D Solute mass diffusivity ( m ) D Mass diffusivity ( m s ) Gc Pr. Sc Sr Re g. q Radiative heat flux. U Reference velocity ( m s )

Transcription

1 ISSN (e): 5 35 Vlme 7 Isse 6 Jne 7 Inernanal Jrnal f mpanal Engneerng Research (IJER) Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared Vercall Inclned Plae: hermal Dffsn and agnec Feld Effecs P.Sresh *. V. Ramana rh S.Harsngh Nak K Sreeram Redd Deparmen f ahemacs& mper Scence nvers llege fr Scences Osmana nvers Hderabad 57 elangana Sae Inda. * Deparmen f ahemacs haana harah Inse f echnlg Gandpe Hderabad Rangaredd Dsrc 575 elangana Sae Inda. ASRA he am f presen nvesgan s sd he nare f assn fld n HD free cnvecve fl f ver an mplsvel sared nfne vercall nclned plae h Sre hermal radan hea and mass ransfer effecs. Assmng he medm be nn-scaered and he fld be nn-gra emngabsrbng and pcall hn radan lm prperes he basc gvernng nnlnear cpled paral dfferenal eqans are slved nmercall sng fne dfference mehd. he relevan phscal parameers appearng n velc emperare and cncenran prfles are analzed and dscssed hrgh graphs. We als cmped he Nssel and Sherd nmbers alng h frcn facr fr he same cases. Fnall he resls fr velc prfles and he redced Nssel and Sherd nmbers are baned and cmpared h prevs resls n he lerare and are fnd be n excellen agreemen. Applcans f he presen sd ld be sefl n magnec maeral prcessng and chemcal engneerng ssems. Nmenclare: Ls f varables: nfrm magnec feld (esla) x rdnae axs alng he plae ( m ) -rdnae axs nrmal he plae ( m ) Velc cmpnen n x ( m s ) Fld emperare ( K ) drecn Fld emperare a he all K Fld emperare aa frm he plae (K) Fld ncenran ( Kg m ) ncenran f he plae ( Kg m ) Dmensnless dsplacemen ( m ) ncenran f he fld far aa frm 3 he plae ( Kg m ) p Specfc hea a cnsan pressre J Kg K f he lcal skn-frcn ( N m ) N he lcal Nssel nmber Sh he lcal Sherd nmber Fld velc ( m s ) 3 3 D Sle mass dffsv ( m ) D ass dffsv ( m s ) Gr Gc Pr Sc Sr Re g s agnec feld parameer Grashf nmber fr hea ransfer Grashf nmber fr mass ransfer Prandl nmber Schmd nmber Sre nmber Renlds nmber Acceleran f grav n magnde s 9.8 ( m ) me (sec) q Radave hea flx r q Velc vecr P J E Fld pressre Elecrc crren dens vecr agnec feld vecr Elecrc feld vecr Reference velc ( m s ) g Acceleran f grav 9.8 ( m ) Q hermal radan parameer s.cernlne.cm Open Access Jrnal Page 4

