International Communications in Heat and Mass Transfer 37 (2010) 393–400

Contents lists available at ScienceDirect

International Communications in Heat and Mass Transfer

j ourna l homepage: www.e lsev ie r.com/ locate / ichmt

Finite element simulation of magnetoconvection inside a sinusoidal corrugatedenclosure with discrete isoflux heating from below☆

Goutam SahaDepartment of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

☆ Communicated by W.J. Minkowycz.E-mail address: ranamath06@gmail.com.

0735-1933/$ – see front matter © 2009 Elsevier Ltd. Aldoi:10.1016/j.icheatmasstransfer.2009.12.001

a b s t r a c t

a r t i c l e i n f o

Available online 8 January 2010

Keywords:MagnetoconvectionSinusoidal enclosurePenalty finite element methodNusselt numberGrashof numberHartmann number

A numerical investigation of the steady magnetoconvection in a sinusoidal corrugated enclosure has beenperformed. In this analysis, two vertical sinusoidal corrugated walls are maintained at a constant lowtemperature whereas a constant heat flux source whose length is varied from 20 to 80% of the referencelength of the enclosure is discretely embedded at the bottom wall. The Penalty finite element method hasbeen used to solve the governing Navier–Stokes and energy conservation equation of the fluid medium inthe enclosure in order to investigate the effect of discrete heat source sizes on heat transfer for differentvalues of Grashof number and Hartmann number. The values of the governing parameters are the Grashofnumber Gr (103 to 106), Hartmann number Ha (0 to 100) and Prandtl number Pr (0.71). The presentnumerical approach is found to be consistent and the solution is obtained in terms of stream functions andisotherm contours.

l rights reserved.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Effective cooling of electronic components has become increas-ingly important as power dissipation and component densitycontinue to increase substantially with the fast growth of electronictechnology. It is very important that such cooling systems aredesigned in the most efficient way and the power requirement forthe cooling is minimized. The electronic components are treated asheat sources embedded on flat surface. In many applications, naturalconvection is the only feasible mode of cooling of the heat source.Further, the shape of the heat transfer surfaces influences thedevelopment of the boundary layer. Therefore, the investigation ofthermal and fluid flow behaviors for different shapes of the heattransfer surfaces is necessary to ensure the efficient performance ofthe various heat transfer equipment.

Rudraiah et al. [1] investigated the effect of a magnetic field on freeconvection in a rectangular cavity. They found that magnetic fielddecreased the rate of heat transfer. Kandaswamy and Kumar [2]studied the natural convection of water near its density maximum inthe presence of a uniformmagnetic field. They observed that the effectof magnetic field on the natural convection was to inhibit the heattransfer rate. A study of magnetoconvection in a cavity with partiallyactive vertical walls was conducted by Kandaswamy et al. [3]. Theyshowed that the heat transfer rate was maximum for the middle–middle thermally active locations while it was poor for the top–

bottom thermally active locations. They found that the averageNusselt number decreased with an increase of Hartmann number andincreased with an increase of Grashof number. Also natural convec-tion flow in the presence of a magnetic field in an enclosure filled witha viscous and incompressible fluid has been studied by Garandet et al.[4], Alchaar et al. [5], Kanafer and Chamka [6], Mahmud et al. [7] andEce and Büyük [8].

Several investigations have been carried out on natural convectionheat transfer and fluid flow with corrugated surfaces. Using controlvolume based finite element method, Ali and Husain [9] investigatedthe natural convection heat transfer and flow characteristics in asquare duct of vee corrugated vertical walls. Ali and Husain [10] alsoinvestigated the effect of corrugation frequencies on natural convec-tion heat transfer and flow characteristics in a square enclosure of veecorrugated vertical walls. This investigation showed that the overallheat transfer through the enclosure increased with the increase ofcorrugation for low Grashof number; but there was a reverse trend forhigh Grashof number. Later Ali and Ali [11] carried out a finite elementanalysis of laminar convection heat transfer and flow of the fluidbounded by vee corrugated vertical plate of different corrugationfrequencies. Noorshahi et al. [12] studied heat transfer mechanism inan enclosure with corrugated bottom surface having uniform heat fluxand flat isothermal cooled top surface and adiabatic sidewalls. Theirresults showed that the pseudo-conduction regionwas increasedwiththe increase of the wave amplitude. Yao [13] studied theoretically thenatural convection along a vertical wavy surface. He found that thelocal heat transfer rate was smaller than that of the flat plate case anddecreased with the increase of the wave amplitude. The averageNusselt number also showed the same trend. Adjlout et al. [14]

