Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity

International Communications in Heat and Mass Transfer 38 (2011) 184–193

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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t

Magnetohydrodynamic mixed convection in a horizontal channel with an open cavity☆ M.M. Rahman a,c,⁎, S. Parvin a, R. Saidur b,c, N.A. Rahim c a b c

Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Centre of Research UMPEDAC, Level 4, Engineering Tower, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia

a r t i c l e

i n f o

Available online 16 December 2010 Keywords: Mixed convection Magnetohydrodynamic Open enclosure Channel Finite element method

a b s t r a c t The development of magnetic field effect on mixed convective flow in a horizontal channel with a bottom heated open enclosure has been numerically studied. The enclosure considered has rectangular horizontal lower surface and vertical side surfaces. The lower surface is at a uniform temperature Th while other sides of the cavity along with the channel walls are adiabatic. The governing two-dimensional flow equations have been solved by using Galarkin weighted residual finite element technique. The investigations are conducted for different values of Rayleigh number (Ra), Reynolds number (Re) and Hartmann number (Ha). Various characteristics such as streamlines, isotherms and heat transfer rate in terms of the average Nusselt number (Nu), the Drag force (D) and average bulk temperature (θav) are presented. The results indicate that the mentioned parameters strongly affect the flow phenomenon and temperature field inside the cavity whereas in the channel these effects are less significant. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The influence of the magnetic field on the convective heat transfer and the mixed convection flow of the fluid are of paramount importance in engineering. A combined free and forced convection flow of an electrically conducting fluid in a cavity or in a channel in the presence of magnetic field is of special technical significance because of its frequent occurrence in many industrial applications such as geothermal reservoirs, cooling of nuclear reactors, thermal insulations and petroleum reservoirs. These types of problems also arise in electronic packages, micro electronic devices during their operations. Papanicolaou and Jaluria [1] carried out a numerical study to investigate the combined forced and natural convective cooling of heat dissipating electronic components, located in rectangular enclosure and cooled by an external flow of air. A computational study on mixed convection in ventilated cavity with uniform heat flux in left wall was performed by Raji and Hasnaoui [2]. Moreover, Raji and Hasnaoui [3] investigated the mixed convection in ventilated cavities where the horizontal top wall and the vertical left wall were prescribed with equal heat fluxes. At the same time, Angirasa [4] numerically studied and explained the complex interaction between buoyancy and forced flow in a square enclosure with an inlet and a vent situated respectively, at the bottom and top edges of the vertical ☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh. E-mail address: [email protected] (M.M. Rahman). 0735-1933/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2010.12.005

isothermal surface, where the other three walls are adiabatic. Also, Gau et al. [5] performed experiments on mixed convection in a horizontal rectangular channel with side heating. A numerical study of mixed convection heat transfer in two dimensional open-ended enclosures were investigated by Khanafer et al. [6] for three different forced flow angle of attack. A numerical analysis of laminar mixed convection in a channel with an open cavity and a heated wall bounded by a horizontally insulted plate was presented in Manca et al. [7], where they considered three heating modes: assisting flow, opposing flow and heating from below. Brown and Lai [8] numerically studied a horizontal channel with an open cavity and obtained correlations for combined heat and mass transfer which covered the entire convection regime from natural, mixed to forced convection. Leong et al. [9] performed the analysis of mixed convection from an open cavity in a horizontal channel. Their findings were that the heat transfer rate was reduced and the flow became unstable in the mixed convection regime. Later on, a finite-volume based computational study of steady laminar forced convection inside a square cavity with inlet and outlet ports was presented in Saeidi and Khodadadi [10]. Recently, Rahman et al. [11] studied numerically the opposing mixed convection in a vented enclosure. They found that with the increase of Reynolds and Richardson numbers the convective heat transfer becomes predominant over the conduction heat transfer and the rate of heat transfer from the heated wall is significantly depended on the position of the inlet port. Oreper and Szekely [12] studied the effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular cavity. They found that the presence of a magnetic field can suppress

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 184–193

adiabatic

Nomenclature B0 D g Ha L Nu P P Pr Ra Re T u, v U, V  V w x, y X, Y

magnetic induction drag force gravitational acceleration (ms− 2) Hartmann number length of the cavity (m) Nusselt number dimensional pressure (Nm− 2) non-dimensional pressure Prandtl number Rayleigh number Reynolds number dimensional temperature (K) velocity components (ms− 1) non-dimensional velocity components cavity volume (m3) height of the channel (m) Cartesian coordinates (m) non-dimensional coordinates

Greek symbols α thermal diffusivity (m2 s− 1) β thermal expansion coefficient (K− 1) μ dynamic viscosity (Kg m− 1 s− 1) ν kinematic viscosity (m2 s− 1) θ non-dimensional temperature ρ density (Kg m− 3) σ electrical conductivity

Subscripts av average h heated wall i inlet state

natural convection currents and that the strength of the magnetic field is one of the important factors in determining the quality of the crystal. Bourich et al. [13] made an analytical and numerical study of combined effects of a magnetic field and an external shear stress on soret convection in a horizontal porous enclosure. They used finitedifference method to derive an analytical solution on the basis of the parallel flow approximation, assuming a large aspect ratio layer. Jalil and Tae'y [14] studied the problem of natural convection with magnetic field effect on laminar flow in a two-dimensional square enclosure subject to a vertical mechanical vibration by using the time marching method. Their result stated that the average Nusselt number at a low value of vibration frequency increased when a moderate magnetic field was applied, but decreased under a stronger magnetic field. Garandet et al. [15] studied natural convection heat transfer in a rectangular enclosure with a transverse magnetic field. Rudraiah et al. [16] investigated the effect of surface tension on buoyancy driven flow of an electrically conducting fluid in a rectangular cavity in the presence of a vertical transverse magnetic field to see how this force damps hydrodynamic movements. Natural convection in an enclosure filled with a fluid-saturated porous medium in a strong magnetic field is investigated numerically by Zeng et al. [17]. They found a significant effect of the magnetic force on the flow field and heat transfer in the fluid-saturated porous medium. The problem of unsteady laminar combined forced and free convection flow and heat transfer of an electrically conducting and heat generating or absorbing fluid in a vertical lid-driven cavity in the presence of a magnetic field was formulated by Chamkha [18]. The

185

ui, Ti

B0

0.5L L, Th

Fig. 1. Schematic diagram of the problem.

fully developed flow of an electrically conducting, internally heated fluid in a vertical square duct under the influence of buoyancy and magnetohydrodynamic forces is studied by Sposito and Ciofalo [19]. They determined the limiting values of pressure gradient and mean velocity for the flow to be unidirectional throughout the duct's section. Mahmud et al. [20] studied analytically a combined free and forced convection flow of an electrically conducting and heatgenerating/absorbing fluid in a vertical channel made of two parallel plates under the action of transverse magnetic field. In the light of the above literature, it has been pointed out that there is no significant information about MHD effect on the mixed convection processes in a channel containing an open cavity. The present study addresses the effects of magnetic field which may increase or decrease the heat transfer on mixed convection in a channel with a bottom heated rectangular cavity. Numerical solutions are obtained over a wide range of Rayleigh number, Reynolds number and Hartmann number. Results are presented graphically in terms of streamlines and isothermal lines. Finally the average Nusselt number at the heated surface, the Drag force and the mean temperature of the fluid in the cavity are calculated. 2. Model specification The geometry of the problem herein investigated is depicted in Fig. 1. The system consists of a two-dimensional channel with an open cavity of length L and width 0.5 L. The base of the cavity is heated at a constant temperature Th while the rest of the cavity walls along with channel walls are well insulated. It is assumed that the height of the channel w = 0.5 L, the incoming flow is at a uniform velocity, ui and at the ambient temperature, Ti. A magnetic field of strength B0 is applied in the perpendicular direction to the side walls of the cavity. 3. Mathematical formulation A two-dimensional, steady, laminar, incompressible, mixed convection flow is considered within the cavity and the fluid properties are assumed to be constant. The radiation effects are taken as negligible. The dimensionless equations describing the flow under Boussinesq approximation are as follows: ∂U ∂V + =0 ∂X ∂Y

U

ð1Þ

∂U ∂U ∂P 1 ∂2 U ∂2 U + +V =− + 2 Re ∂X ∂X ∂Y ∂X ∂Y 2

! ð2Þ

Table 1 Grid sensitivity check at Ha = 10.0, Ra = 105 and Re = 100. Nodes (elements)

48427 (5774)

68476 (7526)

82348 (9158)

104261 (11736)

105026 (11826)

Nu θav Time (s)

