magnetohydrodynamics natural convection in a triangular...

A. M. RashadDepartment of Mathematics,

Faculty of Science,

Aswan University,

Aswan 81528, Egypt

Ali J. ChamkhaMechanical Engineering Department,

Prince Sultan Endowment for

Energy and Environment,

Prince Mohammad Bin Fahd University,

Al-Khobar 31952, Saudi Arabia;

RAK Research and Innovation Center,

American University of Ras Al Khaimah,

P.O. Box 10021,

Ras Al Khaimah, United Arab Emirates

Muneer A. Ismael1Mechanical Engineering Department,

Engineering College,

University of Basrah,

Basrah 61004, Iraq

e-mail: [email protected]

Taha SalahBasic and Applied Sciences Department,

College of Engineering and Technology,

Arab Academy for Science & Technology and

Maritime Transport (AASTMT),

Aswan Branch,

P.O. Box 11,

Aswan, Egypt

Magnetohydrodynamics NaturalConvection in a TriangularCavity Filled With a Cu-Al2O3/Water Hybrid Nanofluid WithLocalized Heating From Belowand Internal Heat GenerationThis study investigates the convective heat transfer of a hybrid nanofluid filled in a trian-gular cavity subjected to a constant magnetic field and heated by a constant heat flux ele-ment from below. The inclined side of the cavity is cooled isothermally while theremaining sides are thermally insulated. The finite difference method with the streamfunction-vorticity formulation of the governing equations has been utilized in the numeri-cal solution. The problem is governed by several pertinent parameters namely, the sizeand position of the heater element, B¼ 0.2–0.8 and D¼ 0.3–0.7, respectively, the Ray-leigh number, Ra¼ 102–106, the Hartmann number, Ha¼ 0–100, the volume fraction ofthe suspended nanoparticles, /¼ 0–0.2, and the heat generation parameter Q¼ 0–6. Theresults show significant effect of increasing the volume fraction of the hybrid nanofluidwhen the natural convection is very small. Moreover, the hybrid nanofluid composed ofequal quantities of Cu and Al2O3 nanoparticles dispersed in water base fluid has no sig-nificant enhancement on the mean Nusselt number compared with the regular nanofluid.[DOI: 10.1115/1.4039213]

Keywords: magneto-hydrodynamics, natural convection, hybrid nanofluid, triangularcavity

1 Introduction

A nanofluid is one of the efficient passive strategies adopted inheat transfer enhancement for the purpose of improving the effi-ciency of many thermal systems like heat exchangers, solar col-lectors, thermal storage, biomedical devices, cooling of electroniccomponents, nuclear reactors, transformers, and engine/vehicle.Using the nanofluids in automobile cooling system (radiator), theheat rejected will be large enough in such a way that the radiatoris not necessary be placed in the automobile front, this enablesredesigning the automobile with less drag front, which in turnreduces fuel consumption up to 5% [1]. A nanofluid is a fluid con-taining nanoparticles (less than 100 nm diameters) dispersed in aregular base liquid (water, ethylene glycol, or oil, etc.).

The masterwork of Choi and Eastman [2] was the key study ofthis field of investigation. Many review papers have been pub-lished to keep abreast of developments in the extent applicationsand progressively developing studies regarding nanofluids as inRefs. [3–7]. Heat transfer and fluid flow by natural convectioninside enclosures have attracted the attention of researchersbecause of its vast applications in industry and environment. Hoet al. [8] studied the natural convection of water–Al2O3 nanofluidsfilled in three sizes of vertically square enclosures. Khanafar et al.[9] studied different models of the thermophysical properties ofCu–water. Their results showed that variants among the modelsfor the nanofluid density have substantial effect on natural heattransfer in a rectangular cavity. Khodadadi and Hosseinizadeh[10] focused on inspecting the enhancement of the thermal

conductivity of phase-change materials. Santra et al. [11] consid-ered the natural convection of a non-Newtonian nanofluid in a dif-ferentially heated square cavity. Their prominent result showed asignificant decrease in heat transfer for the increase of the volumefraction of nanoparticles for any Rayleigh number. Abu-Nadaet al. [12] investigated the aspects of various nanofluids insidehorizontal concentric annuli. They showed that at an intermediatevalue of the Rayleigh number with low thermal conductive nano-particles, the Nusselt number decreased with the nanoparticlesvolume fraction. Arefmanesh et al. [13] investigated the naturalconvection of TiO2–water nanofluid-filled annuli of two differen-tially heated square ducts. Selimefendigil and €Oztop [14] investi-gated the conjugate natural convection–conduction heat transferin an inclined partitioned cavity filled with different nanofluids(Al2O3–water and CuO–water) on different sides of the partition.They found adding nanoparticles with low thermal conductivity(CuO) on the right cavity is effective for the heat transferenhancement compared with that of higher thermal conductivity(Al2O3). Oztop et al. [15] and Oztop and Abu-Nada [16] havereviewed and studied the effects of type and location of local heatsources as well as the effects of the different geometries of cav-ities, boundary conditions, and different types of nanoparticles onthe enclosed fluid or nanofluid flow and heat transfer. Sheikhole-slami et al. [17] studied the flow and heat transfer of CuO–waternanofluid in an enclosure having a sinusoidal wall under constantheat flux in the presence of a magnetic field. Malvandi and Ganji[18] investigated the natural convective heat transfer in a verticalenclosure filled with alumina/water nanofluid in the presence of auniform magnetic field using a two-phase mixture model in thehypothesis that Brownian motion and thermophoretic diffusivitiesare the only significant slip mechanisms between the solid andliquid phases. Sheremet et al. [19] conducted a numerical study onmagnetohydrodynamics free convection in an inclined wavy

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

JOURNAL OF HEAT TRANSFER. Manuscript received October 31, 2016; final manuscriptreceived January 6, 2018; published online March 30, 2018. Assoc. Editor: AntonioBarletta.

Journal of Heat Transfer JULY 2018, Vol. 140 / 072502-1Copyright VC 2018 by ASME

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enclosure filled with a Cu–water nanofluid heated by cornerheater.

The role of nanofluid in natural convection inside porous enclo-sures has also been investigated (Chamkha and Ismael [20],Chamkha et al. [21]). Moreover, a partially porous layered enclo-sure has found its place in the nanofluid field (Ismael and Cham-kha [22]). However, a triangular enclosure filled with nanofluidshas received a relatively little attention (see Ghasemi and Amino-ssadati [23], Sun and Pop [24], Aminossadati and Ghasemi [25],Sheremet and Pop [26], Bondareva et al. [27]) compared to thesquare and rectangular cavities.

In parallel to the continuous developments of the commonnanofluids, recently, a very new class of nanofluids, obtained bysuspending more than one type of nanoparticles in a base fluidand is called a “hybrid” nanofluid, is a slowly growing investiga-tion field. Compromised properties between the advantages anddisadvantages of the properties of individual nanoparticles are adeclared task of hybridization. Moreover, nanoparticle suppliersexhibit noticeable differences in prices of different nanoparticlestypes. For example, the price of copper nanoparticles is about tentimes greater than that of alumina nanoparticles. Hence, it isappropriate if one achieves the properties of expensive nanopar-ticles with minimum quantity.

Indeed, the “hybrid” nanoparticles should be limited to thoseprepared as a single composite substance in a base fluid for whichtheir synthetization requires extra attention [28–30]. However, the“hybrid” nanofluid topic is also used to those prepared by sus-pending dissimilar nanoparticles types in a base fluid. Sarkar et al.[31] have reviewed comprehensive details of hybrid nanoparticlessynthesis. This review shows very limited studies concerning themathematical models of the hybrid nanofluids properties. Theexperiments of Ho et al. [32] showed very good agreementsbetween the measured data of the density and mass fraction (qCp)and those predicted using the mixture theory. Their experimentswere based on the suspension of Al2O3 nanoparticles and particlesof micro-encapsulated phase-change material in water as a basefluid. Nevertheless, the experiments of Botha et al. [33], whichwere conducted on a silver–silica–oil-based hybrid nanofluid,showed that the Maxwell relation [34] underestimates the thermalconductivity with more deviation at higher solid volume fractions.Chamkha et al. [35] have analyzed numerically the unsteady con-jugate natural convection in a semicircular cavity with a solidshell of finite thickness filled with a hybrid water-based suspen-sion of Al2O3 and Cu nanoparticles. They showed that the effectof a hybrid nanofluid is announced at high values of Rayleighnumber and the thermal conductivity of the solid wall.

