Influence of Velocity Slip on the Peristaltic Pumping of a Jeffrey Fluid in a Non Uniform Annulus

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Lionel McCoy
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1 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 Inflence of elociy Sli on he Peisalic Pming of a Jeffey Flid in a Non Unifom Annls Sdhaka Rao.G, S H N Kishna Kmai. P *, Y. K. Ravi Kma 3,. Naveen Ass. Pofesso, Deamen of Mahemaics, Niam College, Osmania Univesiy, Hydeabad, India Pofesso, Deamen of Mahemaics, idya Jyohi Insie of Technoy, Hydeabad, India Ass. Pofesso, Pacice School Division, Bila Insie of Technoy & Science- Pilani, Hydeabad, India 3 Ass Pofesso, Mehodis College of Engineeing & Technoy, Hydeabad, India ABSTRACT: Inflence of velociy sli on he eisalic anso of a Jeffey flid in a non nifom annls is invesigaed. A Sinsoidal ave ih is avelling on he oe be all and he inne be is igid, nifom and moving ih a consan velociy. Exessions fo velociy, volme flo ae and esse ise ae obained. The effec of vaios aamees on he ming chaaceisics is discssed ih he hel of gahs. KEYWORDS: Peisalic ming, Jeffey flid, non nifom annls, emeable all. I. INTRODUCTION Peisalic ming is a fom of maeial anso indced by a ogessive ave of aea conacion o exansion along he lengh of a disensible be. Peisalic moion in a channel/be is no knon as an imoan ye of flo occing in seveal engineeing and hysioical ocesses. In living sysems i is a disincive aen of smooh mscle conacions ha oel he conens of he be, as food sffs hogh oesohags and alimenay canal, in anso of ine fom kidney o bladde, he movemen of chyme in he gaso-inesinal ac, vasomoion of he small blood vessels and in many ohe glandla dcs. Enginees have sed he eisalic mechanism in develoing ms hich have seveal indsial alicaions. The oblem of mechanism of eisalic anso has aaced he aenion of many eseaches since he exeimenal invesigaion of Laham []. Sbseqenly a nmbe of analyical, nmeical and exeimenal sdies of eisalic flo of diffeen flids have been eoed nde diffeen condiions ih efeence o hysioical and mechanical siaions. Ravi Kma e.al [] sdied he nseady eisalic ming in a finie lengh be ih emeable all. Mos of he eseaches assmed ha blood and he mos of he hysioical flids behave like non-neonian flids. They consideed vaios non-neonian flids like iscoelasic flid ( Bohme and Fiedich [3] ), oe la flid ( Ravi Kma e al. [] ), Casson flid ( Kishna Kmai e al. [5] ),mico ola flid ( Sinivasachaya e.al [6] ), Coyigh o IJIRSET DOI:0.5680/IJIRSET

2 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 Heschel-Blkley flid (Ramesh Bab [7] al.) in hei heoeical invesigaions. The Jeffey model is elaively simle linea model sing ime deivaives insead of conveed deivaives fo examle he Oldoyd B model does, i eesens a heoy diffeen fom he Neonian. Kishna Kmai e al.[8 ] sdied he eisalic anso of a Jeffey flid in a oos be by consideing Beaves Joseh condiions and Saffman condiions. Peisalic ming of a Jeffey flid nde he effec of a magneic field in an inclined channel is sdied by Kishna Kmai e al. [9]. Ga and Seshadi [0] discssed he eisalic flo hogh non nifom channels and bes ih aicla efeence o he flo of semaic flid in vas defeens consideing he ineia ems o be small in comaison o he viscos ems. Chaani and Ranganaha [] discssed a mahemaical model fo sole ansfe in la fileing glomela caillaies, hee caillaies ae aken as emeable bes. Fhe i is eoed ha he blood flo in small vessels occs de o eisalsis. Sli effec on he flo of a facional second gade flid hogh a cylindical be is analyed by Rahod. e al.[]. Wih he above discssion in mind, e oose o sdy he eisalic anso of Jeffey flid hogh he ga beeen coaxial bes, hee he oe be is non-nifom and has a sinsoidal ave aveling don is all and he inne one is a igid and nifom be and moving ih a consan velociy. This invesigaion may have alicaion in many clinical alicaions sch as he endoscoe oblem. Jeffey model The consiive eqaions fo an incomessible Jeffey flid ae T I S S ( ) hee T and S ae Cachy sess enso and exa sess enso, P is he esse, I is he ideniy enso, is he aio of he elaxaion o eadaion imes, is he eadaion ime and is he shea ae. II. MATHEMATICAL FORMULATION OF THE PROBLEM Le a be he adis of he inne be and a ( ) be he adis of he oe be a any co-axial disance fom he inle.(fig.) ; The geomey of he all sfaces ae a () a( ) bsin ( c) () Coyigh o IJIRSET DOI:0.5680/IJIRSET

