Pulsatile Flow of Hershel- Bulkley Fluid in Tapered Blood Vessels

Transcription

1 Pulaile Flo of Hehel- Bulley Fluid i Tapeed Blood Veel R. Poalaguamy Depame of Mahemaic Naioal Iiue of Techology Tiuchiappalli-6 5 Tamil Nadu Idia. palagu@i.edu Abac The pee pape hed ome ligh o iveigaig he effec of pulailiy o flo hough a apeed aey. Blood ha bee epeeed by a o- Neoia fluid obeyig Hechel-Bulley equaio. ig he Reyold umbe a he peubaio paamee a peubaio echique i adoped o olve he eulig quai-eady o-liea coupled implici yem of diffeeial equaio. Aalyical expeio fo velociy volumeic flo ae all hea e ad he mea flo eiace have bee obaied. I i obeved ha he all hea e ad flo eiace iceae fo iceaig value of ape agle ad he axial diace. I i peie o poi ou ha he phae lag beee he peue gadie ad all hea e o flo ae i foud o be. degee ad hi phae lag become idepede of he axial diace ad ape agle. The pee appoach ha; i geeal validiy i compaio ih may mahemaical model developed by ohe ad may be applied o ay mahemaical model by aig io accou of ay ype of heological popey of blood. Fially ome bioheological applicaio of he pee model have biefly bee dicued. ha blood ho he behavio chaaceiic of a combiaio Bigham plaic ad Peudoplaic fluid-hechel- Bulley fluid ih he fluid behavio idex geae ha uiy. I vie of he expeimeal obevaio [] ad uggeio made i [9] i i peie o coide he behavio of blood a a Hechel- Bulley fluid. Baed o he foegoig vie i i ohhile o decibe a model aig he faco of pulailiy o-uifom co-ecio of a ube ad o-neoia chaace io he pee aalyi ad udy he flo chaaceiic. II. FORMLATION OF THE PROBLEM Coide a lamia pulaile ad fully developed flo of blood Hechel-Bulley fluid i he diecio hough a lighly apeed ube a ho i Fig.. Keyod o-eoia fluid; pulaile flo apeed aey; all hea e; eiace o flo I. INTRODCTION I i ell o ha he iveigaio o blood flo i apeed aeie ad ube ih o-uifom co-ecio could play a impoa ole i he fudameal udeadig diagoi ad eame of may cadiovacula dieae ube [][][]. Looig a he immee impoace i he fudameal udeadig of blood flo he obecive of he pee aalyi i moivaed o povide a geealied model of blood ad obai ome ifomaio abou he flo. May iveigao[4-8] have udied he flo of blood hough apeed aeie by eaig blood a a Neoia fluid Bigham plaic fluid poe-la fluid ad Cao fluid ad obaied he elaiohip beee he flo ae ad peue dop. co Blai e al. [9] have uggeed ha blood obey Cao model oly fo modeae hea ae flo ad ha hee i o diffeece beee Cao ad Hechel-Bulley plo ove he age hee Cao plo i valid fo blood. Fuhemoe ac e al. [] have expeimeally poied ou Fig.. Geomey of Tapeed Tube The all pofile of he flo geomey may mahemaically be decibed a R R a hee R i he adiu of ube i he apeed ecio R i he adiu of he ube i he omal egio i he L axial diecio ad i he agle of ape. Fuhe epee he axial diace of he co ecio beee = ad he coe apex ad L idicae he axial diace of

2 ay co ecio a fom he apex. We hall ae cylidical coodiae yem hoe oigi i locaed o he ube axi. ove a lee deoe he dimeioal fom of he coepodig quaiy. Le u ioduce he folloig o-dimeioal vaiable: hee vicoiy by he elaio velociy R R y R T R u u T i he chaaceiic ime he Neoia he chaaceiic velociy hich i expeed R = he ime T u he axial compoe of he he adial diecio y he yield i he hea e ad i he dimeiole yield e e. Baed o he dicuio made by Oa [7] Oa ad Muaa [8] ad Poalaguamy [] he adial velociy i egligibly mall ad ca be egleced fo a lo Reyold umbe flo hough a apeed ube ih a agle of ape up o. Keepig hee i vie ad uig o-dimeioal vaiable he momeum equaio goveig he flo ae give a u q f Whee u if R R R q q q q f=+a i i he coiecy idex of blood if 4 5 i he poe la idex q i he peue gadie ad q i he coa peue gadie i he omal ube egio. Equaio 4 ad 5 ae educed o ha fo a Bigham fluid he =. o ha fo a Poe-la fluid he =. ad o ha fo a Neoia fluid he =. ad =.. Hee = R i he ouhal umbe hee i he fequecy of he ocillaio of he flo. The dimeiole paamee i he Reyold umbe of he flo. Taig chaaceiic velociy i expeed a R q 6 Coiecy he equie ha he ime cale be choe a T 7 qr The geomey of he apeed ube i dimeiole fom i give by R a 8 The bouday codiio i dimeiole fom ae: i i fiie a = ad ii u= a = R 9 The volumeic flo ae Qi give by: R Q= u d A meioed elehee i[] e ae R ad L L III. OLTION The flo vaiable G i aumed o poe he folloig fom G G G O Whee G efe he velociy ad hea e. I ha follo fo coveiece e ie oly fucio oaio deleig i vaiable. I i of iee o oe ha he Womeley umbe i obaied a hee R i he iemaic vicoiy. ubiuig equaio 4-5 io equaio ad iegaig ice ih he help of he bouday codiio 9

3 he aalyic expeio fo velociy diibuio may be obaied a ]] log 6 } { } / { [. } / co { / [ c f A u ig equaio ad afe ediou maipulaio he aalyic expeio fo flo ae i obaied a ] }] { } 6 { [ co } { [ 4 c f A Q 4

4 hee /{ } The eady flo ae Q i expeed a Q q { } 4 { } { }{ } q q q The hea e o he all i phyiologically impoa quaiy ad i give by 5 A co { } f I i of impoace o pi poi ou hee ha Chauai ad Poalagaamy [] have explaied he mehod of calculaig he value of eady peue gadie q fo ay value of uig equaio 5. The flo eiace i defied a hee f p 7 Q p i he peue dop. hee λ φ θq φ Q φ log φ / d d Q ad Q Q Q 9 The mea flo eiace ove peiod of he flo cycle i defied a p p Q f d 8 I i udeood ha fo ay value of oe ca umeically compue he value of he flo eiace ad he mea flo eiace epecively fom equaio 7 ad 8 fo diffee value of he paamee ivolved i he pee o. I i mahemaically ad phyiologically impoa o obai a aalyic expeio fo he mea flo eiace. I i ee fom equaio 4 ad 8 ha i i o poible o obai he aalyic expeio fo he mea flo eiace fo ay value of. Fo mall value of he fluid havig lo yield e value ; fo example blood [] he aalyic expeio fo he mea flo eiace fom equaio 4 ad 8 may be obaied a The value of Q ad Q have o be umeically compued afe compuig he value of eady peue gadie q uig equaio 5 fo diffee value of he paamee ivolved i he pee o. The flo eiace fo he eady flo of Hechel- Bulley fluid hough a apeed aey ca be obaied fom Eq. 9 a log Q Q / The eady peue gadie q ad all hea e epecively may be obaied a

5 q Q / { } Q { } / IV. DICION / A caeful obevaio of he o doe by Aoey ad Go [-4] aa ad Hemalaha [5-6] ad Deh e al. [7] eveal ha hey have applied adad peubaio echique ad poduced a appoximae oluio i hich he flo chaaceiic ae expeed a aympoic epeeaio i poe of he Womeley umbe [8 ]. Rohlf ad Tei [9] have ecely agued ha he ue of he Womeley umbe a a peubaio paamee o obai appoximae oluio of he pulaile flo of o-neoia fluid i o appopiae ad coideig he Reyold umbe a a peubaio paamee hey have made a aemp o validae hei eul hough hei peubaio heoy i compaio ih a umeical iegaio of he full mahemaical model. I i fuhe oiced ha he Womeley umbe i o depede o he flo velociy ad hu he ame value of ca epee valy diffee flo codiio ad hece he Womeley umbe i o a uiable peubaio paamee. aa ad Hemalaha [5-6] ad aa[] have aalyed he pulaile flo of Hechel-Bulley fluid hough aeie. I peie o poi ou hee ha he aalyic expeio fo flo vaiable uch a velociy all hea e ad flo ae obaied by aa ad Hemalaha [5-6] ad aa[] ad by he pee iveigaio epecively ae eiely diffee. The eao i aibued o he fac ha hey have egleced he highe ode em i he biomial expaio of he elaiohip beee he velociy gadie ad he hea e ivolved he coiuive equaio of Hechel-Bulley fluid. Hece he eul obaied by aa ad Hemalaha [5-6] ad aa[] ae o epeeig he acual behavio of bioheological flo chaaceiic. Alo hey have o deived he aalyic expeio fo flo eiace. I i of iee o oe fom Fig. ha he flo ae deceae ih he iceae i he axial diace Z ad ape agle. The peceage of deceae i he flo ae a he value of Z iceae i foud o be highe fo highe value of ape agle. Fig.. Vaiaio of flo ae ih axial diace fo diffee value of ape agle The vaiaio of all hea e ih he axial diace ad he ape agle ha bee udied ad illuaed i Fig.. I i clea fom Fig. ha a he ape agle iceae he ae of iceae i he all hea e ih epec o a iceae i he axial diace Z i foud o be vey much igifica. Aohe impoa eul i coceig he vaiaio of mea flo eiace ih epec o he ape fo diffee value of he axial diace Z ad i i ho i Fig.4. The mea flo eiace iceae a he value of he ape agle iceae. The mai effec of pulailiy o he flo i he phae lag beee peue gadie ad flo ae ad all hea e. I i oiced fom ha he phae lag beee peue gadie ad flo aeo all hea e ha bee foud o be. degee ad i value i ualeed hile he iceae o deceae i he value of axial diace ad ape agle. I may be of impoace o oe ha may adad eul egadig eady ad uifom ube flo of Poe-la Bigham ad Neoia fluid ca be obaied a pecial cae of he pee iveigaio. Fig.. Vaiaio of all hea e ih axial diace fo diffee value of ape agle

6 adial all moio a compaed o ha of igid aey. Keepig i vie of impoace of hemodyamic vicoelaic popey of a aey ad heological faco i udeadig of blood flo ad aeiocleoic dieae a mode effo ha o be made o iveigae he flo of blood hough a apeed aey by aig io accou of vicoelaic popeie of blood ad i veel all i addiio o he o-neoia behavio of blood ad pulaile flo effec coideed i he pee mahemaical aalyi hich fom he fuue eeach o. REFERENCE Fig.4. Vaiaio of mea flo eiace ih axial diace fo diffee value of ape agle V. CONCLION The pee iveigaio deal ih he poblem of pulaile flo of blood Hechel-Bulley fluid hough a aey ih mild eoi. A peubaio echique i adoped o udy he flo. The aalyical expeio fo velociy flo ae all hea e ad mea flo eiace have bee obaied. The eul have bee depiced hough gaph. ig he fiie volume echique he quai-eady o-liea coupled implici yem of diffeeial equaio ha umeically bee olved ad he axial velociy i compued. I i veified ha he eo beee he axial velociie obaied by he pee peubaio mehod ad he umeical echique become le ha.5% fo he value of lyig beee. ad..fuhe he eo beee hem become moe ha 9.% he ae he value geae ha.. I i peie o meio hee ha he he flo chaaceiic ae expeed i em of he pee peubaio eul coicide ih he eul foud i he pape [5-6]. Hece ou pedicio coicide ih hei ad fuhe compaio i eedle a log a <.. Fuhe hei pedicio ae valid he he Reyold umbe i mallle ha ice he ouhal umbe i uiy i hei aalye. Bu ou appoach i applicable eve o lage blood veel ad modeae Reyold umbe. Oe of he mo emaable mei of he pee peubaio cheme i ha i i vey much uiable o ay mahemaical model of blood flo i ube ih uifom ad o-uifom co ecio i compaio ih he model developed by ohe [-7 ]. Fy [] ha meioed ha hemodyamic faco example; all hea e flo eiace play a ey ole i he developme ad pogeio of aeial dieae. Cao e al [] have expeimeally demoaed ha duig he iiial age of aeial dieae hee may be a impoa iecoelaio beee aheogeei ad deailed chaaceiic of flo of blood hough he damaged o dieaed o affeced aey. Zami[] ha poied ou ha he aey all ae vicoelaic % vaiaio i he aey adiu ove a cadiac cycle have ypically bee obeved ad he all hea e i foud o be pimaily affeced by he [] Divedi A.P. Pal T.. ad Raeh L. Micopola fluid model fo blood flo hough a mall apeed ube Idia J. Techology pp [] Chauai P. ad Palhad R. Blood flo i apeed ube ih bioheological applicaio Bioheology pp [] Poalagaamy R. ad Kaahaa M. A fiie eleme aalyi of ueady flo of vicoelaic fluid hough chael ih o-uifom co-ecio. I. J. Numeical Mehod of Fluid 9 pp [4] Chaavahy. ad Madal P.K. To dimeioal blood flo hough apeed aeie ude eoic codiio I. J. No-Liea Mech. 5 pp [5] Ho T.V. ad Blac R.A. Peue loe i o-neoia flo hough igid all apeed ube Bioheology 4 pp [6] Madal P.K. A ueady aalyi of o-neoia blod flo hough apeed aeie ih eoi I. J. No-Liea Mech. 4 pp [7] Oa. Peue developme i a o-neoia flo hough a apeed ube Bioheology pp [8] Oa. ad Muaa T. Theoy of he eady lo moio of o- Neoia fluid hough a apeed ube Jp. J. Appl. Phy. 8 pp [9] co Blai G.W. ad pae D.C. A Ioducio o Bioheology Elevie cieific Publihig CompayAmedam Oxfod pp [] ac A.H. Rama K.R. Buell JA ad Tice E.G. Auculaoy Veu Diec Peue Meaueme fo Neoia Fluid ad fo Blood i imulaed Aeie VIDYA Repo #9 Dec. 96. [] Poalaguamy R. Blood Flo Though eoed Tube. Ph.D. Thei IIT Bombay Idia 986. [] Chauai P. ad Poalagaamy R. Pulaile Flo of Cao Fluid Though eoed Aeie ih Applicaio o Blood Flo Bioheology pp [] AoeyJ. ad GoJ.F. The Mechaic of Pulaile Flo i mall Veel-I Cao Theoy Micovac. Reeach 4 pp a. [4] AoeyJ. ad GoJ.F. Pulaile Flo i mall Blood Veel-I Cao Theoy Bioheology 9 pp b. [5] haa D.. ad HemalahaK. Pulaile flo of Hechel-Bulley fluid hough eoed aeie - A mahemaical model I. J. No Liea Mech. 4 pp [6] haa D.. ad HemalahaK. Pulaile flo of Hechel-Bulley fluid hough caheeied aeie- A mahemaical model Applied Mahemaical Modellig pp [7] Dah R.K. Jayaama G. ad Meha K.N. Flo i a caheeied cuved aey ih eoi J.Biomechaic pp [8] WomeleyJ.R. Mehod fo he Calculaio of Velociy Rae of Flo ad Vicou Dag i he Aeie Whe he Peue Gadie i Ko J. Phyiol. 7 pp [9] RollfK. ad Tei G. The Role of he Womeley Numbe i Pulaile Blood Flo: A Theoeical udy of he Cao Model J. Biomechaic 4 pp

7 [] aa D.. To-phae No-Liea Model fo Blood Flo i Aymmeic ad Axiymmeic eoed aeie I. J. No Liea Mech. 46 pp [] Fy D.L. Repoe of he Aeial Wall o Ceai Phyical Faco: I Aheogeei: Iiiaig Faco Ciba Foudaio ymp. pp [] Cao C.G. Figeald J.M. ad choe R.C. Aheoma ad Aeial Wall: Obevaio Coelaio ad Popoal of a hea Depede Ma Tafe Mechaim of Aheogeei Poc. Roy. oc. Lod. B 77 pp [] Zami A. The phyic of pulaile flo pige-velag Ne Yo.