Mass Transfer Effects on Unsteady Hydromagnetic Convective Flow past a Vertical Porous Plate in a Porous Medium with Heat Source

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1 Jounal of lied Fluid Mehanis Vol. No vailable online at ISSN EISSN Mass Tansfe Effets on Unsteady Hydomagneti Convetive Flow ast a Vetial Poous Plate in a Poous Medium with Heat Soue S.S. Das S.R. Biswal U.K. Tiathy and P. Das Deatment of Physis K B D V College Niakau Khuda-75 9 (Oissa) India Deatment of Physis Stewat Siene College Mission Road Cuttak-75 (Oissa) India Deatment of Physis B S College Dasalla Nayagah (Oissa) India Coesonding utho dssd@yahoo.om (Reeived Novembe 5 9; aeted May ) BSTRCT The objetive of this ae is to analyze the effet of mass tansfe on unsteady hydomagneti fee onvetive flow of a visous inomessible eletially onduting fluid ast an infinite vetial oous late in esene of onstant sution and heat soue. The govening equations of the flow field ae solved using multi aamete etubation tehnique and aoximate solutions ae obtained fo veloity field temeatue field onentation distibution skin fition and the ate of heat tansfe. The effets of the flow aametes suh as Hatmann numbe M Gashof numbe fo heat and mass tansfe G G ; emeability aamete K Shmidt numbe S heat soue aamete S Pandtl numbe P et. on the flow field ae analyzed with the hel of figues and tables. It is obseved that a gowing Hatmann numbe o Shmidt numbe etads the mean veloity as well as the tansient veloity of the flow field at all oints. The effet of ineasing Gashof numbe fo heat and mass tansfe o heat soue aamete is to aeleate both mean and tansient veloity of the flow field at all oints. The mean veloity of the flow field ineases with an inease in emeability aamete while the tansient veloity ineases fo smalle values of K ( ) and fo highe values the effet eveses. gowing Hatmann numbe deeases the tansient temeatue of the flow field at all oints while a gowing emeability aamete o heat soue aamete eveses the effet. The Pandtl numbe ineases the tansient temeatue fo small values of P ( ) and fo highe values the effet eveses. The effet of ineasing Shmidt numbe is to edue the onentation bounday laye thikness of the flow field at all oints. The oblem has some elevane in the geohysial and astohysial studies. Keywods: Hydomagneti Mass tansfe Fee onvetion Poous medium Sution Heat soue.. INTRODUCTION The henomenon of hydomagneti flow with heat and mass tansfe in an eletially onduting fluid ast a oous late embedded in a oous medium has attated the attention of a good numbe of investigatos beause of its vaied aliations in many engineeing oblems suh as MHD geneatos lasma studies nulea eatos oil exloation geothemal enegy extations and in the bounday laye ontol in the field of aeodynamis. Heat tansfe in lamina flow is imotant in oblems dealing with hemial eations and in dissoiating fluids. In view of its wide aliations Hasimoto (957) initiated the bounday laye gowth on a flat late with sution o injetion. Soundalgeka (97) showed the effet of fee onvetion on steady MHD flow of an eletially onduting fluid ast a vetial late. Yamamoto and Iwamua (976) exlained the flow of a visous fluid with onvetive aeleation though a oous medium. Mansutti et al. (99) have disussed the steady flow of a non-newtonian fluid ast a oous late with sution o injetion. Jha (998) analyzed the effet of alied magneti field on tansient fee onvetive flow in a vetial hannel. Chandan and his assoiates (998) have disussed the unsteady fee onvetion flow of an eletially onduting fluid with heat flux and aeleated bounday laye motion in esene of a tansvese magneti field. haya et al. (999) have eoted the oblem of heat and mass tansfe ove an aeleating sufae with heat soue in esene of sution and blowing. The unsteady fee onvetive MHD flow with heat tansfe ast a semi-infinite vetial oous moving late with vaiable sution has been studied by Kim (). Singh and Thaku () have given an exat solution of a lane unsteady MHD flow of a non- Newtonian fluid. Shama and Paeek () exlained the behaviou of steady fee onvetive MHD flow ast a vetial oous moving sufae. Singh and his o-

2 S.S. Das et al. / JFM Vol. No wokes () have analyzed the effet of heat and mass tansfe in MHD flow of a visous fluid ast a vetial late unde osillatoy sution veloity. Makinde et al. () disussed the unsteady fee onvetive flow with sution on an aeleating oous late. Saangi and Jose (5) studied the unsteady fee onvetive MHD flow and mass tansfe ast a vetial oous late with vaiable temeatue. Das and his assoiates (6) estimated the mass tansfe effets on unsteady flow ast an aeleated vetial oous late with sution emloying finite diffeene analysis. Das et al. (7) investigated numeially the unsteady fee onvetive MHD flow ast an aeleated vetial late with sution and heat flux. Das and Mita (9) disussed the unsteady mixed onvetive MHD flow and mass tansfe ast an aeleated infinite vetial late with sution. Reently Das and his o-wokes (9) analyzed the effet of mass tansfe on MHD flow and heat tansfe ast a vetial oous late though a oous medium unde osillatoy sution and heat soue. Moe eently Das et al. () investigated the hydomagneti onvetive flow ast a vetial oous late though a oous medium with sution and heat soue. The study of stella stutue on sola sufae is onneted with mass tansfe henomena. Its oigin is attibuted to diffeene in temeatue aused by the non-homogeneous odution of heat whih in many ases an est not only in the fomation of onvetive uents but also in violent exlosions. Mass tansfe etainly ous within the mantle and oes of lanets of the size of o lage than the eath. In the esent study we theefoe oose to analyze the effet of mass tansfe on unsteady fee onvetive flow of a visous inomessible eletially onduting fluid ast an infinite vetial oous late with onstant sution and heat soue in esene of a tansvese magneti field. This ae basially highlights the effet of mass tansfe on hydomagneti flow in esene of sution and heat soue.. FORMULTION OF THE PROBLEM Conside the unsteady fee onvetive mass tansfe flow of a visous inomessible eletially onduting fluid ast an infinite vetial oous late in esene of onstant sution and heat soue and tansvese magneti field. Let the x -axis be taken in vetially uwad dietion along the late and y -axis nomal to it. The hysial sketh and geomety of the oblem is shown in Fig.. Negleting the indued magneti field and the Joulean heat dissiation and alying Boussinesq s aoximation the govening equations of the flow field ae given by: Continuity equation: ' v v ' v ' ' (Constant) () y Momentum equation: u u * v gβ( T T ) gβ ( C C ) t y u σb ν ν u u () y ρ K Fig.. Physial sketh and geomety of the oblem Enegy equation: ( ) T T T ν u v k S T T () t y y C y Conentation equation: C C C v D. () t y y The bounday onditions of the oblem ae: w ε ( w ) ( ) i t i t u v v T T T T e ω C C w ε C w C e ω at y u T T C C as y. (5) Intoduing the following non-dimensional vaiables and aametes yv tv νω u η y t ω u ν ν ν v v ρ σb ν v K T T M K ρ v T ν Tw T C C ν νgβ( T w T ) C P G C w C k v * νgβ ( C w C ) ν S ν G S v S D v v E C ( T w T ) (6) whee g ρ σ ν β β* ω η k T T w T C C w C C D P S G G S K E and M ae esetively the aeleation due to gavity density eletial ondutivity oeffiient of kinemati visosity volumeti oeffiient of exansion fo heat tansfe volumeti oeffiient of exansion fo mass tansfe angula fequeny oeffiient of visosity themal diffusivity temeatue temeatue at the late temeatue at infinity onentation onentation at the late onentation at infinity seifi heat at onstant essue moleula mass diffusivity Pandtl numbe Shmidt numbe Gashof numbe fo heat tansfe Gashof numbe fo mass tansfe heat soue aamete emeability aamete Eket numbe and Hatmann numbe. 9

3 S.S. Das et al. / JFM Vol. No Substituting Eq. (6) in Eqs. () () and () unde bounday onditions (5) we get: GT GC M u (7) u u u t y y K T T T u ST E t y P y y t y S y C C C (8). (9) The oesonding bounday onditions ae: iωt iωt u T εe C εe at y u T C as y. (). METHOD OF SOLUTION To solve Eqs. (7) (8) and (9) we assume ε to be vey small and the veloity temeatue and onentation distibution of the flow field in the neighbouhood of the late as ( iωt ) ( ) ε ( ) ( iωt ) ( ) ε ( ) ( ) ( ) iωt ε ( ) u yt u y e u y () T yt T y e T y () C yt C y e C y. () Substituting Eqs. ()-() in Eqs. (7)-(9) esetively equating the hamoni and non-hamoni tems and negleting the oeffiients of ε we get Zeoth ode: u u M u GT GC () K PS u T PT T P E (5) y C SC. (6) Fist ode: iω u u u M u GT GC K (7) P ( ) T PT i S T P u ω E y y (8) iωs C SC C. (9) The oesonding bounday onditions ae y : u T C u T C y : u T C u T C () Solving Eqs. (6) and (9) unde bounday ondition () we get C e S y () C e my () Using multi aamete etubation tehnique and assuming E << we assume u u Eu () T T ET () u u Eu (5) T T ET. (6) Now using Eqs. ()-(6) in Eqs. () (5) (7) and (8) and equating the oeffiients of like owes of E negleting those of E we get the following set of diffeential equations: Zeoth ode: u u M u GT GC (7) K iω u u M u GT GC (8) K PS T PT T (9) P T PT ( iω S ) T. () The oesonding bounday onditions ae y : u T u T ; y : u T u T. () Fist ode: u u M u GT () K iω u u M u GT () K PS T ( ) PT T P u () P ( ) u T PT iω S T P u. (5) y y The oesonding bounday onditions ae y : u T u T ; y : u T u T. (6) Solving Eqs. (7)-() subjet to bounday ondition () we get u e e e (7) my S y m7y T e m y m5y my m9y u e e e 5 5 T e m y (8) (9). () Solving Eqs. ()-(5) subjet to bounday ondition (6) we get S y my m5y ( m S ) y T e 6 e 7 e 8 e 9 ( m7 S ) y e ( m m7) y e e m y () ( m m5) y ( m m) y T e e ( m m9) y ( m5 S ) y ( m S ) y 5e 6e 7e ( m9 S ) y ( m5 m7) y ( m m7) y 8e 9e e 9

4 S.S. Das et al. / JFM Vol. No ( m 7 m 9) y m 5 y e e () u e e e S y my m7y 5 ( m S ) y ( m7 S ) y 6e 7e ( m m7) y my m7y 8e 9e e () ( m m5) y ( m m) y ( m m9) y u e e e ( m5 S ) y ( m S ) y ( m9 S ) y e 5e 6e ( m5 m7) y ( m m7) y ( m7 m9) y 7e 8e 9e e e. () m5y m9y Substituting the values of C and C fom Eqs. () and () in Eq. () the solution fo onentation distibution of the flow field is given by C e e e S y iωt my ε. (5). Skin Fition The skin fition at the wall is given by u τw y y m S m { 7 E S m m 7 5 ( m S) 6 ( m7 S) 7 ( m m7 ) 8 m 9 m 7 iωt εe m m m E m m ( 5) ( ) ( ) ( ) ( m S) 5 ( m9 S) 6 ( m5 m7 ) 7 ( m m7 ) 8 ( m m ) m m m m m m m S }. Heat Flux (6) The heat flux at the wall in tems of Nusselt numbe is given by N u whee T y y m E S m m ( m S) 9 ( m7 S) ( m m7 ) m iωt εe { m5 E ( m m5 ) ( m m ) ( ) ( ) ( ) ( m9 S ) 8 ( m5 m7 ) 9 ( m m7 ) ( m7 m9 ) m 5 } m m m S m S (7) m S S iωs m S S iωs m P P SP m P P SP m5 P P P ( S iω) m6 P P P ( S iω) m7 M K m M 8 K iω m M 9 K iω m M K G ( m m )( m m ) 7 8 G ( m S )( m S ) 7 8 G ( m m )( m m ) G ( m m )( m m ) ( m m ) 9 PS 6 ( m S)( m S) Pm Pm 7 8 m m7 m m7 mp ( m m S ) PS m7 ( )( ) ( m m S )( m m S ) 7 7 Pm ( m m m ) P 5 ( m m5 m6) Pmm m5 m m m6 m m Pmm m9 m5 m m9 m6 m Pm 5 6 m m S ( )( ) ( )( ) ( ) 6 5 9

5 S.S. Das et al. / JFM Vol. No PS m ( m m S )( m m S ) 5 6 PS m 5 9 ( m m S )( m m S ) Pm 5 ( m m m ) ( m m m )( m m m ) Pmm Pm m ( m m m )( m m m ) G ( m S )( m S ) G ( m m )( m m ) 7 8 G 8 m m m 7 ( ) ( m m S )( m m S ) 7 8 G G S m m S ( ) 8 7 G m m m m ( ) 8 7 G ( m m )( m m ) G m m m m m m ( )( ) G ( m m m )( m m m ) 9 9 G 5 m m m m ( ) 9 G ( m m S )( m m S ) G ( m m S )( m m S ) 7 9 G 8 S m m S ( ) 9 ( m m m )( m m m ) G G ( m m m )( m m m ) G m m m m ( ) G ( m m )( m m ) RESULTS ND DISCUSSIONS The effet of mass tansfe on unsteady fee onvetive flow of a visous inomessible eletially onduting fluid ast an infinite vetial oous late with onstant sution and heat soue in esene of a tansvese magneti field has been studied. The govening equations of the flow field ae solved emloying multi aamete etubation tehnique and aoximate solutions ae obtained fo veloity field temeatue field onentation distibution skin fition and ate of heat tansfe. The effets of the etinent aametes on the flow field ae analyzed and disussed with the hel of veloity ofiles (Figs. -) temeatue ofiles (Figs. -7) onentation distibution (Fig. 8) and Tables -5. To be moe ealisti duing numeial alulations we have hosen the values of P.7 eesenting ai at C S.6 eesenting H O vaou G > oesonding to ooling of the late and S > eesenting heat soue.. Veloity Field The veloity of the flow field is found to hange moe o less with the vaiation of the flow aametes. The majo fatos affeting the veloity of the flow field ae Hatmann numbe M emeability aamete K Gashof numbe fo heat and mass tansfe G G ; Shmidt numbe S heat soue aamete S and Pandtl numbe P. The effets of these aametes on the veloity field have been analyzed with the hel of Figs. -. The veloity ofiles losely agee with those of Das et al. (). Figues and 8 deit the effet of magneti field on mean and tansient veloity of the flow field esetively. It is obseved that a gowing Hatmann numbe deeleates both mean and tansient veloity of the flow field at all oints due to the magneti ull of the Loentz foe ating on the flow field. The effets of emeability aamete on mean and tansient veloity of the flow field ae shown in Figs. and 9 esetively. The emeability aamete is found to enhane the mean veloity of the flow field at all oints while the tansient veloity ineases fo small values of K (K ) and the effet eveses fo highe values. Figues and show the effet of Gashof numbe fo heat tansfe on mean and tansient veloity esetively. The Gashof numbe fo heat tansfe is found to enhane both mean and tansient veloity at all oints due to the ation of fee onvetion uent in the flow field. Figues 5 and esent the effet of Gashof numbe fo mass tansfe on mean and tansient veloity esetively. Both the figues show the aeleating effet of the aamete on the veloity of the flow field at all oints. Figues 6 and disuss the effet of heat soue aamete on the veloity of the flow field. In both mean and tansient veloity of the flow field a gowing heat soue aamete enhanes the effet at all oints. In Figs. 8 and we deit the effet of mass tansfe on mean and tansient veloity of the flow field esetively. The esene of heavie diffusing seies in the flow field is found to deeleate both mean and tansient veloity at all oints. 95

6 S.S. Das et al. / JFM Vol. No Fig.. Mean veloity ofiles against y fo diffeent values of M with G 5 G 5 E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig. 6. Mean veloity ofiles against y fo diffeent values of S with G 5 G 5 M E. ω5. ε. ωtπ/ K S.6 P.7 Fig.. Mean veloity ofiles against y fo diffeent values of K with G 5 G 5 M E. ω5. ε. ωtπ/ S. S.6 P.7 Fig. 7. Mean veloity ofiles against y fo diffeent values of S with G 5 G 5 M E. ω5. ε. ωtπ/ K S. P.7 Fig.. Mean veloity ofiles against y fo diffeent values of G with G 5 M E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig. 8. Tansient veloity ofiles against y fo diffeent values of M with G 5 G 5 E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig. 5. Mean veloity ofiles against y fo diffeent values of G with G 5 M E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig. 9. Tansient veloity ofiles against y fo diffeent values of K with G 5 G 5 E. S. S.6 P.7 M ω5. ε. ωtπ/ 96

7 S.S. Das et al. / JFM Vol. No Temeatue Field The temeatue of the flow field suffes a substantial hange with the vaiation of the flow aametes suh as Pandtl numbe P Hatmann numbe M heat soue aamete S and emeability aamete K. These vaiations ae shown in Figs. -7. The temeatue ofiles ae in good ageement with those of Das et al. (). Figue deits the effet of Pandtl numbe on the temeatue field keeing othe aametes of the flow field onstant. It is inteesting to obseve that fo lowe value of P ( ) it enhanes the tansient temeatue while fo highe values the effet eveses. Figue 5 shows the effet of magneti aamete on the temeatue field. The effet of Hatmann numbe is to etad the temeatue of the flow field at all oints. Cuve with M oesonds to the non-mhd flow. This shows that in absene of magneti field the temeatue fist ises nea the late and theeafte it falls. In othe uves thee is a deease in temeatue at all oints. This shows the dominating effet of the magneti field due to the ation of the Loentz foe in the flow field. The effet of heat soue aamete on the temeatue field is esented in Fig. 6. The heat soue aamete is found to enhane the temeatue of the flow field at all oints. In Fig. 7 we analyze the effet of emeability aamete on the temeatue field. gowing emeability aamete is found to inease the temeatue of the flow field at all oints.. Conentation Distibution The vaiation in the onentation bounday laye of the flow field is shown in Fig. 8 due to the hange in the Shmidt numbe S. Cuves with S...6 and.78 esetively eesent the onentation distibution in esene of H He H O vaou and NH in the flow field. Comaing the uves of the said figue it is obseved that a gowing Shmidt numbe deeases the onentation bounday laye thikness of the flow field at all oints.. Skin Fition The values of skin fition at the wall against K fo diffeent values of Hatmann numbe M and heat soue aamete S ae enteed in Tables and esetively. Fom Table it is obseved that a gowing Hatmann numbe M edues the skin fition at the wall fo a fixed value of the emeability aamete due to the ation of Loentz foe in the flow field. It is futhe obseved fom Table that both emeability aamete K and heat soue aamete S enhane the skin fition at the wall. Ou obsevation fo skin fition agees with those of Das et al. ()..5 Rate of Heat Tansfe The ate of heat tansfe at the wall vaies with the vaiation of Pandtl numbe P Hatmann numbe M emeability aamete K. These vaiations ae enteed in the Tables -5. Fom Table we obseve that a gowing Pandtl numbe o emeability aamete inease the magnitude of the ate of heat tansfe at the wall. Futhe it is obseved fom Table that an inease in Hatmann numbe edues its value fo a given value of Pandtl numbe. gain fom Table 5 we see that fo a given value of emeability aamete it enhanes the magnitude of ate of heat tansfe fo small values of M and fo highe values the effet eveses due to the magneti ull of the Loentz foe ating on the flow field. These vaiations agee with those of Das et al. () with a little deviation fo highe value of M. 5. CONCLUSIONS We summaize below the following esults of hysial inteest on the veloity temeatue and the onentation distibution of the flow field and also on the wall shea stess and ate of heat tansfe at the wall.. gowing Hatmann numbe o Shmidt numbe etads the mean veloity as well as the tansient veloity of the flow field at all oints.. The effet of ineasing Gashof numbe fo heat and mass tansfe o heat soue aamete is to aeleate both mean and tansient veloity of the flow field at all oints.. The mean veloity of the flow field ineases with an inease in emeability aamete while the tansient veloity ineases fo smalle values of K ( ) and fo highe values the effet eveses.. gowing Hatmann numbe deeases tansient temeatue of the flow field at all oints while a gowing emeability aamete o heat soue aamete eveses the effet. 5. The Pandtl numbe P ineases the tansient temeatue of the flow field at all oints fo small values of P ( ) and fo highe values the effet eveses. 6. The effet of ineasing Shmidt numbe is to edue the onentation bounday laye thikness of the flow field at all oints. 7. gowing Hatmann numbe edues the skin fition at the wall while a gowing emeability aamete o heat soue aamete eveses the effet. 8. The effet of ineasing Pandtl numbe o emeability aamete is to inease the magnitude of the ate of heat tansfe at the wall. On the othe hand a gowing Hatmann numbe edues its value fo a given value of Pandtl numbe while fo a given value of emeability aamete it enhanes the magnitude of ate of heat tansfe fo small values of M and fo highe values the effet eveses due to the magneti ull of the Loentz foe ating on the flow field. 97

8 S.S. Das et al. / JFM Vol. No Fig.. Tansient veloity ofiles against y fo diffeent values of G with G 5 M E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig.. Temeatue ofiles against y fo diffeent values of P with G 5 G 5 M E. ω5. ε. ωtπ/ K S. Fig.. Tansient veloity ofiles against y fo diffeent values of G with G 5 M E. ω5. ε. ωtπ/ K S. S.6 P.7 Fig. 5. Temeatue ofiles against y fo diffeent values of M with G 5 G 5 E. ω5. ε. ωtπ/ K S. P.7 Fig.. Tansient veloity ofiles against y fo diffeent values of S with G 5 G 5 E. M K S.6 P.7 ω5. ε. ωtπ/ Fig. 6. Temeatue ofiles against y fo diffeent values of S with G 5 G 5 M E. ω5. ε. ωtπ/ K P.7 Fig.. Tansient veloity ofiles against y fo diffeent values of S with G 5 G 5 E. M K S. P.7 ω5. ε. ωtπ/ Fig. 7. Temeatue ofiles against y fo diffeent values of K with G 5 G 5 M E. ω5. ε. ωtπ/ S. P.7 98

9 S.S. Das et al. / JFM Vol. No Kim Y.J. (). Unsteady MHD onvetive heat tansfe ast a semi-infinite vetial oous moving late with vaiable sution. Int. J. Engng. Si Makinde O.D. J.M. Mango and D.M. Theui (). Unsteady fee onvetion flow with sution on an aeleating oous late. MSE J. Mod. Meas. Cont. B 7 () 9-6. Fig. 8. Conentation ofiles against y fo diffeent values of S with ω5. ε. ωtπ/ REFERENCES haya M. L.P. Singh and G.C. Dash (999). Heat and mass tansfe ove an aeleating sufae with heat soue in esene of sution and blowing. Int. J.Engng. Si Chandan P. N.C. Saheti and.k. Singh (998). Unsteady hydomagneti fee onvetion flow with heat flux and aeleated bounday motion. J. Phys. So. Jaan Das S.S. S.K. Sahoo and G.C. Dash (6). Numeial solution of mass tansfe effets on unsteady flow ast an aeleated vetial oous late with sution. Bull. Malays. Math. Si. So. 9() -. Das S.S.. Sataathy J.K. Das and S.K. Sahoo (7). Numeial solution of unsteady fee onvetive MHD flow ast an aeleated vetial late with sution and heat flux. J. Ulta Si. Phys. Si. 9() 5-. Das S.S. and M. Mita (9).Unsteady mixed onvetive MHD flow and mass tansfe ast an aeleated infinite vetial late with sution. Ind. J. Si. Teh. (5) 8-. Mansutti D. G. Pontelli and K.R. Rajagoal (99). Steady flows of non-newtonian fluids ast a oous late with sution o injetion. Int. J. Num. Methods Fluids Saangi K.C. and C.B. Jose (5). Unsteady fee onvetive MHD flow and mass tansfe ast a vetial oous late with vaiable temeatue. Bull. Cal. Math. So. 97 () 7-6. Shama P.R. and D. Paeek (). Steady fee onvetion MHD flow ast a vetial oous moving sufae. Ind. J. Theo. Phys Singh.K..K. Singh and N.P. Singh (). Heat and mass tansfe in MHD flow of a visous fluid ast a vetial late unde osillatoy sution veloity. Ind. J. Pue l. Math. () 9-. Singh B. and C. Thaku (). n exat solution of lane unsteady MHD non-newtonian fluid flows. Ind. J. Pue l. Math. (7) Soundalgeka V.M. (97). Fee onvetion effets on steady MHD flow ast a vetial oous late. J. Fluid Meh Yamamoto K. and N. Iwamua (976). Flow with onvetive aeleation though a oous medium. Engng. Math. -5. Das S.S.. Sataathy J.K. Das and J.P. Panda (9). Mass tansfe effets on MHD flow and heat tansfe ast a vetial oous late though a oous medium unde osillatoy sution and heat soue. Int. J. Heat Mass Tansfe Das S.S. U.K. Tiathy and J.K. Das (). Hydomagneti onvetive flow ast a vetial oous late though a oous medium with sution and heat soue. Intenational Jounal of Enegy and Envionment () Hasimoto H. (957). Bounday laye gowth on a flat late with sution o injetion. J. Phys. So. Jaan Jha B.K. (998). Effets of alied magneti field on tansient fee onvetive flow in a vetial hannel. Ind. J. Pue l. Math. 9()

10 S.S. Das et al. / JFM Vol. No Table Values of skin fition (τ) at the wall against K fo diffeent values of M with G 5 G 5 E. ω5. ε. ωtπ/ S. S.6 Table Values of ate of heat tansfe (N u ) at the wall against P fo diffeent values of K with G 5 G 5 M E. ω5. ε. ωtπ/ S. S.6 K τ M M. M5 M P N u K. K.5 K K Table Values of skin fition (τ) at the wall against K fo diffeent values of S with G 5 G 5 E. ω5. ε. ωtπ/ S.6 K τ S-.5 S-. S-. S. S. S Table Values of ate of heat tansfe (N u ) at the wall against P fo diffeent values of M with G 5 G 5 M E. ω5. ε. ωtπ/ S. S.6 Table 5 Values of ate of heat tansfe (N u ) at the wall against K fo diffeent values of M with G 5 G 5 M E. ω5. ε. ωtπ/ S. S.6 P N u M M. M5 M K N u M M. M5 M