found that the heaterlocation affected the flow and temperature fields. Anilkumar andJilani [15] investigated the natural convection in an enclosurewith fin utilizing nanofluids. Ghasemi and Aminossadati [16]studied natural convection in an inclined enclosure filled with awater-CuO nanofluid. The results indicated that adding nanopar-ticles improved the heat transfer performance. Ho et al. [17]investigated the natural convection of Al2O3-water nanofluid invertical square enclosures of three different sizes. Ghasemi andAminossadati [18] studied the natural convection of an oscillatingheat source embedded on the left wall of a square enclosure filledwith nanofluids. Abu-Nada et al. [19] investigated the natural con-vection in a differentially heated enclosure using variable thermalconductivity and variable viscosity of Al2O3-water and CuO-water nanofluids. It was found that at high Rayleigh numbers, theaverage Nusselt number was more sensitive to viscosity modelsthan to thermal conductivity models. Corcione [20] investigatedthe buoyancy-driven nanofluids inside rectangular enclosures dif-ferentially heated at the vertical walls. Lin and Violi [21] analyzedthe natural convection in a vertical cavity filled with Al2O3-waternanofluid. Alloui et al. [22] studied natural convection in a shal-low rectangular cavity filled with nanofluids. Mahmoudi et al.[23] presented a numerical study of natural convection cooling ofa heat source horizontally attached to the left vertical wall of asquare cavity filled with copper-water nanofluid. From the otherside, various papers are studied natural convection in a parallelog-rammic enclosure filled with different classical fluids [24–29]. Infact, for various practical applications, the parallelogrammic

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received November 21, 2012; finalmanuscript received April 12, 2014; published online May 9, 2014. Assoc. Editor:William P. Klinzing.

Journal of Heat Transfer AUGUST 2014, Vol. 136 / 082502-1Copyright VC 2014 by ASME

Downloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 08/30/2014 Terms of Use: http://asme.org/terms

enclosures have a strong potential to be used in the assembly ofconstruction elements, since they present an excellent heat trans-fer performance relative to the classical square or rectangularshape. Many studies [24–30] explained that the parallelogrammicenclosure can be used as a heat transfer inhibitor (thermal insula-tion) or act as a heat transfer promoter (heat transfer enhance-ment). But none of them studied this problem when theparallelogrammic enclosure is filled with cooper-water nanofluidfor various aspect geometric ratios and high Cu concentrationratios when the sloping walls are inclined at an angle (U) withrespect to the vertical which is varied as 0 deg, 6 30 deg, 6 60 deg,respectively. This point represents the major original contributionof the present work. Therefore; the purpose of the present work isto examine the effects of Rayleigh number, solid volume fraction,aspect ratio and skew angle on the natural convection in a paralle-logrammic enclosure filled with Cu-water nanofluid.

2 Geometrical Configuration, MathematicalModeling, and Numerical Solution

Figure 1 shows a parallelogrammic enclosure of length (H) andwidth (W).The enclosure is filled with a suspension of coppernanoparticles in water. Both the vertical left and right sidewallsare maintained at constant hot and cold temperatures, respec-tively. The left and right sidewalls are inclined at an angle (U)with respect to the vertical (sometimes called skew angle). Theother two upper and lower walls are assumed adiabatic. The Ray-leigh number (Ra) varying from 104 to 106, while solid volumefractions, (u) have been varied from 0% to 20% with an incrementof 5%. Skew angle is varied as 0 deg, �30 deg, �60 deg, 30 deg,and 60 deg, respectively. The enclosure aspect ratio (AR¼H/W)varying from 0.5 to 4. The thermophysical properties of base fluid(water) and copper nanoparticle are listed in Table 1.The problemis modeled mathematically based on the following assumptions:

(1) Nanofluid is considered single phase, Newtonian, steady,laminar, and incompressible.

(2) Both fluid and nanoparticles are assumed in thermal equi-librium state and have a constant thermophysicalproperties.

(3) No slip occurs between liquid and nanoparticles in terms ofboth temperature and velocity.

The dimensionless governing equations can be written asfollows [32]:

@U

@Xþ @V@Y¼ 0 (1)

U@U

@Xþ V @U

@Y¼ �

qfqnf

@P

@Xþ Pr#nf

#f

@2U

@X2þ @

2U

@Y2

� �(2)

U@V

@Xþ V @V

@Y¼ �

qfqnf

@P

@Yþ Pr#nf

#f

@2V

@X2þ @

2V

@Y2

� �þ RaPr qbð Þnf

qnfbfh

(3)

U@h@Xþ V @h

@Y¼ anf

af

@2h@X2þ @

2h@Y2

� �(4)

where

h ¼ T � TCTH � TC

; X ¼ xH; Y ¼ y

H; U ¼ uH

af; V ¼ vH

af

w ¼ Waf; P ¼ pH

2

qnfa2f

; Pr ¼ #a; Ra ¼

gbf TH � TCð ÞH3

#f af

anf ¼ knf=ðqcPÞnf is nanofluid thermal diffusivity. Effective densityof nanofluid ðqnfÞ at reference temperature can be defined as [32]

qnf ¼ 1� uð Þqf þ uqs (5)

The heat capacitance of nanofluid can be given as [32]

ðqcPÞnf ¼ 1� uð ÞðqcPÞf þ u qcPð Þs (6)

The effective thermal expansion coefficient ðbnfÞ and viscosity ofnanofluid ðlnfÞ were introduced by Brinkman [33] as below:

ðqbÞnf ¼ 1� uð ÞðqbÞf þ u qbð Þs (7)

lnf ¼lf

1� uð Þ2:5(8)

The effective thermal conductivity (knf) of nanofluid is given by[34]

knfkf¼ ks þ 2kf � 2uðkf � ksÞ

ks þ 2kf þ uðkf � ksÞ(9)

The relationships between stream function and velocity compo-nents are

U ¼ @w@Y

; V ¼ � @w@X

(10)

The average Nusselt number at the hot left sidewall (Nuh) is givenby

Nuh ¼ �1

S

ðS0

knfkf

� �@h@n

� �X¼0

dn (11)

The boundary conditions are expressed as follows:

(1) Left sidewall of the parallelogrammic cavity is maintainedat uniform hot temperature (TH) so that: 0�X� cosðUÞ,h ¼ 1;V ¼ 0; and U ¼ 0.

(2) Right sidewall of the parallelogrammic cavity is maintainedat uniform cold temperature (TC) so that: 1�X� 1þ cosðUÞ,h ¼ 0;V ¼ 0; and U ¼ 0.

Fig. 1 Schematic diagram and coordinate system of thephysical domain with boundary conditions

Table 1 Thermo physical properties of base fluid (water) andnanoparticles (Cu) [31]

Physical properties Fluid phase (Water) Cu

cP (J/kg�K) 4179 385q (kg/m3) 997.1 8933k (W/m�K) 0.613 400a� 107ðm2=sÞ 1.47 1163.1b� 105ð1=KÞ 21 1.67

082502-2 / Vol. 136, AUGUST 2014 Transactions of the ASME


(3) Lower and upper horizontal walls are considered adiabatic,so thatY¼ 0 and Y¼ 1, ð@h=@YÞ ¼ 0, U¼V¼ 0.

The dimensionless governing equations (Eqs. (1)–(4)) aresolved numerically using the FVM in addition with hybrid differ-encing scheme as explained in Patankar [35]. Equations for thesolid and fluid flow are solved as a single domain problem. Inorder to verify the numerical code accuracy, a comparison withthe previous published results is necessary. To reach this goal, thepresent numerical results are verified against the results obtainedby [36] as shown in Table 2. The comparison of the average Nus-selt number at the hot left sidewall is made using the following pa-rameters: Ra¼ (104–106), u¼ (0–0.2), U¼ 0 deg and using Cu-water nanofluid. It is clear from this comparison that the presentcomputed results are in excellent agreement with the correspond-ing results of [36].

3 Results and Discussion

The flow and thermal fields characteristics in a parallelogram-mic enclosure filled with a nanofluid are presented in Figures 2–9.Due to the buoyancy force effect, the flow field begins to transferfrom the hot left sidewall and then reaches to the insulated topwall. The insulated top wall leads to make the flow field changesits direction and moves toward the cold right sidewall and then

changes its direction second after hitting the insulated bottomwall. This movement creates the regular single circulation vorti-ces. For low Rayleigh number (Ra¼ 104) as shown in Fig. 2,where the heat is transferred due to conduction, the circulation in-tensity is weak (w varies from �1.23 to �6.52) which reflects aslight flow circulation. The figure demonstrates also that the inten-sity of natural convection circulation increases rapidly as the solidvolume fraction and Rayleigh number increase. This behavior isdue to the increase of the heat transfer from the hot left sidewallto the nanofluid particles as a result of the velocity increasing. Asthe Rayleigh number increases (Ra¼ 106) (Fig. 4), the vorticeschange their shape from the uniform circular shape to the nonuni-form semi-elliptical shape while the symmetry property of thesevortices begins to diminish. Moreover, the circulation intensityincreases significantly (w varies from �12.11 to �24.13) as Ray-leigh number increases. In addition, when the solid volume frac-tion increases from (5% to 20%), the stream function increasesdramatically. This can be indicated by the absence of nanoparticleclustering when the volume fraction becomes too high. For exam-ple when (U¼ 30 deg) it increases about (83%) and when(U¼�30 deg) it increases about (84.62).This increasing when theRayleigh number is low (Ra¼ 104). But, when the Rayleigh num-ber is high (Ra¼ 106), the stream function increases strongly, forexample when (U¼ 30 deg) it increases about (86.51%) and when(U¼�30 deg) it increases about (86.81%).The reason of this sig-nificant increasing because as the solid volume fraction increases,

Table 2 Comparison of the average Nusselt number at the left hot sidewall with those of previous study of Ref. [36] for validationof the present numerical code

Ref. [36] Present work Ref. [36] Present work [36] Present work

Ra 104 104 105 105 106 106 Maximum erroru

0.00 2.274 2.271 4.722 4.719 9.230 9.224 �0.0650.05 2.421 2.428 5.066 5.093 9.962 9.954 0.5230.10 2.553 2.554 5.384 5.383 10.656 10.654 �0.0180.15 2.670 2.667 5.674 5.673 11.310 11.299 �0.1120.20 2.776 2.774 5.937 5.935 11.921 11.906 �0.125

Fig. 2 Variation of streamlines for different solid void fractions (u) and skew angles (U) at Ra 5 104

Journal of Heat Transfer AUGUST 2014, Vol. 136 / 082502-3


the velocity of the nanoparticles increases which causes toincrease flow circulation and the thermal energy transport throughthe fluid. With respect to isotherms, for low Rayleigh number(Ra¼ 104), (Fig. 3) the isotherm lines become approximately ver-tical which indicating that conduction dominates the heat transferprocess. But, the isotherm lines become irregular, random,approximately horizontal in the middle of the enclosure andgreatly deform due to the stronger circulation as the Rayleighnumber increases (Ra¼ 106). This behavior can be clearly seen in

Fig. 5. Moreover, as the solid volume fraction increases, the ther-mal conductivity of nanofluid increases which causes the heatpasses much deeper into the nanofluid, before being carried awayby the convection currents. Figures 2–5 depict also the skew angleeffect on the streamlines and isotherms. It can be observed fromFigs. ((2) and (4)), that as skew angle increases in the positivedirection from (U¼ 0 deg to U¼ 60 deg) or decreases in the nega-tive direction from (U¼ 0 deg to U¼�60 deg), the stream func-tion decreases. This is due to the reduction in the buoyancy force

Fig. 3 Variation of isotherms for different solid void fractions (u) and skew angles (U) at Ra 5 104

Fig. 4 Variation of streamlines for different solid void fractions (u) and skew angles (U) at Ra 5 106



effect. This behavior can be noticed for all the considered rangeof Rayleigh number and solid volume fraction. Also, vorticesbegin to enlarge toward the upper right and the lower left cornersof the enclosure as the skew angle increases from (U¼ 30 deg) to(U¼ 60 deg).While, they begin to enlarge toward the upper leftand the lower right corners of the enclosure as the skew angledecreases from (U¼�30 deg) to (U¼�60 deg). So, it can beconcluded that the skew angle can be used to damp the intensity

of circulation for all the considered range of Rayleigh number andsolid volume fraction. For this reason, the maximum stream func-tion can be seen when skew angle is zero (U¼ 0 deg). Withrespect to isotherms, it is evident from Figs. ((3) and (5)) that asthe skew angle increases in the positive direction from (U¼ 0 degto U¼ 60 deg) or decreases in the negative direction from(U¼ 0 deg to U¼�60 deg), the shape of isotherm lines becomesapproximately similar to the vertical lines indicating that the heat

Fig. 5 Variation of isotherms for different solid void fractions (u) and skew angles (U) at Ra 5 106

