Peristaltic Pumping of a Generalized Newtonian Fluid in an Elastic Tube

Transcription

1 Journal of Appled Flud Mechancs, Vol., No., pp , 7. Avalable onlne at ISSN 75-57, EISSN DOI:.889/acadpub.jafm Perstaltc Pumpng of a Generalzed Newtonan Flud n an Elastc Tube A. N. S. Srnvas, C. K. Selv and S. Sreenadh Department of Mathematcs, School of Advanced Scences, VIT Unversty, Vellore, 4, Tamlnadu, Inda Department of Mathematcs, Sr Venkateswara Unversty, Trupat, 575, A.P., Inda Correspondng Author Emal: anssrnvas@vt.ac.n (Receved Aprl 4, 7; accepted August 9, 7) ABSTRACT The paper nvestgates the perstaltc pumpng of an ncompressble non-newtonan flud n an elastc tube wth long wavelengths and low Reynolds number approxmatons. Carreau flud model s consdered for present study to descrbe the perstaltc flow characterstcs of non- Newtonan flud n an elastc tube. Carreau flud s a generalzed Newtonan flud whch exhbts Newtonan behavour for n and t resembles as a power-law model at hgher shear rates. For n t exhbts shear-thnnng property,.e., the apparent vscosty reduces wth ncreasng shear rate. The equatons governng the flud flow are solved wth usual perturbaton expanson by takng Wessenberg number W as a perturbaton parameter. The expressons for axal velocty, stream functon and volume flow rate as functon of pressure dfference are derved. The effects of varous pertnent parameters on varaton of flux for a Carreau flud flow through an elastc tube along wth perstalss are calculated and nterpreted through graphs. The pressure rse per wavelength and shear stress dstrbuton for dfferent values of physcal parameters are calculated and presented. Trappng phenomenon s presented graphcally to understand the physcal behavour of varous parameters. The dfference n flux varaton s examned by two dfferent models of Rubnow and Keller (97) and Mazumdar (99). It s observed that n elastc tubes, the flux of Carreau flud wth perstalss s more when the tenson relaton s a ffth degree polynomal as compared to exponental curve. When the power-law ndex n or Wessenberg numberw and wthout perstalss, the present results are smlar to the observatons of Rubnow and Keller (97). Further, the relaton between the functon g ( a ) and radus of the elastc tube for both Newtonan, non-newtonan cases are dscussed graphcally and these fndngs are dentcal wth the nvestgatons of Mazumdar (99). The results observed for the present flow characterstcs reports several nterestng behavours that warrant further study of physologcal fluds n elastc tubes wth perstalss. Keywords: Perstaltc flow; Elastc tube; Non-newtonan flud; Wessenberg number; Power-law ndex. NOMENCLATURE a a a b c F L n P p p p radus of the tube wthout elastcty change n the tube radus due to elastcty nature change n the tube radus due to perstaltc nature ampltude of the wave wave speed dmensonless flux n movng frame length of the tube power-law Index pressure gradent external pressure nlet pressure outlet pressure () q ( r, z ) pressure of the flud dmensonal flux n fxed frame movng coordnates ( RZ, ) statonary coordnates t tme t, t, k, A elastc parameters T W ( wu, ) ( WU, ) j tenson of the tube wall wessenberg number velocty components n movng frame velocty components n statonary frame stran rate tensor

2 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. stream functon tme constant wave length of the perstaltc wave wave number ampltude rato azmuthal angle j components of extra stress tensor n statonary frame conductvty densty nfnte-shear-rate vscosty zero-shear-rate vscosty. INTRODUCTION Most of the earler research works were concentrated on perstaltc pumpng of non-newtonan fluds through channels/tubes to understand the flow behavour of physologcal fluds. The present study s modelled by consderng the flow through an elastc tube to descrbe the rheologcal characterstcs of blood flow n a small blood vessel due to ther elastc nature whch has many practcal bomedcal applcatons. In physologcal perstalss, the fluds of practcal nterest are Newtonan and non-newtonan fluds, dependng on the varous condtons. The analyss of perstaltc pumpng of non- Newtonan fluds n dfferent type of geometres has drawn more attenton among researchers due to a varety of potental applcatons n bomedcal and ndustral felds. The vasomoton of small blood vessels for example, the perstalss nature s observed n venules, arteroles and the moton n the lymphatc vessels. In vew of such sgnfcant physologcal and engneerng applcatons, a numerous theoretcal and expermental nvestgatons were attempted to understand perstalss mechansm by varous researchers for dfferent fluds under dfferent condtons. After the frst expermental nvestgaton of Latham (9) on perstaltc pumpng, Shapro et al. (99) presented a detaled analyss of perstaltc flow of Newtonan flud along wth expermental results. The perturbaton soluton n powers of ampltude rato was appled by Burns and Parkes (97) n two dfferent cases, one s the perstaltc moton wthout pressure gradent and another one s flow under prescrbed pressure wth snusodally varyng cross secton n a fxed channel walls. The perodcal change n the dameter of vasomoton of blood vessels nvolvng perstalss was consdered by Fung and Yh (98).The shear- thnnng and shear thckenng flud effects on perstaltc pump by lubrcaton analyss are nvestgated by Rao and Mshra (4).The theoretcal analyss of MHD perstaltc transport of Jeffrey flud along wth endoscope and magnetc effects was presented by Hayat et al. (8).Some nvestgatons on perstaltc flow of dfferent physologcal fluds are reported n earler studes. (See Radhakrshnamacharya 98, 7, Vajravelu et. al. 5a, 5b, Srnvas et al. 9,, Hayat et al., Nadeem and Akbar 9) Poseulle-law s consdered n the study of Newtonan fluds snce t explans the flux and pressure dfference relatonshp. Ths relaton s a lnear n the case of ncompressble vscous flud flow through a tube of constant cross secton. But n the most of the vascular systems, the pressure flow relaton s always nonlnear due to elastc nature of blood vessel. Snce most of the physologcal systems are elastc n nature and non-newtonan flud flow through such complex geometres has drawn some mportant applcatons lke blood flow n a small blood vessel, lymphatc vessel etc. A number of dfferent methods are employed to study flow through the tubes havng elastc nature under dfferent condtons. Roach and Burton (957) conducted an experment on human external lac artery to study the statc pressure-volume relaton as tenson versus length curve and explaned the reasons for dstensblty of shape of arteres. Whrlow and Rouleau (95) consdered that tube as a thckwalled cylnder of vsco-elastc materal. Rubnow and Keller (97) gven the detaled analyss of blood flow applcatons by consderng vscous flud flow through elastc tube. Further, an equlbrum condton to determne tenson as a functon of tube radus was presented. Pandey and Chaube () consdered the flexble tube of changng cross secton to study the Maxwell flud flow characterstcs wth perstalss. Takagh and Balmforth () appled lubrcaton analyss to model the deformaton of the tube wall and they determned the pumpng effcency. Al et al. () examned the perstaltc flow characterstcs of bo rheologcal fluds usng numercal smulatons. Most of the earler nvestgatons were made by consderng blood as Newtonan flud that s vald for fluds wth shear rate more than S - whch occurs n the case of large arteres. The study of varatons n flow characterstcs s of consderable research nterest due to the non-newtonan nature of blood flow through small arteres. The non-newtonan behavour of blood at lower shear rates was analysed by Pedley (98).The expermental attempt was made by Johnson et al. (4) that the Carreau flud s approprate model to understand the nature of blood flows n arteres. Modellng the blood flows through elastc arteres was presented by (Wang et al. 99, Sharma et al. 4). Akbar and Nadeem (4) consdered Carreau flud model to analyze the blood flow through a tapered artery wth a stenoss. The flow characterstcs of Carreau flud n dfferent geometres and elastcty effects of tubes under dfferent condtons have been studed. (See Msery et al. 99, Hakeem et al., Mshra and Ghosh, Hakeem et al., Sankara and Jayaraman ) Vajravelu et al. () consdered the case of nsertng a catheter n to an elastc tube to observe the varatons n blood flow pattern by takng Herschel Bulkley flud. Nahar et al. () presented 78

