Heat Transfer in Unsteady Axisymmetric Rotational Flow of Oldroyd Liquid

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1 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 SSN Ha Tasf Usady Axsymmc Roaoal Flow of Oldoyd Lqud A. Msha, G. S. Ray, S. Bswal Absac - Ths a dals wh h sudy of ha asf usady axsymmc oaoal flow of Oldoyd lqud. Cosuv quaos of h oblm hav b dvlod ad solvd by alyg Lalac asfom chqu. Exssos fo flud vlocy, mau, sk-fco ad a of ha asf a obad. Af comuzao wh h umcal valus of flud aams, h vlocy ad mau ofls a lod. Numcal valus of sk-fco ad a of ha asf a d abls. s obsvd ha lascy of h flud ducs h has of oscllao ad h cas Padl umb dcass h mau of flud. Kywods - Ha asf, oaoal flow, Oldoyd lqud.. NTRODUCTON T HE lau s l wh coous suls of vsgaos cad ou by a lag umb of sachs o ha asf by lama flow of Nwoa ad o-nwoa fluds. Obvously such suds hav gad cosdabl moac h flds of chology ad gg sccs. Schlchg has dscussd sval cass of xac soluos of ha asf vscous fluds. Khamu has aalysd h slow sady moo of a f vscous flud du o h oao of a ccula cyld. Ja 3 has sudd h oblm of ha asf by lama flow of lasco-vscous lquds hough aalll walls wh boh h walls a a ufom mau ad Msha 4 has xdd h oblm o h cas wh h walls a a laly vayg mau. Boh of hm hav obad a aoxma soluo of h dffal quao govg h vlocy fld by cosdg h lascy of h lqud o b small. Eal o hs wok of 965, Msha 5 has also dscussd a oblm of ha asf by lama lasco-vscous flow hough aalll oous walls. Aga, Msha 6 sudd h oblm of ha asf by lama flow of a lasco-vscous lqud a ccula cyld wh laly vayg wall mau. Msha ad Roy 7 hav aalysd h oblm of ha asf by h oaoy flow of a lasco-vscous lqud bw wo coaxal ccula cylds. Fuh, h wo woks 8 hav sudd h oblm of ha asf by lama moo of a lasco-vscous lqud bw wo coaxal ccula cylds du o logudal moo of h cyld. Raju ad Dvaaha 9 hav aalysd h oblm of ha asf o-nwoa fluds ad lama flow hough cocc aul wh o whou suco. Dash ad Bha 0 hav vsgad h oblm of lama flow dvlom ad ha asf of a vsco-lasc lqud a JSER 0 h:// covgg chal. b has cosdd h osaoay la ad ccula-symmc flow of a vscous flud whch foms slf wh as wll as ousd a oag ad fly log cyld ad obad a coml aalycal soluo fo ay bouday ad al codos usg h mhod of Lalac asfoms. Subsquly, Rsma has dvlod a soluo of h oblm of wodmsoal axsymmc oaoal flow of a vscous flud aulus boudd by wo cocc ccls cosdg a suabl modfd gfuco aoach. Rcly, Mukhj ad Bhaachaya 3 hav sudd h oaoal flow of vscous lqud du o h oao of a ccula lama o by h aco of shag sss alog h bouday. Mukhj ad Mukhj 4 hav xdd h oblm of Mukhj ad Bhaachaya o cov a wd class of lqud, amly lasco-vscous lqud, ad acula o obsv h ffcs of lasc lms h lqud. Th sady ad h as comos of h vlocy fld a obad xlcly. Dash ad Bswal 5 hav sudd h ha asf h commcm of Cou flow of Oldoyd lqud wh ha soucs. Dvlom of usady Cou flow ad ha asf Oldoyd lqud wh ha soucs ud dff hyscal suaos has also b aalysd by h sam auhos 6. Bswal ad Paak 7 hav vsgad h oblm of commcm of magohydodyamc usady Cou flow of Oldoyd flud hough wo oous walls wh h low wall suddly sas oscllag hamocally. Th objcv of h s sudy s o aalys h oblm of ha asf usady axsymmc oaoal flow of Oldoyd lqud.

2 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 SSN FORMULATON OF THE PROBLEM Th oaoal flow of a lasco-vscous lqud du o h m-dd oao of a ccula cyld s sudd h. W cosd axsymmc flow of a Oldoyd lqud wh a ccula cyld of adus a. Th m-dd xcao s causd by h scbd m-dd agula vlocy o h ccula cyld (Fg.). s assumd ha h lqud sods ccula moo abou h c ad hc h moo of a lqud acl s a fuco of adal co-oda ad m oly. Ou vsgao s bad o a modl of lasco-vscous lquds whch ma ssally h hologcal os of such lquds ud commo oag codos. Th cosuv quao of such comssbl lquds, oducd by Oldoyd8, s gv by P k D D P k P j k P jk j D = 0 [ k + k j jk ] (.) D Wh k = k + k, h sss so ad k = (v, k + v k, ), h a of sa so, P s a abay sooc ssu, k s h mc so of a fxd co-oda x, 0 s h co-ffc of vscosy, s h D sss-laxao m, (< ) s h a of sa laxao m ad oao dos h covcv m D dvav. Th quaos govg h flow of Oldoyd lqud a (+ ) u (+ ) v v v, (.) Ad v, (.3) Wh u (, ) s h vlocy como of h lqud h dco of casg ad v kmac co-ffc of vscosy. JSER 0 h:// 0 s h Th admssbl bouday codos cosodg o h scfcao of m-dd vlocy o h xal bouday ad a h c of h ccula cyld a gv by > 0 : v = v 0 a = a v s f a = 0 (.4a) wh v 0 s h sav vlocy ad s h mosd oscllao. Fo a lqud a s fo all < 0, may b assumd ha h al sa of sss s zo. Th al codos a

3 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 3 SSN = 0 : v (, ) = v, oducg h followg o-dmsoal aams v v,,, v a a 0 0, fo all, (.4b) =, a =, a a a, Equaos (.) ad (.3) bcom (dog ba) (+ v ) v (+ ) v v v, (.5), (.6) Subjc o h bouday ad al codos > 0 : v = a = ad v s f a = 0 (.7a) v = 0 : v 0 fo all, (.7b) 3. SOLUTON OF THE EQUATONS: Th quao (.5) ad h bouday codo (.7a) a subjcd o Lalac asfom chqu ud h codo (.7b) Thus, w oba Wh d u dv v a = v 0 JSER 0 h:// (3.) v s f a = 0 (3.) 0 v v d, R() > 0, (3.3)

4 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 4 SSN Th soluo of quao (3.) subjc o h bouday codos (3.) ca b w as Wh v = A () A () = (3.4) Th vs Lalac asfom of (3.4) s v = d wh s ga ha h al a of h sgulas of h gad. Evaluao of h abov gal qus a kowldg of h au ad locao of h sgulas of h gad. s cla ha h gad (3.5) (3.6) s a gal fuco of ad has oly sml ol a = ad ols a h zos (= ) of Q() (say) To fd h zos of h abov xsso, w u h xsso ad bcoms j (), wh s a h oos of (3.7) J () = 0, (3.8) Th zos of Q () s h P (=P ) = Pols ( =,, 3,.) a all sml ols wh 4 (+ ) <> 4 sc fo h fs qualy h ols a all gav al umbs ad fo h scod o h ols a comlx cojugas ad all h wh h ccl (3.9). may b makd ha JSER 0 h://

5 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 5 SSN JSER 0 h:// h s oblm suls a s of doubl ols wh 4 whch s foud abs h cas of Nwoa flud. To valua h gal (3.6), w fs cosd h cas wh all h sgulas a sml ols. Th sdu a h sml ol = s Th sdu a h sml = s. J J R Wh R = 4 Th xsso fo h vlocy ofl ca b w as v = J J. R (3.0) O, v = v s + v T, (3.) Wh v s ss h sady sa soluo whch s ssally mad u of h sdu cobuo fom h ol a = ad v T ss h as a asg ou of h sdus a =. w u = = 0 (3.0), h vlocy ofl hus obad s good agm wh h flow of Nwoa flud. Fo h cas of doubl ol occug a P = ( =,, ), h vlocy dsbuo s gv by v = v s + Nx x M, (3.) = v s + T T v v, (3.3)

6 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 6 SSN JSER 0 h:// Wh, M = f d d = N = f () = f () = ) (m)( (m) ad m = ) ( ) ( Th o-dmsoal sk-fco o h wall of h ccula cyld s gv by 0 0 a = ) ( = ) (K ) (K K ) ( ) (K P ) ( J K (3.4) Wh K = ad K = ) ( J R Ha asf :

7 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 7 SSN Th mau fld s govd by h quao T K C T, (3.5) Wh h bouday codo 0 : (, ) = 0 > 0 : (0, ) = (3.6) (a, ) = 0 oducg h followg o-dmsoal quas addo o hos ak vously, T T T T 0, w 0 0C P K Wh T s h mau of h flud a h dsk, T w s h mau of h dsk ad T 0 s h mau of h flud a = a W oba (dog h ba), (3.7) P Wh, P s h Padl umb. Th bouday codos fo mau ma as bfo. Takg Lalac asfom of quao (3.7), w hav d S 0, (3.8) P Wh h asfomd bouday codos fo a S > 0 : (0, S) =, (a, s) = 0, (3.9) S Th soluo of qu. (3.8) s (, s) = s, (3.0) S Now, akg h vs Lalac asfom of (3.0), w g (, ) = fc, (3.) Wh = Th a of ha asf gv by P JSER 0 h://

8 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 8 SSN N u = d d 0 E d d 0 (3.) Dog h ba, w hav N u = d d 0 (3.3) Wh h hl of (3.), w g N u =, (3.4) 4. RESULTS AND DSCUSSON: Flud Flow : Th vlocy xssos gv h quaos (3.0) ad (3.) coa boh sad ad as as. Elasc lms ad fluc boh h sady ad as a of h soluos. Takg lasc lms zo, h flow chaacscs smbls o hos of h m-dd oao of h ccula cyld a vscous flud. Vs ss h oscllaoy a whas vt (3.) ad T (3.3) s h as as. As, h as as d away xoally ad h moo fally aas h oscllaoy moo abou h sady sa. Fg. shows h vlocy ofls of Oldoyd lqud fo dff valus of lasc lms ad (<) ad also fo vscous lqud. s obsvd ha h ffc of lascy s o dmsh h has of oscllao of h vlocy ofl.., h a of chag of has of oscllao s quck lasc-vscous lqud ad h amlud chags a aabolc way as h cas of oday vscous flow. Sk-fco: Th ffc of lascy o h sk-fco a h bouday of h cyld has b sd h Fg. 3. s obsvd ha h lasc lm cass h sk-fco a ay o o h bouday of h ccula cyld. Evdly, fo Nwoa flud ( ad = 0), h valus of h shag sss of h oaoal flow a lss ha hos fo o-nwoa flud ( ad 0). Ha asf: JSER 0 h://

9 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 9 SSN Th mau fld of h o-nwoa flud ud cosdao has b show Fg. 4. s makd h cas Padl umb dcass h mau of h flud. Aga, mau s maxmum a h axal go of h cyld ad gadually falls owads h bouday of h ccula cyld. Ra of ha asf: Th a of ha asf s chaaczd by h Nussl umb Nu. Th valus of h Nussl umb a d Tabl. s obsvd ha h a of ha asf gadually falls wh h assag of m. Tabl : Valus of Nussl Numb Tm Nussl umb N u CONCLUSONS Followg coclusos a daw fom h suls obad. ) Th oaoal flow of Oldoyd lqud has boh oscllaoy ad as as. ) As, h as a d away xoally. ) Boh h sady ad as flow a affcd by h lasc lms ad. v) Elascy of h flud ducs h has of oscllao. v) Th lasc lm cass h sk-fco a ay o o h bouday of h ccula cyld. v) Th cas Padl umb dcass h mau of h flud. v) Th a of ha asf dcass as h m asss o. REFERENCES. Schlchg, H., Z.Agw. Mah. Mch., 3, 78 (95). Khamu, S.R., Bull. Cal. Mah. Soc.,47 (957) 3. Ja, M.K., Al. Sc. Rs., A, 95 (963) 4. Msha, S.P., Poc. d. Acad. Sc., 6A, 9 (965a) 5. Msha, S.P., d.j.pu ad Al. Phys.,, 4-44 (964) 6. Msha, S.P., Al. Sc. Rs., 4A, 8-90 (965b) 7. Msha, S.P., ad Roy, J.S. Poc. Na. s. Sc, da, 35A No.6, 88 (968) JSER 0 h://

10 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 0 SSN Msha, S.P., ad Roy, J.S. Poc. Na. s. Sc., da, 35A, (969) 9. Raju, K.K. ad Dvaaha, R. Rho. Aca, 0, (97) 0. Dash, G.C., ad Bha, S.C. Wa, 79, (98). b, H.E., ZAMM. 54, No. 5, 5-4 (974). Rsma, H., ZAMM, 55, Mukhj, S., ad Bhaachaya, P. d.j. Pu ad ald Mah., 3, No. 0, (98) 4. Mukhj, S., ad Kukhj, S. d. J. Pu al. Mah., 4, No., (983) 5. Dash, G.C., ad Bswal S. d. J. Pu al. Mah., 0, No.3, (989) 6. Dash, G.C., ad Bswal, S. AMSE J. 48, No., 4-64 (993) 7. Bswal, S., ad Paak, B.K. Aca Cca dca, 34P, No., 303 (008) 8. Oldoyd, J.G., Poc. Roy. Soc. Lodo, A00, 53 (950) JSER 0 h://

11 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 SSN Fgu : Skch of h hyscal oblm (al flow) JSER 0 h://

12 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 SSN Fgu : Vlocy ofl of lasc vscous ad vscous lqud wh =. JSER 0 h://

13 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 3 SSN Fgu 3 : Effc of lasc lms o shag sss wh =. JSER 0 h://

14 aoal Joual of Scfc & Egg Rsach Volum, ssu 9, Smb-0 4 SSN Cuv P Fgu 4 : Tas Tmau Pofls fo Dff valus of P, R c = 0.05, G = 5.0. Dam of Physcs, A.D. Mahavdyalaya, Bahmag, da. P.G. Dam of Physcs, G.M. Auoomous Collg, Sambalu, da 3. Rd. Pcal, 93, Jayadva Vha, Bhubaswa, da JSER 0 h://