M. Choudhury 1. and G. C. Hazarika 2

Transcription

1 Jonal of led Fld Means Vol. 6 No alable onlne a.jafmonlne.ne ISSN ISSN e ffes of Vaable Vsos and emal Cond on MHD Osllao Fee Conee Flo as a Veal Plae n Sl Flo Regme Vaable Son and Peod Plae emeae M. Cod and G. C. Hazaa Deamen of Maemas N. N. S. College ssam Inda Deamen of Maemas Dbga Unes Dbga ssam 786 Inda Coesondng o mal: mabg@gmal.om Reeed gs ; aeed Jana 7 ) BSRC e effes of aable sos and emal ond on fee onee osllao flo of a sos nomessble and eleall ondng fld as a eal lae n sl flo egme eod lae emeae en son elo osllaes n me abo a onsan mean s dsssed. e fld sos and emal ond ae assmed o be nese lnea fnons of emeae. e oblem s goened b a oled non-lnea ssem of aal dffeenal eqaons. xl fne dffeene meod s emloed o sole e eqaons. e effes of sos aaon aamee emal ond aaon aamee magne aamee on e elo dsbon dna emeae dsbon fo P =.7 fld and fo aefaon aamee = and =. ae dsssed and son gaall. lso e effes of ese aamees on e sn fon oeffen and on e ae of ea ansfe ae allaed. Keods: MHD naal oneon aable sos aable emal ond. NOMNCLUR a onsans son aamee B nfom magne feld seng onsans C f loal sn-fon oeffen sef ea a onsan esse e nmbe g aeleaon de o ga [ms - ] G Gasof nmbe aefaon aamee emal ond [Wm - K] M Haman nmbe N loal Nssel nmbe fld esse [Pa] P Pandl nmbe me [s] emeae [K] onsan onsan elo omonens n e x- and - deons [ms - ] eal and magna as of sead son elo x axs n e deon along and nomal o e sfae eseel α sfae ae amlde β oeffen of emal exanson [K - ] γ onsan σ eleal ond μ oeffen of dnam sos [ Nsm - ] ξ onsan dens [gm -3 ] ω feqen aamee ε amlde υ e nema sos [m s - ] dmensonless emeae emal ond aaon aamee sos aaon aamee amben fee-seam ales all sfae ondons dmensonal qan

2 M. Cod e al. / JFM Vol. 6 No INRODUCION Unsead osllao fee onee flos la an moan ole n emal engneeng bo mane and aeosae enolog. S flos ase de o ee nsead moon of a bonda o bonda emeae. e nseadness ma also be de o osllao fee seam elo o emeae. One moan lass of me deenden sos flo oblems naed b Lgll 95) deals e effes of nsead flaons of e man seam elo on e flo of an nomessble fld as o dmensonal bodes. Man eseaes ae oed on flang flos of sos nomessble fld as an nfne lae. In all ese aes e lae as assmed o be mananed a onsan emeae. In man aal alaons e ale adjaen o a sold sfae no longe aes e elo of e sfae. e ale a e sfae as a fne emeae of e sondng saona fld. B n man alaons qe ofen e lae emeae sas osllang abo a non-zeo mean emeae. e fee-oneon flo s enaned b semosng osllang emeae on e mean lae emeae. e ansen fee oneon flo as an nfne eal lae eod emeae aaon as nesgaed analall b Das e al. 998) Sondalgea and Wae 977) and 977) sded e nsead fee oneon flo as an nfne eal lae and mass ansfe onsan / aable son. In man aal alaons e ale adjaen o a sold sfae no longe aes e elo of e sfae. e ale a e sfae as a fne angenal elo. I sls along e sfae. e flo egme s alled e sl flo egme and s effe anno be negleed Sama 3)Wen e gas s slgl aefed esls agng e obseed sal enomena an be obaned b solng e sal Nae-Soes eqaons ogee modfed bonda ondons allong fo a elo sl and emeae jm a e sfae. s seme of eoeal nesgaon of e so alled sl-flo egme s alal sable fo sdng e effes of aefaon on an lassal sos flo oblem. Redd 96) exended e o of Sa 955) fo e sos flang flo oe an nfne lae son b nodng e sl flo bonda ondon n lae e of no-sl bonda ondon. He obseed a e sl bonda ondon exbed a sbdng nflene on e esonse of e sn-fon and ea ansfe a e all o e man seam flaon. Usng s assmon Sama and Cada 3) ad dsssed e fee oneon flo as a eal lae n sl-flo egme and also dsssed aos alaons fo engneeng oses. med and Kala 8) nesgaed e effe of e ansese magne feld and son aamee on e osllao fee onee flo as a eal lae n sl flo egme aable son and eod lae emeae. Ve eenl Sng and Kma ) nesgaed e effes of adaon and fs ode emal eaon on an nsead MHD fee onee ea and mass ansfe flo as a eal oos fla 78 lae n sl-flo egme en son elo osllaes n me abo a non-zeo onsan mean. Fld sos and emal ond ene emal dffs) la an moan ole n e flo aaes of lamna bonda lae oblems. Fld oees ae sgnfanl affeed b e aaon of emeae. e nease n emeae leads o a loal nease n e anso enomena b edng e sos aoss e momenm bonda lae and so e ea ansfe ae a e all s affeed. In e oolng of eleon eqmens s elael feqen o fnd msanes n aable oe effes ae sgnfan and anno be negleed. saa and Bomma ) esablsed a le oe emosal oees emeae deenden sos al ole n sfae fon and ea ansfe ae nea e all. Hazaa and Laa 997) obseed a a sgnfan aaon aes lae n elo and emeae dsbon e aaon of e sos and emal ond aamees. Hossan e al ) sded naal oneon of fld aable sos fom a eaed eal a sfae. In all e aboe os e Pandl Nmbe s a fnon of sos and emal ond s assmed o be onsan aoss e bonda lae. Panoaoas 5) and 7) and Raman e al. 9) fond a s assmon lead o neals esl n modelng e emal bonda lae flo a emeae-deenden sos. I ms be eaed as a aable ae an a onsan n e bonda lae. e esls obaned b Panoaoas 5) ae dffeen fom ose exsng n e leae ae been obaned e assmon of onsan Pandl nmbe. e objee of s ae s o sd e effes of aable sos and emal ond on fee onee osllao flo of a sos nomessble and eleall ondng fld as a eal lae n sl flo egme eod lae emeae en son elo osllaes n me abo a onsan mean. e fld sos and emal ond ae assmed o be nese lnea fnons of emeae and ene onsdeng e aaon of Pandl nmbe aoss e bonda lae.. FORMULION OF H PROBLM n nsead fee-onee flo of a sos nomessble fld as an nfne eal oos lae n sl flo egme eod emeae en aable son elo dsbon flang me s of e fom e ω med and Kala 8) s onsdeed. e x - axs s aen along e lae deon oose o e ga and - axs s aen o be nomal o e sfae. e elo omonens of e fld ae and n e x and deons eseel aen aallel and eendla o e lae. magne feld B x) of nfom seng s mosed ansesel o e deon of e flo. e magne Renolds Nmbe of e flo s aen o be small enog so a e nded magne feld an be

3 M. Cod e al. / JFM Vol. 6 No negleed. e fld oe aaons emeae ae lmed o. sos. emal ond and. dens. Follong La and Kla 99) e fld sos s assmed o be nese lnea fnon of emeae as ) ) a ) a ) a and ae onsans and e ales deend on e efeene sae and emal oe of e fld. In geneal a > fo lqds and a < fo gases. γ s a onsan based on emal oe of e fld. Fo γ μ = μ onsan). lso e aaon of emal ond s onsdeed as follos Hazaa and Laa 997) )] ) [ 3) ) ee and ae onsans and e ales deend on e efeene sae and emal oe of e fld. ξ s a onsan based on emal oe of e fld. > fo lqds and < fo gases. e nflene of aaon of dens emeae s esed o e bod foe em onl n aodane e Bossnesq aoxmaon. Sne e lae s onsdeed nfne n e x -deon ene all sal qanes ll be ndeenden of x. Unde s assmon e sal aables ae fnons of and. s e eqaons goen e feeonee nsead domagne flo ae: e 5) ) σ μ β e B g 6) e 7) e bonda ondons ae = e a as 8) ee g s e aeleaon de o ga s e fld emeae σ s e eleal ond s e fld dens s e sef ea of e fld a onsan esse μ s e fld sos s e emal ond of e fld β s e emal exanson oeffen ε << ) s e amlde s e sead son elo nomal) a e lae s e Raefaon aamee s e son aamee ω s e feqen aamee e sbss denoes e fee seam ondon and denoes ondon a e all eseel. We node e follong non-dmensonal qanes μ P B M gβ G 3 9) Usng e ansfomaon q. 9) q. 6) and q. 7) edes o e follong non-dmensonal fom M G e ) P P e ) ee s e nema sos G s e Gasof Nmbe M s e Hamann Nmbe s e dmensonless aefaon aamee s e e nmbe P s e amben Pandl nmbe s e sos aaon aamee and s e emal ond aaon aamee. e Pandl nmbe = P s a fnon of sos as ell as emal ond and as bo e sos and emal ond aes aoss e bonda lae e Pandl nmbe also aes. e assmon of onsan Pandl nmbe ma ode neals esls Raman e al. 9) eefoe e aable Pandl nmbe elaed o e aable sos and aable emal ond s defned as P = P ) Usng q. ) e non-dmensonal emeae q. ) an be en as

4 M. Cod e al. / JFM Vol. 6 No P P e 3) s e oeed non-dmensonal fom of e eneg eqaon fo modelng emal bonda lae flos emeae deenden sos and emal ond. e oesondng bonda ondons n nondmensonal fom ; e ; ) 3. SOLUION OF H PROBLM In ode o sole e dffeenal q. ) and q. 3) e assme a: 5) 6) B sbsng q. 5) and q. 6) no q. ) and q. 3) and eqang eal and magna as e ge e follong ssem of dffeenal eqaon: M G sn ω os ω 7) M G ω ω sn os 8) P P ω ω sn os 9) P P snω os ω ) e bonda ondon q. ) edes o sn os ; ; ) qaons 7) o ) ae oled non-lnea aal dffeenal eqaons ae o be soled b sng e bonda ondons q. )). Hoee exa o aoxmae solons ae no ossble fo s se of eqaons and ene e sole ese eqaons b fne dffeene meods. e enal dffeene s sed fo e dffson ems and e foad dffeene seme s sed fo e oneon ems. fe exemenng a fe se of mes szes e mes szes ee fxed a Δ =. and Δ =.. e oede s eeaed nl =. s ges sffen aa as e bonda ondons ae sasfed.

5 M. Cod e al. / JFM Vol. 6 No e moan aaess of e oblem ae e sn-fon o-effen and ea ansfe ae a e lae. e oeffen of sn-fon a e sfae s gen b C f eal a ee τ - μ. ) e ae of ea ansfe n ems of Nssel Nmbe a e sfae s q N -eal a - ee - q -.. RSULS ND DISCUSSION 3) s a esl of e nmeal allaons e elo and emeae dsbon fo e flo ae obaned fom q. 7) o q. ) ae dslaed n fges fo dffeen ales of Vsos aaon aamee) emal ond aaon aamee) M Hamann nmbe) ae osen abal fo fld Pandl nmbe P =.7 Panoaoas 5) and e nmbe =.. In e esen analss e ales of Gasof nmbe G ae osen abal e ae G =5> oesond o e oolng of e lae. ogo o nesgaon e ae aefaon aamee) = and. Sama 3) son aamee =. med and Kala 8) feqen aamee dmensonless) ω=med and Kala 8) and amlde osllaon ε=. med and Kala 8). lso e sn-fon oeffen C f and e Nssel nmbe N ae allaed fom q. ) and q. 3) and ae ablaed n able o able 3 fo same se of aamees. Fom able s obseed a an neasng ale of e sos aamee = ) leads o deease n e ales of sn fon oeffen C f fo aefaon aamee =. and =.. able des a e Nssel nmbe deeases de o nease n e emal ond aamee = ) fo =.. e esls ae qalael smla o e elaed sd b Raman e al. 8) and so a neasng e emal ond of e fld leads o a deease n e Nssel nmbe. s ma al be exlaned b e fa a a neasng emal ond as e effe of aeleang and neasng e emeae of e fld. able 3 des a sn fon oeffen C f e Haman nmbe M= 3 5) fo =.). I s obseed a C f neases Mmaasam and Vadel ) as M neases. Fom able and able 3 s obseed a C f neases as =.) deeases agees Sama 3). able ffes of sos aaon aamee on Sn-fon oeffen C f fo P =.7 G =5 M=.5 =. =5 =. = able ffes of emal ond aaon aamee Nssel nmbe N fo P =.7 G =5 M=.5 =. =5 = able 3 ffes of Haman Nmbe M on Sn-fon oeffen C f fo P =.7 G =5 =5 =. =5. M =. = ee s no sgnfan aaon n Nssel nmbe N de o aaon of sos aaon aamee and Haman nmbe M. e aaon of elo ofles ae son n Fg. fo M=.5 G =5 ee =5 5and =.. Fg. des e elo ofles fo same se of aamees and aefaon aamee =. 8

6 M. Cod e al. / JFM Vol. 6 No Fge 3 and Fg. eesen emeae ofles fo emal ond aaon aamee fo =. and = eseel. In bo ases emeae feld neases neasng. e emeae ofles ae de fo aefaon aamee =.. Fg.. Velo Pofles fo M=.5 G =5 =5 =. Fg.. emeae Pofles fo =5 G =5 M=.5 = Fge 5 and Fg. 6 eesen e aaon n elo dsbon e aaon of Haman nmbe M fo aefaon aamee =. and = eseel. ese es so a elo deeases n e esene of magne feld agees e exeaons sne e magne feld exes a eadng foe on e fee onee flo Mmaasam and Vadel ). Fg.. Velo Pofles fo M=.5 G =5 =5 =. Fg. 5. Velo Pofles fo =5 G =5 =5 =.. Fg.3. emeae Pofles fo =5 G =5 M=.5 =. Fg. 6. Velo Pofles fo =5 G =5 =5 =. 8

7 M. Cod e al. / JFM Vol. 6 No CONCLUSION e eoeal solon of fee onee osllao MHD as a eal lae n sl flo egme aable son and eod lae emeae s onsdeed. Fld sos and emal ond ae onsdeed as nese lnea fnons of emeae. e esened analss as son a fo modelng emal bonda laes emeae deenden sos and emal ond e Pandl nmbe ms be eaed as aable nsde e bonda lae. e flo feld s aeabl nflened b e sos-emeae and emal ondemeae aaon and also b e magne aamee. CKNOWLDGMNS e aos sneel an e efeee fo e sefl ommens and sggesons n mong e ae RFRNCS med N. and D. Kala 8). MHD osllao feeonee flo as a eal lae n sl-flo egme aable son and eod lae emeae. Hea and enolog 6 ) Das U. N. R. K. Dea and V. M. Sondalgea 998). ansen fee oneon flo as an nfne eal lae eod emeae aaon. Jonal of Hea ansfe SM) 9-9. saa.. and B.. Bomma ). e effe of aable sos on lamna flo de o a on sn. Indan Jonal of Pe and led Maemas. 35 6) Hazaa G.C. and J. Laa 997). ffes of aable sos and emal ond on fee onee mass ansfe flo as a eal soemal one sfae n esene of magne feld. Maemaal Fom x -5. Hossan M.. S. Kab and D..S. Rees ). Naal oneon of fld aable sos fom a eaed eal a sfae. Zesf fü angeande Maema nd Ps ZMP 53) La F. C. and F.. Kla 99). e effe of aable sos on onee ea ansfe along a eal sfae n a saaed oos medm. Inenaonal Jonal of Hea and Mass ansfe 35) 8 3. Lgll M. J. 95). e esonse of lamna sn fon and ea ansfe o flaons n e seam elo. Poeedngs e Roal of Soe 56) -3 Mmaasam R. and V. U. S. Vadel ). Hea ansfe effe on mong eal sfae n e esene of magne feld. Maemas daon XXXVIII ) - 9. Panoaoas. 7). Non- Daan foed oneon ea ansfe oe a fla lae n a oos medm aable sos and aable Pandl nmbe. Jonal of Poos Meda ) -8. Panoaoas. 5). Foed and mxed oneon bonda lae flo along a fla lae aable sos and aable Pandl nmbe ne esls. Hea and Mass ansfe ) Raman M. M. M.. Raman M.. Samad and M. S. lam 9). Hea ansfe n a Moola Fld along a Non-lnea Seng See a emeae-deenden Vsos and Vaable Sfae emeae. Inenaonal Jonal of emoss 35) Raman M. M... Mamn M.. zm and M.. lm 8). ffes of emeae deenden emal ond on MHD fee oneon flo along a eal fla lae ea ondon. Nonlnea nalss: Modelng and Conol 3 ) Redd K. C. 96). Flang flo as a oos nfne lae n sl flo egme. Qael Jonal of Means and led Maemas. 7) Sama P. K. 5). Inflene of eod emeae and onenaon on nsead fee onee sos nomessble flo and ea ansfe as a eal lae n sl-flo egme. Maemaas: nsenanza Unesaa XIII ) 5-6. Sama P. K. and R. C. Cada 3). ffe of aable son on ansen fee onee sos nomessble flo as a eal lae eod lae emeae aable n slee flo egme. JR 8 ) Sng K.D. and R. Kma ). Flang Hea and Mass ansfe on Unsead MHD Fee Coneon Flo of Radang and Reang Fld as a Veal Poos Plae n Sl- Flo Regme. Jonal of led Fld Means ) -6. Sondalgea V. M. and P. D. Wae 977). Unsead fee oneon flo as an nfne eal lae onsan son and mass ansfe. Inenaonal Jonal of Hea and Mass ansfe ) Sondalgea V. M. and P. D. Wae 977). Unsead fee oneon flo as an nfne eal lae aable son and mass ansfe. Inenaonal Jonal of Hea and Mass ansfe ) Sa J.. 955). solon of e Nae-Soes and eneg eqaons llsang e esonse of sn fon and emeae of an nfne lae emomee o flaon n e seam elo. Poeedngs e Roal of Soe 38)