3 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. Fg.. Physcal Model. expermental results on non Newtonan flow characterstcs n collapsble elastc tubes. Soch (4) used lubrcaton approxmaton to understand the flow behavour of Newtonan flud and powerlaw flud n elastc tubes by consderng the pressurearea consttutve relaton. The effect of perstalss on Herschel Bulkley flud flow n an elastc tube was dscussed by Vajravelu et al. (4). Soch (5) derved analytcal expressons for the Newtonan flow characterstcs by consderng cylndrcally shaped elastc tubes. Shen et al. () presented an elastc tube model to study the pulsatle flow characterstcs of blood by takng arteral wall moton n to consderaton. Further Vajravelu et al. () nvestgated the Casson flud flow through an elastc tube wth perstalss and they analyzed that Rubnow and Keller model s better than the Mazumdar model. Motvated by the above studes, t s mportant to study the perstaltc pumpng of generalzed Newtonan flud through elastc tube whch has sgnfcant physologcal applcatons lke flow through elastc arteres etc. The problem s formulated under the assumptons that the wave number s very small and flow s to be of nertal free. The usual perturbaton expanson s appled to solve the governng equatons. The nfluence of dfferent pertnent parameters on flux are evaluated numercally and analyzed through graphs. The varatons of flux for dfferent physcal parameters are calculated usng two models Rubnow and Keller, Mazumdar model. By usng the above two models, the obtaned results are compared graphcally.. MATHEMATICAL FORMULATION The perstaltc pumpng of an ncompressble steady Carreau flud through elastc tube wth radus a( z ) and length L s consdered as shown n Fg.. The flow s produced by an nfnte snusodal wave tran propagatng wth constant wave speed c along the tube walls. The nstantaneous radus of the tube at any axal staton z s represented as R a( z, t ) a bsn ( Z ct ) (.) where a s tube radus n the absence of elastcty, b s ampltude of wave, t s the tme. Here the cylndrcal coordnate system ( R,, Z) s chosen where Z axs s taken along the centre lne of the tube, R s the radus of the tube and s azmuthal angle. By consderng the approxmatons that the length of the tube s an ntegral multple of wavelength, the flow s unsteady n the statonary frame and t s assumed to be steady n the movng frame of reference. The transformaton between statonary coordnates ( RZ, ) and movng coordnates ( r, z) s gven by wwc; u U; z Z ct; r R; (.) Here UW, are the radal and axal velocty components n fxed coordnates. u, ware the radal and axal velocty components n movng coordnates. The contnuty equaton and equatons of moton n movng frame are gven by ( ru ) w r r z p ( r) r r r z r (.) (.4) p ( r) z r r z (.5) The consttutve equaton for Carreau flud s expressed as ( n) ( j ) ( ( ) j (.) Here,, j,, denotes extra stress tensor j components and s defned as j j (.7) j j 787

4 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. here s the second nvarant of stran-rate tensor j. We consder the case and n Eq. (.), so the extra stress tensor component s wrtten as n j j (.8) and the components of stran-rate tensor j are gven as u u w w u,,, r r z r z (.9) The approprate boundary condtons are w at r : r (.a) at r a : w c (.b) The non-dmensonal quanttes are z Z r R ct u z, Z=, r, R, t, u, a a ac U w W a a U w W ac c c c c j,,, j,, a c a p a a,,,,, j j W p a a c a c a a b a q =,, F ; a ac (.) Eqs. (.) - (.5) n a non- dmensonal form are ( ru) w r r z p ( r) r r r z r (.) (.) p ( r ) z r r z (.4) The dmenson less boundary condtons are w at r : (.5a) r at r a : w (.5b) The components of rate of stran tensor and extra stress tensor are u u,, r r (.) w u w, z z r n j W j (.7) (.8) j j j Neglectng the wave number, the equatons of moton and extra stress tensor becomes, ( ru) w r r z p r p ( r ) z r r n w w W r r (.9) (.) (.) (.) elmnatng pressure from Eqs. (.) and (.), we have ( r ) r r r. SOLUTION (.) To obtan the perturbaton soluton the followng quanttes are expanded n powers of perturbaton parameter W as u u W u O W (.) 4 ( ) w w W w O W (.) 4 ( ) p p p z z z 4 W O( W ) () () 4 W O( W ) (.) (.4) F F W F O W (.5) 4 ( ) usng Eqs. (.) - (.4) n the Eqs. (.9), (.), (.) and (.) results two systems of dfferent order. Zero Order System (Newtonan System) ( ru ) w r r z () p ( r ) z r r w r () () ( r ) r r r wth dmensonless boundary condtons (.) (.7) (.8) (.9) 788

