Hydrogen bubble visualization of viscoelastic flow in a model of the carotid artery bifurcation van Kampen, T.F.

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1 Hydrogn bubbl visualization of viscolastic flow in a modl of th carotid artry bifurcation van Kampn, T.F. Publishd: 01/01/1996 Documnt Vrsion Publishr s PDF, also known as Vrsion of Rcord (includs final pag, issu and volum numbrs) Plas chck th documnt vrsion of this publication: A submittd manuscript is th author's vrsion of th articl upon submission and bfor pr-rviw. Thr can b important diffrncs btwn th submittd vrsion and th official publishd vrsion of rcord. Popl intrstd in th rsarch ar advisd to contact th author for th final vrsion of th publication, or visit th DOI to th publishr's wbsit. Th final author vrsion and th gally proof ar vrsions of th publication aftr pr rviw. Th final publishd vrsion faturs th final layout of th papr including th volum, issu and pag numbrs. Link to publication Citation for publishd vrsion (APA): Kampn, van, T. F. (1996). Hydrogn bubbl visualization of viscolastic flow in a modl of th carotid artry bifurcation. (DCT rapportn; Vol ). Eindhovn: Tchnisch Univrsitit Eindhovn. Gnral rights Copyright and moral rights for th publications mad accssibl in th public portal ar rtaind by th authors and/or othr copyright ownrs and it is a condition of accssing publications that usrs rcognis and abid by th lgal rquirmnts associatd with ths rights. Usrs may download and print on copy of any publication from th public portal for th purpos of privat study or rsarch. You may not furthr distribut th matrial or us it for any profit-making activity or commrcial gain You may frly distribut th URL idntifying th publication in th public portal? Tak down policy If you bliv that this documnt brachs copyright plas contact us providing dtails, and w will rmov accss to th work immdiatly and invstigat your claim. Download dat: 27. Sp. 2018

2 Hydrogn bubbl visualization of viscolastic flow in a modl of th carotid artry bifurcation T.F. van Kampn Fbruary 1996 WFW coachs: Dr. Ir. F.N. van d Voss Ir. F. Gijsn Laboratory of Biomchanics Dpartmnt of Mchanical Enginring.Eindhovn Univrsity of Tchnology

3 CONTENTS 1. INTRODUCTION 2. MATERIALS AND METHODS Modl of th bifurcation 2.2. Visualization of th flow 2.3. Blood Analog fluid 2.4. Exprimntal st up 2.5. Exprimntal procdurs 3. RESULTS 4. DISCUSSION AND CONCLUSION 5. REFERENCES

4 1. Introduction Athrosclrosis is a complx artrial disas that is on of th most important causs of dath in wstrn countris. Th disas is charactrizd by intimal thickning of th artris. Th thickning of th artrial walls is followd by th plaqu formation by small clls such as cholstrol, lipids and othr clls. Th formation of plaqus can lad to partial or complt obstruction of th vssls. Furthrmor thrombi can brak loos and can gt stuck downstram in smallr artris causing ischmic attacks. Athrosclrosis can oftn b found at locations whr th flow is disturbd such as bifurcations and bnds,.g. at th carotid artry bifurcation. Th carotid artry bifurcation is locatd in th human nck. Th main blood supply to th had, nck and fac is through th two (lft and right) common carotid artris, which bifurcat in th xtrnal carotid and th intrnal carotid branchs. Th xtrnal branch supplis blood to th fac, nck and top of th had. Th brain and th y ar supplid with blood by th intrnal branch. Th intrnal branch has a widning, calld th carotid sinus or bulbus. From clinical practic it is known that athrosclrosis occurs prfrntially at th so calld non-dividr wall in th carotid sinus (s figur 1). non-aividr wall Inrmai Carotid Artry Conmoa Cacoiid hrry dividr wall f;xtrnai Carotid Figur 1, Schmatic ovrviw of th carotid artry bifurcation It is known that thr ar two procsss that mainly caus th dvlopmnt of stnoss: low or oscillating wall shar strsss and diffusion procsss. Both procsss ar closly linkd to local flow phnomna in th carotid artry bifurcation. Th study of th flow fild in th carotid bifurcation is of grat clinical intrst with rspct to th gnsis and diagnosis of athrosclrotic disas [Ku t al (1989)l. Prvious studis of stady flow in th carotid artry bifurcation hav shown that many flow phnomna, such as flow sparation, rvrsal flow and curvatur ffcts ar important. Nar th non-dividr wall flow sparation occurs, rsulting in a rgion of rvrsd flow. Nar th dividr wall a rgion of high axial shar with high axial vlocitis is sparatd from th rvrsd flow rgion by a shar layr. A scondary flow that shows a hlical structur (figur 2) can b found in th rgion of rvrsal flow. 2

5 Figur 2, Schmatic rprsntation of th flowfild of th human carotid bifurcation (Bharadvaj) Ths hlical flow trajctoris show a grat rsmblanc with Dan vortics found in bnds [Rindt (1989)l. Th high vlocitis at th dividr wall and th low vlocitis at th non-dividr wall ar spratd by a high vlocity gradint in th shar layr. Thrfor it is clar that th flow in th human carotid artry bifurcation is highly complx. Studis of th unstady, pulsatil Nwtonian flow [Palmn (1994)] showd that, although many phnomna ar comparabl with th stady stat rsults, thr ar a fw important diffrncs. In th acclration phas of th flow th flow rvrsal rgion fully disappars. Th vlocity profil is positiv across th ntir lumn. In th dclration phas a rgion with flow rvrsal dvlops at th non-dividr wall of th sinus in th intrnal branch. Du to th ngativ vlocity in this rgion and th high axial vlocity nar th dividr wall thr is a significant shar layr with high vlocity gradints in th boundary layr btwn th two rgions. A small vortx is dvlopd in th shar layr. Blood is a concntratd suspnsion of bloodclls in plasma. Thrfor it xhibits non-nwtonian proprtis such as shar thinning and visco-lasticity. Th influnc of th non-nwtonian proprtis of blood is studid by svral groups. Prktold has shown in his study of non- Nwtonian flow in th carotid artry bifurcation that shar thinning has small influnc on th flow fild. In bnds th non-nwtonian proprtis of blood caus lowr wall shar rats and lss rvrsal flow at th innr sid of th bnd [Mann, Tarbll (1990)l. For that rason it is important to study th influnc of non-nwtonian proprtis of blood on th flow in artry modls. Rsarch in th Athrosclrosis projct at th Eindhovn Univrsity of Tchnology until now was concntratd on th study of Nwtonian flow in carotid bifurcation modls.g. [Rindt (1989), Palmn t al (1992)l. LDA xprimnts in a 1: 1 modl of th carotid bifurcation hav shown a non-ngligibl diffrnc in th stady flow fild causd by th viscolastic proprtis of th blood-lik fluid [Zuidrvaart ( In this study a visualization study of pulsatil flow in a 5: 1 nlargd modl of th human carotid artry is prformd with th hydrogn bubbl visualization mthod. Th purpos of th study is to dtrmin th influnc of th non-nwtonian proprtis of blood on th flow. For that rason, two 3

6 xprimnts hav bn don: th first with watr as Nwtonian fluid and th othr with a Xanthangum solution as non-nwtonian fluid. 2, LVatriaBs and mthods 2.1. Modl of th bifurcation Th xprimnts hav bn prformd in a 5: 1 modl of th carotid artry bifurcation machind out of prspx (PMMA) as shown in figur 1. Th gomtry of th rigid thrdimnsional modl of th carotid artry bifurcation is basd upon th data of Bharadvaj t al (1982). It is th sam modl usd in th study of Palmn (1994) Visualization of th flow Th hydrogn bubbl visualization mthod has bn usd to visualiz th flow in th carotid artry bifurcation. This mthod is vry suitabl to provid qualitativ information about th flow in spac and tim. It givs a global viw of th vlocity profils in th flow. Th principl of th hydrogn bubbl visualisation mthod is that littl bubbls of hydrogn ar convctd by th passing fluid, visualizing th vlocity profils of th flow. Th bubbls ar gnratd by th lctrolysis of watr. Th chmical raction of this procss is dscribd by: cathod: anod: Thr ar som limitations to this mthod: (i) th frquncy of th vlocity fluctuations that can b visualizd proprly is limitd bcaus of th inrtia of th hydrogn bubbls, (ii) th siz of th bubbls dtrmins th spatial rsolution of th mthod, (iii) th bubbls ar displacd from th fluid path lin by th buoyancy forcs. This buoyancy forcs ar proportional to th squar of th diamtr of th bubbls. Th hydrogn bubbls hav to b as small as possibl for th following rasons: 0 to rduc th buoyancy forcs 0 to incras th spatial rsolution 0 to incras th maximum frquncy of th vlocity fluctuations that can b followd by th bubbls In ordr to gnrat th hydrogn bubbls, vry thin platina wirs ar placd in th bifurcation as cathod and a gildd anod is placd downstram th intrnal branch. Bcaus th hydrogn 4

