been assumed to be stationary. He also observed that when

Transcription

1 . INODUCION he flo of a viscos flid cotaied betee to paallel disks, oe of hich is pefomig sisoidal oscillatio i omal diectio is of iteest i may pactical sitatios. he peiodic motio of gea teeth is oe sch illstatio. It also aises he hydodyamic slide beaig ae sbjected to peiodic vaiatios i the load. he vaiatio of pesse o the disk is esposible fo the ea ad tea o the sface. he bodig sface may be damaged extesively, if the pesse distibtio gives ise to cavitatio dig the flo. Kh ad Yates 9 examied the pesse distibtio o a thi liqid film betee to disks, oe of hich pefoms omal oscillatios. He iclded the ietia tems ad compaed the eslts ith expeimetal data. hey have also tilied the method of least sqaes to fit a thid ode polyomial to expeimetal vales of maximm pesse. t it did ot agee ith the theoetical fomlae obtaied by them. eill 99 peseted a aalytical soltio fo this poblem. He has sho that the soltio depeds o to paametes vi. amplitde of the oscillatio of the disk ad the eyolds mbe elated to the maximm velocity of the oscillatig disks. he velocity field obtaied by eill fo small vales of eyolds mbe agees ith the eslts available i the liteate. Hoeve, thee ee vaiatios i the pesse tem. He has also sho that the pesse tem is vey sesitive to the magitde of. ajvashi 97 peseted aalytical soltio fo the flo betee to paallel ifiite disks, both of hich ee sbjected to omal sisoidal oscillatios. He peseted the detailed soltio of Navie-stokes eqatios fo the folloig thee cases: i the eyolds mbe, based o maximm velocity of the oscillatig disks, is so small that ietia tems ae eglected., ii is lage, bt the tems of the ode of hee the o-dimesioal amplitde of oscillatio ae eglected, ad, iii Neithe o ae egligible. He obseved a vey iteestig case he eyolds mbe is vey small o vey lage ad obtaied pesse hich as idetical ith eill 99. he depedece of pesse o as of the same fom as sho i his pape, he oe of the disks has 58

2 bee assmed to be statioay. He also obseved that he, the pesse as maximm o miimm he oscillatig disks ae at exteme positios. Secomb 978 solved the poblem of to-dimesioal flo of icompessible viscos flid i a log chael ith paallel alls of idth a t, hose alls plsate i a sisoidal mode. he velocity field i this case depeds liealy o the axial distace. his leads to the simplificatio of the goveig eqatios of motio ad sbseqet detemiatio of the soltio ove a ide age of flo paametes. He obtaied velocity pofiles fo all vales of the steadiess paamete ad the steady steamig eyolds mbe. S he flo of a icompessible viscos flid betee to ifiite hamoically oscillatig disks as cosideed by Sekhaa ad amaaiah 98. oscillatoy body foce havig the same feqecy as that of the disks as applied i the diectio of motio of the disks. Detailed aalytical expessios fo the velocity field of the flid, volme flo ate ad stess at the disks have bee obtaied. hey coclded that the flid velocity oscillates i time ith the same feqecy as that of the disks o the body foce, bt ith a phase diffeece. Sigh ad ajvashi 99 exteded this stdy by takig oe of the disks to be made of atally pemeable medim of fiite thickess boded by a statioay impemeable sface at the bottom. he flo field as divided ito to egios a the fee flid egio betee the to disks hee the flo is goveed by Navie-Stokes eqatios, ad, b the poos egio iside the pemeable disks hee geealied Dacy s la is valid. t the iteface eaves ad Joseph 97 slip boday coditio o the adial velocity as sed. Soltios have bee obtaied i the folloig to cases: i he the steadiess paamete is small. ii he is abitay bt the disks ae sbjected to small sisoidal oscillatios i the fom a t a cos t. Ski fictio o the ppe disk as evalated ad it as obseved that coefficiet of ski fictio iceases ith icease i. Effect of steadiess paamete o axial velocity as also examied ad it as oted that fo fixed vale of, the axial velocity deceases ith icease i. 59

