HEAT TRANSFER IN BOUNDARY LAYER VISCOLASTIC FLUID FLOW OVER ANEXPONENTIALLY STRETCHING SHEET

Size: px

Start display at page:

Download "HEAT TRANSFER IN BOUNDARY LAYER VISCOLASTIC FLUID FLOW OVER ANEXPONENTIALLY STRETCHING SHEET"

Sibyl Beasley
5 years ago
Views:

1 cinc Foc Vol. 8 () pp - Fclty o Pr nd Applid cinc LAUTECH Printd in Nigri HEAT TRANFER IN BOUNDARY LAYER VICOLATIC FLUID FLO OVER ANEXPONENTIALLY TRETCHING HEET *Okdoy A.. hd A. Ayndokn O. O. Dprtmnt o thmtic Covnnt Univrity P.. B. Ot Ogn tt Nigri Dprtmnt o thmtic Emmnl Alynd Collg o Edction Oyo Oyo tt Nigri Atrct Th ppr prnt th tdy o momntm nd ht trnr chrctritic in vico-ltic ondry lyr lid low ovr n ponntilly trtching contino ht with non-niorm ht orc. Th low i gnrtd olly y th ppliction o two ql nd oppoit orc long th -i ch tht trtching o th ondry rc i o ponntil ordr in nd inlncd y niorm mgntic ild pplid vrticlly. Th non-linr ondry lyr qtion or momntm i convrtd into ordinry dirntil qtion y mn o imilrity trnormtion. Approimt nlyticl imilrity oltion i otind or th dimnionl trm nction nd vlocity ditrition nction tr trnorming th ondry lyr qtion into Riccti typ nd olving it qntilly. Ht trnr qtion i thn olvd ing Rng-Ktt orth ordr mthod. Th ccrcy o th nlyticl oltion i lo vriid y compring th oltion otind to tho in litrtr whn Hrtmnn nmr i zro. Th ct o vrio phyicl prmtr on vlocity kin riction tmprtr nd Nlt nmr proil r prntd grphiclly. Ky word: Eponntil trtching ht Prndtl nmr Non-niorm ht orc/ink imilrity oltion vico- ltic lid ondry lyr low kin-riction.. Introdction omntm nd ht trnr in vico-ltic ondry lyr low ovr trtching ht hv n tdid tnivly in th rcnt pt c o it vr incring g in polymr procing indtry in prticlr in mnctring proc o rtiicil ilm nd rtiicil ir in om ppliction o dilt polymr. Th trnport o momntm ht nd m in lminr ondry lyr on th moving intnil trtching rc h conidrl prcticl rlvnc. For mpl in lctrochmitry (Chin (975) Gorl (978)) polymr procing (Gith (964) Erickon (966)) nd in ir indtri. In viw o incring importnc o non-nwtonin low Rjgopl t l (984) mind pcil cl o Vico-ltic lid known cond ordr lid. h nd iddpp(986) tdid th low o Vico-ltic lid o th typ ltr liqid B pt trtching ht. inc th ltr liqid B i lid tht h hort mmory thy rrivd t th om non-linr qtion tht cn drivd with th ondry lyr pproimtion. h () tdid vico-ltic lid low nd ht trnr ovr trtching ht with vril vicoity. All th tdi dl with th tdi concrning non-nwtonin low nd ht trnr in th nc o mgntic ild t prnt yr w ind vrl indtril ppliction ch polymr tchnology nd mtllrgy ( Chkrrti nd Gpt (979)) whr th mgntic ild i pplid in th vico-ltic lid low. Andron (99) invtigtd th low prolm o lctriclly *Corrponding thor mil: dl.okdoy@covnntnivrity.d.ng cinc Foc: An Intrntionl Jornl o Biologicl nd Phyicl cinc IN

