Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium
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1 Intrntionl Journl o Sintii & Enginring srh Volum 3, Issu 7, July- ISSN Ets o Vribl Fluid Proprtis nd Visous Dissiption on Mid Convtion Fluid Flo pst Vrtil Plt in Porous Mdium P.K.Singh Dprtmnt o Mthmtis, nivrsity o Allhbd, Allhbd-, INDIA Abstrt-In th prsnt ork, r nd ord onvtion boundry lyr lo o n inomprssibl nd visous dissiptiv luid ith vribl thrml ondutivity nd tmprtur dpndnt visosity pst n isothrml vrtil plt is invstigtd. Th onvtiv lo is tking pl in suh porous mdium hos prmbility is ssumd to b sptilly vribl. Th onvtiv lo is du to to tors hih inlun th lo simultnously- () r strm long th plt nd () th buoyny or usd by th vritions in dnsity du to tmprtur dirn. Th govrning qutions or th th boundry lyr lo r onvrtd into to systm o oupld ordinry dirntil qutions by using suitbl similrity trnsormtions. Ths qutions r solvd numrilly nd ts o sho numbr, Ekrt numbr nd prmbility prmtr on th vloity nd tmprtur r disussd nd prsntd grphilly Ky Words- Porous mdium, vribl prmbility, thrml ondutivity, visosity, visous dissiption,vrtil plt. INTODCTION Th ombind r nd ord onvtion lo nd ht trnsr problms in luid sturtd porous mdium hv bn th subjt mttr o tnsiv invstigtions du to thir immns pplitions in numbr o nginring nd industril pplitions in industry, sin nd thnology- or mpl in ptrolum industry, hmil nginring, gothrml rsours nd ooling prosss o nulr rtors t., to nm. Th invstigtions on r onvtion los hih r put on irm thortil oundtions by Polhusn[], r tndd by Mrkin[] nd [3] nd Chng nd Minkoyz [4] t.. ngnthnn nd Visknt [5] nd Chn t l. [6] hv studid th ombind r nd ord onvtion rom vrtil plts in porous mdi. Hr it is mntionorthy tht ll ths studis hv bn rrid out or th luid los hih hv onstnt proprtis. Th physil proprtis o th luids, minly visosity nd thrml ondutivity my hng signiintly ith tmprtur Shlihting [7]. Kys nd Crrd [8] hv dsribd in dtils vrious rltions btn th physil proprtis o luids nd tmprtur. It is to b notd tht tht dirnt luids bhv dirntly ith tmprtur. Choi [9] studid th ts o vribl proprtis on th boundry lyr lo. Li nd Kulki [], Pop t l. [] nd Esr nd Nth [] hv shon tht tmprtur dpndnt visosity hs quit signiint t on momntum nd thrml trnsport in th boundry lyr lo. Elbshbshy [3], Sddk [4] nd Sddk nd Almushigh [5] onsidrd th hydromgnti lo nd ht trnsr pst ontinuously moving porous boundry ith simultnous ts o rdition nd vribl visosity. Most o th studis involving vribl visosity hv onsidrd th thrml ondutivity s onstnts. In ordr to hv lrr nd mor insightul pitur o th thrml trnsport, it ould b hlpul to study th ts o thrml ondutivity vrition on ht trnsr boundry lyr phnomn. Elbshbshy nd Ibrhim [6] nd Khound nd Hzrik [7] hv onsidrd th vrition o visosity prmtr long ith thrml ondutivity/ diusivity prmtr nd thir indings sho signiint inlun on th vloity nd tmprtur distribution. Also, th studis o Tirny t l. [8] nd Bnnti nd Brosilo [9] hv shon tht th porosity o th mdium my hng hih in turn my us hng in th prmbility o th mdium. Chndrskhr t l.[], took into ount th vrition o prmbility in thir invstigtions on mid onvtion los in th prsn o horizontl imprmbl surs in sturtd porous mdi nd hv shon tht th vribility in porosity hs signiint t on th vloity distribution nd ht trnsr. Chndrskhr nd Nmboodiri [] studid th mid onvtion los bout inlind surs in sturtd porous mdi inorporting th vrition o prmbility nd thrml ondutivity du to pking o prtils. Mssour nd El-Shr [] hv onsidrd th ts o vribl prmbility nd thrml ondutivity hil ddy nd ddy [3] hv tkn into ount th ts o vribl visosity nd thrml ondutivity on n ltrilly onduting luid lo pst moving vrtil plt. Hssnin [4] onsidrd th inlun o vribl prmbility nd thrml ondutivity on th mid onvtion lo rom n imprmbl vrtil dg hrin h obtind non similrity solution or th s o vribl sur ht lu. In vi o th inrsing thnologil pplitions o mid onvtiv los, it ould b intrsting to study this typ o lo ith vribl luid proprtis. To my knoldg, simultnous ts o vritions in visosity, thrml ondutivity nd mdium prmbility hv not bn studid so r. Thus th im o th prsnt ppr is to invstigt th ts o vribl prmbility nd thrml ondutivity on th mid onvtion boundry lyr dissiptiv lo ith tmprtur dpndnt visosity pst vrtil plt in porous mdium. drop p. MATHEMATICAL FOMLATION Lt us onsidr stdy to dimnsionl boundry lyr lo o n inomprssibl nd visous dissiptiv luid long vrtil plt in porous mdium. dirtion is tkn long th plt nd y is norml to it,i.., th plt strts t = nd - IJSE
2 Intrntionl Journl o Sintii & Enginring srh Volum 3, Issu 7, July- ISSN tnds prlll to th is nd is o smi ininit lngth. Th plt tmprtur is uniormly mintind t T nd th tmprtur T is highr thn th tmprtur T o th luid r y rom th plt. A stdy lo prlll to th plt ith r strm vloity is ssumd to tk pl. Th mid onvtiv lo is ssumd to tk pl du to th simultnous ts o () th r strm long th plt nd () th buoyny or usd by th vritions in dnsity du to tmprtur dirn. Lt u nd v b th vloity omponnts in th boundry lyr rgion long th nd y- s rsptivly. Th visosity nd th thrml ondutivity o th luid r ssumd to b vribl.also, th prmbility o th porous mdium is supposd to b vribl. Thn undr th usul boundry lyr pproimtions, th govrning qutions or th prsnt Dry typ lo, olloing Nild nd Bjn [8] nd Shlihting [9], r givn by: u v () y u u u u v ( ) g T T u () y y y k k( y) v ( ) (3) T T T u u y u y y y y k( y) Boundry onditions p u, v, T T, t y (4) u, T T, t y W introdu th olloing non-dimnsionl vribls: y ( ), suh tht u, v y No, hv u, And ths T vlus T o th vloity omponnts stisy th qu T T (6) Th vritions in th prmbility nd thrml ondutivity hv bn ssumd, olloing Chndrskhr t l.[], s givn blo - k( ) k ( d ) (7) ( ) [ ( d ) b{ ( d )}] hr v d nd d r onstnts,, k nd (5) r th vlus o th diusivity, prmbility nd porosity rsptivly t th dg o th boundry lyr, b bing th rtio o th thrml ondutivity o th solid to tht o th luid. Th vrition o visosity ith dimnsionlss tmprtur, olloing Slttry (978), is ssumd to b o th orm- hr (8) is th visosity vrition prmtr nd it s vlu dpnds on th ntur o th luid nd luid t th dg o th boundry lyr. is visosity o th Thus, ith ths ssumptions on th physil prmtrs givn by (7) nd (8), th qutions (), () nd (3) ith th hlp o qutions (5) nd (6), rdu to th olloing ordinry dirntil qutions: ( d ) (9) [ ( d ) b{ ( d )}] d( b ) P r E E [ ] ( d g T T ) 3 P r ( T T p () k Th trnsormd boundry onditions r givn by- ' t ' t () () Th physil problm is no mthmtilly rprsntd by th qutions (6) nd (7) nd ths qutions involv iv prmtrs,, P r, E nd. bing non dimnsionl buoyny prmtr givs th r onvtion hil, th ynolds numbr givs th ord onvtion nd, thror, givs th rltiv importn o ord nd r onvtion in dtrmining th ovr ll lo. In th similr shion, th prmtr, bing rtio o non dimnsionl prmbility prmtr nd th ynolds numbr rprsnts th rltiv importn o Dry nd gnrl visous drg. Th Ekrt numbr E pprs in th visous dissiption trm. Th prmtr pprs in th Dry trm. 3 ESLTS AND DISSCSION Figur ( & b) shos th ts o on th proils o th non dimnsionl vloity nd tmprtur. As rportd in Chndrskhr t l.[], th los or th vlus o =.5 nd.35, rprsnt th mid lo ith som mrgin o rror. W s tht or ths vlus, th vloity (grph ) initilly inrss rom th no slip ondition nr th plt quiring pk vlu nd thn grdully drss IJSE
3 Intrntionl Journl o Sintii & Enginring srh Volum 3, Issu 7, July- 3 ISSN nd ttins th r strm vloity. Sm is tru or tmprtur (grph b) lso. W hv lso plottd th boundry lyr bhviour or opposing lo lso, i.., or th ngtiv vlus o, hr hv tkn it s -. For vloity nd tmprtur both, th proils stdily pproh rsptivly th vlus nd, th r nd boundry onditions in th rsptiv ss. Whil solving th problm, notid tht th vloity nd tmprtur proils bom osilltory or th vlus o >.75 nd to sho this kind o bhviour, hv plottd it or =3. Also, hv invstigtd th vloity nd tmprtur distribution tking = hih mns no vrition in th visosity. In this s, th pttrns r lik thos or =3, but ith muh grtr pk. Ths unusul bhviours n b ttributd to th simultnous ponntil vritions o th thrml ondutivity, visosity nd mdium prmbility. Figur shos th inlun o prmbility prmtr on th lo ild. As th vlus o this prmtr r inrsd, obsrv tht vloity nd tmprtur both sho inrsing trnds nd thn thy grdully quir thir rsptiv boundry vlus. Also, hv plottd ths proils or to vlus o Pr, 7 &, hih r shon rsptivly by dshd nd dottd lins. Th ts o Ekrt numbr E r shon in th igurs 3 nd it is obsrvd rom both th igurs tht n inrs in hs th t o inrsing both vloity nd tmprtur. Th ts o Nild modiition in th nrgy qution r shon by th dshd lins nd th t is quit prominnt in tht th tmprtur proil hs signiintly drsd hih, in turn, E hs th t to lor th vloity proil. Th ts o uniorm i.., no vritions, in th prmbility r shon by th thik lins in both th grphs nd, th ts r quit visibl in both th proils. Whil in th tmprtur proil, th t is to inrs th tmprtur, on th othr hnd, it ts th vloity proil dvrsly. Th ts o onstnt visosity r shon by dottd lins nd it is obsrvd tht vloity ild rmins lmost untd but, it hs th ts o inrsing th tmprtur proil. Figur 4 shos th ts o visosity vritions on th vloity nd tmprtur proils. W kno tht or th vlus o >, th visosity drss nd rom th grph it is notid tht vloity proils sho drsing pttrns orrsponding to ny inrs in th vlus o. In both th igurs, th ts o lr mdium i.., th bsn o prous mdium r shon by thik lins shoing drsing trnd in both th proils. Also, hv invstigtd th ts o uniorm thrml ondutivity nd mdium prmbility hih r shon by dottd lins nd th ts r to lor th vloity nd tmprtur vlus nd this rsult, prhps, provids n plntion s to hy got somht unusul pttrn ith ponntil vrition o visosity, thrml ondutivity nd mdium prmbility. 4 CONCLSION obsrv tht th vloity initilly inrss rom no slip ondition nd it quirs pk vlu nd thn grdully drss nd ttins th r strm vloity. It is lso notid tht or som vlus o, th lo bhviour is somht unusul. This my b ttributd to th simultnous vritions in th luid proprtis. Th ts o th prmtr, visosity vrition nd th prmbility vrition on th vloity nd tmprtur proils r lrly visibl. IJSE
4 Intrntionl Journl o Sintii & Enginring srh Volum 3, Issu 7, July- 4 ISSN () (b) () (b) Figur Figur () (b ( (b EFEENCES Figur 3 []. K.Pohlhusn, Zur nhrungsisn Intgrtion dr Dirntilglihung dr lminrn nzshiht, ZAMM, 9, 5-68 []. J.H. Mrkin, Th t o buoyny ors on th boundry lyr lo ovr smi ininit vrtil lt plt in uniorm r strm, J. Fluid Mh., 35, 969, [3]. J.H Mrkin, Fr onvtion ith bloing nd sution, Int. J. Ht Mss Trnsr, 5, 97, [4]. P. Chng nd W.J. Minkoyz, Fr onvtion bout vrtil plt mbddd in porous mdium ith pplition to ht trnsr rom dik, J. Gophys s, 8, 977, Figur 4 [5]. P. ngnthnn nd. Visknt, Mid onvtiv boundry lyr lo long vrtil sur in porous mdium, Num Ht Trnsr, 7, 984, [6]. C. H. Chn, T. S. Chn nd C. K. Chn, Non-Dry mid onvtion long non isothrml vrtil surs in porous mdium, Int. J. Ht Mss Trnsr, 39, 996, [7]. H. Shlihting, Boundry lyr Thory, M Hill, NYork, 979 [8]. W.M. Kys nd M.E. Crrd, Convtiv Ht nd Mss Trnsr, M Hill, NYork, 98. [9]. I. G. Choi, Th t o vribl proprtis o ir on th IJSE
5 Intrntionl Journl o Sintii & Enginring srh Volum 3, Issu 7, July- 5 ISSN boundry lyr or moving ontinuous ylindr, Int. J. Ht Mss Trnsr, 5, 98, []. F. C. Li nd F. A. Kulki, Th t o vribl visosity on onvtiv ht trnsr long vrtil sur in sturtd porous mdium, Int.J. Ht Mss Trnsr, 33, 99, 8-3. []. I. Pop,. S.. Gorl nd M. shid, Th t o vribl visosity on lo nd ht trnsr to ontinuous moving lt plt, Int. J. Engg,S., 3, 99, -6. []. A.T. Esr nd G. Nth, nstdy non similr to dimnsionl nd isymmtri tr boundry lyr ith vribl visosity nd Prndtl numbr, Int. J. Engg,S., 3, 994, [3]. E. M. A. Elbshbshy, Fr onvtion lo ith vribl visosity nd thrml diusivity long vrtil plt in th prsns o th mgnti ild, Int.J.Eng.Si. 38,, 7-3. [4]. M. A. Sddk, Ets o vribl visosity on hydromgnti lo nd ht trnsr pst ontinuously moving porous boundry ith rdition, Int. Comm. Ht Mss Trnsr, 7,, [5]. M. A. Sddk nd A.A. Almushigh, Ets o rdition nd vribl visosity on MHD r onvtiv lo nd mss trnsr ovr strthing sht ith hmil rtion, Applitions nd Applid Mthmtis, 5,, [6]. E. M. A. Elbshbshy nd F.N. Ibrhim, Stdy r onvtion lo ith vribl visosity nd thrml diusivity long vrtil plt, J.Phys.6 (), [7]. P. K. Khound nd G.C. Hzrik, Th t o vribl visosity nd thrml ondutivity on liquid ilm on n unstdy strthing sur, Pro. O 46 th Annul Th. Sssion, Ass. S. So.,, [8]. J. W. Tirny, L.H.S. obl nd.m. Brid, dil porosity vrition in pkd bds, A.I.Ch.E. J., 958, [9]..F. Bnnti nd C.B. Brosilo, Void rtion distribution in bds o sphr, A.I.Ch.E. J., 96, []. B. C. Chndrskhr, A.. Hnumnthpp nd S. Chndrnn, Mid onvtion in th prsn o horizontl imprmbl surs in sturtd porous mdi ith vribl to prmbility, Pro. th Ntl Con Fluid Dynmis Fluid Por, IIT, Dlhi, 983, D.8-. []. B. C. Chndrskhr nd P. M. S. Nmboodiri, Inlun o vribl prmbility nd ombind r nd ord onvtion bout inlind surs in porous mdi,int. J. Ht Mss Trnsr, 8, 985, []. M. Mnsour nd N. El-Shr, Mid onvtion rdition in por l luids long non-isothrml dg in porous mdium ith vribl prmbility, Trnsport in porous Mdium Trnsport in porous Mdium, 57 (3), 4, [3]. M. G. ddy nd N.B. ddy, nstdy MHD onvtiv ht nd mss trnsr pst ssmi-ininit vrtil porous plt ith vribl visosity nd thrml ondutivity, Inrntionl Journl o Applid Mthmtis nd Computtion, (), 9, 4-7. [4]. I. A. Hssnin, Vribl prmbility ts on mid onvtion long vrtil dg mbddd in porous mdium ith vribl sur ht lu, Applid Mthmtis nd Computtion, 38 (), 4-59, 3. IJSE
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