Pulsatile Blood Flow through a Catheterized Artery with an Axially Nonsymmetrical Stenosis

Transcription

1 Applied Mathematical Science, Vol. 4,, no. 58, Pulatile Blood Flow through a Catheterized Artery with an Axially Nonymmetrical Stenoi Devajyoti Biwa Department of Mathematic Aam Univerity, Silchar Aam-788, India Uday Shankar Chakraborty Department of Mathematic Aam Univerity, Silchar Aam-788, India udayhkd@gmail.com Abtract Pulatile flow of blood through a catheterized artery in preence of an axially nonymmetrical mild tenoi with a velocity lip at tenotic wall ha been invetigated in thi paper. Blood ha been repreented by a Newtonian fluid. By employing a perturbation analyi, analytic expreion for the velocity profile, flow rate, wall hear tre and effective vicoity, are derived. The influence of tenoi height, hape, lip velocity and radiu of catheter on axial velocity, wall hear tre and effective vicoity are repreented graphically and dicued. Graphical reult how that wall hear tre and effective vicoity decreae while axial velocity increae with velocity lip at wall. Mathematic Subject Claification: 76Z5 Keyword: Stenoi, Catheter, Newtonian fluid.. Introduction An abnormal growth, formed due to depoit of atheroclerotic plaque in the lumen of an artery i uually called tenoi (atherocleroi) and, it ubequent and evere growth on the artery wall reult in eriou circulatory

2 866 D. Biwa and U. S. Chakraborty diorder [6,, 5]. Thee diorder in circulatory ytem may be included a, narrowing in body paage leading to the reduction and impediment to blood flow in the contricted artery region, the blockage of the artery in making the flow irregular and cauing an abnormality of the blood flow and, the preence of tenoi at one or more of the major blood veel, carrying blood to the heart or brain etc., could lead to variou arterial dieae e.g., myocardial infarction, angina pectori, cerebral accident, coronary thromboi, troke etc.[7, 7, 8]. Catheterization refer to a procedure in which a long, thin, flexible platic tube (catheter) i inerted into an artery [4]. A catheter with a tiny balloon at the end i inerted into the artery in balloon angioplaty to treat atherocleroi. The catheter i carefully guided to the location at which tenoi occur and balloon i inflated to fracture the fatty depoit and widen the narrowed portion of the artery [9]. The inertion of a catheter in an artery will naturally form an annular region between the wall of the catheter and artery. A a reult, thi will alter the flow field, like modifying the preure ditribution and increaing the reitance. Thu it i of immene importance to tudy the flow of blood in a catheterized artery. In recent year, coniderable attention ha been paid to tudy the flow of blood in a catheterized artery. Many reearcher have analyed the flow of blood in an artery, in preence of a catheter by modeling the catheter and artery a rigid co-axial cylinder and blood a either a Newtonian or a non-newtonian fluid. McDonald [5] conidered the pulatile blood flow in a catheterized artery and obtained theoretical etimate for preure gradient correction for catheter. Karahalio [4] ha tudied the effect of catheterization on variou flow characteritic in an artery with or without tenoi. Jayaraman and Dah [3] addreed a numerical tudy of blood flow in catheterized curved artery with contriction. Daripa and Dah [] have analyzed the numerical tudy of pulatile blood flow in an eccentric catheterized artery, uing a fat algorithm and in conidering blood a to behave like a Newtonian fluid. Dah et al. [6] conidered the teady and pulatile flow of blood in a narrow catheterized artery etimated the increae in frictional reitance in the artery due to catheterization, uing a Caon fluid model. Sankar and Hemalatha [9] dicued the teady flow of Herchel Bulkley fluid through a catheterized artery. Sankar [] ha tudied a two-fluid model for the pulatile flow of blood in a catheterized artery, by conidering the core layer a a Caon fluid and the peripheral layer a a Newtonian fluid. Although, blood exhibit a non-newtonian character at low hear rate [], at high hear rate, generally found in larger arterie (diameter nearly above mm), blood behave like a Newtonian fluid [9]. Since, tenoi normally generate and develop in large diameter arterie (in the range of 5 to μm), where blood how a Newtonian behaviour, it appear to be reaonable in auming blood to be homogeneou, iotropic, incompreible, Newtonian continuum, having a contant vicoity and denity for flow though tenoed arterie (having repective radii., 5., 4. and.5mm in aorta, femoral, carotid

3 Pulatile blood flow 867 and coronary arterie [3]). In mot of the aforementioned tudie, traditional no-lip boundary condition [8] ha been employed. However, a number of tudie of upenion in general and blood flow in particular both theoretical [8,, 3,, ] and experimental [7, ], have uggeted the likely preence of lip (a velocity dicontinuity) at the flow boundarie (or in their immediate neighbourhood). The apparent (effective) vicoity will be lowered, a a reult of wall lip []. ecently, Mira and Shit [5], Ponalguamy [4], Biwa and Chakraborty [3] have developed mathematical model for blood flow through tenoed arterial egment, by taking a velocity lip condition at the contricted wall. Thu, it eem that conideration of a velocity lip at the tenoed veel wall will be quite rational, in blood flow modeling. With the above motivation an attempt ha been made to tudy the effect of lip (at the tenotic wall) and the influence of tenoi height and hape, on the flow variable (wall hear tre, velocity profile and effective vicoity) for pulatile blood flow through a catheterized veel with an axially nonymmetrical mild tenoi.. Mathematical Formulation We conider an axially ymmetric, laminar, pulatile and fully developed flow of blood (aumed to be incompreible) through a catheterized circular tube with an axially aymmetric but radially ymmetric mild tenoi a hown in Fig.. It i aumed that wall of the tube i rigid and the body fluid blood i repreented by a Newtonian fluid. Fig..Geometry of an axially nonymmetrical tenoi with an inerted catheter The geometry of the tenoi [4] i given by

4 868 D. Biwa and U. S. Chakraborty ( ) ( ) n n A L z d z d ; d z d + L z ( ) = (), otherwie, where ( z ) i the radiu of the artery in the tenoed region, i the radiu of the normal artery, n( ) i a parameter (called hape parameter) determining the tenoi hape (the ymmetric tenoi occur when n = ), L i tenoi length and d indicate it location. The parameter A i given by n ( n ) δn A =, L n n ( ) where δ denote the maximum height of the tenoi located at n ( n ) z = d + L n, uch thatδ <<. It ha been reported that the radial velocity i negligibly mall and can be neglected for a low eynold number flow in a tube with mild tenoi [8, 3]. The equation of motion governing the fluid flow are given by u p ρ = ( rτ ), t z r r () p =. r (3) where u i the fluid velocity in the axial direction, ρ i the denity and p i the preure. The contitutive equation of Newtonian fluid i given by u τ = μ. (4) r where μ i the coefficient of vicoity andτ i the hear tre. The boundary condition are given by ( ) u = u at r = z (5) u = at r = (6) where the catheter. u i the lip velocity at the tenotic wall [] and ( ) << i the radiu of

5 Pulatile blood flow 869 Since, the preure gradient i a function of z and t, we take p ( z, t ) = q ( z ) f ( t ), z (7) p where q( z) = ( z, ), f ( t ) = + ainωt z, a i the amplitude and ω i the angular frequency of blood flow [6]. Let u introduce the following non-dimenional variable z ( z) r L d δ z =, ( z) =, =, r =, t = tω, L =, d = δ =, n u A= A, u = u, u q 4μ = q μ, ωρ α = (8) μ 4 where α i the pulatile eynold number for Newtonian fluid and q i the negative of the contant preure gradient in a uniform tube without catheter. The non-dimenional form of geometry of tenoi i given by ( ) z ( ) ( ) A L z d z d ; d z d + L =, otherwie, n n (9) Uing non-dimenional variable equation () and (4) reduce to u α = 4q( z) f ( t) ( rτ ), () t r r u τ =. () r With the help of (), equation () become u α u = 4q( z) f ( t) + r t r r r. () The boundary condition in the non-dimenional form are given by u = u at r = z, (3) ( ) u = at r =. (4) The non-dimenional volumetric flow rate i given by

6 87 D. Biwa and U. S. Chakraborty ( ) z Q= 4 ru( r, z, t) dr, (5) where Q() t ( z) Q( t ) = ; ( ) π (,, ) ( ) 4 π q 8μ Q t = ru r z t dr i the volumetric flow rate. The effective vicoity μ e defined a p π ( ( z )) 4 z μe =, (6) Qt ( ) can be expreed in dimenionle form a ( ( z) ) Q() t 4 ( ) ( ) μ = q z f t, (7) e where Q() t i defined in equation (5) 3. Solution Conidering the Womerley parameter to be mall, the velocity u can be expreed in the following form u( zrt,, ) = u( zrt,, ) + α u( zrt,, ) +... (8) Subtituting the expreion of u from equation (8) in (), we have u r = 4 rq( z) f ( t), (9) r r u u = r t r r r. () Subtituting u from equation (8) into condition (3) and (4) we get u = u, u = at r = z and u =, u =, r = () ( ) To determine u andu, we integrate equation (9) and () twice with repect to r and ue the boundary condition () (foru, uing the expreion obtained foru ), we have

7 Pulatile blood flow 87 r log ( ) r u = u + q( z) f ( t) ( r ) log, log log () q( z) f '( t) ( ) ( 4 4 ) r u = 4 r r 3 4r log 3r log r log 4 4 ( ) log 3 3 (3) + log log where = ( z) The expreion for velocityu can eaily be obtained from equation (8), () and (3). The wall hear tre τ w (a a reult of equation () and (8)) become u u τw = + α (4) r r r= ( z) which i determined, by ubtituting velocity expreion () and (3) into the above equation (4), in the form u ( ) τ w = + q( z) f ( t) + log log α 3 ( ) q( z) f '( t) log 4 4 ( ) ( 4 3) 4 log log log (5)

8 87 D. Biwa and U. S. Chakraborty From equation (5), () and (3) the expreion for volumetric flow rate i given by log ( ) Q() t = ( ) + u + log ( ) q( z) f ( t) ( ) log + log α / q( z) f ( t) { 8 8 } 48 + ( ) 4 4 ( ) log X log log 4 4 ( ) { log 8 ( ) (6) + log The effective vicoity μ e can be found out with the help of equation (7) and (6). If teady flow i conidered, then equation (6) reduce to log ( ) ( ) + u + log (7) ( ) q( z) ( ) log Q = log whereq i the teady tate flow rate. Q = [8], the value of ( ) Taking q z can eaily be found out from equation (7). In abence of catheter, i.e. when =, the equation (), (3), (5), (6) reduce to u = u + q z f t r, (8) ( ) ( )( )

9 Pulatile blood flow 873 ( ) '( t) ( ) q z f u = 6 r r, (9) α 3 τ w = q( z) f ( t) q( z) f '( t), 8 (3) α 4 Q() t = ( ( z) ) u + q( z) f ( t) ( ( z) ) q( z) f '( t) ( ( z) ). 6 (3) The equation (8)-(3) are in good agreement with the reult obtained in [3] without body acceleration. 4. eult and Dicuion The preent model ha been developed to analye the effect of tenoi height, hape, catheter radiu and lip velocity on axial velocity, hear tre and effective vicoity. The value.5 i taken for the amplitude a and the pulatile eynold numberα, the range -. i taken for the height of the tenoiδ. adiu of the catheter i taken in the range -.5 and the value of the tenoi hape parameter i taken from to 6. Fig. : Variation of axial velocity with radial ditance for different value of t

10 874 D. Biwa and U. S. Chakraborty The variation of axial velocity with radial ditance for different value of time t and for fixed value ofδ =., =., α =.5, z = 8, n= i preented in Fig.. It i found that velocity increae a the time t increae fromt = to t = 9and then it deceae from t = 9 to t = 7 and then again it increae fromt = 7 to t = 36. Fig.3: Variation of axial velocity with radial ditance for differentδ andu Fig.3 repreent the variation of axial velocity with radial ditance for different value of tenoi height δ, lip velocity u and for fixed value of =., α =.5, z = 8, n=, t = 45. The magnitude of axial velocity i oberved to be more in uniform artery than that in a tenoed tube. Alo increae in velocity lip increae the axial velocity in both uniform and tenoed veel. Fig. 4 depict the variation of wall hear tre with axial ditance for different value of catheter radiu, tenoi hape parameter n, lip velocity u and at α =.5, t = 45, δ =.. It i oberved that the wall hear tre ditribution, in the tenotic region increae with the axial ditance in the uptream of the tenoi throat and attain it maximum magnitude at the throat, wherefrom it decreae with the axial ditance. The wall hear tre decreae with increaing hape parameter, n in the uptream of the throat but thi

11 Pulatile blood flow 875 property revere in the downtream. The magnitude ofτ w i found to be more in catheteried artery than that in uncatheterized one. However, for any value of n and, employment of velocity lip at wall decreae the wall hear tre. The variation of wall hear tre with catheter radiu for variou magnitude of u andδ at z = 8, n=, t = 45 i decribed in Fig. 5. From figure it i noted that wall hear tre increae with catheter radiu in both uniform and tenoed arterie. On the other hand, increae in lip velocity reduce the wall hear tre. Fig. 4: Variation of wall hear tre with axial ditance for different u, and n Fig.6 how the variation of effective vicoity with the radiu of the inerted catheter for different value of hape parameter n and lip velocityu atδ =., t = 45. It i found that effective vicoity increae with catheter radiu ignificantly but decreae with hape parameter and lip velocity. Fig.7 decribe the variation of effective vicoity with catheter radiu for variou magnitude ofδ andu and for fixed value of n=, t = 45. It i noted that μe in uniform artery i le in magnitude than that in tenoed tube. Alo increae in tenoi height increae the effective vicoity. However,

12 876 D. Biwa and U. S. Chakraborty employment of lip velocity reduce the magnitude of μe in both uniform and tenoed arterie. Fig. 6: Variation of wall hear tre with catheter radiu for differentδ and u Concluding emark In thi paper an attempt ha been made to tudy the effect of tenoi height, hape and lip velocity on variou flow variable. The governing equation of motion ha been integrated by employing a perturbation technique conidering very mall Womerely frequency parameter. It i oberved that increae in hape parameter increae the wall hear tre in the uptream of the throat but decreae in the downtream. From the analyi it can be concluded that lip velocity ha a ignificant role to play in reducing wall hear tre and effective vicoity. Since elevation of blood vicoity i conidered a a eriou rik factor in the cardiovacular, hematological, neoplatic and other diorder [6], the preent model may be ued a a tool for reducing blood vicoity by uing lip velocity at tenotic wall. Alo, more intereting model can be tudied by conidering the evere tenoi and permeability of the veel wall. Thee tudie

13 Pulatile blood flow 877 will be done in the near future. Fig. 7: Variation of effective vicoity with catheter radiu for different u and n

14 878 D. Biwa and U. S. Chakraborty Fig.8: Variation of effective vicoity with catheter radiu for different u and δ eference [] A.L. Jone, On the Flow of Blood in a Tube, Biorheology, 3(966), [] D. Biwa, Blood Flow Model: A Comparative Study, Mittal Publication, New Delhi,. [3] D. Biwa and U.S. Chakraborty, Pulatile Flow of Blood in a Contricted Artery with Body Acceleration, Appl. Appl. Math., 4(9), [4] D. Srinivaacharya and D. Srikanth, Effect of couple tree on the flow in a contricted annulu, Arch Appl Mech., 78 (8), [5] D.A. McDonald, Pulatile flow in a catheterized artery, J. Biomech., 9 (986), [6] D.F. Young, Fluid Mechanic of Arterial Stenoi, J. Eng. Ind. Tran. ASME, (979), [7] D. F. Young, Effect of a Time-dependent Stenoi of Flow through a Tube, Journal of Eng. Ind., 9 (968),

15 Pulatile blood flow 879 [8] D.S. Sankar and U. Lee, Mathematical modeling of pulatile flow of non-newtonian fluid in tenoed arterie, Commun Nonlinear Sci Numer Simulat., 4 (9), [9] D. S. Sankar, and K. Hemalatha, A Non-Newtonian fluid flow model for blood flow through a catheterized artery - Steady flow, Applied Mathematical Modeling, 3 (7), [] D.S. Sankar, A two-fluid model for pulatile flow in catheterized blood veel, International Journal of Non-Linear Mechanic, 44 (9), [] E.W. Merrill, heology of Human Blood and Some Speculation on it ole in Vacular Homeotai Biomechanical Mechanim in Vacular Homeotai and Intravacular Thromboi. P.N. Sawyer (ed.). Appleton Century Croft, New York, (965), [] G. Bugliarello and J.W.Hayden, High Speed Micro cinematographic Studie of Blood Flow in vitro. Science,38(96), [3] G. Jayaraman and.k. Dah, Numerical tudy of flow in a contricted curved annulu: An application to flow in a catheteried artery, J. Eng. Math., 4 (), [4] G.T. Karahalio, Some poible effect of a catheter on the arterial wall, Med. Phy., 7 (99), [5] J.C. Mira and G.C. Shit, ole of Slip Velocity in Blood Flow through Stenoed Arterie: A Non-Newtonian Model, Journal of Mechanic in Medicine and Biology, 7(7), [6] L. Dintenfa, heology of Blood in Diagnotic and Preventive Medicine: An Introduction to clinical Hemorheology, Butterworth, London and Boton, 976. [7] L. Bennet, ed Cell Slip at a Wall in vitro, Science, 55(967), [8] M.A. Day, The No-Slip Condition of Fluid Dynamic, Erkenntni, 33(99), [9] M.G. Taylor, The Influence of Anomalou Vicoity of Blood upon it Ocillatory Flow, Phyic in Medicine and Biology, 3(959), [] P. Daripa,.K. Dah, A Numerical tudy of pulatile blood flow in an eccentric artery uing a fat algorithm, J. Eng. Math., 4 (), -. [] P. Brunn, The Velocity Slip of Polar Fluid, heol. Acta.,4 (975), [] P. Chaturani and D.Biwa, A Comparative tudy of Poieuille flow of a polar fluid under variou boundary condition with application to blood flow. heol. Acta., 3(984), [3] P. Nagarani, and G. Sarojamma, Effect of body acceleration on pulatile flow of Caon fluid through a mild tenoed artery, Korea-Autralia heology Journal, (8), [4]. Ponalguamy, Blood flow through an artery with mild tenoi: A Two-layered model, Different hape of Stenoe and lip velocity at the wall, Journal of Applied Science, 7(7), 7-77.

16 88 D. Biwa and U. S. Chakraborty [5]. Bali and U. Awathi, Effect of a Magnetic field on the reitance to blood flow through tenoed artery, Appl. Math. Comput., 88 (7), [6].K. Dah and G. Jayaraman and K.N. Mehta, Etimation of increaed flow reitance in a narrow catheterized artery - a theoretical model, J. Biomech., 9(966), [7] S. Chakraborty, A. Datta and P.K. Mandal, Analyi of Non-linear Blood flow in a Flexible artery, Int. J. Eng. Sci., 33 (995), [8] V. Vand, Vicoity of Solution and Supenion. J. Phy. Colloid Chem., 5(948), [9] V.P. Srivatava and. atogi, Effect of Hematocrit on Impedance and Shear Stre during Stenoed Artery Catheterization, Appl. Appl. Math., 4 (9), [3] V.K. Sud and G. S. Sekhon, Arterial flow under periodic body acceleration, Bulletin of Mathematical Biology, 47(985), [3] Y. Nubar, Effect of Slip on the heology of a Compoite Fluid: Application to Blood Flow, heology, 4(967), eceived: April,