Depending on the available mathematical relations of thehybrid nanofluid properties, natural convection in a triangular cav-ity filled with such nanofluid also still needs more attention toreveal its behavior in this important field. As such, this paper is anattempt to contribute in establishing the foundations of this fieldof investigation. The present triangular cavity is to be assumedheated from below with a constant heat flux in the presence ofinternal heat generation and subjected to a magnetic field.

2 Mathematical Formulation

We consider the steady two-dimensional natural convectionflow in a triangular enclosure of side length H filled with Al2O3–water, Cu–water nanofluids, or hybrid Al2O3-Cu/water nanofluid,as shown in Fig. 1, in the presence of internal heat generation andexternal magnetic field. A heat source is located on a part of thebottom wall with length b, while the remaining parts are thermallyinsulated. The left walls of the convective region are assumed tobe thermally insulated and the inclined surface of the enclosure iscooled at a constant temperature Tc. An external magnetic field isapplied to the hybrid nanofluid in the direction parallel to the xdirection and the induced magnetic field is ignored. The hybridnanofluid used in the analysis is assumed to be incompressiblewith laminar flow and the gravitational acceleration acts parallel

to the vertical surface. The base fluid (water) and the solid spheri-cal nanoparticles (Cu and Al2O3) are assumed to be in thermalequilibrium. The thermophysical properties of the base fluid andthe nanoparticles are given in Table 1 [36,37]. The thermophysi-cal properties of the nanofluid are assumed constant except for thedensity variation, which is determined based on the Boussinesqapproximation. Under these assumptions, the equations for theconservation of mass, momentum, and energy in Cartesian coordi-nate system are given by

@u

@xþ @v

@y¼ 0 (1)

u@u

@xþ v

@u

@y¼ � 1

qhnf

@p

@xþ thnf

@2u

@x2þ @

2u

@y2

!(2)

u@v

@xþ v

@v

@y¼ � 1

qhnf

@p

@yþ thnf

@2v

@x2þ @

2v

@y2

!� rhnfB

20

qhnf

v

þ qbð Þhnf

qhnf

g T � Tcð Þ (3)

u@T

@xþ v

@T

@y¼ ahnf

@2T

@x2þ @

2T

@y2

!þ Q0

qcpð Þhnf

(4)

In Eqs. (1)–(4), x and y are the Cartesian coordinates measuredalong the horizontal and the vertical walls of the cavity, respec-tively, u and v are the velocity components along the x- andy-axes, respectively, T is the fluid temperature, p is the fluid pres-sure, g is the gravity acceleration.

Fig. 1 Schematic diagram of the present geometry and coordi-nate system

Table 1 Thermophysical properties of water, copper, andalumina [36,37]

Property Water Copper (Cu) Alumina (Al2O3)

q (kg m�3) 997.1 8933 3970Cp (J kg�1 K�1) 4179 385 765k (W m�1 K�1) 0.613 401 40b (K�1) 21� 10�5 1.67� 10�5 0.85� 10�5

r (lS/cm) 0.05 5.96� 107 1� 10�10

072502-2 / Vol. 140, JULY 2018 Transactions of the ASME


The boundary conditions imposed on the flow field are taken as

On the bottom wall y ¼ 0ð Þ:

u ¼ v ¼ 0;@T

@y¼ q00

khnf

along d � 0:5bð Þ

� x � d þ 0:5bð Þ and@T

@y¼ 0 otherwise

(5a)

On the inclined wall: u ¼ v ¼ 0;T ¼ Tc (5b)

On the left wall x ¼ 0ð Þ u ¼ v ¼ @T

@x¼ 0 (5c)

2.1 Thermophysical Properties of Regular Nanofluids andHybrid Nanofluids. As previously mentioned, although some lit-eratures studied the determination of the thermophysical proper-ties, the classical models are not certain for nanofluids. Of course,experimental results allow us to select an appropriate model for aspecified property.