3 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 Fig.. Physical Model hee a ( a k (k = 0 coesonds o nifom be) (3) ) 3 Hee a is he adis of he oe be a he inle, k (<<) is a consan hose magnide deends on he lengh of he oe be, b is he ave amlide, λ is he ave lengh and is he ime. We choose a cylindical co-odinae sysem, hee he -axis lies along he cene line of he inne and oe bes and is he disance meased adially. The eqaions of moion of he flo in he annls beeen he inne and he oe bes ae ( ) 0 () (5) (6) hee, ae velociy comonens in and diecions esecively, is he esse, is he viscosiy coefficien and is he Jeffey aamee. The bonday condiions ae 0, a (7), k a (8) e inodce he folloing non-dimensional qaniies, a3 c ca,,,,, Re 3, c a c c a 3 3 a3 k v,, v, c a 3, a3 Coyigh o IJIRSET DOI:0.5680/IJIRSET

4 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 Coyigh o IJIRSET DOI:0.5680/IJIRSET Sin a k a 3 3, hee φ= < (amlide aio), Re is he Reynolds nmbe is he ave nmbe aio and is he consan velociy of he inne be. By sing he non-dimensional qaniies, he eqaions () o (6) become, 0 (9) 3 Re (0) Re () Using he long ave lengh, lo Reynolds nmbe assmions and doing ems of δ and highe ode, he Eqn. (0) o () become, 0 () (3) The dimensionless bonday condiions ae a () a Sin a k., 3 (5) hee is he sli aamee and is he cy nmbe. III. SOLUTION OF THE PROBLEM Solving he Eqns. () and (3) ogehe ih bonday condiions () and (5), e ge he velociy field as (6) The insananeos volme flo ae Q, is given by

5 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 Coyigh o IJIRSET DOI:0.5680/IJIRSET d Q (7) The ime mean flo ove ime eiod T=λ/c is given by d Q T Q T 0 ), ( ), ( (8) Fom eqaion (7), e ge he esse gadien as c c Q (9) The esse ise is given by, d 0 hee L /, L is lengh of he be, is ave lengh I. RESULTS AND DISCUSSIONS The effec of Jeffey aamee on he ming chaaceisics fo nifom and non-nifom cases ae discssed hogh Fig. I is noiced ha fo a give ime mean flo ae Q, he esse ise deceases ih inceasing in all he hee egions i.e. ming egion ( > 0), co-ming egion ( < 0) and fee ming egion ( = 0). I is obseved ha he ming is moe in nifom han in non nifom annls fo any given Q. Also he ming is moe fo a Neonian flid and he ming deceases as he non Neonian nae of he flid inceases. The effec of sli aamee on he ming henomenon is obseved fom Figs 3 &. I is conclded fom hese Figs ha he esse ise deceases ih inceasing fo a fixed Q. Also fo a given, he incease in deceases he flo ae in he ming egion. The vaiaion of affecs he ming henomenon as he ming is moe fo nifom case han non nifom case. Fom figes (5) and (6) he effec of cy nmbe is noiced. Fo a given cy nmbe he esse ise deceases fo inceasing Q. I is obseved ha an incease in he flo ae deceases he esse ise. Also fo fixed flo ae, he esse ise inceases ih inceasing. Fo a given, he esse ise is moe fo non nifom case han he nifom case.the effec of non Neonian behavio is also obseved fom hese figes. Fom eqaion (9), e calclaed he esse ise as a fncion of Q fo diffeen vales of (adis

6 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay 06 of he inne be) and is shon in Figes (7) and (8) consideing nifom and non nifom cases. in he case of nifom annls. I is obseved ha fo fixed flo ae, he esse ise inceases ih inceasing. I is obseved ha fo a given, he incease in inceased he flo ae in all he hee ming egions Fig.. aiaion of esse ise ih Q fo diffeen vales of Jeffey aamee λ nifom annls non nifom annls Coyigh o IJIRSET DOI:0.5680/IJIRSET

7 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay Fig. 3. aiaion of esse ise ih Q fo a nifom annls fo diffeen vales of sli aamee α fo Neonian Non Neonian flids Fig. aiaion of esse ise ih Q fo a non-nifom annls fo diffeen vales of α ih Neonian Non Neonian flids Coyigh o IJIRSET DOI:0.5680/IJIRSET

8 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay Fig. 5. aiaion of esse ise ih Q fo a nifom annls fo diffeen vales of ih Neonian Non Neonian flids Fig.6. aiaion of esse ise ih Q fo a non-nifom annls fo diffeen vales of ih ih eonian Non Neonian flids Coyigh o IJIRSET DOI:0.5680/IJIRSET

9 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay Fig. 7. aiaion of esse ise ih Q fo a nifom annls fo diffeen vales of ih Neonian Non Neonian flids Coyigh o IJIRSET DOI:0.5680/IJIRSET

10 ISSN(Online): ISSN (Pin): Inenaional Jonal of Innovaive Reseach in Science, Engineeing and Technoy (An ISO 397: 007 Ceified Oganiaion) ol. 5, Isse, Janay Fig.8. aiaion of esse ise ih Q fo a non-nifom annls fo diffeen vales of ih Neonian Non Neonian flids REFERENCES [] Laham, T.W.,966, Flid moion in a eisalic m. M.SC, Thesis. Cambidge, Mass: MIT-Pess. [] Y..K.RaviKma, S..H.N.KishnaKmai, M.. Ramana Mhy, S.Seenadh, 00, Unseady eisalic ming in a finie lengh be ih emeable all, Tans. ASME, J. of Flids Engg., 3, [3] Bohme, G., and Fiedich, R., (983), Peisalic flo of viscoelasic liqids, J.Flid Mech., 8,.09-. [] Y..K.RaviKma, S..H.N.KishnaKmai, M.. Ramana Mhy, S.Seenadh, 0, Peisalic Tanso of a oe la flid in an asymmeic channel bonded by emeable alls Advances in Alied Science Reseach, volme isse 3, [5] S..H.N.KishnaKmai, M.. Ramana Mhy, Y..K.RaviKma, Chenna Kishna Reddy, 0, Peisalic ming of magneo hydo dynamic Casson flid in an inclined channel, Advances in Alied Science Reseach, volme isse, [6] Sinivasachaya, D.,Misha, M. and Ramachanda Rao, A.,003, Peisalic ming of a mico ola flid in a be, Aca Mechanica, 6, [7] ajavel,k.,seenadh,s and Ramesh Bab,.,005, Peisalic anso of a Heshel Blkley flid in an inclined be,in.j.non Linea Mech.,0, [8] S..H.N.Kishna Kmai.P.,Y..K.Ravi Kma,M..Ramana Mhy, S.Seenadh,0, Peisalic anso of a Jeffey flid in he oos be, J.of Engg.and Al.Sciences,6, No.3, [9] S..H.N.KishnaKmai.P.,Y..K.RaviKma, M.. Ramana Mhy, S.Seenadh, 0, Peisalic ming of a Jeffey flid nde he effec of magneic field in an inclined Channel, Al.Mah.Sciences, ol.5 No.9, [0] Ga, B.B. and Seshadi,.,976, Peisalic ming in non-nifom bes, J.Biomech., 9, [] Chaani,P and Ranganaha, T.R,993, Sole ansfe in flid flo in emeable bes ih alicaion o flo in glomela caillaies,aca Mechanica, 96, [] Rahod, Ania Tljaa, 05, Sli effec on he eisalic flo of a facional second gade flid ogh a cylindical be Advances in Alied Science Reseach, 6(3): 0-. Coyigh o IJIRSET DOI:0.5680/IJIRSET

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence