Jurnal Teknologi INFLUENCE OF HEAT TRANSFER ON THE MHD STAGNATION POINT FLOW OF A POWER LAW FLUID WITH CONVECTIVE BOUNDARY CONDITION.

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1 Jural Tkologi INFLUENCE OF HEAT TRANSFER ON THE MHD STAGNATION POINT FLOW OF A POWER LAW FLUID WITH CONVECTIVE BOUNDARY CONDITION Shah Jaha * Hazah Sakidi Fudatal Alid Sis Dartt Uivrsiti Tkologi PETRONAS 610 Bar Sri Iskar Prak Malaysia Full Par Artil history Rivd 15 May 015 Rivd i rvisd or 1 July 015 Atd 11 August 015 *Corrsodig author jahashah669@gail.o Grahial abstrat Abstrat I this artil w xaid th iat o hat trasr o th agtohydrodyai (MHD) stagatio oit low o a o-nwtoia owr- law luid with ovtiv boudary oditio. By usig suitabl siilarity trasoratios ould oliar artial dirtial quatios ar trasord to ordiary dirtial quatios. Th solvd th rsultig quatios with Hootoy aalysis thod. Itrstig low aratrs suh as MHD M stagatio aratr ovtiv aratr ar disussd grahially. Covrg is hkd at 0th ordr o aroxiatio. Nurial valus o hysial itrstd aratr suh as loal Nusslt ubr ar also tabulatd. Kywords: Covtiv boudary oditios owr-law luid hat trasr 015 Prbit UTM Prss. All rights rsrvd 1.0 INTRODUCTION A stagatio oit low i trs o luid hais is a oit i a low ild whr th loal vloity o luid is zro. Ets o volutri hat gratio/absortio o ixd ovtio stagatio oit low o a isothral vrtial lat i a orous diu was ivstigatd by Sigh t al. [1]. Hiz [] Hoa [] studid th two disioal axisytri thr disioal stagatio oit lows rstivly. Dirt studis o stagatio oit lows a b oud i th works [4-6]. Th robls o stagatio oit lows a b studid i Nwtoia as wll as o-nwtoia luids. May rsarhrs ousd to th study o o-nwtoia luids suh as owr-law luid baus its quatio o otio hav sial rlva i idustris suh as olt lasti olyr lts xtrusio ross ay othrs. Adrsso Daat [7] irst disussd th low o a o-nwtoia luid obyig owr law odl by xtdig th Nwtoia odl as osidrd by Cra [8]. Agai Adrso t al. [9] rstd th MHD low o a owr law luid ovr a strthig sht. Nurial sris solutio o agtohydrodyais stagatio oit low o a owr law luid towards a strthig sura ar obtaid by Mahaatra t al. [10 11]. Thy oard both th rsults oud to b i good agrts. Agai Mahaatra t al. [1] studid th abov tiod robls ovr a orous lat or sutio or blowig as. Thy lottd th stralis obsrvd that th vloity at a oit irass with a iras i th agti aratr M. Th ovtiv boudary oditio has b usd by ay rsarhrs to rvisit th robls studid with isothral/isolux boudary oditio suh as R. C. Batallr [1] disussd th Siilarity solutios or low hat trasr o a quist luid ovr a oliarly strthig sura. Th A. Ishak [14] xlord th siilarity solutios or low hat trasr ovr a rabl sura with ovtiv boudary oditios. A ovtiv boudary oditio or MHD 77:0 (015) ISSN 180 7

2 54 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 ixd ovtio ro a vrtial lat bddd i a orous diu xaid by O. D. Makid A. Aziz [15]. S. Yao T. Fag Y. Zhog [16] studid th hat trasr o a gralizd strthig/shrikig wall robl with ovtiv boudary oditio. Th ai o this ar is th study o hat trasr o stagatiooit low o MHD owr-law luid with ovtiv boudary oditio. Th stady two disioal stagatio oit low o a abit luid whih is subjtd udr th ostat agti ild is osidrd hr: By takig th alid agti ild uior th ovrgt sris solutios ar obtaid by th HAM. Itrstig low aratrs ar disussd grahially Nurial valus o loal Nusslt ubr is also alulatd i tabulatd or. HAM is a iit thod roosd by Liao [18] This thod has b sussully alid to obtaid solutio o o-liar robls [19-0]. As show i Figur 8 this ar highlightd th robl i this ild orulat th robl roosd th solutio aalysis disussig th rsult..0 PROBLEM FORMULATION W aalyz th ts o hat trasr with a ovtiv boudary oditio istad o ooly usd oditios o ostat sura tratur or ostat hat lux. Hr w osidr th stady two disioal stagatio-oit low hat trasr o a ltrially odutig owr-law luid (old luid at tratur towards a lat strthig sht ( T ) oiidig with th la y 0 th low big oid to th rgio o strgth B 0 y 0. A uior agti ild is iosd oral to th sht (alog th y-axis) wh idud agti ild is gltd udr sall agti Ryolds ubr assutio. It is assud that th vloity o th xtral low is ( x) ax th vloity o th strthd sht is U U ( x) x whr a ar ositiv ostats is th oordiat asurd alog th strthig sht. It is also assud that th botto sura o th sht is hatd by ovtio ro a hot luid at tratur T whih rovids a hat trasr oiit h. I th rst robl w hav u 0 wh a 1 u 0 y wh a 1. Udr ths assutios th otu rgy quatios or th boudary layr low o owr-law luid ar u u u v U x y a wh 1 u v 0 x y U x K u y y B0 ( u U ) y x (1) () u u u v U x y wh a 1 U x K u y y T T k T u v x y y whr K ( 0) B0 ( u U ) () (4) dot th osisty oiit low bhavior idis rstivly ar th ltrial odutivity th k thral odutivity th sii hat rstivly. Th orrsodig boudary oditios ar T u U( x) x v 0 k h T T at y 0 y u U ( x) ax T T as y Itroduig th ollowig siilarity trasoratios 1/( 1) K / /1 x 1 1/ (1 ) /(1 ) T T y x. K / T T I th abov quatios dots th siilarity variabls ( x y) th stra utio th disiolss siilarity utios rlatd to th vloity tratur rstivly. I viw o abov trasoratios w hav 1 1 sg( 1) M M 0 (5) (6) (7) 1 0 (8) Pr (0) 0 (0) 1 (0) 1(0) ( ) ( ) 0 as. Hr th othr aratrs ar did as ollows a h / K 1 K 1 B0 M Pr 1/ x 1. ( ) x K 1 K (9) (10) I th abov quatio M is th agti ild aratr Pr is th Prtl ubr is th stagatio aratr is th ovtiv hat trasr. Th hysial quatity o itrst is th loal Nusslt ubr Nu is did by

3 55 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 xqw Nu (11) k( T T ) q k T dotig th sura hat lux. whr w y 0 Usig variabls whr (7) w gt 1/ R x Nu x (0). (1) R x ( x) x K / is th loal Ryolds ubr..0 THE HAM SOLUTIONS FOR INTEGER POWER- LAW INDEX 1 W aly th Hootoy Aalysis Mthod to solv th ould syst o quatios (7) to (9). Fro th govrig quatios w hoos d d L 1 (1) d d d L (14) d as auxiliary liar orators with th ollowig rortis C C C 1) 0 L 1 x( (15) C ) C x( ) 0 L 4 x( 5 (16) whr C i ( i 1 5) ar arbitrary ostats..1 Th zroth-ordr Doratio Probls W ostrut th zroth ordr doratio robl as ( 1 L [ ˆ( ; 0 ( )] N ˆ( ; (17) ( 1 L [ ˆ( ; 0( )] N ˆ( ; ˆ( ; (18) subjt to th boudary oditios ˆ( ˆ( 0 1 ˆ ( 1 ˆ ˆ( ˆ( 0 (19) whr 1 x 1 sg 1 0 ( ) (0) 0( ) (1) 1 as our iitial aroxiatio o is a bddig aratr. Hr. For [01] ar th auxiliary ozro aratrs or 1 th oliar orators ar ˆ( ; ˆ( ; ) ˆ( ; ) ˆ( ; ) ˆ ; N () ˆ( ; M M ˆ( ; ˆ( ; ) ˆ( ; ) Pr ˆ ;. N () Wh th it is asy to hk that ˆ( ;0) 0( ) ˆ( ;1) ( ) (4) 0 1 ˆ( ;0) 0( ) ˆ( ;1) ( ) (5) So as a bddig aratr irass ro 0 to 1 ˆ ( ; vary ro iitial aroxiatios 0 ( ) 0 ( ) ˆ ( ; to th solutio () () o th origial quatios (7) to (8). Usig Taylor's thor Eqs. (4) (5) w x ˆ ( ; ˆ ( ; i th owr sris o a bddig aratr as ollows. ˆ( ; 0 ( ) ( ) (6) 1 ˆ( ; 0 ( ) ( ) (7) 1 whr 1 ( )! ( ; 1 ( )! ( ; 0 0. (8) Obsrv that th zroth-ordr doratio Eqs. (17) (18) otai o-zro auxiliary aratrs. Assu that ths aratrs ar hos so that th sris (6) (7) ar ovrgt at. H w hav du to (8) that 1 ( ; 0 ( ) ( ) (9) 1 ( ; 0 ( ) ( ) (0) 1. Th th-ordr Doratio Probl Dirtiatig th zroth-ordr doratio robls i qs. (17) (18) -tis with rst to th dividig by.. Fially lttig 0 w obtai th ollowig th-ordr doratio robls or 1 th ollowig robl

4 56 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 L 1 R R L (1) 1 () whr oliar orators or R R ( 0) 0 (0) 0 ( ) 0 (0) ( ) 0 () 1 ar as ollows k 1 k k 1 k 1 k0 k0 M M(1 ) sg 1 (4) 1 1 k 1 Mk 1 k0 1 M(1 ) 4 1 k0 k 1 k k 1 k (5) 1 R 1 Pr k1 k (6) k0 is did by Eq. (5) 0 1 (7) 1. Th gral solutios o Eqs. (1)- () ar ) ( ) C C x( ) C x( ) (8) ( 1 ) ( ) C x( ) C x( ) (9) ( 4 5 i whih ( ) ( ) ar th artiular solutios o Eqs. (1)- (). Not that Eqs. (1)- () a b solvd by as o ay syboli outatioal sotwar lik Mal Mathatia t. o atr th othr i th ordr CONVERGENCE OF THE SERIES SOLUTIONS Whil usig Hootoy Aalysis thod th ovrg dds uo th auxiliary aratr as it hls i adjustig otrollig th radius o ovrg o th sris solutios. Cobi urvs o vrsus ar skthd i (0) Figur 1. Whr (0) is auxiliary aratr or is th auxiliary aratr or (0). (0) By kig th valus o othr aratrs ixd w a obtai th ovrg rgio o Th rasoabl. valus ar 1.5 ħ ħ 0.6. Tabl. 1 rrsts th ovrgt valus o sris solutio. This shows th validity o our sris solutio. Figur 1 Cobid urv or (0) (0). at 0th ordr o aroxiatio Tabl 1 Covrg o th sris solutio at dirt ordr o aroxiatios. Ordr o aroxiatios (0) (0)

5 57 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 Tabl Th urial valus o th loal Nusslt ubr or various valus o aratrs [ ] = 1 γ = 0.1 M = 1 λ = 1 = γ = 1 M = λ = R 1/ Nu R 1/ Nu RESULTS AND DISCUSSION Th ts o all othr aratrs hootoy- Pad aroxiatios rsults hav b disussd i th ar by Mahaatra t al..hr w s th [10] ilu o ollowig low aratrs Pr Prtl ubr ovtiv hat trasr stagatio aratr o th tratur distributio. Figurs ar ad to s th t o aratr or Nwtoia o-nowtoia luid. Hr or w rovr th as o ostat sura tratur but or th ovtiv hat trasr th 0 tratur irass as w iras th valus o lutuats raidly as oard to th Nwtoia luid. Th ilu o Prtl ubr tratur or 1 is show i Figurs 4 5. (). Also urvs or o-nwtoia owr-law luid Pr o Fro ths igurs w obsrv that by irasig th valus o or Nwtoia luid h th boudary layr thikss drass; howvr or o-nwtoia luid it irass. Th stagatio aratr ts o is skthd i Figur 6. It is otd that th boudary layr thikss tratur roils drass as w iras th valus o Figur 7 shows that th vloity rahs to its ak valus i th as o o-nwtoia luids. Tabl is ad to s th urial valus o hysial itrstd loal Nusslt ubr it is oud to b i good agrt. Pr th tratur distributio drass. Figur Tratur roils or dirt valus o kig th valus o othr aratrs ixd at = Figur 4 Tratur roils or dirt valus o Pr by kig th valus o othr aratrs ixd by Figur Tratur roils or dirt valus o by kig th valus o othr aratrs ixd Figur 5 Tratur roils or dirt valus o Pr by kig th valus o othr aratrs ixd at =

6 58 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 Th ts o Prtl ubr Pr o Nwtoia owr-law luids is s to b i oosit bhavior. Fro th rsults w obsrvd a dras i th tratur as wll as th thral boudary layr or irasig valus o. Figur 6 Tratur roils or dirt valus o kig th valus o othr aratrs ixd by Akowldgt Th irst author is gratul to th Uivrsity Thology PETRONAS or rovidig th ud. Rrs Figur 7 Tratur roils or dirt valus o kig th valus o othr aratrs ixd at = Figur 8 Blok diagra o th rosss o th syst 6.0 CONCLUSION I this ar w hav disussd hat trasr aalysis o stagatio-oit low o MHD owr-law luid with a ovtiv boudary oditios i aalytial way usig Hootoy aalysis thod. This thod is iit to solv th boudary valu robl i aalytial way. Fro th urial grahial rsults w olud th ollowig ai obsrvatios: It is otd that th tratur irass or irasig valus o or both Nwtoia owr-law luids. So it irass boudary layr thikss. by [1] Sigh G. Shara P. R. Chakha A. J Et o Volutri Hat Gratio/Absortio O Mixd Covtio Stagatio Poit Flow O A Isothral Vrtial Plat I Porous Mdia. It. J. Idustrial Mathatis. (): [] Hiz K Di Grzshiht A Ei I D Glihorig Fussigkitsstor Eigtauht Gard Kriszylidr. Diglrs Poly. J. 6(1911): [] Hoa F Dr Eilub grobr Zahigkit bi dr Stroug u d Zylidr ud u di Kugl Z. Agw. Math. Mh. 16: [4] Nazar R. Ai N. Fili D. Po I Stagatio Poit Flow O A Miroolar Fluid Towards A Strthig Sht. It. J. No Liar Mh. 9(004): [5] Lok Y. Y. Phag P. Ai N. Po I. 00. Ustady Boudary Layr Flow O A Miroolar Fluid Nar Th Forward Stagatio Poit O A Pla Sura. It. J. Eg. Si. 41: [6] Nad S. Abbasby S. Hussai M Sris Solutios O Boudary Layr Flow O A Miroolar Fluid Nar Th Stagatio Poit Towards A Shrikig Sht. Zitshrit ür Naturorshug A. 64a: [7] Adrsso H. I. Daat B. S Flow o a Powr- Law Fluid Ovr A Strthig Sht Stability Al. Aal. Cotiuous Mdia (SAACM) Italy. 1: [8] Cra L. J Flow Past A Strthig Plat. Zit. Agw. Math. Phys. 1: [9] Adrsso H. I. Bh K. H. Daat B. S Magtohydrodyai Flow O A Powr-Law Fluid Ovr A Strthig Sht. It. J. No-Liar Mh. 7: [10] Mahaatra T. R. Ny S. K. Guta A. S Aalytial Solutio O Magtohydrodyai Stagatio- Poit Flow O A Powr-Law Fluid Towards A Strthig Sura. A. Math. Cout. 15: [11] Mahaatra T. R. Ny S. K. Guta A. S Magtohydrodyai Stagatio-Poit Flow O A Powrlaw Fluid Towards A Strthig Sura. It. J. NoLiar Mh. 44: [1] Mahaatra T. R. Ny S. K. Guta A. S 01. Hat Trasr I Th Magtohydrodyai Flow O A Powr- Law Fluid Past A Porous Flat Plat With Sutio Or Blowig. It. Coui. Hat Mass Tras. 9: 17-. [1] Batallr R. C Siilarity Solutios For Flow Ad Hat Trasr O A Quist Fluid Ovr A Noliarly Strthig Sura. J. Matr. Pross Thol. 0: [14] Makid O. D. Aziz A MHD Mixd Covtio Fro A Vrtial Plat Ebddd I A Porous Mdiu With A Covtiv Boudary Coditio. It. J. Thr. Si. 49: [15] Yao S. Fag T. Zhog Y Hat Trasr O A Gralizd Strthig/Shrikig Wall Probl With Covtiv Boudary Coditios. Cou. Noliar Si. Nur. Siulat. 16:

7 59 Shah Jaha & Hazah Sakidi / Jural Tkologi (Sis & Egirig) 77:0 (015) 5 59 [16] Ishak A Siilarity Solutios For Flow Hat Trasr Ovr A Prabl Sura With Covtiv Boudary Coditios. Al. Math. Cout. 17: [17] Liao S. J. 00. Byod Prturbatio: Itrodutio To Hootoy Aalysis Mthod. Chaa Hall CRC Prss Boa Rato. [18] Liao S. J A Gral Aroah To Gt Sris Solutio O No-Siilarity Boudary Layr Flows. Cou. Noliar. Si. Nur. Siulat. 14: [19] Liao S. J Nots o th Hootoy Aalysis Mthod: So Diitios Thors. Cou. Noliar. Si. Nur. Siulat. 14: [0] Abbasby S Solitary Wav Solutios To Th Modiid For O Caassa--Hol Equatio By Mas O Th Hootoy Aalysis Mthod. Chaos Solitos Fratals. 9: