NUMERICAL SIMULATION OF MIXED CONVECTION WITHIN NANOFLUID-FILLED CAVITIES WITH TWO ADJACENT MOVING WALLS

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1 NUMERICAL SIMULATION OF MIXED CONVECTION WITHIN NANOFLUID-FILLED CAVITIES WITH TWO ADJACENT MOVING WALLS Mohammad Hemmat Esfe 1, Ariyan Zare Ghadi 2 and Mohammad Javad Noroozi 1 1 Department of Mechanical Engineering, Najaf Abad Branch, Islamic Azad University, Isfahan, Iran 2 Department of Mechanical Engineering, Jouybar Branch, Islamic Azad University, Jouybar, Iran m.hemmatesfe@gmail.com Received June 2012, Accepted August 2013 No. 12-CSME-71, E.I.C. Accession 3391 ABSTRACT In this study, nanofluid flow and heat transfer in a cavity with two moving lids are investigated. Governing equations are solved by finite volume approach using SIMPLE algorithm over a staggered gird system. The results show that when the moving lids have opposing effect, the streamlines contain two main vortices. By increasing the Richardson number, intensity of the vortex complying with buoyancy force increases, while intensity of the other vortex decreases. When the moving lids have aiding effect, the streamlines contain one the primary dominant vortex in which its strength increases with increase of the buoyancy force. In this case, rate of heat transfer is more than other cases. Keywords: nanofluid; heat transfer; mix convection; double lid-driven cavity; numerical study. SIMULATION NUMÉRIQUE DE LA CONVECTION MIXTE DANS DES NANO CAVITÉS REMPLIES DE LIQUIDE ET COMPORTANT DEUX PAROIS ADJACENTES EN MOUVEMENT RÉSUMÉ Dans cette recherche, on s intéresse à la circulation d un nanofluide et au transfert de chaleur dans une cavité dont les deux couvercles sont mobiles. Pour résoudre les équations de contrôle, on se sert de la méthode des volumes finis qui utilise l algorithme SIMPLE dans un système alternatif. Ces résultats nous montrent deux grands mouvements rotatifs pour la circulation des fluides lorsque les couvercles mobiles se déplacent dans des directions opposées. En augmentant le nombre de Richardson, l intensité dans le mouvement, qui est en conformité avec la direction de la force propulsive, s intensifie alors que l intensité de l autre est diminuée. Dans le cas où les couvercles mobiles se déplacent dans une même direction, la circulation est composée d un mouvement rotatif principal dont l intensité dépend de celle de la force propulsive. Ici, la chaleur transférée est plus élevée que dans les autres cas. Mots-clés : nanofluide ; transfert de chaleur ; convection mixte ; cavité double couvercle axée ; étude numérique. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

2 NOMENCLATURE C p specific heat at constant pressure (J/kg K) G gravitational acceleration (m/s 2 ) H heat transfer coefficient (W/m 2 K) H cavity height (m) K thermal conductivity (W/m K) L cavity length (m) Nu local Nusselt number, hl/k f Nu m average Nusselt number P pressure (Pa) P dimensionless pressure, pl 2 /ρ n f α 2 f Pr Prandtl number, υ f /α f q w heat flux (W/m 2 ) Ri Richardson number, (Ra/(Pr Re 2 )) T temperature (K) T T h T c u,ν velocity components in x,y direction (m/s) u 0 characteristic velocity U,V dimensionless velocity, u/u 0,v/U 0 x,y Cartesian coordinates (m) X,Y dimensionless Cartesian coordinates, x/l,y/l Greek symbols α thermal diffusivity (m 2 /s) β thermal expansion coefficient (1/K) µ dynamic viscosity (kg/ms) ρ density (kg/m 3 ) θ dimensionless temperature, (T T c )/(T h T c ) ϕ solid volume fraction ψ stream function Subscripts F L M N f pure fluid differential length mean nanofluid 1. INTRODUCTION Analysis of mixed convective flow in a lid-driven cavity has some applications such as: materials processing [1], flow and heat transfer in solar ponds [2], crystal growing [3], float glass production [4], and cooling of electronic components, metal casting, food processing, galvanizing, and metal coating, among others. Fluid used in these applications is mostly water and ethylene glycol which has low heat transfer rate due to its low thermal conductivity and does not meet the rising demand as an efficient heat transfer agent. Because of a need to effective cooling and heating process, heat transfer enhancement in engineering is one of the hottest topics in research. Employing nanofluids is an innovative method to augment heat transfer. Nanofluids are new type of heat transfer fluids containing a small quantity of nanosized particles (generally less than 100 nm) that are uniformly and stably suspended in a liquid. These types of fluids with relatively higher thermal conductivities have received vast interest from researchers owing to their capability in improvement of heat transfer. Many studies have been done on nanoparticles heat transfer inside cavity and 1074 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