Fig. 6 Variation of streamlines for various ARs and skew angles (U) at Ra 5 105, u 5 0, 0.1 with pure fluid (dashed lines - -) andnanofluid (solid lines —)



is transferred by conduction. The isotherm lines are clearly con-fused when the skew angle is zero (U¼ 0 deg).In this case, theflow circulation role is significant and the heat is transferred byconvection. Figures 6 and 7 illustrate, respectively, the variationof streamlines and isotherms for various AR and skew angle (U)for pure fluid (u¼ 0) and nanofluid (u¼ 0.1) when Ra¼ 105. Itcan be seen that the variation in the aspect ratio from (AR¼ 0.5)to (AR¼ 4) causes a clear change in the shape of both streamlinesand isotherms. When the aspect ratio is less than one (AR¼ 0.5)[slender enclosure], the stream function are higher than their cor-responding values at another aspect ratio values for both pure fluidand nanofluid. In this case, the convection effect becomes signifi-cant and the circulation intensity is strong. This is due to theextension in the enclosure width which increases the flow circula-tion area and as a result causes to increase the convection effect. Itis also observed that when (AR¼ 0.5), vortices begin to extendhorizontally inside the enclosure taking approximately the ellip-soidal shape. When the aspect ratio is unity (AR¼ 1) [square en-closure], the stream function begins to decrease slightly for bothpure fluid and nanofluid. Therefore, it can be concluded that as the

aspect ratio increases, the intensity of circulation decreases. Thisis due to the reduction in the flow circulation area when the aspectratio increases from (AR¼ 0.5) to (AR¼ 1). Also, the core ofrecirculating vortices begins to compress inside the enclosure.When the aspect ratio is greater than one (AR¼ 2 and 4) [shallowenclosure], the stream function begins to decrease rapidly. In thiscase, the core of recirculating vortices begins to be more uniformand decreases as aspect ratio increases. It can be seen in this casethat, the convection role becomes slight and the circulation inten-sity diminishes gradually. This is due to the reduction in the en-closure width which decreases the flow circulation area and as aresult causes to damp the convection role. Moreover, the recircu-lating vortices inside the enclosure begin to extend vertically andits shape becomes approximately circular. Moreover, it can benoticed in Fig. 6, that as the skew angle increases in the positivedirection from (U¼ 0 deg to U¼ 60 deg) or decreases in the nega-tive direction from (U¼ 0 deg to U¼�60 deg), the stream func-tion decreases due to the decreasing of the buoyancy force effect.Also, the flow structure considerably changes and begins to spreadalong the inclined enclosure sidewalls with changing the skew

Fig. 7 Variation of isotherms for various ARs and skew angles (U) at Ra 5 105, u 5 0, 0.1 withpure fluid (dashed lines - -) and nanofluid (solid lines —)

Fig. 8 Variation of average Nusselt numbers with skew anglesfor u 5 0, 0.1 Ra 5 104 & 106 and AR 5 1

Fig. 9 Variation of the average Nusselt number with aspectratio for various skew angles at Ra 5 105 and u 5 0, 0.1



angle in positive or negative direction for various aspect ratios.With respect to isotherms, the variation in the aspect ratio causesa clear change in the isotherms. When the aspect ratio is less thanone (AR¼ 0.5), a strong disturbance in isotherms can be seenindicating that the heat is transferred by convection. When the as-pect ratio is greater than one (AR¼ 2 and 4), the isotherms areapproximately linear in shape and nearly parallel one to eachother. This refers that the heat is transferred by conduction. Fur-thermore, it can be noticed from Fig. 7 that the isotherms for thenanofluid are more vertical than the pure water indicating a highconduction heat transfer contribution. This is because the additionof nanofluid with higher thermal conductivity improves signifi-cantly the conduction heat transfer effect. Figure 8 explains thevariation of the average Nusselt number at the hot left sidewallwith skew angle for u¼ 0 and 0.1, AR¼ 1 and Ra¼ 104, 105, and106, respectively. It can be seen that the average Nusselt numberincreases as the Rayleigh number increases. This is due to theincrease in the convection circulation intensity as a result of highbuoyancy forces. From the other side, the average Nusselt numberincreases also with the addition of nanofluid (u¼ 0.1) comparedwith its value of pure fluid (u¼ 0) for all values of skew angleand Rayleigh number. The reason of this behavior is due to thestrong circulation produced by the thermal energy transport withthe addition of nanofluid which accelerates the flow. Also, theeffective thermal conductivity increases with the addition of nano-fluid which eventually leads to enhance the heat transfer rate rep-resented by the average Nusselt number. Figure 8 shows also, thatthe maximum value of the average Nusselt number can beobtained when the skew angle equals zero. This is due to the highamount of the flow velocity when the skew angle equals zero.But, when the skew angle increases in the positive direction from(U¼ 0 deg to U¼ 60 deg) or decreases in the negative directionfrom (U¼ 0 deg to U¼�60 deg), the average Nusselt numberbegins to decrease due to the reduction in the flow velocity whichcauses to reduce the transport of thermal energy through the fluidcausing a clear reduction in the value of average Nusselt number.Figure 9 shows the variation of the average Nusselt number withaspect ratio for various skew angle at Ra¼ 105 and u¼ 0, 0.1.Again, it can be seen that the average Nusselt number is maxi-mum for both pure fluid and nanofluid when the skew angle equalszero. After that, it begins to decrease when the skew angleincreases in the positive direction or decreases in the negativedirection for the same reasons explained previously. In addition,the average Nusselt number increases with the addition of nano-fluid (u¼ 0.1) compared with its value of pure fluid (u¼ 0) forall values of skew angle and aspect ratio. In general, it is clearfrom Fig. 9, that there is a linear variation between the averageNusselt number and the aspect ratio and no remarkable increase ofthe average Nusselt number can be observed when the aspect ratioincreases. This behavior can be seen when the skew angle(U¼ 0 deg and U¼�30 deg). The maximum value of the averageNusselt number occurs adjacent (AR¼ 2).When the aspect ratio isless than (AR¼ 2), or greater than (AR¼ 2) the average Nusseltnumber decreases due to the reduction in the convection effectand the flow circulation area, respectively. In contrast of this gen-eral behavior, when the aspect ratio is greater than (AR¼ 2) for(U¼�60 deg), the average Nusselt number increases continu-ously when the aspect ratio increases. This is because at high val-ues of skew angle (U¼�60 deg) the strong reduction in theconvection role causes a high increase of the conduction role inthe heat transfer process which causes a jump in the average Nus-selt number values. This behavior can be seen for both fluid andnanofluid. However, it can be noticed also from Fig. 9, that themaximum value of the average Nusselt number occurs adjacent(AR¼ 2).