3 Misa et al have stdied the steady flo of a icompessible viscos flid i a plae chael ith plsatig alls. he flid flx eteig the chael oscillates sisoidally ith time so that thee is alays a iflo at the etace. he goveig eqatios ee solved meically by fiite diffeece method. hey obseved that fo lage vales of feqecy of plsatio of the chael alls oscillatig sisoidally ith time ad a back flo occs ea the chael alls. It as also fod that at a sbseqet time thee as flo evesal acoss the etie coss-sectio of the chael. Fo sfficietly small vales of feqecy of plsatio of the chael alls sch back flo did ot occ. he flo betee to paallel disks appoachig o ecedig fom each othe symmetically as aalyed by Sigh et al 99. hey tasfomed the Navie-Stokes eqatios ito odiay diffeetial eqatios sig similaity tasfomatio ad the esltig eqatios ee solved meically. alytical expessios fo the velocity compoets, pesse distibtio ad sheaig stess o the all ee peseted fo vaios positive ad egative vales of eyolds mbe. Sai 99 aalyed the poblem of the sqeeig film betee ectagla disks, he oe of the disks has a vayig pemeability. he goveig eqatios ee deived o the basis of the assmptios, that i the flo i the film egio is slo, ii the flid has costat popeties ad is icompessible, iii the poos facig is of ifom thickess ad iv Dacy s la is satisfied i the poos medim. Dacy s la is iadeqate fo the flo ivolvig lage shea i poos media ith high pemeability. alteate appoach i sch sitatios is to adopt ikma 97 model. his etails coespodig chage i the boday coditios at the iteface of fee flid ad the poos medim. Kim ad ssel 985 aalyed the flid-poos medim iteface of ikma model by itodcig effective viscosity. hese boday coditios have bee fthe discssed by Che ad Che 99. I this chapte the flo of a viscos icompessible flid betee to paallel disks has bee stdied. he ppe disk is solid ad is sbject to a omal sisoidal oscillatio. he loe oe is made p of a atally pemeable mateial ith a impemeable sface at the bottom. he flo field has bee divided ito to egios. he egio betee the disks is filled ith fee flid egio hee the flo is goveed by the Navie-Stokes eqatios, hile i the poos egio iside the pemeable disk ikma

4 eqatios ae valid. Sitable boday coditios at the iteface of the flid ad poos medim have bee sed. he soltio of goveig eqatios has bee obtaied i the folloig cases. i. I this case oly the deivative ith espect to time fom the ietia tems eed be iclded. he esltig diffeetial eqatios ae liea ad soltio has bee obtaied i sectio.. ii Fo small. Fo this case a seies soltio i poes of has bee obtaied i sectio.. Explicit expessios fo the pesse distibtio has bee obtaied fo both the egios i each of the above to cases ad velocity pofiles have bee calclated.. GOVENING EQUIONS Coside to paallel cicla disks iitially at a distace a apat ad sepaated by a viscos icompessible flid. he loe disk is made p of atally pemeable mateial of thickess b, ith a impemeable sface at its bottom ad is statioay fo all times. he ppe impemeable disk is pefomig omal sisoidal oscillatio ith a feqecy ad amplitde a.he ppe face of the pemeable disk is take to be the plae = ith oigi at the cete of this face. he positio of the ppe disk at ay sbseqet time is give by a si t.. Let, be the velocity compoets i fee flid egio i the diectio of ad espectively. he eqatios of motio i this egio, assmig axial symmety ae p t.. p t hee, ad p ae kiematics viscosity, desity ad pesse espectively...

5 he eqatio of cotiity is.. Let, be the velocity compoets i ad diectio espectively, i the poos egio iside the pemeable disk ad p be the pesse i this egio.he flo i this egio is goveed by ikma la. he eqatios of motio i this egio ae k p t..5 k p t.. hee k is pemeability of poos medim. = = -b = a +ε si t Fig... Geometical cofigatio of the poblem Fee-flid egio Poos egio

6 he eqatio of cotiity i poos medim is he boday coditios ae..7, a cos t at a si t..8,, p p at..9 at b.. hee is costat depedig po the pemeability of poos medim. It follos fom eqatios of cotiity.. ad..7 that a soltio of the fom, t,, t, be soght i the fee flid ad poos medim espectively... he assmptio that ad ae idepedet of adial co-odiate ad fom of eqatios.. ad.. imply that p.. p Usig eqatio.. ad.. i eqatio.., e obtai.. t.. he folloig o-dimesioal vaiables ae itodced i the aalysis a, a, t, a W..5 Usig..5 i.., e get.. hee a

7 he positio of ppe disk is time depedet ad boday coditios..8 ae also time depedet. o emove the time depedece fom the boday coditios, the tasfomatio si,,,..7 is itodced. Eqatio.. takes the fom cos si si..8 Eqatio..5,.. ad..5 yield, k t..9 Eqatio..5 ad..9 yield W W W.. hee a k he boday coditios..8 to.. chage to cos,, at.. W.. W.. p p.. W at h..5 hee a b h. SOLUION OF EQUIONS FO he paamete is the atio of the amplitde of the oscillatio of ppe disk to the

8 iitial distace betee the to disks ad theefoe, is less tha oe. If is small, the coditio is atomatically satisfied. he coditio ca also be satisfied fo a abitay lage eyolds mbe by makig the amplitde of oscillatio satisfy. Soltio of eqatios i fee flid egio Fo, the eqatio..8 edces to.. Soltio of.. is soght i the fom, f si g cos.. Use of.. i eqatio.. ad sbseqet eqatig the coefficiets of cos o both sides give si ad f g iv iv g f.. hee pime deotes diffeetiatio ith espect to. he soltio of eqatio.. is give by g f C si Dcos e F cos C cos Dsi e E cos E si e F si e....5 hee / he costats,, C, D, E, F, ad ae eqied to be detemied fom boday coditios.. to... Pesse distibtio i fee flid egio O eglectig tems ith, eqatios..,..,..,..5 ad 5

9 ..7 yield p a.. p Eqatios.. ad.. to..7 give the pesse distibtio i the fee flid egio as p p a L cos si si..7 cos..8 hee the pesse at a chaacteistic adial distace =L fom the oigi o the iteface of the pemeable disk has bee take to be eqal to p. Soltio of eqatios i the poos egio W is assmed as W, F si G cos..9 Eqatios.. ad..9 give iv F F G iv G G F.. oday coditio..5 takes the fom F h ad G h.. he soltio of eqatios.. satisfyig.. is take i the fom F h.. G h.. he costats ad ae eqied to be detemied fom the boday coditios.. to...

