MHD CASSON VISCOUS DISSIPATIVE FLUID FLOW PAST A VERTICALLY INCLINED PLATE IN PRESENCE OF HEAT AND MASS TRANSFER: A FINITE ELEMENT TECHNIQUE

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1 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Glbal Dgtal Cntral ISSN: Frntrs n Hat and Mass Transfr Avalabl at.thrmalfludscntral.rg MHD CASSON VISCOUS DISSIPATIVE FLUID FLOW PAST A VERTICALLY INCLINED PLATE IN PRESENCE OF HEAT AND MASS TRANSFER: A FINITE ELEMENT TECHNIQUE R. Srnvasa Rau a,*, G. Jthndr Rdd b, G. Antha a a Dpartmnt f Mathmatcs, GITAM Unvrst, Hdrabad Campus, Rudraram, 5039, Tlangana Stat, Inda b Dpartmnt f Mathmatcs, VNR Vgnana Jth Insttut f Engnrng and Tchnlg, Hdrabad, Ranga Rdd (Dt), , Tlangana Stat, Inda ASTRACT In th prsnt stud, cnsdr an nflunc f chmcal ractn n an unstad MHD fr cnvctv, vscus dsspatv Cassn flud fl vr a vrtcall nclnd plat n prsnc f magntc fld, hat and mass transfr. Th mdlng quatns ar cnvrtd t dmnsnlss quatns, thn slvd thrugh fnt lmnt tchnqu. Cmputatns r prfrmd t analz th bhavr f flud vlct, tmpratur, cncntratn and nducd magntc fld n th nclnd vrtcal plat th th varatn f mrgng phscal paramtrs. Cmpard th prsnt rsults th arlr rprtd studs fr crrctnss and applcablt f fnt lmnt tchnqu. Ths mdl ma b usful n v f lab prmntal rsults fr crrctnss and applcablt and usful t analz th flud bhavr n thrmal ngnrng ndustrs th th nflunc f th thrmal, magntc and chmcal ractn ffcts tc. rds: Magntc Fld, Cassn Flud Fl, Fr Cnvctn, Hat and Mass transfr, Fnt Elmnt Mthd.. INTRODUCTION Nn-Ntnan flud thr s a part f flud mchancs basd n th cntnuum thr that a flud partcl ma b cnsdrd as cntnuus n a structur. On f th nn-ntnan fluds s a Psud plastc tm ndpndnt flud hs bhavur s that Vscst dcrass th ncrasng vlct gradnt.g. bld, plmr slutns, tc. Cassn flud s n f th psudplastc fluds that mans shar thnnng fluds (Cassn, 959). At l shar rats th shar thnnng flud s mr vscus than th Ntnan flud, and at hgh shar rats t s lss vscus. S, MHD fl th Cassn flud s rcntl ll-knn. Harnath Rdd t al. (06) studd th nflunc f radatn absrptn and chmcal ractn n unstad magnthdrdnamc fr cnvctv hat and mass transfr Cassn flud past an scllatng vrtcal plat mbddd n a prus mdum n th prsnc f cnstant all tmpratur and cncntratn fl usng fnt dffrnc mthd. Vnkatsarlu and Sata Naraana (06) studd th cmbnd ffct f Srt and Dufur n MHD fl f a Cassn flud past a strtchng sht n th prsnc f vscus dsspatn, chmcal ractn and varabl thrmal cnductvt usng shtng mthd. Rammhan Rdd t al. (06) fund th analtcal slutns f MHD cnvctv fl f a ncmprssbl, vsclastc, radatv, chmcall ractv, lctrcall cnductng and rtatng flud thrugh a prus mdum flld n a vrtcal channl n th prsnc f thrmal dffusn usng prturbatn tchnqu. hattachara (03) fund th numrcal slutns f stad bundar lar stagnatn-pnt fl f Cassn flud and hat transfr tards a shrnkng/strtchng sht usng vr ffcnt shtng mthd. Mustafa and han (05) nvstgatd magntc fld ffcts n Cassn nanflud vr nnlnarl strtchng sht. Nadm t al. (0) analzd th stad stagnatn pnt fl f a nn-ntnan flud (Cassn mdl) tards a strtchng surfac th hat transfr and Nan partcls. Nadm t al. (03) plrd thr dmnsnal lctrcall cnductng bundar lar fl f Cassn flud vr strtchng sht saturatd n a prus mdum. hald t al. (05) nvstgatd th ffcts f magntc fld n fr cnvctn fl f Cassn flud vr scllatng plat mbddd n prus mdum. Sharadha and Shankar (05) studd bundar lar fl f Cassn flud vr pnntall strtchng sht. In th bundar lar, th nflunc f thrmal radatn s rlvant n man ngnrng prblms bcaus f ts applcatns spcall n hgh tmpratur ngnrng prcsss. Th ffct f hat radatn s mprtant n cntrllng th qualt f th fnal prduct as t affcts th rat clng. Du t th abv fact, sm f th authrs hav studd th ffct f thrmal radatn n thr rks, vz., Pal t al. (03), Mukhpadha t al. (0), Akbar t al. (03), hattachara and hs c-rkrs (03) and (0), Rashd t al. (0), Su t al. (0). Ahmd and Mahd (06) dlbratd th magnthdrdnamc rtatng fl f a lamnar ncmprssbl vscus lctrcall cnductng flud n th stagnatn rgn f an mpulsvl rtatng sphr n th prsnc f thrmal radatn, hat and mass transfr ffcts. Prakash t al. (0) studd th cmbnd ffcts f nducd magntc fld and radatn n MHD hat and mass transfr fl f vscus, ncmprssbl, Ntnan flud vr a prus vrtcal plat n th prsnc f vscus and magntc dsspatn usng prturbatn tchnqu. Rapts t al. (003) studd th ffct f radatn n an ptcall thn gra gas flng past a vrtcal nfnt plat n th prsnc f a magntc fld. Ahmd (00) fund th analtcal slutns f nducd magntc fld th radatng flud vr a prus vrtcal plat usng Prturbatn tchnqu. Thrfr ths rk can b cnsdrd as tnsn f Prakash t al. (0). S Nvlt f ths papr s dscussn f numrcal slutns * Crrspndng authr. Emal: srvass999@gmal.cm

2 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 usng fnt lmnt tchnqu f th unstad natural cnvctv Cassn flud fl past vr an vrtcall nclnd plat n th prsnc f a thrmal radatn, chmcal ractn, magntc and vscus dsspatns th nducd magntc fld. Als, th stud f grd ndpndnc f fnt lmnt tchnqu s dscussd thrugh tabular frm. Th bhavrs f dffrnt prtnnt paramtrs n vlct, nducd magntc fld, tmpratur and cncntratn prfls as ll as Skn-frctn, Rat f hat and mass transfr cffcnts s dscussd n dtal. Fg. Crdnat sstm and phscal cnfguratn. MATHEMATICAL FORMULATION In ths prblm, cnsdr th ffcts f magntc and vscus dsspatn n stad t-dmnsnal magnthdrdnamc md cnvctv Cassn flud fl f a nn-ntnan and vscus ncmprssbl radatv flud vr a vrtcal plat th nducd magntc fld and hat and mass transfr. Th crdnat sstm and th phscal mdl f th prblm ar shn n Fg.. Fr ths prsnt rsarch rk, mad th fllng assumptns: ). In th flud rgn, th plat s takn alng as n vrtcall upard drctn and as s nrmal t t. ). It s assumd that th all s prsrvd at an unvarng tmpratur T and cncntratn C hghr than th ambnt tmpratur T and cncntratn C rspctvl. ). A unfrm magntc fld f strngth s assumd t b appld nrmal t th drctn f fl. v). Th magntc Rnlds numbr f th fl s nt takn t b small and hnc th nducd magntc fld s nt nsgnfcant. v). Th thrmal dffusn and dffusn thrm ffcts ar nglctd du t th cncntratn f th dffusng spcs s assumd t b vr small n cmparsn th th thr chmcal spcs. v). All th flud prprts ar assumd t b cnstant cpt th ffct f th prssur gradnt n th bd frc trm. v). Thr s a frst rdr chmcal ractn btn th dffusng spcs and th flud. v). Snc th fl s assumd t b n th drctn f as, thrfr all th phscal quantts ar functns f spac crdnats n nl. Glbal Dgtal Cntral ISSN: ). Cnsdr vscus dsspatn n th nrg quatn. Th rhlgcal quatn f stat fr th Cauch strss tnsr f Cassn flud (Dash t al. (996)) s rttn as * 0 () p, c quvalntl () p, c c hr s shar strss, 0 s Cassn ld strss, s dnamc * vscst, s shar rat, and s th, th cmpnnt f dfrmatn rat, s th prduct basd n th nn-ntnan flud, c s a crtcal valu f ths prduct, s plastc dnamc vscst f th nn-ntnan flud, p (3) dnt th ld strss f flud. Sm fluds rqur a graduall ncrasng shar strss t mantan a cnstant stran rat and ar calld Rhpctc, n th cas f Cassn flud (Nn-Ntnan) fl hr c p () Substtutng Eq. (3) nt Eq. (), thn, th knmatc vscst can b rttn as (5) Fnall s th Cassn flud paramtr and as, th gvrnng quatns f th Cassn flud mdl ( ) gvn b Eqs. (8)-() bcm th gvrnng quatns f th Ntnan flud mdl ( ). Thn undr usual ussnsq s apprmatn alng th th assumptns cnsdrd t th fl, th fundamntal gvrnng partal dffrntal quatns that llustrat th phscal stuatn ar gvn b Equatn f Cnsrvatn f Elctrc Charg: J 0 hr J J, J, J z (6) Gauss La f Magntsm: H 0 H (a cnstant) (7) Equatn f Cnsrvatn f Mmntum: u u v g T T cs (8) * H g C C cs Equatn f Cnsrvatn f Enrg: T T qr u v C p C p C p (9) H C p Equatn f Cnsrvatn f Magntc Inductn: H H u v (0)

3 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Spcs Dffusn Equatn: C C v D k C C r () subct t th apprprat bundar cndtns (Prakash t al. 0) u 0, v V, T T, C C, H 0 at 0 () u U T T,, C C, H 0 as In cas f an ptcall thn gra gas, th prssn fr lcal radant (Prakash t al. 0) s gvn b q r * a T T (3) It s assumd that th tmpratur dffrncs thn th fl ar suffcntl small and that T ma b prssd as a lnar functn f th tmpratur. Ths s btand b pandng T n a Talr srs abut T and nglctng th hghr rdr trms, thus gt (Prakash t al. 0) 3 T T T 3T () Usng th fllng nn-dmnsnal quantts (5) u V H T T C C u,,,,, U U T T C C g T T g C C M, Gr, Gm, (5) V U V U V C p U Pr, Sc, R, Pr m D and th th hlp f Eqs. (3) and (), th gvrnng Eqs. (8)-() rduc t d u du d M Grcs cs 0 Gc (6) d d d d d Pr EcPr Pr m d d d d d d R 0 du d Pr EcPr d d M Pr m Pr m 0 du d (7) (8) d d Sc rsc 0 (9) d d th asscatd ntal and bundar cndtns u 0,,, 0 at 0 (0) u, 0, 0, 0 as All th smbls ar dfnd n nmnclatur. Th mathmatcal statmnt f th prblm s n cmplt and mbds th slutn f Eqs. (6), (7), (8) and (9) subct t bundar cndtns (0). Fr practcal ngnrng applcatns and th dsgn f chmcal ngnrng sstms, th lcal skn-frctn, Nusslt numbr and Shrd numbr ar mprtant phscal paramtrs fr ths tp f bundar lar fl. Th Skn-frctn at th plat, hch n th nndmnsnal frm s gvn b u C f () U 0 Th rat f hat transfr cffcnt, hch n th nn-dmnsnal frm n trms f th Nusslt numbr s gvn b T Nu T T 0 Nu R 0 Glbal Dgtal Cntral ISSN: () Th rat f mass transfr cffcnt, hch n th nndmnsnal frm n trms f th Shrd numbr, s gvn b C 0 Sh ShR (3) C C 0 3. NUMERICAL SOLUTIONS Y FEM 3.. Fnt Elmnt Mthd (FEM): Th fnt lmnt mthd (FEM) s a numrcal and cmputr basd tchnqu f slvng a vart f practcal ngnrng prblms that ars n dffrnt flds such as, n hat transfr, flud mchancs (Srnvasa Rau t al. 05; hargava and Rana, 0; Shr and Rau, 05, Shr and Rau, 06, Svaah and Rau, 03; Srnvasa Rau t al. 06), chmcal prcssng (Ln and L, 003)), rgd bd dnamcs (Dttmr and Prc, 006), sld mchancs (Hansb and Hansb, 003), and man thr flds. It s rcgnzd b dvlprs and usrs as n f th mst prful numrcal analss tls vr dvsd t analz cmpl prblms f ngnrng. Th sphstcatn f th mthd, ts accurac, smplct, and cmputablt all mak t a dl usd tl n th ngnrng mdlng and dsgn prcss. It has bn appld t a numbr f phscal prblms, hr th gvrnng dffrntal quatns ar slvd b transfrmng thm nt a matr quatn. Th prmar fatur f FEM s ts ablt t dscrb th gmtr r th mda f th prblm bng analzd th grat flblt. Ths s bcaus th dscrtzatn f th dman f th prblm s prfrmd usng hghl flbl unfrm r nn unfrm patchs r lmnts that can asl dscrb cmpl shaps. Th mthd ssntall cnssts, assumng th pcs cntnuus functn fr th slutn and btanng th paramtrs f th functns n a mannr that rducs th rrr n th slutn. An cllnt dscrptn f fnt lmnt frmulatns s avalabl n ath (996) and Rdd (985). Th stps nvlvd n th fnt lmnt analss aras flls Dscrtzatn f th Dman: Th basc cncpt f th FEM s t dvd th dman r rgn f th prblm nt small cnnctd patchs, calld fnt lmnts. Th cllctn f lmnts s calld th fnt lmnt msh. Ths fnt lmnts ar cnnctd n a nn vrlappng mannr, such that th cmpltl cvr th ntr spac f th prblm Gnratn f th Elmnt Equatns: A tpcal lmnt s slatd frm th msh and th varatnal frmulatn f th gvn prblm s cnstructd vr th tpcal lmnt. Ovr an lmnt, an apprmat slutn f th varatnal prblm s suppsd, and b substtutng ths n th sstm, th lmnt quatns ar gnratd. Th lmnt matr, hch s als knn as stffnss matr, s cnstructd b usng th lmnt ntrplatn functns. Ths stps rsult n a matr quatn f th frm u F hch dfns th fnt lmnt mdl f th rgnal quatn Assmbl f th Elmnt Equatns: Th algbrac quatns s btand ar assmbld b mpsng th ntr lmnt cntnut cndtns. Ths lds a larg numbr f algbrac quatns knn as th glbal fnt lmnt mdl, hch gvrns th hl dman. 3

4 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt Impstn f th undar Cndtns: On th assmbld quatns, th Drchlt's and Numann bundar cndtns (0) ar mpsd Slutn f Assmbld Equatns: Th assmbld quatns s btand can b slvd b an f th numrcal tchnqus, naml, Gauss lmnatn mthd, LU dcmpstn mthd, and th fnal matr quatn can b slvd b a drct r ndrct (tratv) mthd. Fr cmputatnal purpss, th crdnat s vard frm 0 t ma = 9, hr ma rprsnts nfnt.., trnal t th mmntum, nrg and cncntratn bundar lars. Th hl dman s dvdd nt a st f 90 ln sgmnts f qual dth 0., ach lmnt bng t-ndd. 3. Varatnal frmulatn: Th varatnal frmulatn asscatd th Eqs. (6)-(9) vr a tpcal t-ndd lnar lmnt, s gvn b d u du d M Grcs d d d d 0 () Gccs d d Prm PrmPr PrmR Pr d d d 0 (5) du d PrmEcPr EcPr d d d du d 3 M Pr m Pr m 0 d (6) d d d d d Sc rsc 0 d (7) d d Whr,, 3, ar arbtrar tst functns and ma b vd as th varatn n u,, and rspctvl. Aftr rducng th rdr f ntgratn and nn-lnart, arrv at th fllng sstm f quatns du d du d M d d d d d cs cs Gr Gc (8) u 0 Pr mpr Pr m Pr mec Pr Ec Pr Pr mpr R u u Pr m 0 d (9) d d3 M 3 d d d Prm 3 d d d d d rsc Prm Sc du d d d d d 0 3 Glbal Dgtal Cntral ISSN: (30) (3) 3.3 Fnt Elmnt frmulatn: Th fnt lmnt mdl ma b btand frm Eqs. (8)-(3) b substtutng fnt lmnt apprmatns f th frm: u,,,, (3) u Wth 3 (,) hr u,, and ar th Vlct, tmpratur, nducd magntc fld and cncntratn rspctvl at th th nd f tpcal th lmnt, and, and ar takn as: ar th shap functns fr ths lmnt and, Th fnt lmnt mdl f th quatns fr s gvn b 3 u M M M M u 3 M M M M M M M M 3 M M M M b b 3 b b mn mn Whr, M, u,,,, u, m, and b, n,, 3,,, (33) th lmnt thus frmd (3) m ar th st f matrcs dfnd as flls: d d, M 3 d, Gr Gc d, ,

5 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 M M M 3 M M M 3 3 M M M 33 3 M M M Pr mpr M M Pr m d, 3 b d, 3 d Pr mprr d, Pr mecpr M M EcPr d, 3 3 b d M Pr m d, d 3 0, d Pr m d, Sc d, rsc d, b, Pr 3 m, b, In n - dmnsnal spac, lnar lmnt, quadratc lmnt, r lmnt f hghr rdr can b takn. Th hl dman s dvdd nt a st f 90 ntrvals f qual lngth 0.. At ach nd functns ar t b valuatd. Hnc aftr assmbl f th lmnts btan a st f 36 quatns hch ar nnlnar. Thrfr, an tratv schm must b utlzd n th slutn. Aftr mpsng th bundar cndtns, a sstm f quatns has bn btand hch s slvd b th Gauss lmnatn mthd hl mantanng an accurac f A cnvrgnc crtrn basd n th rlatv dffrnc btn th currnt and prvus tratns s mpld. Whn ths dffrncs satsf th dsrd accurac, th slutn s assumd t hav bn cnvrgd and tratv prcss s trmnatd. Th Gaussan quadratur s mplmntd fr slvng th ntgratns. Th cd f th algrthm has bn cutd n MATLA runnng n a PC. Ecllnt cnvrgnc as achvd fr all th rsults.. STUDY OF GRID INDEPENDENCE OF FINITE ELEMENT METHOD In gnral, t stud th grd ndpndnc/dpndnc, th msh sz shuld b vard n rdr t chck th slutn at dffrnt msh (grd) Glbal Dgtal Cntral ISSN: szs and gt a rang at hch thr s n varatn n th slutn. Th numrcal valus f vlct (u), Inducd magntc fld (), tmpratur (θ) and cncntratn (ϕ) fr dffrnt valus f msh (grd) sz ar shn n tabl. Frm ths tabl, th authrs bsrvd that, thr s n varatn n th valus f vlct (u), Inducd magntc fld (), tmpratur (θ) and cncntratn (ϕ) fr dffrnt valus f msh (grd) sz. Hnc, t s cncludd that th rsults ar ndpndnt f msh (grd) sz. Tabl-. Th numrcal valus f u,, θ and ϕ fr varatn f msh szs Msh (Grd) Sz = u θ ϕ Msh (Grd) Sz = 0.00 u θ ϕ Msh (Grd) Sz = 0.0 u θ ϕ CODE VALIDATION Fr prgram cd valdatn, th authr cmpard th prsnt numrcal rsults th analtcal rsults f Prakash t al. (0), Rapts t al. (003) and Ahmd (00) n tabls, 3 and rspctvl hch ar avalabl n ltratur. Frm ths tabls, th authr bsrvd that th rsults ar n gd agrmnt th thr stud. Tabl-: Cmparsn btn th prsnt Vlct, Inducd magntc fld and Tmpratur rsults th th rsults f Prakash t al. (0) fr Prm << M. Rsults f Prakash t al. (0) Prsnt numrcal rsults u θ u θ M = 0.5 M = 0.5 R = 0.5 M = 0.5 M = 0.5 R =

6 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Tabl-3: Cmparsn btn th prsnt vlct rsults th th vlct rsults f Prakash t al. (0), Rapts t al. (003) and Ahmd (00) fr Prm = 0. < M = 0.5, Gr = 5.0, γ = 0.0, ψ = 0.0 and Gc = 0.0. Rsults f Rapts t al. (003) Rsults f Prakash t al. (0) Rsults f Ahmd (00) Prsnt vlct rsults R = 0. R = 0. R = 0. R = Glbal Dgtal Cntral ISSN: th vlct, nducd magntc fld, tmpratur and cncntratn prfls can b apprachd t th rlvant fr stram vlct. Th varatn f numrcal valus f thrmal Grashf numbr r Grashf numbr fr hat transfr (Gr) n vlct prfls at th bundar lar s as shn n th Fg.. It s bsrvd that an ncras n Gr lads t ncras n th flud vlct du t nhancmnt n buanc frc. Hr, th pstv valus f Gr crrspnd t clng f th plat. In addtn, t s bsrvd that th vlct ncrass sharpl nar th all as Gr ncrass and thn dcas t th fr stram valu. Tabl-: Cmparsn btn th prsnt Inducd magntc fld rsults th th rsults f Prakash t al. (0), Rapts t al. (003) and Ahmd (00) fr Prm = 0. < M = 0.5, Gr = 5.0, γ = 0.0, ψ = 0.0 and Gc = 0.0. Rsults f Rapts t al. (003) Rsults f Prakash t al. (0) Rsults f Ahmd (00) Prsnt Inducd Magntc fld rsults R = 0. R = 0. R = 0. R = Fg. 3 Influnc f Gc n vlct prfls 6. RESULTS AND DISCUSSION Fg. Influnc f M n vlct prfls Fg. Influnc f Gr n vlct prfls Numrcal slutns f nn-lnar cupld partal dffrntal quatns (6)-(9) undr bundar cndtns (0) ar btand thrugh fnt lmnt Galrkn tchnqu fr flud vlct, tmpratur, cncntratn and nducd magntc fld. Th bhavur f th flud (.., flud vlct, tmpratur, nducd magntc fld and cncntratn) as analzd and als Skn-frctn, Nusslt numbr and Shrd numbr th varus phscal paramtrs at =.0 thrugh th Fgs. ()-(0). In ths stud th bundar cndtn fr s rplacd b ma = 9 s a suffcntl larg valu f, hr Fg. 3 shs th nflunc f Grashf numbr fr mass transfr (Gc) n th vlct prfl at th bundar lar. Th vlct dstrbutn attans a dstnctv mamum valu n th vcnt f th plat and thn dcrass prprl t apprach a fr stram valu. As pctd, th flud vlct ncrass and th pak valu bcms mr dstnctv du t ncras n th buanc frc rprsntd b Gc. Th ffct f th Hartmann numbr (r) Magntc fld paramtr s shn n Fg.. It s bsrvd that th vlct f th flud dcrass th th ncras f th magntc fld paramtr valus. Th dcras n th vlct as th Magntc fld paramtr ncrass s bcaus th prsnc f a magntc fld n an lctrcall cnductng flud ntrducs a frc calld th Lrntz frc, hch acts aganst th fl f th magntc fld s appld n th nrmal drctn, as n th prsnt stud. Ths rsstv frc sls dn th flud vlct cmpnnt. Fgs. 5 and 6 6

7 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Glbal Dgtal Cntral ISSN: sh th bhavur f flud vlct and flud tmpratur rspctvl th an nflunc f Prandtl numbr. Flud vlct and tmpratur dcrass as ncrasng f Prandtl numbr du t nhancmnt f th thrmal cnductvt. Snc Prandtl numbr s th rat f mmntum dffusvt t thrmal cnductvt. Fr small Prandtl numbr, thrmal cnductvt s hgh and mmntum dffusvt s l, bcaus f that flud vscst s l. Thrfr rduc th hat transfr fr small Prandtl numbr. Fg. 7 Influnc f Sc n vlct prfls Fg. 5 Influnc f Pr n vlct prfls Fg. 8 Influnc f Sc n cncntratn prfls Fg. 6 Influnc f Pr n tmpratur prfls Fgurs 7 and 8 sh that, nflunc f Schmdt numbr (Sc) n vlct prfl and tmpratur prfl. It s bsrvd that, Schmdt numbr (Sc) s n th cncntratn and t s cupld n th mmntum quatn. Incrasng f Schmdt numbr (Sc) th mmntum bundar lar thcknss s ncrasng as ll as flud vlct and flud cncntratn s dcrasng n th ntr bundar f th rgn. Th ffct f th vscus dsspatn paramtr.., th Eckrt numbr Ec n th vlct and tmpratur ar shn n Fgs. 9 and 0 rspctvl. Th Eckrt numbr Ec prsss th rlatnshp btn th kntc nrg n th fl and th nthalp. It mbds th cnvrsn f kntc nrg nt ntrnal nrg b rk dn aganst th vscus flud strsss. Gratr vscus dsspatv hat causs a rs n th tmpratur, as ll as th vlct and crss fl vlct. Ths bhavur s vdnt frm Fgs. 9 and 0. 7 Fg. 9 Influnc f Ec n vlct prfls Fgs. and sh th nflunc f magntc Prandtl numbr Prm n th vlct f flud and tmpratur f flud. th flud vlct and tmpratur dcras as ncrasng f magntc Prandtl numbr Prm. Th ffcts f th thrmal radatn paramtr R n th vlct and tmpratur prfls n th bundar lar ar llustratd n

8 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Fgs. 3 and rspctvl. Incrasng th thrmal radatn paramtr prducs sgnfcant ncras n th thrmal cndtn f th flud and ts thrmal bundar lar. Ths ncras n th flud tmpratur nducs mr fl n th bundar lar causng th vlct f th flud thr t ncras. As pctd, th prsnc f th chmcal ractn sgnfcantl affcts th cncntratn prfls as ll as th vlct prfls frm Fgs. 5 and 6. It shuld b mntnd that th studd cas s bth fr dstructv and gnratv chmcal ractn. In fact, as chmcal ractn ncrass, th cnsdrabl rductn n th vlct prfls s prdctd, and th prsnc f th pak ndcats that th mamum valu f th vlct ccurs n th bd f th flud cls t th surfac but nt at th surfac. Als, th an ncras n th chmcal ractn paramtr, th cncntratn dcrass. It s vdnt that th ncras n th chmcal ractn sgnfcantl altrs th cncntratn bundar lar thcknss but ds nt altr th mmntum bundar lars. Glbal Dgtal Cntral ISSN: Fg. 9 fr ak buanc and arfl. Clarl, as Hartmann numbr ncrass, th nducd magntc fld rducs. Th ffct f magntc Prandtl numbr Prm n th nducd magntc fld s prsntd n th Fg. 0. In ths fgur magntc Prandtl numbr Prm s st as lss than unt, hch mpls that th magntc dffusn rat cds th vscus dffusn rat. As such Prm ncrass, mmntum dffusvt ll ncras. Thrfr, Prm ncrass frm 0. t.5, th nducd magntc fld s fund t ncras abslutl n th bundar lar 0 6, but ths trnd s ppst fr th rgn 6 9. Gratr flu rvrsal arss n th bundar lar rgn 0, 6 and fr Prm = 0. (magntc dffusn rat cds th vscus dffusn 6, 9. rat); but ths trnd s rvrsd fr th rgn Fg. Influnc f Prm n tmpratur prfls Fg. 0 Influnc f Ec n tmpratur prfls Fg. Influnc f Prm n vlct prfls Fg. 7 shs th flud vlct th an nflunc f Cassn flud paramtr. Wth an ncrasng Cassn paramtr, th flud vlct dcrass n th ntr bundar rgn. Snc flud vscst ll b hgh th ncras f Cassn paramtr, mmntum bundar lar thcknss dcrass. An nflunc f angl f nclnatn f th plat n th vlct prfl s shn n th Fg. 8. It s bsrvd that flud vlct dcrass th ncrasng nclnatn paramtr (ψ). Th rspns f nducd magntc fld t Hartmann numbr s shn n th 8 Fg. 3 Influnc f R n vlct prfls Tabls 5 and 6 rprsnt th varatns f Ec, R, γ and Prm, r, ψ n skn-frctn cffcnt rspctvl. Th skn-frctn cffcnt s ncrasng th ncrasng valus f Ec and dcrasng th ncrasng valus f R and γ. Th nfluncs f R, Pr and Ec n rat f hat transfr cffcnt r Nusslt numbr ar dscussd n tabl 7 th th hlp f numrcal valus. Th rat f hat transfr cffcnt s ncrasng th ncrasng valus f Ec and th rvrs ffct s bsrvd th ncrasng valus f R and Pr. Th cmbnd nflunc f Sc and r n rat f mass transfr cffcnt r Shrd numbr s dscussd n tabl 8. Frm ths tabl, th authrs bsrvd that th rat f mass transfr cffcnt s dcrasng th ncrasng valus f Sc and r.

9 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Glbal Dgtal Cntral ISSN: Fg. Influnc f R n tmpratur prfls Fg. 6 Influnc f r n cncntratn prfls Fg. 5 Influnc f r n vlct prfls Fg. 7 Influnc f γ n vlct prfls Tabl-5: Skn-frctn valus fr varatn f Ec, R and γ Ec R γ Cf Tabl-6: Skn-frctn valus fr varatn f Prm, r and ψ Prm r ψ Cf Fg. 8 Influnc f ψ n vlct prfls 9

10 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Fg. 9 Influnc f M n nducd magntc fld Glbal Dgtal Cntral ISSN: Th numrcal slutns hav bn dvlpd fr vlct, nducd magntc fld, tmpratur, cncntratn, Skn-frctn, Rat f hat and mass transfr cffcnts. Th faturs f th fl charactrstcs r analzd b plttng graphs and dscussd n dtal. Th nflunc f Grashf numbr fr hat and mass transfr stablzs th mmntum bundar lar grth. Incrasng th valus f Cassn flud paramtr, Hartmann numbr and Angl f nclnatn f th plat rtards th vlct f th fl fld at all pnts. A grng Prandtl numbr dcrass tmpratur f th fl fld at all pnts and ncrass th ncrasng thrmal radatn and vscus dsspatn paramtrs. Inducd Magntc fld dcrass th th ncrasng Hartmann numbr. Magntc Prandtl numbr dpl nflunc n th nducd magntc fld n th thrmal bundar lar.. nducd magntc fld dcrass th th ncras n th Magntc Prandtl numbr. Th cncntratn rducs th rsng f Schmdt numbr. Th vlct as ll as cncntratn rducs th an nlargmnt n th chmcal ractn paramtr. Th numrcal rsults ar btand and cmpard th frmrl rprtd cass avalabl n th pn ltratur and th ar fund t b n vr gd cncurrnc. Th prsnt stud has bn cnfnd t nn-ntnan vscus mdl. Futur nvstgatns ll cnsdr vsclastc and pr-la rhlgcal flud mdls and ll b cmmuncatd n th nar futur. ACNOWLEDGEMENT Authrs ar gratful t th rvrs fr thr valuabl cmmnts and suggstns hch hlpd thm t mprv th qualt f ths rsarch papr. Als th authrs ar thankful t Dr.. Tasan, GITAM Unvrst, Hdrabad campus, fr hr nputs. Fg. 0 Influnc f Prm n nducd magntc fld Tabl-7: Nusslt numbr valus fr varatn f Ec, R and Pr Ec R Pr Nu Tabl-8: Shrd numbr valus fr varatn f Sc and r Sc r Sh CONCLUSIONS Th prsnt stud addrssd th charactrstcs f hat and mass transfr n unstad magnthdrdnamc fr cnvctv Cassn flud fl f an ptcall thck flud vr an nclnd vrtcal plat th magntc and vscus dsspatns. Th st f fundamntal gvrnng quatns hav bn slvd numrcall usng fnt lmnt mthd. 0 C NOMENCLATURE Cncntratn f th flud far aa frm th plat 3 ( g m ) 3 C Cncntratn f th plat ( g m ) Dmnsnlss dsplacmnt ( m ) Gc Grashf numbr fr mass transfr u Nn-dmnsnal flud vlct ( m s ) Sh Th lcal Shrd numbr T Flud tmpratur aa frm th plat v Vlct cmpnnt ( m s ) u Vlct cmpnnt n drctn ( m s ) g Acclratn f gravt, 9.8 ( m s ) a A Cnstant r Chmcal ractn paramtr Crdnat as alng th plat ( m ) C-rdnat as nrmal t th plat ( m ) J Currnt dnst vctr ( A. s. m ) k r Dmnsnal Chmcal ractn paramtr Ec Eckrt numbr (Vscus Dsspatn paramtr) C 3 Flud Cncntratn ( g m ) T Flud Tmpratur ( ) T Flud tmpratur at th all

11 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 Gr Grashf numbr fr hat transfr Inducd Magntc fld (Tsla) H Inducd Magntc fld alng drctn (Tsla) H Inducd Magntc fld alng drctn (Tsla) M Magntc fld paramtr (r) Hartmann numbr Pr m Magntc Prandtl numbr Pr Prandtl numbr q r Radatv hat transfr cffcnt U Rfrnc vlct at th plat ( m s ) R Rnlds numbr J, J, J ) Scalar Cmpnnts f J ( z Sc Schmdt numbr D Slut mass dffusvt ( m s ) C Spcfc hat at cnstant prssur J g p Nu Th lcal Nusslt numbr C Th lcal skn-frctn ( N m ) f 0 Unfrm magntc fld (Tsla) Grk Smbls: Thrmal cnductvt f th flud ( W m ) Nn dmnsnal flud tmpratur Shar strss ( N m ) * Vlumtrc Cffcnt f thrmal pansn th 3 cncntratn ( m g ) Angl f nclnatn f plat ( dg rs ) Cassn flud paramtr Elctrc cnductvt f th flud ( s m ) nmatc vscst ( m s ) Magntc Prmablt ( N. A ) 3 Spcs cncntratn ( g m ) * Stfan ltzmann Cnstant ( W. m. ) 3 Th cnstant dnst ( g m ) Suprscrpts: / Subscrpts: p Vlumtrc cffcnt f thrmal pansn ( ) Dmnsnlss Prprts Fr stram cndtns Plat Cndtns n th all REFERENCES Ahmd, S., 00, "Inducd Magntc Fld th Radatng Flud vr a Prus Vrtcal Plat: Analtcal Stud," J. Navl f Arch. Marn Eng., 7, 6-7. Glbal Dgtal Cntral ISSN: Ahmd, S., and Mahd, A., 06, "Unstad MHD Dubl Dffusv Cnvctn n th Stagnatn Rgn f an Impulsvl Rtatng Sphr n th Prsnc f Thrmal Radatn Effct," J. Taan Inst. Chm. Eng., 58, Akbar, N.S., Nadm, S., Haq, R.U., and han, Z.H., 03, "Radatn Effcts n MHD Stagnatn Pnt Fl f Nanflud Tards a Strtchng Surfac th Cnvctv undar Cndtn," Chn. J. Arnaut., 6, ath,.j., 996, "Fnt Elmnt Prcdurs," Prntc-Hall, N Jrs. hargava, R., and Rana, P., 0, "Fnt Elmnt Slutn t Md Cnvctn n MHD Fl f Mcrplar Flud alng a Mvng Vrtcal Clndr th Varabl Cnductvt," Int. J. Appl. Math. Mch., 7, 9-5. hattachara,., 03, "undar Lar Stagnatn-Pnt Fl f Cassn Flud and Hat Transfr Tards a Shrnkng/Strtchng Sht," Frnt. Hat Mass Transf.,, hattachara,., 03, "MHD Stagnatn-Pnt Fl f Cassn Flud and Hat Transfr vr a Strtchng Sht th Thrmal Radatn," J. Thrmdn., 03, hattachara,., Mukhpadha, S., Lak, G.C., and Pp, I., 0, "Effcts f Thrmal Radatn n Mcrplar Flud Fl and Hat Transfr vr a Prus Shrnkng Sht," Int. J. Hat Mass Transf., 55, Cassn, M., 959, "A Fl Equatn fr Pgmnt-Ol Suspnsns f th Prntng Ink Tp,". In: C. C. Mlls (d.), Rhlg f dsprs sstms, Nrk, Ofrd: Prgamn Chamkha, A.J., and Camll, I., 000, "Effcts f Hat Gnratn/Absrptn and Thrmphrss n Hdrmagntc Fl th Hat and Mass Transfr vr a Flat Surfac," Int. J. Numr. Mthds Hat Flud Fl, 0, Dash, R.., Mhta,.N., and Jaaraman, G., 996, "Cassn Flud Fl n a Pp Flld th a Hmgnus Prus Mdum," Int. J. Eng. Sc., 3, Dttmr, W., and Prc, D., 006, "A Cmputatnal Framrk fr Flud-Rgd d Intractn: Fnt Elmnt Frmulatn and Applcatns," Cmput. Mthds Appl. Mch. Eng., 95, Hansb, A., and Hansb, P., 00, "A Fnt Elmnt Mthd fr th Smulatn f Strng and Wak Dscntnuts n Sld Mchancs," Cmput. Mthds Appl. Mch. Eng., 93, Harnath Rdd, S., Rau, M. C., and shava Rdd, E., 06, "Radatn Absrptn and Chmcal Ractn Effcts n MHD Fl f Hat Gnratng Cassn Flud Past Oscllatng Vrtcal Prus Plat," Frnt. Hat Mass Transf., 7,.

12 Frntrs n Hat and Mass Transfr (FHMT), 8, 7 (07) DOI: /hmt.8.7 hald, A., han, I., han, A., and Shaf, S., 05, "Unstad MHD Fr Cnvctn Fl f Cassn Flud Past vr an Oscllatng Vrtcal Plat Embddd n a Prus Mdum," Eng. Sc. Tchnl. Int. J., 80, Ln, Y-Y., and L, S-P., 003, "Fnt Elmnt Mdlng fr Chmcal Mchancal Plshng Prcss undr Dffrnt ack Prssurs," J. Matr. Prcssng Tchnl., 0, Mukhpadha, S., hattachara,., and Lak, G.C., 0, "Stad undar Lar Fl and Hat Transfr vr a Prus Mvng Plat n Prsnc f Thrmal Radatn," Int. J. Hat Mass Transf., 5, Mustafa, M., and han, J.A., 05, "Mdl fr L f Cassn Nanflud Past a Nnlnarl Strtchng Sht Cnsdrng Magntc Fld Effcts," AIP Adv., 5, Nadm, S., Haq, R.U., Akbar, N.S., and han, Z.H., 03, "MHD Thr-Dmnsnal Cassn Flud Fl Past a Prus Lnarl Strtchng Sht," Alandra Eng. J., 5, Nadm, S., Mhmd, R., and Akbar, N.S., 0, "Oblqu Stagnatn Pnt Fl f a Cassn Nanflud Tards a Strtchng Surfac th Hat Transfr," J. Cmp. Thr. Nansc.,, Pal, D., Mandal, G., and Varavlu,., 03, "MHD Cnvctn- Dsspatn Hat Transfr vr a Nn-Lnar Strtchng and Shrnkng Shts n Nanfluds th Thrmal Radatn," Int. J. Hat Mass Transf., 65, Prakash, J., Vaa umar, A.G., Rush umar,., and Varma, S.V.., 0, "Th Effcts f Inducd Magntc Fld and Radatn n MHD Md Cnvctv Fl vr a Prus Vrtcal Plat n th Prsnc f Vscus and Magntc Dsspatns," Int. Cnfrnc n Adv. Marn, Industral and Mch. Eng., 5-6. Ramamhan Rdd, L., Rau, M. C., Rau, G.S.S., and Rdd, N.A., 06, "Thrmal Dffusn and Rtatnal Effcts n Magnthdrdnamc Md Cnvctn Fl f Hat Absrbng/Gnratng Vsc-Elastc Flud Thrugh a Prus Channl," Frnt. Hat Mass Transf., 7, 0. Rapts, A.A., Prdks, C., and Lnttss, A., 003, "Effcts f Radatn n an Optcall Thn Gra Gas Flng Past a Vrtcal Infnt Plat n th Prsnc f a Magntc Fld," Hat Mass Transf., 39, Rashd, M.M., Al, M., Frdnmhr, N., Rstam,., and Hssan, M.A., 0, "Md Cnvctv Hat Transfr fr MHD Vsclastc Glbal Dgtal Cntral ISSN: Flud Fl vr a Prus Wdg th Thrmal Radatn," Adv. Mch. Eng., Rdd, J.N., 985, "An Intrductn t th Fnt Elmnt Mthd," McGra - Hll, N Yrk. Sharada,., and Shankar,., 05, "MHD Md Cnvctn Fl f a Cassn Flud vr an Epnntall Strtchng Surfac th th Effcts f Srt, Dufur, Thrmal Radatn and Chmcal Ractn," Wrld J. Mch., 5, Sva Rdd Shr, and Srnvasa Rau, R., 05, "Srt Effct n Unstad MHD Fr Cnvctv Fl Past a Sm-Infnt Vrtcal Plat n th Prsnc f Vscus Dsspatn," Int. J. Cmput. Mthds Eng. Sc. Mch., 6, Sva Rdd Shr, and Srnvasa Rau, R., 06, "Transnt MHD Fr Cnvctv Fl Past an Infnt Vrtcal Plat Embddd n a Prus Mdum th Vscus Dsspatn," Mccanca, 5, Svaah, S., and Srnvasa Rau, R., 03, "Fnt Elmnt Slutn f Hat and Mass Transfr Fl th Hall Currnt, Hat Surc and Vscus Dsspatn," Appl. Math. Mch. Englsh Edtn, 3, Srnvasa Rau, R., Jthndr Rdd, G., Anand Ra, J., Rashd, M.M., and Grla, R.S.R., 06, "Analtcal and Numrcal Stud f Unstad MHD Fr Cnvctn Fl vr an Epnntall Mvng Vrtcal Plat th Hat Absrptn," Int. J. Thrmal Sc., 07, Srnvasa Rau, R., Mahsh Rdd,., Rashd, M.M., and Grla, R.S.R., 06, "Applcatn f Fnt Elmnt Mthd t Unstad MHD Fr Cnvctn Fl Past a Vrtcall Inclnd Prus Plat Includng Thrmal Dffusn and Dffusn Thrm Effcts," J. Prus Mda, 9, Su, X., Zhng, L., Zhang, X., and Zhang, J., 0, "MHD Md Cnvctv Hat Transfr vr a Prmabl Strtchng Wdg th Thrmal Radatn and Ohmc Hatng," Chm. Eng. Sc., 78, Vnkatsarlu,., and Sata Naraana, P.V., 06, "Influnc f Varabl Thrmal Cnductvt n MHD Cassn Flud Fl vr a Strtchng Sht th Vscus Dsspatn, Srt and Dufur Effcts," Frnt. Hat Mass Transf., 7, 6. Zuc, J., ég, O.A., Takhar, H.S., and Prasad, V.R., 009, "Thrmphrtc Hdrmagntc Dsspatv Hat and Mass Transfr th Latral Mass Flu, Hat Surc, Ohmc Hatng and Thrmal Cnductvt Effcts: Ntrk Smulatn Numrcal Stud," Appl. Thrmal Eng., 9,