Asian Journal of Current Engineering and Maths 2: 2 March April (2013) Contents lists available at

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1 Asian Jounal of Cuent Engineeing and Maths : Mah Ail (1) Contents lists available at ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Jounal homeage: htt:// MASS TRANSFER AND RADIATION EFFECT ON MHD FLOW PAST AN IMPULSIVELY STARTED EXPONENTIALLY ACCELERATED INCLINED POROUS PLATE WITH VARIABLE TEMPERATURE IN THE PRESENCE OF HEAT SOURCE AND CHEMICAL REACTION R. N. Baik Deatment of Mathematis, Tident Aademy of Tehnology, Infoity, Bhubaneswa-7514, Odisha, India. ARTICLE INFO ABSTRACT Coesonding Autho R. N. Baik Deatment of Mathematis, Tident Aademy of Tehnology, Infoity, Bhubaneswa-7514, Odisha, India, baik.abinaayan@ediffmail.om Key Wods: MHD flow, themal adiation, heat soue, hemial eation, mass tansfe. INTRODUCTION Study of MHD flow with heat and mass tansfe lay an imotant ole in vaious industial aliations. Some imotant aliations ae ooling of nulea eatos, liquid metals fluid, owe geneation system and aeo dynamis. It is of imotane in onnetion with many engineeing oblems, suh as sustained lasma onfinement fo ontolled themo nulea fusion and eletomagneti asting of metals.mhd finds aliations in eletomagneti ums, ystals gowing, MHD oules and beaing, lasma jets and hemial synthesis. Radiative heat and mass tansfe lay an imotant ole in manufatuing industies fo the design eliable equiment. Nulea owe lants, gas tubines and vaious oulsion devies fo aiaft, missiles, satellites and sae vehiles ae examles of suh engineeing aliations. Hydomagneti inomessible visous flow has many imotant engineeing aliations suh as magnetohydodynami owe geneatos and the ooling of eatos also its aliations to oblems in geohysis, astohysis et. On the othe hand, the study of heat geneation o absotion in moving fluids is imotant in oblems dealing with hemial eations dissoiating fluids. Sine some fluids an emit o absob themal adiation, it is of inteest to study the effets of magneti field on the temeatue distibution and heat tansfe when the fluid is not only an eletial onduto but also when it is aable of emitting and absobing adiation, Hene, heat tansfe by themal adiation is beoming of geate imotane in sae aliations and highe oeating temeatues. In seveal oblems elated to demanding of effiient tansfe of mass ove inlined beds elated to An attemt is made to investigate the mass tansfe and adiation effets on unsteady MHD flow of an eletially onduting adiating, visous inomessible fluid ast an imulsively stated moving exonentially aeleated inlined oous late with vaiable temeatue in the esene of heat soue and hemial eation. The fluid is onsideed is gay, absobing/emitting adiation but a non-satteing medium. The dimensionless govening equations involved in the esent analysis ae solved using the Lalae tansfom tehnique. The veloity, temeatue, onentation and skin fition ae studied gahially fo diffeent hysial aametes. 16 1, AJCEM, All Right Reseved. geohysial, etoleum, hemial, bio-mehanial, hemial tehnology and in situations the visous dainage ove an inlined oous lane is a subjet of onsideable inteest to both theoetial and exeimental investigatos. Eseially, in the flow of oil though oous ok, the extation of geo-themal enegy fom the dee inteio of the eath to the shallow layes, the evaluation of the aability of heat emoval fom atiulate nulea fuel debis that may esult fom hyothetial aident in a nulea eato, the filtation of solids fom liquids, flow of liquids though ion exhange beds, dug emeation though human glands, hemial eato fo eonomial seaation o uifiation of mixtues flow though oous medium has been the subjet of onsideable eseah ativity in eent yeas due to its notable aliations. An imotant aliation in the etoleum industy whee ude oil is taed fom natual undegound esevois in whih oil is entaed sine the flow behavio of fluids in etoleum esevoi ok deends to a lage extent on the oeties of the ok, tehniques that yield new o additional infomation on the haateistis of the ok would enhane the efomane of etoleum esevois. An imotant bio medial aliation is the flow of fluids in lungs, blood vessels, ateies and so on, whee the fluid is bounded by two layes whih ae held togethe by a set of faily egulaly saed tissues. Sluy adhees to the eato vessel and gets onsolidated in many hemial oessing industies, as a esult of whih hemial omounds within the eato vessels eolates though the boundaies. Thus adheed substane within the eato vessel ats as a oous bounday. The oblem assumes geate imotane in all suh situations. The thin film adheing to the sufaes of the ontaine must be taken into

2 aount fo the uose of eise hemial alulations in all suh situations wheein hea and mass tansfe ous. Failue to do so leads to sevee exeimental eos. Hene, thee is a need fo suh an analysis in detail. A mathematial model elated to suh a situation has been studied in detail. Kuma and Vema [1] studied the adiation effets on MHD flow ast an imulsively stated exonentially aeleated vetial late with vaiable temeatue in the esene of heat geneation. They [] have also studied the themal adiation and mass tansfe effets on MHD flow ast a vetial osillating late with vaiable temeatue and vaiable mass diffusion. The foed onvetion flow in a hoizontal hannel emeated by unifom vetial magneti fluid in the esene of adiation was studied by Viskanta []. Takha et al. [4] investigated the effets of adiation on the MHD fee onvetion flow ast a semiinfinite vetial late. The themal diffusion effets on moving infinite vetial late in the esene of vaiable temeatue and mass diffusion wee studied by Muthuumaaswamy and Kuma [5]. England and Emey [6] have studied the themal adiation effets of an otially thin gay gas bounded by a stationay vetial late. Radiation effets on mixed onvetion along isothemal vetial late wee studied by Hossain and Takha [7]. Das et al. [8] have analyzed adiation effets on flow ast an imulsively stated infinite isothemal vetial late. Sobha and Ramakishna [9] esented the effets of magneti field on heat tansfe haateistis in oous medium unde natual onvetion. Effets of the hemial time, eation and adiation absotion on fee onvetion flow though oous medium with vaiable sution in the esene of unifom magneti field wee studied by Sudhee Babu and Satyanaayana [1]. Dulal Pal et al. [11] studied etubation analysis of unsteady magnetohydodynami onvetive heat and mass tansfe in a bounday laye sli flow ast a vetial emeable late with themal adiation and hemial eation. Reently, Ramana Reddy et al. [1] have studied the mass tansfe and adiation effets of unsteady MHD fee onvetive fluid flow embedded in oous medium with heat geneation/absotion. Radiation effets on MHD fee onvetion flow ove a vetial late with heat and mass flux was studied by Sivaiah et al. [1]. Reently, adiation u u Bu g T T os g CC os u t y K T Tq C k S T T tyy C C D K CC t y With bounday onditions t : u, T T, C C fo all y t : u u ex( at), T T TT At, C C C C At at y and u, TT, C C as y w w The loal adiant fo the ase of an otially thin gay gas is exessed by q 4 4 4a T T y and mass tansfe effets on MHD fee onvetion flow though oous medium ast an exonentially aeleated vetial late with vaiable temeatue has been studied by Pattnaik et al. [14]. Reddy et al. [15] have studied the adiation and hemial eation effets on MHD heat and mass tansfe flow inlined oous heated late. The objetive of the esent study is to analyze the effets of themal adiation and hemial eation of an unsteady MHD flow ast an imulsively stated moving exonentially aeleated inlined oous late with vaiable temeatue in the esene of heat geneation unde the ation of a onstant magneti field of onstant essue gadient is subjeted to an extenal magneti field of onstant stength in the dietion to the late and to the dietion to the flow. We have inooated the mass tansfe aset with hemial eation to the wok of Kuma and Vama [1]. 1. Mathematial Fomulation: Themal adiation and mass tansfe effets on unsteady MHD flow of a visous inomessible fluid ast along a vetial osillating late with vaiable temeatue and also with vaiable mass diffusion in the esene of heat soue and hemial eation has been studied. The x axis is taken along the late in the vetial uwad dietion and y axis is taken nomal to the late in the dietion of the alied magneti field. Initially it is assumed that the late and fluid ae at the same temeatue T in the stationay ondition with onentation level C at all the oints. At t the late is given an osillatoy motion in its own lane with veloity u ex a t. At the same time the late temeatue is aised linealy with time t and also mass is diffused fom the late linealy with time. A tansvese magneti field of unifom stength B is assumed to be alied nomal to the late. The indued magneti field and visous dissiation is assumed to be negligible as the magneti Reynolds numbe of the flow is taken to be vey small. The fluid onsideed hee is gay, absobing/emitting adiation but anon-satteing medium. Then by usual Boussinesq s aoximation, the unsteady flow is govened by the following equations. (1) () () (4) (5) 17

3 4 It is assumed that the temeatue diffeenes within the flow ae suffiiently small and that T may be exessed as a 4 linea funtion of the temeatue. This is obtained by exanding T in a Taylo seies about T and negleting the highe ode tems, thus we get 4 4 T4T T T (6) Now, Intoduing the following non-dimensional quantities into equations (1) () u tu yu TT u a C Ku K u, t, y, T, A, a, P,, K K u TT u k u w 16T S CC B R S C M ku C u C C u,,, w g C C g T T G, G, S u u D w w Using the above non-dimensional quantities equations (1) () edue to u u 1 GT os GC os M u t y K T 1 T R S T t P y P C 1 u KC t S y Whee M, K, G, G, P, R, S, S and (7) (8) (9) (1) K denote the magneti aamete, emeability aamete, Gashof numbe, modified Gashof numbe, Pandtl numbe, adiation aamete, heat soue aamete, Shmidt numbe and hemial eation aamete esetively. With the bounday onditions t : u, T, C fo all y t : u ex( at), T t, C t at y and u, T, C at y (11) Using Lalae tansfom we get u, T, C at y s a s s u, T, C as y (1). Solution of the oblem: The solutions of equations (8) (1) subjet to the bounday onditions (1) ae obtained by Lalae tansfom tehnique. Hene, the solutions of equations (8), (9) and (1) satisfying the bounday onditions (1) ae: 1 y SsK C e (1) s 1 y d sp T e (14) s 1 G os G os y s u e s a P 1 s s 1 S 1 s s G os y d sp G os y SsK e e (15) P 1 s s S 1 s s 1 Invese Lalae tansfom of (1), (14) and (15) yields: t y S y SK y S t y S y SK y S C e ef Kt e ef Kt 4 K t 4 K t (16) 18

4 t y P y d y P dt t y P y d y P dt T e ef e ef 4 d t P 4 d t P at e y a y y a y u e ef a t e ef a t t t 1t y 1 y y y 1 e e ef 1 t e ef 1 t t t y y y y e ef t e ef t t t t y y y t y y y 1 e ef t e ef t t t 4 t y y y y e e ef t e ef t t t 4 y y y y e ef t e ef t t t t y y y t y y y 4 e ef t e ef t t t y P d y P d 1 1 t P t P 1t 1 1 e e ef t e ef t (17) y P d y P d y d y P dt y d y P dt e ef e ef t P t P t yp y d y P dt t yp y d y P dt 1 e ef e ef 4 d t P 4 d t P 4 t y S SK y S y S SK y S e e ef Kt e ef Kt t t 4 y SK y S y SK y S e ef Kt e ef Kt t t t y S y SK y S t y S y SK y S 4 e ef Kt e ef Kt 4 K t 4 K t (18) 1 d SK G os G os Whee d R SP, M, 1,,, 4 K P 1 S 1 P 1 S 1 1 Skin fition: Fom veloity field, the skin fition (ate of hange of veloity in flow with eset to y) in non-dimensional fom is given by du dy (19) y Fom equations (18) and (19) we get the skin fition as follows at 1 at 1t 1 1 t e a ef a t e e 1ef 1 t e t t 19

5 1 t 1 t t 4 ef t e 4e ef t e t t t t 1 4 t ef t e ef t d 1 t 1t d P dt P dt P P e d 1P ef 1 t e def e P t P t dt P dt dt tp P 1 ef t def e d P P t S K t S Kt 4e SK Sef K t e 4 SKef Kt e t t 1 S ts Kt 4 ef Kt t SKef Kt e () K Rate of heat tansfe (Nusselt numbe): Fom temeatue field, the Nusselt numbe (ate of hange of heat tansfe in flow with eset to y) in non-dimensional fom is given by dt Nu dy y Fom equations (17) and (1) we get the skin fition as follows dt P dt dt tp P Nu ef t d ef e () d P P Rate of mass tansfe (Shewood numbe): Fom onentation field, the Shewood numbe (ate of hange of mass tansfe in flow with eset to y) in nondimensional fom is given by dc Sh dy () y Fom equations (16) and () we get the skin fition as follows 1 S St Kt Sh ef Kt t SKef Kt e (4) K RESULTS AND DISCUSSION In ode to get the hysial insight into the oblem the veloity, temeatue, onentation, skin fition, the ate of heat tansfe and the ate of mass tansfe ae shown by gahs and tables and some numeial omutations ae also efomed fo diffeent values of of the hysial aametes like Radiation aamete ( R ), Magneti aamete ( M ), Heat soue aamete ( S ), Poosity aamete ( K ), Pandtl numbe P ), Chemial eation aamete ( K ), Gashof numbe ( ( G ), Modified Gashof numbe ( G ), exonential index ( a ). The esent study bings the esult of Kuma and Vema [1] as atiula ase. When K is vey lage but the ase of without hemial eation assoiated with mass tansfe annot be disussed as a atiula ase. Moeove, the ase in the absene of buoyany effet due to onentation diffeene ( G ) an be studied as a atiula ase. (1) Fig.1 exhibits the effets of Pandtl numbe and heat soue aamete on the temeatue field. It is obseved that, an inease in the Pandtl numbe ontibutes to a deease in the temeatue. Futhe, it is obseved that as the heat soue aamete ineases, the temeatue ineases. The effets of adiation aamete and time aamete on the temeatue field ae shown in fig.. It is notied that as adiation aamete ineases the temeatue field deeases but the evese effet is maked in ase of time aamete. It is onluded that an inease in P and R deease the temeatue but with lage time san temeatue ineases. This deease of temeatue may be attibuted to the loss of heat enegy due to adiation as well as low diffusivity. 11

6 Cuves P S R t I.5 4. II III IV V Cuves S K I. II.6 III.78 IV. 5 V. 1 Fig.4 shows the effet of time aamete on onentation field. It is seen that as time san ineases, the onentation field ineases. Cuves P S R t I II III IV V Fig. illustates the effets of Shmidt numbe ( S ) and hemial eation aamete ( K ) on the onentation field. It is notied that as the Shmidt numbe ineases, the onentation of the fluid medium deeases signifiantly in the bounday laye egion and theeafte not muh of vaiation is notied. Futhe, it is seen that as the hemial eation aamete ineases the onentation also deeases. This esult is analogous to temeatue distibution. In ase of mass tansfe, S and K lay the simila ole as P and R in temeatue distibution. The esult is in good ageement with the wok of Pattnaik et al. [14]. Cuves S K t I.. II..4 III..6 Table 1 shows the effet of Pandtl numbe on veloity field. It is notied that as Pandtl numbe ineases, the veloity field ineases. Table & exhibits the effet of heat soue aamete and adiation aamete on veloity field. The inease in S and R deeases the veloity and onsequently ineases. Fom table 4 it is obseved that an inease in magneti aamete ( M ) leads to deease the veloity field. This onlusion meets the logi that the magneti field exets a etading foe on the fee onvetion flow. Futhe, it is notied that as emeability aamete ineases, the veloity ineases. This is due to the ineasing Loentz foe whih edues the veloity with high value of oosity aamete K. Table 5 shows the effets of Gashof numbe and modified Gashof numbe on veloity field. It is seen that as the Gashof numbe ineases the veloity field ineases, but the modified Gashof numbe has no signifiant effet on the veloity field. Table 6 exhibits the effets of time aamete, angle of inlination and exonential index on veloity field. It is obseved that the inease in time aamete ( t ) and exonential index ( a ) ineases the veloity. An inease in the angle of inlination esulted in the deease of the veloity. 111

7 Table 7 shows the effets of Shmidt numbe and hemial eation aamete on the veloity field. It is seen that as Shmidt numbe ineases, the veloity field ineases. Skin fition is a measue of sheaing stess exeiened at the solid sufae. Tables 8, 9 & 1 exhibit the effets of vaious aametes on skin fition. All the enties ae negative and the absolute values of skin fition ineases as the Gashof numbe, oous aamete, Pandtl numbe, adiation aamete, Shmidt numbe, hemial eation aamete, time aamete and exonential index inease. The absolute values of skin fition deease as the magneti aamete, heat soue aamete and the angle of inlination inease. The modied Gashof numbe has no signifiant effet on skin fition. Fom table 11 it is obseved that the absolute values of the ate of heat tansfe ineases as the adiation aamete and time aamete inease. But the absolute values of ate of heat tansfe deeases as the Pandtl numbe and heat soue aamete inease. Fom table 1 it is seen that the absolute values of the ate of mass tansfe ineases fo ineasing values of all the aametes i.e Shmidt numbe, hemial eation aamete and time aamete. P when Table 1 Veloity fo diffeent values of M 1, K 1, S., S, R 5, K, t,, G, G, a.5 y P.5 P.71 P Table 5 Veloity fo diffeent values of Table Veloity fo diffeent values of S when M 1, K 1, S., P.71, R 5, K, t,, G, G, a.5 y S S 4 S Table Veloity fo diffeent values of R when M 1, K 1, S., P.71, S, K, t,, G, G, a.5 y R R 5 R K when Table 4 Veloity fo diffeent values of M and R 5, S., P.71, S, K, t, G, G, a.5 y M, K M 1, K M 1, K G and G when R 5, S., P.71, S, K, t, M 1, K 1, a.5 y G, G G 4, G G 1, G G, G 5 G, G Table 6 Veloity fo diffeent values of, y t and a when R 5, S., P.71, S, K, M 1, K 1, G, G t.,, a.5t.4,, a.5 t.6,, a.5 t.,, a.5 t.,, a Table 7 Veloity fo diffeent values of S and K when R 5, P.71, S, M 1, K 1, G, G, a.5, t t.4,, a.7 t.,, a

8 S and K when R 5, P.71, S, M 1, K 1, G, G, a.5, t y S., K S.6, K S.78, K S., K 4 S., K 7 Table 7 Veloity fo diffeent values of SKIN FRICTION Table 8 Effet of G, G, M, K on skin fition with P.71, S., S, R 5, K, t 1,, a.5. Sl. No G G M K Table 9 Effet of P, S, R on skin fition with G, G, M 1, K 1, K, S., t 1,, a.5. Sl.no P S R Table 1 Effet of S, K, t,, a on skin fition with G, G, M 1, K 1, P.71, S, R 5. S Sl. No 1.. i/ i/ i/ i/ i/ i/ i/ i/ i/ i/ Table 11 Rate of heat tansfe K t a P S R t Nu Sl. No Table 1 Rate of mass tansfe S K t Sh Sl. No CONCLUSION (i) The deease of temeatue may be attibuted to the loss of heat enegy due to adiation as well as low diffusion (ii) The temeatue deeases as the adiation aamete ineases but it ineases with inease of heat soue aamete o time aamete. (iii) The absolute values of skin fition ineases with ineasing of oosity aamete, adiation aamete, Shmidt numbe, hemial eation aamete, time aamete but deeases with inease of magneti aamete, heat soue aamete and the angle of inlination inease. (iv) The Nusselt numbe inease with ineasing values of adiation aamete and time aamete but deeases with ineases heat soue aamete. (v) The Shewood numbe ineases with ineasing of Shmidt numbe o hemial eation aamete o time aamete. (vi) The veloity field ineases with an inease of eithe Pandtl numbe o oosity aamete o time aamete o exonential index but it deeases with an inease of heat soue aamete o adiation aamete. (vii) The magneti field exets a etading foe on the fee onvetion flow. REFERENCES 1. Vijaya Kuma, A.G. and Vama, S.V.K., The adiation effets on MHD flow ast an imulsively stated exonentially aeleated vetial late with vaiable temeatue in the esene of heat geneation, Intenational Jounal of Engineeing Siene and Tehnology, vol.. No (11).. Vijaya Kuma, A.G. and Vama, S., The themal adiation and mass tansfe effets on MHD flow ast a vetial osillating late with vaiable temeatue and vaiable mass diffusion, Annals of Faulty Engineeing Hunedoaa- Intenational Jounal Engineeing, Tome IX. Fasiule. ISSN P (11).. Viskanta, R., The foed onvetion flow in a hoizontal hannel emeated by unifom vetial magneti fluid in the esene of adiation, ZAMP, vol (196). 4. Takha, H.S., Goia, R.S.R. and Soundalgeka, V.M., The effets of adiation on the MHD fee onvetion flow ast a semi-infinite vetial late. Int. J. Numeial methods Heat Fluid Flow, vol (1996). 5. Muthuumaaswamy, R. and Senthil Kuma G., The themal diffusion effets on moving infinite vetial late in the esene of vaiable temeatue and mass diffusion, Theoet. Al. Math. vol. 1 (1). P (4). 6. England, W.G. and Emey, A.F., The themal adiation effets of an otially thin gay gas bounded by a

9 stationay vetial late, Jounal of Heat tansfe, vol. 91. P.7-44 (1969). 7. Hossain, M.A. and Takha, H.S., Radiation effets on mixed onvetion along isothemal vetial late, Heat and Mass Tansfe, vol. 1. P (1996). 8. Das, U.N., Deka, R.K., and Soundalgeka, V.M., The adiation effets on flow ast an imulsively stated infinite isothemal vetial late, J. Theo. Meh.Vol (1996). 9. Sobha,V.V and Rama Kishna, K., Convetive heat tansfe ast a vetial sufae embedded in oous media with an alied magneti field. Jnl. of the Instt. of Enginees (India). 84, P 1-14 (). 1. Sudhee Babu, M. and Satya Naayana, P.V., Effets of the hemial eation and adiation absotion on fee onvetion flow though oous medium with vaiable sution in the esene of unifom magneti field, J.P. Jounal of Heat and mass tansfe,.19-4 (9). 11. Dulal Pal and Babulal Talukda., Petubation analysis of unsteady magneto hydo dynami onvetive heat and mass tansfe in a bounday laye sli flow ast a vetial emeable late with themal adiation and hemial eation. Communiations in Nonlinea Siene and Numeial Simulation (1). 1. Ramana Reddy, G.V., Ramana Muthy, Ch. V. and Bhaska Reddy, N., Mass tansfe and adiation effets of unsteady MHD fee onvetive fluid flow embedded in oous medium with heat geneation/absotion, Jounal of Alied Mathematis and Fluid Mehanis, vol., No.1, (1). 1. Sivaiah, M.,Nagaajan, A.S. and Reddy, P.S., Radiation effets on MHD fee onvetion flow ove a vetial late with heat and mass flux, Emiates Jounal fo Engg. Reseah, vol.15 (1),. 5-4 (1). 14. Pattnaik, J.R., Dash, G.C. and Singh, S., Radiation and mass tansfe effets on MHD fee onvetion flow though oous medium ast an exonentially aeleated vetial late with vaiable temeatue, Annals of Faulty Engineeing Hunedoaa- Intenational Jounal Engineeing, Tome X. Fasiule. ISSN P (1). 15. Reddy, S., Reddy, G.V.R. and Reddy, K.J., the adiation and hemial eation effets on MHD heat and mass tansfe flow inlined oous heated late, Asian Jounal of Cuent Engineeing and Maths 1:, (1). 114