ACCELERATED MICROPOLAR FLUID--FLOW PAST AN UNIFORMLY ROTATING CIRCULAR CYLINDER
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1 ALATD MIOOLA FLID--FLOW AST AN NIFOMLY OTATING ILA YLIND Auz Aid Siddiqui Dpm o i Si oll o. & T.. Z. iviy Mul- ki. A I i pp w omuld o-dy low du o uiomly ld d oi iul ylid om i ioy viou iompil d miopol luid. Ti low polm i xmid umilly y dopi pil m ompii Adm-o Tmpol Foui Si mod d u-u Tmpol Spil Fii-Di mod. Ti umil m om ovi quio o miopol luid o i polm io ym o ii-di quio. Ti ym w u olvd umilly y poi SOmod. T ul w lo u xpold y ido xpolio mod. Ti m i vlid o ll vlu o low d luid-pm d o ll im. Moov oudy odiio o voiiy d pi poi om ylid i impod d oud oo. T ul ompd wi xii ul o o-oi iul ylid i Nwoi luid. T ompio i ood. T m o li d duio i d w ovd i miopoliy iiid. Sm i ppd i oio o ylid i. Fumo vox-pi i wk i dlyd o uivly i im oio pm i. I ddiio oio lp o oly i diolvi voi dj o ylid d dv pu io u lo i diolvi oudy ly pio. Fumo oio du miopol pi oudy ly lo. ywod: Ald oi iul ylid Miopol luid pil Numil m SO-ido xpolio mod. MS : 7D7 5N 5Q5. Ioduio T polm o luid low du o iii iul ylid wi i uiomly ld om i lil o. I 9 o i im oidd l polm o wo o moio om. I udid low p impulivly d iul ylid wi uiom vloiy wil i ivid impulivly d iul ylid wi uiom lio. Fo od owd lo i o umilly. Göl 5 lid i oy o o yp o viio o iiil vloiy o ylid. T umil mod ud o olv Nvi-Sok quio o yp o low w vy ppoxim d vlid o ll yold um oly o ldi m i i oluio wi i xpdd i pow o im. u uqu m i xpio vlid oly o iii yold um. I 97 udid od o vy wll d impovd d olvd m y wo iqu. I i pl mod wil i od di iio o Nvi-Sok quio wi w xio o i mod. xpimlly i polm w lo xmid y 7 i 97. mud im o pio o low d ow o pd wk o yold um = d x. T polm o luid-low du o uiomly ld d oi iii iul i ioy iompil viou luid i o udml i owi o i vlul d l um o ppliio: Fo xmpl i i i li i.. i Fl oo ip w oi vil ylid w mployd o dvlop u oml o y wid lowi p
2 ip; ii i oolli oudy ly pio 9; iii i pplyi kwy pomo 5 o u o oio; iv i oolli oudy ly dvi u o lp o /STOL i d upm o ip udd d o ovom pio i uoi diu. I i mp w udy id polm o miopol luid wo ovi d oiuiv quio w dvlopd y 7. T ul o mioluid vi ix d o dom--- mo i Nwoi luid. T miopol luid xmpliid y lood 9- polym d polymi upio iid-od poxi olloidl upio d liquid yl. T luid i ud o l-o--ip 5 owdy. T iii oiuio i ild o miopol-luiddymi 7 d. O moivi iii ppliio o low p ld d oi ylid w mkd o po miopoliy-. I ddiio oio- w pud owi o miopi o iul olloid. Fo i w omuld iiio-oudyvlu-polm I d olvd i y umil m ompii wo i od omiod mod. Moov w xmid o u o i low viou viio o ody voi d pu pilly i oudy ly o miopol luid. I ddiio i w udid pi-viou d i viio wi p o yold um oio pm d im. W lo pod w will low viou d i ld popi o d impoi o oio o iul ylid. Moov w xmid; w will domio vlidiy d iliy o umil m o miopol luid? I w ovd polm ud oidio i o iiv d omplid i mo umil m do o uppo wll o ll im d yold um. u umil m wi w v dopd dv i ov o ll vlu o d. Alou lulio w md o viou vlu o yold um i 5 d vl vlu o i.5 y w pd ul o ou vlu o { 9.5} d vlu o {.5.5} o diuio pupo. T ul pd i pil wll i ul om i io i wi w v xmid mli viou qui-pi li qui-voiiy li voiiy viio wi im viio o u o wi o di vlu o d pilly vox ddi ov u ylid d pu dvlopm i wk o ylid. T omulio d i lyi o polm ud oidio i iv i io. I io w dom oudy vlu polm io modiid pol o-odi ym. T umil m i diud i io. T dil di lulio d vluio o oudy odiio o voiiy d pi om ylid iv i io 5 d pivly wil i dipio o umil xpim i pd i io 7.. i Alyi d Fomulio T mi o polm ud oidio ily d ollow: low i oml o iii iul ylid wi i uiomly ld d oi om i iii ioy viou iompil
3 ω υ ω x υ x i i luid u iul ylid i oi wi ul pd wil dvi om i o l wi uiom lio. T oio d lio d m im. Oviouly ylidil o-odi ym will ud w m i ixd o ylid u oii oiid wi ylid. owv oodi ym will modiid u y ki =l. Moov low will oidd ody wo-dimiol d lmi o ll im. T oiuiy d ovi quio o iompil miopol luid i dimiol om i o ody o iv ollow: w =ul d i dimiol luid vloiy vo miooiol voo pi diy j mio-ii yio pm d mil o o vioiy oii. Now ov quio.. would k om i dimiol om ollow: w lio w dimiol d dimiol vil i d iii dimiol vil wil d lio d diu o iul ylid pivly ii pm. d ov umpio quio.. i ompo om i ylidil pol oodi ym would.9 v u j -. v u.7 v u.. j - x x -.. x x x x x x -.. ν ν ν ω ν ν ω ω ν ω.. j - x x -..5 x x x x x x.. ν ν ν ω ν ν ω ω ν ω
4 w u v dil d v ompo o vloiy vo pivly wil d xil ompo o mio-oio pi d voiiy vo pivly. I m o m uio d voiiy quio.7.9 would k om w d dimiol ild o wil i yold um. No i w == ov quio would om io ovi quio o Nwoi luid. T oudy odiio o i polm would. i Modiid ol oodi Sym O omi quio io modiid pol oodi ym l u k =l quio.. would dom Oviouly oudy odiio iv i. would dom. o i - w. ψ ψ -. ψ ψ -. ψ - - w d j.5 o i. - w w. } { -. } { -. -
5 w = p u o ylid. w i o voiiy o u o ylid wil uip do poi o ll wy o u o ylid. li umil mp w ld o polm ud oidio wiou oio o iul ylid d w udid y ui oudy ly iqu wi wo mjo diii: o i i i vlid o i yold um low d o o low d odly oluio o iio o oiuiv quio o lo ov o l vlu o im. T l yp o diiuly oud y viou uo o xmpl d 7- o Nwoi luid. To ovom diiuli w dop i od umil m wi i vlid o ll yold um oio pm d pilly o ll im. T dil o i m i iv i ollowi io. Ti umil m du ily o-li ym o pil diil quio o ym o di quio wi will olvd y SO-iiv mod wi piliy o l ov o iiv m. o i umil podu i ii o udi o u yp o iiv low polm d i iowd/ oomil i o o quim o ompu d y o pomm.. Numil Sm W ll u ollowi oio. T id iz lo - - d - diio will k y pivly d poi d - - will pd y uip 5 d pivly. Moov id wi w dopd o diiz wol domi o ompuio i iv i iu. Ti m oi o wo mod wi did ollow: Adm-o Tmpol Foui Si Mod ATFSM I i mod w oid ψ i θ i θ i θ... O ui quio. -. io quio.. muliplyi quio y i d ii wi p o om = o = w oi. Q.5. 5
6 w d Q m m m m d q=m-d m- p i o m- wi i=. I quio.. i k poiiv i d oluio o quio di i oy iii o quio i d. I pi i i quio.. ud o N. All uio wi uip >N k zo i xpio o q.... I m o d oudy odiio iv i. would dom : d oudy odiio iv i quio.5 i m o d om o o d.9 w = wil = o =.. d. d. d. A liv om o oudy odiio iv i. oid y muliplyi quio. y - d ii om = o = d ui.9 d. w Ao xplii om o. oid y ui..9 d. wi i i. Filly oudy odiio o ov u o ylid will k om ui.. o. w Now oudy odiio iv ov uii o olv quio. o.. Nx o om o pil diil quio iv i quio.. io di quio w u od-od Adm-o mpol mod d l di o pil oodi. oquly quio o olvd ppoximi poi m m m q m m m m m q m q m m. m q q q m m.7.5
7 7 N M L J.7 O A d. N M M N A 5 5 I D D A A w Q Q O J I D T L D
8 u-u Tmpol Spil Fii Di Mod L u pli quio. io ollowi quio Now o ollowi podu o mod iv i w wi: Alivly ov quio wi : } { } { d. Z w. 7 5 i Z L i i. d.9 A A. A T O. L } { } {
9 9 Nx o ollowi imil podu o quio. ud o quio. w T ui u-u mpol pil ii-di mod o quio. d. w oud d L d L. L D L D } { D D } { D D.5 Z 5 5 i i i L D D w. Z L ψ d θ ψ - ψ θ ψ -
10 Moov vlu o i quio. d. oid o olvi quio.. 5. oudy odiio o l A i od omul I i io w ll p o iqu o oluio o ovi quio.. l wi i ily iv ollow. L u p poi N+ N N- N- N- N+ + N+ - N+ + d N+ - y uip d d i pivly. O ui oudy odiio o iiiy io quio. w T ov quio will ppoximd poi y mod imil o o pil ii-di mod w oi d d om impliiio w ollowi implii m o oudy odiio iiiy d d Z d Z d Z Z Z Z d Z Z w d d 5. o i
11 w L d S i Q o T viio o i o-zo pi imuld o = d i iv i io d pilly i iu.. oudy odiio o l A i od omul Aloou o io 5 l u u iii oudy odiio o io quio. w Followi imil podu o quio. o pil ii-di mod w. Q - Q d i L Q Q S S Q w. o i Q - Q d d i L Q Q S S Q
12 o Q i S d L T viio o i o-zo pi imuld o = d i iv i io d pilly i iu. 7. ompuiol odu W p i uvy o ompuiol podu : Iiilly quio.5.7 olvd wi p o oudy odiio iv i.9. o. o. d. ill om im w i il im limi wi mod o ov opodi o. Fom i w oluio d. T d uiud i quio.. o d. I od p im = vlu o d o u ud iiil im o i od TSFD mod o olv quio. d.7 wi p o oudy odiio. d.5. W ov i umil m i o ov d du im o xuio o miopol luid lo. So i m i oomil d. I ov podu w u poi S. O. - iiv podu uj o ppopi oudy odiio. Iiilly pomm i xud wi oudy odiio == w om uy i ivd oudy odiio diod d oudy odiio iv i quio 5. d. ud.t ov podu i pd uil ov i oid odi o iio mx m m 5 m mx w upip m p um o iio. T ul oid u id d d up o ix od y ido xpolio mod. oquly i m i vlid o ll d wi i o ov.. luld ul d Diuio m mx T oluio v oid o { 9.5} wi id iz {/ / / / /} d {/ / /} u i iv i l Tl : T o dopi vlu o o. Fo low d mod vlu o vlu o i k / wil o i vlu o i i k / d i umil xpim i do up o = u vy pil io i oud o low i ivl < o low dvlopm io d viio o ody voi i viiiy o 5 d m m 5
