Free Convective Flow of a Visco-Elastic Fluid Bounded By an Oscillating Porous Flat Plate in Slip Flow Regime

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1 IOSR Jornal of ahemaics (IOSR-J) e-issn: p-issn: 9-765X. olme Isse er. (ar - pr. 5) PP Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in Slip Flow Reime r.sshila hand Pradhan Dr.J.P. Panda Dr.R.N. Saapah PrincipalSGI Jr.ollee for WomenackOdisha Prof-in ahemaicsssbrlaodisha PrincipalHIBBSROdisha bsrac: he aim of he presen problem is o sd and brin o he effec of rarefacion parameer R Harman nmber Schmid nmber Sc and Grashof nmber Gr on veloci concenraion and skin fricion of he HD free convecive flow of visco-elasic flid (Walers Bodel)[] hroh a poros medim bonded b an oscillain poros fla plae in slip flow reime. I. Inrodcion he sd of flcain flow is imporan in he paper indsr and man oher echnoloical fields. lso he sd of flow and hea ransfer in slip flow reime is of rea ineres de o is applicaion in hih speed flih in he pper amosphere. herefore flcain flow problems are of rea ineres o man researchers. ooke e al. [] have repored nsead HD free-convecion and mass ransfer flow pas an infinie heaed poros verical plae wih ime dependan scion. Das e al. [] have analzed free convecion flow and mass ransfer of an elasico-viscos flid pas an infinie verical poros plae in a roain poros medim. hadhar and hand [] have sdied hdromaneic flow pas a lon verical channel embedded in poros medim wih ranspiraion coolin. rcebe and Erim [4] have analzed hea ransfer of a visco-elasic flid in a poros channel. Panda e al. [5] have sdied free convecion of condcin viscos flid beween wo verical walls filled wih a poros maerial. Sharma and Yadav [6] have repored sead HD bondar laer flow and hea ransfer beween wo lon verical wav walls. Dash[7] has sdied he effecs of radiaion and chemical reacion in HD flow pas a sreched verical permeable srface hroh a poros medim wih consan scion. Sharma e al. [8] have analaed he sead HD flow and hea ransfer beween wo roain poros disk. Sharma and Sharma [9] have sdied he effec of oscillaor scion and hea sorce on hea and mass ransfer in HD flow alon a verical movin poros plae bonded b poros medim. Das and Panda [] have repored he effec of hea sorce and variable maneic field on nsead hdromaneic flow of a viscos sraified flid pas a poros fla movin plae in he slip flow. Panda and Das[] have analzed he HD free convecion flow of a pariclae sspension pas an infinie poros inclined fla plae wih hea absorpion. he aim of he presen problem is o sd and brin o he effec of rarefacion parameer R Harman nmber Schmid nmber Sc and Grashof nmber Gr on veloci concenraion and skin fricion of he HD free convecive flow of visco-elasic flid (Walers Bodel)[] hroh a poros medim bonded b an oscillain poros fla plae in slip flow reime. II. Formlaion Of he Problem he phsical confiraion consiss of an nsead flow of an elecricall condcin and incompressible elasico-viscos liqid of Walers B model [] wih simlaneos hea and mass ransfer near an oscillain infinie poros fla plae in slip flow reime nder he inflence of a ransverse maneic field of niform srenh. he x-axis is aken alon he fla plae in horizonal direcion and -axis is perpendiclar o i. niform maneic field of srenh B is applied in he direcion of axis. For problems in aeronaical enineerin he Renolds nmber is sall small. nder he condiion he indced maneic field ma be neleced wih respec o he applied maneic field. he pressre P in he flid is assmed o be consan. he represens scion veloci. Iniiall he plae and flid are a res. hen he plae is se o an oscillaor moion. he plae is a consan emperare w and concenraion w. nder he followin assmpions he presen problem has been sdied. he moleclar ranspor properies are assmed o be consan. he densi variaion de o emperare and concenraion difference is approximaed b Bossinesq approximaion.ass fracion of diffsin species is low compared o he oher species in he binar mixre.he viscos dissipaion in ener eqaion is neliible.no chemical reacion akes place in he flid.he permeabili of he medim is niform. DOI:.979/ Pae

2 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. DOI:.979/ Pae s he plae is infinie lenh all he variables in he problem are fncion of and. s i is common in Ralih s problem convecive erms and pressre radien erm in momenm and ener eqaion are neleced. he dissipaion nmber correspondin o viscos dissipaion is small for mos common experimen wih flids for he raviaional field srenh of he earh. For flid like waer and ases a ordinar emperare viscos dissipaion will rejec in rblen flow. iscos dissipaion is neleced becase of small veloci is enconered in free convecion laminar flows. he relaive effec of pressre radien is evalaed b comparin pressre force wih he viscos force in he bondar laer. he raio is a fncion onl of Prandl nmber and varies monoonicall from o. over he Prandl ranes from - o. Hence accordin o Gebhar[] he pressre radien for free convecion laminar flow neleced wih sal bondar laer approximaion. he overnin eqaions for visco-elasic liqid of Walers B model is iven b v v=consan( = - a =) () B () () D (4) Firs order veloci slip bondar condiion of he problem when he plae execes linear harmonic oscillaion in is own plane are iven b = a L e in as (5) where L = ( m ) / P and L m L is he mean free pah and m is he axwell s reflecion coefficien. On inrodcin he followin non-dimensional qaniies p n n R = L (Rarefacion parameer) = (Elasici parameer)

3 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. DOI:.979/ Pae Sc = D (Schmid nmber) Pr = (Prandl nmber) = / B (Harman nmber) Gr = (hermal Grashof nmber) Gm = (ass Grashof nmber) Q = + p in eqaion () () and (4) afer droppin he aserisks () Q Gm Gr (6) Pr (7) Sc (8) Wih he bondar condiions = a R e in as (9) III. ehod Of Solion Eqaion (6) is he hird order and wo bondar condiions are available. De o inadeqae condiion a perrbaion mehod has been applied wih < as he perrbaion parameer. Le = = =... R Sbsiin in eqaion(6)-(8) and eqain he powers of we e he followin zeroh order and firs order eqaions wih he bondar condiions. Zeroh order Q Gm Gr () Pr () Sc ()

4 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. Firs order () Pr (4) Sc Gr Gm (5) wih bondar condiions Q = e in + R a as (6) Frher we inrodce = + e in (7) = + e in (8) = + e in (9) = + e in () = + e in () = + e in () in zeroh order and firs order eqaions harmonic and non-harmonic consideraions separael and obain he he followin eqaions. Q Gr Gm () Q in Gr Gm (4) Q Gr Gm (5) Q in Gr Gm in (6) Pr (7) Pr in Pr (8) Pr (9) Pr in Pr () Sc () Sc insc () DOI:.979/ Pae

5 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. Sc () Sc insc (4) wih bondar condiion R R as = as (5) Hence he veloci emperare and concenraion field can be expressed in erms of he flcain par as Pr Sc e e e n sinn e 4 cos 4 Pr Sc 5 e e 6 e e + 7e e cosn sinn ne cosn (6) 4 sin n P r e (7) e Sc (8) where 4 o are consans iven in ppendix. I. Resls nd Discssion In order o e a clear insih o he phsical problem considered here nmerical resls are displaed wih he help of raphical illsraions. he dimensionless veloci profiles for differen vales of maneic parameer () Rareficaion parameer (R) porosi parameer (p) Elasici parameer () modified Grashof nmber (Gm) schimd nmber (Sc) Prandl nmber (Pr) are presened wih he help of raphs. For nmerical compaion he vale of Gr is aken posiive (i.e. Gr.=5.).his indicaes ha he sd has been carried o nder he inflence of he coolin of he plae. he ineresin aspec of he problem is o sd he combined effec of he flow parameers wih ha of firs order veloci slip bondar condiion when he plae execes linear harmonic oscillaion in is own plane. From Fi. i is observed ha for he heavier species i.e wih increasin Sc he veloci decreases. his is in ood areemen wih he proper of he Schmid nmber presenin he raio of inemaic viscosi and diffsivi of he diffsin species. Frher wih an increasin maneic parameer veloci increases when <.8. hereafer reverse effec is observed. However he effec of he permeabili parameer is js opposie o ha of maneic parameer. Frher i is observed ha nder he inflence of rareficaion parameer he veloci decreases a all poins (rves I and III). oreover boanc effec (Gm) de o mass ransfer enhances he veloci (rves I and I). he sdden rise in he veloci is marked near he plae de o he presence of elasici of he flid b when =. i.e. in case of viscos liqid no sdden increase is marked insead slow and niform variaion is noiced. oreover frher increase in he elasic parameer leads o an increase in he veloci (rves I and I). I is noe worh o record in respec of he represenaive crves sch as X ( =.)II ( =.) and XI ( p = ) represenin he case of viscos flid wiho maneic field and wiho poros medim respecivel. rve II characerizes he dela in aainin he free sream condiion and hinnin of bondar laer hickness when maneic field is wihdrawn. DOI:.979/ Pae

6 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. Fi - eloci profiles when Pr =.7 n = Gr = 5 =. he crve X ( =.) shows ha he veloci is almos linear in he absence of elasic proper of he flid. rve XI ( p =) has no special feare in comparision wih is conerpar. i.e. p =. 4.. onclsion. he effec of he permeabili parameer is js opposie o ha of maneic parameer.. Rarefacion parameer conribes o decelerae he flid paricles in he flow domain. Elasic elemens conribes o sdden rise of he veloci near he plae.. he boanc effec de o mass ransfer enhances he veloci. 4. aneic force enhances he skin fricion as he ime elapses. Reference []. Israel-ooke. and SialoF.B. nsead HD free-convecion and mass ransfer flow pas an infinie heaed poros verical plae wih ime dependan scion SE.od.eas. on. B []. Das S.S.Panda J.P. and Dash G..4 Free convecion flow and mass ransfer of an elasico-viscos flid pas an infinie verical poros plae in a roain poros medim SE. od. eas. on. B []. hadhar R.. and hand.5 Hdromaneic flow pas a lon verical channel embedded in poros medim wih ranspiraion coolin SE.od.eas. on.b [4]. rcebe.and Erim.J.5 Hea ransfer of a visco-elasic flid in a poros channel In. J. of Hea and ass ransfer [5]. Panda J.P. Panaik.B. and. chara.6 Free convecion of condcin viscos flid beween wo verical walls filled wih a poros maerial [6]. Sharma P. R. and Yadav G. R.6 Sead HD bondar laer flow and hea ransfer beween wo lon verical wav walls [7]. Dash Shreekana7 Effecs of radiaion and chemical reacion in HD flow pas a sreched verical permeable srface hroh a poros medim wih consan scion 768. DOI:.979/ Pae

7 Free onvecive Flow of a isco-elasic Flid Bonded B an Oscillain Poros Fla Plae in. [8]. Sharma P. R. Yadav G. R.and Sharma..7 Sead HD flow and hea ransfer beween wo roain poros disk [9]. Sharma P. R. and Sharma..7 Effec of oscillaor scion and hea sorce on hea and mass ransfer in HD flow alon a verical movin poros plae bonded b poros medim []. Das S.S. and Panda J.P.8 Effec of hea sorce and variable maneic field on nsead hdromaneic flow of a viscos sraified flid pas a poros fla movin plae in he slip flow reime In. J. dv. ppl. Flid echanics4 pp.87-. []. Panda J.P and Das S.S.9 HD free convecion flow of a pariclae sspension pas an infinie poros inclined fla plae wih hea absorpion SE. od. eas. on. B 78 pp.-. []. Walers. Qar. J. ech. appl. ah. pp ppendix 4Q 4Q 4Q 6n 4Q 4Q 6n 4Q 4 4Q 6n 4Q Gr 4 4Q 6n 4Q Pr Pr Gm Sc Sc R Pr RSc R 4 R R R Pr R 4 R Sc 6 Pr Pr Sc Sc ( 4 ) ( 4 ) 4 ( 4 ) 4 ( 4 ( ) ( ) 4 4 i 4 4 i ) DOI:.979/ Pae