Two-Layered Pulsatile Blood Flow in a Stenosed Artery with Body Acceleration and Slip at Wall

Transcription

1 Available at htt://vamu.edu/aam Al. Al. Math. ISS: Vol. 5, Issue (December ), (Previously, Vol. 5, Issue,. 4 47) Alications and Alied Mathematics: An International Journal (AAM) Two-Layered Pulsatile lood Flow in a Stenosed Artery with ody Acceleration and Sli at Wall Devajyoti iswas and Uday Shankar Chakraborty* Deartment of Mathematics Assam University Silchar-788, India udayhkd@gmail.com Received: January 3, ; Acceted: July 9, Abstract Pulsatile flow of blood through an artery in resence of a mild stenosis has been investigated in this aer assuming the body fluid blood as a two-fluid model with the susension of all the erythrocytes in the core region as ingham Plastic and the eriheral region of lasma as a ewtonian fluid. This model has been used to study the influence of body acceleration, non- ewtonian nature of blood and a velocity sli at wall, in blood flow through stenosed arteries. y emloying a erturbation analysis, analytic exressions for the velocity rofile, Plug-core radius, flow rate, wall shear stress and effective viscosity, are derived. The variations of flow variables with different arameters are shown diagrammatically and discussed. It is noticed that velocity and flow rate increase but effective viscosity decreases, due to a wall sli. Flow rates and seed are enhanced further due to the influence of body acceleration. Keywords: Pulsatile flow; Periheral Plasma Layer (PPL); Velocity-sli; ody acceleration; ingham Plastic; Stenosis MSC o.: 76Z5, 74G. Introduction An abnormal growth, reducing the lumen of an artery is usually called a stenosis or an atherosclerosis [Young (968); Sankar and Lee (9); iswas (); iswas and Chakraborty *Corresonding Author 33

2 34 D. iswas and U.S. Chakraborty (9a); iswas and Chakraborty (9b)], which is one of the most widesread diseases that can cause serious circulatory disorders, by reducing or occluding the blood suly. For instance, stenosis in arteries sulying blood to brain, can bring about cerebral strokes, likewise, in coronary arteries, it can be myocardial infarction, leading to a heart failure [Sinha and Singh (984)]. Several theoretical and exerimental analyses were erformed to study the blood flow characteristics in the resence of stenosis [McDonald (979); Mandal (5); ali and Awasthi (7); iswas and Chakraborty (9a); iswas and Chakraborty (9b); Sankar and Ismail (9)]. It has been reorted that though at high shear rates blood exhibits ewtonian character in large arteries like aorta [Taylor (959)], blood being a susension of coruscles, at low shear rates and during its flow through narrow vessels, behaves like a non-ewtonian fluid [Merrill et al. (965); Charm and Kurland (974)]. ugliarello and Sevilla (97) and Cokelet (97) have exerimentally roved that for blood flowing through small vessels, there exists a cell-oor lasma (ewtonian fluid) layer and a core region of susension of almost all the erythrocytes. ugliarello and Sevilla (97) resented the flow of blood in small diameter tubes by a two-layered model assuming eriheral and core fluids as ewtonian fluids of different viscosities. Following the exerimentally verified model of ugliorello and Sevilla (97), two-fluid modeling of blood flow has been discussed and used by a good number of researchers. Shukla et al. (98) alied a two-fluid model to discuss the flow of blood through a stenosis. Chaturani and Uadhya (979, 98) addressed the flow of blood in small diameter tubes using the two-layered model of Microolar and Coule stress fluids resectively. Two-fluid model analyses have been carried out by Srivastava () to observe the effects of a non-symmetrical stenosis on blood flow characteristics. It has been ointed out by many investigators that under certain flow situations, blood ossesses a finite yield stress [Fung (98); Kaur et al. (98)]. One interesting and secialized case of material with yield stress, is known as ingham lastic whose consistency curve or, flow behaviour, is shown by a straight line curve [Fung (98); Kaur et al. (98)]. This articular material deforms elastically, until the yield stress is reached, but once this stress is exceeded, it flows as a ewtonian fluid, with shear stress being linearly related to rate of shear strain [Schlichting (968)]. It therefore seems to be realistic, in considering blood behaving as a ingham lastic in the core region of a constricted artery. It is well known that blood flow in the human circulatory system is caused by the uming action of the heart, which in turn roduces a ressure gradient throughout the system [Fung (98); Misra et al. (8)]. Human heart is a muscular um and due to contraction and exansion of heart muscles, there roduces a ressure difference in its systolic and diastolic conditions, oularly known as ressure ulse which hysicians check at the wrist. Flow of blood due to this ressure ulse, is known as ulsatile flow [Chaturani and Samy (985); Guyton and Hall (6)]. Again under excetional circumstances, human body may also be subjected to accelerations (or vibrations). Accelerative disturbances are quite common in normal life, for examle, while riding in a vehicle or while landing, taking off and flying in an aircraft or sacecraft, oerating a jack hammer and, sudden and fast movements of the body, during gymnastics and sorts activities. In such situations, human body may unintentionally be

3 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 35 subjected to external accelerations. The flow of blood in the arteries of such a subject is influenced by such whole body accelerations [Sud and Sekhon (985, 987)]. Though human body can adat to changes, however rolonged exosures to such accelerative disturbances may lead to health roblems, like headache, abdominal ain, loss of vision and increased ulse rate. Sud and Sekhon (985) resented a mathematical model of blood flow in a single artery subject to ulsating ressure gradient as well as body acceleration. agarani and Sarojamma (8) resented a theoretical model of ulsatile blood flow in a stenosed artery under the action of eriodic body acceleration considering blood behaving as a Casson fluid. In all the above mentioned studies, traditional no-sli boundary condition [Day (99)] has been emloyed. However, a number of studies of susensions in general and blood flow in articular both theoretical [Vand (948); Jones (966); uber (967); runn (974); Chaturani and iswas (984)] and exerimental [ugliarello and Hayden (96); ennet (967)], have suggested the likely resence of sli (a velocity discontinuity) at the flow boundaries (or in their immediate neighbourhood). Recently, Misra and Shit (7), Ponalgusamy (7), iswas and Chakraborty (9a, 9b, ) have develoed mathematical models for blood flow through stenosed arterial segment, by taking a velocity sli condition at the constricted wall. Thus, it seems that consideration of a velocity sli at the stenosed vessel wall will be quite rational, in blood flow modeling. With the above motivations, an attemt has been made to study the effects of sli (at the stenotic wall) and the influence of body acceleration, on the flow variables (wall shear stress, velocity rofiles, flow rate and effective viscosity) for two layered ulsatile blood flow through a constricted vessel.. Mathematical Formulation We consider an axially symmetric, laminar, ulsatile and fully develoed flow of blood (assumed to be incomressible) through a circular tube with an axially symmetric mild stenosis as shown in Fig.. It is assumed that the wall of the tube is rigid and the body fluid blood is reresented by a two-fluid model with a core region of susension of all erythrocytes as a ingham lastic fluid and a eriheral layer of lasma as a ewtonian fluid. The artery length is assumed to be large enough as comared to its radius so that the entrance and exit, secial wall effects can be neglected.

4 36 D. iswas and U.S. Chakraborty Figure. The geometry of an axially symmetric arterial stenosis. The geometry of the stenosis in the eriheral region is given by z R cos, for z z, Rz ( ) z R, for z z. () The geometry of the stenosis in the core region is given by c z R cos, for z z, R ( z) z R, for z z, () where artery in the stenosed core region such that R z R z R z is the radius of the stenosed artery with eriheral layer; R z is the radius of the ; R, R are the radii the normal artery and core region of the normal artery resectively; is the maximum height of the stenosis in the eriheral region, is the ratio of the central core radius to the normal artery radius, c is the maximum height of the stenosis in the core region such that c and z is the half length of the stenosis. It has been reorted that the radial velocity is negligibly small for a low Reynolds number flow in a tube with mild stenosis [agarani and Sarojamma (8); Sankar and Lee (9)]. The equations of motion governing the fluid flow are given by u t z r r r Ft, r R z (3)

5 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 37 u t z r r r Ft, R z r Rz (4) in the core and eriheral regions, resectively, where u, u are the fluid velocities in the core region and eriheral region resectively;, are the shear stresses for ingham lastic fluid and ewtonian fluid resectively;, ewtonian fluid resectively; is the ressure and are the densities for ingham lastic fluid and Ft is the body acceleration. The constitutive equations of ingham lastic fluid and ewtonian fluid are resectively given by u y, if y, R r R z, r u, if y, r R, r (5) and u r if R z r R z, (6) where R is the radius of the lug flow region. The eriodic body acceleration in the axial direction is given by F t a cos bt, (7) where a is its amlitude, b fb, f b is its frequency in Hz, is the lead angle of Ft with resect to the heart action. The frequency of body acceleration f b is assumed to be small so that wave effect can be neglected. The ressure gradient at any z and t may be reresented as follows z t A A t z, cos, (8) where A is the steady comonent of the ressure gradient, A is amlitude of the fluctuating comonent and f, where f is the ulse frequency. oth A and A are functions of z. We introduce the following non-dimensional variables

6 38 D. iswas and U.S. Chakraborty z z R, R z R( z) R, R z R ( z) R, r r R, t t, R,, b u u c c R, u, u AR 4 AR 4, us u s AR,,, AR AR 4 e A A, a A, y AR, R, R, (9) where and are the ulsatile Reynolds numbers for ingham lastic fluid and ewtonian fluid resectively. Using non-dimensional variables, equations () and () become Rz ( ) z cos, for z z, z, for z z, () R ( z) z cos, for z z, c z, for z z. () The governing equations of motion given by equations (3) and (4) are reresented in the nondimensional form as u 4 f t r, r R z t r r, () where u 4 f t r, R z r R z t r r cos cos, (3) f t e t t, (4) Using non-dimensional variables equations (5) and (6) reduce to u, if, R r R z, (5) r

7 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 39 u r r u, if, r R, if R z r R z, (6). (7) The boundary conditions in the non-dimensional form are given by (i) is finite at r, (8) (ii) u (iii) u at r Rz, (9) s, u u at r R z. () The non-dimensional volumetric flow rate is given by Rz Q 4 ru( r, z, t) dr, () where Qt ; Qt Q t R 4 A 8 is the volumetric flow rate. The effective viscosity e defined as e z R z 4 Q( t ) can be exressed in dimensionless form as 4 cos, () e R z e t Q t. (3) 3. Method of Solution Since, are time deendent, therefore it is necessary to exand equations (), (3) and (5)-(7) in erturbation series about and. We exand u, R and u as follows:,,, u z t u z t u z t (4)

8 3 D. iswas and U.S. Chakraborty,,, R z t R z t R z t (5),,,,,, u zrt u zrt u zrt (6) Similarly, u,, can be exanded in erturbation series in terms of and. Substituting the erturbation series exansions in equations (), (5) and (6) and equating owers of, the resulting equations of the core region can be obtained as r r f t r u t r r, r, u r u r,. (7) Similarly, using the erturbation series exansions in equations (3) and (7) and equating owers of, the resulting equations of the eriheral region can be obtained as r r f t r u t r r, r u r, u r,. (8) Using erturbation series exansions in equations (8)-() and equating constant terms and terms containing and, we get and are finite at r,,, u u, u u at r R z, u u, s u at r Rz, (9) On solving equations (7) and (8) for unknowns u, u, u, u, u, u,,,, using equation (9), we can obtain, f t r, s f t r, (3) u u f t R z r, (3) u u f t R z r kf t R z r, (3) s u u f t R z R kf t R z R, (33) s o o t / f 3 6Rz r3r 4k3Rzrr 4, (34)

9 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 3 / f t 6Rz r3r 4k 4 3 R z 3 4 u Rz r r Rz k R z 48 / f t 4 3 r ln t r, (35) R z, (36) / f u Rz r 3r 9Rz 4k 6R zr r R z R z R z ln, (37) 3 Rz t / f u Rz R 3R 9Rz 4k 6RzR R R z Rz Rz ln, (38) 3 Rz where k f t. eglecting the terms of o and higher owers of in equation (5), the first aroximation lug-core radius R can be obtained as R f t k. (39) Using equations (3), (3), (36) and (37) the exressions for axial velocities in the core and eriheral regions are obtained as s u u f t R z r / f t Rz r 3r 9Rz 6kR z ln r 48 R z. (4)

10 3 D. iswas and U.S. Chakraborty s / f t u u f t R z r kf t R z r R z r 3r 9Rz 4k 6R zr r R z Rz Rz ln. 3 Rz 4 3 (4) Similarly, using equations (4), (33) and (38), the exression for lug-core velocity can easily be obtained. The exression for wall shear stress w can be obtained by, (4) w r R z t 3 / f 3 f t Rz 3Rz 4k R z 4 R z From equations (), (4) and (4) the volumetric flow rate is given by. (43) R R z R z R Rz Q 4 r u u dr 4 r u u dr 4 r u u dr 4k ur f t R R R R s / f t R 6R R R R 6R R 9R R 4k3RR R R R ln R / f t R 3 5 9RR 6RR 4R R 8kR ln RR R 4 R, (44) where R Rz, R R z. The exression for effective viscosity e can be obtained from equations (3) and (44). The second aroximation lug core radius R can be obtained by neglecting terms of 4 o and higher owers of in equation (8) as

11 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 33 R R = f t = f t k / 6R 5k R. (45) 4 From equations (5), (39) and (45), the exression for lug-core radius can be obtained as f t k. (46) 4 / 4 R k 6R 5k R In absence of yield stress i.e., i.e.,, R z R z the flow system reresented by the equations (4)-(44) reduce to one-layered ewtonian flow system with body acceleration as and eriheral lasma layer 4 4 u us f t R z r f ' t 4 R z r 3 R z r, (47) 6 Q R z u f t R z f t R z 6 4 s ', (48) 3 w f t Rz f ' t Rz, (49) 8 e 6 4 Rz ecost us f trz f ' t Rz. (5) The results given by equations (47)-(5) are the results obtained in iswas and Chakraborty (9a). 4. Results and Discussions The resent model has been develoed to analyze the combined effects of body acceleration; stenosis and velocity sli on the flow variables viz., axial velocity, flow rate, shear stress and effective viscosity of blood, flowing in an artery with a mild constriction. The equations governing the abovementioned flow are integrated by using a erturbation analysis with very small Womersley frequency arameters (.5 ). The ressure gradient arameter e is taken in the range -5, the body acceleration arameter is taken in the range -, magnitude of the lead angle is taken as. and the range -.5 is taken for the height of the stenosis in the

12 34 D. iswas and U.S. Chakraborty eriheral region. The axial velocity sliu s is considered from to., the ratio of the central core radius to the normal radius of the artery is taken as.8, yield stress is taken as,. and the magnitude of z is taken from -4 to 4. The variation of axial velocity, using equations (4) and (4) at the throat of the stenosis (i.e. at z =) with radial distance, for fixed values of eriheral stenosis height, ressure gradient e, timet and for different values of sli velocityu s and body acceleration arameter, are resented in Figure. Figure : Variation of axial velocity with radial distance r fort 45, e,., z,.8 and for different values of z and u s, It is observed from the figure that axial velocity is maximum at r, wherefrom it gradually decreases with the increase in radius of the artery r and attains a minimum value at the stenotic wall ( r Rz) for any value of body acceleration arameter. However, an emloyment of sli at the wall increases the axial velocity. Increase in body acceleration further enhances the axial velocity. Variation of volumetric flow rate with the ressure gradient arameter e for fixed values of eriheral stenosis height,timet and for different values of sli velocityu s, yield stress and

13 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 35 body acceleration arameter, are deicted in Fig. 3. It is observed that flow rate gradually increases with the increase in ressure gradient arameter e for any value of and. However, the magnitude of flow rate in the absence of yield stress ( ) is more than its magnitude in resence of yield stress. It is further noticed that emloyment of body acceleration enhances the flow rate. Variation of Wall shear stress with axial distance z and timet are resented in Fig. 4 and Fig. 5 resectively. Form the figures, it can be clearly observed that wall shear stress w increases from its aroached magnitude (i.e. at z 4 )in the ustream of the throat with the axial distance and achieves its maximal at the throat of the stenosis and then decreases in the downstream and attains a lower magnitude at the end of the constriction rofile (i.e. at z 4 ). Magnitude of wall shear stress w in uniform tube ( ) is lower than its magnitude in stenosed artery ( ). It increases significantly with the height of the eriheral stenosis for any value of body acceleration arameter. However, body acceleration decreases wall shear stress in both uniform ( ) and stenosed ( ) arteries. Figure 3. Variation of flow rate with ressure gradient for,., t 45, z,.8

14 36 D. iswas and U.S. Chakraborty Figure 4. Variation of wall shear stress with axial distance for., t 8, z,.8 Figure 5. Variation of wall shear stress with axial distance for.,.8, z It is also noticed that for any value of body acceleration arameter, wall shear stress w gradually decreases as timet increases until it attains its minimum att 8, wherefrom it gradually increases with time and reaches its aroached magnitude att 36.

15 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 37 Variation of effective viscosity e with axial distance z and eriheral stenosis height are resented grahically in Figure 6 and Figure 7, resectively. Figure 6. Variation of effective viscosity with axial distance fort 45,.8, z t 45,.8, z Figure 7. Variation of effective viscosity with eriheral stenosis height for

16 38 D. iswas and U.S. Chakraborty Effective viscosity increases with the axial distance z from the initiation of a stenosis (i.e. at z 4 ) till it reaches to its maximum value at the throat (i.e. at z ), wherefrom it gradually decreases to its initial value at the termination of the stenosis (i.e., at z 4 ). It is also observed from the figures that effective viscosity e increases with eriheral stenosis height for both the cases of no-sli and sli at wall. However, for any value of body acceleration arameter, emloyment of an axial sli velocity at wall reduces the effective viscosity both in resence and absence of yield stress. For fixed values of andu s, effective viscosity increases with yield stress. 5. Conclusion The resent analysis deals with the two-layered ulsatile blood flow through an artery (Fig.) embedded with an axi-symmetric mild stenosis and an axial velocity sli is emloyed at the constricted wall. The body fluid blood is assumed to behave like a ewtonian fluid in the eriheral lasma layer and it is reresented as ingham Plastic in the core region. The equations of motion, governing the flow are integrated by using a erturbation method. Analytic exressions for flow variables are obtained and their variations with different flow arameters are resented grahically. It is observed that as exected, axial velocity and flow rate increase with the wall sli whereas effective viscosity decreases due to a sli. Also wall shear stress and effective viscosity decrease but velocity and flow rate increase with the body acceleration arameter.effective viscosity e increases as increases. However, e is lowered for both the uniform tube and stenosed artery, as a result of wall sli. Since this study, takes care of ulsatility of the flow and also it incororates the characteristics of non-ewtonian nature of blood (which is rominent in case of flow through arteries with smaller diameter), it is strongly felt that the resent model may rovide a better insight to the study of blood flow. This model indicates that sli at a diseased artery could lay a rominent role in blood flow modeling. From the analysis, it may also be concluded that with sli, the damages to the vessel wall could be reduced. This kind of reduction in wall shear stress and effective viscosity could be exloited for better functioning of the diseased arterial systems. Hence one may look forward for drugs or devices which would roduce sli and use them for treatment of eriheral arterial diseases. Further imrovement in the model could be done by considering ermeability of the blood vessels. Acknowledgements Authors exress sincere thanks to Prof. A. M. Haghighi, the Editor-in-Chief and the reviewers of the journal for their valuable comments. They are also grateful to Dr. Sudi Choudhury, Assam University, Silchar, India for valuable discussions.

17 AAM: Intern. J., Vol. 5, Issue (December ) [Previously, Vol. 5, Issue,. 4 47] 39 REFERECES ali, R. and Awasthi, U. (7). Effect of a Magnetic Field on the Resistance to lood Flow through Stenosed Artery, Al. Math. Comut., Vol. 88, ennet, L. (967). Red Cell Sli at a Wall in vitro, Science, Vol. 55, iswas, D. (). lood Flow Models: A Comarative Study, Mittal Publications, ew Delhi. iswas, D. and Chakraborty, U.S. (9a). Pulsatile Flow of lood in a Constricted Artery with ody Acceleration, Al. Al. Math., Vol. 4(), iswas, D. and Chakraborty, U.S. (9b). Pulsatile Flow of lood in a Constricted Artery with a Velocity Sli, Far East Journal of Alied Mathematics, Vol. 36(3), iswas, D. and Chakraborty, U.S. (). Pulsatile lood Flow through a Catheterized Artery with an Axially onsymmetrical Stenosis, Alied Mathematical Sciences, Vol. 4(58), runn, P. (975). The Velocity Sli of Polar Fluids, Rheol. Acta., Vol. 4, ugliarello, G. and Sevilla, J. (97).Velocity Distribution and Other Characteristics of Steady and Pulsatile lood Flow in Fine Glass Tubes, iorheology, Vol.7, Chaturani, P. and Uadhya, V. S. (979).On Microolar Fluid Model for lood Flow through arrow Tubes, iorheology, Vol. 6, Chaturani, P. and Uadhya, V. S. (98). A Two-Fluid Model for lood Flow through Small Diameter Tubes with on-zero Coule Stress oundary Condition at the Interface, iorheology, Vol.8, Chaturani, P. and iswas, D. (984). A Comarative Study of Poiseuille Flow of a Polar Fluid Under Various oundary Conditions with Alications to lood Flow, Rheol. Acta., Vol. 3, Chaturani, P. and Samy, R.P. (985). A Study of on-ewtonian Asects of lood Flow through Stenosed Arteries and its Alications in Arterial Diseases, iorheology, Vol., Charm, S.E. and Kurland, G.S. (974). lood Flow and Micro Circulation, John Wiley, ew York. Cokelet, G. R. (97). The Rheology of Human lood: In iomechanics (Ed. Y.C. Fung et al.), Prentice Hall, Englewood Cliffs, ew Jersey. Day, M.A. (99). The o-sli Condition of Fluid Dynamics. Erkenntnis, Vol. 33, Fung, Y.C. (98). iomechanics: Mechanical Proerties of Living Tissues, Sringer-Verlag, ew York. Guyton, A.C. and Hall, J.E. (6). Textbook of Medical Physiology, Elsevier. Jones, A.L. (966). On the Flow of lood in a Tube. iorheology, Vol. 3, Kaur, J.., hat,.s. and Sacheti,.C. (98). on-ewtonian Fluid Flows (A Survey Monograh), Pragati Prakashan, Meerut. McDonald, D.A. (979). On Steady Flow through Modeled Vascular Stenosis, J. iomech., Vol.,.3. Mandal, P.K. (5). An Unsteady Analysis of on-ewtonian lood Flow through Taered Arteries with a Stenosis, Int. J. onlinear Mech., Vol. 4,.5-64.

18 3 D. iswas and U.S. Chakraborty Merrill, E.W. (965). Rheology of Human lood and Some Seculations on its Role in Vascular Homeostasis iomechanical Mechanisms in Vascular Homeostasis and Intravascular Thrombosis, P.. Sawyer (ed.), Aleton Century Crofts, ew York, Misra, J.C. and Shit, G.C. (7). Role of Sli Velocity in lood Flow through Stenosed Arteries: A on-ewtonian Model, Journal of Mechanics in Medicine and iology, Vol. 7, Misra, J.C., Adhikary, S.D. and Shit, G.C. (8). Mathematical Analysis of lood Flow through an Arterial Segment with Time-deendent Stenosis, Mathematical Modeling and Analysis. Vol. 3, agarani, P. and Sarojamma, G. (8). Effect of ody Acceleration on Pulsatile Flow of Casson Fluid through a Mild Stenosed Artery, Korea-Australia Rheology Journal, Vol., ubar, Y. (967). Effects of Sli on the Rheology of a Comosite Fluid: Alication to lood Flow, Rheology, Vol. 4, Ponalgusamy, R. (7). lood flow through an artery with mild stenosis: A Two-Layered Model, Different Shaes of Stenoses and Sli Velocity at the Wall. Journal of Alied Sciences, Vol. 7, Sankar, D.S. and Ismail, A.I.M. (9). Two-Fluid Mathematical Models for lood Flow in Stenosed Arteries: A Comarative Study, oundary Value Problems, Vol. 9, Sankar, D.S. and Lee, U. (9). Mathematical Modeling of Pulsatile Flow of on-ewtonian Fluid in Stenosed Arteries. Commun. onlinear. Sci. umer. Simulat., Vol. 4, Schlicting, H. (968). oundary Layer Theory, McGraw- Hill ook Comany, ew York. Shukla, J.., Parihar, R. S. and Guta, S. P. (98). Effects of Periheral Layer Viscosity on lood Flow through an Artery with Mild Stenosis. ulletin of Mathematical iology, Vol. 4, Sinha, P. and Singh, C. (984). Effects of Coule Stresses on the lood Flow through an Artery with Mild Stenosis, iorheology,, Srivastava, V. P. (). lood Flow through Stenosed Vessels with a Periheral Plasma Layer and Alications. Automedica, Vol. 8, Sud, V.K. and Sekhon, G. S. (985). Arterial Flow Under Periodic ody Acceleration, ulletin of Mathematical iology, Vol. 47, Sud, V.K. and Sekhon, G. S. (987). Flow through a Stenosed Artery Subject to Periodic ody Acceleration, Med. iol. Eng. & Comut., Vol. 5, Taylor, M.G. (959). The Influence of Anomalous Viscosity of lood uon its Oscillatory Flow. Physics in Medicine and iology, Vol. 3, Vand, V. (948). Viscosity of Solutions and Susensions, J. Phys. Colloid. Chem. Vol. 5, Young, D. F. (968). Effects of a Time-Deendent Stenosis of Flow through a Tube, Journal of Eng. Ind., Vol. 9,