Available online at ScienceDirect. Procedia Engineering 105 (2015 )

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1 Available online at ScienceDirect Procedia Engineering 105 (2015 ) th BSME International Conerence on Thermal Engineering (ICTE 2014) Eect o tilt angle on pure mixed convection low in trapezoidal cavities illed with water-al 2 O 3 nanoluid Mahmudul Hasan Hasib*, Md. Saddam Hossen, Sumon Saha Department o Mechanical Engineering, Bangladesh University o Engineering and Technology, Dhaka 1000, Bangladesh Abstract A numerical study is carried out to investigate the eect o tilt angle o the cavity on mixed convection heat transer inside two dierent lid-driven trapezoidal cavities; one having heated wall on short base and another having heated wall on long base. In this investigation, the top wall is maintained at isothermal cold temperature, which is moving in its own plane at a constant speed while a constant high temperature is provided at the bottom surace o the cavity. The cavity is assumed to be illed with water- Al 2 O 3 nanoluid. The governing Navier Stokes and thermal energy equations and boundary conditions are non-dimensionalised and are solved using Galerkin inite element method. Attention is paid in the present study on the pure mixed convection regime at Richardson number, Ri = 1 where the natural and the orced convection are equally dominated. Parametric investigations are carried out by taking base wall tilt angle rom 0 o to 45 o with a step o 15 o and also varying Reynolds numbers rom 0.1 to a maximum order o 10 4 with the corresponding Grasho numbers varying rom 0.01 to a maximum order o 10 8 or Ri = 1. Simulations are carried out by considering both plain luid (water) and nanoluid with 10% solid-volume raction o nanoparticles. Flow and heat transer characteristics are explained using streamline and isotherm contours, and the variation o average Nusselt number o the heated wall and average luid temperature o the cavity are analysed or dierent tilt angles The Authors. Published by Elsevier Ltd The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility o organizing committee o the 6th BSMEInternational Conerence on Thermal Engineering ( Peer-review (ICTE 2014). under responsibility o organizing committee o the 6th BSME International Conerence on Thermal Engineering (ICTE 2014) Keywords:mixed convection; nanoluid; Richardson number; trapezoidal cavity; Nusselt number. * Corresponding author. Tel.: ; address: hasib_67@yahoo.com The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility o organizing committee o the 6th BSME International Conerence on Thermal Engineering (ICTE 2014) doi: /j.proeng

2 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Introduction Mixed convection is the process o heat transer including both natural and orced convection, which bears a great importance due to its wide applications such as cooling o electronic devices, lubrication technologies, heating and drying technologies, ood processing, loat glass production, low and heat transer in solar ponds, thermal hydraulics o nuclear reactors, dynamics o lakes, crystal growing, metal coating, reservoirs and cooling ponds, materials processing and so on. There have been numerous studies in the past on mixed convective low in lid-driven cavities. However, majority o these investigations considered the cases o simple geometry like square, rectangular or triangular cavities. Only a ew studies considered the coniguration o lid-driven trapezoidal cavities or mixed convection problems [1-5]. These cavities can be divided into two categories based on the length o the top and the bottom walls. Hossain et al. [1], Chowdhury et al. [2], Hasan et al. [3] and Mamun et al. [4] considered mixed convection inside a lid-driven trapezoidal cavity having heated wall on the short base. On the other hand, Bhattacharya et al. [5] perormed investigation on lid-driven trapezoidal cavity with heated long base. Among these works, only Mamun et al. [4] showed the eect o cavity tilt angle on mixed convection heat transer. Recently, Cheng [6] only pointed out the combined eect o Reynolds and Grasho numbers on pure mixed convection low in a lid-driven square cavity. However, to the best o the authors knowledge, no attention has been paid to investigate the eect o tilt angle on pure mixed convection low inside the lid-driven trapezoidal cavity illed with nanoluid. Nanoluid, ater gaining popularity in various industrial applications, has become one o the eective means or the enhancement o convective heat transer. This is because metallic nanoparticles such as copper, aluminum, silver, silicon, etc., with higher thermal conductivity improve the thermo-physical properties o the mixture o conventional base luids like water, ethylene glycol, etc. Although many researchers such as Tiwari and Das [7], Talebi et al. [8], Abu-Nada and Chamkha [9] and Salari et al. [10] carried out investigations on mixed convection inside a lid-driven square cavity illed with nanoluid, a similar analysis inside a trapezoidal cavity is still missing. Thereore, the main objective o the present study is to present the inluence o the cavity tilt angle o two dierent trapezoidal cavities illed with water-al 2 O 3 nanoluid on pure mixed convective low condition. The combined eects o Reynolds and Grasho numbers on the characteristics o mixed convection heat transer are also revealed via streamline and isotherm plots, and the variation o average Nusselt number and average luid temperature o the cavity. 2. Problem ormulation Two lid-driven trapezoidal cavities with dierent base and top walls but equal domain area are considered in the present study. The schematic diagrams o these cavities along with their co-ordinate system are shown in Figs. 1 (a) Fig. 1. Schematic representation o trapezoids having heated wall on (a) short base and (b) long base.

3 390 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) and (b). The irst trapezoidal cavity has short base wall and long top wall with aspect ratio o L/H = 0.732, whereas the second cavity has long base wall and short top wall with aspect ratio o L/H = Both cavities are illed with water-al 2 O 3 nanoluid and have the adiabatic sidewalls inclined at an angle = 15 o with the y-axis. Constant isothermal high (T h ) and low (T c ) temperatures are maintained at bottom and top walls respectively. In addition, the top wall o the cavities is allowed to move in its own plane along positive x-direction at a constant speed U o, which justiies the name lid-driven trapezoid. Both cavities are inclined at an angle a with the horizontal reerence x-axis and thus arising opposing low condition where the shear low caused by the moving top wall opposes the buoyancy driven low caused by the thermal non-homogeneity o the cavity boundaries. The working luid is assumed to be incompressible and Newtonian. The Boussinesq approximation is adopted to account or the variations o temperature as a unction o density and to couple in this way the temperature ield to the low ield. The dissipation eect due to the viscous term is neglected and no heat generation is considered. Then the governing equations or steady laminar mixed convection can be expressed by using conservation o mass, momentum and energy equations in the dimensionless orm as ollows: U V 0, X Y 2 2 U U P 1 n U U n U V Risin, 2 2 X Y X n ReX Y n 2 2 V V P n 1 V V n U V Ricos, 2 2 X Y Y n ReX Y n U V X Y RePr X Y 2 2 n (1) (2) (3) (4) The dimensionless parameters in the above equations are deined as ollows: x y u v p T T X, Y, U, V, P,, H H U U U T T c n 0 H c (5) where X and Y are the non-dimensional coordinates varying along horizontal and vertical directions respectively, U and V are the non-dimensional velocity components in the X and Y-directions respectively, Θ is the non-dimensional temperature and P is the non-dimensional pressure. The non-dimensional governing parameters appeared in the above equations are Reynolds number (Re), Grasho number (Gr), Prandtl number (Pr) and Richardson number (Ri) respectively and those are deined as ollows: UH g (T T )H Gr Re, Gr, Pr, Ri. (6) 3 0 H c 2 2 Re The thermo-physical properties o the nanoluid (water-al 2 O 3 ) are obtained rom the relations listed in table 1 whereas the individual thermo-physical properties o water and Al 2 O 3 are presented in table 2. Table 1. Thermo-physical relationship or the nanoluid. Property Equation Eective density, ρ n n 1 s Eective viscosity, μ n n Thermal expansion coeicient, β n n 1 / s n Eective thermal diusivity, α n k / n n Cp n Eective thermal conductivity, k n kn k ks 2k 2 k ks / ks 2k k ks C 1 C C Heat capacitance, (ρc p) n p p p n s

4 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Table 2. Thermo-physical properties o water and Al 2O 3 nanoparticles [9]. Fluid / Solid ρ (kg/m 3 ) C p (J/kgK) k (W/mK) β (1/K) μ (Pa.s) Water Al 2O Table 3. Non-dimensional boundary conditions o the present problem. Dependent variables Top wall Bottom wall Inclined side walls Velocity U = 1, V = 0 U = V = 0 U = V = 0 Temperature Θ = 0 Θ = 1 Θ/ N = 0 Here, is the solid-volume raction o the nanoparticles and the subscript s, and n represent the properties o nanoparticles, base luid and nanoluid, respectively. The non-dimensional boundary conditions or the present problem are speciied in table 3. Two important non-dimensional parameters such as average temperature (Θ av ) o the luid inside the cavity and average Nusselt number (Nu) o the heated wall are evaluated in the present investigation in order to describe the characteristics o mixed convection heat transer. Those parameters are deined as ollows: 1 da, av (7) A k LH / n H Nu dx. (8) k L 0 Y 3. Numerical procedure The non-dimensional governing equations (1-4) or the present problem are solved using Galerkin inite element method. The detailed procedures to apply this technique in order to transorm a set o partial dierential equations into a set o nonlinear algebraic equations can be ound in [11]. The computational domain is discretized using nonuniorm eight noded quadrilateral mesh elements and iner mesh is chosen especially near the solid walls to capture the rapid changes in the dependent variables. All eight nodes are associated with velocities as well as temperature, whereas only the corner nodes are associated with pressure. The nonlinear equations are solved iteratively using Broyden s method with an LU-decomposition preconditioner. The relative tolerance or the convergence criteria is set to be Grid independence check Seven sets o mesh elements are selected to perorm the check or grid size independency. The numbers o mesh elements chosen or this test are 50 50, 60 60, 65 65, 70 70, 75 75, and Table 4 shows the comparison o using dierent mesh elements and rom these comparisons, it is conirmed that non-uniorm mesh elements are suicient to produce results within reasonable accuracy and independent o any urther mesh reinement. Hence, we have selected mesh elements to carry out all simulations. Table 4. Grid independence check using the variation o Nu o two dierent trapezoids or Ri = 1, Re = 100, Gr = 10 4, α = 45 o and = 0.1. Mesh elements Nu or trapezoid with short base Nu or trapezoid with long base

5 392 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Fig. 2. Comparison o (a) isotherm and (b) streamline contours o the present computation (dashed blue lines) with those o Bhattacharya et al. [5] (solid black lines) or Re = 100, Pr =10, Ri = 0.1 and α = 30 o Code Validation Both qualitative and quantitative validations o the present numerical code are perormed beore starting the simulation o the present problem. The qualitative comparison is made with the results obtained or mixed convection lows in a trapezoidal enclosure as mentioned by Bhattacharya et al.[5]. Fig. 2 shows the comparison o isotherm and streamline contours obtained by the present code or the trapezoid having hot wall on long base and cavity tilt angle o 0 o with the aorementioned paper [5]. It is ound that the present numerical results are almost in close agreement with the mentioned one [5], in terms o isotherm and streamline plots. Further quantitative validation is carried out in terms o average Nusselt number o the heated wall o a lid-driven square cavity illed with water-al 2 O 3 nanoluid and inclined at an angle 30 o with the horizontal axis as obtained by Abu-Nada and Chamkha [9]. The comparison is shown in table 5 and the agreement is ound to be satisactory. Thereore, both qualitative and quantitative comparisons validate the present computations and lend us conidence or the use o the present code. Table 5. Comparison o Nu rom the present code with the results o Abu-Nada and Chamkha [9] or Gr = 100, α = 30 o and = 0.1. Ri Nu rom present code Nu rom Abu-Nada and Chamkha [9] Results and discussions The present study has carried out investigation o pure mixed convection or dierent tilt angles o two lid-driven trapezoidal cavities. The simulations are perormed by considering both plain luid (water) and nanoluid (water- Al 2 O 3 ) with 10% solid-volume raction o nanoparticles. Reynolds number is varied rom 0.1 to a maximum order o 10 4 with a view to understand the combined eect o Re and Gr at Ri = 1 on the low and the thermal ields. The cavity tilt angle is considered to vary rom 0 o to 45 o with a step-angle o 15 o Eect o cavity tilt angle on low and thermal ields Figs. 3 and 4 are dedicated or the investigation o the eect o cavity tilt angle (α) on the streamline and the isotherm proiles inside the trapezoidal cavities with short and long bases, respectively, or constant Reynolds (Re = 200) and Richardson (Ri = 1) numbers. Careul observations yield that, or all values o α, two circulating cells appear inside the cavity. Upper clockwise rotating cell corresponds to the mechanical eect o lid inducing orced convection, and lower counter-clockwise circulating cell represents natural convection due to buoyancy eect. For horizontal trapezoids (α = 0 o ), primary lid-driven circulation occupies major portion o the cavity, whereas the comparatively smaller secondary eddy due to buoyancy appears at lower right portion (see Figs. 3(a) and 4 (a)). Presence o only vertical component o the buoyancy orce results in aiding core low at α = 0 o, whereas due to the eect o inclination angle o the cavity, the opposing buoyancy driven low against the low o the moving lid gives rise to the secondary eddy. With increasing tilt angle, this circulating cell almost dominantly capture the low region especially at α = 45 0 (see Figs. 3 (g) and 4 (g)) indicating the adverse eect on convection heat transer.

6 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 (2015) Fig. 3. Comparison o streamlines (top row) and isotherm (bottom row) contours inside trapezoidal cavity with short base at Re = 200, Ri = 1, Gr = or (a), (b) α = 0o; (c), (d) α = 15o; (e), () α = 30o and (g), (h) α = 45o. The solid black line represents data or plain luid ( = 0) whereas the red dashed line represents data or nanoluid with φ = 0.1. Fig. 4. Comparison o streamlines (top row) and isotherm (bottom row) contours inside trapezoidal cavity with long base at Re = 200, Ri = 1, Gr = or (a), (b) α = 0o; (c), (d) α = 15o; (e), () α = 30o and (g), (h) α = 45o. The solid black line represents data or plain luid ( = 0) whereas the red dashed line represents data or nanoluid with φ = 0.1. Lid-driven trapezoids with short and long base show almost similar behaviours in terms o streamline contours, except that the buoyancy induced counter clockwise vortex is relatively larger or the trapezoid with long base (see Figs. 4 (c), (e) and (g)). This is because this type o trapezoids possess longer heated wall and shorter mechanical lid which results in reduced lid eect and increased buoyancy eect. While observing the isotherm contours, it reveals that the contours have comparatively uniorm distribution at inclination angle, α = 00 (see Figs. 3 (b) and 4 (b)). Core low due to the mechanical eect o moving lid dominates in the upper zone providing low temperature region, whereas the contours are clustered near the bottom wall 393

7 394 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) providing higher temperature gradient. For α = 15 0, we can see rom Figs. 3 (d) and 4 (d) that the isotherm contour lines are more distorted than those or α = 0 0. Both driving mechanisms give individual isothermal zone, with a high temperature gradient at their interaction region at α = With increasing α, the nonlinearity o the contour lines increases and the isothermal zone is mainly inluenced by the buoyancy eect, which shits high temperature gradient interaction region towards upper wall (see Figs. 3 (), (h) and 4 (), (h)). There is also remarkable variation o streamlines and isotherms between nanoluid and plain luid or all tilt angles o the cavities. However, at moderate inclination angles α = 15 0 and 30 0, inluence o the buoyancy orce and moving lid on nanoluid and base luid is remarkable. Mostly moving lid aects the nanoluid and base luid proiles in these igures. At higher inclination angle (α = 45 0 ), both nanoluid and plain luid is mostly inluenced by the buoyancy orce, and ollow completely dierent proiles or these two types o trapezoids Combined eect o Reynolds and Grasho numbers on low and thermal ields Apart rom the change o tilt angle, both Reynolds and Grasho numbers also play a great role to inluence the temperature and the low ields. Figs. 5 and 6 show the variation o streamline and isotherm contours with the change o Reynolds and Grasho numbers simultaneously, where Richardson number and tilt angle are kept constant at Ri = 1 and α = 45 o, respectively. Three dierent Reynolds numbers (Re = 10, 100 and 10 3 ) are considered here or comparison o low and thermal ields where the corresponding Grasho numbers become 100, 10 4 and 10 6 or ixed Ri = 1. It is easily understood rom these igures that at Re = 10, with corresponding Gr = 100, buoyancy low converges with core low to orm a single clockwise rotating cell o semicircular structure and thus indicates dominating low ield (see Figs. 5 (a) and 6 (a)). With the urther increase o Reynolds and Grasho numbers, a gradually expanding counter-clockwise rotating vortex appears at the bottom o the enclosure and thus opposes the circulating low. At Re = 10 3 and Gr = 10 6, this secondary circulating cell almost dominantly capture the low region (see Figs. 5 (e) and 6 (e)) indicating the dominating eect o buoyancy induced convection heat transer. Like streamlines, the isotherm contours show signiicant change with increasing Re and Gr. At Re = 10 and Gr = 100, isothermal contours possess nearly uniorm distribution (see Figs. 5 (b) and 6 (b)) with steep temperature gradient along the Y-direction. Single circulation exists in the cavity with high temperature gradient and thus having stratiied thermal boundary layer. With increasing Reynolds and Grasho numbers, the buoyancy eect becomes dominant, which ultimately induce reduced temperature gradient. As a result, relatively signiicant nonlinear contours with huge isothermal zone occupy most o the cavity region (see Figs. 5 (d), () and 6 (d), ()). Fig. 5. Comparison o streamlines (top row) and isotherm (bottom row) contours inside trapezoidal cavity with short base at Ri = 1 and α = 45 o or (a), (b) Re = 10, Gr = 100; (c), (d) Re = 100, Gr = 10 4 and (e), () Re = 10 3, Gr = The solid black line represents data or plain luid ( = 0) whereas the red dashed line represents data or nanoluid with φ = 0.1.

8 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Fig. 6. Comparison o streamlines (top row) and isotherm (bottom row) contours inside trapezoidal cavity with long base at Ri = 1 and α = 45 o or (a), (b) Re = 10, Gr = 100; (c), (d) Re = 100, Gr = 10 4 and (e), () Re = 10 3, Gr = The solid black line represents data or plain luid ( = 0) whereas the red dashed line represents data or nanoluid with φ = 0.1. Figs. 5 and 6 are also dedicated or comparing the presence o nanoluid on low and temperature proiles between two dierent types o trapezoids. At low Re and Gr, the proiles o streamline or both nanoluid and plain luid coincide, whereas the isotherm patterns clearly show the inluence o nanoluid due to improved thermophysical properties. Similar observations are also ound or higher values o Re and Gr. However, at moderate values o Re and Gr, streamline and isotherm contours ollow completely dierent proiles. The nanoluid seems to alter the low and the thermal region or both types o trapezoids (see Figs. 5 (c), (d) and 6 (c), (d)). Fig. 7. Variation o (a) average Nusselt number o the heated wall and (b) average temperature o the luid inside cavity with both Reynolds and Grasho numbers or dierent tilt angle o the trapezoidal cavity with hot wall on short base. The results are shown or both plain luid (solid lines) and nanoluid (dashed lines) with = 0.10.

9 396 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Evaluation o the perormance o heat transer Figs.7 and 8 show the eect o increasing both Reynolds and Grasho numbers on the average Nusselt number or the cases o = 0 and 0.1 at Ri = 1or various inclination angles. It can be observed that the average Nusselt number along the bottom wall o the trapezoidal cavity increases continuously with increasing Re and Gr simultaneously. However, three important distinct points can be observed rom this igure. First, the transition rom conduction to convection regime is noticed within the laminar zone, where Nu remains constant or both plain luid and nanoluid. The second and third observations are the beginning and the end o transition rom laminar to chaos within very narrow region. A sudden but gradual drop o average Nusselt number is observed or both plain luid and nanoluid, where the inluence o increasing both Reynolds and Grasho numbers is overwhelmed in the pure mixed convective lows. While observing the eect o inclination angle on heat transer, Figs. 7 and 8 clearly indicate that there is completely opposite behaviour or the variation o Nu with respect to α between these two trapezoidal cavities. Apart rom the conduction dominated region, Nu increases with increasing α or short based trapezoid (see Fig. 7 (a)) while or long based trapezoid, Nu decreases with increasing α (see Fig. 8 (a)). This is mainly due to the length o the heated base wall, which inluence the buoyancy eect with the change o the tilt angle. Since the side-corners o the short based trapezoid have obtuse angle and those or the long based trapezoid have acute angle, better convective low over the bottom wall appears in short based cavity, whereas convective low deteriorates or the long based one. For both types o trapezoids, average temperature o the cavity increases with increasing tilt angle ater the transition rom laminar to chaos (see Figs. 7(b) and 8 (b)). It is evident that all inclined trapezoids reach to the transition point at lower Re than the horizontal one. Moreover, Figs. 7 (a) and 8 (a) also reveal that Nu or nanoluid is greater than base luid or all inclination angles, and all Reynolds and Grasho numbers, which clearly indicates heat transer augmentation by using nanoluid. Observing Figs. 7 (b) and 8 (b), it can be concluded that ater a certain value o Re, average temperature or cavities illed with nanoluid becomes less than those o the base luidor all α implying better heat transer due to the presence o nanoluid. Fig. 8. Variation o (a) average Nusselt number o the heated wall and (b) average temperature o the luid inside cavity with both Reynolds and Grasho numbers or dierent tilt angle o the trapezoidal cavity with hot wall on long base. The results are shown or both plain luid (solid lines) and nanoluid (dashed lines) with = 0.10.

10 Mahmudul Hasan Hasib et al. / Procedia Engineering 105 ( 2015 ) Conclusion The problems o steady mixed convection heat transer o an incompressible, Newtonian and Boussinesq luid inside two dierent 2D lid-driven trapezoidal cavities, one having heated wall on short base and another with heated wall on long base have been investigated numerically. The top wall o the cavity is maintained at the surrounding low temperature and is moving with a constant velocity. Simulations are perormed or dierent tilt angle o these cavities with the variation o both Reynolds and Grasho numbers at ixed Richardson number. One o the important indings is that both the heat transer and low characteristics inside the cavities strongly depend on the choice o Reynolds and Grasho numbers at Ri = 1. Presence o the nanoluid in comparison to the plain luid also signiicantly aects the thermal scenario inside the cavity. Overall mixed convection heat transer characteristics rom the hot bottom wall o the trapezoidal cavities are ound to be inluenced by the mechanical eect o the moving lid and the buoyancy-driven low. Extensive investigation or various tilt angles suggests an interesting eature on the characteristics o mixed convection in these two lid-driven trapezoidal cavities. For short based trapezoids, tilting the enclosure increases heat transer rom heated wall, whereas or long based trapezoid, it decreases. Acknowledgements The authors grateully acknowledge the support provided by the Department o Mechanical Engineering, Bangladesh University o Engineering and Technology (BUET) during this research work. Reerences [1] M.N. Hossain, M.A.H. Mamun, S. Saha, Mixed convection in a trapezoidal cavity with moving lid at top wall and heating rom below, International Conerence on Chemical Engineering, Dhaka, Bangladesh, [2] M.N.H.K. Chowdhury, S. Saha, M.A.H. Mamun, Mixed convection analysis in a lid driven trapezoidal cavity with isothermal heating at bottom or various aspect angles, 8 th International Conerence on Mechanical Engineering, Dhaka, Bangladesh, [3] M.N. Hasan, S. Saha, G. Saha, M.Q. Islam, Eect o sidewall inclination angle o a lid-driven trapezoidal enclosure on mixed convective low and heat transer characteristics, 13 th Asian Congress o Fluid Mechanics, Dhaka, Bangladesh, [4] M.A.H. Mamun, T.R. Tanim, M.M. Rahman, R. Saidur, S. Nagata, Mixed convection analysis in trapezoidal cavity with a moving lid, Int. J. Mech. Mat. Eng., 5 (1) (2010) [5] M. Bhattacharya, T. Basak, H.F. Oztop, Y. Varol, Mixed convection and role o multiple solutions in lid-driven trapezoidal enclosures, Int. J. Heat Mass Trans., 63 (2013) [6] T.S. Cheng, Characteristics o mixed convection heat transer in a lid-driven square cavity with various Richardson and Prandtl numbers, Int. J. Therm. Sci., 50 (2011) [7] R.K. Tiwari, M.K. Das, Heat transer augmentation in a two-sided lid-driven dierentially heated square cavity utilizing nanoluids, Int. J. Heat Mass Trans., 50 (9) (2007) [8] F. Talebi, A.H. Mahmoudi, M. Shahi, Numerical study o mixed convection lows in a square lid-driven cavity utilizing nanoluid, Int. Comm. Heat Mass Trans., 37 (1) (2010) [9] E. Abu-Nada, A.J. Chamkha, Mixed convection low in a lid-driven inclined square enclosure illed with a nanoluid, Euro. J. Mech.- B/Fluids, 29 (6) (2010) [10] M. Salari, M.M. Tabar, A.M. Tabar, H.A. Danesh, Mixed convection o nanoluid lows in a square lid-driven cavity heated partially rom both the bottom and side walls, Numer. Heat Trans., Part A: Applications, 62 (2) (2012) [11] O.C. Zienkiewicz, R.L. Taylor, The inite element method, 1 (1973)