Thermo-Diffusion Effect on Free Convection Heat and Mass Transfer in a Thermally Linearly Stratified Non-Darcy Porous Media

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1 The Open Tanspot Phenomena Jounal, 2011, 3, Open Aess Themo-Diffusion Effet on ee Convetion Heat and Mass Tansfe in a Themally Linealy Statified Non-Day Poous Media P.V.S.N. Muthy *,1 and M.. El-Amin 2,3 1 Depatment of Mathematis, Indian Institute of Tehnology Khaagpu, West Bengal, , India 2 Computational Tanspot Phenomena Laboatoy (CTPL), Division of Physial Sienes and Engineeing (PSE), King Abdullah Univesity of Siene and Tehnology (KAUST), Thuwal , Kingdom of Saudi Aabia 3 Depatment of Mathematis, Aswan aulty of Siene, South Valley Univesity, Egypt Abstat: Themo-diffusion effet on fee onvetion heat and mass tansfe fom a vetial sufae embedded in a liquid satuated themally statified non - Day poous medium has been analyzed using a loal non-simila poedue. The wall tempeatue and onentation ae onstant and the medium is linealy statified in the vetial dietion with espet to the themal onditions. The fluid flow, tempeatue and onentation fields ae affeted by the omplex inteations among the diffusion atio Le, buoyany atio N, themo-diffusion paamete S and statifiation paamete ε. Non-linea inteations of all these paametes on the onvetive tanspot has been analyzed and vaiation of heat and mass tansfe oeffiients with themo-diffusion paamete in the themally statified non-day poous media is pesented though ompute geneated plots. Keywods: ee onvetion, heat tansfe, mass tansfe, bounday laye, themo-diffusion effet, themal statifiation, non- Day poous media. INTRODUCTION Coupled heat and mass tansfe phenomenon in a liquid satuated poous medium is gaining attention due to its inteesting appliations. The flow phenomenon in this ase is elatively omplex than that in pue themal/solutal onvetion poess. Poesses involving heat and mass tansfe in poous media ae often enounteed in the hemial industy, in esevoi engineeing in onnetion with themal eovey poess, in the study of dynamis of hot and salty spings of a sea. Undegound speading of hemial waste and othe pollutants, gain stoage, evapoation ooling and solidifiation ae few othe appliation aeas whee ombined themo/ solutal onvetion in poous media an be obseved. Due to the oupling of tempeatue and onentation, new paametes like buoyany atio and Lewis numbe (diffusion atio) aise and they influene the onvetive tanspot to a geate extent. A eview of both natual and mixed onvetion bounday laye flows in fluid satuated poous media is given in Nield and Bejan [1]. In poesses involving vigoous fee onvetion heat and mass tansfe in lea fluids as well as in poous media, thee may be inease in the heat tansfe oeffiient beause of the onentation gadients and a simila inease in the mass tansfe oeffiient due to the themal gadients. Soet (themo-diffusion) effet oesponds to speies diffeentiation developing in an initially homogeneous mixtue submitted to a themal gadient and Dufou (diffusion-themo) effet o- *Addess oespondene to this autho at the Depatment of Mathematis, Indian Institute of Tehnology Khaagpu, West Bengal, India; Tel: (Offie); ax: ; pvsnm@maths.iitkgp.enet.in esponds to diffusion of heat aused by onentation gadients. In the ealie studies, Soet and Dufou effets ae negleted, on the basis that they ae of smalle ode magnitude than the effets desibed by ouie s and ik s laws. These effets ae onsideed as seond ode phenomenon, but these may beome signifiant in aeas suh as hydology, petology and geo-sienes et. Stability of Soet diven themo-solutal onvetion in lea fluids with lage themal gadients has been analyzed theoetially and expeimentally by Hule and Jakeman [2] and it has been shown that Soet effet ould give aise to ovestable solutions of the themo-solutal onvetion. Platten and Chavepeye [3] have analyzed the osillatoy motion in Benad ell due to Soet effet. Themo-solutal onvetion in liquids with lage negative Soet oeffiient has been analyzed by Caldwell [4]. Soet effet in inompessible fluids has been analyzed by Benano-Melly et al., [5]. Kafoussias and Williams [6] studied the themo-diffusion (Soet) and diffusion-themo (Dufou) effets on foed, fee and mixed onvetion with tempeatue dependant visosity. Anghel et al., [7] analyzed both the Soet and Dufou effets on fee onvetion bounday-laye ove a vetial sufae embedded in a poous medium and notied an appeiable hange in the flow field. Postelniu [8] analyzed the influene of magneti field onsideing Soet and Dufou effets fom a vetial sufae in poous medium. Patha et al., [9] studied the effet of magneti field and double dispesion on fee onvetive heat and mass tanspot onsideing the Soet and Dufou effets in non-day poous medium. Postelniu [10] analyzed the influene of hemial eation on flow field onsideing Soet and Dufou effets fom a vetial sufae in poous medium. The effet of Soet and Dufou paametes on fee onvetion heat and mass tansfe fom a vetial sufae in a / Bentham Open

2 50 The Open Tanspot Phenomena Jounal, 2011, Volume 3 Muthy and El-Amin doubly statified Daian poous medium has been epoted by Lakshmi Naayana and Muthy [11]. Reently, Muthy and Lakshmi Naayana [12] analyzed the influene of Soet and Dufou effets on fee onvetive heat and mass tanspot fom a hoizontal plate in non-day poous medium by onsideing powe law vaiations fo the tempeatue and onentation at the wall. Moeove, Kaii and Muthy [26] studied the effet of melting and themo-diffusion on natual onvetion heat mass tansfe in a non-newtonian fluid satuated non-day poous medium. Joo [13] studied the instabilities in Maangoni onvetion in liquid mixtues with Soet effet theoetially and pointed out that a oss-oupling in the tempeatue field, namely Dufou effet, ould exist, but is usually insignifiant fo liquid mixtues. Also, the tems Soet effet and Dufou oeffiient give ise to inteation between the themal and solute fields even when the fluid is at est. But it is well known fom the liteatue that the Soet oeffiient has a onsideable effet on onvetion poess in liquids wheeas the Dufou effet an be negligible in liquids but it plays a pominent ole in gaseous mixtues. Statifiation is a haateisti of all fluid bodies suounded by diffeentially heated and salted side walls. An impotant engineeing appliation an be found in the poblem of natual onvetion in an enlosed etangula ell filled with fluid satuated poous medium with one wall heated and the othe ooled. Heated fluid aising fom the hot wall ovelays the top of the ell and the ool fluid falling fom the old wall lies along the bottom theeby eating statifiation inside the ell. Although the effet of statifiation of the medium on the heat emoval poess in a poous medium is impotant, faily little wok has been epoted in the liteatue. Bejan [14], Singh and Shama [15] and Kalpana and Singh [16] studied the poblem of bounday laye fee onvetion along an isothemal vetial plate immesed in a themally statified fluid satuated poous medium using integal and seies solution tehniques. The ase of powe law vaiation of wall tempeatue with themal statifiation of the medium was disussed at length by Nakayama and Koyama [17], and by Lai et al., [18]. Takha and Pop [19] investigated the fee onvetive tanspot fom a vetial flat plate in a themally statified Daian fluid satuated poous medium whee the ambient tempeatue vaies as x 1/3 using the similaity solution tehnique. Muthy et al., [20] extended the wok of Takha and Pop [19] to unove the effet of double statifiation on fee onvetion heat and mass tansfe in a Daian fluid satuated poous medium using the similaity solution tehnique fo the ase of unifom wall heat and mass flux onditions. Chamkha and Khaled [21] epoted the oupled mixed onvetion fom a vetial plate embedded in statified poous medium with themal dispesion effets. Hossain et al., [22] investigated the visous dissipation effets on natual onvetion with unifom sufae heat flux plaed in a themally statified media. Also, natual onvetion flow with ombined buoyany effets due to themal and mass diffusion in a themally statified media was intodued by Saha and Hossain [23]. The pesent investigation is undetaken to see the effet of themo-diffusion paamete on the fee onvetion heat and mass tansfe in a themally statified poous medium. Quite often we enounte themal statifiation in a losed envionment and also in the semi infinite poous media. In the pesent investigation, the medium is assumed to be linealy themally statified whih is physially moe ealisti athe than onsideing an abitay powe law vaiation fo the themal statifiation. GOVERNING EQUATIONS Conside the natual bounday-laye flow nea a vetial impemeable sufae embedded in a poous medium satuated with Newtonian liquid. The x-oodinate is taken along the plate, in the asending dietion, the y-oodinate is measued nomal to the plate, while the oigin of the efeene system is onsideed at the leading edge of the vetial plate. The wall is maintained at onstant tempeatue and onentation, T w and C w espetively, while the tempeatue in the ambient medium vaies linealy on the x- oodinate (themally linealy statified). Visous esistane due to the solid bounday is negleted unde the assumption that the medium is with low pemeability. o this boundaylaye poblem in poous media, we assume the flow is govened by non-day s law, Boussinesq appoximation is valid, the fluid and the poous medium ae in loal themodynami equilibium. Unde these assumptions the govening equations fo the flow in the medium may be witten as,!u!x +!v!y = 0 (1)!u!y + K "!u 2!y = K g " $ % # T!T!y + # C!C!y u!t!x + v!t!y = "!2 T!y 2 (3) u!c!x + v!c!y = D!2 C!y + D k 1! 2 T 2 T m!y 2 (4) ( ) The bounday onditions ae given by y = 0: v = 0, T = T w (onstant), C = C w (onstant) (2) (5a) y! " : u! 0, T! T ",0 + Ax, C = C! (onstant) (5b) Hee x and y ae the Catesian oodinates along and nomal to the plate espetively, u and v ae the aveaged veloity omponents in x and y dietions espetively, T and C ae tempeatue and onentation espetively, β T and β C ae the oeffiient of themal and solutal expansion espetively, v is the kinemati visosity of the fluid, ( 0) is the ohheime (non-day) oeffiient, = 0 lead the above system of equations epesent the heat and mass tanspot in the Day poous medium. K is the pemeability, g is the aeleation due to gavity, α, D ae the themal and solutal diffusivities of the medium, k is the themal diffusion

3 Themo-Diffusion Effet The Open Tanspot Phenomena Jounal, 2011, Volume 3 51 atio, D 1 is the Soet oeffiient, T m is the mean fluid tempeatue. The suffix! indiates the onditions in the ambient medium. The veloity omponents u and v ae witten in tems of steam funtion!, using the following tansfomation,! = y x Ra 1/2, Ax x! =, T w " T #!(",#)= T $ T %,0 $ T w $ T %,0! (x, y)=" Ra 1/2 x f (#,$), Ax, C $ C!(",#) = %, T w $ T %,0 C w $ C % the govening equations and the bounday onditions (1) (5) ae tansfomed as: f + 2 f f =! + N " (6)! f! % = " f #! #" $! #f ( #" ) * + " f! + Le S " Le f! = Le # f $! $# %! $f ) ( $# * + f (!, 0) = 0, "(!, 0) = 1#!, $(!, 0) = 1 at η = 0 f (!,") # 0, $(!,") # 0, %(!,") # 0 as! " # (7) (8) (9a) (9b) In the above, epesents diffeentiation with espet to the vaiable!. The seond-level non-similaity method is used to onvet the non-simila equations into a system of odinay diffeential equations. This poedue is given in Spaow et al., [25], fo bevity, it is not epeated hee. Witing =!f,g =!#!"!", H =!$ and diffeentiation of!" Eqs. (6)-(9) with espet to!, the size of the system of equations will inease and we have the seond level nonsimila equations along with the bounday onditions as: f + 2 f f =! + N " (10)! f! $ = " f G #! % ( ) + " f (11)! + Le S " Le f! % = Le # f H $! ( (12) ) * (1+ 2 f ) + 2 f = G + N! (13) G (! + fg ) = f G "! + # + f H + Le S G Le(! + fh ) # = Le f H "! $ % ( (14) (15) f (!, 0) = 0, "(!, 0) = 1#!, $(!, 0) = 1, (!, 0) at η = 0 = 0, G(!, 0) = #1, H (!, 0) = 0 f (!,") = #(!,") =$(!,") = (!,") as! " # = G(!,") = H(!,") = 0 In the above,!w = T w " T #,0 and!w (16a) (16b) = C w " C #. The non-dimensional paametes ae Ra x = Kg! T " w x the #$ Rayleigh numbe defined based on the themal onditions, inetia paamete = K Kg! T (T w " T # ) $ 2, the diffusivity atio Le =! D and the buoyany atio N =! " w! T # w. Positive values of N indiate aiding buoyany, whee both the themal and solutal buoyanies ae in the same dietion and negative values of N indiate opposing buoyany whee the solutal buoyany is in the opposite dietion to the themal buoyany. Also S = D k! 1 w is the Soet paamete. This T m " # w paamete an be both positive and negative as! w an be positive o negative. The fundamental inteest of the pesent investigation is to examine the vaiations in the heat and mass tansfe oeffiients with vaious values of these physial paametes that influene the system. These non-dimensional heat and mass tansfe oeffiients ae witten as, Nu x Ra x 1/2 =! " / (#,0) and RESULTS AND DISCUSSION Sh x Ra =!#"($, 0) espetively. 1/2 x The system of odinay diffeential equations (10)-(15) subjet to the bounday onditions (16a,b) have been solved numeially by means of the fouth-ode Runge-Kutta method with shooting tehnique. The step size!" = 0.05 is used while obtaining the numeial solution with! max = 12 and five-deimal auay as the iteion fo onvegene. Extensive alulations have been pefomed to obtain the flow, tempeatue and onentation fields fo 0!! 1.0, 0 < Le! 10,! 0.5 " N " 1.5,! 0.1 " S " 1.0 and 0 "! " 0.7. The numeial esults ae analyzed and pesented fo aiding and opposing buoyanies in the pesene and absene of the Soet paametes in both Day and the non-day themally statified poous media. = 0 oespond to Day poous media while the esults fo non-day ase ae pesented fo

4 52 The Open Tanspot Phenomena Jounal, 2011, Volume 3 Muthy and El-Amin = 0.2 ; and also! = 0 epesents that the medium is unstatified poous medium. o the suounding fluid to be stable, A should be positive, and the ange of! should fom 0 to 1, fo the wall tempeatue to exeed the suounding ambient tempeatue. Seeking veifiation, ig. (1) shows a ompaison between the uent esults and the published esults by Lakshmi Naayana and Muthy [26] of the mass tansfe oeffiient against statifiation paamete ε, at N=2, =0.0, Le=1 and S=0.1. The ompaison shows a good ageement between the pesent alulations and the esults in [26]. onentation ae beoming negative in the bounday laye egion. Also, by onsideing the Soet and Dufou effets in the medium, Postelniu [15], and Patha et al., [16] epoted that fo etain ombinations of values of these paametes, the non-dimensional heat and mass tansfe oeffiients beome negative. ig. (3). Effets of statifiation (ε= 0.0, 0.1, 0.5, 0.7) on tempeatue pofiles of the non-day flow at N=1, =0.2, Le=1 and S=0.1. ig. (1). Compaison between the uent esults and the published esults by Lakshmi Naayana and Muthy [25] of the mass tansfe oeffiient against statifiation paamete ε, at N=2, =0.0, Le=1 and S=0.1. Vaiations of non-dimensional veloity, tempeatue and onentation fields against the similaity vaiable η fo diffeent values of the statifiation paamete ε ae shown in igs. (2-4). The non-dimensional veloity f! and tempeatue θ deease with ineasing values of the statifiation paamete!. On the othe hand the non-dimensional onentation ineased with the value of ε. o elatively lage values of! " 0.5 the non-dimensional tempeatue attains negative values inside the bounday laye, aftewads it tends ig. (4). Effets of statifiation (ε=0.0, 0.1, 0.5) on onentation pofiles of the non-day flow at N=1, =0.2, Le=1 and S=0.1. The non-dimensional heat and mass tansfe oeffiients ae plotted against the themal statifiation paamete in the Day and ohheime poous media in igs. (5 and 6) espetively. It is lea that Nusselt and Shewood numbes deease with ε fo all. It is evident that the effet of on the Nusselt and Shewood numbes is moe signifiant fo small values of ε. ig. (2). Effets of statifiation (ε = 0.0, 0.1, 0.5, 0.7) on veloity pofiles of the non-day flow at N=1, =0.2, Le=1 and S=0.1. to zeo as η appoahes to the edge of the bounday laye. These esults fo linea statifiation ae onsistent with those epoted in Muthy et al., [20] fo non-linea statifiation. It was epoted in the ealie study by Lakshmi Naayana and Muthy [11] that due to the double statifiation of the medium, the non-dimensional tempeatue and ig. (5). Vaiations of Nusselt numbe against statifiation paamete ε fo vaious values of the paamete at S=0.1, N=1 and Le=1.

5 Themo-Diffusion Effet The Open Tanspot Phenomena Jounal, 2011, Volume 3 53 ig. (6). Vaiations of Shewood numbe against statifiation paamete ε fo vaious values of the paamete at S=0.1, N=1 and Le=1. In igs. (7-9) the non-dimensional veloity, tempeatue and onentation pofiles ae plotted by vaying! and fo fixed S, N and Le. The veloity pofile deeases with ineasing and!. Also a deease in the tempeatue is seen with!, while it ineases with in the bounday laye. It is easy to undestand that ineasing the value of the themal statifiation paamete edues the tempeatue diffeene between the vetial sufae and ambient medium whih auses edution in the tempeatue distibution. The onentation pofile ineases with the non-day paamete and also with the statifiation paamete! in the bounday laye. ig. (9). Vaiations of onentation pofiles fo vaious values of paametes ε and at S=0.1, N=1 and Le=1. igs. (10-12) depit vaiation of non-dimensional veloity, tempeatue and onentation pofiles against the similaity vaiable η fo diffeent values of N and S. It is obseved that the non-dimensional veloity ineases, while tempeatue and onentation distibutions deease with the buoyany paamete N. A aise in the veloity and onentation pofiles is seen with ineasing S fo all values of N (<, =, > 0) in the bounday laye. On the othe hand tempeatue pofile deeases with the ineasing S fo aiding buoyany while evese phenomenon is noted in the ase of opposing buoyany. It is inteesting to note that the non-dimensional onentation takes negative values fo lage negative value of S (! -1) inside the bounday laye and finally it attains its oute edge bounday ondition! = 0 as the value of! appoahes the edge of the bounday laye. ig. (7). Vaiations of veloity pofiles fo vaious values of paametes ε and at S=0.1, N=1 and Le=1. ig. (10). Vaiations of veloity pofiles fo vaious values of paametes N and S at ε=0.1, =0.2 and Le=1. ig. (8). Vaiations of tempeatue pofiles fo vaious values of paametes ε and at S=0.1, N=1 and Le=1. ig. (11). Vaiations of tempeatue pofiles fo vaious values of paametes N and S at ε=0.1, =0.2 and Le=1.

6 54 The Open Tanspot Phenomena Jounal, 2011, Volume 3 Muthy and El-Amin ig. (12). Vaiations of onentation pofiles fo vaious values of paametes N and S at ε=0.1, =0.2 and Le=1. The effet of diffusivity atio Le on the Nusselt and Shewood numbes is shown in igs. (13 and 14), espetively, with fixed! and. As Le ineases the Nusselt numbe deeases in aiding buoyany while it ineases fo opposing buoyany. The Nusselt numbe ineases with ineasing S in aiding buoyany while it deeases in ase of opposing buoyany, this is due to the vaiations in the thikness of the bounday layes. The Shewood numbe ineases with Le and deeases with S as ineasing S ineased onentation bounday laye thikness fo all N, whih is evident fom the ealie figues, fo both aiding and opposing buoyanies. Also the effet of S on the Shewood numbe ineases as the value of the buoyany paamete ineased. The impat of S on the Nusselt and Shewood numbe is moe signifiant fo lage values Le. CONCLUSIONS Themo-diffusion effet on fee onvetion heat and mass tansfe fom a vetial sufae embedded in a themally statified non - Day poous medium has been analyzed using a loal non-simila poedue. The wall tempeatue and onentation ae onstant and the medium is linealy statified in the vetial dietion with espet to the themal onditions. Both the aiding and opposing buoyanies ae onsideed fo the analysis, non-dimensional veloity, tempeatue, onentation distibution and the heat and mass tansfe oeffiients ae obtained fo vaious values of flow influening paametes. It is epoted that the nondimensional veloity ƒ and tempeatue θ deease with ineasing values of the statifiation paamete!. On the othe hand the non-dimensional onentation ineases with the value of ε. o elatively lage values of "! 0. 5 the nondimensional tempeatue attains negative values inside the bounday laye, aftewads it tends to zeo as η appoahes to the edge of the bounday laye. A aise in the veloity and onentation pofiles is seen with the ineasing values of S fo all values of N (<, =, > 0) in the bounday laye. On the othe hand tempeatue pofile deeases with the ineasing S fo aiding buoyany while evese phenomenon is noted in the ase of opposing buoyany. It is inteesting to note that the non-dimensional onentation takes negative values fo lage negative value of S (! -1) inside the bounday laye and finally it attains its oute edge bounday ondition φ = 0 as the value of η app-oahes the edge of the bounday laye. The Shewood numbe ineases with Le and deeases with S. The impat of S on the Nusselt and Shewood numbes is moe signifiant fo lage values Le in the themally statified medium. ig. (13). Vaiations of Nusselt numbe against Le fo vaious values of paametes N and S at ε=0.1 and =0.2. ig. (14). Vaiations of Shewood numbe against Le fo vaious values of paametes N and S at ε=0.1 and =0.2. NOMENCLATURE d D D 1 ohheime onstant Poe diamete [m] Solutal diffusivity [m 2 /s] Soet oeffiient! themal statifiation paamete whih depends on the loal oodinate x. f Dimensionless steam funtion 2 g Aeleation due to gavity [ m/s ] 2 K Pemeability of the poous medium [ m ] N Nu Sh S Le T C x, y u, v non-day (inetia) paamete Buoyany atio Nusselt numbe Shewood numbe Themal-diffusion o Soet paamete Lewis numbe Tempeatue Conentation Axial and nomal o-odinates [m] Veloity omponents in x and y dietions [m/s]

7 Themo-Diffusion Effet The Open Tanspot Phenomena Jounal, 2011, Volume 3 55 GREEK SYMBOLS µ Visosity [kg/(ms)]! " Refeene density at some point [kg/m 3 ]! Themal diffusivity [m 2 /s]! T Coeffiient of themal expansion [1/K]! Coeffiient of solutal expansion [1/K]! Dimensionless steam funtion! Similaity vaiable! Dimensionless tempeatue! Dimensionless onentation! w = T" # Tm # = " C w C w! SUBSCRIPTS w,! Conditions on the wall and at the ambient medium ACKNOWLEDGEMENT None delaed. CONLICTS O INTEREST None delaed. REERENCES [1] D.A. Nield, and A. Bejan, Convetion in poous media. Spinge- Velag, NewYok, [2] D.T. Hule, and E. Jakeman, Soet diven themo solutal onvetion, J. luid Meh., vol. 47, pp , [3] J.K. Platten, and G. Chavepeye, Osillatoy motion in Benad ell due to Soet effet, J. luid Meh., vol. 60, pp , [4] D.R. Caldwell, Themosolutal onvetion in a solution with lage negative Soet oeffiient, J. luid Meh., vol. 293, pp , [5] L.B. Benano-Melly, J.P. Caltagione, B. aissat, E. Montel, and P. Casteseque, Modeling Soet oeffiient measuement expeiments in poous media onsideing themal and solutal onvetion, Int. J. Heat Mass Tansf., vol. 44, pp , [6] N.G. Kafoussias, and E.W. Williams, Themal-diffusion and diffusion-themo effets on mixed fee-foed onvetive and mass tansfe bounday laye flow with tempeatue dependent visosity, Int. J. Eng. S., vol. 33, pp , [7] M. Anghel, H.S. Takha, and I. Pop, Dufou and Soet effets on fee onvetion bounday-laye flow ove a vetial sufae embedded in a poous medium, Studia Univesitatis Babes- Bolyai, Mathematia. XLV, pp , [8] A. Postelniu, Influene of a magneti field on heat and mass tansfe by natual onvetion fom vetial sufaes in poous media onsideing Soet and Dufou effets, Int. J. Heat Mass Tansf., vol. 47, pp , [9] M.K. Patha, P.V.S.N. Muthy, and G.P.R. Sekha, Soet and Dufou effets in non-day poous media, Tans. ASME, J. Heat Tansf., vol. 128, pp , [10] A. Postelniu, Influene of hemial eation on heat and mass tansfe by natual onvetion fom a vetial sufaes in poous media onsideing Soet and Dufou effets, Heat Mass Tansf., vol. 43, pp , [11] P.A.L. Naayana, and P.V.S.N. Muthy, Soet and Dufou effets in a doubly statified Day poous medium, J. Poous Media, vol. 10, pp , [12] P.V.S.N. Muthy, and P.A.L. Naayana, Soet and Dufou effets on fee onvetion heat and mass tansfe fom a hoizontal plate in non-day poous media, Int. J. luid Meh. Res., vol. 37, 70-84, [13] S.W. Joo, Maangoni instabilities in liquid mixtues with Soet effets, J. luid Meh., vol. 293, pp , [14] A. Bejan, Convetion Heat Tansfe. John Wiley Sons, [15] P. Singh, and K. Shama, Integal method fo fee onvetion in themally statified poous medium, Ata Mehania., vol. 83, pp , [16] T. Kalpana, and P. Singh, Natual onvetion in a themally statified fluid satuated poous medium, Int. J. Eng. Si., vol. 30, pp , [17] A. Nakayama, and H. Koyama, Effet of themal statifiation on fee onvetion with a poous medium, AAIA J. Themophys. Heat Tansf., vol. 1, pp , [18].C. Lai, I. Pop, and.a. Kulaki, Natual onvetion fom Isothemal plates in themally statified poous media, AAIA J. Themo-physis. Heat Tansf., vol. 4, pp , [19] H.S. Takha, and I. Pop, ee onvetion fom a vetial flat plate to a themally statified Daian fluid, Meh. Res. Commun., vol. 14, pp , [20] P.V.S.N. Muthy, D. Sinivasahaya, and P.V.S.S.S.R. Kishna, Effet of double statifiation on fee onvetion in Daian poous medium, Tans. ASME, J. Heat Tansf., vol. 126, pp , [21] A.J. Chamkha, and A.-R.A. Khaled, Hydo-magneti simultaneous heat and mass tansfe by mixed onvetion fom a vetial plate embedded in a statified poous medium with themal dispesion effets, Heat Mass Tansf, vol. 36, pp , [22] E.M. Spaow, H. Quak, and C.J. Boene, Loal non-similaity bounday laye solutions, AIAA J., vol. 8, pp , [23] M.A. Hossain, S.C. Saha, and R.S.R. Gola, Visous dissipation effets on natual onvetion fom a vetial plate with unifom sufae heat flux plaed in a themally statified Media, Int. J. luid Meh. Res., vol. 32, pp , [24] S.C. Saha, and M.A. Hossain, Natual onvetion flow with ombined buoyany effets due to themal and mass diffusion in a themally statified media, Non-linea Anal: Model Contol, vol. 9 pp , [25] P.A.L. Naayana, and P.V.S.N. Muthy, ee onvetive heat and mass tansfe in a doubly statified non-day poous medium, Tans. ASME, J. Heat Tansf., vol. 128, pp , [26] R.R. Kaii, and P.V.S.N. Muthy, Effet of melting and themodiffusion on natual onvetion heat mass tansfe in a non- Newtonian fluid satuated non-day poous medium, Open Tanspot Phenomena J., vol. 1, pp. 7-14, Reeived: June 13, 2011 Revised: Otobe 10, 2011 Aepted: Otobe 10, 2011 Muthy and El-Amin; Liensee Bentham Open. This is an open aess atile liensed unde the tems of the Ceative Commons Attibution Non-Commeial Liense ( lienses/by-n/3.0/), whih pemits unestited, non-ommeial use, distibution and epodution in any medium, povided the wok is popely ited.