*B. Shankar Goud 1, M.N Rajashekar 2

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1 IOSR Jorna of Mathematcs (IOSR-JM) e-iss: , p-iss: X. Vome, Isse 6 Ver. IV(ov. - Dec. 6), PP Fnte Eement Method Appcaton of Effects on an Unsteady MHD Convectve Heat and Mass Transfer Fow In a Sem- Infnte Vertca Movng In a Poros Medm wth Heat Sorce and Scton B. Shanar God, M. Raashear Department of Mathematcs, JTUH Coege of Engneerng Katpay, Hyderabad- 8, TS, Inda. Department of Mathematcs, JTUH Coege of Engneerng achpay, Karmnagar -555,TS, Inda. Abstract: The obectve of ths paper s to anayze the effects of Soret, chemca reacton and radaton absorpton on an nsteady MHD convectve heat and mass transfer fow of a sem-nfnte vertca movng n a poros medm wth heat sorce and scton. The pate s assmed to move wth a constant veocty n the fd fow drecton. The eqatons of contnty, momentm, energy and dffson, whch govern the fow fed, are soved by sng Gaeran fnte eement method. merca sotons of veocty, temperatre and concentraton profes are dscssed wth the hep of the graphs for dfferent parameters. Keywords: Vertca pate, MHD, Radaton, FEM, Chemca reacton, sset nmber, Soret nmber. I. Introdcton The stdy of magnetohydrodynamcs (MHD) fow has receved a dea of research nterest de to ts mportance n many engneerng appcatons sch as heat- treated materas traveng between a feed ro or materas manfactred by gass and paper prodcton. Varos aspects of the probem have been nvestgated by many athors. Dash and Das [] anayzed the effect of Ha crrent MHD free convecton fow aong an acceerated poros heated pate wth mass transfer and nterna heat generaton. Sattar [] dscssed free convecton and mass transfer fow throgh a poros medm past an nfnte vertca pate wth tme dependent temperatre and concentraton. Acharya et.a [3] stded the effect of chemca and therma dffson wth Ha crrent on nsteady hydro magnetc fow near an nfnte vertca poros pate. Das et.a [4] wored ot nsteady free convecton and mass transfer bondary ayer fow past an acceerated nfnte vertca pate wth scton. B.Vas [5] et.a anayzed radaton effect and mass transfer on transent free convecton fow of a dsspatve fd past sem- nfnte vertca pate wth nform heat and mass fx.a.mythreye and J.P.Promoda [6] presented chemca reacton on nsteady MHD convectve heat and mass transfer past a sem - nfnte vertca permeabe movng pate wth heat absorpton. D.Srnvasacharya and B.Maarna [7] stded Soret and Dfor effects on mxed convecton aong a vertcay away srface n poros medm wth varabe propertes. The stdy of heat generaton or absorpton effects n movng fds s mportant n vew of severa physca probems, sch as fds ndergong exothermc or endothermc chemca reacton. Radaton effects on an nsteady MHD convecton convectve heat and mass transfer fow past a sem - nfnte vertca permeabe movng pate embedded n poros medm was stded by Ramachandra Prasad et.a [8]. F.S.Ibrahm et.a [9] stded the effect of the chemca reacton and radaton absorpton on the nsteady MHD free convcton fow past a sem nfnte vertca permeabe movng pate wth heat sorce and scton. Unsteady MHD free convectve fow past a movng vertca pate wth tme dependent scton chemca reacton n a sp fow regme was stded by K.S Baamrgan et.a []. Md.Abds Sattar [] nvestgated free convecton and mass transfer fow throgh a poros medm past an nfnte vertca poros medm past an nfnte vertca pate tme dependent temperatre and concentraton. Da pa and Bab Taadar [] nvestgated pertrbaton anayss of nsteady MHD convectve heat and mass transfer n a bondary ayer sp fow past a vertca permeabe pate wth therma radaton and chemca reacton. G.S.Seth et.a [3] nvestgated MHD natra convecton fow wth radatve heat transfer past an mpsvey movng pate wth ramped temperatre. Stdy of effects of radaton and magnetc fed on the mxed convecton mcro poar fd fow towards a stagnaton pont on a heated vertca permeabe pate sng fnte eement method was stded by G Swapna et.a [4]. S.Svaah.et.a [5] examned the effects of therma dffson and radaton on nsteady MHD free convecton fow past an nfnte heated vertca pate n a poros medm. Tanvr Ahmed and Md.Mahmad [6] nvestgated a fnte dfference soton of MHD mxed convecton fow wth heat generaton and chemca reacton. M.K Mazmdar and R.K Dea [7] have anayzed MHD fow past an mpsve started nfnte vertca pate n the presence of therma radaton. Mohmed Abd E-Azz [8] presented nsteady mxed convecton heat transfer aong a vertca stretchng srface wth varabe vscosty and vscos dsspaton. DOI:.979/ Page

2 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass Ebashbeshy [9] nvestgated heat and mass transfer aong a vertca pate wth varabe srface temperatre and concentraton n the presence of magnetc fed. Ja Sngh [] stded vscos dsspaton and chemca reacton effects on fow past a stretchng poros srface n a poros medm. II. Mathematca Formaton The x - axs s taen n the vertcay pward drecton aong the sem-nfnte pate and y axs s taen norma to t. Snce the moton s two dmensona and ength of the pate s arge, therefore a the physca varabes are ndependent of x. Let and v be the components of veocty n x and y drectons, respectvey, taen aong and perpendcar to the pate. The governng bondary ayer eqatons are as foows: Contnty eqaton: v y - () Momentm eqaton: p B v g T T g C C t y x K y Energy eqaton: T T T q Q r T T Q C C t y C p Cp y y Mass dffson eqaton: C C C T D K C C D t y r T t y Where g s the gravtatona acceeraton, s the fd densty, and concentraton expanson coeffcents respectvey, - () - (3) - (4) are the therma and K s the Darcy permeabty, B s the magnetc ndcton, C s the correspondng concentraton, C s the specfc heat at constant pressre, D and T s the therma temperatre nsde the therma bondary ayer and s the eectrc condctvty, and moecar dffsvty, p D are the dffson coeffcent K r s the chemca reacton parameter, Q s the dmensona heat absorpton coeffcent, Q s the coeffcent of proportonaty of the radaton. The bondary condtons are: n t n t p, T T Tw T e, C C Cw C e at y n t U U e, T T, C C, as y Where respectvey, p s the veocty of the fd, U s the free stream veocty and Tw and U Cw T - (5) are the temperatre and concentraton of the wa, n are the constants. From eqaton (), t s cear that the scton veocty at the pate s ether a constant and or a fncton of tme. Hence the scton veocty norma to the pate s assmed n the form: n t v v Ae - (6) Where A s a rea constant, and s sma sch that, A and constant, the negatve sgn ndcates that scton s towards the pate. Otsde the bondary ayer, eqaton () gves p du B x dt K U U v s a non-zero postve - (7) DOI:.979/ Page

3 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass By sng the Rosseand dffson approxmaton the radoactve heat fx, q r 4 ' T 3K y s 4 q r s gven by Where ' and K s are the Stefan - Botzmann constant and the Rosseand mean adsorpton coeffcent 4 respectvey. We assme the temperatre dfference wthn the fow s sffcenty sma sch that T may be expressed as a near fncton of temperatre. T 4T T 3T - (9) Usng (8) and (9), n the ast term of eqaton (3) we obtan 3 qr 6 ' T T y 3K s y Introdcng the foowng non- dmensona qanttes, y U Up T T C C y,,, U, U p,, C U U U Tw T Cw C gvtw T g vcw C v vq vq C w C Gr, Gm, Sc, Q, Q 3 3 v v D Cpv v Tw T 3 vn B v K v Krv 4 ' T vc D T Tw T p n, M, K, Kr, R, Pr, So v v v v Ks Cw Cv Where Gr, Gm, Sc, Q, Q, Kr, R, Pr and So are the therma Grashof nmber, Sota Grashof nmber, Schmdt nmber, heat absorpton parameter, absorpton of radaton parameter, Chemca reacton nmber, and therma radaton parameter, Prandt nmber. Wth the hep of non- dmensona qanttes, eqatons (), (3) and (4) becomes nt U A e ( U ) Gr GmC t y t y nt 3 4R A e Q Q C t y Pr 3 y C nt C C A e KrC S t y Sc y y Where M s Magnetc fed parameter, permeabty parameter. K - (8) - () - () - () - (3) - (4) The modfed bondary condtons for t are: nt nt U p, e, C e at y nt U e,, C at y - (5) III. Method of Soton By appyng Gaern fnte eement method for eqaton () over the eement ) (e, y y s: y y T T T T T B R dy y y t - (6) y nt nt Where R ne ( Gr) ( Gm) C U, B Ae, M K DOI:.979/ Page

4 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass Integratng the frst term n eqaton (6) by parts, we obtan T y ( ) y T T T e T T T T y y y t y - (7) y y B R dy egectng the frst term n eqaton (7), the foowng s obtaned. y T T T T T T B R dy y y t y y Let be the near pecewse approxmaton soton over the eement ) where T, and and are the veocty components at We obtan the foowng. th and y y y y, y y y y th nodes of the typca eement ) (e,( y y y ) are the bass fnctons, (e,( y y y ). dy dy B dy y y dy R dy y y y y y y y y Smpfyng above eqaton we get, ( ) ( ) ( ) e e e B R 6 6 Where prme and dot denotes dfferentaton wth respect to y and tme t respectvey. Assembng the eement eqatons for two consectve eements y y y and y y y the foowng s obtaned: B 4 ( ) ( e ) e 6 - (8) R 4 6 ow eqate row correspondng to the node to zero, from eqaton (8) the dfference schemes wth s as foows. B h h 6 4 R 6 4 Appyng Cran choson method to the eqaton (9), we obtan foowng system of eqatons: A A A A A A R - () n n n n n n ( e ) h - (9) DOI:.979/ Page

5 Where Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass A 6r 3Brh A 6r 3Brh 4 A 8 r 4, A 8 r 4 5 A 6r 3Brh A 6r 3Brh 3 6 nt n n R R n e Gr Gm C U ( ( ) ( ) ), From eqatons (3) and (4), the foowng eqatons are obtaned: B B B B B B P - () n n n n n n C C C C C C C C C C C C R - () n n n n n n Where B P 6r 3P Brh PQ B P 6r 3P Brh PQ 4 B 8P 4PQ r B 8P 4PQ r 5 B P 6r 3P Brh PQ B P 6r 3P Brh PQ 3 6 C Sc 6r 3 rbh. Sc ScKr C Sc 6r 3 rbh. Sc ScKr 4 C 8Sc r 4ScKr C 8Sc r 4ScKr C Here, r h Sc 6r 3 rbh. Sc ScKr C Sc 6r 3 rbh. Sc ScKr n y R R PQ C R ScS and h, are mesh sze aong the y drecton and the tme drecton respectvey. Index refers to the space, and n refers to the tme. In the Eqatons () (), tang,..., n condtons (5), the foowng system of eqatons s obtaned: and sng bondary A X B,,... n - (3) Where A s are matrx of order n and X, B s comn matrces havng n components. The sotons of above systems of eqatons are obtaned by sng the Thomas agorthm for veocty, temperatre and concentraton. Aso the nmerca sotons are obtaned by exectng the C-program wth the smaer vaes of h and. o sgnfcant change was observed n, and C, then the Gaern fnte eement method s stabe and convergent. The sn frcton, sset nmber and Sherwood nmber are mportant physca parameters for ths type of bondary ater fow. Knowng the veocty, temperatre and concentraton feds, the Sn-frcton at the pate, ths n the nondmensona form s gven by y y The rate of heat transfer coeffcent can be obtaned, whch n the non- dmensona form n terms of the sset nmber s gven by T y y The rate of mass transfer coeffcent can be obtaned, whch n the non- dmensona form n terms of the Sherwood nmber s gven by S b C y y IV. Rests and Dscsson merca rests are compted for varos physca parameters whch are presented wth the hep of graphs. The rests obtaned are sed to strate the nfence of the therma Grashof nmber( Gr ), Sota Grashof nmber( Gm ), Magnetc fed parameter( M ), Schmdt nmber( Sc ),Chemca reacton parameter( Kr ), heat absorpton parameter( Q ), absorptons of radaton parameter( Q ), therma radaton parameter ( R ) and Soret nmber( So ), Prandt nmber( Pr ). The effects of the above parameters on veocty, temperatre and concentraton profes are presented graphcay throgh fgres -9. DOI:.979/ Page

6 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass The veocty profes are potted n fgre for varos vaes of M, t s cear from the fgre that the exstence of the magnetc fed s to decrease the veocty n the momentm bondary ayer becase the appcaton of the transverse magnetc fed s a resstng type of force nown as Lorentz force, whch acts aganst fow, f the magnetc fed s apped n the drecton norma to the fow. Fgre shows that the veocty decreases wth an ncrease n Kr. It s cear from the fgre 3 that the bondary ayer thcness w decreases wth an ncrease n the heat absorpton parameter Q. Fgres 4 and show that an ncrease n the Prandt nmber gve rase to a decrease n the veocty as we as the temperatre, the reason s that smaer vaes of Pr are eqvaent to ncreasng the therma condctvtes, and therefore heat s abe to dffse away from the heated pate more rapdy than for hgher vaes of Pr. Fgre 5 shows the effect of Gr on veocty, t s observed that an ncrease n Gr eads to an ncrease n the vaes of veocty de to enhancement n the boyancy force. Here, the postve vaes of Gr correspond to coong of the pate. Fgre 6 depcts veocty profe n the bondary ayer for the varos vaes of Sota Grashof nmber Gm, the fd veocty ncreases and pea vae s more dstnctve de to an ncrease n the speces boyancy force. The veocty dstrbton attans a dstnctve maxmm vae n the regon of the pate and then decreases propery to approach a free stream vae. We observe from the fgre 7 that at very ow vaes of the Schmdt nmber (e.g. Sc =), there s an ncrease n the pea veocty near the pate, where as for hgher vaes of Schmdt nmber (e.g. Sc =4), the pea shfts coser to the pate. Fgre 8 shows the veocty profe for dfferent vaes of permeabty parameter K, ceary, as K ncreases the pea vaes of veocty tends to ncrease. Fgre 9 shows the veocty profes for dfferent vaes of the radaton parameter R, as R ncreases the pea vaes of the veocty tends to ncrease. Fgre shows that the veocty profe for dfferent vaes of absorptons of radaton parameter ( Q ), we observe that an ncrease n the vae of absorptons of radaton parameter ( Q ) there s an ncrease n the boyancy force, whch acceerates the fow rate. Fgre and 6 depct the veocty and concentraton profes for dfferent vaes of the Soret nmber (S ). We observe that an ncrease n the Soret nmber S rests to an ncrease n veocty and concentraton wthn the bondary ayer. Fgre3 dspays the temperatre decrease wth an ncrease n the heat sorce parameter Q becase as heat s absorbed, the boyancy force decreases the temperatre profes. Fgre4 shows the temperatre profe for dfferent vaes of radaton parameter R, ceary, as R ncreases the temperatre profe ncreases. It s seen from fgre 5 that the effect of absorpton of radaton s to ncrease temperatre n the bondary ayer as the radated heat s absorbed by fd whch n trn ncreases the temperatre of the fd very cose to the poros bondary ayer and ts effect dmnshes far away from the bondary ayer. The effect of ncreasng the vae of Schmdt nmber (Sc) s as shown n fgre7. As the Schmdt nmber (Sc) ncreases, the concentraton profe decreases. Fgre8 dspays the effect of the chemca reacton Kr on concentraton, we observe that concentraton profe decreases wth an ncrease n Kr. Fgre 9 shows the temperatre profe for dfferent vaes of Soret nmber S, ceary, as S ncreases the temperatre profe ncreases. Fg.. Effect of M on veocty profe Fg..Effect of Kr on veocty profe. DOI:.979/ Page

7 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass Fg.3.Effect of Q on veocty profe Fg.4.Effect of Pr on veocty profe Fg.5.Effect of Gr on veocty profe Fg.6.Effect of Gm on veocty profe Fg. 7.Effect of Sc on veocty profe Fg.8.Effect of K on veocty profe Fg. 9.Effect of R on veocty profe Fg..Effect of Q on veocty profe DOI:.979/ Page

8 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass Fg.Effect of So on veocty profe Fg..Effect of Pr on Temperatre profe Fg.3. Effect of Q on Temperatre profe Fg. 4.Effect of R on Temperatre profe Fg.5. Effect of Q on Temperatre profe Fg. 6.Effect of So on Concentraton profe Fg.7. Effect of Sc on Concentraton profe Fg. 8.Effect of Kr on Concentraton profe DOI:.979/ Page

9 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass Fg. 8.Effect of So on Temperatre profe V. Concsons In ths paper we nvestgated the effects of Soret, chemca reacton and radaton absorpton on an nsteady MHD convectve heat and mass transfer fow a sem - vertca movng n a poros medm wth heat sorce. merca cacatons are carred ot for varos vaes of the dmensoness parameters. A fnte eement method has been empoyed to evaate and sove for veocty, temperatre, concentraton, sn frcton, Sherwood nmber and sset nmber. The concsons of the stdy are as foows: The veocty ncreases wth the Sota Grashof nmber, therma Grashof nmber and permeabty parameter. The veocty decreases wth an ncrease n magnetc fed, Prandt nmber and Chemca reacton parameter. The temperatre decreases wth an ncrease n the heat sorce parameter, Prandt nmber and Schmdt nmber. The temperatre ncreases wth an ncrease n radaton parameter. An ncrease n the Soret nmber eads to an ncrease n veocty and concentraton. The concentraton decreases wth an ncrease n chemca reacton. References [] Dash, G.C and Das, S.S., Ha effect on MHD fow aong an acceerated poros fat pate wth mass transfer and nterna heat generaton, Math.Engg. Indst., 7(4), pp , 999. [] Satter,M.A., Free convecton and mass transfer fow throgh a poros medm past an nfnte vertca poros pate wth tme dependent temperatre and concentraton, Ind.J.Pre Appe.Math.,3, pp , 994. [3] Achary.M.,Dadh, G.C. and Sggh,L.P., Effect of chemca and therma dffsn wth Ha crrent on nsteady hydromagnetc fow near an nfnte vertca poros pate, J.Phys.D App.Phys.8, pp , 995. [4] Das, S.S. Shoo.S.K. and Dash. G.C merca soton of mass transfer effects on nsteady fow past an acceerated vertca poros pate wth scton, B.Maays MathScence Soc.(), 9, pp.33-4, 6. [5] B.Vas.,V. Ramchandra.,.Bhaser Reddy., Radaton and and Mass Transfer effects on transent free convecton fow of dsspatve fd past sem-nfnte pate wth nform heat and mass fx, Jorna of Apped mathematcs and fd Mechancs, 4(), pp.5-6, 6. [6] Mythreye., J.P.Pramoda., K.S. Baamtgan., Chemca Reacton on Unsteady MHD Convectve Heat and Mass Transfer Past a Sem-Infnte Vertca Permeabe Movng Pate wth Heat Absorpton, Preeceda Engneerng, 7, pp. 63-6, 5. [7] D.Srnvasacharya. B. Maarna., R.Bhvanavaya. Soret and Dfor effects on mxed convecton aong a vertcay away srface n poros medm wth varabe propertes An Shams Engneerng Jorna, 6, pp , 5. [8] Ramachandra prassad V and Bhaser reddy,. Radaton effects on an nsteady MHD convecton convectve heat and mass transfer fow past a sem- nfnte vertca permeabe movng pate embedded n poros medm,jorna of energy Heat and mass transfer, 3, pp.57-68, 8. [9] F.S.Ibrahm., A.M.Eaw.,A.A.Bar., stded effect of the chemca reacton and radaton absorpton on the nsteady MHD free concvecton fow past a sem nfnte vertca permeabe movng pate wth heat sorce and scton, Commncatons n nonnear Scence and nmerca smaton,3, pp.56-66, 8. [] K.S. Baamrgan,. J.L. Ramaprasad., and S.Vyaya Kmar Varma, Unsteady MHD free convectve Fow past a movng vertca pate wth tme dependent Scton chemca reacton n a sp fow regme, Proceda Engneerng 7, pp.56 53, 5. [] Md.Abds sattar.free convecton and mass transfer fow throgh a pors medm past an nfnte vertca poros medm past an nfnte vertca pate tme dependent temperatre and concentraton, Indan J.pre app.ath.,5(7), pp , 994. [] Da pa and Bab Taadar., pertrbaton anayss of nsteady MHD convectve heat and mass transfer n a bondary ayer sp fow past a vertca permeabe pate wth therma radaton and chemca reacton, Commn onnear Sc nmer smate,5, pp.83-83,. [3] G.S.Seth,.Md.S.Ansar and R.andeoyar, MHD natra convecton fow wth radatve heat transfer past an mpsvey movng pate wth ramped temperatre, Heat Mass Transfer, 47, pp.55 56,. [4] G. Swapna, L. Kmar and. Bhardwa, Stdy of effects of radaton and magnetc fed on the mxed convecton mcropoar Fd fow towards a stagnaton pont on a heated vertca permeabe pate sng fnte eement method, Int. J. of Mechanc Systems Engneerng, 5(), pp. -3, 5. [5] Svaah., K. Antha and S. Venataramana., Effects of therma dffson and radaton on nsteady MHD free convecton fow past an nfnte heated vertca pate n a poros medm, Internatona schoary research networ IRS thermodynamcs,, pp.-8. DOI:.979/ Page

10 Fnte Eement Method Appcaton on Effects on an Unsteady MHD Convectve Heat and Mass [6] Tanvr Ahmed and Md.Mahmad Fnte dfference soton of MHD mxed convecton fow wth heat generaton and chemca reacton, Proceda Engneerng 56, pp.49 56, 3. [7] M.K Mazmdar and R.K Dea MHD fow past an mpsve started nfnte vertca pate n the presence of therma radaton, Rom. Jorn. Phys., 5, os. 5 7, pp , 7 [8] Mohmed Abd E-Azz Unsteady mxed convecton heat transfer aong a vertca stretchng srface wth varabe vscosty and vscos dsspaton, Jorna of the Egyptan Mathematca Socety,, pp , 4. [9] Ebashbeshy E.M.A Heat and mass transfer aong a vertca pate wth varabe srface temperatre and concentraton n the presence of magnetc fed, Int.J.Eng.Sc., 34, pp.55-5, 997. [] Ja Sngh Vscos dsspaton and chemca reacton effects on fow past a stretchng poros srface n a poros medm, Adv. Theor. App. Mech., 5(8), pp.33 33, DOI:.979/ Page