Convection in Superposed Fluid and Porous Layers in the Presence of a Vertical Magnetic Field

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1 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab Conecion in Seose Fi n oos Les in e esence o Veic Mgneic Fie ANAI M. BANJER n ABULLA A. ABULLA een o Meic Sciences U A- Uniesi. O. Bo 67 M SAUI ARABIA o@q.e.s Absc:- A ine sbii nsis is ie o sse consising o oizon i e oeing e o oos ei se i e se i i nio eing o beo in e esence o eic gneic ie. e o in oos ei is sse o be goene b c s. e Bees-Jose coniion is ie e inece beeen e o es. Neic soions e obine sing ension o Cebse onois. is sec eo s song bii o soe iee obes n os s o obin ig cce eigenes in e eicien nne. Neic ess e obine o ieen es o e ee ˆ ( e o i e/e o oos e) n ieen es o e gneic ee. e eec o gneic ie is inesige n i is on e ine sbii ces o e onse o conecion oion e bio een o ige es o ˆ. Ke-Wos: -Seose oos n i es- e conecion- c's - Mgneic ie- Sion conecion Cebse eo. Inocion e onse o conecion o oizon i e seose oos e en e sse is ee o beo s been consiee is b Sn [] o soe e ciic Reig nbe in e oos e eceses coninos s e icness o e i e is incese. e se sooing eo o obin neic soions o e ine sbii eqions. Nie [] oe e obe i sce-ension eecs n obine soic soions o s e nbes o consn e bon coniion. Eeien obseions o e onse o conecie oion in be o incie ee ice en iqi e eiss oe e be e been obine b Ree e. []. ei obseions soe incesing e e o e iqi e oe e be ene o oe e ciic inen Reig nbe ic e onse o conecion occe. Sn [] n Nie [] se c s in oing eqions o oos e ie Soeon n Con [] se e Binn s eension o c s o s e e insbii o seose oos n i es en inen eing is oe in e oos ei. e soe ge c nbe es o ess sbe i e n is e o e incese eeo o i oion in e oos e oe b e incese in eebii. A neic s s been oce b oios e. [] o ise e occence o conecion in i e ic os on o o i-se oos ei. ei s oie ie o e in ees o e o in e conecie egie i.e. Reig nbes ge n e ciic e neee o e onse o conecion. Cen n Cen [6] oce cssic e in ic e sie e conecion in o-e sse coose o oos e se i i oe ic s e o e se i. e e s ee o beo n e consiee e boo o e oos e s e s e e sce o e i o be ie. e eoe e nen oe eeoe b Nie [] n e soe e ine insbii ces o e onse o conecie oion i.e. e Reig nbe gins e nbe ces be bio in e ces ossess o oc ini. e ineee ei ining b soing o ˆ ( e o i e/e o oos e) s (. ) e insbii s iniie in e oos ei ees o ˆ ge n is e ecnis cnge n insbii s conoe b e i e. ISSN: Isse Voe J

2 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab ese ess ee qiie n qniie eiie b eeien o o Cen n Cen [7]. Cen n Cen [8] eene e o o Cen n Cen [6] b ssing e oion o e i in oos e is goene b c s eqion i e Binn es o iscos eecs n e Foceie e o inei eecs. Neic ess ee obine sing cobinion o Gein n inie-ieence eo. e obe o Cen n Cen [6] s been soe neic b Bi [9] sing is n secon oe Cebse eo. McK [] sie sii oos-i e obe o Cen n Cen [6] b e oe ceic ecions in e es. Venci-Loez n Oco-i [] esene coison o o oes c n Binn o s e bonc-ien-o in conine i oeing oos e. Sgn [] se Cebse eo o obin ess o e sce-ension-ien conecion n o e eec o oe n oeing esecie in i oeing oos e. Fe eeences o o on e oos-i conecion obe cn be on in [-]. e insbii eo s been enge b e inees in onic os o eecic concing is in e esence o gneic ie. e esence o sc ie in n eecic concing i s s e eec o inibiing e eeoen o insbiiies. oson [] Cnse [] Agg & Ve [] n oes e eine e eec o gneic ie o e o i. oee in e iee so no esec e been one o iscss e eec o gneic ie on e e conecion o oe sse consiss o oizon i e oeing e o oos ei se i e se i i nio eing o beo. e ieenion o Cebse eo is ig se in obining cce eigenes o one n o e obes. e icion o Cebse eos o ie o sbjec es is iscsse b Ab & Lins [-6] Ab [7] A-Aios & Ab [8] Sgn [-] C n Sgn [] C [] n oes. Bi [9] se is oe Cebse eo o obin ess o i-ee conin n e soe e is oe Cebse eo is oe cce n e Cebse eo. Conecie e nse in oos ei s been sbjec o ge inees o e s see eces. is inees s oie b neos e engineeing icions in ios isciines. e enoos oe o o eoe o is ie is e ocene in e boo o Nie n Bejn [9] n ecen o in is ie is consiee b Ceng [] n Rii []. e obe o conecion in oos-i sse s n insi n geosic icions. A oog nesning o e sics o bonc-ien o in is oeing oos ei is esseni en esigning e insion sses e eneg soge sses n gin soes. ee e neos enionen n engineeing cicsnces in ic i es n i-se oos es e ee ogee o beo(e.g. oizon es o ibos insion gin soge insions os-ccien cooing o nce ecos). In connecion i e obe o os-ccien cooing o nce ecos Soeon & Con [] n Ree e. [] eo e ine sbii nsis o n conecion in be o e geneing ices cooe o boe b e o i. e e cicion in es so cos es n oe eseois is inence b e inecion beeen e bo o e n e oe e-se sbse. e oses o is e e oo: one is o s e eec o gneic ie on e e conecion o o-e sse consiss o oizon i e oeing e o oos ei se i e se i i nio eing o beo n e oe is o eie is oe Cebse eo o ie eigenes cce o e o e obe. is obe is sii o iscsse b Cen n Cen [6] b ee e eec o gneic ie is ince n e ine sbii eqions e soe neic sing is oe Cebse eo. Meic Foion Consie o ininie oizon es L n L conine beeen o e oizon bonies sc e o o e L oces e boo o e L. e ne inece beeen e o es is e e bon o L is n e oe bon o L is ee e sbscis n enoe i e n oos ei e esecie. Sose e e e L is ie b n incoessibe e n eecic concing iscos i n is goene b Nie-Soes eqions ees e oe e L is occie b oos ei eee b e se i ic is goene b c s. Bo es e sbjece o ISSN: Isse Voe J

3 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab consn eic gneic ie. Gi g cs in e negie iecion o n e oos ei is ee is oe bon (see Fig. ). Fig. : Sceic ig o e obe. e Bossinesq oiion s been ie o bo es s e ensi is consn eeee ece in e bo oce e. Le be e Kein eee o e i be consn eeence Kein eee ρ be e ensi o e i n α be e coeicien o oe ension o e i (consn) en e i ensi ρ is ooione o sc ρ ρ [ α( )]. () e goening eqions o e i e e V ρ ( ρ c ( V ) ) V µ V ρ g µ ( ) π V ( ) V ( V ) η () ee V n e eoci gneic ie n esse in i e n µ µ η c e ineic iscosi gneic eebii e concii eecic esisii n seciic e consn esse in i e. e goening eqions o e oos ei e L e ρ V φ ( ρ c ) ( ρ c ) µ V K µ ρ g π ( ) ( ) V ( V ) η Fi e L oos ei e L V g () ee V n e eoci gneic ie n esse in oos ei e n K µ η φ e eebii o oos ei gneic eebii eecic esisii n oosi in e oos ei e n ee X ( ) φ X φ X ee in () X is ece b o ρ c. e Bon Coniions Sose n e igi n inine consn eee n esecie. In es o n e i eoci coonens o e i in L n L ese eqieens e o e ooing coniions ( ) ( ) ( ) ( ) on e o bon o L () n e coniions ( ) ( ) ( ) () on e oe bon o L ee n e e no coonens o e gneic ie in L n L esecie n e sbscis n enoe e n oe bonies esecie. e i/oos ei inece bon coniions e bse on e ssion eee e n no i eoci e coninos. s e inece ne e e ( ) ( ) ( ) ( ) ( ) ( ) ( ) µ ( ) ( ). (6) ese ineci coniions en esecie e no i eoci eee e n no coonen o e sess enso e coninos. I is ion o e en sing e c eqion e conini o no sess oes no ince e iscos conibion in e oos egion. is ees ee in coniions o be seciie on o o ese e ee o e gneic ie in L n L esecie ic e ISSN: Isse Voe J

4 ( ) ( ) (7) n e in coniion is e o Bees & Jose [] ic s e o ( ) K α ( ) K α (8) ee n e e iiing ngeni coonens o e i eoci s e inece is oce o e es L n L esecie n α is Bees & Jose consn. e se se soion in ose sbii e e ineese is one o ic ee is no i oion in eie e e gneic ie is consn in e eic iecion in bo es n e eees e e n oe bonies e e ie consn eees n esecie i >. Eqions () n () e se se soion o e o V V ( ) ( ) ( ) ( ) (9) ee is e eee e inece ic is on b eqiing conini o eee n e s. Since e esses e no eqie eici in e ensing insbii nsis e o no gie e. e ebion Eqions Le e eqiibi soion be ebe b e ooing ine ebion qniies ( ) ( ) ( ) ( ) e V e V n eie ineize eqions o ese qniies in e s nne. o obin e non-iension eqions o e i e L e coose eng o be ie o be / eoci o be / esse o be / µ eee o be n gneic ie o be. / η Fo e oos e L e coose eng o be ie o be / eoci o be / esse o be K / µ eee o be n gneic ie o be η / ee ) ( c ρ is e e isii o e i se. s e ineize esion o eqions () n () cn be ien in e non-iension o R e β () G R e β φ () ee en eing o beo. en eing o boe ) ( ) ( β Noice n R e noniension nbes ic enoe esecie e iscos n nbe gneic n nbe Cnse nbe n Reig nbe o e i e n n R e e coesoning non-iension nbes in e oos ei e. ese non-iension nbes e gien b K g R g R ν α πρνη µ η ν ν α πρνη µ η ν WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab ISSN: Isse Voe J

5 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab n G K is e c nbe ( ρc ( ρ c ) ) n ρ ) is e e ( c isii o e oos ei se. We s iscss e obe in e cse o eing e oos ei e o beo so e s sse β. We no e no oe ension o e o ψ e[ ] ( ) ψ( ) i( q ) σ () o e ncions n. s e i n oos es eqions becoe σ ( ) σ ( ) σ σ φ σ ( ) ( ) σ R G σ ( ) ( ) σ ( ) ( ) R () () ee q n q e noniensionise e nbes in e i e n oos ei e esecie σ n σ e e go es n ˆ ˆ σ σ ˆ ( ). ( ) e bon coniions in e in o e on () ε ε ε σ ˆ on (6) σ φ ε ˆ ˆ α on (7) ee ˆ ˆ n ε e gien b ˆ ˆ We noice ˆ ˆ ˆ ˆ R ˆ R. ˆ ε. ˆ (8) e coee eigene sse o be soe coises ()-(7). is sse consiss o n eig oe OE in e i e n si oe OE in e oos ei e i bon coniions. e eo o soion se o soe is sse is e is oe Cebse eo. e ieenion o is eo is ion since i is ig cce eo n e e be o cce s n eigenes s e nee ic is se in cses en e eigenes o inees e cnging in ee sce. We cn so cce e eigenncions esi. We is ( ) n ( ) ino e Cebse oin ( ) b e nsoions z n z esecie en e sose M ( z) ( z) α z ( ) o be e Cebse ension o e ibes ic e eine b ISSN: Isse Voe J

6 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab en eqions () n () cn be ien s sse o is oe oin ieeni eqions o e o Y EY σfy o ie n eigene z obe o e o A σb. is geneize i eigene obe is soe sing e Z goi ic e eoe i e NAG oine FF. Ress n iscssion e neic ess obine so e onse o insbii is in e o o sion conecion een in e esence o gneic ie. is eine e ssion o Cen n Cen [6] o e beginning e incie o ecnge o insbiiies os. Neic ess n sbii ces e obine o e esen obe i e concii io -6 ˆ.7 α. φ G n o ie o ecioc e io ˆ nging o. o n o ios es o nging o o. Fig. sos e eion beeen n e ciic Reig nbe in e oos ei e R o ieen es o ˆ (see be. n be.b). I is ce o e ige R inceses coninos s inceses ic ens e gneic ie s sbiizing eec in is sse. Moeoe R eceses coninos s e icness o e i e inceses n is es is in geeen i e es obine b Sn [] in e bsence o gneic ie. e gin sbii ces nbe o e ios e son in Fig. n Fig. o n esecie. e ess in Fig. e consisen i ose o Cen n Cen [6] in ic e soe e gin sbii ces e bio eibiing o eie ini en ˆ is ess n cein ciic e. In e esence o gneic ie s in e esen obe e gin sbii ces e so bio b o ge e ios. Fig. sos en ˆ. n e gin sbii ces e bio. In c s ecees cein ciic e in oe o e o ini es o R. As n ee is e s been cce neic en ˆ. n e on i so be gee n 7. Moeoe in Fig. o ˆ. e ong e bnc is e os nsbe ees o ˆ >. e so e bnc is e os nsbe. In e esence o gneic ie s in Fig. o ˆ. e so e bnc is e os nsbe. Fige () sos e eion beeen ˆ n ciic e nbe o oos ei e o ieen es o. I is ce o is ige s ˆ inceses ee is i cnge o e ciic e nbe. Fige (6) sos e eion beeen ˆ n ciic R o ieen es o. I is ce o is ige ee is eciiion o o e es o e ciic R s ˆ inceses. is eciiion eceses s inceses. e neic ess ee o is ige is ise in be (). e ciic Reig nbe in e i e R cn be cce o eqion (8) n e eions beeen ˆ n e ciic e Reig nbes in e oos n i es en e ise in Fig. 7. As ˆ inceses R inceses n i qic ecees e ciic e n conecion enses. In c en ˆ is ge e obe eces o oizon e o i ee o beo n e ciic Reig nbe o R oces e e ic is e e o e ciic Reig nbe i one igi n one ee bon in e cssic Ben obe. ee e oos inece bees oe ie ee bon n igi one. Sii in e esence o gneic ie en ˆ is ge e obe eces o oizon e o i ee o beo n sbjec o eic gneic ie n e ciic Reig nbe R oces e e o e ciic Reig nbe i one igi n one ee bon e cssic gneic Ben obe ccoing o e e o. is ige sos so e eciios o o e ciic Reig nbe R i incesing ˆ. e ISSN: Isse Voe J

7 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab neic ess ee o is ige is ise in be (). e eciios o o e ciic Reig nbe R i incesing ˆ n e i cnge o e ciic e nbe e son in Fig. 8 en. e gin sbii ces o ieen es o α en ˆ... e son in Fig.9- esecie en. I es o e iges R is sensiie o α o s es o ˆ ( ˆ. ecees cein e n is e o Gene ) en is ecese en R inceses b incesing α is incese. α. Moeoe i ˆ>. R is no onge sensiie o α. In e esence o gneic ie e gin sbii ces o ieen es o α en ˆ... e son in Fig.- esecie en. I es o e iges R is sensiie o α n e eec o α is ieen en gneic ie is ie. ee R inceses s α inceses o s es o b en ecees cein e R eceses s α inceses. e obe is so soe en e e sce is ee. e coniions on e e e is ece b e ooing coniions on. Neic ess e obine o e se es o e ees sggese in e cse en e e bon is igi. e eecs o e ees in is cse e sii o e cse en e e e is igi b e neic ess e qniie ess. oos ei o e i e eening on e e o e e io ˆ. e is oe Cebse eo is se o obin e neic soions o e obe. is eo is ion o is css o obes since i is ig cce. See neic ess e obine eing e ciic Reig nbes o i n oos es e e nbe e e io e gneic ee n e Bees n Jose consn. be.: e eion beeen n ciic ciic R ˆ be.b: e eion beeen n ciic ciic R ˆ R. R. 6 Concsion In is e e e ieene eie ine sbii nsis in seose i n oos es in e esence o eic gneic ie. e eec o gneic ie s been inesige in ei. A ie nge o ees e been se o o o e ossibii o oesbii n e neic ess soe insbii occs on i sion conecion. We e son een in e esence o gneic ie e onse o conecion bio ne in ic conecion be oine b e be. e eion beeen ˆ n ciic ciic R ˆ R. ISSN: Isse Voe J

8 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab be. e eion beeen ˆ n ciic ˆ ciic R R R ˆ. ˆ. ˆ. ˆ. Fig.: e eion beeen n R ( ). R ˆ. ˆ. ˆ. ˆ. ˆ. ciic Fig.: e eion beeen n R. ˆ Fig. : e eion beeen ˆ n ciic R ˆ. ˆ. Fig.: e eion beeen n ˆ. ˆ. R ( ). ciic R ˆ Fig. 6 : e eion beeen ˆ n ciic R ISSN: Isse Voe J

9 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab R R R R R R R Fig.7: e eion beeen ˆ ˆ R n R. Fig.: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n R Fig.8: Viion o ˆ R R R R n i ˆ. 6 R Fig.: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n Fig.9: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n R. Fig.: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n. ISSN: Isse Voe J

10 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab 8 6 Fig.: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n R R Fig.: e eion beeen e nbe n Reig nbe in e oos e R o ieen es o α en ˆ. n. Reeences [] W. J. Sn Conecie Insbiiies in Seose oos n Fee Les.. isseion Uniesi o Minneso 97. []. A. Nie Onse o Conecion in Fi Le Oeing Le o oos Mei J. Fi Mec. Vo [] S. Ree V. i n I. Con N Conecion e nse in Bes o Incie ee ices J. e nse Vo [] C. W. Soeon n I. Con On e e Insbii o Seose oos n Fi.. Les. ASME J. e nse A. Vo []. oios A. Bjn B. Seios n K. Be ig Reig Nbe Conecion in Fi Oeing oos Be In. J. e & Fi Fo Vo.7 No [6] F. Cen n C. F. Cen Onse o Finge Conecion in oizon oos Le Uneing Fi Le J. e nse. Vo [7] F. Cen n C. F. Cen Eeien Inesigion o Conecie Insbii in Seose Fi n oos Le en ee o Beo J. Fi Mec. Vo [8] F. Cen n C. F. Cen Conecion in Seose Fi n oos Les J. Fi Mec. Vo [9] A. F. Bi A Sec Eigene Meo o Mi-ee Conin.. esis Uniesi o Gsgo 997. [] G. McK Onse o Bonc-ien Conecion in Seose Recing Fi n oos Les J. Eng. M. Vo [] J. Venci-Loez n J. Oco-i A S o Bonc-ien Fo in Conine Fi Oeing oos Le In. J. e n Mss ns. Vo [] B. Sgn Sce-ension-ien Conecion in Fi Oeing oos Le J. Co. s. Vo [] B. Sgn Eec o oe Viion n Moeing on Conecion in Fi Oeing oos Le In. J. Ne. An. Me. Geoec. Vo [] M. C n B. Sgn eneie Conecion in Fi Oeing oos Le A. We Res. Vo [] M. C eneie Conecion in Seose oos-ei-i Le Vi Inen eing J. Fi Mec. Vo [6] M.-. Cng e Conecion in Seose Fi n oos Les Sbjece o oizon ne Coee Fo s. Fis. Vo [7] M.-. Cng F. Cen n B. Sgn Insbii o oiseie Fo in Fi Oeing oos Le J. Fi Mec. Vo [8] S. i B. Goe. obin M. C n R. Co Line Sbii o N Conecion in Seose Fi n oos Les : Inence o ineci oeing In. J. e n Mss ISSN: Isse Voe J

11 WSEAS RANSACIONS on FLUI MECANICS ni M. Bnje Ab A. Ab n. Vo [9] A. i n B. Sgn oiseie Fo in Fi Oeing oos Mei J. Fi Mec. Vo [] S. i B. Goe. Gobin M. Cnesis n. Je Sbii o N Conecion in Seose Fi n oos Les : Eqience o e One- n o- oin Aoces In. J. e n Mss n. Vo [] W. oson e Conecion in Mgneic Fie. i. Mg. Se. Vo7 No [] S. Cnse onic n ogneic Sbii oe 98. [] A. Agg n A. Ve e Insbii o Roing Viscoesic Fi eee i Ssene ices in ogneics oc. o e 7 Wses In. Con. on Fi Mec. -6. [] A. A. Ab n K. A. Lins Ben Conecion in Non-ine Mgneic Fi Ac Mecnic. Vo [] A. A. Ab n K. A. Lins Ben Conecion in Non-ine Mgneic Fi Une e Inence o Non-eic Mgneic Fie Conin Mec. eon. Vo [6] A. A. Ab n K. A. Lins Soe Res on e Coion o e Eigenes o Line Sses M. Mo. Me. A. Sci. Vo. No [7] A. A. Ab eoso Conecion in Non-ine Mgneic Fi In. J. e. Sci. Vo [8] A. A-ios n A. A. Ab Mngoni Insbii in Non-ine Mgneic Fi Wses ns. M. Vo. No.. -. [9].A. Nie A. Bejn Conecion in oos Mei n e. Singe-Veg Ne Yo 999. [] Cing-Yng Ceng Mgneic Fie Eecs on Coe e n Mss nse b Mie Conecion ong Veic Sce Ebee in oos Mei b Ineg Meos oc. o e In. Con. on e n Mss ns. Asi []. Rii Conecion in oizon oos Le oc. o e 7 IASME/WSEAS In. Con. on e ns e Eng. An Enionen Rssi 9-8. [] G. S. Bees n.. Jose Bon Coniions N eebe W J. Fi Mec. Vo ISSN: Isse Voe J