The effective properties of the Al2O3/water nanofluid andAl2O3-Cu/water hybrid nanofluid are defined as follows

qnf ¼ ð1� /Þqbf þ /qp (6)

Equation (6) was originally derived according to the mixturetheory. So, the density of the hybrid nanofluid is specified by [32]

qhnf ¼ /Al2O3qAl2O3

þ /CuqCu þ ð1� /Þqbf (7)

where / is the overall volume concentration of two different typesof nanoparticles dispersed in hybrid nanofluid and is calculated as

/ ¼ /Al2O3þ /Cu

The heat capacity of the nanofluid is given by Khanafer et al. [9]and Chamkha and Abu-Nada [38] as

ðqCpÞnf ¼ /ðqCpÞp þ ð1� /ÞðqCpÞbf (8)

According to (8), the heat capacity of the hybrid nanofluid can bedetermined as follows:

ðqCpÞhnf ¼ /Al2O3ðqCpÞAl2O3

þ /CuðqCpÞCu þ ð1� /ÞðqCpÞbf

(9)

The thermal expansion coefficient of the nanofluid can be deter-mined by

ðqbÞnf ¼ /ðqbÞp þ ð1� /ÞðqbÞbf (10)

Fig. 2 The uniform mesh grid (81 3 81) used in this study

Fig. 3 Comparison of the present isotherms (right) and those obtained by Aminossadatiand Ghasemi [40] (left) at B 5 0.4 and Ra 5 105;/ 5 0:1;D 5 0:5

Table 2 Grid independency test for B 5 D 5 0.5, Ra 5 104,Ha 5 10, / 5 0.05, and Q 5 1.0

Grid size Mean Nusselt number Num

41� 41 5.132361� 61 5.534281� 81 5.546991� 91 5.5469101� 101 5.5469

Table 3 Comparison of the average Nusselt number Num forB 5 0.4, / 5 0.1, D 5 0.5

Ra Aminossadati and Ghasemi [40] Present

103 5.451 5.450104 5.474 5.475105 7.121 7.204106 13.864 14.014

Journal of Heat Transfer JULY 2018, Vol. 140 / 072502-3


where bbf and bp are the coefficients of thermal expansion of thefluid and of the solid fractions, respectively.

Hence, for a hybrid nanofluid, the thermal expansion can bedefined as follows:

ðqbÞhnf ¼ /Al2O3ðqbÞAl2O3

þ /CuðqbÞCu þ ð1� /ÞðqbÞbf (11)

The thermal diffusivity, anf , of the nanofluid is defined as

anf ¼knf

qcpð Þnf

(12)

In Eq. (11), knf is the thermal conductivity of the nanofluidwhich, for spherical nanoparticles and according to theMaxwell–Garnetts model [34], is given by

knf

kbf

¼ kp þ 2kbfð Þ � 2/ kbf � kpð Þkp þ 2kbfð Þ þ / kbf � kpð Þ

(13)

Thus, the thermal diffusivity, ahnf of the hybrid nanofluid can bedefined as

ahnf ¼khnf

qCpð Þhnf

(14)

Fig. 4 (a) Streamlines and (b) isotherms for hybrid suspension at Ha 5 10, / 5 0.05, D 5 0.5, Ra 5 104, Q 5 1

Fig. 5 Profiles of the local Nusselt number for hybrid suspen-sion at Ha 5 10, / 5 0.05, D 5 0.5, Ra 5 104, Q 5 1

Fig. 6 Variation of the average Nusselt number for hybrid sus-pension at Ha 5 10, D 5 0.5, Ra 5 104, Q 5 1



If the thermal conductivity of the hybrid nanofluid is definedaccording to the Maxwell model, Eq. (13) must be employed forthis purpose so that

khnf

kbf

¼/Al2O3

kAl2O3þ /CukCu

� �/

þ2kbf þ 2 /Al2O3kAl2O3

þ /CukCu

� �� 2/kbf

�

��

/Al2O3kAl2O3

þ /CukCu

� �/

þ2kbf � /Al2O3kAl2O3

þ /CukCu

� �þ /kbf

��1

(15)

The effective dynamic viscosity of the nanofluid based on theBrinkman model [39] is given by

lnf ¼lbf

1� /ð Þ2:5(16)

where lbf is the viscosity of the fluid fraction; then the effectivedynamic viscosity of the hybrid nanofluid is defined as

lhnf ¼lbf

1� /Al2O3þ /Cu

� �� 2:5(17)

The effective electrical conductivity of the nanofluid was pre-sented by Maxwell [34] as

Fig. 7 Variation of the average Nusselt number for hybrid sus-pension at Ha 5 10, / 5 0.05, D 5 0.5, Q 5 1

Fig. 8 (a) Streamlines and (b) isotherms for hybrid suspension at Ha 5 10, / 5 0.05, B 5 0.5, Ra 5 104, Q 5 1

Fig. 9 Profiles of the local Nusselt number for hybrid suspen-sion at Ha 5 10, / 5 0.05, B 5 0.5, Ra 5 104, Q 5 1



rnf

rbf

¼ 1þ3

rp

rbf

� 1

� �/

rp

rbf

þ 2

� �� rp

rbf

� 1

� �/

(18)

and therefore, the effective electrical conductivity of the hybrid nanofluid is given by

rhnf

rbf

¼ 1þ3

/Al2O3rAl2O3

þ /CurCu

� �rbf

� /Al2O3þ /Cu

� � !

