Velocity of Blood flow through uniform rigid artery of human body

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1 IOS Joual of athmatics (IOS-J) -ISSN: ISSN: X. Volum 3 Issu 3 V. IV (ay - Ju 7) PP Vlocity of loo flow though uifom igi aty of huma boy D.Amajyoti Goswami* Safa Ahm* D.Kisha Goal Sigha# *Kaiaga ivaity Assam Iia #Kaaga Gils H.S.S. Assam Iia. Abstact: A attmt is ma i this aalysis to stuy th flow of bloo though a uifom igi aty with a axial vlocity sli coitio at vssl wall has b cosi by assumig bloo a ow flui. It has b obsv that th ffct of th Hatma umb a th yols umb o th vlocity fil as wll as o th wall sha stss is vy omit a wh th Hatma umb a yols umb icass th flui vlocity cass. It also iclus Poisuill flui mols of bloo fo both sli a o-sli at a wall a olay ow flui mol a o sli at tub wall as its scial cass. Th ol of sli i iflucig th flow vaiabls a hysiological imlicatios of this thotical molig a iscuss i bif. Kywos: Pow flui mol stos vssls agtic ffct yols umb Hatma umb loo flow. I. Itouctio Huma bloo is a sussio of clls i a cotiuous a aquous substac call lasma (oy [5]).Plasma bhavs as Nwtoia flui whas whol bloo shows o-nwtoia chaactistic (Fug [9]) cdoal [3] has oit out that i a cas of bloo vssls with iamt abov 5 micomts bloo may b cosi as a Nwtoia flui.stosis is fom by substac ositig o vssl walls. Stosis is fom by substac ositig o vssl walls. A stosis may la to atial o total vssl blockag i som istacs a thfo oss a sious mical oblm. I a huma boy hysiological fluis st i huma systms iclu igstiv juics swatig bloo saliva ui tc. Amog th boy fluis th most imotat is obviously bloo which is ga as a sussio of ifft clls i a cotiuous aquous solutio call lasma (oy [5]). Plasma bhavs as Nwtoia flui whas whol bloo shows o- Nwtoia chaactistics (Fug [9]) cdoal [3] has oit out. ut at low sha ats bloo xhibits o- Nwtoia bhavio.it is wll kow that o-nwtoia atu of bloo sigificatly iflucs th flows aticulaly i th cass wh bloo vssls a cuv bachig o aow tc a xlai by.ustafa.ahma Kha[4].ustafa T.HayatI.Po a A. Ai also stui ustay bouay lay flow of a casso flui u to a imulsiv stat movig flat lat[]. Futh bloo has a fiit yil stss a asso s quatio ca tak ca of this oty which has b ot by may ivstigatos (oklt t al. [8] il t al. [5] huya. a Haaika []). Owig to th itmittt umig of hat muscls it oucs a ssu gait a as a cosquc bloo flows though th ciculatoy systms (Su V.K [7] atho V.P [8]). Atial systm is also subjct to cotiuous sha stss chags. O of th ciculatoy isass is athosclosis which has u to lii o lim osits (athoma) o th i coat of th atis. About 75% of all aths i th iustiali wol a caus u to ciculatoy isas. Th accumulatio of lii fat a oth aticls alog th isi aty wall la to aowig of th vssl. I som cass it is obsv that a so-call stosis of th vssl with ag that mas lastic fibs i vssl wall a lac by mo igi collag fibs a this ocss is accomai by th wiig of th atial lum (alik []). Also th isas aas of bloo vssls a chaacti by ahsivss haig loss of lasticity a cotactio i th coss-sctio fou omiatly i th viciity of bas bifucatios a oth lacs. As a cosquc th may ais sious comlicatios a isos i bloo suly u to aowig i th bo of th vssls o wiig of th i wall. To sto omal bloo suly i isas vssls it sms is a itouctio of vlocity sli at tub wall will b maigful (Nuba [6] u [7] iswas a Nath [6]). I o to stuy th hology of bloo alog with its ol i th fuamtal ustaig of may caiovascula isass lik myocaial ifactio stok thombosis sickl cll isass tc. I thi mols flow has b cosi stay o ulsatil flui as a Nwtoia o a th bouay coitio o o-sli at wall tc. As bloo shows som viatios fom Nwtoia bhavio it sms cosiatio of bloo i bhavig lik a o-nwtoia flui alog with sli at vssl wall tc as is o h will b aoiat a sigificat fom th hysiological oit of viw. Th alicatio of magto hyoyamics icils i mici giig is of gowig itst.e.ali[]. ay ivstigatios hav b ma o bloo flow u th ffct of magtic fil. aothy [] has show that by th alicatio of a xtal magtic fil th biological systms a gatly affct. Vaaya [9] show DOI:.979/ Pag

2 Vlocity of loo flow though uifom igi aty of huma boy DOI:.979/ Pag that th alicatio of magtic fil ucs th flow of bloo. huya a Haaika [] hav ivstigat th oblm of bloo flow with ffcts of sli i atial stosis u to sc of tasvs magtic fil a thy hav also obsv that th ali magtic fil ucs th vlocity of bloo though atis. athmatical Fomulatio Of Th Poblm A two-flui mol fo bloo flow i a uifom tub has b vlo i th st oblm. Th mol basically cosists of a co of a cll sussio i th mil lay a th ihal lasma i th out lay. Th stay lamia flow of a icomssibl flui assumig it to bhav as a ow flui (Fug [9] huya. Haaika [][3][4]) though a igi cicula tub is cosi. Th axial cooiat a vlocity a ẑ a v sctivly. is th ali magtic fil i ictio. Flow is gov by th cotiuity a Navi-Stoks quatios a i aitio ow costitutiv quatio th magtic fil ali i a ictio th quatios i th axial a aial ictios i imsiolss fom. Fom th ot of Youg [] a Su [7] cosiig th axisymmtic lamia stay flow of bloo th gal costitutiv quatio i th cas of mil stosis subjct to th aitio may thfo b witt as: () u () Fom which w obsv that ssu os ot vay i th aial ( cicumftial ( ) a axial ictio a that ssu mai costat acoss ay coss-sctio of th tub a Is a fuctio of th oly that is a so ssu gait tm i th last quatio abov bcoms Th () v (3) No-imsioal fom V P P v (4) SOLVING THE EQATION (4): k v v Wh K Solutio: Lt D 4 V (5) Also sha stss comot at ay istac fom th tub axis is giv by

3 Vlocity of loo flow though uifom igi aty of huma boy v v (6) Exss fo wall sha stss ca b obtai fom th fomula ( ) (7) w w sig Equatio (6) a c y I btw Y a ( c ) y xss fo will la to th fom w th may ais two cass wall sha stss is gat a that yil stss. I cas Y w that is if th th will occu o flow accoigly vlocity fuctio will bcom w w w Agai ow flui th costitutiv quatio may b ouc i th followig fom k y K is a costat a is th scala stai at fo th flow th stai at ca b xss as = wh is th axial vlocity v k Itgat with sct to w w( ) c w wc ( ) c c Th volum flux Q of th flow though th i is giv by Q w II. sults A Discussio Th objctiv of this aalysis is to stuy th flow chaactistics of a ow flui mol fo bloo flow with vlocity sli i sc of magtic ffct. Th oblm is solv umically usig Shootig mtho. Numical calculatios hav b o fo vaious cosiatios of aamts i.. th Hatma Numb a th yols umb...numical sults a show gahically. It has b obsv that th ffct of th Hatma umb a th yols umb o th vlocity fil as wll as o th wall sha stss is vy omit. If sha stss is gat tha yil stss y th a flow of bloo is ossibl othwis th will b o bloo flow. Fig () shows that th atu of vlocity ofil is sam if th is o sli. Fig () Illustat that th vlocity ofils with a ffct of a magtic fil fo vaious valus of yols umb. It is s that th vlocity ofil cass as th yols umb icass with sli. Fig (3).Illustat that th vlocity ofil v i atu is ot chag if o sli. Th aalysis vlo h is bas o ctai assumtios which may la to som hysiological imlicatios vi. (a) Flow is assum stay which is i tu fo vy thi atis i VS wh th ulsatility ffcts a small. (b) Th assumtio that DOI:.979/ Pag

4 Vlocity of loo flow though uifom igi aty of huma boy vlocity vaiatio i a axial ictio is gligibl as coma to its vaiatio i aial ictio may la to th imlicatio that th lgth of th aty is too lag as coma to th aius. Fig-4 Illustat that th vlocity ofil v cass with icass of Hatma umb i sc of sli. It is also obsv that Hatma Numb icass th ffct of magtic fil is cass DOI:.979/ Pag

5 Vlocity of loo flow though uifom igi aty of huma boy fcs []. aothy.f. (.) iological ffcts of agtic fil Vol-I & II Plum Pss (969). []. huya. a Haaika G. io-scic sach ullti Vol.7() PP.5-(). [3]. huya. a Haaika. G. oc. Natioal cofc o ali athmatics -7(). [4]. huya.. a Haaika G.. loo flow i chals of vayig coss sctio with mabl bouais i sc of tasvs of magtic fil io scic sach ullti Vol. 8 (No. ) () [5]. oy W. Txt ook of Pathology: Stuctu a Fuctios i Disass La a Fbiga Philalhia (963). [6]. iswas D a Nath J. Oscillatig loo flow though a uifom Aty with wall sli Joual of Assam ivsity 4(). 4-54(999). [7]. u P. Th Vlocity Sli of Pola Fluis hol. Acta (975). [8]. ocklt G.. Th hology of huma bloo iomchaics. Y.. Fug. 63. (97). [9]. Fug Y.. iomchaics: chaical Potis of Livig Tissus Sig-Vlag Nw Yok Ic. (98). []..E. Ali N. Sa attao-histov mol fo aioactiv hat tasf of magto hyoyamics asso-fo flui: A umical stuy s.hy. 7(7) -3. []..ustafa T.Hayat I.Po A.Ai stay bouay lay flow of a asso flui u to a imulsivly stat movig flat lat Hat Tasf. Asia s4() []..Y. alik. Nas S. Nam Abul hma Th bouay lay flow of assomooflui ov a vtical xotial sttchig cyli Al. Naosci.4(4). [3]. acdoal D.A. O Stay Flow Though oll Vascula Stoss J. iomchaics. 3- (979) [4].. ustafa J. Ahma Kha ol fo flow of cassoaoflui ast a o-lialysttchig sht cosiig magtic fil ffcts AIP Av. 5 (5). [5]. ill F.W. hology of Huma loo a Som Sculatios o its ol i Vascula Homostatics iomchaical chaisms i Vascula Homostatics a Itavascula Thombosis. P.N. Sawy Alto tuy ofts Nw Yok (965). [6]. Nuba Y. loo Flow Sli a Viscomty iohys. J (97). [7]. Su V.K. a Skho G.S. Phy.. iol. Vol (989). [8]. atho V.P. Gayati Ali Scic ioical vol.() PP 5-58(). [9]. Vaaya V.A. iohysics Vol. 8 (3) (973) []. YougD.F J.Egg I.Tas.Am.Soc.ch.Egs.Vol (968) DOI:.979/ Pag