3 effect of thermophysical properties such as thermal conductivity, dynamics viscosity and thermal expansion coefficient. In the following section, previous studies are reviewed. Khanafer et al. [5] investigated a 2-D enclosure filled with nanoparticles. The nanofluid used in their study was water-copper. They observed that in a specific Grashof number, increasing volume fraction of nanoparticles increases heat transfer rate. Another numerical study has been conducted by Jou et al. [6]. This study also confirms results of the previous study, which states that increasing the volume fraction of nanoparticles causes the heat transfer rate to increase. Oztop et al. [7] investigated the heat transfer in a cavity by adding different types of nanoparticles to the base fluid. The results of this study show that increasing the volume fraction of nanoparticles increases the average Nusselt number. The above-mentioned studies are examples of studies that have been conducted to investigate natural convection inside cavities filled with nanofluids. However, some studies are also done for the case of mixed convection inside cavities which occurs due to moving lids as well as presence of buoyancy force. Muthtamilselvan et al. [8] studied a cavity with one moving lid for different aspect ratios. Their findings include that the aspect ratio and nanoparticles concentration have great effect on heat transfer process and fluid flow. Talebi et al. [9] investigated combined convection flow inside a lid-driven cavity filled with water-copper nanofluid. They observed that for a specific Reynolds number, flow pattern and thermal behavior are completely dependent on the copper particles concentration. Abu-nada et al. [10] studied mixed convection flow inside a tilted lid-driven cavity. The results obtained in their study demonstrate that increasing nanoparticles causes a considerable increase in the rate of heat transfer. Mahapatra et al. [11] investigated a cavity with two opposite moving lids. They studied effect of interaction of the lids for both cases of opposing or aiding movement of the lids. They also considered effect of radiation in their study. Hakan et al. [12] studied mixed convection flow inside a cavity with two opposite moving lids in which they considered three different cases for movement of the lids. They investigated effect of both opposing flow with the buoyancy force and the aiding flow with the buoyancy force. Oueslati et al. [13] studied aspect ratio in a 3-D cavity with two adjacent moving lids. Their results were only limited to forced convection. Recently, Sebdani et al. [14] have done a research on heat transfer inside a cavity when the two adjacent lids are moving. They chose the side walls to be the cold source and part of the lower wall to be the heat source. Their observations demonstrate that at a specific Reynolds number and for high Rayleigh numbers, rate of heat transfer decreases by increasing nanoparticles. Furthermore, the results show that by increasing Rayleigh number, rate of decease in heat transfer also increases. They obtained the results based on variable viscosity and thermal conductivity models. Previous studies confirm that lid-driven differentially heated cavities have various interesting applications in many fields. However, so far a study of cavities with two adjacent moving lids filled with nanofluid as the working fluid in which the effect of direction of lids movement are also considered, has never been carried out. Depending on the problem, the interaction between forced convection and natural convection should be known and thus in this study a mixed convection inside a cavity with two moving lids is investigated. Then the effect of parameters such as direction of movement of the lids and nanoparticles concentration in different Richardson numbers, which includes forced convection-dominated and mixed convection-dominated on the stream pattern and thermal behavior, are analyzed. 2. MATHEMATICAL MODEL Schematic diagram of the double lid-driven cavity for this study including boundary conditions and coordinate are shown in Fig. 1. The fluid is considered to be water copper nanofluid. Size and form (shape) of the particles are assumed to be uniform with a diameter equal to 47 nm. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

4 Table 1. Thermophysical properties of water and copper. Property Water Copper c p ρ κ β Particle diameter 47 nm Fig. 1. Schematic diagram of the double lid-driven cavity considered in the present study. Case 1: Upper wall moves to right and left wall moves to down. Case 2: Upper wall moves to left and left wall moves to up. Case 3: Upper wall moves to right and left wall moves to up. Case 4: Upper wall moves to left and left wall moves to down. The boundary conditions of problem are as follows: the right lid and the left moving lid are considered to be insulated while the upper moving lid is kept at a low temperature. Also, heat sources are mounted on the lower lid. As shown in the figure, the left and the upper lids are moving with constant velocity and gravitation force is also acting downward. The nanofluid is considered to be Newtonian and the flow is assumed to be laminar and incompressible. Also, the base fluid and the particles are in thermal equilibrium and no friction exists between them. The only body force acting on the system is buoyancy force which is included in momentum equation after using Boussinesq approximation. Thermophysical properties of water as the base fluid and properties of the copper nanoparticles are tabulated in Table 1. The left and the upper lids are moving and four cases are considered for direction of their movement (as shown in Fig. 1). For a steady, two-dimensional laminar and incompressible flow, the governing equations are u u x + ν u y = 1 ρ n f u ν x + ν ν y = 1 ρ n f u x + ν = 0, (1) y p x + υ n f 2 u, (2) p y + υ n f 2 ν + (ρβ) n f ρ n f g T, (3) 1076 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

5 The following dimensionless parameters are introduced: The Reynolds number and other parameters are u T x + ν T y = α n f 2 T. (4) X = x L, Y = y L, V = ν U 0, U = u U 0, T = T h T c, θ = T T c T, P = p ρ n f U0 2. (5) Re = ρ fu 0 L µ f, Ri = Ra Pr Re 2, Ra = gb f T L 3 υ f α f, Pr = υ f α f. (6) Equations (1 4) in dimensionless form are as follows: U X + V Y = 0, (7) U U U +V X Y = P X + υ nf 1 υ f Re 2 U, (8) U V V +V X Y = P Y + υ n f 2 V + Ri β n f θ, (9) υ f β f The general boundary conditions for all cases are as follows: U θ θ +V X Y = α n f 2 θ. (10) α f { θ/ X = 0, X = 0, 0 < Y < 1, θ/ X = 0, X = 1, 0 < Y < 1, { U = V = 0, Y = 0, 0 < X < 1, θ = 0, Y = 1, 0 < X < 1. The special boundary conditions for each case are as follows: { U = 0, V = 1, X = 0, 0 < Y < 1 Case 1 : U = 1, V = 0, Y = 1, 0 < X < 1 { U = 0, V = 1, X = 0, 0 < Y < 1 Case 2 : U = 1, V = 0, Y = 1, 0 < X < 1 { U 0, V = 1, X = 0, 0 < Y < 1 Case 3 : U = 1, V = 0, Y = 1, 0 < X < 1 { U = 0, V = 1, X = 0, 0 < Y < 1 Case 4 : U = 1, V = 0, Y = 1, 0 < X < 1. The thermal diffusivity and effective density and other properties of the nanofluid are expressed by the following relations: α n f = k n f, (11) (ρc p ) n f Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

6 ρ n f = ϕρ s + (1 ϕ)ρ f. (12) The heat capacitance and thermal expansion coefficient of the nanofluid can be given as (ρc p ) n f = ϕ(ρc p ) s + (1 ϕ)(ρc p ) f, (13) (ρβ) n f = ϕ(ρβ) s + (1 ϕ)(ρβ) f. (14) The effective viscosity of the nanofluid which was proposed by Brinkman [15] is as follows: µ n f = µ f. (15) (1 ϕ) 2.5 The effective thermal conductivity of the nanofluid is calculated by the Hamilton Crosser model [16], which is k n f = k s + 2k f 2ϕ(k f k s ) k f k s + 2k f + ϕ(k f k s ). (16) The Nusselt number can be calculated as where heat transfer coefficient h is defined as and the thermal conductivity may be expressed as Nu = hl k f, (17) h = k n f = The Nusselt number for hot wall can be written as Nu = q w T h T C (18) q w T/ Y. (19) ( kn f The average Nusselt number is calculated over hot surface by Eq. (18): Nu m = 1 L k f L 0 )( ) θ. (20) x Nu dx. (21) Also, for computing the stream function in the Cartesian coordinate system, we can use u = ψ y or ν = ψ x. (22) 3. NUMERICAL METHOD The governing equations including continuity, momentum and energy equations associated with the boundary conditions are calculated numerically based on finite volume method and with staggered grid system. The SIMPLE algorithm introduced by Patankar [17] is implemented to solve coupled system of the governing equations. The convection terms are approximated by a blend of central difference scheme and upwind scheme (hybrid-scheme) which results in a stable solution. Besides, a second-order central differencing scheme is served for the diffusion terms. The algebraic system arising from numerical discretization is 1078 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

7 Table 2. Grid independence study for case 1 for a nanofluid with Ri = 1,ϕ = %2. Grid size Nu ave Table 3. Assessment of solutions for mixed convection in an enclosure. Re Ri Present study Ref. [19] Ref. [20] Ref. [21] Ref. [22] Ref. [23] computed using Tridiagonal Matrix Algorithm (TDMA) [18]. The solution process is repeated until an acceptable convergence criterion is reached. A FORTRAN computer code has been developed to solve the equations as described above. Error = M j=1 N i=1 φ n+1 φ n M j=1 N i=1 φ n+1 < Here, M and N refer to the number of grid points in x and y directions, respectively. N is the number of iteration and Φ denotes any scalar quantity. To prove grid independence, numerical practice was executed for nine different grid sizes, i.e ,31 31,41 41,51 51,61 61,71 71,81 81,91 91 and Average Nusselt number for the bottom hot wall is attained for each mesh size as shown in Table 2. As can be seen, an uniform grid size gives the accuracy needed in this work. All simulations in this study have been performed using this grid sizes. To validate our numerical approach, conditions of physical domain of present code was mimicked with the conditions as invoked in [19 23]. Then the average Nusselt number for top heated moving lid extracted from our code was compared with the above-mentioned references, for different values of Reynolds and Richardson (Table 3). This comparison shows that our results are in high quality conformity with other works reported in the literature. 4. RESULTS AND DISCUSSION In this study, fluid flow and heat transfer inside a 2-D cavity are investigated. The cavity contains two adjacent moving lids. Four cases are considered for direction of the lids movement and the results are obtained based on the effect of direction of the lids movement on flow pattern and heat transfer mechanism. In this paper, the Richardson number is set to be between and 10 and the Grashof number is considered constant and equal to Nanoparticles concentration also varies in range of 0 to Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

8 Figure 2 shows the results obtained for case 1 in the form of streamlines and isotherms. The solid line denotes the pure fluid while the dashed line denotes the nanofluid with volume fraction of For case 1, the upper moving lid moves to right and the left moving lid moves downward. These movements of the lids cause two primary vortices to be formed near the lids. According to the direction of the lids movement, the vortex which is formed by the force resulting from the movement of the upper lid is clockwise and the vortex which is formed by the force resulting from movement of the left lid is counterclockwise. Richardson number in Figs. 2(a b) is set to This means that the force due to movement of the lids dominates over the buoyancy force. Thus, the formed vortices are due to the shear forces and the dominant mechanism of heat transfer is forced convection. In addition, according to the figure, a small vortex is formed in the right corner of the cavity. However, another obvious point in the figure is the effect of nanoparticles on the flow pattern. By increasing the nanoparticles, intensity of the vortices increases. The isotherms for Richardson number of are shown in the Fig. 2(b). The isotherms demonstrate that temperature gradient near the cold wall and particularly the hot wall is very high which shows high rate of heat transfer in these areas. In fact, temperature distribution inside the cavity is highly dependent on the forced convection flow caused by movement of the lids. In Figs. 2(c d), Richardson number is assumed to be equal to unity. The same as the preceding case, streamlines contain two primary vortices being formed near the moving lids. A small vortex is also formed at lower right corner of the cavity. In this range of Richardson number, the effect of buoyancy force increases in comparison to the previous case. Therefore, vortex formed by the shear force of the upper lid is reinforced by the buoyancy force, while the force created due to the movement of the left lid decreases because its direction is opposite to the buoyancy force. In this figure, increase of strength of the vortex due to increase of nanoparticles is obvious. The heat transfer in this case is more balanced in comparison to the previous case, thus temperature is distributed throughout the entire cavity. It can be understood from the figure that the temperature gradient on the lower wall is high which means there is a high heat transfer rate in this area. However, the temperature gradient is weaker compared to the previous case. Figures 2(e f) shows the streamlines and the isotherms for Richardson number equal to 10. In this range of Richardson number, the buoyancy force is the dominant force acting on the system. Effect of the buoyancy force (which is the dominant force in this case) and the force caused by the upper lid (which is comply with the buoyancy effect) reinforce each other, thus a powerful vortex is formed which occupies bulk of the cavity. Contrary to the upper vortex, strength of the left vortex decreases in a way that it occupies a very small area near the left lid. The figure also demonstrates that the third vortex is completely disappeared. Using nanofluid instead of pure fluid increases strength of the vortices. The isotherms also mark a more uniform temperature distribution in comparison to the previous case. In case 2, the left lid moves upward and the direction of movement of the upper lid is leftward. This means that direction of movement of both lids is now opposed and as a result, forces caused by these movements are opposite of each other. In Figs. 3(a b), Richardson number is low and the forced convection caused by movement of the lids is dominant. Two primary vortices are formed inside the cavity and is caused by the shear forces of the moving lids. Besides, a small vortex is formed in lower section of the cavity. With increasing amount of nanoparticles, flow pattern does not change and only strength of the vortices increases. The isotherms also show a great gradient in the upper and the lower areas of the cavity which means there is high amount of heat transfer in the upper and lower sections. Figures 3(c d) deal with mixed convection flow. Contrary to case 1, force caused by movement of the left lid aids the buoyancy force, thus the left vortex is reinforced and the upper vortex is weakened. The isotherms also show temperature distributions throughout the entire cavity. In Fig. 3(e) since Richardson number is high enough, the buoyancy force is dominant and by augmenting the force caused by movement of the left lid, the left vortex is reinforced and the upper vortex which moves in 1080 Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

9 Fig. 2. Stream lines and isotherms for case 1 at (a b): Ri = 0.001, (c d): Ri = 1, (e f): Ri = 10. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

10 Fig. 3. Stream lines and isotherms for case 2 at (a b): Ri = 0.001, (c d): Ri = 1, (e f): Ri = Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

11 opposite direction of the buoyancy force is weakened. The isotherms (Fig. 3f) are also completely dependent on the streamlines in a way that profile of temperature distributions inside the cavity is divided into two distinct regions in which each region is located in an area of the formed vortices. Figure 4 shows the results for case number 3. In this case, the left and upper lids move upward and rightward, respectively. This means that movements of the lids are in a way that support each other and agree with the buoyancy force. Thus in Fig. 4(a), a strong primary vortex is formed due to agreement of the two forces caused by the movement of the lids. Two small vortices are also formed in the lower section of the cavity. Figure 4(c) deals with mixed convection flow and thus a primary vortex caused by forced and natural convection is formed. This figure also shows that small vortices disappear in this case. Temperature distribution inside the cavity is affected by both forced and natural convection. However, movement of the lids is in a way that reinforces the buoyancy force, thus a strong vortex is caused by the buoyancy force. Isotherms in this case show a uniform temperature distribution throughout the entire cavity. In the last case, the upper lid moves leftward while the left lid moves downward. When forced convection is dominant (Figs. 5a b), a primary vortex is formed due to conformity of forces caused by the moving lids. Two small vortices are also formed in the right section of the cavity. In mixed convection flow (Figs. 5c d), primary vortex is formed by the moving lids and the buoyancy force. In this case, small vortices are disappeared and strength of the primary vortex increases. In Figs. 5(e f), Richardson number is set to 10 and the buoyancy force is dominant, thus a more powerful primary vortex than the previous cases is formed. Isotherms for case 4 for different flow regimes are also shown. It can be deduced from the figure that increasing Richardson number decreases temperature gradient in the cold and hot walls, and for high Richardson numbers, a uniform temperature distribution forms inside the cavity. The average Nusselt number diagrams for all the above discussed cases for different values of volume fraction and Richardson numbers of 0.001, 1 and 10 are shown in Fig. 6. The diagrams show that for different Richardson number, increasing volume fraction of nanoparticles causes average Nusselt number to increase in all cases. This can be a criterion to analyze rate of heat transfer in the system. Moreover, cases 4 and 3 have the maximum Nusselt number, respectively. This is because that forces caused by the moving lids amplify each other in these two cases. Cases 1 and 2 have lower Nusselt number in comparison to cases 3 and 4, because the forces due to the moving lids are opposite in these cases. As shown in Figs. 2 and 3, since the temperature gradient in case 1 is higher than that of case 2, the heat transfer rate is also higher for case 1. Figure 6 which is drawn for different values of Richardson number demonstrates that increasing Richardson number, which consequently increases the buoyancy force with respect to the shear forces, causes average Nusselt number and consequently heat transfer rate to decrease. Figure 7 shows vertical velocity in mid-section of the cavity for all the four cases for volume fraction of In case 1 (Fig. 7a), there are velocity fluctuations in mid-section area of the cavity. In fact, this happens due to presence of two oppositely-directed vortices. In the upper section of the cavity, velocity decreases rapidly and then approaches to the lower section of the cavity with lesser gradient and finally tends to zero at faster rate. For case 2 (Fig. 7b), in which the upper lid moves leftward and the left lid moves upward, there still exist some velocity fluctuations in mid-section of the cavity. First, velocity approaches to 0.5 at high gradient and then decreases due to effect of the second vortex and finally tends to zero at high rate near the lower wall. For cases 3 and 4 (Figs. 7c d), since effects of two moving lids reinforce each other and also there is one primary vortex inside the entire cavity, velocity fluctuations are expected to be lower than the previous cases. In cases 3 and 4, quantity of velocity is expected to be maximum at the upper lid and to decrease linearly to zero at the lower lid. The results from the figures also confirm this trend. Table 4 shows maximum stream function (ψ max ) for cases 1 and 4. It is evident from the table that by adding the nanoparticles to the fluid, the strength of the vortices increases. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

12 Fig. 4. Stream lines and Isotherms for case 3 at (a b): Ri = 0.001, (c d): Ri = 1, (e f): Ri = Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

13 Fig. 5. Stream lines and Isotherms for case 4 at (a b): Ri = 0.001, (c d): Ri = 1, (e f): Ri = 10. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

14 Fig. 6. Average for all cases Nu at (a): Ri = 0.001, (b): Ri = 1, (c): Ri = Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

15 Fig. 7. U velocity profile in centerline of cavity at different Richardson numbers for (a): case 1, (b): case 2, (c): case 3, (d): case CONCLUSIONS In this paper, mixed convection flow of water copper nanofluid inside a cavity with two adjacent moving lids has been investigated. The following results have been obtained from an analysis of this study: a. When movement of the lids are in a way that resulting forces augment each other, only one primary vortex is formed and temperature lines distribute uniformly inside the cavity. Rate of heat transfer is also higher than the condition when movement of the two lids counterbalances effect of each other. b. When forces resulted from movement of the lids opposes each other, two primary vortices are formed inside the cavity. In this case, increasing Richardson number or in other word, increasing the buoyancy force, causes the vortex complying with the buoyancy force to be reinforced and the other vortex is weakened. Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4,

16 Table 4. Vortices strength for cases 1 and 4 in different values of solid volume fraction. c. By adding nanoparticles to the fluid, strength of the vortices increases in all cases. Also, by increasing volume fraction of nanoparticles, average Nusselt number increases which also shows increase of heat transfer rate. d. Velocity distribution in the mid-section of the cavity demonstrates that when effects of movement of the two lids agree with each other, velocity fluctuations are low in mid-section, while the effects are opposing with each other, there are more fluctuations in the mid-section. For all four cases, velocity gradient is high near the upper and lower walls. The present theoretical effort is hoped to be helpful for the empirical works to examine combined convection heat transfer within a nanofluid-filled enclosure with more than one moving wall. Future work is recommended to develop the current study for other nanofluids such as Al 2 O 2 -Water, TiO 2 -Water, ZnO- Water and comparing all findings with this paper. It would be also interesting to further study the fluid motion and heat transfer inside a cavity with three or four moving walls for time-dependent flows. REFERENCES 1. Jaluria, Y., Fluid flow phenomena in materials processing the 2000 Freeman Scholar Lecture, Journal of Fluids Engineering, Vol. 123, No. 2, pp , Cha, C.K. and Jaluria, Y., Recirculating mixed convection flow for energy extraction, International. Journal of Heat and Mass Transfer, Vol. 27, No. 10, pp , Roy, B.N., Crystal Growth from Melts Applications to Growth of Groups 1 and 2 Crystals, Wiley, Pilkington, L.A.B., Review lecture: the float glass process, Proceedings of the Royal Society of London A, Vol. 314, pp. 1 25, Khanafer, K., Vafai, K. and Lightstone, M., Buoyancy driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, International Journal of Heat and Mass Transfer, Vol. 46, No. 19, pp , Jou, R.Y. and Tzeng, S.C., Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures, International Communications in Heat and Mass Transfer, Vol. 33, No. 6, pp , Oztop, H.F. and Abu-Nada, E., Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, International Journal of Heat and Fluid Flow, Vol. 29, No. 5, pp , Muthtamilselvan, M., Kandaswamy, P. and Lee, J., Heat transfer enhancement of Copper-water nanofluids in a lid-driven enclosure, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 6, pp , Talebi, F., Mahmoudi, A.H. and Shahi, M., Numerical study of mixed convection flows in a square lid-driven cavity utilizing nanofluid, International Communications in Heat and Mass Transfer, Vol. 37, No. 1, pp , Oztop, H.F. and Abu-Nada, E., Effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid, International Journal of Heat and Fluid Flow, Vol. 30, No. 4, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 37, No. 4, 2013

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