13 iul ylid. A low i o ylid o xpi om x o wi my du im p u i viou i di oio pm i id. T i wy w k = - w iio i d i ll d o low vlu o im. Ti umil xpim i pomd o viou vlu o i.5 io i oud o ou vlu o mly {.5.5} o imulio d diuio pupo. L u iodu ou pm mx mi mi d mi o ompio o wo omiod mod muully d wi o o Nwoi o y ki == i ovi quio o miopol luid. mx mi mi d mi p o mximum vlu o m uio mximum vlu o voiiy miimum vlu o voiiy d miimum vlu o pi pivly. T ompio o ul i pd i l d. W pli diuio i wo di ymmi d ymmi low oiuio i oio o iul ylid u ymmi viou o low.. Symmi Flow I i low i uppod o ymmi o ll d d o w oid upp l o ylid domi o ompuio. O o ii u o p polm i quio o ow o di voi id ylid yold um oio pm d im vy. I iu 9 mli diplyd o low p upp l o ylid o di vlu o d miod ov ou ld im lvl mly { }. Fiu ow mpol dvlopm o low o = =.5 y ul oid o olvi quio y ATFSM. T omio o ody low wi lod mli i pp =.99 o u o ylid v w i vy low i.. = d w ylid i o oi ow i iu d. Ti i oly ld o miopol luid o i polm i i i o o o Nwoi o. W i mod d = ody ddy will pp i wk o ylid. Ti ddy will lo pp li ov o i vlu d i ow dily i. I d p l o i vox pi w ov i i i y i o im. No i w w i o oio o ylid. I oio o ylid i oidd d ylid o o lo i ddiio o ld lio low-viou i lily di. T pp o ddy i wk i dlyd o uivly i im i d i pp >. u om poio o luid wi i d o u o ylid will o o lo wi oio o ylid d i io o luid lod mli i viiiy o ylid i iu 9. T lod mli ii y diolv vox pi d m vox pp oud ylid i ly o low o ll vlu o d. T lod mli oud ylid oi oulokwi oud ylid d i ik d i wll im p. T mi o oio o ylid i i u dly i pp o ddi u lo i ool oudy ly pio ow i iu 9. Fo
14 i vlu o o d iuio i di. T i pd o ld low domi d du oio d voi i wk d will pp o io poi wii lod mlid iuli ody low io i vy ly o low dvlopm ow i iu 7. T ddi will ovlp d om l dim ddy d i o o io poi oio pd i u id ow i iu 9. T ow o oudy ly ik d oudy ly pio lo xmid y udyi qui-voiiy li viio o voiiy ov u o ylid wi p o viio o mx d mi o viou vlu o d. mpl qui-voiiy li o {.5.5} w = ou im lvl diplyd i iu d. T iu ow oudy ly ik d i d i im p w i ylid i oi o o. Moov i lo d i oio i id. Fiu 7 d idi i pd io i im i o oio u i d oio i; ii pio poi ou omim = w i zo wil i im-limi will dly pio w i o qul o zo. qui-pi li iz moio o idividul pil o luid. qui-pi li dw o ll miod ov u om iu d iv o { } d {.5.5}. T iu ow oio du miopol pi oudy ly. Fiu p viio o qui-pi li o = w ylid i o oi im i miod ov. T umou pk i i iu opod o mulipliio o pi voi d xi mo o pio poi. u w oio i impod muli-pio poi oiid o il poi d i luuio i pi i lo udid y ploi mi vu im i wi i pd i iu. Ti iu idi w = umou pk o mi uv opod o udd i pi ppd i ly o low u i i o o d i om moo d lm viio w i o qul o zo. Ti viio mi m o ll o-zo vlu o o ixd vlu o. T viio i u o lo xmid o ll miod ov u iu d 5 pd o mpol dvlopm i d d li ov u o ylid o {.5.5} w = pivly. O udyi i will o low vlu o d mi o d vy low u o mod d lily i vlu o i luu d oill v i vy ly o o low o ll ow i iu. Fiu 5 ow mpol viio o li oii wi i vi v o viio o d oii. T mpol viio i pu ov u o ylid o d oio o ylid i lo udid d i pd i iu. Ti pu i imd y omul iv ollow:
15 - α θ θ ξ ξ ψ ψ d θ θθ Fo.5 pu ov u o u o ylid d im p. i pu i dily i diio om io poi o o io poi o ylid. miimum pu ov u o ylid xi io poi o ylid. mximum pu ov u ou o io poi o ylid. ow i iu -. dθ - θθ Fo >.5 pu ov u o ylid luu im p. I xm vlu vy wi im ow i iu -d.. Aymmi Flow I i low i uppod o ymmi o ll d d o w oid ompl domi o ompuio wi wi i ow i iu. T lulio id ou o ll vlu o low luid d omi pm wi miod i. ll imulio will o diplyd u mpl w p iu 9 d. Fo low vlu o yold um.. = d low vlu o wi <.5 i i low i ymmi. Fo =.5 low om ymmi. T mpol dvlopm o low viou o = =.5 i ow i iu 9. Moov mpol ow o qui-voiiy li o i i lo imuld i iu. 9. oluio I i udy m o li d duio i d w ovd i miopoliy iiid. Sm i ppd i oio o ylid i. Fumo vox-pi i wk i dlyd o uivly i im oio pm i. I ddiio oio lp o oly i diolvi voi dj o ylid d dv pu io u lo i diolvi oudy ly pio. Fumo oio du miopol pi oudy ly lo. liu. 9. Gzi i Flüiki mi kli iu Z. Aw. M. y. 5 7 li lio i NAA-TM-5. Goldi S. 9. Som -dimiol diuio polm wi iul ymmy. o. Lodo M. So. i 5. Goldi S. d od L.9. oudy ly ow. o. m. il. So. 9. d 5
16 Göl. 9. duuiku d lmi Gzi ud ukuidd. I. A 5. 5 Göl.9. Giu d Zylid i Ai u d u. A.d.M 7. olli W.M. d Di S Symmil low p uiomly ld iul ylid. J. Fluid M Tmd S. 97. o udy oudy ly 5. Qu: Lvl iviy. Ak J. 95. D ooi ud Si yikli Gudl dok ud up Göi dok ud upp. 9 dl L. 95. T Mu d widpowd ip Nuwi 9. dl L. d Tij O.G.A. 9. Applid ydo-d Ao-mi. MGw ill ul. Nw Yok. T J.S. Joo W.S. d oplli A. 97. oi ylid o iulio ool o ioil. J. ydou 5. Moo D.W T low p pidly oi iul ylid i iii m. J. Fluid M ily N dy lmi oudy ly. SIAM v T J.S. 97. A uoi diu wi movi wll o oudy-ly ool. AIAA J.. 5 Wlk J.D.A. 97. T oudy ly du o ili vox. o.. So. Lodo S. A iy D.. i J.W. d My J Fli o oi ylid lp o No Ami okwll YO- i NASA 5. 7 i A..9. Toy o miopol luid. J. M. M... i A..9. Simpl mioluid. I. J.. Si Mi J.. d Go S... A mmil modl o udy o iiil luid movm vi--vi o-nwoi viou o lood i oid y. ompu. M. Appl. 7. Tuk M. A. Sylv N. D. d Aim T. 97. O pulil lood low. Joul o oloy 7. puky I. zzl J. Aml T. d Fzi A Lmi luid vio i miol ui miopol luid oy. So d Auo A 7. Gimii M. Amdol. d Lily olikd liquid ylli poxy i: o iid-od l d pplid o o od o ud mo. Momolul miy d yi i A... Mioouuum Fild Toi Flu Mdi Spi Nw Yok. io D. J. Mz A. d Glvi. G. 99. loomoi pumpi wii mil o ym id o ilio. odi o Iiol o o Solid-S So d Auo Tdu 9 Di o Til p S Fio ijkl J.. T. d v d A.. T pomi o ooloy o pio dvi om op-dow ppo o u-ipid pio dvi. lopoi owi S.. d io. J. 97. O dy oiol moio o pol luid ol. A owi S..97. T oy o pol luid. Adv. Appl.M
17 Gol. S.. d Tk. S.99. dy mixd ovio oudy ly low o miopol luid low io poi o ylid. I. J.. Fluid M Gum G. S. d ml M. A.979. Sdy low o miopol luid du o oi dik. J.. M.. o. A. d ik M.99. A lvio o miopol modl o lood low ou idlizd oi. J. iomi.. ml M. A. d Siddiqui A..A.. Miopol luid low du o oi d oilli iul ylid: od umil udy ZAMM 9. ml M. T. 95. Flow o pol luid w wo i oi ylid. J. ol ml M. T. d M Fo. F. 99. Miopol luid low w wo i oxilly oi p A M mkioo. d Mjumd S D o xilly ymmi ody i Sok' low o miopol luid. y. Fluid 9. 5 Töz A. Sklk.977. Miopol luid modl o upio o iid p. I. J.. Si Tuk M. A. Aim T. d Sylv N. D. 97. O pulil lood low. J. Appl. M.. 7 i T.979. T lmi oudy ly o iul ylid d impulivly om. J. omp. y 57. Di S Fii di oid wi od-od diil quio. J. M. Appl. M Domml L. L. S S.F. 9. T poou io o iuliy i pi lmi oudy ly. J. omp. y. 5. Domml L. L. 9..D. i oll iviy. S W.. d Tlioi D oudy-ly pio i udy low. SIAM J. Appl. M. 55. ildd F..97.Ioduio o Numil Alyi T MGw ill ulii o. Ld. Ji M Numil mod o iii d ii ompuio. Nw A Iiol Limid Nw Dli. 7
18 Fi.. T mi o ompuiol domi o ymmi low ymmi low.
19 Fi.. Smli o = =.5 oid y ATFS mod im lvl =. =. =.5 d =.99. Fi.. Smli o = = oid y dopd umil m im lvl =. =. =.5 d =.99. 9
20 Fi.. Smli o = =.5 oid y dopd umil m im lvl =. =. =.5 d =.99. Fi. 5. Smli o = =.5 oid y dopd umil m im lvl =. =. =.5 d =.99.
21 Fi.. Smli o = =. oid y dopd umil m im lvl =. =. =.5 d =.99. Fi. 7. Smli o = =. oid y dopd umil m im lvl =. =. =.5 d =.99.
22 Fi.. Smli o = =.5 oid y dopd umil m im lvl =. =. =.5 d =.99. Fi. 9. Smli o = =.5 oid y dopd umil m im lvl =. =. =.5 d =.99.
23 Fi.. qui-voiiy li o = = im lvl =. =. =.5 d=.99. Fi.. qui-voiiy li o = =.5 im lvl =. =. =.5 d=.99.
24 Fi.. qui-pi li o = = im lvl =. =. =.5 d =.99. Fi.. qui-pi li o = =.5 im lvl =. =. =.5 d=.99.
25 Fi.. T viio o d ov u o ylid o = w = =. 5 = d =. 5. Fi. 5. T viio o li ov u o ylid o = w = =. 5 = d =. 5. 5
26 Fi.. T viio o pu ov u o ylid o = w = =. 5 = d =. 5. Fi. 7. T viio o voiiy ov u o ylid o mll im o = w = =. 5 = d =. 5. Fi.. T viio o voiiy ov u o ylid o l im o = w = =. 5 = d =. 5.
27 Fi. 9. Smli o = =.5 o im lvl =.=. =.5d=.99. Fi.. quivoiiyli o = =.5 o im lvl =. =. =.5 d=.99. 7
28 Fi.. T viio o voiiy pi o ou oudy o domi o = =. Fi.. T viio o mi o = w = =. 5 = d =. 5. =. =.5 =. =.5 mx mx mx mx y y y y y y y y ATFSM SFDM ATFSM SFDM ATFSM SFDM ATFSM SFDM Tl. T viio o mx wi d y wo mod. =. =.5 =. =.5 mi mi mi mi Nwoi Miopol Nwoi Miopol Nwoi Miopol Nwoi Miopol luid luid luid luid luid luid luid luid Tl. T viio o mi wi d o wo luid.
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