5 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. w r at r : (.a) at r a : w (.b) Frst Order System (Non- Newtonan System) ( ru) w r r z () p ( r ) z r r w nw r r () () ( r ) r r r wth dmensonless boundary condtons w r at r : (.) (.) (.) (.4) (.5a) at r a : w (.5b) solvng the Eqs. (.) - (.9) and Eqs. (.) - (.4) usng the dmensonless boundary condtons Eq. (.) and Eq. (.5) results dp w r a 4 dz 4 4 dp n dp r a (.) w r a 4 dz dz (.7) usng Eq. (.) the axal velocty w s gven as 8F 8 r a w 4 a a ( n) W 8F 8 a ( r a ) ( r a ) 4 4 a a (.8) From Eq. (.8), the expresson for stream functon s obtaned by usng w, u and r r r z at r 4 r 8F r a r 4 a a 4 4 ( n) W a 8F 8 r a r 4 (.9) 48 a a 4 5 ( n) W 8F 8 r a r 4 4 a a The nstantaneous volume flow rates for zeroth and frst order F and F through any cross secton are a F rw dr (.) a F rw dr (.) substtutng Eqs. (.) and (.7) n Eqs. (.) and (.), we get 4 a dp F a 8 dz F a 8 dz 9 dz 4 a dp n dp dp solvng Eq. (.) and Eq. (.) for dz respectvely gves, dp 8F 8 dz a a 4 dp 8F n dp a 4 dz a dz usng Eq. (.), the pressure gradent expressed by (.) (.) dp and dz (.4) (.5) dp dz dp 8F 8 8 FW ( n ) W 8F 8 a dz a a a a a (.) replacng F F W F n Eq. (.) and neglectng the terms greater than OW ( ), we get dp 8F 8 ( n ) W 8F 8 a 4 4 (.7) dz a a a a From Eq. (.7), we determne the volume flow rate F through any cross secton by consderng only up to frst order, whch s gven by a 4a 8( n) W F Pa 8a 8( n) W 4a 84( n) W (.8) dp where P dz Equaton (.8) gves the volume flow rate for perstaltc flow of a Carreau flud through elastc tube wth radus az () n the absence of elastcty. 4. THEORETICAL DETERMINATION OF FLUX - APPLICATION TO BLOOD FLOW THROUGH ARTERY In ths secton, the deformaton of the tube wall due to elastcty s taken n to consderaton along wth perstalss to determne the varaton of flux. Consder the perstaltc pumpng of a steady ncompressble Carreau flud through an elastc tube of length and radus a( z) a a as shown n Fg.. s 789

6 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. Here a( z) s the varyng radus whch consstng both perstalss and elastcty effects. To calculate the flux of Carreau flud through the tube havng elastc nature, we use the Rubnow and Keller (97) model. Let p and p represents the pressure of flud at the entrance and ext respectvely and p s the external pressure. Here the nlet pressure p s assumed to be greater than outlet pressure p. As a result of nsde and outsde pressure dfference, the tube wall may expand or contract. Due to ths elastc property of the tube wall there exst changes n the shape of cross secton of tube. Hence, the conductvty of the tube at z depends on the pressure dfference. Therefore the conductvty pz ( ) p s a functon of ( pz ( ) p).we assume that flux and the pressure gradent are related by the expresson dp F ( p p) (4.) dz from Eqs. (.8) and (4.) we have a ( p p) (4.) 8a 8( n) W By takng elastc property n to consderaton n addton to the perstaltc movement, the above Eq. (4.) can be wrtten as ( a a) ( p p) (4.) 8( aa) 8( n) W here a and a denotes the tube radus wth perstalss and elastcty respectvely. Snce the flow s of Poseulle type, the radus a s a functon of ( p p) at each cross secton. The tube wall deformaton due to perstaltc wave s a( z) sn z. Integratng Eq. (4.) wth respect to z from z and applyng nlet condton p() p, we get p p 4a 8( n) W Fa dz ( p) dp 4a 84( n) W pz ( ) p (4.4) here p p( z) p. The above Eq. (4.4) determnes p( z ) mplctly n terms of F and z. In Eq. (4.4), we take z and P() p to fnd the flux F as p p W ( n) F ( p) dp substtutng Eq. (4.) n Eq. (4.5) W ( n) F p p ( a a) dp 8( a a) 8( n) W p p p() p (4.5) (4.) We can evaluate the Eq. (4.), f the functon of the form a( p p) s known. If the tenson n the tube wall T( a ) s a known functon of a then a( p) can be obtaned from the equlbrum condton usng Rubnow and Keller (97) model. Ta ( )/ a p p (4.7) 4.. Rubnow and Keller Model The statc pressure volume relaton s determned by Roach and Burton (957) whch s converted n to a tenson versus length curve. Ths relaton s represented by the followng equaton usng Rubnow and Keller model (97) Ta ( ) t( a ) t( a ) 5 (4.8) where t and t Now substtutng Eq. (4.7) n Eq. (4.8) we have, t dp t 4a 5a a da a a (4.9) substtutng Eq. (4.9) n Eq. (4.), we evaluated the ntegral numercally from p p to p p usng Mathematca software and neglectng the terms greater than OW ( ). The flux s gven as, W ( n) F ga ( ) ga ( ) 8 (4.) where ( t t) a ga () ( W ( n) a ) a t8 W ( n) a t( W ( n)(88 a ) a a ( 5 a ) a8( W ( n))( a) a ta t a 4 a 5a 4 a 8a 5a tt a 8 W ( n) 8a a 8 W ( n) 5a a 4 t a a a 5a a (4.) 5 4a45a 8a 5 t a 5 8 W ( n ) 5 8a t 5 W ( n) 5aa 7 8 a t5a a ta t t a 4 W ( n) a log( a) 4 a W ( n) a 79

7 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. We observe that Eq. (4.) reduces to the correspondng results of Rubnow and Keller (97) when W or nand wthout perstalss. 4.. Mazumdar model: From Mazumdar (99), the tenson relaton can be expressed as ka k Ta ( ) Ae ( e) (4.) where A.745 and k 5.5 Substtutng Eq. (4.) n (4.7), we get ka k p p p A e e a k ka k e dp Ae da a a a From Eq. (4.) and Eq. (4.4), we have a (4.) (4.4) W ( n) F a k Aa ( a) ka k e e da 8( a a) 8( n) W a a a (4.5) The above Eq. (4.5) evaluated numercally to obtan the flux for Carreau flud n elastc tube. 5. PUMPING CHARACTERISTICS The pressure rse per wavelength for Carreau flud flow through elastc tube wth perstalss s calculated usng the Eqs. (4.) and (4.) whch s gven by numercal computaton, the choce of parameters for Carreau flud s consdered from Brd et al. (977) and Tanner (985).The volume flow rate of a non- Newtonan Carreau flud flow through an elastc tube n the presence of both Perstalss and elastcty nature s calculated from Eq. (4.) usng numercal computaton. Fgs. descrbe the varaton of flux along wth z axs by usng the model of Rubnow and Keller (97). It s notced from Fg. that, the flux enhances as the ampltude rato ncreases. The varaton of flux wth z axs for varous values of Wessenberg number s llustrated n Fg.. It s clear that the volume flow rate n elastc tube for Carreau flud s more as compared to Newtonan flud ( W ). The varaton of flux along the z axs for varous values of flud behavour ndex n s llustrated n Fg z..5. Fg.. varaton F vs. z for dfferent values of ampltude rato wth n.98, W., t, t, a., a. (by Rubnow and Keller model). p F8( aa) 8( n) W dp dz dz a a (5.) The non dmensonal shear stress at the tube wall r a s calculated usng the Eqs. (.) and (.8) whch s gven by 8 4 W = W =. W =. 4 8F 8 a a a a a a n W a a F 4 j 4 ( ) a a a a (5.). RESULTS AND DISCUSSION In the present analyss, the non-newtonan Carreau flud flow n an elastc tube n the presence of perstalss s nvestgated. The effects of varous pertnent parameters lke flud behavour ndex n, ampltude rato, Wessenberg number W, nlet elastc radus a, and outlet elastc radus a on volume flow rate F are dscussed graphcally. For..5. z.5. Fg.. varaton F vs. z for dfferent values of Wessenberg number W wth n.98,.4, t, t, a., a. (by Rubnow and Keller model). It s notced that the there s a small varaton at the maxmum value of flux n an elastc tube due to non- Newtonan behavour of Carreau flud. That s flux s more for Carreau flud as compared to Newtonan case ( n ). The effects of nlet and outlet elastc radus a and a on volume flow rate are presented n Fgs. 5 and respectvely. It s notced from Fg. 5 that, wth a gven fxed value for outlet elastc radus, the flux of Carreau flud n an elastc tube decreases wth ncreasng values of nlet elastc radus of the 79