2 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared K Absrpn ceffcen e Plank's fncn b Greek Smbls: assn fld parameer s Knemac vscs ( m ) Speces cncenran ( Kg m ) he cnsan dens ( Kg m ) Vlmerc ceffcen f hermal expansn ( K ) * Vlmerc effcen f hermal expansn h cncenran ( m ) Elecrc cndcv f he fld ( s m ) Kg Fld emperare K Fld ncenran ( Kg m ) Angle f nclnan f plae ( deg rees ) Shear sress ( N m ) hermal cndcv f he fld W / mk effcen f vscs m. Pa. s Dsspan f energ per n vlme de vscs Sperscrps: / Dmensnless Prperes Sbscrps: ndns n he all Free sream cndns p Plae 3 I. INRODION he her f nn-nenan fld s a par f fld mechancs based n he cnnm her ha a fld parcle ma be cnsdered as cnns n a srcre. Psed plasc me ndependen fld s ne f he nnnenan flds hse behavr s ha vscs decreases h ncreasng velc graden e.g. plmer slns bld ec. assn fld s ne f he psedplasc flds ha means shear hnnng flds. A l shear raes he shear hnnng fld s mre vscs han he Nenan fld and a hgh shear raes s less vscs. S HD fl h assn fld s recenl fams. assn [] presened assn fld mdel fr he predcn f he fl cndc f pgmen-l sspensns. khpadha [] dscssed hea ransfer fl f assn fld ver nnlnearl srechng shee. he nmercal slns are carred b shng mehd. She cnclded ha skn frcn and emperare graden are ncreasng fncns f assn fld parameer. he sead sae elecrcall cndcng fl f assn fld ver a srechng shee n a prs medm s repred b Shak [3]. In he sbseqen ear Nadeem e al. [4] explred hree dmensnal elecrcall cndcng bndar laer fl f assn fld ver srechng shee saraed n a prs medm. Khald e al. [5] nvesgaed he effecs f magnec feld n free cnvecn fl f assn fld ver scllang plae embedded n prs medm. Prnma e al. [6] cnsdered he velc slp a all fr assn fld ver a prs srechng srface. he fnd ha slp parameer decreases fld velc and enhances shear sress a he all. he mechansm f slp cndn n sagnan pn fl f assn fld has been repred b Haa e al. [7]. Nadeem e al. [8] explred he cmbned effecs f paral slp and magnec feld n sagnan pn fl f assn fld ver srechng srface. he cnclded ha slp parameer redces he velc f fld n he bndar regn. khpadha e al. [9] analzed he hea ransfer n assn fld fl n a smmerc edge. Oelakn e al. [] sded nsead cnvecve assn nanfld fl ver a srechng shee n he presence f hermal radan and slp bndar cndns. Salaa e al. [] sded dble dffsve effecs n HD mxed cnvecn cassn fld fl ards a vercall nclned plae flled n prs medm n presence f nmber sng fne elemen echnqe. Srnvasa Ra e al. [] dscssed he nflence f angle f nclnan n nsead HD cassn fld fl pas a vercal srface flled b prs medm n presence f cnsan hea flx chemcal reacn and vscs dsspan. he assmpn f Sre r hermal dffsn effec s vald hen he cncenran level s ver l. he hermal dffsn effec (als knn as Sre effec) s merel de he effec f mass flx ha ccrs nder a emperare graden and hs phenmenn s lzed fr spe separan. he expermenal nvesgan n hs effec as frs perfrmed b harles Sre n 879. In mxres beeen gases h ver lgh mleclar egh (H He) and medm mleclar egh (N Ar) he hermal dffsn s fnd be f a magnde sch ha cann be negleced as as emphaszed b Ecker and Drake [3]. Srnvasa Ra [4] sded he effecs f Sre and Dfr n naral cnvecve fld fl pas a vercal plae embedded n prs medm n presence f hermal radan va fne elemen mehd. Anand Ra e al. [5] sded fne elemen analss f nsead HD free cnvecn fl pas an nfne vercal plae h Sre Dfr hermal radan and hea srce. Ahmed [6] fnd exac slns f HD ransen free cnvecn and mass ransfer fl f a vscs ncmpressble and elecrcall cndcng fld pas a sddenl sared nfne vercal plae akng n accn he hermal dffsn as ell as he hermal radan sng Laplace ransfrm echnqe. he nflence f Sre and Dfr effecs n fl feld n free cnvecn bndar laer frm a vercal srface.cernlne.cm Open Access Jrnal Page 43

3 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared embedded n a Darcan prs medm has been sded b Pselnc [7]. Pal and ndal [8] sded he HD nn-darc mxed cnvecve dffsn f speces ver a srechng shee embedded n a prs medm h nn-nfrm hea srce/snk varable vscs and Sre effec. hermphress parcle depsn n a nn-darc prs medm nder he nflence f Sre Dfr effecs sded b Parha [9]. Haa e al. [] examned he Sre and Dfr effecs n hree-dmensnal fl ver an expnenall srechng srface h prs medm chemcal reacn and hea srce/snk. he sead and nsead mxed cnvecn h Sre and Dfr effecs n chemcall reacng HD fl pas a vercal plae embedded n prs medm sded b aknde ([]-[4]). Pal and ndal [5] analzed he chemcal reacn and hermal radan n mxed cnvecn hea and mass ransfer ver a srechng shee n Darcan prs medm h Sre and Dfr effecs. rch e al. [6] sded analcall and nmercall he Sre effec n he nse f cnvecn n a vercal prs laer sbeced nfrm hea flx. he nvel f he presen paper s sd he effecs f Sre and hermal radan n nsead free cnvecn nn-nenan assn fld fl pas an mplsvel sared nfne vercall nclned plae n he presence f magnec feld hea and mass ransfer effecs. Nn-dmensnal qanes are nrdced n he gvernng eqans. he nnlnear f he basc eqans and addnal mahemacal dffcles asscaed h have led s se nmercal mehd. he ransfrmed dmensnless gvernng eqans are slved nmercall b sng fne dfference mehd. he effecs f vars phscal parameers n velc emperare cncenran prfles as ell as n skn-frcn Nssel nmber and Sherd nmber are analzed. Fg.. Schemac ve f fl cnfgran II. AHEAIAL FORLAION: he fndamenal gvernng eqans n vecr frm f he mn f an ncmpressble vscs elecrcall cndcng radang assn fld fl pas an mplsvel sared nfne vercall nclned plae n presence f a magnec feld are nn Eqan:. q () menm Eqan: q q. q P J g q () Energ Eqan: J q r q. (3) p Speces ncenran Eqan:.cernlne.cm Open Access Jrnal Page 44

4 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared Ohm's La: J E q q. D D We n cnsder an nsead free cnvecn assn fld fl f an ncmpressble vscs and elecrc cndcng pas an mplsvel sared nfne vercall nclned plae n presence f a nfrm ransverse magnec feld f srengh hermal radan and Sre effecs. Gemer f he prblem s presened n Fg.. Or nvesgan s resrced he fllng assmpns:. rdnae ssem s chsen n sch a a ha x axs s cnsdered alng he plae n pard drecn and axs nrmal he plae n he fld.. A nfrm ransverse magnec feld s appled n a drecn hch s parallel axs.. Inall.e. a me bh he fld and plae are a res and are mananed a a nfrm v. emperare. Als speces cncenran a he srface f he plae as ell as a ever pn hn he fld s mananed a nfrm cncenran. v. he emperare a he srface f he plae s rased nfrm emperare and speces cncenran a he srface f he plae s rased nfrm speces cncenran and s mananed hereafer. v. All he fld prperes excep he dens n he banc frce erm are cnsans. v. he vscs dsspan and Ohmc dsspan f energ are neglgble. v. he magnec Renlds nmber s s small ha he ndced magnec feld can be negleced. x. Als n appled r plarzed vlages exs s he effec f plarzan f fld s neglgble. he rhelgcal eqan f sae fr he ach sress ensr f assn fld [7] s ren as (6) eqvalenl here s shear sress p e c p e c c.cernlne.cm Open Access Jrnal Page 45 (5) (4) * s assn eld sress s dnamc vscs s shear rae e e (7) * and e s he h cmpnen f defrman rae s he prdc based n he nn-nenan fld s a c crcal vale f hs prdc s plasc dnamc vscs f he nn-nenan fld p (8) dene he eld sress f fld. Sme flds reqre a gradall ncreasng shear sress manan a cnsan sran rae and are called Rhepecc n he case f assn fld (Nn-Nenan) fl here c p (9) Sbsng Eq. (8) n Eq. (9) hen he knemac vscs can be ren as () nder he abve fregng assmpns and ssnesq's apprxman he eqans gvernng he fl and ranspr redce he fllng eqans: nn Eqan: () x menm Eqan:

5 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared.cernlne.cm Open Access Jrnal Page 46 g g * cs cs () Energ Eqan: q r p (3) Speces Dffsn Eqan: D D (4) geher h nal and bndar cndns as a all fr : : (5) As menned earler n he case f pcall hn lm he fld cann absrb s n emed radan b absrbs he radan emed b bndares. Fllng gle e al. [8] he rae f flx f radan n he pcall hn lm fr a nn-gra gas near eqlbrm s gven b I q r 4 (6) Where d d de K I b. We n nrdce he fllng nn-dmensnal varables and parameers: Re ) ( 4 Pr ) ( 3 * 3 x D Sr I Q D Sc g Gc g Gr p (7) he abve defned nn-dmensnless varables n Eq. (7) n Eqs. ()-(4) and e ge cs cs Gc Gr (8) Pr Q (9) Sr Sc Sc () h cnneced nal and bndar cndns as a all fr : : () Fr he desgn f chemcal engneerng ssems and praccal engneerng applcans he lcal skn-frcn Nssel nmber and Sherd nmber mpran phscal parameers fr hs pe f bndar laer fl. he Skn-frcn a he plae hch n he nn-dmensnal frm s gven b

6 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared.cernlne.cm Open Access Jrnal Page 47 v v f () he rae f hea ransfer ceffcen hch n he nn-dmensnal frm n erms f he Nssel nmber s gven b x N Re N (3) he rae f mass ransfer ceffcen hch n he nn-dmensnal frm n erms f he Sherd nmber s gven b x Sh Re Sh (4) III. NERIAL SOLIONS Y FINIE DIFFERENE EHOD: he nn-lnear mmenm and energ eqans gven n eqans (8) (9) and () are slved nder he apprprae nal and bndar cndns () b he mplc fne dfference mehd. he ranspr eqans (8) (9) and () a he grd pn ( ) are expressed n dfference frm sng alr s expansn. he mmenm eqans reads. cs cs Gc Gr (5) Pr Q (6) Sc Sr Sc (7) Where he ndces and refer and respecvel. he nal and bndar cndns () eld: & a all fr (8) hs he vales f θ and ϕ a grd pn = are knn; hence he emperare and cncenran prfles have been slved a me sng he knn vales f he prevs me fr all... N. hen he velc feld s evalaed sng he alread knn vales f emperare and cncenran prfles baned a. hese prcesses are repeaed ll he reqred sln f θ and ϕ s ganed a cnvergence crera. 3 nmercal exac abs (9)

7 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared IV. VALIDAION OF ODE: able-: Varan f he velc prfles agans he agnec feld parameer. Velc prfles Presen velc resls Velc resls f Ahmed [6] Gr =. Gr =. and Gr =. and Gr =. and Gr =. and Gc = Gc =. Gc =. Gc = 5. and Gc =.. Gr =. and Gc = able-: Varan f he Nssel nmber agans hermal radan parameer Q. Q Presen Nssel nmber resls Nssel nmber resls f Ahmed [6] Pr =.7 Pr = 7. Pr =.7 Pr = able-3: Varan f he Sherd nmber agans Sre nmber Sr a Pr =.7. Sr Presen Sherd nmber resls Sherd nmber resls f Ahmed [6] Q =. Q = 3. Q =. Q = V. RESLS AND DISSSIONS: Fg.. Gr effec n velc prfles he frmlan f he prblem ha accns fr he effecs f hermal radan and Sre n he assn fld fl pas an mplsvel sared nfne vercall nclned plae n he presence f magnec feld as accmplshed n he precedng secns. he gvernng eqans f he fl feld ere slved nmercall.cernlne.cm Open Access Jrnal Page 48

8 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared sng b fne dfference mehd. In rder ge phscal nsgh n he prblem he velc emperare and cncenran felds have been dscssed b assgnng nmercal vales f agnec parameer Grashf nmber fr hea ransfer Gr Grashf nmber fr mass ransfer Gc Prandl nmber Pr Schmd nmber Sc assn fld parameer γ Angle f nclnan parameer α hermal radan parameer Q and Sre nmber Sr. hrgh he calclans he paramerc vales are fxed be Gr =. Gc =. =.5 Pr =.7 Sc =. Sr =.5 Q =.5 γ =.5 and α = 45 nless herse ndcaed. Fg. 3. Gc effec n velc prfles Fg. 4. effec n velc prfles he velc prfles fr dfferen vales f Grashf nmber fr hea ransfer (Gr) s defned n Fg.. I s nced ha ncrease n Grashf nmber fr hea ransfer cnrbes he ncrease n velc f he fld. Frher s nced n he bndar laer regn he velc frs ncreases a maxmm vale and hen decreases. Als far aa frm he plae n mch f sgnfcan effec f Grashf nmber fr hea ransfer s nced. Fr he case f dfferen vales f he Grashf nmber fr mass ransfer (Gc) he velc prfles n he bndar laer s shn n Fg. 3. he velc dsrbn aans a dsncve maxmm vale n he vcn f he plae and hen decreases prperl apprach a free sream vale. As expeced he fld velc ncreases and he peak vale becmes mre dsncve de ncrease n he banc frce represened b Gc here as Gc ncreases he velc and he emperare ncreases. Fg. 4 llsraes he effec f magnec feld parameer n velc prfles n he bndar laer. I s neresng ne frm Fg. 4 ha he effec f agnec feld s decrease he vale f he velc prfles hrgh he bndar laer. he peak vale drascall decreases h ncrease n he vale f he magnec feld becase he presence f magnec feld n an elecrcall cndcng fld nrdces a frce called he Lrenz frce hch acs agans he fl f he magnec feld s appled n he nrmal drecn. hs pe f ressng frce sldns he fld velc as shn n he graph. he nflence f Prandl nmber Pr n he velc prfles s llsraed n Fg. 5. I s bserved ha ncrease n Pr cnrbes he decrease n he fld velc. he cnrbn b Pr s n ha sgnfcan a he bndar. he effec appears be mre sgnfcan as e mve far aa frm he bndng srface..cernlne.cm Open Access Jrnal Page 49

9 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared Fg. 5. Pr effec n velc prfles Fg. 6. Pr effec n emperare prfles Fg. 6 llsraes he nflence f Prandl nmber n he emperare feld. I s bserved ha he Prandl nmber has sgnfcan cnrbn ver he emperare feld. Frm he llsrans s seen ha as he Prandl nmber ncreases he emperare decreases. Frm hs pl s evden ha emperare n he bndar laer falls ver qckl fr large vale f he Prandl nmber becase f he fac ha hckness f he bndar laer decreases h ncrease n he vale f he Prandl nmber. Fr dfferen vales f he Schmd nmber Sc he velc prfles are pled n Fg. 7. I s bvs ha he effec f ncreasng vales f Sc resls n a decrease velc prfles. Fg. 8 dsplas he effec f Schmd nmber Sc n cncenran prfles. Phscall he ncrease f Sc means decreases f mleclar dffsn. Hence he cncenran f he speces s hgher fr small vales f Sc and ler fr larger vales f Sc.herefre as he Schmd nmber ncreases he cncenran decreases. hs cases he cncenran banc effecs decrease and here s a redcn n he fld velc. Fg. 7. Sc effec n velc prfles.cernlne.cm Open Access Jrnal Page 5

10 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared Fg. 8. Sc effec n cncenran prfles he cnrbn f Sre nmber Sr n he velc prfles s nced n Fg. 9. he ncrease n Sre nmber Sr cnrbes he ncrease n he velc prfles. Frher s nced ha he velc decreases as e mve aa frm he plae hch s fnd be ndependen f Sre nmber Sr. he nflence f Sre nmber Sr n he cncenran f he fld medm s seen n Fg.. In general s ned ha ncrease n Sre nmber Sr cnrbes ncrease n cncenran f he fld medm. Frher he effec s fnd be dmnshng as e mve aa frm he plae. he effec f he hermal radan parameer Q n he dmensnless velc and emperare prfles are shn n Fgs. and respecvel. Fg. 9. Sr effec n velc prfles Fg.. Sr effec n cncenran prfles.cernlne.cm Open Access Jrnal Page 5

11 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared Fg.. Q effec n velc prfles Fg. shs ha velc prfles decreases h an ncrease n he hermal radan parameer Q. Frm Fg. s seen ha he emperare decreases as he hermal radan parameer Q ncreases. hs resl qalavel agrees h expecans snce he effec f radan s decrease he rae f energ ranspr he fld hereb decreasng he emperare f he fld. he velc prfles n he Fg. 3 shs ha rae f mn s sgnfcanl redced h ncreasng f assn fld parameer γ. Als s bserved frm hs Fg. 3 he bndar laer mmenm hckness decreases as ncrease f assn fld parameer γ. he effec f angle f nclnan f he plae α n he velc feld has been llsraed n Fg. 4. I s seen ha as he angle f nclnan f he plae α ncreases he velc feld decreases. Fg.. Q effec n emperare prfles Fg. 3. γ effec n velc prfles.cernlne.cm Open Access Jrnal Page 5

12 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared he effecs f agnec parameer Grashf nmber fr hea ransfer Gr Grashf nmber fr mass ransfer Gc Prandl nmber Pr Schmd nmber Sc assn fld parameer γ Angle f nclnan parameer α hermal radan parameer Q and Sre nmber Sr n he skn-frcn ceffcen are prraed n able 4. Frm hs able s bserved ha he skn-frcn ceffcen ncreases as he Grashf nmber fr hea ransfer Gr Grashf nmber fr mass ransfer Gc and Sre nmber Sr ncreases hereas decreases as he agnec parameer Prandl nmber Pr Schmd nmber Sc assn fld parameer γ Angle f nclnan parameer α and hermal radan parameer Q. he effecs f he hermal radan parameer Q and Prandl nmber Pr n rae f hea ransfer ceffcen r Nssel nmber are presened n able 5. Frm hs able s nced ha he Nssel nmber decreases as hermal radan parameer Q and Prandl nmber Pr ncreases. he effecs f Sre nmber Sr and Schmd nmber Sc n rae f mass ransfer ceffcen r Sherd nmber are presened n able 5. Frm hs able s bserved ha he Sherd nmber ncreases as Sre nmber Sr ncreases hereas decreases as Schmd nmber Sc ncreases. Fg. 4. α effec n velc prfles able-4: Nmercal vales f Skn-frcn ceffcen Gr Gc Pr Sc Sr Q γ α f able-5: Nmercal vales f rae f hea and mass ransfer ceffcens Pr Q N Sc Sr Sh VI. ONLSIONS he effecs f hermal radan and Sre n an nsead free cnvecn bndar laer fl f an elecrcall cndcng assn fld pas an mplsvel sared vercall nclned plae b akng magnec feld hea and mass ransfer n accn. he reslan gvernng paral dfferenal eqans are hen slved nmercall sng fne dfference mehd. he effecs f gvernng herm phscal parameers n he velc emperare and cncenran as ell as Skn-frcn Nssel nmber and Sherd nmber are cmped and.cernlne.cm Open Access Jrnal Page 53

13 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared presened n graphcal and ablar frms. he resls and dscssn f he presen sd leads he fllng bservans.. he velc prfles are ncreasng h ncreasng vales f Grashf nmber fr hea ransfer Gr and Grashf nmber fr mass ransfer Gc.. he effec f ncreasng he Prandl nmber Pr Schmd nmber Sc he assn fld parameer γ he Angle f nclnan parameer α he hermal Radan parameer Q he agnec feld parameer deceleraes he velc f he fl feld a all pns hle a grng he Sre nmber Sr acceleraes he velc f he fl feld a all pns. 3. A grng he Prandl nmber Pr and hermal radan parameer Q are reard he emperare f he fl feld a all pns. 4. he ncenran f he fl decreases h ncreasng vales f he Schmd nmber Sc here as ncreases h ncreasng vales f Sre nmber Sr. 5. he skn-frcn ceffcen decreases hen Prandl nmber Pr Schmd nmber Sc he assn fld parameer γ he Angle f nclnan parameer α he hermal Radan parameer Q he agnec feld parameer are ncreases here as ncreases h ncreasng vales f Sre nmber Sr. 6. As he Prandl nmber Pr and hermal radan parameer Q ncreases he rae f hea ransfer ceffcen r Nssel nmber decreases. 7. Increase n he Schmd nmber Sc resls n a decrease n he rae f mass ransfer ceffcen r Sherd nmber here as ncreases h ncreasng vales f Sre nmber Sr. 8. he nmercal resls baned and cmpared h frmerl repred cases avalable n he pen lerare and he are fnd be n ver gd cncrrence. REFERENES []. N. assn A fl eqan fr he pgmen l sspensns f he prnng nk pe Rhelg f Dsperse Ssems Pergamn Ne Yrk (959) pp. 84. []. S. khpadha assn fld fl and hea ransfer ver a nnlnearl srechng srface hn. Phs. Vl. (3) hp://d.rg/.88/674-56//7/747. [3]. H.. Shak agnehdrdnamc assn fld fl h hea and mass ransfer hrgh a prs medm ver a srechng shee J. Prs eda 5 () pp [4]. S. Nadeem R.. Haq N.S. Akbar Z.H. Khan HD hree-dmensnal assn fld fl pas a prs lnearl srechng shee Alexandra Eng. J. 5 (3) pp [5]. A. Khald I. Khan A. Khan S. Shafe nsead HD free cnvecn fl f assn fld pas ver an scllang vercal plae embedded n a prs medm Eng. Sc. echnl. In. J. 8 (5) pp. 9. [6].. Prnma P. Sreenvasl N.. Redd Slp fl f cassn rhelgcal fld nder varable hermal cndcv h radan effecs Hea ransf. Res. 44 (4) pp [7].. Haa. Farq A. Alsaed hermall srafed sagnan pn fl f assn fld h slp cndns In. J. Nmer. ehds Hea Fld Fl 5 (5) pp [8]. S. Nadeem R. ehmd N.S. Akbar mbned effecs f magnec feld and paral slp n blqel srkng rhelgcal fld ver a srechng srface J. agn. agn. aer. 378 (5) pp [9]. S. khpadha I. ndal A.J. hamkha assn fld fl and hea ransfer pas a smmerc edge Hea ransf Asan Res 4 (3) pp []. Oelakn S. ndal P. Sbanda nsead assn nanfld fl ver a srechng shee h hermal radan cnvecve and slp bndar cndns Alexandra Eng J 55 () (6) pp []. S. V. Salaa. Shanker R. Srnvasa Ra Dble Dffsve Effecs On HD xed nvecn assn Fld Fl ards A Vercall Inclned Plae Flled In Prs edm In Presence Of Nmber: A Fne Elemen echnqe Jrnal f Nanflds Vl. 6 pp []. R. Srnvasa Ra. ahesh Redd G. Jhender Redd Inflence Of Angle Of Inclnan On nsead HD assn Fld Fl Pas A Vercal Srface Flled Prs edm In Presence Of nsan Hea Flx hemcal Reacn And Vscs Dsspan Jrnal f Nanflds Vl. 6 pp [3]. Ecker E. R. G. and Drake R.. 97 Analss f Hea and ass ransfer cgra-hll Ne Yrk. [4]. R. Srnvasa Ra Effecs Of Sre And Dfr On Naral nvecve Fld Fl Pas A Vercal Plae Embedded In Prs edm In Presence Of hermal Radan Va FE Jrnal f he Krean Sce fr Indsral and Appled ahemacs Vl. N. 4 pp [5]. J. Anand Ra P. Ramesh ab R. Srnvasa Ra Fne elemen analss f nsead HD free cnvecn fl pas an nfne vercal plae h Sre Dfr hermal radan and Hea srce ARPN Jrnal f Engneerng and Appled Scences Vl. N. pp [6]. N. Ahmed Sre and Radan Effecs n ransen HD Free nvecn Frm an Implsvel Sared Infne Vercal Plae Jrnal f Hea ransfer Vl. 34 pp [7]. A. Pselnc Inflence f magnec feld n hea and mass ransfer frm vercal srfaces n prs meda cnsderng Sre and Dfr effecs In J Hea ass ransf 47 (4) pp [8]. D. Pal H. ndal HD nn-darc mxed cnvecve dffsn f speces ver a srechng shee embedded n a prs medm h nn-nfrm hea srce/snk varable vscs and Sre effec mmn Nnlnear Sc Nmer Smla 7 () pp [9].. K. Parha hermphress parcle depsn n a nn-darc prs medm nder he nflence f Sre Dfr effecs Hea ass ransf 44 (8) pp [].. Haa. hammad S.A. Shehzad A. Alsaed Sre and Dfr effecs n hree-dmensnal fl ver an expnenall srechng srface h prs medm chemcal reacn and hea srce/snk In J Nmer ehds Hea Fld Fl 5 (4) (5) pp cernlne.cm Open Access Jrnal Page 54

14 Nare f assn Fld n ransen Free nvecn Fl Pas ards an Implsvel Sared []. O.D. aknde P.O. Olanrea nsead mxed cnvecn h Sre and Dfr effecs pas a prs plae mvng hrgh a bnar mxre f chemcall reacng fld hem Eng mmn 98 (7) () pp []. O.D. aknde On HD mxed cnvecn h Sre and Dfr effecs pas a vercal plae embedded n a prs medm Lan Am Appl Res 4 () pp [3]. P.O. Olanrea O.D. aknde Effecs f hermal dffsn and dffsn herm n chemcall reacng HD bndar laer fl f hea and mass ransfer pas a mvng vercal plae h scn/necn Arab J Sc Eng 36 () pp [4]. O.D. aknde K. Zmba O. Anar eg Nmercal sd f chemcall-reacng hdrmagnec bndar laer fl h Sre/Dfr effecs and a cnvecve srface bndar cndn In J hermal Envrn Eng 4 () () pp [5]. D. Pal H. ndal Inflence f chemcal reacn and hermal radan n mxed cnvecn hea and mass ransfer ver a srechng shee n Darcan prs medm h Sre and Dfr effecs Energ nvers anage 6 () pp. 8 hp://dx.d.rg/.6/.encnman [6].. rch. Hasna A. Amahmnd Sre cnvecn n a shall prs cav sbmed nfrm flxes f hea and mass In mmn Hea ass ransf 3 (4) pp [7]. Dash R. K. eha K. N. Jaaraman G.: assn Fld Fl n a Ppe Flled h a Hmgenes Prs edm. In. J. Eng. Sc (996). [8]. gle A.. Vncenne W.. and Glles S. E. 968 Dfferenal Apprxman fr Radave ransfer n a Nn- Gra Gas Near Eqlbrm AIAA J. Vl. 6 pp cernlne.cm Open Access Jrnal Page 55