Heat And Mass Transfer On Mhd Oscillatory Slip Flow In A Channel Filled With Porous Medium

Transcription

1 IOSR Jonal of Mahemaics IOSR-JM e-issn: p-issn:39-765x. Volme Isse 4 Ve. II Jl-Ag Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim Dis Akeem B. Ishola hisie Y. & Olonnishola oin School of Science an echnolog Naional Open nivesi of Nigeia Vicoia Islan Nigeia Absac: his s was conce o invesigae hea an mass ansfe of nsea MHD oscillao slip flow of opicall hin fli hogh a channel fille wih saae poos meim an non-nifom walls empeae. he govening eqaions of oscillao fli flow wee non-imensionalise simplifie an solve. he close-fom solions wee obaine fo he veloci empeae an concenaion. he nmeical compaions wee pesene gaphicall o show he salien feaes of he fli flow hea an mass ansfe chaaceisics. he skin ficion Nssel nmbe an Shewoo nmbe wee also analze. e wos: hemical eacion oos meim MHD oscillao flow Navie slip I. Inocion he s of oscillao fli flow in a poos channel has been eceiving consieable aenion in he las ecen eas e o is applicaions in soil mechanics gon wae holog iigaion ainage wae pificaion pocesses absopion an fillaion pocesses in chemical engineeing. Also he Navie slip flow egime has been eceiving aenion of man eseaches becase of is applicaions in moen science echnolog an insializaion []. Owing o hese applicaions Makine an Osalsi [] pesene he effecs of slip coniions on he homagneic sea flow in a channel wih pemeable bonaies. he effecs of Navie slip coniion on he lowe wall of nsea MHD of a viscos fli in a planne channel wih fille wih saae poos was epoe b Mehmoo an Ali [3]. I was assme ha he Navie slip bona coniion effec epens on he shea sess of boh lowe an ppe walls of a channel b Eegnjobi an Makine []. Recenl Aesana an Makine [4] invesigae he navie slip coniion on he ppe an lowes of plsaile flow of oscillao fli hogh a channel fille wih poos meia. he s of combine hea an mass ansfe wih chemical eacion on oscillao fli flow is of gea pacical applicaions in esign of chemical pocessing eqipmens fomaion an ispesion of flog foo pocessing an cooling of owe. he effecs of chemical eacion on MHD oscillao flow pas hogh a poos plae wih viscos issipaion an hea sink was pesene b Vijaa an Viswannah [5]. Rabi Goange an Apia [6] invesigae he effecs of chemical eacion on MHD oscillao flow hogh a poos meim bone b veical poos plae wih hea soce an soe effec. In view of hese ses we pesen chemical eacion effecs on MHD oscillao slip flow in an opicall hin fli hogh a channel fille wih saae poos meim an non-nifom walls empeae. II. Mahemaical Fomlaion onsie he nsea lamina slip flow of an elecicall concing hea geneaing opicall hin an chemical eacing oscillao fli in a channel fille wih saae poos meim ne he inflence of an exenall applie homogenos magneic fiel an aiaive hea ansfe. g g H X Y k c Y c q Y D R 3 Y he bona coniions elevan o he poblem ae aken as follows: 69 age

2 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim 7 age ay ay 4 whee is he axial veloci he ime he fli empeae fli concenaion he pesse g he gaviaional foceq he aiaive hea flx he coefficien of volme expansion e o empeae he coefficien of volme expansion e o concenaion he specific hea a consan pesse D hemal iffsivi k he hemal concivi he kinemaic viscosi H he nifom magneic fiel he poos meim pemeabili coefficien an he ensi of he fli.i is assme ha boh walls empeae ae high enogh o ince aiaive hea ansfe. I is assme ha he fli is opicall hin wih a elaivel low ensi an he aiaive hea flx is given b cogle e al [7] as follow: q 5 whee is he mean aiaion absopion coefficien. he following imensionless vaiable an paamees ae inoce Re Da s Da g Gc g G D Sc Y X x R 6 k N k c e H H 4 whee is he flow mean veloci G he Gasshof nmbe Gc he solal Gasshof nmbe H he Hamann nmbe N is he aiaion paamee e he ecle nmbe Re he Renol nmbe Da he Dac nmbe Da s poos meim shape faco paamee an is he imensionless feqenc of oscillaion he eqaions o 5 in imensionless fom ece o G Gc H s x Re 7 N e 8 Sc 9 he bona coniion becomes on

3 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim on III. Meho Of Solion In oe o solve eqaions 7 9 wih espes o he bona coniions fo pel oscillao flow i i i i e e e e x whee is a consan an is he feqenc of oscillaion. Sbsiing eqaion ino 7 we obain: m G Gc m m3 4 ogehe wih bona coniions on on 5 whee m s H i m N ie an m3 i Sc Eqaions 5 ae solve an he solion fo fli veloci empeae an concenaion ae given as follows: sin m i e 6 sin m sinh m i e 7 sinh m cosh m sinh m A 8 4A sinh m cosh m sinh m m m cosh m sinh m he shea sess he coefficien of he ae of hea ansfe an he ae of mass a poin in he fli can be expesse b N Sh 9 he skin ficion he Nssel nmbe N an he Shewoo nmbe Sh ae expesse as follow: A cosh m sinh m m sinh m m cosh m 4 sinh m A m m cos m m3 cosh m3 N Sh sin m sinh m age

4 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim IV. Discssion Of Resls We obaine he exac solions of eqaions - 4 sbjec o bona coniion 4. As an accae check seing Gc = an neglecing eqaion 4 he wok of Aesana an Makine [7] is fll ecovee. heefoe s in [7] is a special fom of he pesen pape. Nmeical compaion was pefome on he exac solions an he esls wee illsae gaphicall in Fig -6 o show he ineesing feaes of significan paamee on veloci empeae an concenaion isibion.hogh he compaions we emploe efal vales = H = s = k = e =.7 G = Gc = Sc = an = nless ohewise sae on he gaph. Fige illsaes ha he pesence of ansvese magneic fiel poces a esisive foce on he fli flow. his foce is calle Loenz foce which slows own he moion of he fli. I is obseve ha he inceases in he feqencies of oscillaion ecease he veloci as shown in Fige. Inceases in hemal an solal Gashof nmbes significanl incease he bona lae hickness which esle ino api enhancemen of fli veloci as pesene in Figes 3 an 4. Fige 5 epics he vaiaion of fli veloci wih invese of he Dac paamee he gaph shows ha an incease in invese of he Dac paamee eceases he fli veloci. Fige 6 epics ha he inceases in he aiaion paamee inceases he fli veloci. Figes 7 8 an 9 illsae he vaiaion of chemical eacion paamee Renols nmbe Re an elec nmbe especivel. An incease in an of he paamee poces a ecion on he fli veloci. Fige shows ha he veloci pofile vaiaion wih col wall slip paamee.he gaph shows ha fli veloci inceases wih an incease in he col wall slip paamee which enhancing he fli flow mean ha he slip paamee has he enenc o ece ficion foces which inceases he fli veloci. I is obseve in Fige ha incease in he heae wall slip paamee eceases he fli veloci. Fige illsaes incease in he Schmi nmbe inceasing he fli veloci slighl. he empeae pofile is ploe in Fige 3 an 4 i can be obseve ha he fi empeae inceases ansvesel wih maximm vale a he lowe wall an maximm vale a he ppe wall. Howeve a geneal incease in he fi empeae is obseve wih an incease in an of aiaion o ecle nmbe. Fige 5 illsaes he ecease in concenaion fo inceasing chemical eacion paamee which inicaes ha he iffsion ae which can be change emenosl b chemical eacion. Fige 6 epics vaiaion of concenaion pofle an Schmi nmbe. his shows ha incease in Schmi nmbe inceases he mass ansfe. ables& Show he effecs of col an heae wall on he skin ficion. hese show ha an incease in an of he hea wall inceases he shea sess an he ansfe wihin he channel. able 3 shows he effec of elec nmbe on he Nssel nmbe. I is obseve ha an incease in elec nmbe enhancing he ae of hea ansfe on he wall. ables 4 an 5 show ha effec of Schmi nmbe an chemical eacion paamee on Shewoo nmbe.i can been seen ha an incease in eihe Schmi nmbe o chemical eacion paamee inceases he Shewoo nmbe. Fige : Veloci pofile fo iffeen vales of H 7 age

5 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim Fige : Veloci pofile fo iffeen vales of Fige 3: Veloci pofile fo iffeen vales of G Fige 4: Veloci pofile fo iffeen vales of Gc Fige 5: Veloci pofile fo iffeen vales of s 73 age

6 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim Fige 6: Veloci pofile fo iffeen vales of N Fige 7: Veloci pofile fo iffeen vales of Fige 8: Veloci pofile fo iffeen vales of Re Fige 9: Veloci pofile fo iffeen vales of e 74 age

7 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim Fige : Veloci pofile fo iffeen vales of Fige : Veloci pofile fo iffeen vales of Fige : Veloci pofile fo iffeen vales of Sc Fige 3: empeae pofile fo iffeen vales of e 75 age

8 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim Fige 4: empeae pofile fo iffeen vales of N Fige 5: oncenaion pofile fo iffeen vales of Fige 6: oncenaion pofile fo iffeen vales of Sc able : Effecs of col wall on he skin ficion age

9 Hea An Mass ansfe On Mh Oscillao Slip Flow In A hannel Fille Wih oos Meim able : Effecs of heae wall on he skin ficion able 3: Effecs of ecle nmbe on he skin ficion e N e N e N able 4: Effecs ofscmi nmbe on Shewoo nmbe sc Sh sc Sh sc Sh able 5: Effecs of chemical eacion paamee on Shewoo nmbe sc Sh Sh Sh V. onclsion his pape invesigae he hea an mass ansfes on MHD oscillao slip flow in a channel wih non-nifom wall empeae ha is fille wih poos meim. Exac solions of he velociempeae an concenaion ae obaine. hee is a goo ageemen beween he pesen s an he effec of slip on MHD oscillao pesene b Mehmoo an Ali [6] when Gc an he pesene b Aesana an Makine when Gc =. Howeve Nssel nmbeinceases a he col plae b eceases a he heae plae wih an incease in ecle nmbe. Refences []. A.S. Eegnjobi an O.D. Makine ombine effec boanc foce an Naive slip on enop geneaion in a Veical poos channel Enop []. O.D.Makine an E. Osalsi MHD sea flow in a channel wih slip a he pemeable bonaies Rom. J.hsis [3]. A. Mehmoo an A. Ali he effec of slip coniion on nsea MHD oscillao flow of a viscos fli in a planne channel Rom.J.hsis [4]. S.O. Aesana an O.D. Makine MHD oscillao slip flow an Hea ansfe in a channel fille wih poos meia Rom.J.hsics [5]. D. Vijaa an G.R. Viswanah Effecs of chemical eacion on MHD fee convecive oscillao flow pas a poos plae wih issipaion an hea sink Avance in Applie science Reseach [6]. N.B. Rabi.D. Goange an M. Apia hemical eacion effecs on MHD oscillao slip flow hogh a poos meim bone b wo veical poos plaes wih hea soce an soe effec Jonal of Applie Analsis an compaions [7]. A..L. ogle W.G Vinven an E.S. Giles Ameican Insie of Aeonaics an Asonaics age