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1. INTRODUCTION

Nanofl uids are a mixture of nanosized particles suspended in a base fl uid such as eth- ylene glycol, water or propylene glycol. Choi (1995) was the fi rst to coin the "nanofl u- ids" for these fl uids. The existence of high thermal conductivity metallic nanoparticles in a pure fl uid improves the thermal conductivity of such mixtures; hence convention- al fl uids have a rather low thermal conductivity with respect to nanofl uids. So differ- ent types of nanofl uids are used to enhance the rate of heat transfer in many practical engineering applications.

MIXED CONVECTION FLUID FLOW AND HEAT TRANSFER

OF THE Al

₂

O

₃

–WATER NANOFLUID WITH VARIABLE PROPERTIES

IN A CAVITY WITH AN INSIDE QUADRILATERAL OBSTACLE

Mohammad Hemmat Esfe,

^*

Amir Hossein Refahi, Hamid Teimouri, Mohammaj Javad Noroozi, Masoud Afrand & Arash Karimiopour

Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran

*Address all correspondence to Mohammad Hemmat Esfe E-mail: M.hemmatesfe@gmail.com

This investigation is focused on mixed convection fl uid fl ow and heat transfer of Al₂O₃–water inside a square enclosure containing a hot rectangular obstacle at its bott om wall. The govern- ing equations have been solved using the fi nite volume method. The SIMPLER algorithm was employed to couple the velocity and pressure fi elds. Utilizing the developed code, a parametric study was conducted and the impact of important parameters such as the solid volume fraction, Richardson number, size of the hot obstacle on the fl uid fl ow and thermal fi elds and heat transfer inside the enclosure were investigated. The results show that for all Richardson numbers, the av- erage Nusselt number increases with increase in the volume fraction of nanoparticles. Moreover, at all values of the Richardson number, heat transfer decreases when the height of the heated obstacle increases.

KEY WORDS: nanofl uid, nano heat transfer, mixed convection, lid-driven cavity

During recent years, in many papers the infl uence of nanoparticles on convective heat transfer was considered theoretically, numerically, and experimentally. Some of them examined a nanofl uid in an enclosure with different geometry and bound- ary conditions. Numerical simulation of natural convection in a cavity subjected to a nanofl uid was made by Khanafer et al. (2003) who considered a nanofl uid in the enclosure to be in a single phase.

Oztop and Abu-Nada (2008) investigated heat transfer and fl uid fl ow due to buoy- ancy forces in a partially heated cavity subjected to a nanofl uid. Aminossadati and Ghasemi (2009) studied natural convection in a cavity through nanofl uid cooling of a localized heat source at the bottom. Numerical simulation to examine the effects of the inclination angle on free convection in enclosures subjected to a nanofl uid were performed by Abu-Nada and Oztop (2009). In another investigation, Ogut (2009)

NOMENCLATURE

Greek symbols specifi c heat, J·kg^–1·K^–1

c_p

thermal diffusivity, m²·s α

Grashof number Gr

thermal expansion gravitational acceleration, m·s^–2 β

g

coeffi cient, K^–1 heat transfer coeffi cient, W·m^–2·K^–1

h

dimensionless temperature θ

enclosure length, m L

dynamic viscosity, thermal conductivity, W·m^–1·K^–1 μ

k

kg·m^–1·s^–1 Nusselt number

Nu

kinematic viscosity, pressure, N·m^–2 ν

p

m²·s^–1 dimensionless pressure

P

density, kg·m^–3 ρ

Prandtl number Pr

volume fraction φ

heat fl ux, W·m^–2 q

of nanoparticles Reynolds number

Re

Subscripts Richardson number

Ri

dimensional temperature, K T

cold c

dimensional velocity components u, v

effective eff

in the x and y direction, m·s^–1

fl uid f

dimensionless velocity components U, V

hot h

in the X and Y direction

nanofl uid nf

lid velocity U₀

solid particles s

dimensional Cartesian coordinates, m x, y

wall w

dimensionless Cartesian coordinates X, Y

studied numerically natural convection heat transfer of water-based nanofl uids in an inclined square cavity where the left vertical side is heated with a constant heat fl ux, the right side is cooled, and the other sides are kept adiabatic.

Nikfar and Mahmoodi (2012) also studied natural convection in a square cavity fi lled with an Al₂O₃–water nanofluid. The horizontal walls of the cavity were insulat- ed while the left and right wavy side walls of the cavity were maintained at high and low constant temperatures. They demonstrated that on increase in the volume fraction of the nanoparticles, the average Nusselt number of the hot wall also increased. Also another work about free convection fl uid fl ow and heat transfer was investigated nu- merically by Mahmoodi and Sebdani (2012).

Mixed convection, which is a combination of both natural and forced convection, is an important heat transfer mechanism that occurs in many industrial applications. The most common application of mixed convection fl ow with the lid-driven effect is to in- clude the cooling of electronic devices, lubrication technologies, drying technologies, chemical processing equipment, etc. Due to the interaction of buoyancy and shear forces, mixed convection heat transfer is a complex phenomenon. With regard to the importance of this phenomenon, there are a large number of recent investigations on mixed convection heat transfer in cavities especially with the use of nanofl uids.

Muthtamilselvan et al. (2010) investigated numerically a mixed convection fl ow in a lid-driven rectangular enclosure subjected to a copper–water nanofl uid. They sur- veyed the effects of the aspect ratio and nanoparticle concentration on the fl uid fl ow and heat transfer in the cavity. A numerical survey of mixed convection heat trans- fer in a lid-driven cavity subjected to copper–water nanofl uid was made by Talebi et al. (2010). They reported that the heat transfer was enhanced inside the cavity on increase in the volume fraction of nanoparticles. The numerical study of mixed con- vection in a square lid-driven cavity partially heated from below and fi lled with a water-based nanofl uid was studied by Mansuor et al. (2010). Their results show that the average Nusselt number increases with the volume fraction of nanoparticles. In their recent work, Arani et al. (2012) investigated mixed convection fl ow in a lid-driv- en square cavity. A Cu–water nanofl uid was inside the cavity whose horizontal walls were adiabatic while the side walls were heating sinusoidally. They proved that a decrease in the Richardson number and increase in the volume fraction of nanoparti- cles cause an increase in the rate of heat transfer. Anyway, in recent years numerous fruitful investigations such as (Talebi et al., 2010; Abu-Nada et al., 2010; Mahmoodi, 2011; Ghasemi and Aminossadati, 2010; Arefmanesh and Mahmoodi, 2011; Sebdani et al., 2012) were carried out studying mixed convection heat transfer in cavities by various techniques whose description is beyond the scope of this paper.

In this study for the fi rst time, a nanofl uid-fi lled lid-driven cavity containing a hot obstacle has been analyzed with a new formulation of variable properties. These new models are used to evaluate the thermal conductivity and dynamic viscosity of the nanofl uid inside the cavity.

2. PHYSICAL MODEL AND GOVERNING EQUATIONS

Figure 1 demonstrates a lid-driven square cavity subjected to a nanofl uid containing a hot quadrilateral obstacle. The bottom and vertical walls of the cavity are insulated while the top wall is kept at constant low temperatures. The fl uid in the enclosure is a water-based nanofl uid containing Al₂O₃ nanoparticles. The nanofl uid in the enclo- sure is Newtonian, incompressible, and laminar. In addition, it is assumed that both the fl uid phase and nanoparticles are in the thermal equilibrium state and they fl ow with the same velocity. The density variation in the body force term of the momentum equation is satisfi ed by Boussinesq's approximation. In this study, the temperature of the nanofl uid and the diameter of nanoparticles are assumed to be constant and equal to 25°C and 47 nm, respectively. The thermophysical properties of the nanoparticles and water as the base fl uid are presented in Table 1.

The thermal conductivity and the viscosity of the nanofl uid are taken into con- sideration as variable properties; both of them change with the volume fraction and

TABLE 1: Thermophysical properties of water and nanoparticles at T = 25^oC

Physical properties Fluid phase (water) Solid (Al₂O₃)

C_p (J/kg·K) 4179 765

ρ (kg/m³) 997.1 3970

K (W·m^–1·K^–1) 0.6 25

β·10^–5 (1/K) 21. 0.85

μ·10^–4 (kg/ms) 8.9 ……..

d_p (nm) …….. 47

FIG. 1: Boundary conditions, geometry, and coordinate system

temperature of nanoparticles. Under the above assumptions, the system of governing equations is

∂ ∂ 0,

+ =

∂ ∂

u v

x y ⁽¹⁾

1 2

∂ ∂ ∂ ,

+ = − + υ ∇

∂ ∂ ρ_nf ∂ ^nf

u u p

u v u

x y x ⁽²⁾

2 ( )

1 ρβ ,

∂ ∂ ∂

+ = − + υ ∇ + Δ

∂ ∂ ρ ∂ ρ

nf nf

nf nf

v v p

u v v g T

x y y ⁽³⁾

and

2 .

∂ ∂

+ = α ∇

∂ ∂ ^nf

T T

u v T

x y ⁽⁴⁾

The dimensionless parameters may be presented as

0 0

02

, , ,

, , .

= = = =

Δ = − θ = − =

Δ ρ

h c c

nf

x y v u

X Y V U

L L u u

T T p

T T T P

T u

(5)

Hence,

0 3

Re , 2 , , Pr .

Pr Re

ρ β Δ υ

= = = =

μ υ α α

f f f

f f f f

u L Ra g TL

Ri Ra (6)

The dimensionless form of the above governing equations from (1) to (4) becomes

∂ ∂ 0

+ =

∂ ∂

U V

X Y ^{, (7)}

1 2

Re

∂ ∂ ∂ υ

+ = − + ∇

∂ ∂ ∂ υ

nf f

U U P

U V U

X Y X ^{, (8)}

1 2

Re Pr

υ β

∂ ∂ ∂

+ = − + ∇ + Δθ

∂ ∂ ∂ υ β

nf nf

f f

V V P Ri

U V V

X Y Y ^{, (9)}

and

α 2

∂θ ∂θ

+ = ∇ θ

∂ ∂ α

nf f

U V

X Y ^{. (10)}

2.1 Thermal Diffusivity and Effective Density

The thermal diffusivity and effective density of the nanofl uid are

( )

α =

ρ

nf nf

p nf

k

c ^{, (11)}

(1 )

ρ_nf = ϕρ +_s − ϕ ρ_f . (12)

2.2 Heat Capacity and Thermal Expansion Coeffi cient

The heat capacity and thermal expansion coeffi cient of the nanofl uid are

(ρc_{p nf}) = ϕ ρ( c_{p s}) + (1− ϕ ρ)( c_{p f}) , (13) ( )ρβ _nf = ϕ ρβ +( )_s (1− ϕ ρβ)( )_f . (14)

2.3 Viscosity

The effective viscosity of the nanofl uid was calculated by

( )

23

eff f(1 2.5 ) 1 1

⎡ ⎛ ⎞−2ε ⎤

⎢ ⎥

μ = μ + ϕ ⎢⎣ + η⎜⎝ ⎟⎠ ϕ ε + ⎥⎦ dp

L ^{. (15)}

This well-validated model was presented by Jang et al. (2007) for a fl uid containing a dilute suspension of small rigid spherical particles. It accounts for the slip mechanism in nanofl uids. The empirical constants ε and η are equal to 0.25 and 280 for Al₂O₃, respectively.

It is worth mentioning that the viscosity of the base fl uid (water) is considered to vary with temperature and the fl owing equation is used to evaluate the viscosity of water:

H O2

5 4 3

2 6

(1.2723 8.736 33.798

246.6 518.78 1153.9) 10 ,

rc rc rc

rc rc

T T T

T T

μ = × − × + ×

− × + × + × (16) where T_rc = log (T – 273).

2.4 Dimensionless Stagnant Thermal Conductivity

The effective thermal conductivity of the nanoparticles in the liquid is calculated by the Hamilton and Crosser model (H–C model) (1962):

stationary 2 2 ( )

2 ( )

+ − ϕ −

= + + ϕ −

s f f s

f s f f s

k k k k k

k k k k k ^{. (17)}

2.5 Total Dimensionless Thermal Conductivity of Nanofl uids The total dimensionless thermal conductivity of nanofl uids is

( )

( )

stationary

1 2 max min

2 2

max min

2 2 ( )

2 ( )

2 1 1

.

Pr 1 1

−

−

+ − ϕ −

= + =

+ + ϕ −

⎡⎛ ⎞ ⎤

⎢⎜ ⎟ − ⎥

⎢⎝ ⎠ ⎥

− ⎣ ⎦

+ − ⎛⎜ ⎞⎟ −

⎝ ⎠

nf c s f f s

f f f s f f s

Df

p f f f

Df p

f

k k k k k k k

k k k k k k k

d Nu d D D d

c D d d

d

(18)

This model was proposed by Xu et al. (2006) and it has been chosen in this study to describe the thermal conductivity of nanofl uids. The quantity c is an empirical con- stant (e.g., c = 85 for a deionized water and c = 280 for ethylene glycol) but it is inde- pendent of the type of nanoparticles, Nu_p is the Nusselt number for the liquid fl owing around a spherical particle and equal to two for a single particle in this work. The fl uid molecular diameter is d_f = 0.45 nm for water in the present study. The fractal dimension D_f is determined by

, min , max

2 ln ln

= − ϕ

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎝ ⎠

p p

Df d

d

, (19)

where d_p,max and d_p,_min are the maximum and minimum diameters of nanoparticles, respectively. The ratio of the minimum to maximum diameters of nanoparticles d_p,min/d_p,max is R. In this study R = 0.007:

1 , min , max

, max

, min

1

1

⎛ ⎞−

= − ⎜⎜ ⎟⎟

⎝ ⎠

= −

f p

p p

f p

p p f

f

D d

d d

D d

d d D

D

. (20)

3. NUMERICAL APPROACH

The governing equations for continuity, momentum, and energy associated with the boundary conditions in this investigation were calculated numerically based on the fi -

nite volume method and associated staggered grid system, using the FORTRAN com- puter code. The SIMPLER algorithm is used to solve the coupled system of govern- ing equations. The convection terms are approximated by a hybrid-scheme which is conducive to a stable solution. In addition, a second-order central differencing scheme is utilized for the diffusion terms. The algebraic system resulting from numerical dis- cretization was calculated utilizing TDMA applied in a line going through all volumes in the computational domain.

To verify the grid independence, the numerical procedure was carried out for nine different mesh sizes. An average Nu number of the hot body wall at w = 0.2, Ri = 0.1, h = 0.2, and φ = 0.01 for d_p = 47 nm is obtained for each grid size as shown in Fig. 2.

As can be observed, an 111 × 111 uniform grid size yields the required accuracy and was hence applied for all simulation exercises in this work as presented in the next section.

To ensure the accuracy and validity of the new model, we analyze a square cavity fi lled with a base fl uid at Pr = 0.7 and different Ra numbers. Table 2 shows the com- parison between the results obtained with our new model and the values presented in the literature. The quantitative comparisons made for the average Nusselt numbers indicate an excellent agreement between them.

Also, the proposed numerical scheme is validated by comparing the present code results against the numerical simulation published by Talebi et al. (2010) in Fig. 3. It is clear that the present code is in excellent agreement with another work reported in the literature, as shown in Fig. 3.

FIG. 2: Mesh validity

TABLE 2: The comparison between the results obtained in the present study and other research works

Hadjisophocleous et al. (1998) Tiwari and Das

(2007) Lin et al. (2010)

Present study

(b) Ra = 10⁴

15.995 16.1439

16.158 16.052

u_max

0.814 0.822

0.819 0.817

Y

18.894 19.665

19.648 19.528

v_max

0.103 0.110

0.112 0.110

X

2.29 2.195

2.243 2.215

Nu_ave

(c) Ra = 10⁵

37.144 34.30

36.732 36.812

u_max

0.855 0.856

0.858 0.856

Y

68.91 68.7646

68.288 68.791

v_max

0.061 0.05935

0.063 0.062

X

4.964 4.450

4.511 4.517

Nu_ave

(d) Ra = 10⁶

66.42 65.5866

66.46987 66.445

u_max

0.897 0.839

0.86851 0.873

Y

226.4 219.7361

222.33950 221.748

v_max

0.0206 0.04237

0.03804 0.0398

X

10.39 8.803

8.757933 8.795

Nu_ave

FIG. 3: Comparison between the current study and that of Talebi et al. (2010) at Re = 1, Ra

= 1.47e4, φ = 0.03

4. RESULTS AND DISCUSSION

This paper investigates numerically the heat transfer and mixed convection fl ow char- acteristics in a square cavity fi lled with a water–aluminum oxide nanofl uid having a four-sided hot obstacle inside. The effects of the parameters such as the Richard- son number, nanofl uid solid volume fraction, hot obstacle height, and the length have been studied and discussed.

Figure 4 shows the effects of the changes in the nanofl uid solid volume fraction on the streamlines and isotherms at Ri = 100, h = 0.1, w = 0.1, and d_p = 47 nm. Since the buoyancy force caused by temperature changes aids in the shear forces resulting from the lid movement, the fl ow pattern depicts the formation of a primary central cell which occupies the entire cavity space. The isotherm lines, too, show the intensity around the hot obstacle walls, while the distribution of such lines is negligible inside the cavity. The fi gures show that the streamlines and cell fl ow are intensifi ed inside the cavity as the nanofl uid solid volume fraction increases. However, an increase in the nanofl uid solid volume fraction does not result in fundamental changes in the fl ow pattern.

With increase in the nanoparticles solid volume fraction, the intensity of the iso- therm lines decreases slightly around the hot walls, which is caused by the increase in the thermal conductivity of the nanofl uid. Despite the decrease in the temperature gradient, we are not able to make a precise prediction about heat transfer in the cavity because such a decrease in the temperature gradient is followed by an increase in the thermal conductivity coeffi cient, and heat transfer inside the cavity depends on both of these factors.

Figure 5 shows the effects of changes in the Richardson number on the stream and isotherm lines at φ = 0.03, w = 0.1L, h = 0.1L, and d_p = 47 nm. In this range of parameters, the streamlines indicate the formation of a central cell as a result of the co-directionality of the buoyancy and shear forces while small vortices are formed beside the hot obstacle. The streamline intensity at higher Richardson values shows the strength of buoyancy over the shear force and the amplifi cation of the central cell inside the cavity. An increase in the Richardson number causes a decrease in the intensity of isotherm lines and decrease in the temperature gradient near the nonadia- batic walls. It is, therefore, expected that with increase in the Richardson number, heat transfer between the hot obstacle and nanofl uid decreases.

Changes in the hot obstacle length and its effects on the thermal behavior and the fl ow of the fl uid are displayed in Fig. 6 at φ = 0.03, h = 0.1 L, Ri = 100, and d_p = 47 nm. As previously, the streamlines display the co-directionality of the buoyancy and shear forces with formation of the central primary cell. As it could be seen in this fi gure, an increase in the obstacle length intensifi es the streamlines and strength- ens the central cell in the cavity. Additionally, the isotherm lines display the inten- sity of temperature lines in areas close to the circular hot obstacle. With increase

FIG. 4: Streamlines and isotherms for different solid volume fractions of nanoparticle at Ri = 100, h = 0.1L, w = 0.1L, and d_p = 47 nm

FIG. 5: Streamlines and isotherms for different Richardson numbers at φ = 0.03, w = 0.1L, h = 0.1L. and d_p = 47 nm

FIG. 6: Streamlines and isotherms for different lengths of the heated obstacle at φ = 0.03, h = 0.1L, Ri = 100, and d_p = 47 nm

in the length of the obstacle, the intensity of the isotherm lines slightly decreases around the obstacles’ walls, which indicates the decrease in the temperature gradient in such a position. Regarding the slight decrease in the temperature gradient through the increase in the obstacle length in this parametrical interval, it is expected that heat transfer inside the cavity is slightly decreased on increase in the obstacle length.

Figure 7 displays streamlines and isotherms for different obstacle heights at Ri = 100, φ = 0.03, w = 0.1L, and d_p = 47 nm. In these fi gures, too, the reaction of buoy-

ancy and shear forces resulted in the formation of a clockwise central cell inside the cavity. Occupying the majority of the cavity space, this cell, its severity and strength indicate the co-directionality of the buoyancy and shear forces. By carefully inspecting the streamline values in different positions, it could be understood that with increase in the hot obstacle height, the intensity of streamlines decreases and the severity and strength of the clockwise central cell slightly decreases inside the cavity.

As it was the case in the previous fi gure, the isotherm lines display higher intensi- ty near the hot walls. The fi gure clearly shows that with increase in the hot obstacle FIG. 7: Streamlines and isotherms for different heights of the heated obstacle at φ = 0.3, w = 0.1L, Ri = 100, and d_p = 47 nm

height inside the cavity, the thermal intensity decreases near the isothermal walls and the temperature gradient decreases. It is accordingly expected that despite the higher heat transfer rate in higher obstacles, the Nusselt average number decreases as a result of the decrease in the temperature gradient that is caused by the increased contact surface between the fl uid and obstacle.

Figure 8 displays the average Nusselt number with respect to the Richardson num- ber for a basefl uid and a nanofl uid with solid volume fractions for an obstacle of dif- ferent lengths at h = 0.1L. At all of the lengths investigated, an addition of nanopar- ticles to the fl uid resulted in the increase in the Nusselt number and heat transfer rate in the cavity compared to the basefl uid. It needs to be mentioned that an increase in FIG. 8: Average Nusselt number with respect to the Richardson number for different solid volume fractions at h = w = 0.1L and d_p = 47 nm

the solid volume fraction of nanoparticles also increases the Nusselt number in all the intervals investigated. Paying close attention to the Nusselt diagrams, one could infer that an increase in the obstacle length for a particular volume fraction and Richardson value, causes a minor decrease in the Nusselt number and hence in the heat transfer rate. A slight decrease in the Nusselt number as a result of the increase in the obstacle length could be predicted based on isotherm lines and decrease in the temperature gradient in isotherms in Fig. 6. These fi gures also display a decrease in the energy transactions with an increasing Richardson number at a particular volume fraction and obstacle length inside the cavity.

Figure 9 displays changes in the average Nusselt number depending on the Rich- ardson number for different heights of the hot obstacle inside the cavity at w = 0.2L and φ = 0.05.

As it was predicted before in analyzing Fig. 7, the Nusselt number and according- ly the heat transfer rate decrease with increase in the hot obstacle height inside the cavity. As these fi gures clearly show, an increase in the Richardson number and the overcoming of buoyancy over shear force results in the decrease in the heat transfer rate inside the cavity.

FIG. 9: Average Nusselt number with respect to the Richardson number for different heights of the hot obstacle at w = 0.2L and φ = 0.005

5. CONCLUSIONS

In the present paper the problem of mixed convection of a nanofl uid of variable prop- erties (Al₂O₃–water) inside a lid-driven cavity with a hot obstacle located on the bot- tom wall was investigated numerically utilizing the fi nite volume method and SIM- PLER algorithm. A parametric study was undertaken and the effects of the Richardson number, the volume fraction of the Al₂O₃ nanoparticles (with d_p = 47 nm), and the length and height of the obstacle on the fl uid fl ow, temperature fi eld, and the rate of heat transfer were studied and the following results were obtained:

1. Adding nanoparticles to the base fl uid increases the Nusselt number and heat transfer rate in all ranges of the parameters in this study.

2. An increase in the volume fraction at different values of Richardson number and body size causes an increase in heat transfer between the hot body and the nanofl uid.

3. Increasing height and width of the cavity reduce the temperature gradient and conse- quently decrease the heat transfer rate.

4. In all ranges of the parameters in this study, a decreasing Richardson number makes the isotherm lines near the nonadiabatic walls to become more intense which causes an increase in the heat transfer rate.

REFERENCES

Abu–Nada, E. and Chamkha, A. J., Mixed convection fl ow in a lid driven square enclosure fi lled with a nanofl uid, Eur. J. Mech. B, Fluid, vol. 29, pp. 472–482, 2010.

Abu-Nada, E. and Oztop, H., Effects of inclination angle on natural convection in enclosures fi lled with Cu–water nanofl uid, Int. J. Heat Fluid Flow, vol. 30, pp. 669–678, 2009.

Aminossadati, S. M. and Ghasemi, B., Natural convection cooling of a localized heat source at the bottom of a nanofl uid-fi lled enclosure, Eur. J. Mech. B, Fluid, vol. 28, pp. 630–640, 2009.

Arani, A. A., Sebdani, M. S., Mahmoodi, M., Ardeshiri, A., and Aliakbari, M., Numerical study of mixed convection fl ow in a lid-driven cavity with sinusoidal heating on sidewalls using nanofl u- id, Superlattice Microst., vol. 51, pp. 893–911, 2012.

Arefmanesh, A. and Mahmoodi, M., Effects of uncertainties of viscosity models for Al₂O₃–water nanofl uid on mixed convection numerical simulations, Int. J. Thermal Sci., vol. 50, pp. 1706–

1719, 2011.

Choi, S. U. S., Enhancing thermal conductivity of fl uids with nanoparticles, in: D. A. Siginer and H. P. Wang (Eds.), Developments and Applications of Non-Newtonian Flows, FEDvol. 231/MD, vol. 66, New York: ASME, pp. 99–105, 1995.

Ghasemi, B. and Aminossadati, S. M., Mixed convection in a lid-driven triangular enclosure fi lled with nanofl uids, Int. Commun. Heat Mass Transfer, vol. 37, pp. 1142–1148, 2010.

Hadjisophocleous, G. V., Sousa, A. C. M., and Venart, J. E. S., Predicting the transient natural con- vection in enclosures of arbitrary geometry using a nonorthogonal numerical model, Numer.

Heat Transfer A, vol. 13, pp. 373–392, 1998.

Hamilton, R. L. and Crosser, O. K., Thermal conductivity of heterogeneous two component systems, Ind. Eng. Chem. Fund., vol. 1, pp. 187–191, 1962.

Jang, S. P., Lee, J. H., Hwang, K. S., and Choi, S. U. S., Particle concentration and tube size de- pendence of viscosities of Al₂O₃–water nanofl uids fl owing through micro- and minitubes, Appl.

Phys. Lett., vol. 91, pp. 24–31, 2007.

Khanafer, K., Vafai, K., and Lightstone, M., Buoyancy-driven heat transfer enhancement in a two-di- mensional enclosure utilizing nanofl uids, Int. J. Heat Mass Transfer, vol. 46, pp. 3639–3653, 2003.

Lin, K. C., and Violi, A., Natural convection heat transfer of nanofl uids in a vertical cavity: Effects of non-uniform particle diameter and temperature on thermal conductivity, Int. J. Heat Fluid Flow, vol. 31, pp. 236–245, 2010.

Mahmoodi, M. and Sebdani, M. S., Natural convection in a square cavity containing a nanofl uid and an adiabatic square block at the center, Superlattice Microst., vol. 52, pp. 261–275, 2012.

Mahmoodi, M., Mixed convection inside nanofl uid fi lled rectangular enclosures with moving bottom wall, Thermal Sci., vol. 15, pp. 889–903, 2011.

Mansour, M. A., Mohamed, R. A., Abd-Elaziz, M. M., and Ahmed, S. E., Numerical simulation of mixed convection fl ows in a square lid-driven cavity partially heated from below using nanofl u- id, Int. Commun. Heat Mass Transfer, vol. 37, pp. 1504–1512, 2010.

Muthtamilselvan, M., Kandaswamy, P., and Lee, J., Heat transfer enhancement of copper–water nanofl uids in a lid-driven enclosure, Commun. Nonlinear Sci. Numer. Simul., vol. 15, pp. 1501–

1510, 2010.

Nikfar, M. and Mahmoodi, M., Meshless local Petrov–Galerkin analysis of free convection of nanofluid in a cavity with wavy side walls, Eng. Anal. Bound. Elem., vol. 36, pp. 433–445, 2012.

Ogut, E. B., Natural convection of water-based nanofl uids in an inclined enclosure with a heat source, Int. J. Thermal Sci., vol. 48, pp. 2063–2073, 2009.

Oztop, H. F. and Abu-Nada, E., Numerical study of natural convection in partially heated rectangular enclosures fi lled with nanofl uids, Int. J. Heat Fluid Flow, vol. 29, pp. 1326–1336, 2008.

Sebdani, M. S., Mahmoodi, M., and Hashemi, S. M., Effect of nanofl uid variable properties on mixed convection in a square cavity, Int. J. Thermal Sci., vol. 52, pp. 112–126, 2012.

Talebi, F., Mahmoudi, A. H., and Shahi, M., Numerical study of mixed convection fl ows in a square lid-driven cavity utilizing nanofl uid, Int. Commun. Heat Mass Transfer, vol. 37, pp. 79–90, 2010.

Tiwari, R. K. and Das, M. K., Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofl uids, Int. J. Heat Mass Transfer, vol. 50, pp. 2002–2018, 2007.

Xu, J., Yu, B., Zou, M., and Xu, P., A new model for heat conduction of nanofl uids based on fractal distributions of nanoparticles, J. Phys. D, vol. 39, pp. 4486–4490, 2006.