Nomenclature

A internal domainB0 magnetic field [T]g gravitational acceleration [m/s2]Gr Grashof numberH height of the enclosure [m]Ha Hartmann numberJ JacobianL length of the heat source [m]Ni standard six-noded shape functionNu Nusselt numberp pressure [N/m2]P dimensionless pressurePr Prandtl numberq heat flux [W/m2]Ra Rayleigh NumberRi residual equationsT dimensionless temperatureTs local dimensionless surface temperatureu, υ velocity component in x and y-direction [m/s]U, V dimensionless velocity component in X, Y-directionW width of the enclosure [m]x, y Cartesian co-ordinates [m]X, Y dimensionless Cartesian co-ordinates

Greek symbolsγ penalty parameterκ thermal conductivity of fluid [W/m2k]α thermal diffusivity [m2/s]β coefficient of volumetric expansion [1/K]ε discrete heat source size ratioθ temperature [K]θc temperature of the cold surface [K]ν kinematic viscosity [m2/s]ρ fluid density [kg/m3]σe electrical conductivity [S/m]

Subscriptsc Cold wall

Fig. 1. Schematic diagram of the physical domain.

394 G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

reported a numerical study of the effect of a hot wavy wall in aninclined differentially heated square cavity. Tests were performed fordifferent inclination angles, amplitudes and Rayleigh numbers for oneand three undulation. The trend of the local heat transfer was found tobe wavy in nature. It may also be noted that the sinusoidal walltemperature variationmay produce uniformmelting ofmaterials suchas glass (Sarris et al. [15]). Saha et al. [16] investigated the effect ofdiscrete heat source length and angle of inclination on naturalconvection inside a sinusoidal corrugated enclosure.

In this investigation, magnetoconvection problem has been solvedfor sinusoidal corrugation geometry and air has been taken as theworkingfluid. The corrugation geometry and thecoordinate systems areshown in Fig. 1. It consists of a sinusoidal corrugated enclosure ofdimensions, W×H, two sidewalls are maintained at a constanttemperature θc, a constant flux heat source, q is discretely embeddedat the bottom wall, and the remaining parts of the bottom surface andthe upper wall are considered to be adiabatic. The enclosure has thesame height and width, H=W with single corrugation frequency andthe corrugation amplitude has beenfixed at 10% of the enclosure length.The ratio of the heating element to the enclosure width, ε=L/W is

varied from 0.2 to 0.8. The Grashof number, Gr is varied from 103 to 106,Hartmann number is varied from 0 to 100 and Prandtl number, Pr istaken as 0.71.

2. Mathematical formulation

A steady two-dimensional magnetoconvection flow in a sinusoidalcorrugated enclosure of dimensions, W×H filled with an electricallyconducting fluid is considered. The gravity acts vertically downwards.The uniform external magnetic field B0 is applied parallel to gravity. Itis assumed that the induced magnetic field is negligible compared tothe applied magnetic field. Under the above assumptions, theconservation equations of mass, momentum and energy in a two-dimensional Cartesian coordinate system can be expressed in thedimensionless form as follows:

∂U∂X +

∂V∂Y = 0 ð1Þ

U∂U∂X + V

∂U∂Y = −∂P

∂X +∂2U∂X2 +

∂2U∂Y2

!−Ha2U ð2Þ

U∂V∂X + V

∂V∂Y = −∂P

∂Y +∂2V∂X2 +

∂2V∂Y2

!+ Gr T ð3Þ

U∂T∂X + V

∂T∂Y =

1Pr

∂2T∂X2 +

∂2T∂Y2

!ð4Þ

where U and V are the velocity components in the X and Y directions,respectively, T is the temperature, P is the pressure, and Ha, Gr and Prare the Hartmann number, Grashof number and Prandtl number,respectively, and they are defined as:

Ha =B20W

2σe

μ;Gr =

gβΔθW3

ν2 and Pr =υα

ð5Þ

395G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

The dimensionless parameters in the equations above are definedas follows:

X =xW

;Y =yW

;U =uWν

;V =υWν

;

P =pW2

ρν2 ; T =θ−θcΔθ

andΔθ =qWκ

ð6Þ

The corresponding boundary conditions for the above problem aregiven by:

All walls: U=V=0, Top wall: ∂T∂Y = 0, Right and left side walls:

T=0

Bottom wall :∂T∂Y = 0 for

−1 for0 b X b 0:5ð1−εÞ and 0:5ð1 + εÞb Xb 1

0:5ð1−εÞ≤ X≤0:5ð1 + εÞ�

ð7Þ

The average Nusselt number (Sharif and Mohammad [17]) can bewritten as,

Nu =1ε∫ε

0

1TSðXÞ

dX ð8Þ

3. Finite element formulation

The continuity Eq. (1) will be used as a constraint due to massconservation and this constraint may be used to obtain the pressuredistribution. In order to solve Eqs. (2)–(4), the Penalty finite elementmethod (Basak et al. [18]) is used where the pressure P is eliminatedby a penalty parameter γ and the incompressibility criteria given byEq. (1) which results in:

P = −γ∂U∂X +

∂V∂Y

� �ð9Þ

The continuity Eq. (1) is automatically satisfied for large values ofγ. Using Eq. (9), the momentum Eqs. (2) and (3) reduces to

U∂U∂X + V

∂V∂Y = γ

∂∂X

∂U∂X +

∂V∂Y

� �+

∂2U∂X2 +

∂2U∂Y2

!−Ha2U ð10Þ

U∂V∂X + V

∂V∂Y = γ

∂∂Y

∂U∂X +

∂V∂Y

� �+

∂2V∂X2 +

∂2V∂Y2

!+ GrT ð11Þ

Expanding the velocity components (U, V) and temperature (T)using basis set {Nk}k=1

N as

U≈ ∑N

k=1UkNk; V≈ ∑

N

k=1VkNk; and T≈ ∑

N

k=1TkNk ð12Þ

Then the Galerkin finite element method yields the followingnonlinear residual equations for Eqs. (10)–(12) respectively at nodesof internal domain A:

Rð1Þi = ∑

N

k=1Uk ∫

A

∑N

k=1UkNk

!∂Nk

∂X + ∑N

k=1VkNk

!∂Nk

∂Y

" #NidXdY

+ γ ∑N

k=1Uk ∫

A

∂Ni

∂X∂Nk

∂X

� �dXdY + ∑

N

k=1Vk ∫

A

∂Ni

∂Y∂Nk

∂Y

� �dXdY

" #

+ ∑N

k=1Uk ∫

A

∂Ni

∂X∂Nk

∂X +∂Ni

∂Y∂Nk

∂Y

� �dXdY + Ha2 ∫

A

ð∑N

k=1UkNkÞNidXdY

ð13Þ

Rð2Þi = ∑

N

k=1Vk ∫

A

∑N

k=1UkNk

!∂Nk

∂X + ∑N

k=1VkNk

!∂Nk

∂Y

" #NidXdY

+ γ ∑N

k=1Uk ∫

A

∂Ni

∂Y∂Nk

∂X

� �dXdY + ∑

N

k=1Vk ∫

A

∂Ni

∂Y∂Nk

∂X

� �dXdY

" #

+ ∑N

k=1Vk ∫

A

∂Ni

∂X∂Nk

∂X +∂Ni

∂Y∂Nk

∂Y

� �dXdY−Gr ∫

A

∑N

k=1TkNk

!NidXdY

ð14Þ

Rð3Þi = ∑

N

k=1Tk ∫

A

∑N

k=1UkNk

!∂Nk

∂X + ∑N

k=1VkNk

!∂Nk

∂Y

" #dXdY

+1Pr

∑N

k=1Tk ∫

A

∂Ni

∂X∂Nk

∂X +∂Ni

∂Y∂Nk

∂Y

� �dXdY

ð15Þ

Three points Gaussian quadrature formula is used to evaluate theintegrals in the residual equations. The nonlinear residualEqs. (13)–(15) are solved using Newton's method to determine thecoefficients of the expansions in Eq. (12). Then the followingrelationships are introduced:

X = ∑6

k=1XkNkðξ;ηÞ and Y = ∑

6

k=1YkNkðξ;ηÞ ð16Þ

where Ni (ξ, η) are the local six-noded triangular basis functions onthe ξ–η domain. The integrals in Eqs. (13)–(15) can be evaluated inξ–η domain using following relationships:

∂Ni

∂X∂Ni

∂Y

2664

3775=

1J

∂Y∂η −∂Y

∂ξ

−∂X∂η

∂X∂ξ

26664

37775

∂Ni

∂ξ∂Ni

∂η

26664

37775anddX dY = J dξdη ð17Þ

where

J =∂ðX;YÞ∂ðξ;ηÞ = j ∂X∂ξ ∂X

∂η∂Y∂ξ

∂Y∂η

j ð18Þ

4. Numerical procedure

An iterative scheme is adopted to obtain the solution of theEqs. (1) to (4) with boundary conditions (7) numerically. The relativetolerance for the error criteria is considered to be 10−6. The sixnodded triangular elements are used in this paper for the develop-ment of the finite element equations. Non-uniform grids of triangularelement are employedwith denser grids clustering in regions near theheat sources and the enclosed walls. The nonlinear equations aresolved iteratively using Broyden's method with an LU-decompositionpreconditioner. To test and assess grid independence of the presentsolution scheme, many numerical runs are performed for higherGrashof number as shown in Fig. 2. These experiments reveal that anon-uniform spaced grid of 4928 elements is adequate to describecorrectly the flow and heat transfer process inside the enclosure. Inorder to validate the numerical model, the results are compared withthose reported by Corvaro and Paroncini [19], for square straightenclosure with different Raleigh number (Ra) and ε=0.2 as shown inTable 1. The agreement is found to be excellent which validates thepresent computations.

Fig. 2. Convergence of average Nusselt number with grid refinement for Gr=103,ε=0.2 and Ha=25.

396 G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

5. Results and discussion

In the present work, the working fluid inside the sinusoidalenclosure is chosen as air with Prandtl number, Pr=0.71. The Grashofnumber (Gr) is varied from 103 to 106, Hartmann number (Ha) isvaried from 0 to 100 and discrete heat source size, ε is varied from 0.2to 0.8. Flow and temperature fields are presented in terms ofstreamline and isotherm contours respectively. Effects of the Grashofnumber, Hartmann number and heat source size on the heat and fluidflow phenomenon are observed. Later, heat transfer performance isexamined in terms of average Nusselt number (Nu) to predict thecharacteristics of magnetoconvection.

5.1. Effect of Grashof number

The evolution of the flow and thermal fields in the sinusoidalenclosure for Gr=103, 104, 105 and 106 are shown in Fig. 3 for arepresentative case of Ha=0 and ε=0.2. Because of the symmetricalboundary conditions on the sinusoidal corrugated sidewalls, the flowand the temperature fields are symmetric about the vertical centerlineof the enclosure. The symmetrical boundary conditions in the verticaldirection result in a pair of rotating cells in the left and the right halvesof the enclosure. It can be seen that left cell forms an elliptical shapeand right cell forms a circular shape at the centre line of the enclosure.Due to the symmetry, the flows in the left and the right halves of theenclosure are identical except for the sense of rotation. In each case,the flow rises along the vertical symmetry axis from the middleportion of the bottom wall and gets blocked at the top adiabatic wall,which turns the flow horizontally towards the isothermal cold walls.