2 Okdoy t l. Nomncltr dimnionl trm nction p clr prr k ltic prmtr min trm vlocity/r trm w k vicoltic prmtr v th vlocity componnt l rrnc lngth y th coordint i R non-dimnionl Rynold nmr C p pciic ht T tr tnor Q Ht orc/ink prmtr imilrity vril Grk lttr th norml tr modli cclrtion/dclrtion prmtr A nd A kinmticl tnor; dynmic vicoity B pplid mgntic ild kinmtic vicoity HD prmtr Flid dnity Ec Eckrt nmr lctricl condctivity c dimnionl kin-riction coicint Ht orc/ink prmtr Pr Prndtl nmr condcting vico-ltic lid pt lt nd imprml ltic ht. Lwrnc nd Ro (99) tdid th non-niqn o th HD low o cond ordr lid pt trtching ht. A nw dimnion w ddd to thi invtigtion y Elhhy () who mind th low nd ht trnr chrctritic y conidring n ponntilly trtching contino rc. h t l ( ) in thir ppr rport ht trnr in HD Vico-ltic lid ovr trtching rc. Kmri nd Nth () tdid HD low o Non-Nwtonin lid ovr continoly moving rc with prlll r trm. h t l (4) invtigt Non-Nwtonin gnto hydrodynmic low ovr trtching rc with ht nd m trnr. Rcntly iddhhwr nd hlhwr (5) mind th ct o rdition nd ht orc on HD low o vico-ltic liqid nd ht trnr ovr trtching ht. In hi work jit (6) otind irt nd cond ordr imilrity oltion o ondry lyr vico-ltic lid low ovr n ponntilly trtching ht with niorm ht orc. In idy prnt th tdy o diion o rctiv pci ndrgoing irt-ordr chmicl rction in ondry lyr low o n incompril homogno cond ordr lid ovr linrly hrinking ht in th prnc o trnvr mgntic ild. Th tdy rvl tht th vlocity i gtting clor towrd wll or incring mgntic prmtr whr it i going wy rom th wll or incring vico-ltic prmtr. It i lo ond tht th diion o rctiv pci i conidrly rdcd with incring vl o chmidt nmr mgntic nd rction rt prmtr whr it i incrd or nhncd vl o vico-ltic prmtr. Ngtiv concntrtion i orvd in om c which my not hv rl world ppliction. In rlity mot o th lid conidrd in indtril ppliction r mor non-nwtonin in ntr pcilly o vico-ltic typ thn vico typ. otivtd y ll th tdi w intnd to tdy th vicoltic lid low nd ht trnr ovr trtching ht in th prnc o niorm mgntic ild in th ondry lyr rgion. inc or HD low th ct o intrnl ht gnrtion i importnt nd hnc it i tkn into conidrtion nd comind ct o vico-lticity mgntic ild nd Rynold nmr on th kin riction coicint r conidrd.. Formltion o th prolm.. Prliminri Th contittiv qtion o n incompril cond ordr lid i givn y T pi A A A Kinmticl tnor A nd A r dind y A grd q grd q da A A dt T A T grd q grd q A (.) (.)

3 Ht trnr in ondry lyr vicoltic lid low Eqtion (.) w drivd y Colmn nd Noll (96) ing th potlt o grdlly ding mmory. Uing om primntl dt vriiction Fodick nd Rjgopl (979) gv th rng o vl o nd. (.) king o Eq.(.) Brd nd ltr (964) drivd th tdy tt two-dimnionl ondry lyr qtion or vico-ltic lid low in th orm v k v (.4) y y y y y y y Thi qtion h n drivd with th mption tht th norml tr i o th m ordr o mgnitd tht o th hr tr in ddition to th l ondry lyr pproimtion... Flow govrning qtion In ormlting th prolm w conidr th ollowing mption. (i) (ii) Th ondry ht i md to moving illy with vlocity o ponntil ordr in th il dirction nd gnrting th ondry lyr typ o low. A tdy two-dimnionl lminr low o n incompril lctriclly condcting vico-ltic liqid (ltr liqid B modl) d to n ponntilly trtching ht i conidrd. B B B B Imprml trtching lit w p ; v ; T Tw T A( ) t y l L Figr : Bondry lyr ovr n imprml ponntilly trtching ht. Th ht li in th pln y = with th low ing conind to y >. Th coordint i ing tkn long th trtching ht nd y i norml to th rcd two ql nd oppoit orc r pplid long th -i o tht th ht i trtchd kping th origin id. Thi low oy th rhologicl qtion o tt drivd y Brd nd ltr (964).For o nlyi w m tht diiption d to ltic tr i ngligil. It i lo md tht ondry lyr pproimtion cn d to impliy th ic qtion govrning th trnport o momntm nd ht. Frthr thi low pod ndr th inlnc o niorm trnvr mgntic ild. Th t o ic qtion togthr with th continity qtion thn com: v (.5) y

4 Okdoy t l. v k y y y c p T T v y T k y v y y y y y Q T T B B (.6) (.7) whr ll prmtr hv thir l mning. Th pproprit ondry condition or thi prpo r: t y ; w p ; v ; T Tw T A( ) l L t y ; ; T T (.8) whr A i poitiv prmtr. Aming tht mtril proprti ppring in th ov qtion r contnt th momntm qtion cn olvd rgrdl o th nrgy qtion.. oltion o th ondry lyr qtion At thi tg w try to ind itl imilrity vril ch tht th prolm cn rdcd to t o ODE intd o PDE. And it trn ot tht th ollowing imilrity vril cn mt or prpo: w l y l Uing thi imilrity vril th trm nction ( y) nd th tmprtr ild T (y) cn md dimnionl nd rpctivly: l y) lw ( ) T T ( ; () (.) Tw T Aming th low to two-dimnionl th vlocity componnt cn rltd to th trm nction : w l l w v (.) y l titting th dimnionl prmtr into th govrning qtion (.) to (.4) on wold otin: v k (.4) Pr Pr Pr Ec (.5) Pr c p Q k B w l Ec c k c p p w c p wk In trm o dimnionl prmtr th ondry condition rqird to olv Eqtion(.4) nd (.5) r: () () () ( ) Intgrting qtion (.4) w otin whr. For w gt ( ) v k d (.) (.6) (.7)

5 Ht trnr in ondry lyr vicoltic lid low 4 v k d intgrt qtion (.7) onc gin nd pply ondry condition (.6). Thi yild v k d d (.9) Now th oltion procdr o th qtion (.9) my rdcd to th qntil oltion o th Riccti-typ qtion n n n n n v n RH Thi itrtion lgorithm h to olvd y titting itl zro-ordr pproimtion on th R.H. o qtion (.9). m zro-ordr pproimtion o p or (.8) (.) (.) which tii th ondry condition t ininity. Intgrting th Eq.(.) nd mking o ondry condition t o (.6) w gt p (.) titting ll th drivtiv o zro-ordr pproimtion into R.H. o qtion (.9) nd ming tht irt-ordr itrtion on th L.H. o qtion (.9) tiying ll th ondry condition t (.6) w otin th vl o 4 k (.) Hr th qtion or irt-ordr itrtion tk th orm k k 4 (.4) Eqtion (.4) i th non-linr Riccti qtion nd thi cn olvd or hypr-gomtric hittkr nction (Armowitz nd tgn 964): Ltting in trm o conlnt k k k 4 4 / 5 c 4 w hv 6 C / C

6 Okdoy t l. 5 hitt C hitt hitt hitt hitt C hitt hitt hitt C hitt C C hitt hitt hitt C 6 5 kr kr kr kr kr kr kr kr kr kr kr kr (.5) whr kr kr hitt hitt kr kr hitt hitt nd C C with / Th dimnionl kin-riction coicint c i prd l B y v y y k y c w p t y R 7 k c Hr w l R i th non-dimnionl Rynold nmr.

7 Ht trnr in ondry lyr vicoltic lid low 6 Figr : Vrticl vlocity proil otind rom zro nd irt ordr oltion whn 5 nd k. Figr : Horizontl vlocity proil otind rom zro nd irt ordr oltion whn 5 nd k. 4. Ht Trnr Th govrning ondry lyr ht trnport qtion in th prnc o pc- nd tmprtr- dpndnt intrnl ht gnrtion/ orption or two-dimnionl low i Pr Pr Pr Ec (4.) (4.) Th nrgy qtion (4.) togthr with th ondry condition (4.) i linr cond ordr ordinry dirntil qtion with vril coicint (η) which i known rom th oltion o th low qtion (.) nd (.5) nd th Prndtl nmr Pr i md contnt. From Figr () nd () ov it i clr tht oth zro nd irt ordr oltion r vry clo. Hnc hv n md o zro ordr oltion in olving qtion (4.) nmriclly ndr th ondry condition (4.) ing cntrl dirn c or th drivtiv nd Thom lgorithm or th oltion o th t o dicritizd qtion. Th rlting ytm o qtion h to olvd in th ininit domin < η <. A init domin in th η-dirction cn d intd with η chon lrg nogh to nr tht th oltion r not ctd y impoing th ymptotic condition t init ditnc. Grid-indpndnc tdi how tht th compttionl domin cn dividd into intrvl ch o niorm tp iz.. Thi rdc th nmr o point twn withot criicing ccrcy. Th vl w ond to dqt or ll th rng o prmtr tdid hr. Convrgnc i md whn th rtio o vryon o or or th lt pproimtion 5 dird rom nity y l thn t ll vl o η in. 5. Rlt nd Dicion: In th ppr w invtigt th ondry lyr low nd ht trnr in vico-ltic liqid ovr n ponntilly trtching ht in prnc o non-niorm ht orc. imilrity oltion i d to otin th vlocity ditrition which i govrnd y non-linr dirntil qtion. Figr 4: Vrition o vrticl vlocity or vrio vl o Figr 5: Vrition o vrticl vlocity or vrio vl o k

8 Okdoy t l. 7 Figr 4 nd 6 i grphicl rprnttion which dpict th ct o gntic ild Prmtr on th vlocity proil rpctivly. It i ond tht th ct o gntic ild Prmtr i to nd rdc th vlocity igniicntly in th vico-ltic low in comprion with th vico low thi i d to th ct tht incr o ignii th incr o Lorntz orc which oppo th low in th rvr dirction. Th grph or th non-dimnionl vlocity proil or dirnt vl o th vico-ltic nd prmtr k r hown in Figr 5 nd 7 rpctivly. Th nlyi o th igr dmontrt tht th ct o th vico- ltic prmtr ki to dcr vlocity throghot th ondry lyr low ild which i qit ovio. Figr 6: Ect o th prmtr on th horizontl vlocity Figr 7: Ect o th prmtr k on th horizontl vlocity Figr 8: Tmprtr ditrition or vrio vl o Pr Figr 9: Tmprtr ditrition or vrio vl o k Figr : Tmprtr ditrition or vrio vl o Figr : Tmprtr ditrition or vrio vl o Figr (8) nd (9) i plottd or th tmprt r ditrition or vrio Prndtl nmr nd vico-ltic prmtr k rpctivly it i n intrting not tht thr i igniicnt nhncmnt o tmprtr in oth c. On comprion o th crv it cold n tht thr wold n incr in tmprtr in th low

9 Ht trnr in ondry lyr vicoltic lid low 8 rgion or lowr vl o Prndtl nmr which rlt in incr o thrml ondry lyr thickn Prndtl nmr incr. Alo incr in vico-ltic prmtr k ring ot n incr in tmprtr in th low rgion In Figr w diplyd th ct o tmprtr ditrition. On comprion o th crv it i n tht tmprtr incr in th low rgion d to th ppliction o mgntic ild. Hr incr o mgntic orc c igniicnt incr o thrml ondry lyr thickn in th lid low. Figr how th ct o ht orc/ink on th thrml ondry lyr o th low ild. orv tht or ht gnrtion tmprtr ditrition dcr throghot th ondry lyr o th low ild ht orption nd incr with ht gnrtion. Figr : kin riction ditrition or vrio vl o Hrtmnn nmr Tl : Ect o Eckrt nmr on horizontl vlocity y Ec=. Ec=. Ec= E-5.7E-5 8 4E-5.6E-5.E-5 9.7E E-6 4.8E E-6.4E-6.9E-6

10 Okdoy t l. 9 Tl how th ct o Eckrt nmr on th tmprtr ditrition. It cold n tht incr in Eckrt nmr ring ot dcr th tmprtr ditrition throghot th ondry lyr. Th grph o th non-dimnionl kin-riction prmtr c gint vico-ltic prmtr or dirnt vl o th Hrtmnn nmr i hown in Figr. Th igr how tht th kin riction prmtr incr on th wll with th ppliction o mgntic ild. Thi i c o th mgntic orc ct rtrding orc nd c th incr o hr tr. Th comind ct o vicolticity nd imprmility o th wll i to incr th kin riction t th wll lrgly. Hr dditionl introdction o hr tr t th wll y mgntic ild non-nwtonin ntr o vicoltic low nd imprmility o th wll thry dcr th ondry lyr thickn ld to incr th kin riction o th low. Tl : Ect o low prmtr on wll r tr k N Ec Pr N Th ht trnr phnomn i lly nlyzd rom th nmricl rlt o phyicl prmtr nmly wll tmprtr grdint nd th m r docmntd in tl. Anlyzing th tl rvl tht th ct o incring th vl o Prndtl nmr Eckrt nmr nd ht ink i to incr th wll tmprtr grdint. hil vico-ltic prmtr Hrtmnn nmr nd ht orc rdc th Nlt nmr. Th rlt r in tn with wht hppn in rgion wy rom th ht. Conclion A mthmticl prolm h n ormltd or th ht nd m trnr in vico-ltic lid low ovr n ponntilly trtching imprml ht. In th oltion procdr th non-linr dirntil qtion i convrtd into n ordinry dirntil qtion y pplying imilrity trnormtion. qntil imilrity oltion o th trnormd momntm qtion r otind nlyticlly y olving th non-linr Riccti typ qtion. Eprion r lo otind or th dimnionl kin-riction coicint c nd wll r tr N. Th importnt inding o th grphicl nlyi o th rlt o th prnt prolm r ollow.. Incr in oth ht gnrtion nd Hrtmnn nmr incr th tmprtr throghot th ondry lyr.. Th ct o incring th vl o th vico-ltic prmtr ki to incr th tmprtr ditrition throghot th ondry lyr nd dcr th Nlt nmr.. Th ct o incring th vl o th vico-ltic prmtr ki to dcr th vlocity throgh ot th ondry lyr.

