Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt b)

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Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR Vle Ie Obe 4 844-854 ISSN 347-37X (in & ISSN 347-34 (Online www.jnl.g lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel Nbil.M. El-be Mhe Y. Ab-zei He A. Sye b Deen f Mhei Fly f Ein Ain Sh Univeiy Rxy Ci Egy b Deen f Mhei Fly f Siene Zgzig Univeiy Zgzig Egy Deen f Mhei Fly f Siene b Univeiy b KSA Ab: he lile flw f nn-newnin fli whih beying biviiy fli ( Cn el e-in elin i ie. We e in niein he iy f ei n he ney in wih he n nfe ne he effe f gnei fiel n he e. he eqin f en enegy n nenin hve been lve by ing Lighill eh. he gvening eqin e lve nlyilly by ing Mhei ge. he veliy eee n nenin iibin e bine. he effe f vi ee f he ble n hee iibin e ie n eie ghilly hgh e f fige. Keyw: lile flw; Nn-Newnin fli; ei; Mgnei fiel; He nfe; M nfe; Diffin he.. INRODUCION he ble f lile flw hve gine ine e hei ieie il liin in biehnil n engineeing iene. In hyilgy lile ehni i invlve in ine n f iney ble hgh he ee veen f hie in he gineinl n f ez in he effeene f he le eive n in he evil nl in veen f v in he fllin n f lyh in he lyhi veel. In iin lie vel in vehile if eing jhe n en veen f by ing iviie he hn by exeiene exenl by elein. lnge exe f helhy hn by exenl elein y e ei helh ble lie hehe inee f le e n l f viin n n f ibne in bl flw []. he nlyi f he ehni enible f lile n hve been ie by ny h. he ble f lile flw wih efeene eni in iilin w nlye by Bin n Belle []. R n Rhn Devnhn [3] n Shne n O[4] ie lile flw hgh il be vying -ein lw Reynl nbe. Yng n i[5-6] ie he ey n ney flw eni exeienlly whih n be fn l in Siffi e l. [7]. Elbe e l. [8-9] ie lile gne hyyni vieli flw hgh hnnel bne by eeble le n he effe f le ee n lile hy gnei ieille flw. In hee ie he be wll i en be ieeble. Mey[] ie he ey flw f vi fli hgh il be wih eeble wll. he lile flw f nn-newnin fli in ie ie by Ew e l. []. he ble f lile flw f MHD nn-newnin fli beying we lw el wih nvein he nfe hgh nn Dy ei beween w xil yline i ie by Ab-zei[3]. Flw hgh ei i vey evlen in ne n heefe he y f flw hgh ei h bee f inile inee in ny engineeing liin. Mny h hve ie he effe f ei n he in f he fli. Se f hee ie hve been e by Abel-hy n Kel [4] ie he ey MHD flw f vieli fli hgh ei. Neil y f lile ARC ge 844

Nbil.M. El-be e l. MHD nn-newnin fli flw wih he n nfe hgh ei beween w eeble llel le w ie by Elbe e l. [5]. In hi e he in i i bin neil lin f he ble f ney lile flw wih he n nfe. he fli e i biviiy fli hgh nif ei in li ylinil ie in he eene f gnei fiel. he gvening eqin wee lve neilly by ing Mhei. he veliy eee n nenin iibin e bine. he veliy eee n nenin iibin e lle f iffeen vle f Re K M ω q Sn S.. FORMULAION OF HE ROBLEM Cnie he ney flw f nn-newnin fli beying Cn el hgh eeble infinie ylinil be ne he in f exenl nif gnei fiel f engh B. Che ylinil ine ( z whee z i he xi f he be. he niive eqin f he Cn fli el i τ ij (μ (μ y y πe π ij e ij π π π π Whee y i he yieling e π e ij eij whee e ij i he (i j nen f he efin e n μ π i he ieninle e lii f en viiy effiien. F iny Newnin fli ( y. y Fig. Shei f he ble he gvening eqin e in hi ble n be wien fllw: Cniniy eqin: ρ ρ.v ( Men eqin in ei ne he effe f nif exenl gnei fiel ρ V V. V eee eqin wih he genein.τ μ V J B ( ρ Q (3 Cnenin eqin wih hel iffin C D D C (4 Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 845

lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel Whee J σ(e V B i he en eniy ve σ i he eleil niviy B μe H i he gnei fiel engh μ e i he gnei eebiliy H i he lie gnei fiel E i he elei fiel V i he veliy fli ρ i he eniy τ (τ ij i he e en i he ee i he ie μ i he li viiy f he fli i he eebiliy ei i he fli eee i he hel niviy f he fli i he eifi he nn ee Q i he he genein C i he fli nenin D i he effiien f iffiviy en fli eee. i he hel iffin e i he Sine he flw ee e ineenen f he zihl ine he veliy i given by V ( n he gnei fiel ve i B (B. Nw we hll nie he gnei Reynl nbe i vey ll heefe he ine gnei fiel n exenl elei fiel e neglee. Une hee in he eqin (-4 n be wien : μ ( ( ( ρ z ρ ρ μ σ B (5 Q ( (6 ρ C C C D D ( ( (7 he xie bny niin e n n C C C (8 Le ine he fllwing ieninle qniie fllw: ρ z z C C ω ω (9 Afe biing f eqin (9 eqin (5 (6 n (7 y be wien in ieninle f fe ing. ( ( ( M ( z K ( q ( ( S S( he ieninle bny niin e ( Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 846

Nbil.M. El-be e l. n n (3 Whee σ M (he gnei fiel ee K (he iy ee ρ (nl nbe Q q (he e ρ S ( Shi nbe D S D C (Se nbe. 3. MEHOD OF SOLUION F lin ee gien le z e iω Whee ω i he feqeny n ee gien eeively (4 e he ey nen n he illy nen f he eqin ( ( n ( n be lve by ing he fllwing ebin ehniqe: iω e e iω e iω (5 Sbiing f (4 n (5 in ( ( n ( n eqing he lie e n bh ie we ge he fllwing ye eqin: ( K M ( ( (6 (iω K ( M ( (7 q (8 (iω q (9 S S ( S i C S ( Sbje he fllwing bny niin finie finie n finie n finie ( Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 847

lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel 4. NUMERICAL RESULS AND DISCUSSION he ye f eqin h gven he nn-newnin fli flw in li ylinil ie in he eene f gnei fiel n ei e lve nlyilly by ing Lighill eh. he fl f he veliy eee n nenin iibin e bine n lle f iffeen vle f Re K M ω q Sn S in fige (-4. he effe f hyil ee n he veliy iibin e inie hgh fige -8. In hee fige he veliy iibin i le ve he ine. Fige ( n (3 hw he behvi f he veliy f iffeen vle f he ey nen f ee gien n he illy nen f ee gien eeively. I i een f Fig. ( n (3 h he veliy inee wih inee bh f n i.e. he inee f he ee gien e ineen in he veen f fli. Fige (4 ille he hnge f he veliy f evel vle f he nn-newnin fli ee. I i beve f hi fige h he veliy inee wih he inee f. he el in fige (4 i in geeen wih he el whih i bine by Abelnby e l. [5]. he effe f he gnei ee M n he veliy i hwn in Fig. (5. I i le h he veliy eee wih he inee f he gnei ee M bee he nif gnei fiel i in ein iein he veen f he fli. Siilly if we w he veliy f iffeen vle f he eebily ee K we will bin fige in whih he behvi f he ve e he e he bine in fige (6. he veliy f vi vle f he ie n he feqeny f he illing ω e exhibie in Fig. (7 n (8 eeively. F hee fige we beve h he veliy eee wih inee bh f n ω. he effe f he he genein q n he eee i hwn in Fig. (9 n i hwn h he eee inee by ineing q. Bee he fli i exe he n hi i hwn f he he eqin n h hi el i geeen wih he bine by Ab-zei [3]. In fige ( he effe f nl nbe n he eee i eene. F hi fige we beve h he effe f n i iil he effe f q n ille in Fig. (. Fige ( give he hnge f he nenin f evel vle he he e ee q. I i ne f hi fige h he nenin eee wih ineing q. We beve h hi el i ven f bee he he e i inveely inl nenin. he effe f Shi nbe S Se nbe S n nl nbe n he nenin e elie in fige ( (3 n (4 eeively. We beve f hee fige h he effe f hee ee e fn be iil he effe f he he e ee q n ille in Fig. (. Fige( Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 848

Nbil.M. El-be e l. Fige(3 Fige (4 Fige (5 Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 849

lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel Fige (6 Fige (7 Fige (8 Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 85

Nbil.M. El-be e l. Fige (9 Fige ( Fige ( Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 85

lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel Fige ( Fige (3 Fige (4 Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 85

Nbil.M. El-be e l. 5. CONCLUSION In hi e we hve ie he ney lile in f nn-newnin Cn fli hgh ei in eeble ylinil ie in he eene f gnei fiel. he eqin f en enegy n nenin hve been lve by ing ebin ehniqe. he gvening eqin wee lve neilly by ing Mhei. he veliy eee n nenin iibin e bine. he effe f vi ee f he ble n hee iibin e ie n eie ghilly hgh e f fige. Hene hi e el wih n in bnh f fli ehni whih h ny in liin in ny fiel h bilgy eiine n heiy n l in he e iene. F exle:. he helgy f bl h eeive h y. Bl i helgilly lex n w n: i i enin bee eyhye wih heii ienin f evel iee e een in exe f 4 vl. n he enin fli ielf exhibi nn- Newnin behvi bee f he eene f high lel weigh ein. he ine f helgil eie f he by fli i nw egnize. In il he helgil ene f i eiy ye f bh infn n l e n in f f e eiy ye f bh infn n l e n in f f e eiy behvi. F engineeing e ne i e ineee in he vle f he veliy n he nfe hn in he he f he veliy n eee file. REFERENCES [] S. N jhi n V. R Ni lile flw f hi ge fli ne by elein- Melling bl flw: In. J. Eng. Si. 3 (994 839-846. [] J.. Bin n D. Belle lile flw wih efeene eni in iilin: Bihelgy 3 (986 5. [3] A. Rhn R n Rhn Devnhn lile flw in be f vying ein:z. A. M.. 4(9733. [4] D. J. Shne n S. Oh lile bl flw in hnnel f ll exnenil ivegene-. he line xiin f lw en Reynl nbe: J. Fli Eng. 6(975 353. [5] D. F. Yng n F. Y. i Flw heii in el f eil eni-i. Sey Flw J. Biehni 6 (973 395-4. [6] D. F. Yng n F. Y. i Flw heii in el f eil eni-ii. Uney Flw J. Biehni 6 (973 554-559. [7] M. Siffi R. elie D. Fhif n R. Rie he effe f neinen he flw hgh ene n Bifin J. Biehni 7 (984 99-35. [8] N.. Elbe n S. M. Elhni lile gnehyyni vieli flw hgh hnnel bne by w eeble llel le J. hy. S. Jn 64( (995 465. [9] N.. Elbe n S. M. Elhni Effe f le ee n lile ieille flw Fli Dyni Reeh 5 (995 33. [] R. I. Mey he ey flw f vi fli hgh il be wih eeble wll I: Bll. Mh. Bihy. 5 (963. [] R. I. Mey he ey flw f vi fli hgh il be wih eeble wll II: Bll. Mh. Bihy. 7 (9657. [] M. F. Ew D. A. Neli n W. L. Wilinn lile flw f nn-newnin fli in ie Che. Eng. Si. 7(97 545. [3] M. Y. Ab-zei Neil lin f he genein effe n MHD lile nn- Newnin fli flw wih nvein he nfe in nn-din ei beween w ing yline Bll. Cl. Mh. J. (9 53-55. [4] C. L. Vhney he ey MHD flw f vieli fli hgh ei: J. e Ale.Mh. (979 558. Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 853

lile Min f Nn-Newnin Fli wih He n M nfe hgh Mei in Sli Cylinil ie in he eene f Mgnei Fiel [5] M. A. Abelnby N.. M. Elbe n M. Y. Ab-zei Neil y Neil y f lile MHD nn-newnin fli flw wih he n nfe hgh ei beween w eeble llel le In. J. Meh.Cn.& Mh Si (6-5. [6] M. J. Lighhill Inin fie nlyi n genelize fnin Bll. Ae. Mh. S. 65 (4 (959 48-49. Ineninl Jnl f Sienifi n Innvive Mheil Reeh (IJSIMR ge 854

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