Grc. Pr Sc. Du Sr. coefficient of thermal expansion kinematic viscousity. Subscripts

Transcription

1 Journal of Naval Arctecture and Marne Engneerng December, 14 ttp://dx.do.org//1.339/jname.v ttp://.banglajol.nfo UNSEADY MHD MIXED CONVECION FLOW PAS AN OSCILLAING PLAE WIH HEA SOURCE/SINK A. M. Okedoye Department of Matematcs and Computer Scence, Federal Unversty of Petroleum Resources, P. M. B. 11, Effurun, Delta State Ngera, Emal: Abstract: s paper study unsteady MHD mxed convecton flo past an nfnte vertcal oscllatng plate troug porous medum, takng account of te presence of free/forced convecton and mass transfer. Usng smlarty transformaton, te coupled non lnear governng equatons are solved numercally by applyng te combnaton of te base sceme sub-metods mdpont, and a metod enancement sceme Rcardson extrapolaton tecnque togeter t Felberg fourt-fft order Runge-Kutta sootng teraton metod t degree four nterpolant. e results are obtaned for velocty, temperature, concentraton. e effects of varous materal parameters are dscussed on flo varables and presented by graps. Keyords: Mxed convecton, magnetoydrodynamc flos, porous medum, mass transfer, oscllatng plate generatve reactons,convergence, Runge Kutta, Dufour effect, Soret effect, sotope separaton NOMENCLAURE u, v C Dm u C termal conductvty velocty components along x- and y- axes, respectvely concentraton of te flud dffuson coeffcent flud temperature free steam velocty free stream concentraton free stream temperature Dmensonless Group mass Grasof number termal Grasof number Prandtl number Scmdt number Hartmann Number Dufour number Soret number Ha Hartmann number Greek symbols Q eat generaton coeffcent non - dmensonal flud temperature c p C B t Q g k' b v specfc eat at constant pressure surface concentraton surface temperature magnetc nducton me coeffcent of eat transfer acceleraton due to gravty s te Darcy permeablty s te emprcal constant s te sucton/blong parameter Grc Grt Pr Sc M Du Sr c Subscrpts rato of free stream velocty parameter to stretcng seet parameter eat source/snk coeffcent coeffcent of concentraton expanson coeffcent of termal expanson knematc vscousty electrcal conductvty densty condton on te all ambent condton 1. Introducton e study of Magneto-ydrodynamc (MHD) flos ave stmulated consderable nterest due to ts mportant applcatons n cosmc flud dynamcs, meteorology, solar pyscs and n te moton of Eart s core (Cramer and Pa, 1973)]. In a broader sense, MHD as applcatons n tree dfferent subject areas, suc as astropyscal, geopyscal and engneerng problems. Convecton flo drven by temperature and concentraton dfferences (Prnt), (Onlne) 14 ANAME Publcaton. All rgts reserved. Receved on: November, 1