/Al2O3rAl2O3

þ /CurCu

� �/rbf

þ 2

!�

/Al2O3rAl2O3

þ /CurCu

� �rbf

� /Al2O3þ /Cu

� � ! (19)

Introducing the following dimensionless set

X ¼ x

H; Y ¼ y

H; U ¼ uH

af; V ¼ vH

af;

P ¼ pH2

qhnf a2f

; h ¼ T � Tc

DT; DT ¼ q00 H

kf; Q ¼ H

q00Q0;

B ¼ b

H; D ¼ d

H(20)

into Eqs. (1)–(5) yields the following dimensionless equations:

@U

@Xþ @V

@Y¼ 0 (21)

U@U

@Xþ V

@U

@Y¼ � @P

@Xþ Pr

qf

qhnf

� �lhnf

lf

� �@2U

@X2þ @

2U

@Y2

� �(22)

U@V

@Xþ V

@V

@Y¼ � @P

@Yþ Pr

qf

qhnf

� �lhnf

lf

� �@2V

@X2þ @

2V

@Y2

� �

�qf

qhnf

� �rhnf

rf

� �Ha2PrV þ qbð Þhnf

qhnfbf

RaPrh (23)

U@h@Xþ V

@h@Y¼ ahnf

af

@2h@X2þ @

2h@Y2

� �þ

q cpð Þfq cpð Þhnf

Q (24)

where Pr ¼ tf =af ; Ra ¼ gbf q00 H4=tf af kf ;Ha ¼ B0Hffiffiffiffiffiffiffiffiffiffiffiffirf =lf

qare the Prandtl number, Rayleigh number, and the magnetic Hart-mann number, respectively. Q is the dimensionless heat genera-tion coefficient. B and D are the dimensionless heat source lengthand the dimensionless heat source location, respectively.

Fig. 10 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, B 5 0.5, Ra 5 104, Q 5 1

Fig. 11 Variation of the average Nusselt number for hybridsuspension at / 5 0.05, B 5 0.5, Ra 5 104, Q 5 1

Fig. 12 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, / 5 0.05, B 5 0.5, Q 5 1



The dimensionless boundary conditions for Eqs. (21)–(24) areas follows:

On the bottom wall Y ¼ 0ð Þ:

U ¼ V ¼ 0;@h@Y¼ kf

khnf

for D� 0:5Bð Þ

� X � Dþ 0:5Bð Þ and@h@Y¼ 0 otherwise (25a)

On the inclined wall: U ¼ V ¼ 0; h ¼ 0 (25b)

On the left wall X ¼ 1ð Þ: X ¼ V ¼ @h@X¼ 0 (25c)

Considering the following definitions of the dimensionless streamfunction w and vorticity X

U ¼ @W@Y

; V ¼ � @W@X

; X ¼ @V

@X� @U

@Y

� �(26)

Equations (21)–(24) can be rewritten, after eliminating the pres-sure gradient terms by cross-differentiation, in Cartesian coordi-nates X and Y as

@2W@X2þ @

2W@Y2¼ �X (27)

Fig. 13 (a) Streamlines and (b) isotherms for hybrid suspension at / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1

Fig. 15 Variation of the average Nusselt number for hybridsuspension at D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1

Fig. 14 Profiles of the local Nusselt number for hybrid suspen-sion at / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1



@W@Y

@X@X� @W@X

@X@Y¼

qf

qhnf

� �lhnf

lf

� �Pr

@2X@X2þ @

2X@Y2

� �

þqf

qhnf

� �rhnf

rf

� �Ha2Pr

@2W@X2

þ qbð Þhnf

qhnfbf

RaPr@h@X

(28)

@W@Y

@h@X� @W@Y

@h@Y¼ ahnf

af

@2h@X2þ @

2h@Y2

� �þ Q

q cpð Þfq cpð Þhnf

h (29)

The conditions of h on the outer boundaries remain as mentionedabove and the conditions of W and X on the outer boundaries areas follows:

W¼ 0 on the solid fixed boundary

X ¼ �ð@2W=@X2Þ along X ¼ 0

X ¼ �ð@2W=@Y2Þ along Y ¼ 0

X ¼ �fð@2W=@X2Þ þ ð@2W=@Y2Þg along the inclined wallThe local Nusselt number is defined as

Nus ¼1

hð Þheat source

(30)

and the average Nusselt number is defined as

Num ¼1

B

ðDþ0:5B

D�0:5B

NusdXjY¼0 (31)

3 Numerical Solution and Validation

The numerical algorithm used to solve the dimensionless gov-erning Eqs. (27)–(29) is founded on the finite difference

methodology. Central difference quotients have been employed toapproximate the second derivatives in both X- and Y-directions.Thus, the obtained discretized equations have been solved byemploying the successive under-relaxation method.

The solution procedure is iterated until the next convergencenorm is convinced

Xi;j

vnewi;j � vold

i;j

�� 10�7 (32)

where v is the general dependent variable. The appropriate valueof the relaxation parameter is found to be equal 0.7. In addition,Eq. (31) is calculated using the trapezoidal rule. The numericalmethod is implemented in a FORTRAN software. Accuracy test is

Fig. 16 (a) Streamlines and (b) isotherms for hybrid suspension at Ha 5 10, D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1

Fig. 17 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, D 5 0.5, B 5 0.5, Q 5 1



made for grid independence test at B¼D¼ 0.5, Ra¼ 104,Ha¼ 10, /¼ 0.05, and Q¼ 1.0 using five sets of grids: 41� 41,61� 61, 81� 81, 91� 91, and 101� 101. Table 2 shows thatmean Nusselt number demonstrates good accuracy between(81� 81) and (101� 101) grids; therefore, the numerical compu-tations are carried out for (81� 81) grid nodal points, which areshown graphically in Fig. 2.

In order to check the accuracy of the present method, theobtained results are compared for the special case (B ¼ 0:4,/ ¼ 10%;D ¼ 0:5) with the results obtained by Aminossadatiand Ghasemi [40]. These comparisons are presented clearly inFig. 3 in terms of the streamlines and in Table 3 in terms of theaverage Nusselt number at the heat source. A very good agree-ment is found between the results.

4 Results and Discussion

We utilize the streamlines and isotherms contour maps togetherwith the local and average Nusselt numbers in displaying the pres-ent numerical results. The present problem is analyzed by study-ing the effects of the following ranges of the governingparameters: the size and position of the heater elementB¼ 0.2–0.8 and D¼ 0.3–0.7, respectively, the ratio of buoyancyto viscous forces, which is governed by the Rayleigh numberwithin the laminar range Ra¼ 102–106, the magnetic body force,which is governed by the Hartmann number Ha¼ 0–100, the vol-ume fraction of the suspended nanoparticles /¼ 0–0.2, and theheat generation parameter Q¼ 0–6. The Prandtl number has beenfixed at 6.26 corresponding to water base fluid. The effects of

Fig. 20 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, D 5 0.5, B 5 0.5, Q 5 1

Fig. 18 (a) Streamlines and (b) isotherms for hybrid suspension at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5, Q 5 1

Fig. 19 Profiles of the local Nusselt number for hybrid suspen-sion at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5, Q 5 1



these parameters are reported as separately as possible, unlessstated.

4.1 The Effects of Heater Size and Position. In all cases,the heater is centered along the horizontal side of the enclosure.Its size varies from 0.2 to 0.8 while the other parameters are fixedat Ha¼ 10, /¼ 0.05, D¼ 0.5, Ra¼ 104, Q¼ 1. The magnitude of/ composes of equal quantities of Cu and Al2O3 nanoparticles.The resulting contour maps of the streamlines and the isothermsare shown in Fig. 4. The streamlines show that adding heat to ananofluid can generate an upward movement, which is surroundedby impermeable walls; thus, the flow turns to the cold inclinedwall and falls down to form a single-core vortex. As shown inFig. 4, the heater size has no effect on the streamlines patterns,while the isotherms, which are almost parallel to the inclinedwall, change from high to low gradients patterns with increasing

values of B. The concentration of the streamlines along the heaterdecays with increasing values of B. The local Nusselt number Nus

(Fig. 5) and the mean Nusselt number Num (Figs. 6 and 7) showthat the convective heat transfer decreases with the increase in theheater size. This refers to the dominance of the conductive heattransfer for high values of B. Figure 6 depicts the well-knowneffect of the Rayleigh number, which illustrates that increasing Rapredominates the buoyancy effect over the viscous effect, whichyields higher mean Nusselt numbers. However, we postpone therandom effect of / on Num (Fig. 7) to the relevant category.