5.098613 0.378798 226.265

5.138278 0.376889 292.594

5.262424 0.374889 388.157

5.272548 0.372291 421.328

5.55241 0.37204 627.375

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∂V ∂V ∂P 1 ∂2 V ∂2 V U + +V =− + 2 Re ∂X ∂X ∂Y ∂Y ∂Y 2 ∂θ ∂θ 1 ∂2 θ ∂2 θ U + +V = 2 RePr ∂X ∂X ∂Y ∂Y 2

! +

Ra Ha2 V θ− 2 Re Re Pr

ð3Þ

! ð4Þ 3

−Ti ÞL are Reynolds number, where Re = uνi L, Pr = αν and Ra = gβðThνα Prandtl number and Rayleigh number respectively and Ha is 2 2 Hartmann number which is defined as Ha2 = σBμ0 L . The above equations were non-dimensionalized by using the following dimensionless quantities

X=

ðT−Ti Þ x y u v p ;θ= ;Y = ;U = ;V = ;P = ðTh −Ti Þ L L ui ui ρ u2i

where X and Y are the coordinates varying along the horizontal and vertical directions respectively, U and V are the velocity components in the X and Y directions respectively, θ is the dimensionless temperature and P is the dimensionless pressure. The boundary conditions for the present problem are specified as follows: at the inlet: U = 1, V = 0, θ = 0 at the outlet: convective boundary condition P = 0

at all solid boundaries other than bottom wall: U = 0; V = 0; ∂θ =0 ∂N at the bottom wall: U = V = 0, θ = 1 where N is the non-dimensional distances either X or Y direction acting normal to the surface. The average Nusselt number at the heated surface is calculated as 1

Nu = − ∫0

∂θ ∂Y

1

dX, Drag force, D = − ∫0

∂U dX ∂Y

and the average tem   perature of the fluid is defined as θav = ∫θ d V = V, where V is the cavity volume.

4. Computational procedure The momentum and energy balance equations are the combinations of mixed elliptic–parabolic system of partial differential equations that have been solved by using the Galerkin weighted residual finite element technique [21]. The continuity equation has been used as a constraint due to mass conservation. The basic unknowns for the above differential equations are the velocity components U, V the temperature, θ and the pressure, P. The six node triangular element is used in this work for the development of the finite element equations. All six nodes are associated with velocities as well as temperature; only the corner nodes are associated

Fig. 2. (a) Streamlines and (b) Isotherms for different values of Hartmann number Ha, while Re = 100 and Ra = 103.

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 184–193

with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through continuity equation. The velocity component, the temperature distributions and the linear interpolation for the pressure distribution according to their highest derivative orders in the differential Eqs. (2)–(4) are as follows: U ðX; Y Þ = Nβ Uβ ; V ðX; Y Þ = Nβ Vβ ; θðX; Y Þ = Nβ θβ ; P ðX; Y Þ = Hλ Pλ where β = 1, 2, … …, 6; λ = 1, 2, 3. Substituting the element velocity component distributions, the temperature distribution and the pressure distribution from Eqs. (2)–(4) the finite element equations can be written in the form  1  Sαβxx + Sαβyy Uβ = Qαu ð5Þ Kαβγx Uβ Uγ + Kαβγy Vβ Uγ + Mαμ x Pμ + Re Kαβγ Uβ Vγ + Kαβγ Vβ Vγ + Mαμ x

y

y

 1  Pμ + Sαβxx + Sαβyy Vβ Re

2

Ra Ha K V = Qαv − 2 Kαβ θβ + Re αβ β Re Pr Kαβγx Uβ θγ + Kαβγy Vβ θγ +

 1  Sαβxx + Sαβyy θβ = Qαθ Re:Pr

187

where the coefficients in the element matrices are in the form of the integrals over the element area A and along the element edges S0 and Sw as Kαβx = ∫A Nα Nβ; x dA; Kαβy = ∫A Nα Nβ; y dA; Kαβγy = ∫A Nα Nβ Nγ; y dA;

Kαβγx = ∫A Nα Nβ Nγ; x dA;

Kαβ = ∫A Nα Nβ dA; Sαβxx = ∫A Nα; x Nβ; x dA ;

Sαβyy = ∫A Nα; y Nβ; y dA; Mαμ x = ∫A Hα Hμ; x dA; Mαμ y = ∫A Hα Hμ; y dA; Qαu = ∫S0 Nα Sx dS0 ; Qαv = ∫S0 Nα Sy dS0 ; Qαθ = ∫Sw Nα qw dSw : The set of non-linear algebraic Eqs. (5)–(7) are solved by using reduced integration technique [22,23] and Newton–Raphson method [24]. 4.1. Grid refinement check

ð6Þ

ð7Þ

In order to determine the proper grid size for this study, a grid independence test is conducted with Ha = 10.0, Ra = 105 and Re = 100. The following five types of mesh are considered for the grid independence study. These grid densities are 48427 nodes, 5774 elements; 68476 nodes, 7526 elements; 82348 nodes, 9158 elements; 104261 nodes, 11736 elements and 105026 nodes, 11826 elements.

Fig. 3. (a) Streamlines and (b) Isotherms for different values of Hartmann number Ha while Re = 100 and Ra = 104.

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The extreme value of the average Nusselt number Nu, that relates to the heat transfer rate of the heated surface and average temperature θav of the fluid in the cavity are used as a sensitivity measure of the accuracy of the solution and are selected as the monitoring variables for the grid independence study. Table 1 shows the dependence of the quantities Nu and θav on the grid size and the computational time. Considering both the accuracy of the numerical values and the computational time, the following calculations are performed with 68476 nodes, 7526 elements grid system. 4.2. Code validation A description of the code and its validation are available in Rahman et al. [25] and are not repeated here. The existing code in [25] has been modified for the current problem. 5. Results and discussion A numerical computation has been carried out through finite element method to analyze the magnetohydrodynamic (MHD) mixed convection in a horizontal channel containing an open cavity at the bottom wall. Effects of the controlling parameters such as Hartmann number (Ha), Rayleigh number (Ra) and Reynolds number (Re) on heat transfer and fluid flow inside the cavity have been studied. The changes in flow and temperature fields in terms of streamline and isotherms, average Nusselt number, the Drag force and average fluid temperature for different cases are focused in the following sections. The ranges of Ha, Ra and Re for this investigation vary from 0 to 20, 103

to 105 and 100 to 500 respectively while the Prandtl number Pr (= 0.71) is kept as constant. The influence of Hartmann number Ha (= 0.0, 5.0, 10.0, 20.0) on streamlines as well as isotherms for the present configuration at Ra = 103 and Re = 100 has been demonstrated in Fig. 2(a)–(b). The flow with Ha = 0.0 creates a small vortex at the right bottom part of the cavity due to the buoyancy force. This vortex is disappeared with increasing Hartmann number as shown in Fig. 2(a). It is also observed from the figure that the magnetic field affects only the flow inside the cavity whereas in the channel it remains almost similar. Corresponding temperature field shows that in the absence of magnetic field the isothermal lines form a thin thermal boundary layer near the hot bottom wall of the cavity which are drastically changed with increasing values of Ha. Moreover, the isothermal lines become almost parallel to the horizontal channel for the highest value of Ha (= 20). This is because the magnetic field tends to retard the flow. Fig. 3(a)–(b) illustrate the flow and temperature field for different Ha at Ra = 104 and Re = 100. A buoyancy induced large rotating cell is developed in the cavity at Ha = 0 which loses its strength as Ha increases and finally vanished at the highest value of Ha. On the other hand, the isotherms are attracted more to less towards the hot wall as well as the channel with escalating Ha. The changes in fluid flow and isotherm patterns at Ra = 105 and Re = 100 for various Hartmann number have been depicted in Fig. 4 (a)–(b). A large whirling cell in the fluid motion occupying almost the whole cavity is seen for all cases while the cell becomes more strengthened as Ha devalues as shown in Fig. 4(a). However, the channel flow remains unaffected. The stratified isotherms become

Fig. 4. (a) Streamlines and (b) Isotherms for different values of Hartmann number Ha while Re = 100 and Ra = 105.

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189

denser adjacent to the hot wall and the channel with decreasing Ha. At the lowest value of Ha the cavity centre contains no isothermal lines because the dominant natural convection is capable to carry more heat through the strong recirculation. It is noteworthy that for a particular Ha, the appearance and size of vortex in the cavity flow increase with increasing Rayleigh number. It is also noticed that the strength of the circulating cell in the cavity is enhanced with rising Ra. This is because at a large Rayleigh number, buoyancy effects become more significant. However, the flow in the channel is not affected significantly by the magnetic field. On the other hand, the change in temperature profile for growing values of Ra with a fixed Ha is remarkable. The thermal boundary layer thickness near the hot wall and the channel is reduced as Ra grows up. Variation in the mean Nusselt number, the Drag force and average bulk temperature for governing parameters Ha and Ra are shown Fig. 5(a)–(c). A careful observation indicates that all three figures are similar in pattern that is increasing Hartmann number decreases the heat transfer, the Drag force and the mean temperature of the fluid for all values of Rayleigh number whereas these quantities increase with increasing of Rayleigh number for all values of Hartmann number. It means that the magnetic field can be used to control the heat transfer phenomena. Fig. 6(a)–(b) show that for a negligible buoyancy effect (Ra = 103) and dominant inertial effects (Re ≥ 102), the flow remains steady and no significant change in the streamlines but the isothermal layer become concentrated near the heated surface for large Re. Fig. 7(a)–(b) correspond to flow and thermal field at Ra = 104 with different Re (= 100, 150, 300, 500). An eddy appears in the cavity for all cases whose size is reduced with higher values of Re. On the other hand, the isotherm patterns are analogous to the above case. There are significant changes in the streamlines and isotherms at Ra = 105 with various Re (= 100, 150, 300, 500) as shown in Fig. 8(a)–(b). A large spinning cell appears in the cavity for Re = 100 which loses its strength and finally creates two cells of lower strength with rising Re. The isotherms are attracted more towards the hot wall as well as the channel with larger Re. For a given Rayleigh number, the heat transfer regime changes from natural convection, through mixed convection, to forced convection as the Reynolds number increases. At a large Reynolds number, buoyancy effects become insignificant and are confined to the cavity only. As a result, the flow in the channel is not at all affected by the cavity. Similarly for a particular Re, the role of natural convection becomes more significant as Ra increases. The average Nusselt number, Drag force and average bulk temperature for governing parameters Re and Ra have been plotted in Fig. 9(a)–(b). Increasing Re increases the heat transfer rate but decreases the Drag force and the average fluid temperature. In addition, increase in Ra raises all these quantities. 6. Conclusion A computational study has been performed to investigate the mixed convection in a channel with a bottom heated open enclosure in presence of magnetic field. Results are obtained for wide ranges of Hartmann number Ha, Reynolds number Re and Rayleigh number Ra. The following conclusions may be drawn from the foregoing investigations: • The influence of Hartmann number on streamlines and isotherms are remarkable for the different values of Ra. Buoyancy-induced vortex in the streamlines is diminished and thermal layer near the heated surface becomes thick and less concentrated with growing values of Ha. • The natural convection parameter Ra and forced convection parameter Re have significant effects on the flow and temperature fields. The eddy due to buoyancy force enhances with increasing Ra

Fig. 5. Effect of Hartmann number on (a) average Nusselt number, (b) Drag force and (c) average fluid temperature in the cavity for varying Ra, while Re = 100.

and reduces with increasing Re. Isothermal lines become denser near the hot surface for rising values of both Ra and Re. • Average Nusselt number at the heated surface enhances as Ra and Re increase and lessen with increasing Ha. The Drag force and the average fluid temperature change from highest to lowest as Ha and Re increase while they grow up with larger values of Ra. References [1] E. Papanicolaou, Y. Jaluria, Mixed convection from an isolated heat source in a rectangular enclosure, Numer. Heat Transfer A 18 (1990) 427–461. [2] A. Raji, M. Hasnaoui, Mixed convection heat transfer in a rectangular cavity ventilated and heated from the side, Numer. Heat Transfer A 33 (1998) 533–548.

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Fig. 6. (a) Streamlines and (b) Isotherms for different values of Reynolds number Re while Ha = 10.0 and Ra = 103. [3] A. Raji, M. Hasnaoui, Mixed convection heat transfer in ventilated cavities with opposing and assisting flows, Int. J. Comput-Aided Eng. Softw. 17 (5) (2000) 556–572. [4] D. Angirasa, Mixed convection in a vented enclosure with an isothermal vertical surface, Fluid Dyn. Res. 26 (4) (2000) 219–233. [5] C. Gau, Y.C. Jeng, C.G. Liu, An experimental study on mixed convection in a horizontal rectangular channel heated from a side, ASME J. Heat Transfer 122 (2000) 701–707. [6] K. Khanafer, K. Vafai, M. Lightstone, Mixed convection heat transfer in twodimensional open-ended enclosure, Int. J. Heat Mass Transfer 45 (2002) 5171–5190. [7] O. Manca, S. Nardini, K. Khanafer, K. Vafai, Effect of heated wall position on mixed convection in a channel with an open cavity, Numer. Heat Transfer A 43 (2003) 259–282. [8] N.M. Brown, F.C. Lai, Correlations for combined heat and mass transfer from an open cavity in a horizontal channel, Int. Commun. Heat Mass Transfer 32 (8) (2005) 1000–1008. [9] J.C. Leong, N.M. Brown, F.C. Lai, Mixed convection from an open cavity in a horizontal channel, Int. Commun. Heat Mass Transfer 32 (2005) 583–592. [10] S.M. Saeidi, J.M. Khodadadi, Forced convection in a square cavity with inlet and outlet ports, Int. J. Heat Mass Transfer 49 (2006) 1896–1906. [11] M.M. Rahman, M.A. Alim, M.A.H. Mamun, M.K. Chowdhury, A.K.M.S. Islam, Numerical study of opposing mixed convection in a vented enclosure, ARPN J. Eng. Appl. Sci. 2 (2) (2007) 25–36. [12] G.M. Oreper, J. Szekely, The effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular cavity, J. Cryst. Growth 64 (1983) 505–515. [13] M. Bourich, M. Hasnaoui, A. Amahmid, M. Er-Raki, M. Mamou, Analytical and numerical study of combined effects of a magnetic field and an external shear stress on soret convection in a horizontal porous enclosure, Numer. Heat Transfer A 54 (11) (2008) 1042–1060. [14] J.M. Jalil, K.A. Al-Tae'y, Thermovibrational convection in an enclosure with magnetic field damping, Numer. Heat Transfer A 53 (7) (2008) 766–786.

[15] J.P. Garandet, T. Alboussiere, R. Moreau, Buoyancy driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer 35 (1992) 741–748. [16] N. Rudraiah, M. Venkatachalappa, C.K. Subbaraya, Combined surface tension and buoyancy-driven convection in a rectangular open cavity in the presence of magnetic field, Int. J. Non-linear Mech. 30 (5) (1995) 759–770. [17] M. Zeng, Q.W. Wang, Z.P. Huang, G. Wang, H. Ozoe, Numerical investigation of natural convection in an enclosure filled with porous medium under magnetic field, Numer. Heat Transfer A 52 (10) (2007) 959–971. [18] A.J. Chamkha, Hydromagnetic combined convection flow in a vertical lid-driven cavity with internal heat generation or absorption, Numer. Heat Transfer A 41 (2002) 529–546. [19] G. Sposito, M. Ciofalo, Fully developed mixed magnetohydrodynamic convection in a vertical square duct, Numer. Heat Transfer A 53 (9) (2008) 907–924. [20] S. Mahmud, S.H. Tasnim, M.A.H. Mamun, Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect, Int. J. Therm. Sci. 42 (2003) 731–740. [21] M.M. Rahman, M.A. Alim, M.A.H. Mamun, Finite element analysis of mixed convection in a rectangular cavity with a heat-conducting horizontal circular cylinder, Nonlinear analysis: Modeling and Control 14 (2) (2009) 217–247. [22] J.N. Reddy, An Introduction to Finite Element Analysis, McGraw-Hill, New-York, 1993. [23] O.C. Zeinkiewicz, R.L. Taylor, J.M. Too, Reduced integration technique in general analysis of plates and shells, Int. J. Numer. Methods Eng. 3 (1971) 275–290. [24] S. Roy, T. Basak, Finite element analysis of natural convection flows in a square cavity with nonuniformly heated wall(s), Int. J. Eng. Sci. 43 (2005) 668–680. [25] M.M. Rahman, M.A. Alim, M.M.A. Sarker, Numerical study on the conjugate effect of joule heating and magneto-hydrodynamics mixed convection in an obstructed lid-driven square cavity, Int. Commun. Heat Mass Transfer 37 (2010) 524–535.

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Fig. 7. (a) Streamlines and (b) Isotherms for different values of Reynolds number Re while Ha = 10.0 and Ra = 104.

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Fig. 8. (a) Streamlines and (b) Isotherms for different values of Reynolds number Re while Ha = 10.0 and Ra = 105.

M.M. Rahman et al. / International Communications in Heat and Mass Transfer 38 (2011) 184–193

Fig. 9. Effect of Reynolds number on (a) average Nusselt number, (b) Drag force and (c) average fluid temperature in the cavity for varying Ra, while, Ha = 10.

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