7 bubbls ar much smallr than th oxygn bubbls, th cathod is placd in th masuring sction and th anod is placd downstram in th intrnal branch. Th lctrods ar placd in th plan of symmtry, bcaus th vlocity prpndicular to this plan is zro. From Palmn (1994) it follows that in ordr to gt good rsults with this mthod, th cathod wir s diamtr has to b vry small (20 pm) and th voltag has to b low (30 V.). Th voltag is supplid in form of squar wav pulss with a frquncy of Hz. and a duration of O. 10 sc. In this way th bubbls will form discrt lins and show th vlocity profils of th flow. To gnrat th squar wav pulss Labviw is usd. Th gnration of th bubbls is tïiggrd with th flowpuls: at th bginning of ach Ecwpuh i: triggrpuls is snd to Labviw to start th gnration of th bubbls Blood Analog fluid Blood is a complx non-nwtonian fluid, having shar rat and tim dpndnt viscous and lastic proprtis. Looking at th microscopic structur of blood th shar thinning and viscolastic bhavior is xplainabl. Blood is a concntratd suspnsion of rd blood clls (RBC), whit blood clls (WBC), platlts and othr clls as cholstrol in a fluid calld plasma. Th RBC ar th most importd rason for th non-nwtonian proprtis of blood, du to thir high volum concntration (45%). Although th WBC ar largr thn th RBC, and th platlts ar quit numrous, thir volum concntration is much lss thn that of th RBC (1 %). It can b said that blood bhavs lik a Nwtonian fluid (plasma) with flxibl asymmtrical particls (RBC) suspndd in it. Th dformation and orintation of th flxibl RBC in flow dirction at high shar rats causs shar thinning. At low shar rats shar thinning is causd by aggrgation of th blood clls. Thrfor also tim constants rlatd to viscolasticity will charactriz th flow bhavior. Th fluid usd in th xprimnts as non-nwtonian blood analog fluid has to rsmbl th rhological proprtis of blood. Furthrmor it has to b transparnt with rfrnc to th visualization mthod usd. In his rport Zuidrvaart has found a fluid that satisfis this dmands. On th ground of his rsults th xprimnts ar don with a solution of watr with 250 ppm Xanthangum. Xanthangum is a natural polysaccharid, with a thr-dimnsional structur that looks lik a doubl hlix which aggrgats to a stiff rod lik macromolcul. Th viscosity of th solution has bn tstd on a viscomtr. Th rsults ar plottd in th figurs blow with th rsults of blood [Thurston (1979) and Coklt (1980). 5

8 x * * 1 1( o shar rat [lis] Figur 3, Elastic componnt of th complx viscosity-frquncy of blood and Xanthangum (lfi), th viscous componnt-frquncy of blood and Xanthangum (right} with - = Xanthangum, -. = blood and stady shar-viscosity of blood and Xanthangum (abov) with - = Xanthangum, + = blood (Mrrit t al), O =blood (Coklt) )[Thurston ( 1979),Coklt (1980)] As shown in th figur abov th non-nwtonian fluid usd is sharthinning and viscolastic. Th viscosty at low shar rats of Xanthangum is lss than th viscosity of blood, although th trnd is followd. Th viscosity of Xanthangum at high shar rats (9 + m) gos to th viscosity of watr. Th lastic componnt of th viscosity is also lss than that of blood, just lik th viscous componnt of th viscosity, but th trnd is still followd. This mans that th non-nwtonian proprtis of blood ar ovrstimatd in th fluid usd, but th fluid can b usd to giv qualitativ information about th non-nwtonian flow. 6

9 2.4. Exprimntal st up A schmatic rprsntation of th xprimntal stup is givn in figur 4. constant bad tank constant had rank flox* sttling chambr masuiiná snsor rsrvoir Figur 4, schmatic rprsntatation of th xprimntal stup Th flow systm provids stady flow btwn two constant had tanks. An adjustabl valv is controlld by a stppnmotor and a PC, to obtain th rquird flowpuls. From th upstram tank th fluid passs a sttling chambr and a numbr of flowstraightnrs to nsur a stabl flow at h ntranc of th inlt tub. To b sur that th stady and unstady vlocity profils ar fully dvlopd at th masuring sction, th lngth of th inlt tub is approximatly 30D. D rprsnts th diamtr of th cornmon carotid artry, bing 40 mm. Th flow is masurd with lctromagntical flowmtrs (Skalar instrumnts transflow 601), placd aftr th valv and in th intrnal carotid artry. For that rason a littl salt is addd to th fluid. Th flow ratio btwn th intrnal and th xtrnal branch can b adjustd by mans of a tap in th xtrnal branch. Aftr th fluid passs th downstram tank it is collctd in a larg rsrvoir from which th upstram tank is rfilld again. Th platina cathod wirs ar placd in th masuring sction as shown in figur 5. intrnal carotid xtrnal carotid Figur 5,Position of th cathod wirs in th masuring sction Th hydrogn bubbls ar mad bttr visibl by illuminating th masuring sction from bhind. Th vlocity profils shown by th bubbls ar rcordd on a vido tap. 7

10 2.5. Exprimntal procdurs Th physiological flow conditions can b dfind by thr dimnsionlss numbrs, bing th Rynolds numbr (R), th Womrsly numbr (a) and th flow division ratio (y). Thy ar dfind as follows: With D rprsnting th diamtr of th common carotid artry (CCA), U th avrag vlocity in v th kinmatic viscosity ( for th viscosity of th non-nwtonian fluid is takri th viscosity at. 27c + 00, which is qual to th viscosity of th Nwtonian fluid), R= D/2, o = - T with T th priod tim of th flow puls and QE, Qc th flow through th xtrnal carotid artry and th common carotid artry. For visco-lastic flows not only th Rynolds numbr, but also th Dborah numbr (D) and th Wissnburg numbr (W) ar important. Th linar viscolastic bhavior of th non- Nwtonian fluid can b modlld with a Maxwll modl as follows: h&+ C= 2v0D To compar th flow of th non-nwtonian fluid in th carotid artry modl with th flow of biood in a human carotid artry th dimnsionlss form of th Maxwll quation is usd: D~$+c= = 2D* with o*, D" rsp. th dimnsionlss strss and dformation To scal th non-nwtonian xprimnts corrctly th Dborah numbrs and th Wissnbrg numbrs of blood and th Xanthangum solution hav to b qual. Thy ar dfind as h D=-, U W=h- D with h th charactristic tim of th fluid, 8 th charactristic tim of th flow, U th charactristic vlocity of th flow and D th diamtr of th artry. Th charactristic tim of th non-nwtonian fluid is [Bird t al] h=- "/ó with 77' and q" following from th viscomtr masurmnts (Figur 3). v' W-10 blood U 15 cds o 6.3 raas D 1 cm I tabl 1, Charzctristics of blood [Sharp] and xanthum giim Xanthangum 2.5 cds rad/s 4 cm 0.33 s 8

11 With th data in tabl 1 is found: Dblood = 0.38 and Wbiood = 0.9 D,, = and W,, = 0.2 If th Dborah numbr is high, th lastic forcs dominat and if th Dborah numbr is small th viscous forcs dominat. In this cas th Dborah numbr of blood is largr than th Dborah numbr of Xanthangum, thus th lastic forcs hav at th flow of blood in th human carotid artry bifurcation mor influnc than at th non-nwtonian flow in th carotid bifurcation modl usd in th xprimnts. During th xprimnts th flow division ratio y is kpt constant (y = 0.45). Th Rynolds niùmbï varis duïifig th flow priod: R = 390 during th diastol, R = 1000 at th pak of th systol. Th Womrsly paramtr is a = 6. Ths paramtrs ar in th xprimnts with watr th sam as in th xprimnts with th Xanthangum solution, to nsur th flow conditions ar th sam in th xprimnts. That way it is possibl to dtrmin th influnc of th non-nwtonian proprtis. Th flowpuls usd in th xprimnts is a simplification of th physiological flowpuls [Palmn (1994)l. Th flowpuls is shown in figur 6. I 0 ' tim [SI Figur 6, Flowpuls Th xprimnts ar don with th Nwtonian fluid and th non-nwtonian fluid. Th flow is masurd and th data is procssd with Labviw. Th pulss for th hydrogn bubbls gnration ar also gnratd by Labviw. At ach lctrod wir th vlocity profils ar rcordd with a camra during on priod. Th vlocity profils of th Nwtonian fluid ar compard with th profils of th non-nwtonian fluid. For th non-nwtonian xprimnts th polarity of th lctrods had to b switchd bcaus th hydrogn bubbls wr vry larg compard with th Nwtonian xprimnts and stickd to th wirs. Aftr switching th polarity th bubbls showd th vlocity profil vry good. Thr is no xplanation of this ffct. Picturs of th vidotap ar mad with TIM. Bcaus th vlocity profils wr not wll visibl at th picturs mad with TIM, th vlocity profils ar printd dirctly from th vido scrn, 9

12 a. Rsults Th vlocity profils ar a function of tim and will b dscribd for th Nwtonian and th non-nwtonian xprimnts. First th rsults of th Nwtonian xprimnts will b dscribd. Th lft hand sid of figur 7 rprsnt th visualization rsults for th Nwtonian fluid. Th picturs rprsnt svral phass of th flowpuls. At th nd of th diastolic phas (t/t = O) th vlocity profil is almost th sam as in th stady flow situation. Th vlocitis ar positiv ovr th ntir profil. At th pak of th systol (t/t = O. 10) th vlocitis ar positiv ovr th ntir diamtr. Th vlocity at th dividr wall is highr than at th non-dividr wall. Th shar rats ar high at th sid of th dividr wall and low at th non-dividr sid. During th dclration of th systol (t/t = 0.15) an ara of low shar rats occurs at th nondividr sid of th sinus. At t/t= O. 15 th high vlocitis at th dividr sid of th sinus bcom strongr and in th cntr of th sinus a shar layr occurs. At th non-dividr wall th axial vlocitis ar ngativ. At th nd of th systol (t/t= 0.20)in th low shar rgion th axial vlocitis bcom mor ngativ. Th shar lay: bccms mor pronouncd and shifts to th dividr sid. In th initial phas of th diastol ( t/t = 0.23) th shar layr bcoms instabl and downstram a small vortx occurs. In th cntral part of th low shar rgion small positiv vlocitis can b sn. Halfway th diastolic phas th shar layr bcoms stabl again and th vortx is downstram th sinus. Th right hand sid picturs in figur 7 shows th non-nwtonian situation. At som points th flow shows significant diffrncs with th Nwtonian rsults. Th stady stat of th non-nwtonian flow is mor uniform distributd ovr th lumn. At th pak of th systol th non-nwtonian flow profil shows littl diffrncs with th flow profil of th Nwtonian fluid. Ovr th ntir lumn th axial vlocitis ar positiv, at th dividr wall th vlocitis ar highr than th vlocitis nar th non-dividr wall. During th dclration phas of th systol th vlocitis in th low shar rgion ar vry small and th rgion of high vlocitis nar th dividr wall is small. During th dclration th vlocitis ar still positiv although small ovr th ntir lumn. Th flow is mor uniform ovr th lumn than during th Nwtonian situation. At th nd of th systol th diffrnc btwn th two fluids bcom mor vidnt. A rgion of rvrsd flow occurs nar th non-dividr wall. In th cntr of th diamtr th so calld shar layr is lss clar and thr is no onst of a vortx. Th shar layr shows a smallr shift towards th dividr wall during th nd of th systol and th initial phas of th diastol. At th rgion of rvrsd flow th flow is mor uniform. Th transition to th stady flow situation in th diastolic phas occurs in th first half of th diastolic phas. Although not shown in figur 7 th viscolastic flow in th communis diffrs from th Nwtonian flow. During th diastol th small rgions of rvrsd flow nar th walls ar not prsnt in th non-nwtonian flow. 10

13 6 t/t= O t/t = 0.10 t/t = 0.15 Figur 7, Visualization rsults of th Nwtonian fluid (lft) and th non-nwtonian fluid (right) in th carotid bifurcation modl 11

14 t/t = 0.2 t/t = 0.23 Figur 7, continud 4. Discussion and conclusion Th dominant forc on th hydrogn bubbls in th fluid is th drag forc du to th local fluid motion. Du to th buoyancy forcs th hydrogn bubbls hav also a ris vlocity. This vlocity is dpndnt of th dnsity of th particl, th dnsity of th fluid and th dynamic viscosity of th fluid, xprssd by Straub t al (1965) as (Pf - Pp)g u. = m 18v with pp and pf rprsnting th dnsity of th particl and th fluid rspctivly, Dp th diamtr of th particl (stimatd smallr than 100 pm), g th gravitational acclration and v th dynamic viscosity of th fluid ( v N ~ = ~ ~ Pa.s, Vnon-NwF 2~~ Pas). Substituting ths 12

15 valus, it follows that uns = 5 mms-' for th Nwtonian fluid and uns = 2.5 mms-' for th non- Nwtonian fluid. This ris vlocity causs an rror in th axial vlocitis found in th xprimnts.th ris vlocity of th bubbls in th non-nwtonian fluid is smallr than in th Nwtonian fluid. Th rror in th axial vlocity du to th ris vlocity is lss in th non- Nwtonian fluid. Th influnc of th buoyancy forcs is gratr at low vlocitis as occur in th low shar rgion during th diastolic phas. This mans that thr is a diffrnc btwn th motion of th hydrogn bubbls and th local fluid motion. Dspit of this rror th rsults of th visualizations still giv qualitativ information about th larg scal flow phnomna. Sinc th flow conditions ar th sam for both xprimnts (th f?ow division rztic is constant) it is possibl to compar th xprimnts and dtrmin th influnc of th viscolastic non-nwtonian proprtis. In th Nwtonian visualizations vortx formations in th sinus occurs in th initial phas of th diastol. In th dclration phas of th systol an ara of rvrsd flow can b sn which dvlops furthr during th rst of th systol. This rsult shows rsmblanc's with th rsults of th visualization xprimnts of Ku t al (1983). Also othr xampls of vortx formation in th sinus ar found in th litratur (Palmn, 1994). Th non-nwtonian visualizations show no vortx formation in th dclration phas. Th vlocityprofils ar mor uniform distributd ovr th lumn and th shar rats at t nondividr wall ar highr during th diastolic phas. This corrsponds with th rsults of Zuidrvaart (1995). This rsult is in contradiction with th litratur (.g. Frktold, 1991). Th hydrogn bubbl mthod only visualizs th flow in th plan of symmtry. Th scondary flow in th plan of symmtry opposs th influnc of th buoyancy forcs. Th visualization with th hydrogn bubbls will giv qualitativ information about th flow in th plan of symmtry, dspit th fact that th flow in th carotid sinus has a hlical structur (B haradvaj, 1 982). It can b concludd that thr is a significant diffrnc btwn th flow of a Nwtonian fluid and a non-nwtonian fluid through a bifurcation. ' Sinc th lastic proprtis of th non-nwtonian fluid ar ovrstimatd, it is ncssary to do mor rsarch on th influnc of th non-nwtonian proprtis of blood on th flow through th bifurcation. 13

16 5. Rfrncs Bharadvaj B.K., Mabon R.F., Giddns D.P. (1982). Stady flow in a modl of th human carotid artry bifurcation, J. Biomchanics, vol. 15, Bird R.B., Armstror,g R.C. md Eassagr O. (1987). Dynmics of p ~ l y liquids ~ ~ r 1 Coklt G.R.( 1980). Rhology and hmodynamics. Ann. Rv Physiol42: Ku D.N., Giddns D.P., Zarin C.K. and Glagov S. (1985). Pulsatil flow and athrosclrosis in th human carotid bifurcation, Athrosclrosis 5, Lipsch D. And Moravc S. (1984). Pulsatil flow of non-nwtonian fluid in distnsibl modls of human artris, Biorhology 2 1, Mann D.E. and Tarbll J.M. (1990). Flow of non-nwtonian blood analog fluids in rigid curvd and straight artry modls, Biorhology 27, Palmn D.E.M., van d Voss F.N, Janssn J.D. and van Dongn M.E.H. (1 994). Analysis of th flow in stnosd carotid artry bifurcation modls-hydrogn-bubbl visualization, J. Biomchanics Vol. 27, Palmn D.E.M. (1994). Th influnc of minor stnoss on carotid artry flow, Ph.D. Thsis Eindhovn Univrsity of Tchnology Prktold K., Rsch M., Florian H. (1991). Pulsatil non-nwtonian charactristics in a thr-dimnsional human carotid bifurcation modl, Journal of Biomchanical Enginring Vol. 113, Rindt C.C.M. (1989). Analysis of th thr dimnsional flow fild in th carotid artry bifurcation, Ph.D. Thsis Eindhovn Univrsity of Tchnology Schraub F.A., Klin S.J., Hnry J., Rundstadlr P.W. jr., Littll A. (1965). Us of hydrogn bubbls for quantitativ dtrmination of tim-dpndnt vlocity filds in low-spd watr flows, ASME J. Basic Enginring 87, Sharp M.K. (1994). Th ffct of blood viscosity on pulsatil flow in straight tubs. Thurston G.B. (1 979) Rhological paramtrs for th viscosity, viscolasticity and thixotropy of blood. Zuidrvaart J. (1995). Influnc of viscolastic fluid bhavior on th stady flow fild in a thr dimnsional modl of th human carotid artry bifurcation. 14