The effect of the heater position on the contour maps (Fig. 8)also depicts no significant effect on the behavior of the stream-lines. When D increases, the associated moving up nanofluidbecomes narrower to the inclined wall resulting in a constrictionof the movement. Thus, the strength of streamlines decreasesobviously with increasing values of D (Fig. 8(a)). On the other

Fig. 23 Profiles of the local Nusselt number for hybrid suspen-sion at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104

Fig. 21 Variation of the average Nusselt number for hybridsuspension at / 5 0.05, D 5 0.5, B 5 0.5, Q 5 1

Fig. 22 (a) Streamlines and (b) isotherms for hybrid suspension at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104



hand, the isotherms become denser close to the inclined side withincreasing values of D (Fig. 8(b)). This means that the convectiveheat transfer enhances when the heater position is pushed to theright as can be clearly seen from the local Nusselt number (Fig. 9)and from the mean Nusselt numbers (Figs. 10–12).

4.2 Effects of Hartmann and Rayleigh Numbers. The mag-netic field effect opposes the buoyancy forces; as such, the vortexassociated with Ha¼ 0 circulates freely within most of the trian-gular cavity, whereas with the increase in the value of Ha, themagnetic force quantifies and weakens the vortex, which becomeslimited close to the heating element as shown in Fig. 13. The iso-therms uniformly scatter with fewer gradients with increasing val-ues of Ha (Fig. 13(b)). Another demonstration to the drag effect ofHa is the reduction of the local and average Nusselt numbers asshown in Figs. 14 and 15. However, Fig. 15 depicts a marginalinfluence of the Hartmann number Ha on the mean Nusselt num-ber for very low Rayleigh numbers. This is an expected result,where the buoyancy force is already negligible in this range of theRayleigh number.

Increasing Rayleigh number causes the nanofluid to circulaterobustly and occupies all the cavity space as shown in Fig. 16(a).The isotherms bifurcate with an initiation of the plume-likebehavior when Ra increases more than 105 as shown in Fig. 16(b).This is due to the augmentation of the buoyancy force over the

viscous force, which also can be seen in the augmentation of theaverage Nusselt numbers presented in Fig. 17.

4.3 Effect of Volume Fraction. As mentioned above, thevolume fraction / of the hybrid nanofluid comprises of equalquantities of the two different Cu and Al2O3 nanoparticles. Figure18(a) presents little influence on the streamlines behavior with /,where their intensification decreases while their strength notice-ably decays. This is due to the enhancement of the viscosity of thehybrid nanofluid. The enhancement of the thermal conductivityreflects denser isotherms as an indication to the enhancement ofnatural convection as shown in Fig. 18(b). Accordingly, theenhancement in the local and mean Nusselt numbers is recordedfor a wide range of the Rayleigh number as shown in Figs. 19–21.

Now, let us turn back to discuss the left maps of the previous /effect. In Fig. 6, we see that the mean Nusselt number at Ra¼ 104

decreases with / when the heater size is less than B¼ 0.6. Whileit remains unchanged when B¼ 0.6, Fig. 11 shows a retardation ofNum when the heater position D is closer to the vertical side of thecavity (D � 0.5), while it augments when D¼ 0.6. Figure 15shows that when Ha � 25, the mean Nusselt number augmentswith /. Eventually, Fig. 20 shows that the hybrid nanofluidbecomes active only when Ra � 103. From these examinations, itis very reasonable to conclude that the impact of the hybrid nano-fluid becomes active when the natural convection is already small.

4.4 Effect of Heat Generation. Figure 22 depicts the effectof the heat generation parameter Q on the contour maps of thestreamlines and the isotherms for Ha¼ 10, /¼ 0.05, D¼ 0.5,B¼ 0.5, and Ra¼ 104. The streamlines strengthen with theincrease in the rate of internal heat generation. Upon increasingthe values of Q, the isotherms tend to be mostly vertical withinthe lower part of the cavity (Fig. 22(b)). This gives us a notifica-tion to the dominance of the conductive heat transfer over the con-vective one, which is clearly shown in the distribution of the localNusselt number of Fig. 23, and the mean Nusselt number plottedin Figs. 24–26.

4.5 Regular Nanofluid Versus Hybrid Nanofluid. Figures27 and 28 depict the effect of using two different nanoparticles(hybrid) over the single nanoparticles (regular) nanofluid. Thestreamlines and isotherms are shown in Fig. 27 for each nanofluidindividually namely, Cu–water, Al2O3–water, Cu-Al2O3–water,and pure water. It is seen that they are identical in patterns. A littlereduction in the streamlines strength accompanies with eithernanofluid. However, the mean Nusselt number plot in Fig. 28shows that the hybrid nanofluid can retard the natural convection

Fig. 24 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, D 5 0.5, B 5 0.5, Ra 5 104

Fig. 25 Variation of the average Nusselt number for hybridsuspension at / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104

Fig. 26 Variation of the average Nusselt number for hybridsuspension at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5



phenomenon. In other words, we can conclude that the mixingadvantage and disadvantage of properties found in copper and alu-mina nanoparticles may attenuate the convective heat transfer.Hence, this result stimulates us to examine other combination ofdifferent nanoparticles type, other volume fraction ratios, andother models for the thermos-physical properties.

5 Conclusions

Magnetohydrodynamics natural convection in a triangular cav-ity with internal heat generation and filled with a hybrid Cu-Al2O3–water nanofluid is investigated numerically. The cavity is

subjected to a horizontal magnetic field, while a constant heat fluxis applied on a part of the horizontal wall. Varying several perti-nent parameters have led to the following conclusions:

� Compared with a regular nanofluid, a hybrid nanofluid com-posed of equal quantities of Cu and Al2O3 nanoparticles dis-persed in a water base fluid has no significant enhancementon the mean Nusselt number.

� The Nusselt number increases with the decrease in the heat-ing element size. The Nusselt number increases when theheating element brought to right, closer to the inclined sideof the cavity.

� The effect of increasing the volume fraction of the hybridnanofluid becomes significant in the situations where the nat-ural convection is very small. These situations are higherheater size, farther heater position, lower Rayleigh number,and higher Hartmann number.

� For higher values of the Rayleigh number, the Hartmannnumber significantly suppresses the Nusselt number. On theother hand, when Ra � 103, the magnetic field is inactive.

� Increasing the rate of internal heat generation increases theconductive heat transfer and decreases the convective heattransfer.

Nomenclature

B ¼ heat source/sink length (m)D ¼ heat source/sink position (m)g ¼ gravitational field (m s�2)H ¼ cavity side length (m)

Ha ¼ Hartmann number Ha ¼ BoHffiffiffiffiffiffiffiffiffiffiffiaf =lf

qNus ¼ local Nusselt number

Num ¼ average Nusselt numberp ¼ pressure (N/m2)

Fig. 27 (a) Streamlines and (b) isotherms at Ha 5 10, / 5 0.05, D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1

Fig. 28 Variation of the average Nusselt number at Ha 5 10,D 5 0.5, B 5 0.5, Ra 5 104, Q 5 1



Pr ¼ Prandtl number Pr ¼ �f =af

Q ¼ heat generation or absorptionRa ¼ Rayleigh number Ra ¼ g bDT H3=�f af

T ¼ temperature (K)u ¼ velocity component along x-direction (m s�1)U ¼ dimensionless velocity component along x-direction

Uo ¼ velocity of the moving wall (m/s)v ¼ velocity component along y-direction (m s�1)V ¼ dimensionless velocity component along y-direction

x, y ¼ Cartesian coordinates (m)X, Y ¼ dimensionless Cartesian coordinates

Greek Symbols

a ¼ thermal diffusivity (m2 s�1)b ¼ thermal expansion coefficient (K�1)/ ¼ nanoparticles volume fractionl ¼ dynamic viscosity (Pa�s)� ¼ kinematic viscosity (m2 s�1)h ¼ dimensionless temperatureq ¼ density (kg m�3)r ¼ electrical conductivity (S m�1)

w, W ¼ stream function (m2 s�1), dimensionless stream function

Subscripts

Al2O3 ¼ aluminabf ¼ base fluidc ¼ cold

Cu ¼ copperf ¼ fluidh ¼ hot

hnf ¼ hybrid nanofluidm ¼ averagenf ¼ nanofluid

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