8 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. tube. The opposte behavour s observed n the case of ncreasng outlet elastc radus for fxed gven nlet elastc radus whch s shown n Fg.. That s ncreasng outlet radus ncreases the flux of Carreau flud flow n an elastc tube. 8 4 n =.98 n =.49 n = numercally usng the method of Mazumdar for dfferent pertnent parameters whch are graphcally descrbed n Fgs. 7-. It s clear that the flux enhances n the case of Rubnow and Keller model (97) when compared to the Mazumdar model (99). That s the enhancement of flux s observed when the tenson relaton s a ffth degree polynomal rather than that of an exponental curve. 8 k = k =. 5 k = z.5. Fg. 4. varaton F vs. z for dfferent values of power-law ndex n wth.4, W., t, t, a., a. (by Rubnow and Keller Model) z =. =.4 =.7 Fg. 5. varaton F vs. z for dfferent values of nlet elastc radus a wth n.98,.4, t, t, W., a. (Rubnow and Keller model). =. =.4 = Fg.. varaton F vs. z for dfferent values of outlet elastc radus a wth.4, W., t, z t, a., n.98 (by Rubnow and Keller model). The varaton of flux along the z axs s calculated..5. z.5. Fg. 7. varaton F vs. z for dfferent values elastc parameter k wth n.98,.4, A.745, W., a., a. (by Mazumdar model) z.5. W = W =. W =. 4 Fg. 8. varaton F vs. z for dfferent values Wessenberg number W wth.4, A.745, a., a., n.98, k 5.5 (by Mazumdar model). The Eq. (4.) corresponds to the relatonshp between the functon g ( a ) and non-dmensonal radus of the tube a( zwth ) perstalss and elastcty effects. Fg. demonstrates the effect of flud behavour ndex n on the functon g ( a ) n the absence of perstalss. We found that the values of the functon g ( a ) ncreases as power-law ndex n ncreases. In partcular g ( a ) values are hgher for Newtonan flud case n when compared to Carreau flud case n.98,.49. Mazumdar (99) examned the same relatonshp for a flow of a power-law flud through an elastc tube. Our present results are smlar to the analyss of Mazumdar (99) n the absence of perstalss. Further, the sgnfcant observaton s that for 79

9 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. Newtonan flud n wthout perstalss, present results are n good agreement wth the nvestgatons of Rubnow and Keller (97). The varaton of g ( a) for varous values of Wessenberg number W n the absence of perstalss s depcted n Fg.. It s notced that the values of g ( a ) reduces wth ncreasng values ofw. It s clear that the functon g ( a) takes hgher values for Newtonan fludw as compared to Carreau fludw., n =.98 n =.49 n =..5. z.5. Fg. 9. varaton F vs. z for dfferent values power - law ndex n wth k 5.5, A.745,.4, W., a., 8 4 a. (by Mazumdar model)...5. z =. =.4 =. Fg.. varaton F vs. z for dfferent values nlet elastc radus a wth.4, A.745, k 5.5, W., n.98, a. (by Mazumdar model). ncreases and t s maxmum when ( W ). For a gven pressure rse flux decreases as W ncreases. Fg.. llustrates that for a gven flux, the pressure rse ncreases as ampltude rato ncreases. The varaton n pressure rse for dfferent values of elastc radus s shown n Fg.7. It s observed that for gven flux, the pressure rse decreases wth ncreasng values of elastc radus a z.5. =. =.4 =. Fg.. varaton F vs. z for dfferent values outlet elastc radus a wth k 5.5,.4, A.745, W., g(a) 4 a., n.98 (by Mazumdar model). 4 n =. 9 8 n =. 4 9 n = Fg.. The functon ga ( ) vs. a for dfferent values of power law ndex n wth t, t, W., a. W = W =. W =.4 The effect physcal parameters on varaton n pressure rse p along wth flux for elastc tube are calculated usng Eq. (5.) and presented n Fgs From Fg. 4 t s observed that for a gven flux, the pressure rse per wavelength ncreases wth ncreasng values of power-law ndex n and the maxmum pressure rse s notced for Newtonan case ( n ). Also for a gven pressure rse, the flux ncreases wth ncreasng n. The varaton n pressure rse for dfferent values of Wessenberg number W s shown n Fg. 5. It s clear that for a gven flux, the pressure rse decreases as W g(a) Fg.. The functon ga ( ) vs. a for dfferent values of Wessenberg number W wth t, t, n.98, a. 79

10 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. 5 n=.98 n=.49 n= parameter a ncreases. p F -5 a ll =. a ll =.5 a ll = Fg. 4. The pressure rse vs. flux for dfferent values of power-law ndex n wth a., W.,.4. p F W = W =. W =.4 Fg. 7. The pressure rse vs. flux for dfferent values of elastc radus a wth.4, n.98, W.. p F Fg. 5. The pressure rse vs. flux for dfferent values of Wessenberg number W wth a., n.98, n=.98 n=.49 n= z..4 Fg. 8. The shear stress vs. z for dfferent values of power-law ndex n wth.4, a., W.4. p F Fg.. The pressure rse vs. flux for dfferent values of ampltude rato wth a., n.98, W.. The shear stress dstrbuton at the wall for dfferent physcal parameters s presented form Fgs. 8-. From Fg. 8 t s observed that shear stress ncreases as power-law ndex n ncreases where the opposte behavour s observe n the case of Wessenberg number W. That s shear stress reduces for ncreasng values of W s shown n Fg.9. The effect of ampltude rato on shear stress dstrbuton s llustrated n Fg.. It s found that shear stress ncreases wth ncreasng values of.the varaton n shear stress dstrbuton for dfferent values of elastc radus s shown n Fg. and t s seen that the shear stress decreases as elastc radus W = W =. W = z...4 Fg. 9. The shear stress vs. z for dfferent value of Wessenberg number W wth.4, a., n.98. Trappng s the other nterestng phenomenon observed n perstalss mechansm. The effects of dfferent pertnent parameters on the sze of trapped bolus are presented n Fgs. -. The varaton n the sze of bolus due to the Wessenberg number W 794

11 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , a ll =. a ll =.5 a ll = z..4 Fg.. The shear stress vs. z for dfferent values of ampltude rato wth W.4, a., n z...4 Fg.. The shear stress vs. z for dfferent values of elastc radus a wth W.4,.4, n.98. (a) (b) (c) Fg.. Streamlnes wth.4, n.49, a., a. and ( aw ) ( bw ). ( cw ).4. (a) (b) (c) Fg.. Streamlnes wth.4, W., a., a. and ( an ).98 ( bn ).49 ( cn ). 795

12 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. (a) (b) (c) Fg.4. Streamlnes wth n.98, W.4, a., a. and ( a).4 ( b).5 ( c).. (a) (b) (c) Fg.5. Streamlnes wth n.98, W.4,.4, a. and ( aa ). ( ba ).4 ( ca ).5. (a) (b) (c) Fg.. Streamlnes wth n.98, W.4,.4, a. and ( aa ). ( ba ).5 ( ca ).7. for fxed values of nlet and outlet elastc radus s llustrated n Fg.. It s clear that the bolus sze reduce due to the ncreasng values of W whch means that the bolus sze decreases due to the non-lnearty nature of Carreau flud for W.,.4. The effect of power-law ndex n for Newtonan flud n and non-newtonan case n.98,.49 are analysed from Fg.. It s clear that the sze of the bolus s large for Newtonan flud when compared to the non- Newtonan case through elastc tube. Fgure 4 depct the effect of ampltude rato on sze of trapped bolus for fxed gven values of Wessenberg number W, power-law ndex n and elastc radus parameters a and a. The sze of the bolus ncreases as ampltude rato ncreases. The bolus sze ncreases wth ncreasng nlet and outlet elastc radus are presented n Fgs. 5 and respectvely. 7. CONCLUSIONS The present study deals wth the perstaltc transport of a generalzed Newtonan flud n an elastc tube under the approxmatons of long wavelength and low Reynolds number. Carreau flud model s consdered as a non-newtonan flud due to ts shear thnnng behavour. The pressure gradent, axal velocty, flow rate and shearng stress are expanded n a usual perturbaton seres wth a Wessenberg number that contaned the non-newtonan coeffcents approprate to shear thnnng. The effects of physcal parameters on volume flow rate are 79

13 A. N. S. Srnvas et al. / JAFM, Vol., No., pp , 7. calculated by Rubnow and Keller model and Mazumdar model. The trappng phenomenon s explaned graphcally. The mportant observatons are summarzed as follows. () The flux enhances wth ncreasng values of ampltude rato for fxed values of nlet and outlet radus parameters and flux varaton s more for Carreau flud when compared to Newtonan case. () For a gven fxed value of outlet elastc radus, the flux of Carreau flud n an elastc tube decreases wth ncreasng values of nlet elastc radus of the tube. The opposte behavour s observed n the case of ncreasng outlet elastc radus for fxed gven nlet elastc radus. ()The varaton of flow pattern n the presence of perstalss and elastc nature s observed by two dfferent models namely, Rubnow and Keller model and Mazumdar model, that s the flux s much more enhanced n the Rubnow and Keller model as compared to Mazumdar model. (v) In the absence of perstalss, g ( a) as a functon of elastc tube radus takes the maxmum values for Newtonan flud n or W when compared to the Carreau flud. (v) For a gven flux, The pressure rse ncreases for ncreasng values of power-law ndex and ampltude rato where t decreases wth ncreasng values of Wessenberg number and elastc radus. (v) The shear stress dstrbuton at the wall ncreases wth ncreasng values of power-law ndex and ampltude rato where the opposte behavour s notced for ncreasng values of Wessenberg number and elastc radus parameter. (v) The bolus sze s large for Newtonan flud case as compared to Carreau flud. The sze of the tapered bolus ncreases wth ncreasng values of ampltude rato, nlet elastc radus and outlet elastc radus. ACKNOWLEDGEMENT The authors thank the referees for ther constructve comments whch lead to betterment of the artcle. REFERENCES Abd El, H. and A. E. M. El Msery (). Effects of an endoscope and generalzed Newtonan flud on perstaltc moton. Appl. Math. Comp. 8, 9-5. Abd El, H., A. E. M. El Msery and I. E. Shamy (). Hydrodynamc flow of generalzed Newtonan flud through a unform tube wth perstalss. Appl. Math. Comp. 7, Akbar, N. S. and S. Nadeem (4). Carreau flud model for blood flow through a tapered artery wth a stenoss. An Shams Engneerng Journal. 5, 7-. Al, N., K. Javd, M. Sajd and O. Anwar Beg (). Numercal smulaton of Perstaltc flow of a bo rheologcal flud wth shear - dependent vscosty n a curved channel. Computer Methods n Bomechancs and Bomedcal Engneerng. 9, 4-7. Brd, R.B., Armstrong, R.C. and O. Hassager (977). Dynamcs of Polymerc Lquds., New York: John Wley & Sons. Burns, J.C., and T. Parkes (97). Perstaltc moton. J. Flud Mech.9, Fung, Y.C. and C.S. Yh (98). Perstaltc transport. J. Appl. Mech. Trans ASME. 5, Hayat, T. and S. Hna (). The nfluence of wall propertes on the MHD Perstaltc flow of a Maxwell flud wth heat and mass transfer. Nonlnear Anal: Real World Appl., Hayat, T., N. Ahmad and N. Al (8). Effects of endoscope and magnetc feld on the Perstalss nvolvng Jeffrey flud. Comm. Nonlnear. Sc.Numer.Smul., Hua Shen, Yong Zhu and K. R. Qn (). A theoretcal computerzed study for the electrcal conductvty of arteral pulsatle blood flow by an elastc tube model. Medcal Engneerng and Physcs. 8, Johnston, B. M., P. R. Johnston, S. Corney and D. Klpatrck (4). Non-Newtonan blood flow n human rght coronary arteres: Steady state smulatons. J. Bo Mech. 7, Latham, T. W. (9). Flud motons n a perstaltc pump. MS thess, Massachusetts Insttute of Technology, Cambrdge. Mazumdar, N. J. (99). Bo flud Mechancs, World Scentfc, chapter.5, Sngapore. Msery, A. M. E., Elsayed F. Elshehawey and A. A. Hakeem (99). Perstaltc moton of an ncompressble generalzed Newtonan flud n a planar channel. Journal of physcal Socety of Japan 5, Mshra, J. C. and S. K. Ghosh (). Pulsatle flow of a vscous flud through a porous elastc vessel of varable cross-secton -A mathematcal model for Hemodynamc flows. Computers and Mathematcs wth Applcatons. 4, Nadeem, S. and N. S. Akbar (9). Influence of heat transfer on a perstaltc transport of Herschel Bulkley flud n a non unform nclned tube. CNSNS. 4, 4-4. Nahar, S., S. A. K. Jeelan and E. J. Wndhab (). Predcton of velocty profles of shear thnnng fluds flowng n elastc tubes. Chemcal Engneerng Communcatons., Pandey, S. K. and M. K. Chaube (). Perstaltc 797

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