A Mathematical Model of Two Layered Pulsatile Blood Flow through Stenosed Arteries

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1 A Mathematical Model of Two Layered Pulatile lood Flow through Stenoed Arterie S. U. Siddiqui, S.. Shah, Geeta Deartment of Mathematic, Harcourt utler Technological Intitute, Kanur-8, India Abtract In the reent aer a two fluid model for ulatile blood flow through tenoed artery ha been develoed. The model conit of a core region of uenion of all the erythrocyte aumed to be a ingham latic fluid and a eriheral layer of lama a ewtonian fluid. The analytic exreion for blood flow characteritic namely velocity, wall hear tre, flow rate, lug core radiu and effective vicoity are obtained by uing erturbation method. The effect of body acceleration and li velocity have been dicued. We have hown the variation of flow variable with the different arameter and dicued with the hel of grah. It i oberved that the velocity and flow rate increae but effective vicoity decreae, due to a li velocity. ody acceleration further enhance the velocity and flow rate. Keyword: Sli velocity; ody acceleration; eriheral lama layer; ingham latic. Introduction The behavior of blood flow i highly affected by arterial tenoi. Due to the reence of arterial tenoi normal blood flow i diturbed. Partially occluion of blood veel due to deoition of liid, uch a choleterol commonly known a tenoi, i one of the mot frequently occurring dieae in cardiovacular ytem of the human. The ewtonian behavior may be true in larger arterie, but blood being a uenion of cell in lama exhibit a non-ewtonian behavior at the low hear rate in mall arterie. umerou invetigator have cited hydrodynamic factor laying an imortant role in the formation of tenoi and hence the mathematical modeling of blood flow through a tenoed tube i very imortant. Many author have dealt with thi roblem, treating blood a a ewtonian fluid and auming the flow to be teady. Several author have tudied the ulatile nature of blood flow conidering blood a a ewtonian fluid and non-ewtonian fluid alo. They found that under normal condition, blood flow in the human circulatory ytem deend uon the uming action of heart. The heart um roduce the reure gradient throughout the arterial and venou network. Thi reure gradient conit of two comonent, one of which i contant or non fluctuating and other fluctuating or ulatile. [-], have tudied the ulatile flow of non ewtonian fluid in tenoed artery by conidering blood a Caon fluid and Herchel ulkley fluid. Chaturani & Sumy [] have invetigated the effect of non-ewtonian nature of blood and ulatality on flow through a tenoed tube. Along with the ulatality, ody acceleration i an imortant factor in the tudy of blood flow. It lay a very imortant role in the normal a well a dieaed arterial ytem. Shahed [4] tudied the ulatile flow of blood through a tenoed orou htt://e-jt.teiath.gr 7

2 medium under eriodic body acceleration. They reent a model of blood flow in a artially occluded tube ubjected to both the ulatile reure gradient due to the normal heart action and eriodic body acceleration. Cloed form exreion have been obtained for the axial velocity, flow rate, fluid acceleration and hear tre. Sud and Sekhon [7] have tudied the ulatile flow of blood through rigid circular tube ubject to body acceleration, treating blood a a ewtonian fluid. Under excetional circumtance, however human may alo be ubject to whole body acceleration (or vibration) For eg. while riding in a vehicle or while flying in an aircraft or acecraft, man may unintentionally be ubjected to acceleration. Though human body can adat to change but rolonged exoure to high level unintended external acceleration may caue eriou or even fatal ituation on account of diturbance in blood flow. Headache, abdominal ain, increae in ule rate, lo of viion, venou ooling of blood in the extremitie and hemorrhage in the face, neck, eye ocket, lung and brain are ome of the ymtom which reult from rolonged acceleration. Variou invetigator [7, 9, ] have analyzed the effect of body acceleration on ulatile flow of caon fluid through mild tenoed artery. They tudied the effect of ulatality and body acceleration through tenoed artery treating blood a caon and H- fluid. Several author carried out the role of li velocity in the blood flow through tenoed arterie and uggeted the reence of red blood cell occurring in the li condition at the veel wall. [5,8] have develoed mathematical model for blood flow through tenoed arterial egment, by taking a velocity li condition at the contricted wall. Pulatile flow of blood through a catheterized artery in the reence of different geometry of tenoi with a velocity li at the tenotic wall ha been invetigated by [, 8]. In all the above mentioned tudie, only ingle layered blood flow ha been conidered but ome theoretical and exerimental analyi were erformed to tudy the two layered blood flow under different hyiological condition. Srivatava, atogi & Vihnoi [6] have conidered the two layered uenion of blood flow through overlaing tenoi. [4, 9, ] have conidered two-hae arterial blood flow through a comoite tenoi. Some author alo conidered the two layered flow with the effect of magnetic field. They tudied the two fluid nonlinear mathematical model for ulatile blood flow through tenoed arterie. He tudied the ulatile flow of a two fluid model for blood through tenoed narrow arterie at the low hear rate, auming the uenion of all the erythrocyte in the core region of blood veel at a Caon fluid and lama in the eriheral layer a a ewtonian flow. In the reent model we have conidered the two layered blood flow through tenoed artery. With the above motivation an attemt ha been made to tudy the effect of li velocity and body acceleration on the different flow variable auming the body fluid blood a a model with the uenion of all the erythrocyte in the core region a ingham latic and the eriheral a a ewtonian fluid. We have een the effect of variou arameter on the different flow variable and comared with the exiting model. Mathematical formulation Conider an axially ymmetric, laminar, ulatile and fully develoed flow of blood in the axial direction through circular tube with an axially ymmetric mild tenoi. It i aumed that the body fluid blood i rereented by a two fluid model with a central layer of uenion of all erythrocyte a a ingham latic fluid and a eriheral (),, 5 8

3 layer of lama a a ewtonian fluid. The artery length i aumed to be large enough a comared to it radiu o that the entrance and exit, ecial wall effect can be neglected. The geometry of the arterial tenoi i hown in Fig.. Since the tenoi reent in the artery i conidered to be mild, the radial tranort of blood i negligible and thu we have neglected the radial velocity of the blood in thi tudy. Hence in the reent tudy, the flow of blood i conidered to be unidirectional and i in the axial direction. The geometry of tenoi in the eriheral region and core region are hown in Figure. & given by P L co z d in d z d L z L () in the normal artery region. z L co z d in d z d L L in the normalartery region () Where z and reectively uch that z z are the radii of the with eriheral layer and core region ( z), and are the radii of the normal artery and core region of the normal artery reectively; i the maximum height of the P tenoi in the eriheral region, i the ratio of the central core radiu to the normal artery radiu, i the maximum height of the tenoi in the core region uch that and P z i the half length of the tenoi. L i the length of tenoi; d indicate the location of the tenoi. The law of conervation of ma for one dimenional fluid flow in the deformable tube give the following equation of continuity in the core region and eriheral region. htt://e-jt.teiath.gr 9

4 u u in r ( z) t z z (3)\ u u in ( z) r ( z) t z z (4) Where u the velocity of the ingham latic fluid in the core region and u i the velocity of the ewtonian fluid in the eriheral layer region. Since the blood flow through the narrow arterie at the low hear rate, therefore flow i low and the vicou force dominate over the inertial force and thu, the magnitude of the convective term are negligibly mall. Therefore in the reent model, we have neglected the convective term in the momentum equation and body acceleration i taken a external force. u r F( t) r ( z) t z r r (5) u r F( t) ( z) r ( z) t z r r (6) and are the hear tree for the caon fluid and ewtonian fluid reectively, and are the denitie for caon fluid and ewtonian fluid reectively, P i the reure and Ft () i the body acceleration. The conecutive equation for ingham Platic fluid and ewtonian fluid are reectively given by u ( ) y if y and r z r u if y and r r u ( ) ( ) if z r z r (7) (8) Where and are the vicoitie of the ingham latic fluid and ewtonian fluid, reectively; y i the yield tre; i the radiu of the lug flow region. The eriodic body acceleration in the axial direction i given by F( t) a co( t ) (9) b (),, 5 3

5 Where a i it amlitude, b f, b f i it frequency in Hz., b i the lead angle of Ft () with reected to the heart action. The frequency of body acceleration f i b aumed to be mall o that wave effect can be neglected. The reure gradient at any z and t may be rereented a follow. ( z, t ) A A co( t ) z () Where A the teady comonent of the reure gradient i, A i amlitude of the fluctuating comonent and f where f i the ule frequency. oth A and A are function of z. we introduce the following non-dimenional variable. z ( z) ( z) r w z z z r t tw w w u b, ( ), ( ),,,, u u,,,, / 4 / 4 c c u u A A u,,, e, A /4 A / A / A a A A y,,, / A () Where and are ulatile eynold number for ingham latic fluid and ewtonian fluid reectively. Uing non-dimenional variable, equation () and () become z P L co z d in d z d L L in the normal artery region. L co z d in d z d L z L in the normal artery region () (3) The governing equation of motion given by equation (5) and (6) are imlified in the non dimenional form a u 4 f ( t) ( r) r ( z) t r r (4) htt://e-jt.teiath.gr 3

6 u 4 f ( t) ( r) ( z) r ( z) t r r (5) Where f ( t) ( eco t) co( t ) (6) The conecutive equation for ingham Platic fluid and ewtonian fluid are reectively given by equation (5) & (6) uing non dimenional variable reduce to u, if, r ( z) r (7) u r, if, r (8) u, z r z r (9) The boundary condition are u i finite and at r r u at r () and u u at r The boundary condition in the dimenionle form are u i finite and at r r u u at r and u u at r The non-dimenional volumetric flow rate i given by () Where ( z) Q 4 ru( z, r, t) rdr () 4 Qt () Q( t) ; Q( t) i the volumetric flow rate. A 8 The effective vicoity e defined a 4 e / Q t z Can be exreed in the non dimenional form a, (3) (),, 5 3

7 4 co / e e t Q t (4) 3. Method of olution Since it i not oible to find an exact olution to the ytem of nonlinear equation (4)-(9), the erturbation method i ued to obtain the aroximate olution to the unknown u, u, and.when we non-dimenionalize the momentum equation (5) and (6) and occur naturally and hence it i more aroriate to exand the equation (4)-(9) about and. Let u exand the lug core velocityu, the velocity in the core region u in the erturbation erie of a below (where <<) u ( z, t) u ( z, t) u ( z, t)... (5) u ( z, r, t) u ( z, r, t) u ( z, r, t)... ( z, t) ( z, t) ( z, t)... (6) (7) u ( z, r, t) u ( z, r, t) u ( z, r, t)... (8) Subtituting the erturbation erie exanion in equation (4), (7) and (8) and equating the ower of and contant term. u u u r f ( t) r, r,, r t r r r r (9) Similarly, uing the erturbation erie exanion in equation (5) & (8) and equating the ower of, the reulting equation of the eriheral region can be obtained a u u u r f ( t) r, r,, r t r r r r (3) Uing erturbation exanion in the given boundary condition and equating contant term and the term containing & we get and are finite at r,,, u u, u u at r z, u u, u at r z (3) On olving equation (9) & (3) uing equation (3) we can obtain the value for unknown u, u, u, u, u, u,,,,. htt://e-jt.teiath.gr 33

8 f t r, f t r (3) u u f t r (33) u u f t r kf t r (34) u u f t kf t (35) t f ' 3 6 r 3r 4k 3r r 4 3 f ' t 3 6 r 3r 4k 4 r f ' t r u r 3r 9 6k ln 48 f ' t u r 3r 9 4k 6r r r ln f ' t u 3 9 4k 6 ln Where z z k f() t (36) (37) (38) (39) (4) ( ), ( ),. (4) eglecting the term of o( ) and higher ower of in equation (7), the firt aroximation lug core radiu can be obtained a f () t k The exreion for axial velocitie in the core and eriheral region are obtained a f ' t r u u f t r r 3r 9 6k ln r 3r 9 4k f ' t 3 3 u u f t r r r ln (4) (43) The exreion for wall hear tre w can be obtained a w r 3 '( ) 3 ( ) f t f t 3 4k 4 (44) (),, 5 34

9 From equation (), (4) and (43) the volumetric flow rate i given by Q 4 r u u dr 4 r u u dr 4 r u u dr 4k Q u f ( t) f '( t) k 3 o ln 5 5 f '( t) k ln 4 (45) The exreion for effective vicoity e can be obtained from equation (4) and (45) 4 4 4k 3 3 u f ( t) f '( t) e ecot k 3 o ln 5 5 f '( t) k ln 5 4 (46) The econd aroximation lug core radiu can be obtained by neglecting term 4 of o( ) and higher ower of in equation (3) a f() t f '( t) k k (47) From equation (7), (4) & (47), the exreion for lug core radiu can be obtained a 4 f '( t) k k 6 5k (48) 4 4. eult and Dicuion In order to etimate the effect of body acceleration and li velocity by modeling blood a two layered flow. Perturbation technique are ued to olve the equation governing the fluid flow. The effect of ulatality, tenoi, yield tre of the fluid, htt://e-jt.teiath.gr 35

10 Axial Velocity reure gradient, hear tre, lug velocity, effective vicoity are invetigated and dicued briefly with the hel of grah u. u adial Ditance and u Figure. Velocity ditribution for different value of with.5,.,.8,.,.5, 45 c t The velocity rofile for different value of body acceleration arameter and li velocity u & for fixed value of eriheral tenoi height, reure gradient e, time t wth radial ditance r are hown in figure. The variation of axial velocity ha been een at the eak value of the tenoi i.e. at z=. It i oberved that the maximum velocity i attained at r=, after that it decreae gradually with the increae in the radiu of the artery r. At the tenotic wall i.e. at r ( z) axial velocity i minimum for any value of body acceleration arameter. However velocity i more when we conider the li velocity at the wall than the no li condition. Axial velocity i further increaed with the increae in body acceleration. Figure 3. Show the variation of volumetric flow rate with the reure gradient e for different value of yield tre and for fixed value of L,.5,.,.8, z. We have een the effect of both ody acceleration and li velocity on the volumetric flow rate. It i found that the flow rate increae with the increae in reure gradient arameter e for any value of,. However it can be eaily een from the figure that flow rate i highly affected by the body acceleration and li velocity. Flow rate i increae with the increae in body acceleration a well a with the increae in li velocity. It i oberved from the figure that the magnitude of flow rate in the reence of yield tre (i. e..) i le than it magnitude in the abence of yield tre (i.e. ). (),, 5 36

11 Flow ate u. u Preure Gradient Figure 3. Variation of flow rate with the reure gradient e for.5,.,.4 c Figure 4. deict the variation of effective vicoity with the eriheral tenoi height for different value of body acceleration and yield tre.it i oberved that effective vicoity decreae with the increae in body acceleration.it i alo found that effective vicoity increae with the eriheral tenoi height for both the e cae of no-li and li at wall. However increae in the value of yield tre increae the value of effective vicoity.the effective vicoity reduce when we aly li at the wall for both the cae in reence and abence of yield tre. Figure 5. and Figure 6. Show the variation of wall hear tre with time t and eriheral tenoi height.from the figure 6. It i oberved that wall hear tre o decreae with the time t for the range t 8 o and after that it increae for the o range8 t 36 o. Wall hear tre increae in magnitude with increae in eriheral tenoi height. htt://e-jt.teiath.gr 37

12 Wall Shear Stre Effective Vicoity Periheral Stenoi Height Figure 4. Variation of effective vicoity with the eriheral tenoi height for.,.5, u, c Time t Figure 5. Variation of wall hear tre with the time t for.5,.,.95 (),, 5 38

13 Wall Shear Stre Wall Shear Stre Periheral Stenoi Height Figure 6. Variation of wall hear tre with eriheral tenoi height for L,.., c time t Figure 7. Variation of wall hear tre with the time for L,.5,.,.8, e From Figure 6. It i found that wall hear tre i highly affected by the body acceleration; in the abence of body acceleration wall hear i low in comarion to the htt://e-jt.teiath.gr 39

14 reence of body acceleration. It i deicted that a we increae the body acceleration wall hear tre increae. Figure 7. Show the variation of wall hear tre with time t. the variation ha been hown for a full cale of time t( t 36) in degree. It i found that the body acceleration arameter highly influencing the wall hear tre w in a tenoed artery, a body acceleration increae wall hear tre decreae. However, w attain it minimum at t 8 Concluion o and the maximum at o t, t 36 The reent mathematical model bring out the many intereting reult due to the two layered blood flow i. e. due to eriheral layer. Effective vicoity increae a eriheral tenoi ize increae It i found that lug core radiu, wall hear tre, reure dro and flow rate increae a the yield tre or tenoi ize increae while keeing all other arameter contant. It i deicted that axial velocity and flow rate increae with the wall li wherea effective vicoity decreae due to the reence of li velocity. ody acceleration alo lay a very imortant role in blood flow modeling. Velocity and flow rate increae but wall hear tre decreae with the body acceleration. However e i lowered for both the cae with or without tenoi due to the li velocity. The reent tudy can be extended by conidering other fluid. In future one can conider the magnetic effect alo for further reearch. o eference. iwa, D. and Chakraborty, U. S. () Pulatile blood flow through a catheterized artery with an axially non ymmetrical tenoi. Alied Mathematical Science, 4: Chaturani, P., Sumy,. P., (986), Pulatile flow of Caon fluid through tenoed arterie with alication to blood flow, iorheology, Vol. 3, McDonald, D. A. (979) On teady flow through modeled vacular tenoi. J. iomech., Vol Medhavi A., Srivatav ueh K., Ahmad Q. S. and Srivatava V. P., () Twohae arterial blood flow through a Comoite tenoi,e-journal of Science and Technology, Vol.4(7), Mira J.C. and Shit, G.C., (7). ole of li-velocity in blood flow through tenoed arterie: A non-ewtonian model. Journal of Mechanic in Medicine and iology, Vol. 7, Morgan,. E., Young, D. F., (974) An integral method for the analyi of flow in arterial tenoi, ull. Math. iol., Vol. 36, agarani, P., Sarojamma,G., (8), Effect of body acceleration on ulatile flow of caon fluid through a mild tenoed artery, Korea-Autralia heology J., Vol., Ponalaguamy,., (7) lood flow through an artery with mild tenoi: A two layered model, different hae of tenoi and li velocity at the wall. Journal of Alied Science, Vol. 7, Iue 7, Sankar A.., Gunakala S.. and Comiiong D. M.G., (3) Two-layered lood Flow through a Comoite Stenoi in the Preence of a Magnetic Field. (),, 5 4

15 International Journal of Alication or Innovation in Engineering & Management, Vol., Iue, Sankar D. S. and Hemlatha K. (6) Pulatile flow of Herhel-ulkley fluid through tenoed arterie-a mathematical model, Int. J. of non-linear Mechanic, Vol. 4, Sankar, D. S., (9), A two fluid model for ulatile flow in catheterized blood veel, Int. J. of on-linear Mechanic Vol. 44, Sankar, D. S., Lee, U., (9), Mathematical modeling of ulatile flow of non- ewtonian fluid in tenoed arterie, Commun onlinear Sci umer Simulat., Vol. 4, Sarojamma G., amana., (), Pulatile flow of blood in a catheterized artery under external acceleration, International Journal of Advanced Engineering Technology (IJAET),Vol. III Iue IV, Shahed, M. E., (3), Pulatile flow of blood through a tenoed orou medium under eriodic body acceleration, Al. Math. and Comutation, Vol. 38, Siddiqui S.U., Mihra Shaileh and Verma.K., 3. Particulate uenion blood flow through an overlaing tenoi in an artery with ermeable wall. Alied Mathematic and Comutation, Vol. 7, (etherland) 6. Srivatava V. P., atogi, Vihnoi.() A two-layered uenion blood flow through an overlaing tenoi Comuter and Mathematic with Alication, Vol Sud, V.K. and Sekhon, G.S., (985), Arterial flow under eriodic body acceleration, ull. Math. iol., Vol. 47, Verma,. K., Siddiqui, S. U. and Guta,. S., Mihra S. () Effect of li velocity on blood flow through a catheterized artery. Alied Mathematic, Vol. () Verma,. K., Siddiqui, S. U. and Guta,. S., Mihra S. () A mathematical model for ulatile flow of Herchel ulkley fluid through tenoed arterie. Journal of cience and technology, 5(4) : Young, D. F. (979) Fluid mechanic of arterial tenoi. J. iomech, Engg., Tran, ASME, : Young, D. F., Tai, F. Y., (973) Flow charactertic in model of arterial tenoi-i teady flow, J.iomechanic, Vol. 6, htt://e-jt.teiath.gr 4