11 Ht trnr in ondry lyr vicoltic lid low 4. Th ct o incring th vl o th vico-ltic prmtr ki to dcr th kin-riction prmtr c nd th ct o incring vl o th hrtmnn nmr i to incr th kin riction coicint c nd dcr th Nlt nmr. Acknowldgmnt Th thor wih to thnk th thority o Covnnt Univrity or nling nvironmnt providd wll nintrrptd intrnt rvic. Th morl pport o mmr o clty i lo pprcitd. lo wih to cknowldg th ort o th ditor nd rviwr or thir thorogh work nd vll ggtion which ld to improvmnt o thi work. Rrnc: Armowitz. nd tgn I. (964): Hndook o thmticl Fnction. Nw York: Dovr. AndronH. I (99): HD low vicoltic lid pt trtching rc. Act.ch95 7 Brd D.. nd ltr K. (964): Eltico-vico ondry lyr low.. Two dimnionl low nr tgntion point. Proc. Com. Phil. oc. vol.6 pp.667. Chkrrti A nd Gpt.A. (979): Hydromgntic low nd ht trnr ovr trtching ht. Q. Appl. th.77 ChinD. T. (975): trnr to contino moving ht lctrod J. Elctrochm. oc Colmn B.D. nd Noll. (96): An pproimtion thorm or nctionl with ppliction in continmmchnic. Arch. Rt. ch. Anl. vol.6 pp.55. Elhhy E..A. (): Ht trnr ovr n ponntilly trtching contino rc with ction. Arch. ch. vol.5 No.6 pp ErickonL. E. FnL. T. nd Fo V. G. (966): Ht nd trnr on moving Contino lt plt with ction or injction Indt.Engg. Chm. Fnd. Vol5No Fodick R.L nd Rjgopl K.R. (979): Anomlo tr in th modl o cond ordr lid. Arch. Rtion. ch. Anl. vol.7 pp.45. GithR.. (964): Vlocity tmprtr nd concntrtion ditrition dring ir pinning. J. Ind. Eng. Chm.Fndmntl Gorl R..R (978): Untdy trnr in th ondry lyr on contino moving ht lctrod J. Elctrochm. oc Kmri. ndnthg.() HD low o Non-Nwtonin lid ovr continoly moving rc with prlll r trm. Act. ch Lwrnc P.. nd Ro B.N. (99): Ht trnr in th low o vico-ltic lid ovr trtching ht. Act. ch. vol.9 pp.5-6. idy C. (): Diion o chmiclly rctiv pci in Vicoltic low ovr hrinking ht in th prnc o mgntic ild.int. J. o Appl. th.nd ch. 8(8): Rjgopl K. R N T. Y Gpt A.. (984): Flow o Vicoltic Flid ovr trtching ht.rhologicl Act pp. -5. iddhhwrp. G. ndhlhwru..(5)ect o rdition nd ht orc on HD low o vico-ltic liqid nd ht trnr ovr trtching ht. Int. J. Non-linr ch h A nd iddpp.b (986): Vico-ltic ondry low pt trtching plt with ction nd ht trnr RholAct5 79 h A Vn P.HRjgopl K. ndprvin V.K Non-Nwtonin gnto hydrodynmic low ovr trtching rc with ht nd m trnr Int.J.Non-Linr.ch. 9(4) h A. Johi A ndonth R. () Ht trnr in HD Vico-ltic lid ovr trtching rcchnicvol8 No jit K. K. (6): Bondry lyr vico-ltic lid low ovr n ponntilly trtching ht: Int. J. O pplid mchnic nd nginring vol. No. pp.-5

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion