CHAPTER-2 MIXED CONVECTION FLOW OF MICROPOLAR FLUID OVER A POROUS SHRINKING SHEET WITH THERMAL RADIATION 2.1 INTRODUCTION

Transcription

1 CHAPTER- MIXED CONVECTION FLOW OF MICROPOLAR FLUID OVER A POROUS SHRINING SHEET WITH THERMAL RADIATION. INTRODUCTION Bounary layr flow of ncomprssbl flu ovr a shrnkng sht has attract th ntrst of many rsarchrs u to ts applcatons n polymrc matrals procssng. Shrnkng sht flows ar mportant n th manufactur of crtan polymrs an hghprformanc matrals for arospac coatngs, as ocumnt by Bar an Bar. Furthr scussons of shrnkng sht flows an thr mportanc n th procssng of varous non-nwtonan matrals hav bn prov by Zhong t al. 4 for cramc suspnsons, Parng an Yang 55 for supr-plastc polymrc shts, Gupta an War 87 for thrmal shrnkng of polythn shts an Chrmsnoff 5 for vscolastc mmbrans us n ptrolum applcatons. Many thortcal an numrcal stus of such flows hav bn rport. Ishak t al. 5 nvstgat th stagnaton pont flow of mcropolar flu ovr a shrnkng sht. Th flow an hat transfr ovr a shrnkng sht mmrs n a mcropolar flu was consr by Yacob an Ishak 9. Das 59 nvstgat th slp ffcts on MHD mx convcton flow of a mcropolar flu towars a shrnkng vrtcal sht. Bhattacharyya an Layk 39 consr sucton/blowng ffcts on stagnaton-pont flow towars a shrnkng sht wth thrmal raaton. Oblqu stagnaton-pont flow towars a shrnkng sht wth thrmal raaton was xamn by Mahapatra t al. 3. Ahma t al. 3 stu th ffct of thrmal raaton on MHD axsymmtrc stagnaton pont flow an hat transfr of mcropolar flu ovr a shrnkng sht. Th obctv of th prsnt stuy s to analyz th stay bounary layr mx convcton flow an hat transfr of mcropolar flu wth sucton ovr a shrnkng sht n th prsnc of thrmal raaton. Ths flow rgm arss n th mchancal procss ngnrng systms nclung shrnk packagng, shrnk wrappng, shrnk flm an 7

2 tmpratur controllng of fnal proucts. By sutabl smlarty transformatons th govrnng partal ffrntal quatons ar transform nto a st of coupl nonlnar ornary ffrntal quatons. Ths quatons ar solv numrcally subct to physcally-ralstc bounary contons usng a varatonal formulaton of th fnt lmnt mtho. Th rsults ar prsnt graphcally for vlocty, mcrorotaton an tmpratur functons wth th varous valus of physcal paramtrs such as sucton, raaton an buoyancy paramtrs. Atonally, th skn frcton coffcnt, local coupl strss an th local Nusslt numbr hav also bn comput.. MATHEMATICAL MODEL Lt us consr a stay two-mnsonal lamnar flow of an ncomprssbl mcropolar flu of tmpratur T rvn by a porous shrnkng sht wth prscrb surfac hat flux. It s assum that th vlocty U (x) an th surfac hat flux q (x) of th sht vary proportonal to th stanc x from th fx pont on th sht,.. U( x) a x an q( x) bx, whr a an b ar constants. Th x axs s takn along th sht an th y axs s normal to t. A unform transpraton (sucton) vlocty V w s appl normal to th sht. Th mcropolar flu s assum to b a gray, mttng an absorbng, but non-scattrng mum. Th mcropolar flu also has constant proprts xcpt for th nsty changs whch prouc a thrmal buoyancy forc. Th flow confguraton an th coornat systm ar shown n th Fg... x,u g V w U(x) q(x) T T O y,v Fgur.: Physcal mol an coornat systm 8

3 Th govrnng bounary layr quatons for th flow problm ar as follows Mass Consrvaton (Contnuty) u v, x y Lnar Momntum Consrvaton (.) u u u N u v S S g ( T T ), (.) x y y y Angular Momntum (Mcro-rotaton) Consrvaton N N N u u v S N, (.3) x y y y Enrgy Consrvaton T T u v x y c p T y c p qr, (.4) y wth th bounary contons u T q u U ( x), v Vw, N. 5, at y, y y (.5a) u, N, T T as y. (.5b) At th sht, th bounary conton for angular momntum (mcrorotaton) mpls that th mcrorotaton s qual to th flu vortcty. In accoranc wth ths, n th nghborhoo of a rg bounary, th ffct of mcrostructur s nglgbl snc th suspn partcls cannot gt closr to th bounary than thr raus. Hnc n th nghborhoo of th bounary, th only rotaton s u to flu shar an thrfor, th gyraton vctor must b qual to th flu vortcty. Th spn grant vscosty s gvn by S /, whr / a s th rfrnc lngth. Ths assumpton s nvok to allow th fl of quatons to prct th corrct bhavor n th lmtng cas whn th mcrostructur ffcts bcom nglgbl an th total spn N rucs to th angular vlocty as suggst by Ahma 4. Th raatv hat flux q r s fn by th Rosslan ffuson approxmaton, whch s val for bounary layr flows 3, as gvn blow q r 4 T 3k y 4, (.6) 9

4 whr an k ar th Stfan Boltzmann constant an th Rosslan man absorpton coffcnt, rspctvly. It s assum that th tmpratur ffrncs wthn th flow ar suffcntly small such that xpanng 4 T may b xprss as a lnar functon of tmpratur by 4 T n a Taylor srs about T an nglctng hghr-orr trms. Thus T 4T T 3T. (.7) By usng (.6) an (.7), nrgy qn. (.4) rucs to T u x T v y c p T y 3 6 T 3 c k p T. y (.8) Th vlocty componnts u an v can b xprss n trms of th stram functon as follows u y, v x. So th contnuty qn. (.) s satsf automatcally. Usng th smlarty transformatons 3 a a y, a x f, N x g, (.9) T T a, (.) q th govrnng partal ffrntal qns. (.), (.3) an (.8) ruc to th followng systm of coupl nonlnar ornary ffrntal quatons f g, ( ) f ff (.) ( ) g fg f g 4 Pr f, 3R g f, an th bounary contons transform to (.) (.3) f, f, g.5 f, at, (.4a) f, g, as. (.4b) Hr prm nots th ffrntaton wth rspct to an S th mcropolar couplng constant paramtr, Gr x Gr R R 5 / x buoyancy paramtr, 5 x g b x local Grashof numbr, x a x local Rynols numbr,

5 3 4 R k T raaton paramtr, Pr c Prantl numbr an p V w a transpraton paramtr. In th prsnt stuy only sucton s consr at th sht for whch >. Th ngnrng paramtrs of rlvanc to matrals procssng ar th skn frcton coffcnt, local coupl strss an th local Nusslt numbr, whch ar fn rspctvly as C f w, U M x M w, Nu U x qw x ( T T ), (.5) w whr th wall shar strss w, plat coupl strss M w an th hat flux q ar gvn by 4 u N T 4 T w S SN, M w, q. y y y 3 w y y k y y y (.6) Usng th smlarty transformatons (.), w obtan C f R / f,.5 g x M x Nu 4 3 R x an. R / x (.7).3 FINITE ELEMENT SOLUTION Th st of ffrntal qns. (.)-(.3) wth th bounary contons (.4) has bn solv numrcally by usng fnt lmnt mtho 7. In orr to apply fnt lmnt mtho frst w assum f h. (.8) Wth ths substtuton, th qns. (.)-(.3) bcom ( ) h fh h g, (.9) ( ) g fg hg 4 Pr f, 3R g h, an th corrsponng bounary contons ruc to (.) (.) f, h, g.5 h, at, (.a)

6 h, g, as. (.b) It has bn obsrv that for larg valu of ( 8), thr s no apprcabl chang n th rsults. Thrfor, for th computaton purpos nfnty has bn fx at 8. Th whol oman s v nto a st of n ln lmnts of wth.3. VARIATIONAL FORMULATION h 8 n. Th varatonal form assocat wth qns. (.8)-(.) ovr a two-no lmnt, s gvn by w w f h, (.3) ( ) h fh h g, w ( ) g (.4) g h, 3 fg hg (.5) 4 w4 Pr, 3 f R (.6) whr w, w, w3 an w 4 ar wght functons whch may b vw as th varaton n f, h, g an rspctvly..3. FINITE ELEMENT FORMULATION Th fnt lmnt mol can b obtan from qns. (.3)-(.6) by substtutng fnt lmnt approxmatons of th form f f, h h, g g,, (.7) wth w w w3 w4 (, ), an an ar th shap functons for a typcal lmnt, whch ar takn as,,. (.8) Th fnt lmnt mol of th quatons thus form can b xprss n th form

7 3, } { } { } { } { } { } { } { } { b b b b g h f (.9) whr m n an m b ),,3, 4, ( n m ar th matrcs of orr an rspctvly. All ths matrcs ar fn as follows,, 4 3,,, ) ( h f, 3, 4, 3, 3, 33 h f, 34, , Pr f R (.3) an, b, ) ( h b, ) ( 3 g b R b (.3) whr f f an h h ar assum to b known. Aftr assmbly of th lmnt quatons, a systm of algbrac nonlnar quatons s obtan, whch s solv tratvly. Th functon f an h ar assum to b known an ar us for lnarzng th systm. Th vlocty, mcrorotaton an tmpratur functons ar st qual to. for th frst traton an global quatons ar solv for th noal valus of ths functons. Ths procss s rpat untl th sr accuracy of four sgnfcant fgurs s obtan. W hav carr out calculatons for n, 4,..., 6 an th fnal rsults ar rport for n 6 only whch ar shown n tabl. (a).

8 For th cas of vscous flu ( ) an n th absnc of buoyancy forc ( ), th xact soluton for f ( ) subct to th bounary contons (.) s gvn as follows f xp z z, (.3) whr 4, z an s ntcal to that obtan by Fang an Zhang 77 wth M an by Yacob an Ishak 9 wth. Th comparson of th flow vlocty f obtan by fnt lmnt mtho an by analytcal mtho from (.3) s tabulat n tabl. (b). It s monstrat from th tabl that th numrcal rsults so obtan ar n full agrmnt wth th analytcal rsults an thus confrm th valty of th prsnt FEM computatonal solutons. Tabl. (a): Convrgnc of rsults wth th varaton of numbr of lmnts (,Pr, 3, R, 5) Numbr of lmnts h (.) g (.) (.) Tabl. (b): Comparson of th flow vlocty (,, 3,Pr, R ) f obtan by analytcal mtho an FEM f Analytcal Mtho FEM , 6, 7, 8.. 4

9 .4 RESULTS AND DISCUSSION Th numrcal computatons ar prform for varous valus of sucton paramtr, th raaton paramtr R an th buoyancy paramtr. Th othr paramtrs such as Prantl numbr Pr an couplng constant paramtr ar kpt fx at.. Th rsults so obtan, ar prsnt n Fgs..-. an th corrsponng valus ar shown n tabls.-.. Th skn frcton coffcnt, local coupl strss an local Nusslt numbr hav also bn comput for ths paramtrs an ar tabulat n tabls.-.3. Hghr valus of sucton paramtr ar takn so as to sustan stay flow nar th sht by confnng th gnrat vortcty ns th bounary layr. Som ntrstng obsrvatons can b ma for vlocty from Fgs.. (a)-. (f). For consr by Bhattacharyya an Layk 39 an Muhamn t al. 45, ncras n ncrass th vlocty. Our Fg.. (a) compltly agrs wth thm. But as ncrass, th ffcts of ncras n ar ffrnt n th rgons clos to th bounary an away from t as pct n Fgs.. (b) through. (f). Nar th bounary, vlocty ncrass whl away from t th ffct s ust oppost. For larg, th ffct of s nglgbl nar th bounary. Th sucton (mass rmoval from th bounary layr) ffct n fact nucs flow rvrsal vry clos to th sht, as ncat by ngatv valus of th vlocty n ths rgon. Fg..3 pcts th varaton of mcrorotaton.. mcro-lmnt angular vlocty, wth th sucton paramtr. It s obsrv that nar th sht surfac th mcrorotaton ncrass wth th ncras n. Th ngatv valus of mcrorotaton show th rvrs rotaton only nar th bounary. Thus th rvrs rotaton can b ruc by ncrasng th sucton. Furthr from th bounary (sht surfac) th rvrs trn s obsrv.. ncrasng sucton acts to cras mcrorotaton. Howvr postv valus for mcrorotaton ar sustan all th way to th fr stram ncatng that thr s no rvrsal n spn furthr from th wall. Ths trn has also bn ntf by Bhattacharyya an Layk 39. Th ffct of sucton on th tmpratur s pct n Fg..4. Tmpratur s obsrv to cras as sucton ncrass. Mcropolar flu partcls nar th hat surfac absorb hat from th sht va thrmal conucton, u to whch th tmpratur of th flu s largr nar th sht. As sucton s appl, ths flu partcls ar rmov from th sht an as a consqunc th tmpratur of th flu falls. Th sucton paramtr provs an ffctv mans of controllng th flow an hat transfr 5

10 charactrstcs. Ths has also bn ocumnt by Bhargava t al. 37 an furthrmor by Yao t al.. Confnc n th prsnt FEM computatons s thrfor hgh. Fg..5 llustrats that th vlocty of th mcropolar flu s ruc wth 3 ncrasng raaton paramtr R. R k T rprsnts th rlatv contrbuton of 4 thrmal conucton hat transfr to thrmal raaton hat transfr. It s somtms rfrr to as th Boltzmann-Rosslan numbr n th ltratur 3. Larg R valus wll thrfor mply wakr thrmal raaton contrbuton an vc vrsa for low R valus. Whn R s unty both mos of hat transfr ar xpct to hav th sam contrbuton. An ncras n R wll mply gratr thrmal conucton contrbuton an causs flow rvrsal nar th sht surfac. Flow vlocty s thrfor maxmz wth lowst R valus, for whch thrmal raaton has a gratr ffct. Th maxmum vlocty comput corrspons to R.5. In lght of ths, hgh-tmpratur matrals procssng opratons ar foun to bnft from thrmal raaton whch tns to oppos flow rvrsal an sustans a mor stabl flow rgm n shrnkng shts. From Fg..6 t s obsrv that th mcrorotaton ncrass as R ncrass n th vcnty of th sht. Th strong mcrorotaton rvrsal nar th sht surfac s progrssvly ruc wth ncrasng R.. wth wakr thrmal raaton contrbuton. Aftr a small stanc from th sht th ffct s rvrs an thraftr mcro-rotaton s foun to b mor postvly affct by th strongr thrmal raaton cas ( R.5). Howvr mcrorotaton magntus ar obsrv to b much lowr as w progrss from th sht surfac towars th fr stram. Fg..7 llustrats th nflunc of th raaton paramtr on th tmpratur strbuton. Tmpratur s vry strongly ncras wth a cras n R. As laborat arlr, thrmal raaton flux has a progrssvly gratr ffct as th valu of R s ruc, wth a smultanous cras n thrmal conucton contrbuton. Wth lowr R valus, thrfor th thrmal raatv flux supplmnts nrgy n th bounary layr an ths lvats tmpraturs. Th concomtant acclraton n th flow (Fg..5) mpls that raaton has a vry promnnt an bnfcal ffct on th ynamcs of th shrnkng sht an th tmpratur n th vcnty of th sht, whch as n manufacturng control. Fg..8 shows that th vlocty ncrass wth th ncras n buoyancy paramtr. Ths paramtr, 5 / Gr x R x an largr valus corrspon to gratr thrmal buoyancy forc. Ths acts to a momntum vlopmnt n th bounary layr rgm an 6

11 ffctvly acclrats th flow. Nar th sht th flow s strongly rvrs. Th maxmum vlocty comput s assocat thrfor wth th strongst buoyancy paramtr cas... A vlocty pak s comput nar th sht for 3, 5,7, but vanshs for th wakst buoyancy cas of (buoyancy an vscous forc qual). Fg..9 rprsnts th mcrorotaton strbuton wth th varaton of buoyancy paramtr. As ncrass th mcrorotaton crass markly nar th bounary. Furthr from th surfac, th oppost bhavour s obsrv an mcro-rotaton s foun to b nhanc wth ncrasng buoyancy paramtr. Howvr wth furthr progrsson nto th bounary layr thr s vanshng n mcro-rotaton. Fg.. shows that tmpratur crass wth ncras n buoyancy paramtr. Pak tmpratur always arss at th sht surfac. Th profls cay smoothly from th wall to th fr stram. In th bounary layr, th nflunc of buoyancy s foun to b strongst at ntrmat stancs from th wall. Th prsnc of buoyancy acclrats th flow.. nhancs vscous ffuson but nhbts thrmal ffuson n th bounary layr.. hat s ffus lss ffctvly. Ths manfsts n supprsson n tmpratur valus n th mcropolar flu. Ths bhavour has also bn obsrv by Hayat t al. 9 an furthrmor by Ishak t al. both stus concrnng mcropolar flows. In fact th prsson n tmpraturs unr buoyancy forcs s also obsrv xprmntally n many othr non-nwtonan fr convcton flows (plastcs) as laborat by Bar an Bar. W furthr not that n Fgs..8-., R ncatng that thrmal conucton an thrmal raaton hav an qual contrbuton an also 3 corrsponng to strong sucton at th wall. Tabl. gvs skn frcton for Pr, an for ffrnt valus of, R an. It s clar that th skn frcton crass numrcally wth ncras n both sucton paramtr an raaton paramtr R, whl t ncrass wth an ncras n buoyancy paramtr. Th postv valus of th skn frcton ncat that th flu xrt a rag forc on th sht. Thus th skn frcton can b ffctvly ruc by ntroucng th sucton an raaton. From tabl. coupl strss.. mcrorotaton grant, s foun to cras wth ncras n raaton whras t ncrass wth a rs n th sucton an buoyancy paramtrs. Thrfor a fast rat of mcro-lmnt rotaton can b achv by th sucton an buoyancy ffcts. Also th rotaton of th mcropartcls can b ruc by ncrasng th raaton paramtr (crasng thrmal raatv flux contrbuton). It s clar from tabl.3 that th hat transfr rat ncrass wth th ncras n sucton, 7

12 buoyancy an raaton paramtrs. Thus by applyng sucton, raaton an buoyancy paramtrs th hatng of th sht can b controll n actual manufacturng opratons..5 CONCLUSIONS Th prsnt stuy has arss thortcally an numrcally th stay flow an hat transfr of an ncomprssbl mcropolar flu ovr a porous shrnkng sht n th prsnc of thrmal raaton. Usng a smlarty transformaton, th govrnng partal ffrntal quatons hav bn normalz nto a st of nonlnar, coupl, mult-gr ornary ffrntal quatons. A robust, valat varatonal fnt lmnt mtho (FEM) has bn mploy to solv th rsultng wll-pos two-pont bounary valu problm. Numrcal rsults obtan hav clarly monstrat that th skn frcton can b ruc ffctvly by mposng sucton an crasng raaton contrbuton. It has also bn obsrv that a fast rat of coolng can b achv wth ucous slcton of sucton, raaton an buoyancy paramtrs. Th rsults of ths nvstgaton play an mportant rol n thrmal control of th synthss of packagng unts such as shrnk wrappng, bunl wrappng, shrnk packagng an shrnk flm. 8

13 Tabl. (a): Vlocty strbuton for ffrnt ( R, ) f ' -.4 λ=.5, 3, 3.5, Fgur. (a): Vlocty strbuton for ffrnt ( R, ) 9

14 Tabl. (b): Vlocty strbuton for ffrnt ( R,.5) f ' λ=.5, 3, 3.5, Fgur. (b): Vlocty strbuton for ffrnt ( R,.5) 3

15 Tabl. (c): Vlocty strbuton for ffrnt ( R, ) f ' λ=.5, 3, 3.5, Fgur. (c): Vlocty strbuton for ffrnt ( R, ) 3

16 Tabl. (): Vlocty strbuton for ffrnt ( R, ) f ' λ=.5, 3, 3.5, Fgur. (): Vlocty strbuton for ffrnt ( R, ) 3

17 Tabl. (): Vlocty strbuton for ffrnt ( R, 3) λ=.5, 3, 3.5, 4 -. f ' Fgur. (): Vlocty strbuton for ffrnt ( R, 3) 33

18 Tabl. (f): Vlocty strbuton for ffrnt ( R, 5) f ' -. λ=,.5, 3, 3.5, Fgur. (f): Vlocty strbuton for ffrnt ( R, 5) 34

19 Tabl.3: Mcrorotaton strbuton for ffrnt ( R, 5) g λ =,.5, 3, 3.5, 4 Fgur.3: Mcrorotaton strbuton for ffrnt ( R, 5) 35

20 Tabl.4: Tmpratur strbuton for ffrnt ( R, 5) θ.6.4. λ =,.5, 3, 3.5, Fgur.4: Tmpratur strbuton for ffrnt ( R, 5) 36

21 Tabl.5: Vlocty strbuton for ffrnt R ( 3, 5) R f '. -. R =.5,.7,,.5, Fgur.5: Vlocty strbuton for ffrnt R ( 3, 5) 37

22 Tabl.6: Mcrorotaton strbuton for ffrnt R ( 3, 5) R g R =.5,.7,,.5, Fgur.6: Mcrorotaton strbuton for ffrnt R ( 3, 5) 38

23 Tabl.7: Tmpratur strbuton for ffrnt R ( 3, 5) R θ.6.4 R =.5,.7,,.5, Fgur.7: Tmpratur strbuton for ffrnt R ( 3, 5) 39

24 Tabl.8: Vlocty strbuton for ffrnt ( 3, R ) f ' σ =, 3, 5, 7, Fgur.8: Vlocty strbuton for ffrnt ( 3, R ) 4

25 Tabl.9: Mcrorotaton strbuton for ffrnt ( 3, R ) g σ =, 3, 5, 7, Fgur.9: Mcrorotaton strbuton for ffrnt ( 3, R ) 4

26 Tabl.: Tmpratur strbuton for ffrnt ( 3, R ) θ σ =, 3, 5, 7, Fgur.: Tmpratur strbuton for ffrnt ( 3, R ) 4

27 Tabl.: Th skn frcton coffcnt f () for ffrnt valus of, R an Pr, R =, =5 = 3, =5 3, R = f () R f () f () Tabl.: Th local coupl strss g for ffrnt valus of, R an Pr, R =, =5 = 3, =5 3, R = g () R g () g () Tabl.3: Th local Nusslt numbr / () for ffrnt valus of, R an Pr, R =, =5 = 3, =5 3, R = / () R / () / ()