Journal of Applied Fluid Mechanics, Vol. 9, No. 1, pp , Available online at ISSN , EISSN

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1 Jounal of Applied Fluid Mehanis, Vol. 9, No., pp. 3-7, 6. Available online at ISSN , EISSN Heat and Mass Tansfe Effets on Unsteady MHD Natual Convetion Flow of a Chemially Reative and Radiating Fluid though a Poous Medium Past a Moving Vetial Plate with Abitay Ramped Tempeatue G. S. Seth, R. Shama and B. umbhaka Depatment of Applied Mathematis, Indian Shool of Mines, Dhanbad 864, INDIA Coesponding Autho gsseth_ism@yahoo.om, gsseth.ism@gmail.om (Reeived August, 4; aepted Novembe, 4) ABSTRACT Investigation of unsteady hydomagneti natual onvetion flow with heat and mass tansfe of a visous, inompessible, eletially onduting, hemially eative and optially thin adiating fluid past an exponentially aeleated moving vetial plate with abitay amped tempeatue embedded in a fluid satuated poous medium is aied out. Exat solutions of momentum, enegy and onentation equations ae obtained in losed fom by Laplae tansfom tehnique. The expessions fo the shea stess, ate of heat tansfe and ate of mass tansfe at the plate fo both amped tempeatue and isothemal plates ae deived. The numeial values of fluid veloity, fluid tempeatue and speies onentation ae displayed gaphially wheeas those of shea stess, ate of heat tansfe and ate of mass tansfe at the plate ae pesented in tabula fom fo vaious values of petinent flow paametes. It is found that, fo isothemal plate, the fluid tempeatue appoahes steady state when t.5. Consequently, the ate of heat tansfe at isothemal plate appoahes steady state when t.5. eywods: Natual onvetion; Magneti field; Chemial eation; Radiation; Abitay amped tempeatue. NOMENCLATURE a * a B C p sufae aeleation paamete absoption oeffiient unifom magneti field speies onentation speifi heat at onstant pessue D M hemial moleula diffusivity g aeleation due to gavity G G k M N themal Gashof numbe solutal Gashof numbe themal ondutivity of the fluid pemeability of poous medium pemeability paamete hemial eation oeffiient hemial eation paamete magneti paamete adiation paamete P q T t Pandtl numbe adiating flux veto fluid tempeatue itial time fo ampedness t non-dimensional itial time fo ampedness haateisti veloity u fluid veloity in x dietion S Shmidt numbe U oeffiient of themal expansion * oeffiient of expansion fo speies onentation eletial ondutivity fluid density * Stefan-Boltzmann onstant kinemati oeffiient of visosity

2 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6.. INTRODUCTION Theoetial/expeimental investigation of poblems of unsteady hydomagneti natual onvetion flow of an eletially onduting fluid within poous and non-poous media has eeived onsideable attention of seveal eseahes duing past few deades due to its ovewhelming and impotant appliations in many aeas of siene and engineeing whih inludes geophysis, astophysis, eletonis, aeonautis, metallugy, hemial and petoleum engineeing et. eeping in view the impotane of this fluid flow, seveal eseahes investigated unsteady hydomagneti natual onvetion flow of an eletially onduting fluid past bodies with diffeent geometies unde diffeent initial and bounday onditions. Mention may be made of the eseah studies of Gupta (96), Pop (969), Chamkha (997, ), Helmy (998), im (), Raptis et al. (3), Makinde and Tshela (4) and Seth et al. (3, 4). Hydomagneti natual onvetion flow of adiating and non-adiating fluid with heat and mass tansfe in poous and non-poous media is studied by seveal eseahes due to its vaied and wide appliations in astophysis, geophysis, aeonautis, eletonis, metallugy, hemial and petoleum industies. Hydomagneti natual onvetion flow of an eletially onduting fluid in a fluid satuated poous medium has also been suessfully exploited in ystal fomation. Oepe and Szekely (983) have found that the pesene of a magneti field an suppess natual onvetion uents and the stength of magneti field is one of the impotant fatos in eduing nonunifom omposition theeby enhaning quality of the ystal. In addition to it, hydomagneti poblems with heat and mass tansfe is of muh signifiane in MHD flow-metes, MHD enegy geneatos, MHD pumps, ontolled themo-nulea eatos, MHD aeleatos et. Hossain and Mandal (985) investigated mass tansfe effets on unsteady hydomagneti fee onvetion flow past an aeleated vetial poous plate. Jha (99) studied hydomagneti fee onvetion and mass tansfe flow past a unifomly aeleated vetial plate though a poous medium when magneti field is fixed with the moving plate. Elbashbeshy (997) disussed heat and mass tansfe along a vetial plate in the pesene of magneti field. Chamkha and haled () investigated oupled heat and mass tansfe by natual onvetion fom a vetial, semi-infinite flat plate embedded in a poous medium in the pesene of an extenal magneti field. Chen (4) analyzed hydomagneti natual onvetion flow with heat and mass tansfe ove a pemeable inlined sufae with vaiable wall tempeatue and onentation. Ibahim et al. (4) disussed unsteady magnetohydodynami flow of mio-pola fluid and heat tansfe past a vetial poous plate though a poous medium in the pesene of themal and mass diffusions with a onstant heat soue. Pasad and Reddy (7) analyzed an unsteady, two-dimensional, hydomagneti, lamina fee onvetive bounday-laye flow of an inompessible, Newtonian, eletially-onduting and adiating fluid past an infinite heated vetial poous plate with heat and mass tansfe by taking into aount the effet of visous dissipation. Chaudhay and Jain (7) studied hydomagneti natual onvetion flow with heat and mass tansfe past an infinite vetial osillating plate embedded in a fluid satuated poous medium. Makinde and Sibanda (8) studied MHD mixed onvetion heat and mass tansfe flow past a vetial poous plate embedded in a poous medium with onstant heat flux. Rajesh and Vama (9) investigated the effets of themal adiation on unsteady fee onvetion flow past an exponentially aeleated infinite vetial plate with mass tansfe in the pesene of magneti field. Sangapatnam et al. (9) studied the effets of themal adiation on the natual onvetive heat and mass tansfe of a visous, inompessible, gay absobing, emitting fluid flowing past an impulsively stated moving vetial plate with visous dissipation. Eldabe et al.() disussed unsteady MHD flow of a visous and inompessible fluid with heat and mass tansfe in a poous medium nea a moving vetial plate with time-dependent veloity. Pakash et al. (3) investigated diffusion themo and adiation effet on unsteady MHD fee onvetion flow though poous medium past an impulsively stated infinite vetial plate with vaiable tempeatue and unifom mass diffusion. In most of the hemial engineeing poesses, hemial eation ous between a foeign mass and the fluid. Chemial eations an be lassified as eithe heteogeneous o homogeneous poesses. This depends on whethe they ou at an intefae o as a single phase volume eation. These poesses take plae in numeous industial appliations viz. polyme podution, manufatuing of eamis o glasswae, food poessing et. Chamkha (3) investigated MHD flow ove a unifomly stethed vetial pemeable sufae in the pesene of heat geneation/absoption and hemial eation. Afify (4) studied the effet of adiation on fee onvetive flow and mass tansfe past a vetial isothemal one sufae with hemial eation in the pesene of a tansvese magneti field. Ibahim et al. (8) analyzed the effet of hemial eation and adiation absoption on the unsteady MHD fee onvetion flow past a semi-infinite vetial pemeable moving plate with heat soue and sution. Bak () disussed the effets of hemial eation on MHD fee onvetion and mass tansfe flow of a mio pola fluid with osillatoy plate veloity and onstant heat soue in a otating fame of efeene. Chamkha et al. () disussed the effets of Joule heating, hemial eation and themal adiation on unsteady hydomagneti natual onvetion bounday laye flow with heat and mass tansfe of a mio pola fluid fom a semi-infinite heated vetial poous plate in the pesene of a unifom tansvese magneti field. Bhattahaya and Layek () 4

3 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. obtained similaity solution of MHD bounday laye flow with mass diffusion and hemial eation ove a poous flat plate with sution/blowing. ishoe et al. (3) studied the effets of adiation and hemial eation on unsteady hydomagneti natual onvetion flow of a visous fluid past ove an exponentially aeleated vetial plate. Mohamed et al. (3) investigated unsteady hydomagneti natual onvetion flow with heat and mass tansfe of a visous, inompessible and eletially onduting and adiating fluid past though a poous medium nea an impulsively moving hot vetial plate in the pesene of homogeneous hemial eation of fist ode and tempeatue dependent heat sink. In fat, natual onvetion flows ae geneally modelled by the eseahes unde the onsideation of unifom sufae tempeatue o unifom sufae heat flux. But, in many physial situations, the tempeatue of the bounding sufae may equie non-unifom o abitay wall onditions. Moeove, thee may be step disontinuities in the sufae tempeatue o amped sufae tempeatue. eeping in view this fat, seveal eseahes investigated natual onvetion fom a vetial plate with step disontinuities in the sufae tempeatue onsideing vaious aspets of the poblem. Mention may be made of the eseah investigations of Hayday et al. (967), ellehe (97), ao (975) and Lee and Yovanovih (99). In eent yeas, seveal eseahes investigated unsteady hydomagneti fee onvetion flow past a vetial plate with amped tempeatue onsideing vaious vaiations in the poblem. Mention may be made of the eseah studies of Rajesh (), Samiulhaq et al. (), Das (), Nandkeolya and Das (3, 4), Ahmed and Das (3a, b), Nandkeolya et al. (3a, 3b), Naahai and Sulaiman (3), Das et al. (4), undu et al. (4). Reently, Seth et al. (4, 5a, b) onsideed hydomagneti natual onvetion heat and mass tansfe flow with Hall uent past an infinite moving vetial plate with amped tempeatue in a otating medium onsideing diffeent aspets of the poblems. Howeve, in all these investigations, eseahes have onsideed the time inteval fo ampedness in the plate tempeatue as fixed i.e. t t, ( t and t being time and haateisti time espetively). Reently, Ahmed and Dutta (4) investigated magnetohydodynami tansient flow of a visous, inompessible, eletially onduting and optially thin adiating fluid past an impulsively moving infinite vetial plate with tempoay amped wall tempeatue. They onsideed time t as an abitay onstant in plae of haateisti time. But they have non-dimensionalized physial vaiables, namely, y and t in thei poblem using t. This is possible only when t is onsideed as a haateisti time. With this nondimensionalization poess they have defined Reynolds numbe Ut, ( U and being the R e unifom veloity of the plate at time t and kinemati oeffiient of visosity espetively) whih is not oet. If t is haateisti time then its dimension in tems of haateisti length L Ut U L L is then is equal to whih is equal to R e. Due to this eason, the definition of Reynolds numbeis also not taken oetly. Also inteval of amped pofile edues to t (t being non-dimensional time) only when t is onsideed haateisti time. In ou opinion Ut is a non-dimensional abitay time onstant whih we have named as itial time fo ampedness. This is tue if t is not a haateisti time. It is to be noted that time inteval fo amped pofiles vaies fom mateial to mateial depending upon the speifi heat apaity of the mateial. Abitay amped pofiles appea in eal wold situation in building ai-onditioning systems, fabiation of thin-film photovoltai devies, phase tansition in poessing of mateials, tubine blade heat tansfe, heat exhanges et. Pupose of pesent investigation is to study unsteady hydomagneti natual onvetion heat and mass tansfe flow of an eletially onduting, visous, inompessible, hemially eative and optially thin heat adiating fluid past an exponentially aeleated moving vetial plate though fluid satuated poous medium with abitay amped wall tempeatue. This poblem is totally diffeent fom the eseah pape investigated by Ahmed and Dutta (4). In ou pape, we have onsideed medium as poous, hemially eative and optially thin adiating fluid when fluid flow is indued due to exponentially aeleated moving vetial plate along with themal and onentation buoyany foes. We have appoximated gadient of adiating q flux veto i.e. by appoximation suggested by y Raptis () in plae of Cogley et al. (968) whih was adopted by Ahmed and Dutta (4). This eseah study may have stong beaings on numeous poblems of patial inteest whee initial tempeatue pofiles ae of muh signifiane in designing of so many hydomagneti devies and in seveal industial poesses ouing at high tempeatues whee the effets of themal adiation play a vital ole in the fluid flow haateistis. We have ompaed numeial values of shea stess at the plate with those of Das et al. () as a speial ase of ou esults whih ae in exellent ageement with the numeial values of Das et al. ().. FORMULATION OF THE PROBLEM AND ITS SOLUTION Conside unsteady magnetohydodynami natual onvetion flow with heat and mass tansfe of an eletially onduting, visous, inompessible, hemially eative and optially thin heat adiating 5

4 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. fluid though fluid satuated poous medium past an infinite moving vetial plate with abitay amped tempeatue. Choose the oodinate system in suh a way that x -axis is along the length of the plate in the upwad dietion and y - axis nomal to plane of the plate in the fluid. The fluid is pemeated by a unifom tansvese magneti field B applied in a dietion paallel to y -axis. Initially i.e. at time t, both the fluid and plate ae at est and maintained at a unifom tempeatue T. Also speies onentation at the sufae of the plate as well as at evey point within the fluid is maintained at unifom onentation C. At time t, plate is exponentially aeleated with at veloity Ue in x - dietion ( U being haateisti veloity). Tempeatue of the plate is T T T t/ t when aised o loweed to w t t ( t being itial time fo ampedness) and unifom tempeatue T w when t t. Also at time t, speies onentation at the sufae of the plate is aised to unifom speies onentation C w and is maintained theeafte. The fluid is onsideed as a gay, emitting-absobing adiation but non satteing medium. It is assumed that thee exists a homogeneous hemial eation of fist ode with onstant ate between diffusing speies and the fluid. Physial model of the poblem is pesented in Fig.. Fluid is metalli liquid o patially ionized whose magneti Reynolds numbe is vey small. Hene, the indued magneti field geneated by fluid motion is negligible in ompaison to the applied one (Camme and Pai, 973). Thus, the magneti field B,B,. Sine, no extenal eleti field is applied into the flow-field, the effet of polaization of fluid beomes negligible (Meye, 958). With these assumptions, the govening equations fo unsteady magnetohydodynami natual onvetion flow of an eletially onduting, visous, inompessible, hemially eative and optially thin heat adiating fluid though fluid satuated poous medium ae given by Fig.. Physial model of the poblem. u u B u u g ( T T ) t y g * (C C ), () T k T q, t y y () p p C (C C ) C DM, (3) t y whee u, T,,, k, p, C, DM, q,,,, g, * and ae, espetively, fluid veloity in x - dietion, fluid tempeatue, pemeability of poous medium, hemial eation oeffiient, themal ondutivity, speifi heat at onstant pessue, speies onentation, hemial moleula diffusivity, adiating flux veto, kinemati oeffiient of visosity, eletial ondutivity, fluid density, aeleation due to gavity, oeffiient of themal expansion and oeffiient of expansion fo speies onentation. Appopiate initial and bounday onditions fo the fluid flow poblem ae given by u, TT, C C fo y and t, (4a) at u U e, CC w at y fo t, (4b) T T ( T T ) t/ t at yfo tt, w (4) T T at y fo t t, (4d) w u, w, T T, CC as y fo t. (4e) Fo an optially thin gay fluid the loal adiant absoption (Raptis, ) is expessed as q y a T T * * ( ), (5) * whee a is absoption oeffiient and * is Stefan-Boltzmann onstant. It is assumed that the tempeatue diffeene between the fluid in the bounday laye and fee steam is suffiiently small so that T 4 may be expessed as a linea funtion of T. This is 4 aomplished by expanding T in a Taylo seies about fee steam tempeatue T. Negleting 4 seond and highe ode tems, T is expessed as T 4T T 3 T. (6) Using equations (5) and (6), in equation (), we obtain * * 3 T k T 6a T T T. (7) t y p p In ode to epesent equations (), (3) and (7) along with initial and bounday onditions (4a) to (4e) in non-dimensional fom, we ae intoduing the following non-dimensional vaiables and paametes 6

5 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. ' U U u T T y y, t t, u, T, U T w T C C gt w T C, a a, G, 3 C w C U U * gcw C U G, 3,, U U * B 6aT p M, N, P, U Up k Ut S and t, DM (8) whee a, G, G,,, M, N, P, S and t ae, espetively, sufae aeleation paamete, themal Gashof numbe, solutal Gashof numbe, pemeability paamete, hemial eation paamete, magneti paamete, adiation paamete, Pandtl numbe, Shmidt numbe and non-dimensional fixed time whih vaies fom mateial to mateial of plate depending on speifi apaity of mateial whih we have named as non-dimensional itial time fo ampedness. The equations (), (3) and (7), in non-dimensional fom, edue to u u u MuGT, GC (9) t y T T NT, () t P y C C C, () t S y Initial and bounday onditions (4a) to (4e) in nondimensional fom, ae pesented below u, T, C fo y and t, (a) at u e, C at y fo t, (b) t T at y fo t t, t () T at y fo t t, (d) u, T, C as y, t. (e) Exat solution fo fluid veloity u y, t, fluid tempeatue T y, t and speies onentation C y, t is obtained by solving equations (9) to () subjet to the initial and bounday onditions (a) to (e) using Laplae tansfom tehnique and ae pesented in the following fom afte simplifiation: at e y t u(y,t) f( y,, a, t) e f(y,,,t) f ( y,,, S, t) f ( y,,, t) f ( y,,, S, t) G y, t Ht ( - t) Gyt (, - t ), (3) T( y, t) P( y, t)- H( t- t) P( y, t- t), (4) t y S y S C( y, t) e ef t t y S y S e ef t, (5) t whee 3t Gyt (, ) e f3( ypn,,, 3, t) t 3 - f ( y,,,, t) f ( y, P, N,, t) f4( y,,, 3, t), y P y P N y P P( yt, ) t e ef Nt N t y P y P N y P t e ef Nt, N t G G whee,, M, P S S P N and 3. S P The expessions fo fi (i,,3,4) ae povided in Appendix. Ht ( t) and ef( x) ae unit step funtion and omplementay eo funtion espetively.. Solution fo Unit Pandtl and Unit Shmidt Numbe Solution (3) fo fluid veloity u y,t is not valid fo unit Pandtl numbe and unit Shmidt numbe. Sine Pandtl numbe is a measue of elative stength of visosity to themal diffusivity of fluid and Shmidt numbe is a measue of elative stength of visosity to moleula (mass) diffusivity of fluid, the ase P and S oesponds to those fluids fo whih the visous, themal and onentation bounday laye thiknesses ae same ode of magnitude. Thee ae some fluids of patial signifiane whih belong to this ategoy (Chen, 4). Setting P and S in equations () and () and following the same poedue adopted ealie, exat solution fo fluid veloity u y,t, fluid tempeatue T y, t and fluid onentation C y, t is obtained and pesented in the following simplified fom : at uy, t e fy,, a, tfy,,, t f ( y,,, t) G ( y, t) H( t t ) G y, t t, (6) T( yt, ) P( yt, ) Ht ( t) P( yt, t), t (7) 7

6 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. y C y t e ef t e t y y, y ef t, (8) t whee G ( y, t) f ( y,, t) f ( y, N, t), 5 5 t y y N y P (y,t) t e ef Nt N t y y N y t e ef Nt, N t G whee and N expession fo f5 is povided in Appendix. G The. Solution when Fluid is in Contat to Isothemal Plate Solution (3) to (5) is the solution fo fluid veloity, fluid tempeatue and speies onentation fo natual onvetion flow with heat and mass tansfe of an eletially onduting, visous, inompessible, hemially eative and optially thin adiating fluid past an exponentially aeleated moving vetial plate with abitay amped tempeatue. In ode to emphasize the effets of amped tempeatue distibution within the plate on fluid flow, it may be wothwhile to ompae suh flow with the one nea an exponentially aeleated moving vetial plate with unifom tempeatue. eeping in view the assumptions made ealie, the solutions fo fluid veloity and fluid tempeatue fo the flow past an exponentially aeleated moving isothemal vetial plate is obtained and pesented in the following fom at e 3t uy, t fy,, a, t e fy,, 3, t 3,,,,,,, f y N P t f y t 3 t f y,,n,p,t e f y,,,t f y,,,p,t f y,,,t f y,,,s,t, y P Ty, t e ef Nt e t y P N y PN (9) y P ef Nt. () t.3 Shea Stess and Rate of Heat Tansfe at the Plate The expessions fo shea stess and ate of heat tansfe N at the plate ae obtained and pesented u in the following fom fo both amped tempeatue and isothemal plates. (i) Fo amped tempeatue plate: u y y at e f a t e f t f S t 6,, t 7,, 8,,,,,, G t H tt G t t () T Nu P(, t) H( t t) P(, tt), y y () whee G(, t) f9( 3, P, Nt, ) f(, 3, P, Nt, ), 3 t Nt Pt P P (, t) e t PN t N ef( Nt ). (ii) Fo isothemol plate: u y y f, at, f, 3, P, Nt, 3 f,,p,n,t f,,s,,t f,,s,,t, (3) T Nt P Nu e PN ef Nt, y t y (4) whee the expessions fi(i 6,7,...,) ae povided in Appendix..4 Rate of Mass Tansfe The expessions fo ate of mass tansfe S h at the plate is obtained and pesented in the following fom C t S Sh e S ef t y t y (5) 3. VALIDATION OF RESULT In ode to validate ou numeial esults, we have ompaed, the numeial values of shea stess at the plate of a speial ase of ou numeial esults with those of Das et al. (), fo vaious values of and N, taking M, t, a, t, G, G, and P.7. Ou numeial esults ae in exellent ageement with those of Das et al () whih is lealy evident fom Tables and. 8

7 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. Table Shea stess at the amped tempeatue plate when M, t, a, t,, G, G and P.7. N Das et al.() Pesent Value Table Shea stess at isothemal plate when M, t, a, t,, G, G and P.7. N Das et al.() Pesent Value T fo evey value of time t i.e. t. This means that plate tempeatue is oole up to the itial time fo ampedness t in ase of amped tempeatue plate than that fo isothemal plate. Fig. depits the effet of itial time fo ampedness on fluid flow in the bounday laye egion fo amped tempeatue plate. It is notied fom Fig. that u deeases on ineasing t. This implies that, fo amped tempeatue plate, fluid flow is getting deeleated in the bounday laye egion with the inease in itial time fo ampedness. 4. RESULTS AND DISCUSSION Fig.. Veloity pofiles when In ode to analyze the effets of itial time fo ampedness, sufae aeleation paamete, time, adiation, pemeability of poous medium, mass diffusion, solutal buoyany foe and hemial eation on the flow-field, the numeial values of fluid veloity within bounday laye egion, omputed fom the analytial solutions (3) and (9), ae displayed gaphially vesus bounday laye oodinate y fo vaious values of dimensionless fixed time t, a, time t, adiation paamete N, pemeability paamete,shimidth numbe S, solutal Gashof numbe G and hemial eation paamete in figues to 9 taking magneti paamete M 4, Pandtl numbe P.7 (ionized ai) and themal Gashof numbe G. It is obseved fom figues to 9 that, fo both amped tempeatue and isothemal plates, fluid veloity u attains a distintive maximum value in the viinity of the plate and then deeases popely on ineasing bounday laye oodinate y to appoah fee steam value. Fig. 3 illustates the effet of sufae aeleation paamete on fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 3 that u ineases on ineasing a fo both amped tempeatue and isothemal plates. This implies that, fo both amped tempeatue and isothemal plates, fluid flow is getting aeleated in the bounday laye egion with the inease in sufae aeleation paamete. Fig. 4 depits the effet of time on fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 4 that u ineases on ineasing t fo both amped tempeatue and isothemal plates. This implies that fluid flow in the bounday laye egion is getting aeleated with the pogess of time fo both amped tempeatue and isothemal plates. This is due to fat that themal and solutal buoyany foes have signifiant ole on the fluid flow in the egion nea the plate and its effet is nullified in the fee steam. Also, it is evident fom figues 3 to 9 that fluid veloity is faste in ase of isothemal plate than that of amped tempeatue plate.this may be due to eason that in the ase of amped tempeatue plate, plate tempeatue T ineases with espet to time t and attains value T when t t wheeas in ase of isothemal plate, plate tempeatue is Fig. 3. Veloity pofiles when9

8 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. diffusivity. S deeases on ineasing mass diffusivity. This implies that mass diffusion tends to aeleate the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. Fig. 4. Veloity pofiles whenfig. 5 illustates the effet of adiation on fluid veloity fo both amped tempeatue and isothemal plates. Fig. 6. Veloity pofiles whenfig. 5. Veloity pofiles whenit is notied fom Fig. 5 that, fo both amped tempeatue and isothemal plates, u deeases on ineasing N. This implies that adiation has a tendeny to deeleate the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates fo optially thin adiating fluid. This is due to fat that fluid tempeatue is getting edued due to themal adiation and fluid flow in the bounday laye egion is getting etaded. Fig. 6 eveals the effet of pemeability of poous medium on fluid flow in bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 6 that, fo both amped tempeatue and isothemal plates, u ineases on ineasing. An inease in pemeability of medium implies that thee is a deease in the esistane of the poous medium whih in tun aeleate fluid flow in bounday laye egion fo both amped tempeatue and isothemal plates. Fig. 7 depits the effets of mass diffusion on the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 7 that, fo both amped tempeatue and isothemal plates, fluid veloity u deeases on ineasing S. S epesents the atio of momentum diffusivity and moleula (mass) Fig. 7. Veloity pofiles whenfig. 8. Veloity pofiles whenfig. 8 eveals the effets of solutal buoyany foe on the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 8 that, fo both amped tempeatue and isothemal plates, fluid veloity u ineases on ineasing G. G signifies the elative stength of solutal buoyany foe to visous foe. G ineases when solutal buoyany

9 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. foe ineases. This implies that solutal buoyany foe tends to aeleate the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. Fig. 9 exhibits the effet of hemial eation on the fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. It is notied fom Fig. 9 that u deeases on ineasing fo both amped tempeatue and isothemal plates. This implies that hemial eation tends to deeleate fluid flow in the bounday laye egion fo both amped tempeatue and isothemal plates. Figs. and demonstate the influene of time and adiation on fluid tempeatue T fo both amped tempeatue and isothemal plates. It is notied fom Figs. and that fluid tempeatue T ineases on ineasing t wheeas it deeases on ineasing N fo both amped tempeatue and isothemal plates. This implies that adiation has a tendeny to edue fluid tempeatue in the bounday laye egion fo both amped tempeatue and isothemal plates. Fluid tempeatue is getting enhaned in the bounday laye egion with the pogess of time fo both amped tempeatue and isothemal plates. It is inteesting to note fom Fig. that fluid tempeatue at isothemal plate attains steady state when t.5. Fig. 9. Veloity pofiles when Fig.. Tempeatue pofiles when The numeial values of fluid tempeatue T, omputed fom analytial solutions (4) and (), ae depited gaphially in Figs. to fo vaious values of itial time fo ampedness t, time t and adiation paamete N taking P.7. It is obseved fom Figs. to that fluid tempeatue T is maximum at the sufae of the plate and deeases popely on ineasing bounday laye o-odinate y to appoah fee steam value. It is also notied fom Figs. and that fluid tempeatue is highe in ase of isothemal plate than that of amped tempeatue plate. Fig. depits the effet of itial time fo ampedness on fluid tempeatue fo amped tempeatue plate. It is notied fom Fig. that T deeases on ineasing t fo amped tempeatue plate. An inease in itial time fo ampedness implies that thee is a deease in fluid tempeatue in the bounday laye egion fo amped tempeatue plate. The numeial values of speies onentation C, omputed fom analytial solution (5), is depited gaphially in Figs. 3 and 4 fo vaious values of and S taking t.. It is obseved fom Figs. 3 and 4 that speies onentation C is maximum at the sufae of the plate and it deeases popely on ineasing bounday laye o-odinate y to appoah fee steam value. It is notied fom Figs. 3 and 4 that C deeases on ineasing eithe o S. This implies that hemial eation tends to edue speies onentation wheeas mass diffusion has a evese effet on it. Fig.. Tempeatue pofiles when Fig.. Tempeatue pofiles when

10 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. Table 4 Shea stess at the plate whenfig. 3. Conentation pofiles when a t..4 Ramped Tempeatue Isothemal plate Fig. 4. Conentation pofiles when The numeial values of shea stess at the plate, omputed fom analytial expessions () and (3), ae pesented in tabula fom in Tables 3 to 6 fo vaious values of S, t, a, t, N,, and G taking M 4, G and P.7 (ionized ai). It is analyzed fom Table 5 that, fo amped tempeatue plate, ineases on ineasing eithe N o wheeas it deeases on ineasing fo isothemal plate. This implies that adiation and hemial eation tend to enhane the shea stess at amped tempeatue plate wheeas it have evese effet on the shea stess at isothemal plate. It is evealed fom Table 6 that deeases on ineasing eithe o G fo amped tempeatue plate wheeas it ineases on ineasing eithe o G fo isothemal plate. This implies that pemeability of poous medium and solutal buoyany foe tend to edue the shea stess at amped tempeatue plate wheeas these agenies have evese effet on the shea stess at isothemal plate. Table 5 Shea stess at the plate whentable 3 Shea stess at the plate whenn.5.5 S t Ramped Tempeatue Ramped Tempeatue Isothemal plate It is obseved fom Table 3 that shea stess at the plate ineases on ineasing eithe S o t fo amped tempeatue plate. This implies that mass difusion tends to edue shea stess at amped tempeatue plate. An inease in itial time fo ampedness implies that thee is an enhanement in shea stess at amped tempeatue plate. It is notied fom Table 4 that ineases on ineasing eithe a o t fo amped tempeatue plate wheeas it deeases on ineasing eithe a o t fo isothemal plate. An inease in sufae aeleation paamete and time thee is an enhanement in shea stess at amped tempeatue plate wheeas thee is a edution in the shea stess at isothemal plate. Table 6 Shea stess at the plate when G Ramped Tempeatue Isothemal plate The numeial values of ate of heat tansfe at the

11 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. plate i.e. N u, omputed fom the analytial expessions () and (4), ae pesented in tabula fom in Tables 7 and 8 fo vaious values of NtP,, and t wheeas those of ate of mass tansfe at the plate i.e. S h, omputed fom the analytial expession (5), is pesented in tabula fom in Table 9 fo diffeent values of and S taking t.. It is notied fom Table 7 that, fo both amped tempeatue and isothemal plates, N u ineases on ineasing N. Nu ineases on ineasing t fo amped tempeatue plate wheeas it deeases on ineasing t fo isothemal plate. This implies that adiation tends to enhane the ate of heat tansfe at both the amped tempeatue and isothemal plates. As time pogesses, ate of heat tansfe is getting enhaned at amped tempeatue plate wheeas it is getting edue at isothemal plate. Table 7 Rate of heat tansfe N u at the plate when N t Ramped Tempeatue Isothemal plate N ua at the plate Table 8 Rate of heat tansfe when P t Table 9 Rate of mass tansfe at the plate. S It is inteesting to note fom Table 7 that ate of heat tansfe at isothemal plate attains steady state when t.5. It is evident fom Table 8 that, fo amped tempeatue plate, N u ineases on ineasing P wheeas it deeases on ineasing t. Sine P is a measue of the elative stength of momentum diffusivity to themal diffusivity of fluid, P deeases on ineasing themal diffusivity of fluid. This implies that themal diffusivity tends to edue ate of heat tansfe at amped tempeatue plate. An inease in itial time fo ampedness thee is a edution in ate of heat tansfe at amped tempeatue plate. It is evealed fom Table 9 that S h ineases on ineasing eithe o S. This implies that hemial eation tends to enhane ate of mass tansfe at the plate wheeas mass diffusion has a evese effet on it. 5. CONCLUSIONS Investigation of unsteady hydomagneti natual onvetion flow with heat and mass tansfe of a visous, inompessible, eletially onduting, hemially eative and optially thin adiating fluid past an exponentially aeleated moving vetial plate with vaiable amped tempeatue embedded in a fluid satuated poous medium is aied out. Signifiant finding ae as follows: Citial time fo ampedness tends to edue fluid fluid flow as well as fluid tempeatue in bounday laye egion. Fo both amped tempeatue and isothemal plates: Sufae aeleation paamete, pemeability of poous medium, mass diffusion, solutal buoyany foe and time tend to aeleate fluid flow in bounday laye egion wheeas adiation and hemial eation have evese effet on it. Radiation tends to edue fluid tempeatue and fluid tempeatue is getting enhaned with pogess of time. It appoahes steady state when t.5 fo isothemal plate. Chemial eation tends to edue speies onentation wheeas mass diffusion has a evese effet on it. We have ompaed numeial values of shea stess at the plate of Das et al. () as a speial ase of ou esult whih ae in exellent ageement with the numeial values of Das et al. (). Citial time fo ampedness tends to enhane shea stess at the plate wheeas mass diffusion has a evese effet on it fo amped tempeatue plate. Chemial eation tends to enhane ate of mass tansfe at the plate wheeas mass diffusion has a evese effet on it. REFERENCES Afify, A. A. (4). The effet of adiation on fee onvetive flow and mass tansfe past a vetial isothemal one sufae with hemial eation in the pesene of a tansvese 3

12 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. magneti field. Canad. J. Phys. 8, Ahmed, N. and.. Das (3a). Unsteady MHD mass tansfe flow past a suddenly moving vetial plate in a poous medium in otating system. Int. J. Eng. Si. Teh. 5, Ahmed, N. and.. Das (3b). Hall effets on tansient mhd flow past an impulsively stated vetial plate in a poous medium with amped tempeatue, otation and heat absoption. Appl. Math. Si. 7, Ahmed, N. and M. Dutta (4). Analytial analysis of magnetohydodynami (MHD) tansient flow past a suddenly stated infinite vetial plate with themal adiation and amped wall tempeatue, Asme J. Heat Tansfe 36 Bak, A. A. (). Effets of hemial eation on MHD fee onvetion and mass tansfe flow of a miopola fluid with osillatoy plate veloity and onstant heat soue in a otating fame of efeene. Commun. Nonlinea Si. Nume. Simulation 6, Bhattahayya,. and G. C. Layek (). Similaity solution of MHD bounday laye flow with diffusion and hemial eation ove a poous flat plate with sution/blowing. Meania 47, Chamkha, A. J. and A. R. A. haled (). Hydomagneti ombined heat and mass tansfe by natual onvetion fom a pemeable sufae embedded in a fluidsatuated poous medium. Int. J. Nume. Methods Heat Fluid Flow, Chamkha, A. J. (997). Tansient MHD fee onvetion fom a poous medium suppoted by a sufae. Fluid/Patile Sepaation J.,, -7. Chamkha, A. J. (). Themal adiation and buoyany effets on hydomagneti flow ove an aeleating pemeable sufae with heat soue o sink, Int. J. Eng. Si. 38, Chamkha, A. J. (3). MHD flow of a unifomly stethed vetial pemeable sufae in the pesene of heat geneation/absoption and a hemial eation. Int. Comm. Heat Mass Tansf. 3, Chamkha, A. J., R. A. Mohamed and S. E. Ahmed (). Unsteady MHD natual onvetion fom a heated vetial poous plate in a miopola fluid with Joule heating, hemial eation and adiation effets. Meania 46, Chaudhay, R. C. and A. Jain (7). Combined heat and mass tansfe effets on MHD fee onvetion flow past an osillating plate embedded in poous medium. Rom. J. Phys. 5, Chen C. H. (4). Heat and mass tansfe in MHD flow by natual onvetion fom a pemeable, inlined sufae with vaiable wall tempeatue and onentation. Ata Mehania 7, Came,. R. and S. I. Pai (973). Magnetofluid dynamis fo enginees and applied physiists. MGaw Hill Book Company, New Yok. Das,. (). Magnetohydodynamis fee onvetion flow of a adiating and hemially eating fluid past an impulsively moving plate with amped wall tempeatue. J. Appl. Meh., 79, 67. Das,., S. Jana and P.. undu (4). Unsteady MHD fee onvetion flow nea a moving vetial plate with amped wall tempeatue. Int. J. Fluid Meh. Res. 4, 7-9. Das, S., Jana, M. and R. N. Jana (). Radiation effet on natual onvetion nea a vetial plate embedded in poous medium with amped wall tempeatue. Open J. Fluid Dynamis, -. Elbashbeshy, E. M. A. (997). Heat and mass tansfe along a vetial plate with vaiable sufae tension and onentation in the pesene of the magneti field. Int. J. Eng. Si. 35, Eldabe, N. T. M., E. M. A. Elbashbeshy, W. S. A. Hasanin and E. M. Elsaid (). Unsteady motion of MHD visous inompessible fluid with heat and mass tansfe though poous medium nea a moving vetial plate. Int. J. Enegy Teh. 3,. Gupta, A. S. (96). Steady and tansient fee onvetion of an eletially onduting fluid fom a vetial plate in the pesene of a magneti field. Appl. Sientifi Res. 9, Hayday, A. A., D. A. Bowlus and R. A. MGaw (967). Fee onvetion fom a vetial plate with step disontinuities in sufae tempeatue. ASME J. Heat Tansf. 89,44 5. Helmy,. A. (998). MHD Unsteady fee onvetion flow past a vetial poous plate. J. Appl. Math. Meh. (ZAMM) 78, Hossain, M. A. and A. C. Mandal (985). Mass tansfe effets on the unsteady hydomagneti fee onvetion flow past an aeleated vetial poous plate. J Phys. D: Appl. Phys. 8, Ibahim, F. S., A. M. Elaiw and A. A. Bak (8). Effet of the hemial eation and adiation absoption on the unsteady MHD fee onvetion flow past a semi-infinite vetial pemeable moving plate with heat soue and 4

13 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. sution. Comm. Nonlinea Si. Nume. Simulation 3, Ibahim, F. S., I. A. Hassanien and A. A. Bak (4). Unsteady magnetohydodynami mio-pola fluid flow and heat tansfe ove a vetial poous plate though a poous medium in the pesene of themal and mass diffusion with a onstant heat soue. Canadian J. Phys. 8, Jha, B.. (99). MHD fee onvetion and mass tansfom flow though a poous medium. Astophys. Spae Si. 75, ao, T. T. (975). Lamina fee onvetive heat tansfe esponse along a vetial flat plate with step jump in sufae tempeatue. Lett. Heat Mass Tansf., ellehe, M. (97). Fee onvetion fom a vetial plate with disontinuous wall tempeatue. ASME J. Heat Tansf. 93, im, Y. J. (). Unsteady MHD onvetive heat tansfe past a semi-infinite vetial poous moving plate with vaiable sution. Int. J. Eng. Si. 38, ishoe, P. M., N. V. R. V. P. S. Rao, V.. Vama and S. Venkataamana (3). The effets of adiation and hemial eation on unsteady MHD fee onvetion flow of visous fluid past an exponentially aeleated vetial plate. Int. J. Math. Si. 4, undu, P..,. Das and N. Ahaya (4). Flow featues of a onduting fluid nea an aeleated vetial plate in poous medium with amped wall tempeatue, J. Meh. 3, Lee, S. and M. M. Yovanovih (99). Lamina natual onvetion fom a vetial plate with a step hange in wall tempeatue. ASME J. Heat Tansf. 3, Makinde, O. D. and M. S. Tshehla (4). Unsteady hydomagneti flow of adiating fluid past a onvetively heated vetial plate with the navie slip. Adv. Math. Phys., Atile ID Makinde, O. D. and P. Sibanda (8). Magnetohydodynami mixed onvetive flow and heat and mass tansfe past a vetial plate in a poous medium with onstant wall sution. Asme J. Heat Tansf, 3. 8 Meye, R. C. (958). On eduing aeodynami heat-tansfe ates by magnetohydodynami tehniques. J. Aeo Spae Si. 5, Mohamed, R. A., A. N. A. Osman and S. M. A. Dahab (3). Unsteady MHD doublediffusive onvetion bounday laye flow past a adiate hot vetial sufae in poous media in the pesene of hemial eation and heat sink, Meania 48, Nandkeolya, R. and M. Das (3). Unsteady MHD fee onvetion flow of a heat absobing dusty fluid past a flat plate with amped wall tempeatue. Afika Matmateika, 5, Nandkeolya, R. and M. Das (4). MHD fee onvetive adiative flow past a flat plate with amped tempeatue in the pesene of an inlined magneti field. Comp. Appl. Math, 34, 9-3. Nandkeolya, R., G. S. Seth, O. D. Makinde, P. Sibanda and M. d. S.Ansai (3a). Unsteady hydomagneti natual onvetion flow of a dusty fluid past an impulsively moving vetial plate with amped tempeatue in the pesene of themal adiation. J. Appl. Meh. 8, 63. Nandkeolya, R., P. Sibanda and M. d. S. Ansai (3b). Unsteady hydomagneti adiative flow of a dusty fluid past a poous plate with amped wall tempeatue. Int. Meh. Eng. Congess Exposition, DOI: /5/IMECE Naahai, M. and S. A. Sulaiman (3). Themal adiation effets on unsteady MHD natual onvetion flow past an infinite inlined plate with amped tempeatue. Adv. Si. Lett. 9, Oepe, G. M. and J. Szekely (983). The effet of an extenally imposed magneti field on buoyany diven flow in a etangula avity. J. Cystal Gowth 64, Pop, I. (969). On the unsteady hydomagneti fee onvetion flow past a vetial infinite flat plate. Indian J. Phys. 43, 96. Pakash, J., D. Bhanumathi, A. G. V. uma and S. V.. Vema (3). Diffusion-themo and adiation effets on unsteady MHD flow though poous medium past an impulsively stated infinite vetial plate with vaiable tempeatue and mass diffusion. Tansp. Poous Media 96, Pasad, V. R. and N. B. Reddy (7). Radiation and mass tansfe effets on an unsteady MHD fee onvetion flow past a heated vetial plate in a poous medium with visous dissipation. Theo. Appl. Meh. 34, Rajesh, V. (). Radiation effets on MHD fee onvetive flow nea a vetial plate with amped wall tempeatue. Int. J. Appl. Math. Meh. 6, Rajesh, V. and S. V.. Vama (9). Radiation and mass tansfe effets on MHD fee onvetion flow past an exponentially 5

14 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. aeleated vetial plate with vaiable tempeatue. ARPN J. Eng. Appl. Si. 4, -6. Raptis, A. (). Fee onvetive osillatoy flow and mass tansfe past a poous plate in the pesene of adiation fo an optially thin fluid. Themal Si. 5, Raptis, A., C. Pedikis and A. Leontitsis (3). Effets of adiation in an optially thin gay gas flowing past a vetial infinite plate in the pesene of a magneti field. Heat Mass Tansf. 39, Samiulhaq, I. han, F. Ali and S. Shafie (). MHD fee onvetion flow in a poous medium with themal diffusion and amped wall tempeatue. J. Phys. So. Japan 8, 444, 9 Pages. Sangapatnam, S., N. B. Reddy and R. P Vallampati (9). Radiation and mass tansfe effets on mhd fee onvetion flow past an impulsively stated isothemal vetial plate with dissipation. Themal Si. 3, 7-8. Seth G. S., R. Shama and S. Saka (5a). Natual onvetion heat and mass tansfe flow with Hall uent, otation, adiation and heat absoption past an aeleated moving vetial plate with amped tempeatue. J. Appl. Fluid Meh. 8, 7-. Seth, G. S., R. Nandkeolya and M. d. S. Ansai (3). Effets of themal adiation and otation on unsteady hydomagneti fee onvetion flow past an impulsively moving vetial plate with amped tempeatue in a poous medium. J. Appl. Fluid Meh. 6, Seth, G. S., S. Saka and S. M. Hussain (4). Effets of Hall uent, adiation and otation on natual onvetion heat and mass tansfe flow past a moving vetial plate. Ain Shams Eng. J., 5, Seth, G. S., S. Saka, S. M. Hussain and G.. Mahato (5b). Effets of Hall uent and otation on hydomagneti natual onvetion flow with heat and mass tansfe of a heat absobing fluid past an impulsively moving vetial plate with amped tempeatue. J.Appl. Fluid Meh. 8, Seth, G. S., S. M. Hussain and S. Saka (4a). Hydomagneti natual onvetion flow with adiative heat tansfe past an aeleated moving vetial plate with amped tempeatue though a poous medium. J. Poous Media 7, Appendix 3 f,, 3, 4e ef e ef f 4,, 3, 4, 5 e ef e ef f 3,, 3, 4, 5 e ef e ef f4,, 3, 4,5 5 e ef ef e f5,, 3 3 e ef 3 3 e ef 3 3 e f6,, 3 ef f,, ef ef f,,, ef ef f9,, 3, ef e f,, 3, 4,5 e 3 4 ef 45 t 6

15 G. S. Seth et al. / JAFM, Vol. 9, No., pp. 3-7, 6. ef 5 ef 5 5 e ef 3 f,, e 3 e 5 5 f,, 3, 4,5 e ef ef 4 5 7