2 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) as been te objectve of extensve researc because suc processes exsts n nature and as engneerng applcatons. e process occurrng n nature ncludes poto-syntetc mecansm, calm-day evaporaton and vaporzaton of mst and fog, le te engneerng applcaton ncludes te cemcal reacton n a reactor camber consstng of rectangular ducts, cemcal vapour deposton on surfaces and coolng of electronc equpment. Sarma and Sng (9) report te effects of Varable ermal Conductvty and Heat Source/Snk on MHD flo near a stagnaton pont on a lnearly stretcng seet. e analyss of propagaton of termal energy troug mercury and electrolytc soluton n te presence of external magnetc feld and eat absorbng snks as de range of applcatons n cemcal and aeronautcal engneerng, atomc propulson, space scence etc. Recently, Amed and Amed (4) analyzed to-dmensonal MHD oscllatory flo along a unformly Movng nfnte vertcal porous plate bounded by porous medum. Furter, Ogulu and Prakas (6) nvestgated te effects magnetc feld on eat transfer unsteady flo past an nfnte movng vertcal plate t varable sucton. Later on, te effects of unsteady free convectve MHD flo troug a porous medum bounded by an nfnte vertcal porous plate ere nvestgated by Amed (7). Amed (1) studed mxed convecton ydro-magnetc oscllatory flo and perodc eat transfer of a vscous ncompressble and electrcally conductng flud past an nfnte vertcal porous plate. It as reported tat te mean and transent velocty decreases t te ncrease n Prandtl number. Pyscally ts s true because te ncrease n te Prandtl number s due to ncrease n te vscosty of te flud, c makes te flud tck and ence a decrease n te velocty of te flud. Zueco and Amed (1) studed combned eat and mass transfer by mxed convecton MHD flo along a porous plate t cemcal reacton n presence of eat source. It as reported tat as te cemcal reacton parameter Kncreases, te temperature profles decrease. Sarma et al. (11) nvestgated te nfluence of cemcal reacton and radaton on an unsteady MHD free convectve flo and mass transfer troug a vscous ncompressble, electrcally conductng flud past an nfnte vertcal eated porous plate t sucton, embedded n porous medum n te presence of a unform transverse magnetc feld, oscllatng free stream and eat source by takng nto account te vscous dsspaton. ey dscovered tat te magntude of flud temperature decreases t te ncrease of cemcal reacton parameter, vscous dsspaton effect and molecular dffusvty; le t ncreases t an ncrease of ntensty of magnetc feld and eat source. Also an ncrease n te Grasof number, leads to a rse n te magntude of flud velocty due to enancement n buoyancy force. e peak value of te velocty ncreases rapdly near te porous plate as buoyancy force for eat transfer ncreases and ten decays te free stream velocty. Sarma, Cand and Caudary (11) report an approxmate analyss of unsteady mxed convecton flo of an electrcally conductng flud past an nfnte vertcal porous plate embedded n porous medum under constant transversely appled magnetc feld, ere e transent velocty ncreases t ncrease n dstance from plate untl t attans ts maxmum value (nearly y = 1), after c t decreases. Osman et al. (11) studed termal radaton and cemcal reacton effects on te unsteady MHD convecton troug a porous medum bounded by an nfnte vertcal plate t eat source/snk. ey observed tat te presence of porous meda ncreases te resstance flo resultng n a decrease n te flo velocty. Several oter researcer ave orked on MHD convecton flo, Okedoye et al (8) as good reve of t. Due to te mportance of Soret (termal-dffuson) and Dufour (dffusontermo) effects for te fluds t very lgt molecular egt as ell as medum molecular egt many nvestgators ave studed and reported results for tese flos. For te problem of coupled eat and mass transfer n MHD mxed convectve flo of a conductng flud troug a porous medum n te presence of cemcal reacton, te effect of bot Dufour and Soret effects ere neglected, on te bass tat tey are of a smaller order of magntude tan te effects descrbed by Fourer s and Fck s las. ere are, oever, exceptons te Soret effect, for nstance, as been utlzed for sotope separaton and n mxture beteen gases and t very lgt molecular egt (H, He), and for medum molecular egt (H, ar) te Dufour effect as found to be of consderable magntude suc tat t cannot be neglected. Hence, as a complementary study to tat of Postelncu (4) and Alam and Raman (5), e propose to study te above-mentoned unsteady free - forced convecton flo past an oscllatng plate n a porous medum under te nfluence of transversely appled magnetc feld.e am of ts paper s to present te numercal analyss of unsteady MHD mxed convectve flo of a conductng flud t varable propertes troug a porous medum n te presence of cemcal reacton and eat source or snk en te plate s made to oscllate n tme about a non-zero constant mean t a specfed velocty. Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 168

3 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) Matematcal Formulaton We consder te mxed convecton flo of an ncompressble and electrcally conductng vscous flud along an nfnte non conductng vertcal flat plate troug a porous medum. e x axs s taken along te plate n te vertcally upard drecton and y axs s taken normal to te plate. A magnetc feld of unform strengt B s appled n te drecton of flo and te nduced magnetc feld s neglected. Intally, te plate and te flud are at same temperature n a statonary condton t concentraton level C at all ponts. At tme t te plate starts oscllatng n ts on plane t a velocty u cos t and te. Its temperature s rased to concentraton level at te plate s rased to C. e coordnates system and te confguraton are son n Fg. 1 B, C U g In ve of tese, e consder tat: () All te flud propertes except densty n te buoyancy force term are constant; () e nfluence of te densty varatons n oter terms of te momentum and energy equatons and te varaton of te expanson coeffcent t temperature s neglgble; () e Eckert number and te magnetc Reynolds number are small so tat te nduced magnetc feld can be neglected. (v) All te pyscal varables are ndependent of x, except possbly te pressure. (v) e plate s subjected to a constant sucton velocty. (v) ere exsts a frst-order omogeneous cemcal reacton t a constant rate K beteen te dffusng speces and te flud. Wt foregong assumptons usng te Boussnesq approxmaton, te governng equatons for te flo are gven by: v y u u u v t y y B b g g c C C u u u () k k Dmk c cp v Q t y y cscp y (3) C C C Dmk v Dm A C C t y y m y (4) e boundary condtons are gven by u y,, y,, C y, C y, t (5) u, t U cos t,, t, C, t C at y, t u, Fg. 1: Flo Confguraton, C C as y, t (1) (6) Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 169

4 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) No ntegratng (1) e ave, vy t const,. en an approprate value of constantfor te problem under consderaton s takng to be, v v, (7) ere v > s te sucton parameter and v < s te njecton parameter and s te scale parameter. Let us ntroduce te non-dmensonal varables y c c, u u f ( ),, c c Wt te elp of (7) and (8), te governng equatons () (4) reduce to d 1 f v f f Ha f f Grt Grc dt k d 1 Du v dt Pr Pr d 1 Sr v 1 dt Sc Sc (8) (9) (1) (11) Equatons (9) to (11) are smlar except for te term condton requred tat at s us ct d dt must be a constant. d dt Wtout loss of generalty, e take parameter for unsteady boundary layer problem. c d dt c, and so t ere t appears explctly. us te smlarty, c defne te ell establsed scalng Hence equatons (9) (11) togeter t te boundary and ntal condtons (5) and (6) becomes 1 f cf H f f Grt Grc (1) a k Pr c Du Pr (13) Were Sc c Sr 1Sc f cos, 1, 1 f,, (14) g Grt g c u, Grc c Q k P Pr,, Sr c p m c u c k, k A, 1, Sc Dm Dmk c c bu B Du,, H a, c c S P k 4 e pyscal varables ave ter usual meanngs as defned n nomenclature. c (15) Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 17

5 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) Numercal Computaton e set of equatons (1) (14) under te boundary condtons (15) ave been solved numercally by applyng te combnaton of te base sceme sub metods mdpont (Harer and Wanner (1996)), and a metod enancement sceme Rcardson extrapolaton (for detal dscusson of te metod see Ascer, Mattej, and Russell (1995) and Ascer, and Petzold (1998)) tecnque togeter t Felberg fourt-fft order Runge- Kutta sootng teraton metod t degree four nterpolant. e results are presented n Fgs. () (17) 3.1. Convergence. In ts paper, e consdered RK metods t s and coeffcents satsfyng te ypoteses H1: a j for j 1,, s; H1: tesubmatrx A : ( a j ), j s nvertble; H : b a s for j 1,, s,.e., te metod s stffly accurate. In order to so te convergence crtera oever, for convenence, e present ere te teorems te detal proof s analogous to te one n Ascer and Petzold (1991) HEOREM 1..Let Y, satsfyng s Y Z be te soluton of (1, 13, 14) subject to (15) and consder perturbed values Y, a j f ( Y j, Z j ) (1.1) j1 1,, s g( Y ) t Z :. In addton to te assumptons of eorem 4.1, suppose tat 1 O( ), Z O( ), O( ), O( (1.) ). en e ave for te estmates (1.3) Y Y C, C ere ( 1,, ), (1.4) Z Z s max and smlarly for. Remarks. 1) e condtons (1.) ensure tat all terms O () n te proof belo are small. ) We ntroduce te notaton,, (,, ) Y Y 1, Y Y Y, Y max Y and smlarly for te z-component. Over a multple-vector a tlde '' ndcates te removal Y ( Y,, ) of ts frst subvector, e.g.. Y s Y s Z HEOREM..In addton to te assumptons of eorem 1., suppose tat te condtons C(q), D(r) and te k ypotess H3 old, and tat ( g y f )(, ) O( ) t k 1. en e ave (.1) (, )( ) m Y Y P O, s s (.) Z Z R( )( ) O ere m mn( k 1, q 1, r), R s te stablty functon, and P s te projectordefned under te condton (1.3) by (.3) P : I Q, Remarks. : ( ) 1 Q f g f g. n e mportant result conssts n te factor z y z y m n front of n (.1) - (.4). Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 171

6 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) eorem. yelds te man component for te convergence proof of RK metods t sngular RK matrx A. 4. Results and Dscusson Here e ave nvestgated numercally MHD mxed convecton flo past an oscllatng plate t eat source or snk. In order to pont out te effects of varous parameters on flo caracterstc, te follong dscusson s set out. In smulaton, te values of te Prandtl number are consdered to be.7, 1.7,.97 and 4.34 tat corresponds to elum, sulfur doxde, metyl clorde and ater, respectvely. e values of te Scmdt number are cosen to represent te presence of speces ater vapour (.6). Numercal results ave been obtaned for dfferent values of flo condton and are presented grapcally. Specal cases () In te absence of magnetc feld.em=, te results of te present paper are reduced to tose obtaned by Pop, Grosan, Pop (4) and Maapatra and Gupta (). ()In te absence of magnetc feld, eat source/snk and varable termal conductvty, te results of te present paper are reduced to tose obtaned by Pop, Grosan, Pop (4) nte absence of radaton effect t constant termal conductvty and Maapatra and Gupta () n absence of vscous dsspaton and constant termal conductvty. () In te absence of cemcal reacton, te result of te present paper are reduced to tose obtaned by Sarma and Sng (8) (v) In te absence of Dufour and Soret effect, te results of te present paper are reduced to tose obtaned by Osman, Abo-Daab, and R. A. Moamed1 (11) Fg. : Velocty dstrbuton for varous Fg. 3: Velocty dstrbuton for varous c Fg. 4: emperature dstrbuton for varous c Fg. 5: Concentraton dstrbuton for varous c In Fg. represents te velocty profles due to te varatons n ωt. It s evdent from te fgure, under te cosen condton tat te maxmum velocty s at te plate. Furtermore, te magntude of te velocty decreases t ncreasng pase angle (ωt). Fgs. 3, 4 and 5 reveal te velocty, temperature and speces concentraton varatons t sucton/njecton parameter. It s observed tat n case of coolng of surface (an Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 17

7 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) ncrease n Gr), decrease n njecton rate decreases velocty, temperature and speces concentraton dstrbuton, le ncrease n sucton rate ncreases te felds. It s found tat te velocty decreases t ncrease n magnetc parameter. It s because tat te applcaton of transverse magnetc feld ll result a resstve type force (Lorentz force) smlar to drag force c tends to resst te flud flo and tus reducng ts velocty. e presence of a porous medum ncreases te resstance to flo resultng n decrease n te flo velocty, snce 1/Ks an addton to Ha. s beavour s depcted by te decrease n te velocty as K decreases and en K = (.e. te porous medum effect s vansed) te velocty s greater n te flo feld as son n Fg. 6. In Fgs. 7 and 8, t s observed tat greater coolng of surface (an ncrease n Grt) and ncrease n Grc results n an ncrease n te velocty, respectvely. It s due to te fact ncrease n te values of termal Grasof number and mass Grasof number as te tendency to ncrease te termal and mass buoyancy effect. s gves rse to an ncrease n te nduced flo. Furtermore, te velocty near te plate s greater tan at te plate. e maxmum velocty attans near te plate and s n te negbourood of pont η =.4. After η >.4, te velocty decreases and tends to zero as η. Fgs. 9 and 1 dsplay te effects of Du (Dufour number) on te velocty and temperature felds for te case Grt> and Grc>. It s seen tat ncrease n Dufour number brngs about ncrease n bot te velocty and temperature dstrbuton n te flo feld, le n Fg.11, t s son tat concentraton dstrbuton ncreases t an ncrease n Sr (Soret number) t maxmum concentraton near te plate for ger value of Sr. Fg. 6: Velocty dstrbuton for varous Hartmann number Fg. 7: Velocty dstrbuton for varous ermal Grasof number Fg. 8: Velocty dstrbuton for varous mass Grasof number Fg. 9: Velocty dstrbuton for varous values of Dufor number Fgs. 1, 13 and 14 depct te effect of nternal eat generaton/absorpton on velocty, temperature and concentraton felds respectvely. It s observed tat nternal eat generaton ncreases te velocty, temperature and concentraton dstrbuton le nternatonal eat absorpton reduces te velocty and temperature dstrbuton. Furtermore te magntude of temperature s maxmum at te plate ereas for nternal eat generaton te maxmum velocty and concentraton s near te all and ten decays to zero asymptotcally. Effect of cemcal reactvty on velocty, temperature and concentraton felds are son n Fgs. 15, 16 and 17 respectvely. It sould be noted ere tat, 1 correspond to destructve cemcal reacton and 1 Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 173

8 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) correspond to generatve cemcal reacton. It s observed tat ncrease n generatve cemcal reacton ncreases te velocty and concentraton dstrbuton but t reduces te flo temperature dstrbuton. Wle te reverse s te case for destructve cemcal reacton. Furtermore te magntude of temperature s maxmum at te plate ereas for generatve cemcal reacton, te maxmum velocty and concentraton s near te all and ten decays to zero asymptotcally. Fg. 1: emperature dstrbuton for varous values of Dufor number Fg. 11: Concentraton dstrbuton for varous values of Soret number Fg. 1: Velocty dstrbuton for varous values of reacton parameter Fg. 13: emperature dstrbuton for varous values of reacton parameter Fg. 14: Concentraton dstrbuton for varous values of reacton parameter Fg. 15: Velocty dstrbuton for varous values of eat parameter Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 174

9 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) Fg. 16: emperature dstrbuton for varous values of eat parameter Fg. 17: Concentraton dstrbuton for varous values of eat parameter 5. Concluson In ts paper effect of temperature dependent termal conductvty on MHD free convecton flo along a vertcal flat plate ave been studed numercally. Implct fnte dfference metod togeter t Keller box sceme s employed to ntegrate te equatons governng te flo. Comparson t prevously publsed ork s performed and excellent argument as been observed. From te present numercal nvestgaton, follong conclusons may be dran: For ncreased value of magnetc parameter, te velocty profle decreases but te temperature profle ncreases slgtly. In case of coolng of te plate (Grt> ), te velocty decreases t an ncrease n pase angle, njecton and magnetc parameter. On te oter and, t ncreases t an ncrease n te value of termal Grasof number and mass Grasof number, sucton parameter, Dufour number, nternal eat generaton and generatve cemcal reacton. e local skn frcton coeffcent decreases as ell as te surface temperature dstrbuton ncrease t te ncrease n values of te magnetc parameter. e concentraton decreases t an ncrease n njecton and ncreases t ncrease n Soret number, nternal eat generaton and generatve cemcal reacton. e temperature decreases t an ncrease n njecton and ncreases t ncrease n Dufour number, sucton, nternal eat generaton and generatve cemcal reacton. References Amed, S. (1): Free And Forced Convectve MHD Oscllatory Flo Over an Infnte Porous Surface n an Oscllatng Free Stream,Latn Amercan Appled Researc,Vol. 4, pp Amed, S. (7): Effects of Unsteady Free Convectve MHD Flo roug a Porous Medum Bounded by an Infnte Vertcal Porous Plate, Bulletng of CalcuttaMatematcal Socety, Vol. 99, pp Amed, S. and Amed, N. (4): o-dmensonal MHD Oscllatory Flo along a Unformly Movng Infnte Vertcal Porous Plate Bounded by Porous Medum, Internatonal Journal of Pure and Appled Matematcs,Vol.35, pp Alam, M. S. and Raman, M. M. (5): Dufour and Soret Effects on MHD Free Convectve Heat and Mass ransfer Flo Past a Vertcal Flat Plate Embedded n a Porous Medum, Journal of Naval Arctecture and Marne Engneerng, Vol., No. 1, pp Ascer, U. M. and Petzold, L. R. (1991): Projected Implct Runge-KuttaMetods for Dfferental-Algebrac Equatons,Socety for Industral and Appled Matematcs, Journal of Numercal Analyss, Vol. 8, pp ttp://dx.do.org/1.1137/7859 Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 175

10 A. M. Okedoye/ Journal of Naval Arctecture and Marne Engneerng 11(14) Ascer, U., Mattej. R. and Russell, R. (1995): Numercal Soluton of Boundary Value Problems for Ordnary Dfferental Equatons, Socety for Industral and Appled Matematcs, Classcs n Appled Matematcs, Vol. 13. Ascer, U. M. and Petzold, L. (1998): Computer Metods for Ordnary Dfferental Equatons and Dfferental- Algebrac Equatons,Socety for Industral and Appled Matematcs, Pladelpa.ttp://dx.do.org/1.1137/ Cramer, K. R. and Pa, S. I. (1973): Magneto-flud Dynamcs for Engneers and Appled Pyscsts, McGra- Hll Co., Ne York. Harer, E. and Wanner, G. (1996): Solvng Ordnary Dfferental Equatons II. nd Ed. Ne York: Sprnger.ttp://dx.do.org/1.17/ Maapatra,. R. and Gupta, A. S. (): Heat ransfer n a Stagnaton Pont Flo oard Stretcng Seet, Journal of Heat and Mass ransfer,vol. 38, pp ttp://dx.do.org/1.17/s31115 Ogulu, A. and Prakas, J. (6): Heat ransfer to Unsteady Magneto-Hydrodynamc Flo Past an Infnte Movng Vertcal Plate t Varable Sucton, Royal Seds Academy of Socety, Vol. 74, pp Okedoye, A. M. Faryol,a, P. I. and Bello, O. A. (8): MHD Flo of a Unformly Stretced Vertcal Permeable Surface under Oscllatory Sucton Velocty, J. Ngera Assocaton of Matematcal Pyscs, Vol. 13,pp Osman, A. A., Abo-Daab, S. M. and Moamed, R. A. (11): Analytcal Soluton of ermal Radaton and Cemcal Reacton Effects on Unsteady MHD Convecton troug Porous Meda t Heat Source/Snk, Matematcal Problems n Engneerng, Vol. 11, pp.1-18.ttp://dx.do.org/1.1155/11/5181 Pop, S. R., Grosan,. and Pop, I. (4):.Radaton Effect on te Flo Near te Stagnaton Pont of a Stretcng Seet, ecnscemecank, Vol. 5, pp Postelncu, A. (4): Influence of a Magnetc Feld on Heat and Mass ransfer ay Natural Convecton from Vertcal Surfaces n Porous Meda Consderng Soret and Dufour Effects, Internatonal Journal of Heatand Mass ransfer, Vol. 47, No. (6 7), pp Sarma, B. K., Cand,. and Caudary, R. C. (11): Hydro-magnetc Unsteady Mxed Convecton Flo past an Infnte Vertcal Porous Plate,Appled Matematcs,Vol. 1, No.1,pp ttp://dx.do.org/1. 593/j.am Sarma, P. R. and Sng, G. (9): Effects of Varable ermal Conductvty and Heat Source/Snk on MHD Flo Near a Stagnaton Pont on a Lnearly Stretcng Seet, Journal of Appled Flud Mecancs,Vol., No. 1, pp Sarma, P. R., Kumar, N. and Sarma, P. (11): Influence of Cemcal Reacton and Radaton on Unsteady MHD Free Convectve Flo and Mass ransfer roug Vscous Incompressble Flud Past a Heated Vertcal Plate Immersed n Porous Medum n te Presence of Heat Source, Appled Matematcal Scences, Vol. 5, No. 46, Zueco,J. and Amed, S. (1): Combned Heat and Mass ransfer by Mxed Convecton MHD Flo along a Porous Plate Wt Cemcal Reacton n Presence of Heat Source,Appled Matematcs and Mecancs,Engls Edton, Vol. 31, No. 1, 17 3, DOI 1.17/s ttp://dx.do.org/1.17/s Unsteady MHD mxed convecton flo past an oscllatng plate t eat source/snk. 176