HEAT AND MASS TRANSFER ON UNSTEADY IN TWO NON-CONDUCTING INFINITE VERTICAL PARALLEL PLATES WITH INCLINED MAGNETIC FIELD

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1 Inrnaional Jornal of Mchanical Enginring and chnolog (IJME Volm 9 Iss 3 Dcmbr 8 pp. 36 Aricl ID: IJME_9_3_ Availabl onlin a hp:// am.com/ijm/isss.asp?jpijme&vp 9&Ip3 ISSN Prin: and ISSN Onlin: IAEME Pblicaion Scops Indxd HEA AND MASS RANSFER ON UNSEADY MHD FLOW IN WO NON-ONDUING INFINIE VERIAL PARALLEL PLAES WIH INLINED MAGNEI FIELD P..Hmamalini Prof Dparmn of Mahmaics Karpagam Acadm of Highr Edcaion amilnad India M.Shanhi Rsarch Scholar Dparmn of Mahmaics Karpagam Acadm of Highr Edcaion amilnad India ASRA h nsad magnohdrodnamic flow bwn wo non condcing infini vrical plas in h prsnc of niform inclind magnic fild has bn sdid. On of h plas is assmd o b in moion wih consan vloci whras h ohr pla is considrd o b adiabaic. sing ransformaion associad wih dca facor w hav ddcdd a ssm of ordinar diffrnial qaions. And hs qaions ar solvd analicall for h vloci flow indcd magnic fild mprar and concnraion for varios phsical paramrs. h rsls hav bn discssd hrogh graphs. Kwords: MHD Flow Ha ransfr Mass ransfr Inclind Magnic Fild Adiabaic. i his Aricl: P..Hmamalini and M.Shanhi Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild Inrnaional Jornal of Mchanical Enginring and chnolog 9(3 8 pp. 36. hp:// /isss.asp?jpijme&vp9&ip 3. INRODUION MHD flow has sn a wid rang of applicaions in h rcn pas and has gaind considrabl anion d o hos in cosmic and gophsical flid dnamics. h rsarch of MHD flow wih inclind magnic fild has aracd h anion of man rsarchrs d o is s for xampl in xrsion of plasics in h manfacring indsris. h phnomnon of ha and mass ransfr has a wid significanc in chmical procssing gohrmal ssms hp:// IJME/indx.asp dior@iam.com

2 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild air condiioning c. I is hrfor inrsing o invsiga h problm of h flow wih h ha and mass ransfr in prsnc of indcd magnic fild. Hazm Ali Aia al (4 [] considrd Hall ffc of nsad Magnohdrodnamic oq flow wih h ha ransfr of a ingham flid along wih injcion and scion. Magnohdrodnamic mixd convcion in a vrical channl was discssd b Umavahi and Malash( [].h sd of an nsad fr convciv MHD flow of dissipaiv flid along wih a vrical pla and consan ha flx was analzd nmricall b Joaqin Zco Jordan(6 [3].Singha (8 [4] drmind h inflnc of ha ransfr on nsad hdromagnic flow in a paralll plas of an lcricall condcing incomprssibl and viscos flid. Singha & Dka(9 [] invsigad on h magnohdrodnamic wo-phas flow wih ha ransfr in h prsnc of niform inclind magnic fild. Analical Solion o h problm of MHD fr convciv flow of an lcricall condcing flid bwn had paralll plas along wih an indcd magnic fild was larnd b Singha (9 [6]. Marnni Narahari& ina.k.da( [7] nmrad h fr convcion flow bwn wo vrical plas wih variabl mprar a on bondar. Magnic fild ffc on fr convciv oscillaor flow bwn wo vrical plas wih priodic mprar and dissipaiv ha is concnrad b Ahmd al( [8]. ishwaram Sharma al(3 [9] invsigad h impac of magnic fild and mprar gradin on sparaion of a binar flid mixr in nsad coq flow. Unsad MHD co flow bondd bwn wo paralll poros plas wih ha ransfr and inclind magnic fild was considrd b Josph al (4 []. Kir & Sra ahadr(4 [] prsnd h sad poisill flow bwn wo paralll poros plas in prsnc of inclind magnic fild. SrKala Ksava Rdd (4 [] discssd h MHD co flow of incomprssibl viscos flid hrogh a poros mdim bwn wo paralll plas ndr h inflnc of inclind magnic fild. Panrslvi and Mohswari( [3] larnd h Sor ffc on nsad convciv flow of a ds viscos flid bwn infini paralll plas mbddd b a poros mdim wih inclind magnic fild. Effc of mass and ha ransfr on nsad MHD Poisill flow bwn wo paralll poros plas in prsncs of an inclind magnic fild is sdid b Josph al ( [4]. Unsad MHD poisill flow wih ha and mass ransfr bwn wo infini paralll plas hrogh poros mdim in an inclind magnic fild was invsigad b Rajp & Garav Kmar ( []. h inflnc of h inclind magnic fild and variabl hrmal condcivi on MHD plan poisill flow pas non-niform pla mprar was considrd b Gpa al ( [6]. Agns Mbr Jackson Kwanza and homson Onango (6 [7] discssd h MHD flid flow bwn infini paralll plas sbjcd o an inclind magnic fild and prssr gradin. MHD of incomprssibl flid hrogh paralll plas in inclind magnic fild wih poros mdim wih mass and ha ransfr was sdid b Rishard Richard Hanv al(7 [8]. Nariki al (7 [9] analsd h nsad Hdro magnic coq flow in h prsnc of variabl inclind magnic fild. Unsad magnohdrodnamic flow of viscos lcricall condcing flid bondd bwn wo non-condcing vrical plas wih inclind magnic fild was sdid b AmarjoiGoswami al (7 []. In his papr w hav sdid h ffc of mass ransfr and ha ransfr on h nsad MHD flow of lcricall condcing flid bwn wo paralll vrical plas wih niform magnic fild. hp:// IJME/indx.asp dior@iam.com

3 P..Hmamalini and M.Shanhi. FORMAION OF HE PROLEM In h prsn work h nsad flow of an incomprssibl lcricall condcing flid bondd b wo non condcing vrical infini paralll plas a a disanc h apar is considrd. h coordina ssm is chosn sch ha X axis is akn in pward dircion and Y- axis is chosn prpndiclar o h plans of h plas. h vrical plas of h channl ar ± h and also hr is no prssr gradin in h flow fild. An xrnal magnic fild maks an angl θ wih h posiiv X-axis which indcs a magnic fild ( and also maks an angl θ o h fr sram vloci.h lf wall -h is kp a consan mprar and h righ wall h is ssaind a consan mprar sch ha >. h magnic fild disribions and vloci ar givn b { } { ( x z λ λ } V { ( } whr λ cosθ and is h im. Figr Schmaic rprsnaion of h problm h following assmpions ar mad o fram h govrning qaions of h problm: (i h lcricall condcing flid is considrd viscos dissipaion and Jol ha ar nglcd. (ii Polarizaion ffc and Hall ffc ar ngligibl. (iii No prssr gradin. (ivh plas ar infinil long along X and Z dircions. hrfor h vloci concnraion and mprar filds ar h fncions of and onl. 3. GOVERNING EQUAIONS Eqaion of onini div V is h vloci. ( Whr V hp:// IJME/indx.asp 3 dior@iam.com

4 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild hp:// IJME/indx.asp 4 dior@iam.com Eqaion of Momnm [ ] ( ( ( (.. ( * g g j V V V V β β µ ρ Eqaion of Magnic Indcion (3 ( V σµ whr μ prmabili of h mdim. Enrg Eqaion (4 k d d ρp ρ dnsi and p - Spcific ha a consan prssr oncnraion Eqaion ( D D Mass diffsion cofficin. Hr J is h lcric crrn dnsi. h gnralizd Ohm s law is ( V E J σ whr σ Elcrical condcivi is h magnic indcion (6 Nglcing h lcric fild E qaion (6 rdcs o ( V J σ whr μ h cofficin of viscosi k h hrmal condcivi h im β co-fficin of hrmal xpansion of h flid and β * co-fficin of h mass ransfr of h flid. Using magnic fild disribion and vloci as mniond abov h qaions ( o ( ar: (8 σµ λ (9 α (7 ( ( ( * g g v β β λ ρ σ

5 P..Hmamalini and M.Shanhi hp:// IJME/indx.asp whr p k ρ α ( D h bondar condiions ar: ( > > a h a h a Using h following dimnsionlss paramrs in h govrning qaions of moion (7 ( ( b h h Nglc h asrisks h dimnsionlss qaion ar: (4 b P R R b r m λ ( P (6 ScR h dimnsionlss form of bondar condiions: (7 > a > a a b b b (3 ( R G R G R H R m r a λ

6 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild o solv h qaion (3 (6 b sing qaion (7 considr h following variabl ransformaion F( xp( n b g( xp( n G( xp( n f ( xp( n whr n dnos h dca consan. Sbsiing (8 in qaions (3(4( and (6w hav { H R ( n} G ( ( r f R a λ f F( GmR G( R ( ( ( λ g nr RmPr g RRm Pr [ f ( ] λ (8 (9 ( F ( np [ F( ] ( G ( nscr G( ( h rlvan bondar condiions ar f g F G a f xp( n g xp( n F xp( n G xp( n a > (3 F f xp( n g xp( n G a > h qaions (9 ( ar solvd wih h bondar condiions qaion (3.h vloci disribion of h fild mprar and spcis concnraion ar: ( A cos( A sin( 3 3 A A (4 b( A 7 A ( A 7 A A 4 sin( A 4 A cos( A ( A A 6 A A A ( 9 3 sin A 3 4 cos A 3 ( cos( A cosa (6 sin( sin (7 Whr { H R ( n} A R a λ hp:// IJME/indx.asp 6 dior@iam.com

7 P..Hmamalini and M.Shanhi A G R r nscr G R m A nr R 3 m P r A 4 RRm Pr λ A np A 6 A cos c(a A A cosc 3 A A 4 A A3 7 3 ( A A A4 3 ( ( A A ( A A ( A 8 A A3 3 4 A ( A A A A 9 A A3 A A A 4 3 A A A 4 4 A ( A3 A ( A3 A A 3 A sc(a A A 6 4 A A3 ( A3 A ( A A A A A 4 A 6 A A RESULS AND DISUSSION Eqaion (4 - (7 ar solvd nmricall for diffrn vals of λ whr θ varis as θ Ploing of all sch cass is carrid o b sing MALA. hp:// IJME/indx.asp 7 dior@iam.com

8 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild From figr i is nod ha h vloci profil incrass wih incras in h angl of inclinaion θ. h vloci profil variaions wih h hrmal Grashof nmbr G r ar shown in figr 3.h magnid of h vloci dgrads h flow wih h incras in h Grashof nmbr G r. Figr 4 dpics h ffc of modifid Grashof nmbr G m and i is clarl sn ha h vloci nhancs h flow whn h modifid Grashof nmbr G m is incrmnd. Figr o figr 7 xplains h ffc of Prandl nmbr P r Harmann nmbr H a and Schmid nmbr Sc on h vloci profil. I is drmind ha h vloci profil shows progrss b incrasing hs prinn paramrs. Figr 8 commnicas h ffc of magnic Rnolds nmbr R m. I has bn noicd ha h vloci gradall incrass whn h magnic Rnolds nmbr R m is incrasd. Figr 9 rvals ha h ffc of dca facor n. h incras in dca facor rdcs h vloci of h flow. Figr o figr 7 illsras h ffc of indcd magnic fild. Figr indicas h incras of angl of inclinaion θ of h flow incrass h innsi of h magnic fild. Figr commnicas h impac of hrmal Grashof nmbr G r on h indcd magnic fild. I is obsrvd ha whn G r is incrasd h innsi of h magnic fild is also incrasd. Figr dpics h ffc of modifid Grashof nmbr G m.wih h incras in modifid Grashof nmbr G m hr appars a sharpl incrasing chang in h innsi of h magnic fild. Figr 3 indicas h cas of Harmann nmbr H a. h indcd magnic fild incrass gradall whn h Harmann nmbr H a is incrasd. From figrs 4 and 6 h ffc of Schmid nmbr Sc Prandl nmbr P r magnic Rnolds nmbr R m and Prandl nmbr P r ar obsrvd. I illsras ha h innsi of h magnic fild is mor whn h paramrs Sc R m and P r ar incrasd. Figr 7 commnicas ha h incras in h dca facor n dgrads h magnic fild innsi. In figr 8 i is obsrvd ha h progrss in h Prandl nmbr P r lads o h incras in h flid mprar. In figr 9 i is nod ha h concnraion fild incrass wih h sp forward in h Schmid Figr : Vloci Profil for disinc vals of θ [nr m.r.pr.7 H a 7G m G r and S c. 3] hp:// IJME/indx.asp 8 dior@iam.com

9 P..Hmamalini and M.Shanhi Figr 3:Vloci Profil for diffrn vals of G r [nλ. R m.r.h a 7 G m S c.3 and Pr.7] Figr 4: Graph of vloci Profil for diffrn vals of G m [ nr m.λ.r.h a 7Pr.7S c.3and G r ] Figr : Vloci Profil for nlik vals of P r [G m R. nr m.h a 7 λ. G r S c.3 and R m.] hp:// IJME/indx.asp 9 dior@iam.com

10 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild Figr 6: Vloci graph for varios vals of H a [ R. R m. λ. G m G r S c.3 nand Pr.7] Figr 7:Vloci Profil for diffrn vals of Sc[R. nh a 7λ.R m.pr.7g r S c.3 and G m ] Figr 8: Vloci Profil for divrs R m vals[nr.g m H a 7 λ. S c.3 G r and Pr.7] hp:// IJME/indx.asp 3 dior@iam.com

11 P..Hmamalini and M.Shanhi Figr 9: Vloci graph for diffrn vals of n [R m.pr.7r.s c.3 λ. G m G r and H a 7] Fig : Magnic Fild graph for diffrn vals of [R m S c.3 R.G m H a 7 λ.npr.7 and G r ] Figr:Magnic Fild Profil for disinc vals of G r [Pr.7H a 7 R. λ. R m. S c.3g m and n] hp:// IJME/indx.asp 3 dior@iam.com

12 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild Figr: Magnic Fild Skch for disinc vals of G m [Pr.7 nr.h a 7 λ. G r.g m S c.3 and R m.] Figr3: Magnic Fild Graph for siabl vals of h [S c.3 Pr.7 G m nr.h a 7 λ. R m. G r S c.3 and R.] Figr 4: Magnic Fild Graph for varios vals of Pr[H a 7 nr.r m. G r G m S c.3 and λ.]. hp:// IJME/indx.asp 3 dior@iam.com

13 P..Hmamalini and M.Shanhi Figr: Magnic Fild Profil for disinc vals of R m [ng r R.H a 7Pr.7 G m S c.3 and λ.] Figr 6: Graph of Magnic Fild for diffrn vals of Sc[P r.7h a 7G m R.G r λ.n and R m.] Figr7:Magnic Fild Profil for varios vals of n[r.s c.3p r.7h a 7λ.R m. G r G m and H a 7]. hp:// IJME/indx.asp 33 dior@iam.com

14 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild mprar Profil Figr8: A Graph of mprar Fild Profil for siabl vals of Pr[R. and n] oncnraion Profil Figr 9: Skch of oncnraion for varios vals of Sc [n R.]. ONLUSION h problm of h ffc of inclind magnic fild on nsad magnohdrodnamic vrical flow bwn wo infini non condcing vrical plas wih ha and mass ransfr has bn invsigad. From h analsis h following conclsions wr mad. h vloci pgrads wih incras of angl of inclinaion θ of h magnic fild. h vloci for h flow incrass wih h incras in G m P r H a and Sc whil i rards wih incras in hrmal Grashof nmbr G r and h dca facor n. h indcd magnic fild incrass wih h incras in h angl of inclinaion θ hrmal Grashof nmbr G r modifid Grashof G m Harmann nmbr H a Prandl nmbr P r magnic Rnolds nmbr R m and Schmid S c and also i rdcs wih h incras of dca facor n. h mprar disribion was sn o progrss d o ris in h Prandl nmbr P r. oncnraion profil incrass wih h incrmn vals of Schmid nmbr Sc. hp:// IJME/indx.asp 34 dior@iam.com

15 P..Hmamalini and M.Shanhi REFERENES [] Hazm Ali Aia and Mohamd Eissa Sd-Ahmd Hall ffc on nsad MHD o flow and ha ransfr of a ingham flid wih scion and injcon Applid Mahmaical Modlling 8( [] UmavahiJ. and Malash.M.S Magnohdrodnamic mixd convcion in a vrical channlinrnaional Jornal of Non-Linar Mchanics 4( 9-. [3] Joaqin ZcoJordan Nmrical sd of an nsad fr convciv magnohdrodnamic flow of dissipaiv flid along a vrical pla sbjc o a consan ha flx Inrnaional Jornal of Enginring Scinc 44( [4] SinghaK.G.h ffc of ha ransfr on nsad hdromagnic flow in a paralll plas channl of an lcricall condcing viscos incomprssibl flid In.J.Flid Mch.Rsarch3( [] SinghaK.G.& DkaP.N. Magnohdrodnamic ha ransfr in wo-phas flow in prsnc of niform inclind magnic fild ll.al.mah.soc (9-6. [6] SinghaK.G..Analical Solion o h problm of MHD fr convciv flow of an lcricall condcing flid bwn wo had paralll plas in h prsnc of an indcd magnic fild Inrnaional Jornal of Applid Mahmaics and ompaion( [7] MarnniNarahariina.K.Da.Fr convcion flow and ha ransfr bwn wo vrical paralll plas wih variabl mprar a on bondaracachnica [8] AhmdN.SharmaK.&araD.P.Magnic fild ffc on fr convciv oscillaor flow bwn wo vrical paralll plas wih priodic pla mprar and dissipaiv ha Jornal of Applid mahmaical Scincs 6( [9] ishwaramsharma RamNiroj Singh & RpamKr.Gogi.Effc of magnic fild and mprar gradin on sparaion of a binar fild mixr in nsad co flowinrnaional Jornal of Mahmaical Archiv 4( [] Josph K.MDanilS. & Josph G.M.Unsad MHD co flow bwn wo infini paralll poros plas in an inclind magnic fild wih ha ransfr Inrnaional Jornal of Mahmaics and saisics Invnion( [] KirD.R. & Sra ahadr.effc of an inclind magnic fild on sad poisill flow bwn wo paralll poros plas IOSR Jornal of Mahmaics( [] L.SrKalaE. Ksava Rdd.Sad MHD co flow of an incomprssibl viscos flid hrogh a poros mdim bwn wo infini paralll plas ndr ffc of inclind magnic fildh Inrnaional Jornal Of Enginring And Scinc3( [3] PanrslviR.MohswariP.Sor ffc on nsad fr convcion flow of a ds viscos flid bwn wo infini fla paralll plas filld b a poros mdim wih inclind magnic fild Inrnaional Jornal of Scinific &Enginring Rsarch 6( [4] Josph K.M.AbaP.NiorL.N.Mohammd S.M. Effc of ha and mass ransfr on nsad MHD poisill flow bwn wo infini paralll poros plas in an inclind magnic fild Inrnaional Jornal of Scinific Enginring and Applid Scinc( [] RajpU.S. Garav Kmar.Unsad MHD poisill flow bwn wo infini paralll plas hrogh poros mdim in an inclind magnic fild wih ha and mass ransfrinrnaional Jornal of Mahmaical Archiv 6( hp:// IJME/indx.asp 3 dior@iam.com

16 Ha and Mass ransfr on Unsad MHD Flow In wo Non-ondcing Infini Vrical Paralll Plas wih Inclind Magnic Fild [6] GpaV.G.Aja Jain &Abha Kmar Jha.h ffc of variabl hrmal condcivi and h inclind magnic fild on MHD plan poisill flow pas non-niform pla mprar Global Jornal of Scinc Fronir Rsarch ( -8. [7] AgnsMbrJackson Kwanza and homsononango Magnohdrodnamic flid flow bwn wo paralll infini plas sbjcd o an inclind magnic fild ndr prssr gradin Jornal of Mlidisciplinar Enginring Scinc and chnolog 3( [8] Rishard Richard HanvRajv Kmar Khar AjiPal.MHD of incomprssibl flid hrogh paralll plas in inclind magnic fild having poros mdim wih ha and mass ransfrinrnaional Jornal of Scinific and Innovaiv Mahmaical Rsarch( [9] NarikiE.M.KinanjiM.N.KiogoraP.R.Unsad Hdro magnic co flow in prsnc of variabl inclind magnic fildinrnaional Jornal of Enginring Scinc and Innovaiv chnolog6( 7 -. [] Dr.AmarjoiGoswami MrinmoGoswami & Dr.KrishnaGopalSingha A Sd of Unsad Magnohdrodnamic Flow of An Incomprssibl Viscos Elcricall ondcing Flid ondd wo Non-ondcing Vrical Plas in Prsnc of Inclind Magnic Fild Inrnaional Jornal of Enginring Scinc Invnion6( hp:// IJME/indx.asp 36 dior@iam.com

Unsteady Magnetohydrodynamic Stokes Flow of Viscous Fluid with Radiative Heat Transfer

Unsteady Magnetohydrodynamic Stokes Flow of Viscous Fluid with Radiative Heat Transfer IOSR Jornal o ahmacs IOSR-J -ISS: 78-578 -ISS: 39-765X. Volm Iss Vr. VI ar - Ar. 5 PP 6- www.osrjornals.org Unsad agnohdrodnamc Soks Flow o Vscos Fld wh Rada Ha ransr.sharmlaa *.Gaahr * Darmn o ahmacs