International Journal of Pure and Applied Sciences and Technology

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1 In. J. Pre Appl. Sci. echnol. () (4) pp. -8 Inernaional Jornal of Pre and Applied Sciences and echnolog ISSN 9-67 Aailable online a.ijopaasa.in Research Paper Effec of Variable Viscosi on hird Grade Flid Flo oer a Radiaie Srface ih Arrhenis Reacion Amos W. Ognsola and Benjamin A. Peer * Deparmen of Pre and Applied Mahemaics Ladoke Akinola Uniersi of echnolog Ogbomoso Nigeria * Corresponding ahor (pbenjamin8@ahoo.ca) (Receied: --4; Acceped: 7-4-4) Absrac: In his ork fecs of ariable iscosi and Arrhenis reacion on he hird grade flid oer a radiaie srface hrogh a poros medim is considered. he goerning parial differenial eqaions ere ransformed ino ordinar differenial eqaions in erms of siable similari ariable. We emploed Galerkin eighed residal mehod o sole he resling non-linear eqaions. he fecs of ariable iscosi parameer Frank-Kameneskii parameer Brinkman nmber Renolds nmber Prandl nmber and Darc nmber on he ssem of flo and he resls ere repored graphicall. Keords: hird grade flid eighed residal mehod Arrhenis reacion radiaie srface and poros medim. Inrodcion Non-Neonian flids hae receied mch aenion han Neonian flids in he recen ears de o is pracical imporance rapid deelopmen of modern indsrial maerials and echnological applicaions. I has gien insigh in he reseroir engineering hea echange beeen soil and amosphere flo of moisre hrogh poros indsrial maerials hea echangers ih flid beds preheaing coal-aer mire ceramic processing caalic reacors polmer solion molen plasics oil recoer o menion b js a fe applicaions. Some maerials behae non-neonianl namel md pasa cheese personal care prodc asphal ice cream oils and he hos of ohers. he rheological characerisics of polmer mels and solions ogeher ih he aribes of some polmeric sbsances hich has lead o he recen orld-ide processing of polmer. he rheological behaior of all he non-neonian flids canno be deermined b a single consiie eqaion.

2 In. J. Pre Appl. Sci. echnol. () (4) -8 Hea ransfer problem of hird grade flids has been sdied b seeral ahors: Makinde and Aziz [] considered he bondar laer flo of iscos nanoflid bonded b a sreching shee. Mkhopadha [] eamined he bondar laer flo and hea ransfer oards an eponeniall sreching poros shee embedded in a poros medim. Krishnend e al [] considered similari solion of mied conecie bondar laer slip flo oer a erical plae. Fosdick and Rajagopal [4] performed a complee hermodnamic analsis of consiie eqaions for he hird grade flid inoling hea ransfer process. Massodi and Chrisie [5] analzed nmericall he flo of a hird grade flid in a pipe iho hea sorce here he shear iscosi as assmed o be emperare dependen. Olajon [6] eamined he flo and naral conecion hea ransfer in a poer-la flid pas a erical plae ih hea generaion. Yrso e al [7] eamined enrop analsis for hird grade flid flo ih Vogel s models of iscosi in annlar pipe. Nadeem e al [8] analicall considered he fecs of parial slip on a forh grade flid ih ariable iscosi and Makinde [9] emploed Hermie-Pade approimaions o ealae hermal radiaion fec of inheren irreersibili in a ariable iscosi channel flo. Massodi and Chrisie [] sdied he fecs of ariable iscosi and iscos dissipaion on he flo of hird grade flid in a pipe. Nadeem e al [] sdied analical solions for pipe flo of a forh grade flid ih Renolds and Vogel s models of iscosiies. Lazars [] eamined he fecs of ariable iscosi on he eloci flid and emperare flid sing semi-implici finie difference scheme of Laminar flo in a channel filled ih saraed poros media. he resls sho ha he eloci flid and emperare flid increases as ariable iscosi parameer increases. Moiaed b he ork of Haa e al [] he considered he fec of jole heaing and hermal radiaion in flo of hird grade flid oer a radiaie srface. Folloing Szeri and Rajagopal [4] an incompressible homogeneos flid of hird grade is characerized b Cach sress τ of he folloing form: τ pi β ( ) A α ( ) A α ( ) ( ) A β ( )[ A A A A ] β ( )( ra ) A. A () here pi denoe he indeerminae par of he sress de o he consrain of incompressibili ( ) is he coficien of iscosi and α ( )α ( ) are maerial modli sall rerred o as normal sress coficiens. he kinemaic ensors A A are dined b [] hrogh ( grad ) ( grad ) τ A () d An An An ( grad ) ( grad ) An n. () d d Here denoes maerial ime deriaie and is he eloci ecor. he aboe model conains as a d special sbclass he classical linearl iscos model (he case hen all he coficiens epec are se eqal o zero). Flo of a hermodnamicall compaible flid of hird grade is gien as: ( ) i (4) here i denoes he ni ecor in he -coordinae direcion he direcion ha is chosen parallel he eernal pressre gradiens. In he absence of bod forces he balance of linear momenm d di τ b (Eqaion of Moion) (5) K d

3 In. J. Pre Appl. Sci. echnol. () (4) -8 here K is he poros medim permeabili Q r K di q L d de τ. (6) Here e denoes he inernal energ L is he eloci gradien r is he radian energ boh per ni mass and Q denoes he reacing erm. Goerning Eqaions and Mehod of Solion Folloing aaser e al [] e consider he nsead o-dimensional bondar laer flo of a magneo hdrodnamic (MHD) hrogh a poros medim. he basic goerning eqaions are as follos: (7) K B B E σ β α α 6 (8) c Q K E B k p σ β α 4 (9) he appropriae iniial and bondar condiion are as follos as as o V U () here σ is he San-Bolzmann consan k is he mean absorpion coficienq is he hea release per nis mass < V is he injecion > V is he scion c V c a c a U b a are consans E is he aciaing energ R is he niersal gas consan θ -is he dimensionless emperare k is he permeabili of he poros media K is he hermal condcii is he densi p C is he specific hea a consan pressre is he dnamic iscosi r is he iscos heaing fec direcion ψ is he Frank-Kameneskii parameer η is he similari ariable is he fecie iscosi e is he hermal epansion

4 In. J. Pre Appl. Sci. echnol. () (4) -8 4 is he flid iniial emperare or all emperare r is he rerence emperare is he absole emperare ihin he bondar laer... - emperare a he plae α α β are he flid parameers is he dimensionless eloci in he direcion Ψ is he sream fncion is he dimensionless eloci in he direcion is he free sream emperare. From Eqaions (8) and (9) e seek Renolds model of he form Mθ ( ) e () Mθ α e α β 6 σ ( EB B ) K Mθ α k e 4 qr ( ) β σ B E QC Ae K cp We inrodce he folloing ariables and parameers E R () () Ψ U f Ψ Ψ ( η) ( ) θ ( η) and η U E EQC A R 4σ r k ψ e R θ Da. (4) lo Ulo d ( R E) k K Sbsiing Eqaion (4) ino Eqaions () and () e obain d dη α f Mθ ( i ) ( i ) ( e f ) S{ f ηf } α f f ff f S f ηf 6β Re f f m ( E f ) Da 4 S Mθ θ Rd f θ f θ { ηθ 4θ } Bre ( f ) αec Pr S ( εθ ) { η } β ψ Da 4 f f f f f ff f Ecf m Ec f E e θ (5) (6)

5 In. J. Pre Appl. Sci. echnol. () (4) -8 5 Where ψ EQC A E R e 4σ α a α a β a R α α β d ( R E) k K ( c) ( c) ( c) he ransformed bondar condiion is as follos ( ) A f ( ) f ( ) f ( ) θ ( ) θ ( ) S f (8) We proceed o sole Eqaions (5) and (6) sbjec o (8) nmericall sing Galerkin-Weighed Residal Mehod as follos: c a. (7) Le f Ae i i i i i 4 θ Be (9) he resls are presened in Figres -5 Figre : Graph of he eloci fncion f for arios ales of Br.5 Pr Re A S β. Ec.5 α α m E ψ..

6 In. J. Pre Appl. Sci. echnol. () (4) -8 6 Figre : Graph of he emperare fncion θ for arios ales of Br.5 Pr Re A S β. Ec.5 α α m E ψ.. Figre : Graph of he eloci fncion f for arios ales of Br.5 Pr Re A S β. Ec.5 α α m E ψ..

7 In. J. Pre Appl. Sci. echnol. () (4) -8 7 Figre 4: Graph of he emperare fncion θ for arios ales of Br.5 Pr Re A S β. Ec.5 α α m E ψ.. Figre 5: Graph of he emperare fncion θ for arios ales of Br.5 Pr Re A S β. Ec.5 α α m E ψ Discssion of Resls/Conclsion he sd of hird grade flids is eremel imporan de o is ide arie of pracical applicaions in processes sch as filraion of polmer solions and soil remediaion hrogh he remoal of liqid pollans. From Figres -5 he resls sho ha he flid eloci and emperare increases..

8 In. J. Pre Appl. Sci. echnol. () (4) -8 8 increase in each parameers. I is noiced from Figres & 5 ha he emperare profile reach he maimm poin as Darc nmber Frank Kameneskii parameer increases. We obsered from Figre 4 ha he eloci profile has a minimm poin as Frank Kameneskii parameer increases. Conclsion I is conclded ha eloci flid and he emperare flid decreases as Frank Kameneskii parameer and ariable iscosi parameer increases. A ransien decrease in boh he flid eloci and emperare is obsered ih increase in each β α & α non-neonian parameers E Elecric parameer Br Brinkman nmber and Darc nmber hich decreases he porosi in he ssem of flo. For engineering prpose he flo model of or problem represens he oils ell and as he ψ Frank Kameneskii parameer is increasing here is qick recoer of oil from he oils ell. Also he resls of his problem are of grea ineres in prodcion processing aomobile engine for he safe of life and proper handling of he maerials dring processing. Rerences [] O.D. Makinde and A. Aziz Bondar laer flo of a nanoflid oer a non-linearl sreching shee ih a conecie bondar condiion In. J. hermal Sciences 5() 6-. [] S. Mkhopadha MHD bondar laer flo and hea ransfer oer an eponeniall sreching shee embedded in a hermall sraified medim Aleandra Eng. J. 5() [] K. Bhaachara S. Mkhopadha and G.C. Laek Similari solion of mied conecie bondar laer slip flo oer a erical plae Ain Shams Engineering Jornal 4() [4] R.L. Fosdick and K.R. Rajagopal hermodnamics and sabili of flids of hird grade Proc. R. Soc. Lond. A69(98) [5] M. Massodi and I. Chrise Hea ransfer and flo of hird grade flid in a pipe Mah. Modeling Sci. Comp. (99) [6] B.I. Olajon Flo and naral conecion hea ransfer in a poer-la flid pas a erical plae ih hea generaion In. J. Non-Linear Science 7() (9) [7] M. Yrso H. Barakoekem M. Kapc and F. Akso Enrop analsis for hird grade flid flo ih Vogel s models of iscosi in annlar pipe In. J. Non-Linear Mech. 4(8) [8] S. Nadeem and M. Ali Analical solions for pipe flo of a forh grade flid ih Renolds and Vogel s models of iscosiies Commnicaions in Non-Linear Sci. and Nmerical Simlaion 4(5) (9) 7-9. [9] O.D. Makinde Hermie-Pade approach o hermal radiaion fec on inheren irreersibili in a ariable iscosi channel flo Compers and Mahemaics ih Applicaions 58(9) -8. [] M. Massodi and I. Chrise Effecs of ariable iscosi and iscos dissipaion on he flo of hird grade flid in a pipe In. J. Non-Linear Mech. (5) (995) [] S. Nadeem. Haa N.S. Akbar and M.Y. Malik On he inflence of hea ransfer in perisalsis ih ariable iscosi In. J. Hea and Mass ransfer 5(9) [] L. Rndora Laminar flo in a channel filled ih saraed poros media Ph.D. hesis Deparmen of Mechanical Engineering Cape Peninsla Uniersi of echnolog (). [] H. asaar A. Shafig and A. Alsaedi Effec of jole heaing and hermal radiaion in flo of hird grade flid oer radiaie srface Nonlinear Analsis and Applied Mahemaics Research Grop Facl of Science Sadi Arabia 9() (4) - hp:/.plosone.org Accessed on March (4).