Table 1Comparison between the experimental and numerical average Nusselt number forε=0.2.

Ra Experimental dataCorvaro and Paroncini [19]

Numerical data present Error (%)

7.56×104 4.8 5.31 −10.631.38×105 5.859 6.07 −3.601.71×105 6.3 6.37 −1.111.98×105 6.45 6.58 −2.012.32×105 6.65 6.82 −2.562.50×105 6.81 6.94 −1.91

The flow then descends downwards along the corrugated sidewallsand turns back horizontally to the central region after hitting thebottomwall. Thus, the flowing fluid forms two symmetrical rolls withanticlockwise and clockwise rotations inside the enclosure. ForGr=103 to 105, viscous forces are more dominant than the buoyancyforces and hence, heat transfer is essentially diffusion dominated andthe shape of the streamline tends to follow the geometry of theenclosure. For Gr=106, the left circulating cell spread inside thesinusoidal enclosure resulting higher recirculation strength and hencethe expansion of the left circulating cell inside the enclosuresuppresses the right circulating cell at the right vertical wall. Thisflow scenario indicates that the convective is responsible for the heattransport mechanism. Also left circulating cell becomes dominating inthe enclosure while the right circulating cell is squeezed thinner and itlooks like that it is going to divide into two minor corner cells locatednear the top and bottom of the right vertical wall. In addition, the coreof the left circulating rolls moves upward and forms a circular shapeand the maximum velocity is observed for Gr=106 indicatingsignificant increase of the intensity of convection. The isothermprofiles as indicated in Fig. 3 remain invariant up to Gr≤104. ForGrN104, the circulation near the central regimes are stronger andconsequently, the isotherms profiles starts getting shifted towards theside wall and they break into two symmetric contour lines shown inFig. 3. It can be seen that with the increase of Grashof number, thedeveloping thermal boundary layer thickness at the bottom wallbecomes thinner and thus indicates high heat transfer rate. However,the corrugated walls are cold enough signifying the heat is able to lesspenetrate into that region.

5.2. Effect of discrete heat source size

The fluid flow and heat transfer behaviors with the change ofdiscrete heat source size are investigated by performing numericalsimulations for sinusoidal enclosure at different discrete heat sourcelengths of 0.2 to 0.8 are shown in Fig. 4 for a representative case ofGr=105 and Ha=50. From the streamline plots, very insignificantchange is found for different discrete heat source lengths as shown inFig. 4. It is observed that the buoyancy force is acting only in y direction,two recirculation cells are formed and the streamline plot is symmetricabout the vertical midline owing to the symmetry of the problemgeometry and theboundary conditions. Theonly changed is seen, that is,the two counter rotating elliptical core of the recirculation cells areformed for all the discrete heat source sizes as expected and theymovesup towards the centre of the sinusoidal enclosure as the increase of heatsource sizes. The isotherm plots with temperature contours,T=0.01–0.155 occur symmetrical about the vertical centre line forε=0.2 and 0.4. These smooth symmetric curves indicate low temper-ature gradient results less circulation of fluid flow inside the sinusoidalenclosure. For the heat source size ε=0.6 and 0.8, the isothermplots aresymmetric about the vertical centre line and isotherm with T=0.053breaks into two symmetric contour lines. In addition, the temperaturecontours (0.0023≤T≤0.053) are compressed towards the verticalsidewalls and the compression of isotherm contours still continues.Since the cooled fluid has a downward motion with an increase ofcircular flow, convection become the first mechanism for heat transferand this phenomenon is visualized in Fig. 4.

5.3. Effect of Hartmann number

Due to the symmetrical boundary conditions of the sinusoidalsidewalls, the flow and temperature fields are symmetrical about thevertical central axis of the enclosure. As expected, hot fluid rises upfrom the central region as a result of buoyancy forces, after that owingto the cold sinusoidal walls, flows down along the walls forming twosymmetric rolls as shown in Fig. 5, with clockwise and anticlockwiserotations inside the enclosure. At Ha=25, viscous forces are more

Fig. 3. Streamlines and isotherms for different Grashof number with Ha=0 and ε=0.2.

397G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

dominant than the buoyancy forces and hence, heat transfer isessentially diffusion dominated and the shape of the streamline tendsto follow the geometry of the enclosure. For higher Hartmannnumber, it is interesting to observe that the visual examination ofstream functions is almost same. For Ha=25 as can be expected, heat

transfer characteristics are essentially diffusion dominant as furtherindicated by the isotherm patterns shown in Fig. 5. During diffusiondominant heat transfer, the temperature contours with T≤0.058occur symmetrically near the sidewalls of the enclosure. The otherisotherm lines with TN0.058 are smooth curves symmetric with

Fig. 4. Streamlines and isotherms for different heat source length with Ha=50 and Gr=105.

398 G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

respect to the vertical symmetric line. The distortion of the isothermfield increases with enhanced buoyancy as Ha increases, where theheat transfer becomes increasingly advection dominant. Due to theinitiation of advection, the isotherms are significantly distorted andpushed near the sidewalls for higher Ha. Moreover, for Ha=75 and100, isotherm with T=0.0855 breaks into two symmetric contours.

Due to greater circulation at the top half of the enclosure, there aresmall gradients in temperature at the central regime whereas a largestratification zone of temperature is found at the vertical symmetryline due to the stagnation of flow. As the nonlinearity of the isothermsincreases with the increase of Ha (75 and 100), a mushroom profile isobserved in the Fig. 5.

Fig. 5. Streamlines and isotherms for different Hartmann number with Gr=106 and ε=0.8.

399G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

5.4. Heat transfer characteristics

The influence of the Hartmann number (Ha) and the discrete heatsource length (ε) for the variations of average Nusselt number (Nu) ofthe heated surface with the Grashof number (Gr) are shown in Fig. 6.

Fig. 6 indicates that the average Nusselt number remains invariantup to a certain value of Gr and then increases rapidly with increasingGr. At lower Gr, the curves maintain a flat trend indicates lowtemperature gradients. From Fig. 6(a), it is observed that Nu remainsinvariant for Gr≤5×104, but for increasing value of Gr, average

Fig. 6. Variation of average Nusselt number (Nu) with Grashof number (Gr) fordifferent (a) discrete heat source length (ε) with Ha=50 and (b) Hartmann number(Ha) with ε=0.6.

400 G. Saha / International Communications in Heat and Mass Transfer 37 (2010) 393–400

Nusselt number increases. From Fig. 6(b), it is seen that Nu changesvery little for Gr≤104 and then Nu increases monotonically for104bGr≤105 and when GrN105 then Nu increases rapidly with Gr fordifferent Ha. Maximum heat transfer is obtained for ε=0 withHa=50 (Fig. 6a) and Ha=0 at Gr=8×105 with ε=0.6 (Fig. 6b).

6. Conclusion

The effects of Grashof number, discrete isoflux heat source size andHartmann number on natural convection inside a sinusoidal corru-

gated enclosure have been investigated and analyzed numerically.The penalty finite elementmethod helps to obtain smooth solutions interms of stream functions and isotherm contours for ϵ is varied from0.2 to 0.8, Gr=103 to 106, Ha=0 t0 100 and Pr=0.71. From theabove discussion the following conclusions can be drawn:

i. As the Grashof number increases, left circulating cell in thestreamline becomes dominating. For lower magnitude of theGrashof number, the left vortex spreads over the space but forthe higher magnitude of the Grashof number, the rightcirculating cell tends to split with the major vortex.

ii. As the heat source surface area increases the average Nusseltnumber decreases, indicating that the heat source size hassignificant effect on the heat transfer rate.

iii. For lower magnitude of Hartmann number, heat transfer isessentially diffusion dominated and the shape of the streamlinetends to follow the geometry of the enclosure.

References

[1] N. Rudraiah, R.M. Barron,M. Venkatachalappa, C.K. Subbaraya, Effect of amagnetic fieldon free convection in a rectangular enclosure, Int. J. Eng. Sci. 33 (1995) 1075–1084.

[2] P. Kandaswamy, K. Kumar, Buoyancy-driven nonlinear convection in a squarecavity in the presence of a magnetic field, Acta Mech. 136 (1999) 29–39.

[3] P. Kandaswamy, S. Malliga Sundari, N. Nithyadevi, Magnetoconvection in anenclosure with partially active vertical walls, Int. J. Heat Mass Transfer 51 (2008)1946–1954.

[4] J.P. Garandet, T. Albussoiere, R. Moreau, Buoyancy driven convection in arectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer35 (1992) 741–748.

[5] S. Alchaar, P. Vasseur, E. Bilgen, Natural convection heat transfer in a rectangularenclosure with a transverse magnetic field, J. Heat Transfer 117 (1995) 668–673.

[6] K. Kanafer, A.J. Chamkha, Hydromagnetic natural convection from an inclinedporous square enclosure with heat generation, Numer. Heat Transfer A 33 (1998)891–910.

[7] S. Mahmud, S.H. Tasnim, M.A.H. Mamun, Thermodynamic analysis of mixedconvection in a channel with transverse hydromagnetic effect, Int. J. Therm. Sci. 42(2003) 731–740.

[8] M.C. Ece, E. Büyük, Natural-convection flow under a magnetic field in an inclinedrectangular enclosure heated and cooled on adjacent walls, Fluid Dyn. Res. 38(2006) 564–590.

[9] M. Ali, S.R. Husain, Natural convection heat transfer and flow characteristics in a squareduct of V-corrugated vertical walls, J. Energy Heat Mass Transfer 14 (1992) 125–131.

[10] M. Ali, S.R. Husain, Effect of corrugation frequencies on natural convective heattransfer and flow characteristics in a square enclosure of vee-corrugated verticalwalls, Int. J. Energy Research 17 (1993) 697–708.

[11] M. Ali, M.N. Ali, Finite element analysis of laminar convection heat transfer andflow of the fluid bounded by vee-corrugated vertical plates of differentcorrugation frequencies, Indian J. Eng. Mater. Sci. 1 (1994) 181–188.

[12] S. Noorshahi, C. Hall, E. Glakpe, Effect of mixed boundary conditions on naturalconvection in an enclosure with a corrugated surface, ASME–JSES–KSES,Proceedings of Int. Solar Energy Conference Part 1, Maui, HI, 1992, pp. 173–181.

[13] L.S. Yao, Natural convection along a vertical wavy surface, J. Heat Transfer 10(1983) 465–468.

[14] L. Adjlout, O. Imine, A. Azzi, M. Belkadi, Numerical study of the natural convectionin a cavity with undulated wall, Proceeding of Third Int Thermal EnergyEnvironment Congress, Merrakech, Maroc, 1997 9–12.

[15] I.E. Sarris, I. Lekakis, N.S. Vlachos, Natural convection in a 2D enclosure withsinusoidal upper wall temperature, Numer Heat Transfer, Part A 42 (2002) 513–530.

[16] S. Saha, T. Sultana, G. Saha, M.M. Rahman, Effects of discrete heat source size andangle of inclination on natural convection heat transfer flow inside a sinusoidalcorrugated enclosure, Int. Comm. Heat Mass Transfer 35 (2008) 1288–1296.

[17] A.R. Sharif, T.R. Mohammad, Natural convection in cavities with constant fluxheating at the bottom wall and isothermal cooling from the sidewalls, Int. J.Therm. Sci. 44 (2005) 865–878.

[18] T. Basak, S. Roy, A.R. Balakrishnan, Effects of thermal boundary conditions onnatural convection flowwithin a square cavity, Int. J. Heat Mass Transfer 49 (2006)4525–4535.

[19] F. Corvaro, M. Paroncini, A numerical and experimental analysis on the naturalconvection heat transfer of a small heating strip located on the floor of a squarecavity, Appl. Therm. Eng. 28 (2008) 25–35.