4 Conclusions

Laminar natural convection of a nanofluid consists of water andcopper in a differentially heated parallelogrammic enclosure has

been studied numerically using the finite volume method. It isfound that when solid volume fraction and Rayleigh numberincrease, the intensity of the circulation increases. Also, thestream function decreases when the skew angle increases in thepositive direction from (U¼ 0 deg to U¼ 60 deg) or decreases inthe negative direction from (U¼ 0 deg to U¼�60 deg). The aver-age Nusselt number at the hot left sidewall increases with theaddition of the nanofluid and increasing the Rayleigh number andreaches a maximum value when the skew angle equals zero. But,it begins to decrease gradually as the skew angle increases in thepositive direction or decreases in the negative direction. Also, it isfound that when the enclosure aspect ratio is less than one(AR¼ 0.5), the convection effect becomes strong and the circula-tion strength increases for both pure fluid and nanofluid. A reversebehavior can be seen when the aspect ratio is greater than one(AR¼ 2 and 4).

Nomenclature

AR ¼ aspect ratio (H/W)cP ¼ specific heat at constant pressure (kJ/kg�K)g ¼ gravitational acceleration (m/s2)H ¼ height of the parallelogrammic cavity (m)K ¼ thermal conductivity of fluid (W/m�K)N ¼ outward flux normal to boundary

Nu ¼ average Nusselt numberP ¼ dimensionless pressureP ¼ pressure (N/m2)

Pr ¼ Prandtl numberRa ¼ Rayleigh number

S ¼ nondimensional length along the inclined heated surfaceT ¼ temperature (K)

TC ¼ temperature of the cold surface (K)TH ¼ temperature of the hot surface (K)

u ¼ velocity component in x-direction (m/s)U ¼ dimensionless velocity component in x-directionv ¼ velocity component in y-direction (m/s)V ¼ dimensionless velocity component in y-directionW ¼ width of the parallelogrammic cavity (m)x ¼ Cartesian coordinate in horizontal direction (m)X ¼ dimensionless coordinate in horizontal directiony ¼ Cartesian coordinate in vertical direction (m)Y ¼ dimensionless coordinate in vertical direction

Greek Symbols

a ¼ thermal diffusivity (m2/s)b ¼ volumetric coefficient of thermal expansion (K�1)h ¼ dimensionless temperaturel ¼ dynamic viscosity of the fluid (kg.s/m)q ¼ density (kg/m3)u ¼ nanoparticle volume fractionU ¼ sidewall inclination angle from vertical or skew angle

(deg)# ¼ kinematic viscosity of the fluid (m2/s)w ¼ dimensionless stream functionW ¼ dimensional stream function (m2/s)

Subscripts

C ¼ coldf ¼ fluid (pure)

H ¼ hotnf ¼ nanofluids ¼ solid (nanoparticle)

Abbreviations

FVM ¼ finite volume methodmin ¼ minimum



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