10 Pesse distibtio i poos egio Eqatio..5,..,.. ad..5 give p W W W a.. p W W W..5 he pesse distibtio i the poos egio is obtaied fom eqatio..9 ad.. to..5 i the folloig fom p p L h + cos cos si si.. he boday coditios.. to.. take the folloig fom f, f, g, g, f F, g G, f F, g G..7 Usig boday coditios.. ad..7 the costats have bee evalated. hese ae ecoded i the PPENDIX-III.. SOLUION FO SMLL Soltio i fee flid egio Whe is small f ad g ae expessed i poes of i the folloig fom f f g g.. 7

11 8 Eqatio.. ad.. yield the folloig eqatios iv g.. g f iv.. f g iv.. Soltio of eqatio.. to.. is give by 5 g f C C C C g..7 he abitay costats occig hee ae ecoded i PPENDIX-III. Eqatios..,..7,.. ad..7 give the pesse i the fee flid egio as C L a p p si cos + C C cos } { 5 + si 5..8 Soltio i poos egio F ad G ae assmed i poes of as F F G G..9 Eqatios.. ad..9 yield the folloig set of eqatios G G iv.. G F F iv..

12 iv G G F.. he boday coditio.. is modified as G h, G h, F h.. he paticla soltios of eqatios.. de the boday coditio.. ae take i the folloig fom G D h, F D h, G D h.. he pesse distibtio i the poos egio is obtaied fom eqatio..,..5,..9 ad.. as p p a D D L h D cos D D D si D D cos D si he boday coditios..7 take the folloig fom f, f, g, g, g, g,..5 f F, g G, g G.. Vaios costats occig hee ae obtaied by sig the boday coditios.. ad.. ad ae ecoded i the PPENDIX-III..5 DISSCUSION OF ESULS Pesse dop a o distict cases have bee discssed i detail. Let s defie a o-dimesioal pesse dop p i fee flid egio as p a.5. p p Eqatio..8 ad.5. give pesse dop at i the fom p cos si ].5. [ 9

13 Fo exteme vales of a hee p p.5. L p Exteme vales of ta.5. p ae give by ].5.5 [ p b is small Eqatios..8 ad.5. give p at as follos cos O p.5. Exteme vales of si p occs at Exteme vales of ta p ae give by.5.7 [ p].5.8 Velocity Pofile a he velocity pofiles have bee evalated by takig m=.5,. ad h=.5. Fig.. depicts the pofile of f ad g agaist fo =.,. fo. he fig.. shos that at, f is egative. f vaishes at.8 ad.8 fo =. ad. espectively. It attais maximm positive vale i the cetal egio. he compaiso of vales fo =. ad. shos that the pofile fo =. cosses the coespodig pofile fo =. ad maitais its 7

14 positio fo the emaiig pat. he pofile of g shos miimm vale at the iteface ad iceases gadally, ad the attais its maximm vale at. Fig.. epesets f ad g fo =. ad. espectively. Pofile of f shos positive vale i the fist half ad egative vale i the secod half. It attais maximm positive vale at the iteface ad maximm egative vale at.75. t g is positive thoghot. g has maximm positive vale at ad shos ealy costat behavio pto the cetal egio ad the deceases apidly to attai its miimm vale at. he pofile g fo =. cosses the coespodig pofile fo =. ad its magitde emais less tha fo =.. Simila pheomeo is obseved fo f. 7

15 b < he pofile of f ad g fo small have bee sho i fig... he pofiles have bee da fo =. ad.5. oth f ad g have miimm ad maximm vale at ad espectively. Compaiso of fig.. ad fig.. fo ad < espectively shos simila behavios of g i both the cases, bt f shos maked diffeece i both the cases. I this case it shos maximm vale at. I fig..5, both f ad g sho maximm vale at the iteface. hese decease gadally ad attai miimm vale at. Fig.. ad fig..5 sho maked diffeece i the pofile of f ad g fo ad <. I this case the pofile of f ad g do ot itesect ith icease i vale of, hich is a distict deviatio fom the case a. he axial velocity, has bee plotted i fig.. fo diffeet vales of. he magitde of axial velocity deceases ith icease i fom to /. 7

16 7

17 . CONCLUSION It is coclded that pofiles of f fo decease ith icease i the vale of eyolds mbe ad attai maximm vale i the cetal egio, hile fo the case, pofiles of f decease ith decease i, ad pofiles attai maximm vale at. Pofiles of g sho simila behavio i both the cases he ad. 7