MODELLING OF ROTATIONAL FLUID WITH AN ACCELERATED VERTICAL PLATE EMBEDDED IN A DARCIAN POROUS REGIME

Size: px

Start display at page:

Download "MODELLING OF ROTATIONAL FLUID WITH AN ACCELERATED VERTICAL PLATE EMBEDDED IN A DARCIAN POROUS REGIME"

Maryann Jones
5 years ago
Views:

1 MODELLING OF ROAIONAL FLUID WIH AN AELERAED VERIAL PLAE EMEDDED IN A DARIAN POROUS REGIME. Sahin AHMED. HEA RANSFER AND FLUID MEHANIS RESEARH DEPARMEN OF MAHEMAIS GOALPARA OLLEGE GOALPARA- 78 ASSAM INDIA ASRA: A mahemaical sdy is condced fo he nseady hydomaneic flo of a iscos incompessible elecically condcin and Neonian oain flid pas a nifomly acceleaed infinie eical plae immesed in a Dacian poos eime nde he acion of ansesely applied maneic field. he copled non-linea paial diffeenial eqaions fo he pimay and seconday flo ae soled sin Laplace ansfom echniqe. An incease in Dacian nmbe he pimay elociy is deceleaed hile he seconday elociy acceleaed. I is obseed ha an incease in he oaion paamee leads o decease boh he pimay and seconday elociy pofiles in he Dacian poos eime. Applicaions of he sdy aise in he field of sella and planeay maneosphees aeonaics chemical enineein and eleconics. EYWORDS: Dacian poos eime Roaional flid Isohemal eical plae Maneic field hemical Enineein INRODUION he hydodynamic oain flo of elecically condcin iscos incompessible flids has ained consideable aenion becase of is nmeos applicaions in physics and enineein. In eophysics i is applied o mease and sdy he posiions and elociies ih espec o a fixed fame of efeence on he sface of eah hich oae ih espec o an ineial fame in he pesence of is maneic field. he sbjec of eophysical dynamics noadays has become an impoan banch of flid dynamics de o he inceasin inees o sdy enionmen. In asophysics i is applied o sdy he sella and sola sce ine planeay and ine sella mae sola soms and flaes ec. In enineein i finds is applicaion in MHD eneaos ion poplsion MHD beains MHD pmps MHD bonday laye conol of eeny ehicles ec. Pesen eseach field has aaced many eseaches in ecen yeas de o is asondin applicaions. Fee conecion effecs on flo pas an acceleaed eical plae ih aiable scion and nifom hea flx in he pesence of maneic field as sdied by Rapis al 98. Mass ansfe effecs on flo pas a nifomly acceleaed eical plae as sdied by Sondaleka 98. Sinh 984 sdied MHD flo pas an implsiely saed eical plae in a oain flid. MHD effecs on flo pas an infinie eical plae fo boh he classes of implse as ell as acceleaed moion of he plae as sdied by Rapis and Sinh 985. Ahmed 8 inesiaed he effec of ansese peiodic pemeabiliy oscillain ih ime on he hea ansfe flo of a iscos incompessible flid hoh a hihly poos medim bonded by an infinie eical poos plae by means of seies solion mehod. Ahmed sdied he effec of ansese peiodic pemeabiliy oscillain ih ime on he fee conecie hea ansfe flo of a iscos incompessible flid hoh a hihly poos medim bonded by an infinie eical poos plae sbjeced o a peiodic scion elociy. he ansien naal conecion-adiaion flo of iscos dissipaion flid alon an infinie eical sface embedded in a poos medim by means of neok simlaion mehod inesiaed by Zeco 8. Ahmed inesiaed he effec of peiodic hea ansfe on nseady MHD mixed conecion flo pas a eical poos fla plae ih consan scion and hea sink hen he fee seam elociy oscillaes in abo a non-eo consan mean. he seady maneohydodynamic MHD mixed conecion sanaion poin flo oe a eical fla plae is inesiaed by Ali e. al.. Sahin e. al. inesiaed he effec of he ansese maneic field on a ansien fee and foced conecie flo oe an infinie eical plae implsiely held fixed in fee seam akin ino accon he indced maneic field. Seen e. al. sdied he maneic hydodynamic fee conecie flo pas an infinie eical poos plae ih he effec of iscos dissipaion sbjec o copyih FAULY of ENGINEERING HUNEDOARA ROMANIA 69

2 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein a consan scion elociy. he poblem of a seady maneohydodynamic fee conecie bonday laye flo oe a poos eical isohemal fla plae ih consan scion hee he effecs of he indced maneic field as ell as iscos and maneic dissipaions of eney sdied by Ahmed and ain. Haona pesened a nmeical simlaion on nseady hydomaneic fee conecion nea a moin infinie fla plae in a oain medim. Mhcmaasamy e. al. sdied he effecs of oaion on he hydomaneic fee conecion flo of an incompessible iscos and elecically condcin flid pas a nifomly acceleaed infinie eical plae in he pesence of aiable empeae and nifom mass diffsion. Ahmed and alia pesened he maneohydodynamic ansien conecie adiaie hea ansfe one-dimensional flo in an isoopic homoenos poos eime adjacen o a ho eical plae. Ahmed and alia inesiaed he effecs of chemical eacion as ell as maneic field on he hea and mass ansfe of Neonian o-dimensional flo oe an infinie eical oscillain plae ih aiable mass diffsion. Ahmed and ain sdied he effecs of hemal adiaion and poosiy of he medim on flo pas an implsiely saed infinie isohemal eical plae in he pesence of maneic field. he ppose of he pesen analysis is o sdy an nseady hydomaneic fee conecion flo of an elecically condcin iscos incompessible flid pas a nifomly acceleaed infinie eical plae embedded in a Dacian poos eime in he pesence of aiable empeae and nifom mass diffsion. he dimensionless oenin eqaions ae soled sin he Laplace ansfom echniqe. he solions ae in ems of exponenial and complemenay eo fncion. Sch a sdy is fond sefl in maneic conol of molen ion flo in he seel indsy liqid meal coolin in nclea eacos maneic sppession of molen semi-condcin maeials and meeooloy. MAHEMAIAL ANALYSIS onside he nseady hydomaneic flo of an elecically condcin flid indced by iscos incompessible flid pas a nifomly acceleaed moion of an isohemal eical infinie plae in a Dacian poos eime hen he flid and he plae oae as a iid body ih a nifom anla elociy abo -axis in he pesence of an imposed nifom maneic field nomal o he plae. Iniially he empeae of he plae and concenaion nea he plae ae assmed o be and. A ime > he plae sas moin ih a elociy in is on plane and he empeae fom he plae is aised linealy ih ime and he concenaion leel nea he plae ae also aised o. Since he plae occpyin he plane is of infinie exen all he physical qaniies depend only on and. I is assmed ha he indced maneic field is neliible so ha. Fo nseady moion of a hydomaneic Neonian flid occpyin Dacian poos space in a oain sysem ien by Sheman and Son 965 and Maxell's eqaions he fndamenal eqaions ae in eco fom: q q q q kˆ q p q q J κ q P q D 4 μe J Ampee s La 5 E Faaday s La 6 Solenoidal elaion i.e. maneic field coniniy 7 J E q Ohm s La fo a moin condco 8 hee q E J and μ e ae especiely he elociy eco maneic field eco elecic field eco and cen densiy eco he maneic pemeabiliy of he flid. 7 ome XI Yea. Fascicle. ISSN

3 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein ome XI Yea. Fascicle. ISSN hen he nseady flo is oened by fee-conecie flo of an elecically condcin flid in a oain sysem filled ih poos medim nde he sal ossinesq s appoximaion in dimensionless fom ae as follos: x p 9 y p P κ D p 4 Since hee is no lae elociy adien hee he iscos em in Eqaion 9 anishes fo small and hence fo he oe flo beside hee is no maneic field alon x-diecion adien so e hae x p y eliminain he pesse em fom Eqaions 9 and 4 e obain 5 he ossinesq appoximaion ies 6 On sin 6 in he eqaion 5 and noin ha is appoximaely eqal o he momenm eqaion edces o 7 he iniial and bonday condiions ae: all fo as a > A 8 On inodcin he folloin non-dimensional qaniies: P 4 κ φ θ D Sc P A M G G m In ie of he non-dimensional qaniies he eqaions o 6 become: M G G m φ θ 7 M 8

4 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein θ θ P φ φ Sc Wih he folloin iniial and bonday condiions: θ φ fo all θ φ a > θ φ as he hydomaneic oain fee-conecion flo pas an acceleaed eical plae embedded in a Dacian poos medim is descibed by copled paial diffeenial eqaions 7 o ih he pescibed bonday condiions. o sole he eqaions and e inodce a complex elociy f i eqaions 7 and 8 can be combined ino a sinle eqaion: f f Gθ Gmφ m f he folloin iniial and bonday condiions: θ φ fo all f θ φ a > f θ φ as m M i MEHOD OF SOLUION he dimensionless oenin eqaions 9 and sbjec o he iniial and bonday condiions 8 ae soled by he sal Laplace-ansfom echniqe and he solions ae deied as follos: f c ca d [ expη m efc η m exp η m efc η m ] η m ac [ exp m efc m exp m efc m ] η η η η c efc η P c exp a [ expη m a efc{ η m a } exp η m a efc{ η m a } d efc η Sc [ expη a P efc η P a exp η a P efc P a ] d exp b [ exp{ η m b } efc η m b exp{ η m b } efc η m b ] [ expη b Sc efc η Sc b exp η b Sc efc Sc b ] c exp a η d exp b η η P ac η P efc η P exp η P 4 π η P θ η P efc η P exp η P 5 π 9 φ efc η Sc 6 In ode o e he physical insih ino he poblem he nmeical ales of hae been comped fom 4. While ealain his expession i is obseed ha he amen of he eo fncion is complex and hence e hae sepaaed i ino eal and imainay pas by sin he folloin fomla: exp a ef a i b ef a [ cosab i sinab ] a π 7 ome XI Yea. Fascicle. ISSN

5 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein exp a exp n / 4 n n π n n 4a hee h n a b a a cosh nb cosab nsinh nb sinab n a b a cosh nb sinab nsinh nb cosab [ h a b i a b ε a b ] 6 ε a b ef a i b RESULS AND DISUSSION o descibe he flo behaio he sysem of non-linea diffeenial eqaions 7- oehe ih he bonday condiions has been soled analyically by sin Laplace ansfom echniqe. he nmeical esls ae displayed in he fom of non-dimensional pimay and seconday elociy pofiles fo aios ales of he paamees ha epesen he flo. Hee e conside G G m 5 > coolin of he plae i.e. fee conecion cens coney hea aay fom he plae in o he bonday laye Sc.6 hoho he discssion. he Pandl nmbe P is aken fo ai a o P.7 elecolyic solion P. and ae P 7.. o jde he accacy of he pesen nmeical daa a compaison of he empeae disibion ih he peiosly pblished pape of Mhcmaasamy e. al. [4] fo diffeen ales of he Pandl nmbe he esls ae obseed o be in ood aeemen. able : ompaison of ales of he empeae disibion θ fo he pesen esls ih [4] hen G G m 5 Sc.6 M 5.5: Pesen ok Peios ok [4] P P y In able i is seen ha as P he inceases he hickness of he hemal bonday laye deceases. I is infeed ha he hickness of hemal bonday laye is eae fo ai P.7 and hee is moe nifom empeae pofile acoss he hemal bonday laye as compaed o ae P 7. and seam P.. he eason is ha smalle ales of Pandl nmbe ae eqialen o inceasin hemal condciiy and heefoe hea is able o diffse aay fom he heaed sface moe apidly han fo hihe ales of Pandl nmbe. hs empeae falls moe apidly fo ae han ai and seam. Fies a-b indicae he behaio of Pimay and Seconday elociies ih he aiaions in Dacy nmbe. he Pimay elociy shos a decemen ih he inceases in Dacy nmbe fo...5. I is becase ha he pesence of a poos medim inceases he esisance o flo and hs edces he flid elociy. Fhe he seconday elociy inceases ih he incease in Dacy nmbe. Effecs of M ae ploed in Fi. a and Fi. b. As M inceases fom. hoh. o. a consideable edcion in he pimay elociy occs. As a mae of fac he pesence of a ansese maneic field ill esl in a esisan ype of foce knon as Loen foce hich inceases he esisance o he flo leadin o decease in he flid elociy. As he maneic paamee inceases he hickness of he momenm bonday laye inceases fo seconday elociy. Also obsee ha senh of maneic field is mch hihe in seconday elociy pofiles. Fi. a and Fi. b ie he dimensionless elociy pofiles pimay and seconday elociy fo aios ales of oaion paamee i is obseed ha an incease in he oaion paamee leads o decease boh he pimay elociy and seconday elociy pofiles in he poos eime. he pimay elociy pofile shos a mino deceasin effec hile he seconday elociy pofile has qie lae deceasin effec. Fi. 4a and Fi. 4b epesen he aiaion of Pandl nmbe P in he pimay and seconday elociy pofiles. I is obseed ha ih he incease of Pandl nmbe deceases he pimay elociy hile he seconday elociy inceases. I is menioned ha fo ae 7. he elociy pofiles is mch hihe in boh cases han ha of ohe. I also shos he lae deceasin effec on he pimay elociy and mino inceasin effec on he seconday elociy. ome XI Yea. Fascicle. ISSN

6 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein Fi. a: Pimay elociy pofile fo diffeen ales of Fi. b: Seconday elociy pofile fo diffeen ales of Fi. a: Pimay elociy pofile fo diffeen ales of M Fi. b: Seconday elociy pofile fo diffeen ales of M Fi. a: Pimay elociy pofile fo diffeen ales of Fi. b: Seconday elociy pofile fo diffeen ales of Fi. 4a: Pimay elociy pofile fo diffeen ales of P 74 Fi. 4b: Seconday elociy pofile fo diffeen ales of P ome XI Yea. Fascicle. ISSN

7 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein ONLUSIONS he aboe analysis bins o he folloin esls of physical inees on he pimay and seconday elociy pofiles of he flo field. he Dacian paamee has he inflence of deceasin he pimay elociy and i has he inflence of inceasin he manide of he seconday elociy pofiles. he oaion paamee has he effec of deceasin he pimay elociy pofiles as ell as he manide of he seconday elociy pofiles. he oaion paamee has lae effec on elociy pofiles and hence can effeciely be sed o conol he flid moion. We hae fond saisfacoy aeemen of o pesen esls ih hose of [4]. he Hamann nmbe has he effec of deceasin he flo field Pimay elociy a all he poins de o he maneic pll of he Loen foce acin on he flo field. So maneic field can effeciely be sed o conol he flo A compaison of empeae pofiles shos ha he empeae deceases hoho he Dacian eime. NOMENLAURE x - Spaial coodinae alon he plae y - oodinae axis nomal o he plae - Dimensionless coodinae axis nomal o he plae - ime - Dimensionless ime A - onsan - Species concenaion. m - p - Specific hea a consan pesse J. k -. - Species concenaion in he fee seam. m - - Species concenaion a he sface. m - D - hemical molecla diffsiiy m.s - - Acceleaion de o aiy m.s - G - hemal Gashof nmbe G m - Mass Gashof nmbe M - Hamann nmbe/maneic paamee - Unifom maneic field p - Pesse Pa - Dacy nmbe o poosiy paamee P - Pandl nmbe Sc - Schmid nmbe - Dimensional empeae - Dimensional Flid empeae a he sface - Dimensional Flid empeae in he fee seam - Dimensionless elociy componen in x- diecion m. s - - Velociy of he plae m. s - efc-omplemenay eo fncion ef-eo fncion exp-exponenial fncion Geek symbols - oefficien of olme expansion fo hea ansfe - - oefficien of olme expansion fo mass ansfe - η - Similaiy paamee θ - Dimensionless flid empeae k - hemal condciiy W. m -. - μ - oefficien of iscosiy k. m - - inemaic iscosiy m.s - - Densiy k. m - - Elecical condciiy Φ - Dimensionless species concenaion. m - Sbscips - ondiions on he all - Fee seam condiions REFERENES [] Sinh A Hydomaneic fee-conecion flo pas an implsiely saed eical plae in a oain flid. Inenaional ommnicaions in Hea and Mass ansfe ol. pp [] Rapis A. ianidis GJ. and Pedikis P. 98. Hydomaneic fee conecion flo pas an acceleaed eical infinie plae ih aiable scion and hea flx Lees in hea and mass ansfe. ol.8 pp.7-4 [] Sondaleka VM. 98. Effecs of mass ansfe on flo pas a nifomly acceleaed eical plae Lees in hea and mass ansfe ol.9 pp [4] Rapis A. and Sinh A Roaion effecs on MHD fee conecion flo pas an acceleaed eical pla Mechanics Reseach ommnicaions ol. pp.-4 [5] Ahmed S. 8. ansien hee dimensional flo hoh a poos medim ih ansese pemeabiliy oscillain ih ime Emiae Jonal fo Enineein eseach ol. pp. -7. [6] Ahmed S.. Fee conecie ansien hee-dimensional flo hoh a poos medim oscillain ih ime in pesence of peiodic scion elociy Inenaional Jonal of Applied Mahemaics and Mechanics ol. 6 pp. -6. [7] Zeco J. 8. Unseady fee conecion-adiaion flo oe a eical all embedded in a poos medim ommnicaions in Nmeical Mehods in Enineein ol. 4 pp ome XI Yea. Fascicle. ISSN

. MHD Mixed onecion onday Laye Flo oad a Sanaion Poin on a Veical Sface Wih Indced Maneic Field ansacions of he ASME Jonal of Hea ansfe ol. pp. -6. [] Ahmed S. é O. Ana Vedad S. Zeinalkhani M.

8 ANNALS OF FAULY ENGINEERING HUNEDOARA Inenaional Jonal Of Enineein [8] Ahmed S.. Fee and foced conecie MHD oscillaoy flos oe an infinie poos sface in an oscillain fee seam Lain Ameican Applied Reseach ol. 4 pp [9] Ali FM. Naa R. Aifin NM. and Pop I.. MHD Mixed onecion onday Laye Flo oad a Sanaion Poin on a Veical Sface Wih Indced Maneic Field ansacions of he ASME Jonal of Hea ansfe ol. pp. -6. [] Ahmed S. é O. Ana Vedad S. Zeinalkhani M. and Heidai A.. Mahemaical modellin of maneohydodynamic ansien fee and foced conecie flo ih indced maneic field effecs Inenaional Jonal of Pe and Applied Sciences and echnoloy ol. no. pp [] Seen N. Fancis G. and Johana S.. Maneic hydodynamic fee conecie flo pas an infinie eical poos plae Applied Mahemaics ol. pp. -6. [] Ahmed S. and ain A.. Maneohydodynamic hea and mass ansfe flo ih indced maneic field and iscos dissipaie effecs Lain Ameican Applied Reseach In Pess. [] Haona N.. Modelin of nseady MHD fee conecion flo ih adiaie hea ansfe in a oain flid Naal Science ol. no. pp [4] Mhcmaasamy R. Lal. and Rananayakl. Roaion effecs on MHD flo pas an acceleaed eical plae ih aiable empeae and nifom mass diffsion. Inenaional jonal of Enineein [5] Ahmed S. and alia. Maneohydodynamic ansien flo hoh a poos medim bonded by a ho eical plae in pesence of adiaion: A heoeical analysis Jonal of Enineein Physics and hemophysics ol. 86 no. pp. -9. [6] Ahmed S. and alia. Analyical nmeical sdy fo MHD adiain flo oe an infinie eical plae bonded by poos medim in pesence of chemical eacion Jonal of Applied Flid Mechanics ol. 6 no. 4 In Pess. [7] Ahmed S. and ain A.. onecie lamina adiain flo oe an acceleaed eical plae embedded in a poos medim ih an exenal maneic field Inenaional Jonal of Enineein and echnoloy ol. no. pp [8] Son GW and Sheman A Enineein maneohydodynamics MacGa-hill Ne Yok ANNALS of Facly Enineein Hnedoaa Inenaional Jonal of Enineein copyih UNIVERSIY POLIEHNIA IMISOARA FAULY OF ENGINEERING HUNEDOARA 5 REVOLUIEI 8 HUNEDOARA ROMANIA hp://annals.fih.p.o 76 ome XI Yea. Fascicle. ISSN

ME 3560 Fluid Mechanics

ME 3560 Fluid Mechanics ME3560 Flid Mechanics Fall 08 ME 3560 Flid Mechanics Analsis of Flid Flo Analsis of Flid Flo ME3560 Flid Mechanics Fall 08 6. Flid Elemen Kinemaics In geneal a flid paicle can ndego anslaion, linea defomaion