BIBECHANA Vol. 21, No. 3, December 2024,300-310 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University) Biratnagar Effect of blood flow through mild stenosed artery with effective viscosity Bishnu Prasad Bhandari1,2, Jeevan Kafle1, Chudamani Pokharel1,3,∗ 1Central Department of Mathematics, Tribhuvan University, Kathmandu, Nepal 2Nepal Engineering College, Pokhara University, Changunarayan, Bhaktapur, Nepal 3Dhawalagiri Multiple Campus, Tribhuvan University, Kathmandu, Nepal ∗Corresponding author: Email: chuda.pokherel@dmc.tu.edu.np Abstract Normal blood flow is disrupted by aortic stenosis, which raises risks and affects the cardio- vascular system. This work analyzes blood viscosity from the central core line to the arterial wall in order to look at flow parameters in arteries with minor stenosis. In order to ac- count for effective viscosity at radial distances, fluid dynamics in axisymmetric directions are analyzed using the Navier-Stokes equation. Additionally, analytical expressions for shear stress, pressure drop, velocity profile, and volumetric flow rate are investigated. These results contribute to our growing knowledge of vascular physiology in stenosis and emphasize the intricacy of blood flow dynamics. Keywords Arterial Stenosis, Viscosity, Hematocrit, Hemodynamic parameters. Article information Manuscript received: April 2, 2024; Revised: June 23, 2024; Accepted: June 24, 2024 DOI https://doi.org/10.3126/bibechana.v21i3.64409 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction Atherosclerotic plaque is formed by the accumu- lation of cholesterol, fat, and other foreign parti- cles [1]. Arteriosclerosis, specifically the abnormal thickening and hardening of the artery walls, signifi- cantly impacts the cardiovascular system. This con- dition can result from various poor lifestyle choices, such as smoking, physical inactivity, and an un- healthy diet. The progression of arteriosclerosis leads to notable changes in hemodynamic param- eters, including flow resistance, wall shear stress, pressure distribution, and blood flow [2, 3]. Steno- sis in an artery often results from the accumula- tion of cholesterol-rich particles, leading to the for- mation of atherosclerotic plaques. These plaques build up on the interior walls of arteries, causing several detrimental effects [4]. Heart diseases and stroke are significant global health issues, contribut- 300 http://nepjol.info/index.php/BIBECHANA chuda.pokherel@dmc.tu.edu.np https://doi.org/10.3126/bibechana.v21i3.64409 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 301 ing to high mortality rates [5]. When analyzing medical imaging results like angiograms or ultra- sound scans, different types of stenosis shapes can indeed have specific hemodynamic implications, af- fecting the severity and consequences of blood flow obstruction [6]. Theoretical and experimental in- vestigations into the effects of restrictions on blood flow parameters are crucial for understanding the hemodynamics of blood flow in both physiological and pathological conditions [7–9]. Hematocrit is a significant factor that influences blood viscosity. Additional features that can raise blood viscosity include decreased red blood cell de- formability, high red blood cell aggregation, and increased plasma viscosity [10–12]. Blood velocity and hematocrit percentage are two significant fac- tors that have been shown to influence wall shear stress in the blood flow through a tapered artery [13]. A complete blood count (CBC) is a com- prehensive examination that identifies and tracks various medical disorders by analyzing hematocrit concentration, white blood cell count, and platelet count [2,14]. Examined the effects of pressure gradi- ents, wall shear stress, blood velocity, and volumet- ric flow rate on human carotid arteries. An increase in the hematocrit and viscosity is accompanied by a decrease in the artery wall shear stress, indicat- ing an increase in heart rate [15]. These results highlight the way that hematocrit, catheter size, and stenosis interact to influence blood flow hemo- dynamics. It has been found that the impedance changes with the size of the stenosis, hematocrit, and catheter [12]. Blood clotting in the human heart can be fatal, as evidenced by the correla- tion between hematocrit and blood pressure gradi- ent [14]. If platelets are activated by exceptionally high shear stress near the top of the stenosis, such atherosclerosis damages the cardiovascular system by entirely blocking blood flow to the heart [16]. Figure 1: Schematic illustration of a stenotic artery. Mandal and Chakravarty [17] have shown that blood behaves like a non-Newtonian fluid in arteries with small radii and at low shear rates. According to Chaturani and Ponalagusamy [18] blood exhibits complex rheological behavior, particularly at low shear rates, where it demonstrates a non-zero yield stress. This phenomenon is primarily due to the interactions between erythrocytes (red blood cells), which can aggregate to form structures known as rouleaux. Singh et al. [16] assumed a little asym- metric stenosis along the radial direction. Biswas and Chakraborty [13] examined the pulsatile blood flow across a moderately stenotic tapering artery with slip velocity at the arterial wall. Haldar et al. [19] have investigated the effects of blood viscos- ity, wall shear stress, velocity, and hematocrit per- centage on blood flow in the stenosed artery. Bali and Awasthi [20] determined how blood viscosity varies with hematocrit and distance from the cen- ter, and how the external magnetic field affects the flow. Onitilo and Usman [14] examined the theory that arterial wall shear stress would be decreased by raising hematocrit and viscosity. The literature review that was previously discussed offers proof of the impact that stenosis has on blood flow. In this study, blood flow characteristics with mild stenosis were examined in relation to blood vis- cosity in plasma, hematocrit, and the site of steno- sis. The cylindrical polar form of the Navier-Stokes equation has been applied in axisymmetric direc- tions. 2 Methods Considering the constant blood flow in an axially symmetrical artery that has stenosis. Let R0 and R represent the artery’s radius in the absence and presence of stenosis, respectively. The artery is viewed as a circular, inelastic tube. With steno- sis, the radial blood flow is disregarded. Blood is thought to flow exclusively in an axial direction. 2.1 Model Equation Considering the three components of velocity and pressure at (x, y, z) and at time t are u(x, y, z, t), v(x, y, z, t), w(x, y, z, t) and p(x, y, z, t) respectively. The continuity equation in a steady-state form is [4, 21] ∂u ∂x + ∂v ∂y + ∂w ∂z = 0. (1) Here, a fluid’s density is represented by ρ(x, y, z, t). An incompressible viscous fluid has a constant den- sity, ρ. The Newtonian, viscous, and incompressible fluid N-S equations are [4, 21] Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 302 ρ ( ∂u ∂t + u ∂u ∂x + v ∂u ∂y + w ∂u ∂z ) = ρgx − ∂p ∂x + µ ( ∂2u ∂x2 + ∂2u ∂y2 + ∂2u ∂z2 ) (2) ρ ( ∂v ∂t + u ∂v ∂x + v ∂v ∂y + w ∂v ∂z ) = ρgy − ∂p ∂y + µ ( ∂2v ∂x2 + ∂2v ∂y2 + ∂2v ∂z2 ) (3) ρ ( ∂w ∂t + u ∂w ∂x + v ∂w ∂y + w ∂w ∂z ) = ρgz − ∂p ∂z + µ ( ∂2w ∂x2 + ∂2w ∂y2 + ∂2w ∂z2 ) . (4) In this case, t represents the time, and the external body forces, namely gravity, are f = ρ(gx, gy, gz). The forces resulting from pressure differences are denoted by the terms ∂p/∂x, ∂p/∂y, ∂p/∂z, and the viscous forces with constant viscosity coefficient µ in the x, y, and z- direction are represented by the last term on the right. The system (1)-(4) is closed for four unknown functions u, v, w and p. If gx = 0, gy = 0, and gz = 0 are the only external body forces acting on the motion, and if the motion is steady—that is, not changing over time—then ∂u/∂t = 0, ∂v/∂t = 0, ∂w/∂t = 0, and ∂p/∂t = 0, and the motion is two-dimensional. The Navier-Stokes equations for fluid flow inside a cylinder can be used to predict the flow of blood through arteries. We will use r and p to represent the artery’s blood flow’s radius and pressure drop. The velocities’ components along the radial, angular, and axial directions are vr, vθ, and vz, individually. The system (1)-(4) can be expressed in cylindrical form using x = r cos θ, y = r sin θ, r2 = x2 + y2 and θ = tan−1(y/x) from Cartesian coordinate (x, y, z) to polar coordinate (r, θ, z). The equation of continuity and the equation of motion are [3, 21] 1 r ∂ ∂r (rvr) + ∂ ∂z (vz) = 0 (5) ρ ( ∂vr ∂t + vr ∂vr ∂r + vz ∂vr ∂z ) = −∂p ∂r +µ ( ∂2vr ∂r2 + ∂2vr ∂z2 + 1 r ∂vr ∂r − vr r2 ) (6) ρ ( ∂vz ∂t + vr ∂vz ∂r + vz ∂vz ∂z ) = −∂p ∂z + µ ( ∂2vz ∂r2 + ∂2vz ∂z2 + 1 r ∂vz ∂r ) (7) We assume vθ = 0 in the axisymmetric flow, and p, vr, and vz are independent of θ. For a steady flow of blood, viscosity µ and density ρ are considered to be constant. v is the velocity component parallel to the z-axis. Only axially symmetric flow along the z-axis has been examined, therefore if vr = 0, vθ = 0, and vz = v, then equations (6) - (7) become ∂v ∂z = 0, 0 = −∂p ∂r , 0 = −∂p ∂z + µ ( ∂2v ∂r2 + ∂2v ∂z2 + 1 r ∂v ∂r ) (8) Suppose pressure term as P (z) = −∂p/∂z, equation (8) reduces to −P (z) r µ = ∂ ∂r ( r ∂v ∂r ) . (9) At a radial distance r, the effective viscosity of blood is expressed as [2, 22]. µr = µp [1 + αHr] (10) where µp is a plasma viscosity, α is a constant that characterizes the dependence of viscosity on hema- tocrit, Hr is the hematocrit at radial distance r, and αHr reflects the increased viscosity due to the presence of red blood cells. The formula describes this relationship. Hr = H [ 1− ( r R0 )m] (11) here, m is the number of stenosis, m = 1 is used from (10) and (11) µr = µp [ 1 + αH− αH ( r R0 )] Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 303 put b2 = αH, b1 = 1 + b2 µr = µp [ b1 − b2 ( r R0 )] (12) 2.2 Geometry of Stenosis Figure 1 describes the form of the stenosis that results from the layer being deposited inside a cylindrical artery. R = R0 ( 1− β 2R0 ( 1 + cos πz z0 )) (13) where β is the greatest thickness and R and R0 are the radii with and without stenosis [21]. The boundary condition as stated by [3, 21] v = { 0 at r = R, 0 at r = R0, and ∂v ∂r = 0 at r = 0 2.3 Velocity Profile From equations (9) and (12) Pr µp ( b1 − b2 ( r R0 )) + ∂ ∂r ( r ∂v ∂r ) = 0 Pr µpb1 ( 1− b2r b1R0 )−1 + ∂ ∂r ( r ∂v ∂r ) = 0. Binomial expansion is used, and neglect the higher power of r, Pr µpb1 ( 1 + b2r b1R0 ) + ∂ ∂r ( r ∂v ∂r ) = 0. Apply boundary conditions after integration, ∂v ∂r = − P µpb1 ( r 2 + b2r 2 3b1R0 ) . Again integration, v = − P µpb1 ( r2 4 + b2r 3 9b1R0 ) +A(z) (14) used boundary condition, v = 0, at r = R A(z) = P µpb1 ( R2 4 + b2R 3 9b1R0 ) then equation (14) becomes, v = P µpb1 [ 1 4 ( R2 − r2 ) + b2 9b1R0 ( R3 − r3 )] . (15) Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 304 P = 8µpb1Q πR4 0 ( 1− β 2R0 ( 1 + cosπz z0 ))4 [ 1− 8b2 15b1 ( 1− β 2R0 ( 1 + cos πz z0 ))] Binomial expansion is used, and neglect the higher power of β then it becomes, P = 8µpb1Q πR4 0 [ 1 + 2β R0 ( 1 + cos πz z0 ) − 8b2 15b1 ( 1 + β R0 − 3β2 2R2 0 +( β R0 − 2β2 R2 0 ) cos πz z0 − β2 2R2 0 cos 2πz z0 )] . (19) Pressure drop on the stenosed region is ∆P = ∫ z0 −z0 Pdz from the equation (19) and then integration, we have ∆P = 16µpb1Qz0 πR4 0 ( 1 + β R0 + 3β2 2R2 0 − 8b2 15b1 ) (20) if there is no stenosis i.e., β = 0 then equation (20) becomes (∆P )p = 16µpb1Qz0 πR4 0 ( 1− 8b2 15b1 ) . (21) Ratio of pressure drop is ∆P (∆P )p = ( 1− 8b2 15b1 + β R0 + 3β2 2R2 0 ) ( 1− 8b2 15b1 ) (22) 2.6 Ratio of Shear Stress Kapur and Pokharel et al. [4, 21] have mentioned the formula τ = PR 2 . From the equation 14, τ = 4µpb1Q πR3 ( 1− 8b2R 15b1R0 ) with the help of equation (13), Binomial expansion is used, and neglect the higher power of δ, we get τ = 4µpb1Q πR3 0 ( 1 + 3β 2R0 ( 1 + cos πz z0 ))[ 1− 8b2 15b1 ( 1− β 2R0 ( 1 + cos πz z0 ))] (23) if there is no stenosis i.e., β = 0, then equation (23) becomes, τp = 4µpb1Q πR3 0 ( 1− 8b2 15b1 ) (24) the ratio of shear stress τ τp = ( 1 + 3β 2R0 ( 1 + cosπz z0 )) [ 1− 8b2 15b1 ( 1− β 2R0 ( 1 + cosπz z0 ))] ( 1− 8b2 15b1 ) (25) Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 305 3 Results and Discussion Computational approaches provide a detailed anal- ysis of blood flow characteristics in stenosed arter- ies and offer valuable insights into the physiological impacts and potential treatment choices. 3.1 Velocity of blood flow through a stenotic artery Figure 2(A), explains the relationship between ve- locity and radius for both effective and plasma vis- cosity. The effective viscosity is 33.05 mm s−1 at the beginning, when r = 0, and the plasma viscosity is 47.67 mm s−1. For effective and plasma viscos- ity receptively, the velocities are 29.82 mm s−1 and 43.40 mm s−1 for r = 1 mm. In a similar vein, an artery’s radius is 2 mm, and its effective and plasma viscosity velocities are 19.23 mm s−1 and 27.12 mm s−1, respectively. Additionally, it has been noted that velocities stop at an artery’s inner wall. According to the above, velocity reaches its maximum in the center and progressively declines towards the wall for every value of plasma viscos- ity and effective viscosity. This simulation’s output demonstrates how effective viscosity has a signifi- cant impact on velocity, as seen in the figure. This investigation concludes that the effective viscosity has a greater impact on blood flow velocity than plasma viscosity. Figure 2(B), shows how the hematocrit affects the velocity profile under the assumption that the pres- sure drop is constant. We’ve assumed an artery’s radius of 3 mm and pressure of 80 Pa. The veloc- ity drops from 34.96 mm s−1 to 19.73 mm s−1, or roughly 15.23 mm s−1 at r = 0, when the hemat- ocrit rises from 0.2 to 0.8. The velocity falls from 31.34 mm s−1 to 17.84 mm s−1, or roughly 13.5 mm s−1 at r = 1 mm, when the hematocrit rises from 0.2 to 0.8. In a similar manner, the velocity drops from 20.01 mm s−1 to 11.51 mm s−1, or roughly 8.42 mm s−1 at r = 2 mm, when the hematocrit rises from 0.2 to 0.8. Finally, as the picture illus- trates, the hematocrit increases somewhat, followed by a quick initial reduction in velocity and a sub- sequent steady decrease. Near the inner wall of the artery, the velocity is almost zero, and it gradually increases as we move into the center. A low hema- tocrit causes a quicker rate of velocity rise. Figure 2: Velocity variations with radial distance for A: comparison viscosity, B: various hematocrit, C: various viscosity, D: increased viscosity due to the presence of red blood cells. Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 306 Figure 2(C), shows how viscosity affects the ve- locity profile under the assumption that the pres- sure drop is constant. We’ve assumed that an artery’s radius is 3 mm. The velocity drops from 32.89 mm s−1 to 14.1 mm s−1, or roughly 24.79 mm s−1 at r = 0, when the viscosity increases from 0.3 to 0.7. The velocity drops from 29.73 mm s−1 to 12.74 mm s−1, or roughly 16.99 mm s−1 at r = 1 mm, as viscosity increases from 0.3 to 0.7. In a similar vein, the velocity drops from 19.31 mm s−1 to 8.278 mm s−1, or around 11.032 mm s−1 at r = 2 mm, when viscosity increases from 0.3 to 0.7. At last, as the image illustrates, viscosity increases slightly before velocity first drops quickly and then gradually. As we move toward the center, the veloc- ity gradually increases from zero on the inner wall. A low viscosity causes the velocity to increase more quickly. Figure 2(D), describes the distribution of velocity for different values of increased viscosity due to the presence of red blood cells. Here βH = b2 takes vlues (0.5, 1, 1.5, 2) and 1 + βH = b1 has val- ues (1.5, 2, 2.5, 3). The velocity v at b1 = 1.5 and b2 = 0.5 is 10.57 mm/s. As the b1 and b2 increases the velocity decreases and becomes 8.637 mm/s at b1 = 2 and b2 = 1. The velocity at b1 = 2.5 and b2 = 1.5 is 7.863 mm/s for the constant viscosity of plasma 0.5 gram/mm s. The velocity at b1 = 3 and b2 = 2 is (7.186 mm/s for the constant viscos- ity of plasma (0.5 gram/mm s respectively. It is found that the blood velocity gradually diminishes with increasing increased viscosity due to the pres- ence of red blood cells i.e the flow velocity becomes smaller and smaller as one proceeds away from the center. For equal amount of increases increased vis- cosity due to the presence of red blood cells of an artery, the velocity in center has maximum and in the inner wall of an artery has minimum. As we can see from the above figures, effective vis- cosity has a significant impact on lowering velocity, so include the effective term and increased viscosity due to the presence of red blood cells will yield bet- ter results than only considering plasma viscosity. 3.2 Volumetric flow rate in a stenotic artery Figure 3: Volumetric flow rate A: at different position of stenosis, B,C: at different height of stenosis, D: increased viscosity due to the presence of red blood cells. Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 307 Figure 3(A) explains the volumetric flow rate at various stenosis sites, and several lines are drawn to illustrate the hematocrit values rising. Stenosis positions vary from 0 to 0.5 mm. The volumetric flow rate rises to 494.3 mm3 s−1 from 481 mm3 s−1, it is about 13.3 mm3 s−1 with hematocrit is 0.2, for the position of stenosis from 0 to 0.5. Once more, there is an increase in the volumetric flow rate from 357.8 mm3 s−1 to 343.7 mm.3 s−1, the difference is aroud 14.1 mm3 s−1 at 0.5 for the position of steno- sis changes from 0 to 0.5. The volumetric flow rate in the final scenario increases from 266.3 mm3 s−1 to 279 mm3 s−1 at 0.7, meaning that the difference in this case is 12.7 mm3 s−1. We see that there is an inverse relationship between the hematocrit and the volumetric flow rate. Stenosis position and hematocrit together carry a significant danger. Figure 3(B) is employed to illustrate how, for changing hematocrit, the relationship between vol- umetric flow rate and height of stenosis varies. Dif- ferent lines are created for each hematocrit in order to display the influence of hematocrit individually. The highest hematocrit value is 0.7, and the maxi- mum height of stenosis is 0.5 mm. About the time the hematocrit is 0.2 and the stenosis grows from 0 to 0.5 mm, the volumetric flow rate drops from 507 mm3 s−1 to 216.8 mm3 s−1. In a similar manner, when the hematocrit value is 0.5, the volumetric flow rate drops from 372 mm3 s−1 to 148.7 mm3 s−1. For the hematocrit value of 0.8, the volumetric flow rate drops from 291.8 to 113 mm3 s−1. While analyzing the variations in the volumetric flow rate decrease, we observe that the variation diminishes as the hematocrit increases. The volumetric flow rate lines are nearly parabolic when the hematocrit values are 0.2, 0.5, and 0.8. This suggests that the stenosis-related change is greatest when hematocrit values are compared. Hematocrit has a greater ef- fect when the stenosis height is less than 0.2 mm. When the stenosis height is greater than 0.4 mm, all of the volumetric flow rate lines fall between 216.8 mm3 s−1 and 113 mm3 s−1 for all hematocrit val- ues, indicating a limiting circumstance. This sug- gests that greater area is required to flow a greater amount, and that in such situation, hematocrit is more effective. Figure 3(C), explains the relationship, given a range of plasma viscosity values, between volumetric flow rate and thickness of stenosis (δ). To show the impact of varying plasma viscosity, three lines are drawn. In this instance, the constants are hemat- ocrit 0.5, pressure 80 Pa, and radius 3 mm. The volumetric flow rate drops from 524.20 mm3 s−1 to 204.7mm3 s−1, when the height of stenosis changes from 0 to 0.5 mm for the viscosity 0.3 gram mm−1 s−1. The volumetric flow rate decreases from 314.5 mm3 s−1 to 122.8 mm3 s−1 for the viscosity value of 0.5 gram mm−1 s−1, and from 224.6 mm3 s−1 to 87.75 mm3 s−1 approximately for the viscosity value of 0.7 gram mm−1 s−1. In this sense, when viscos- ity increases, the volumetric flow rate drops. This figure also shows that, when other parameters re- main constant and the plasma’s viscosity grows uni- formly, the volumetric flow rate eventually drops. Figure 3(D) describes the distribution of volumetric flow rate for increased viscosity due to the presence of red blood cells. Here b2 = βH and b1 = 1 + βH takes values (0.5, 1, 1.5, 2) and (1.5, 2, 2.5, 3). The volumetric flow rate at b1 = 1.5 and b2 = 0.5 is 473 mm3/s. As the increased viscosity due to the presence of red blood cells increases the volumetric flow rate decreases for the constant viscosity and becomes 347.8 mm3 s−1 at b1 = 2 and b2 = 1. The volumetric flow rate at b1 = 2.5 and b2 = 1.5 is 281.7 mm3/s for the constant viscosity of plasma 0.5 gram/mm s and becomes 249.3 mm3/s at b1 = 3 and b2 = 2 for the same viscosity.It is found that the volumetric flow rate gradually diminishes with increasing increased viscosity due to the presence of red blood cells i.e the volumetric flow rate becomes smaller as one proceeds away from the center. For equal amount of increases in increased viscosity due to the presence of red blood cells, the volumetric flow rate at b1 = 1.5, b2 = 0.5 has maximum and at b1 = 3, b2 = 2 has minimum. 3.3 Pressure gradient across the stenosis artery Figure 4 describes how the ratio of pressure drop to δ varies for varying hematocrit values. Since the gap used to draw these lines is so narrow, the lines appear to be practically linear curves. The pres- sure drop value ratio is calculated when δ is in the range of 0 and 0.5. The picture illustrates that with a hematocrit of 0.8, the pressure ratio reaches its maximum of approximately 1.323 at δ = 0.5 mm. For the hematocrit value of 0.5, the approximate ratio of pressure drop is 1.296. In the same way, 1.253 for 0.3. When the hematocrit rises within the specified range, the ratio of pressure drop increases steadily. This suggests that as the hematocrit and height of stenosis increase, the ratio of pressure drop increases progressively. The hematocrit measure- ment may benefit from this knowledge. Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 308 Figure 4: Relation between ratio of Pressure drop for different values of hematocrit with different height of stenosis. 3.4 Shear Stress ratio acroos stenotic artery Figure 5(A) explains the relationship between the shear stress ratio at various stenosis points, and sev- eral curves are produced to illustrate the increasing hematocrit levels. In this case, the stenosis length ranges from −0.5 to 0.5, and its maximal height is found at z = 0. Due to the stenosis’s symmet- ric structure, all of the lines symmetrically increase from −0.5 to 0 and decrease after 0. It demon- strates that when stenosis position increases, the ratio of shear stress falls. The ratio of shear stress increases parabolically from the common point to z = 0, where it reaches its maximum values 2, 2.096, and 2.144 for the hematocrits of 0.2, 0.5, and 0.8, respectively. When the hematocrit increases shear stress ratio increases gradually. The conclu- sion from this figure is that the shear stress ratio gets increasingly parabolic and inclines gradually as the hematocrit increases. Figure 5(B) shows the shear stress ratio at vari- ous stenosis heights, with distinct lines being drawn to indicate rising hematocrit values. The highest hematocrit value is 0.8, and the maximum height of stenosis is 0.5 mm. For this reason, every line in- creases from 0 to 0.5 because the stenosis is thought to have a symmetrical shape. This demonstrates that a uniform rise in the hematocrit also results in a uniform increase in the ratio of shear stress. Shear stress increases and reaches 1.734 at δ = 0.5 mm when the hematocrit is 0.2. At 0.5, the shear stress increases and reaches 1.956 at the maximum height δ = 0.5 mm. At 0.8, the shear stress ratio increases and reaches 2.098, as depicted in the fig- ure. This analysis concludes that as hematocrit in- creases, the shear stress ratio increases consistently with a uniform quantity. The ratio of shear stress falls as hematocrit decreases, and the curve takes on a parabolic shape at different positions of stenosis. On the other hand, linear curves for various stenosis heights are produced when there is a consistent rise in hematocrit levels along with a uniform increase in the shear stress ratio. Figure 5: Relation between ratio of the shear stress A: with various position of stenosis for different values of hematocrit, B: at different height of stenosis for different values of viscosity of plasma. 4 Conclusion Atherosclerotic plaque accumulation and abnormal tissue development lead to stenosis, which restricts blood flow and contributes to cardiovascular dis- eases like ischemia and stroke. The Einstein co- efficient of blood viscosity, which quantifies the resistance to flow in blood vessels, is crucial for understanding hemodynamics and vascular health. Navier-Stokes equations have been applied to ana- Bishnu Prasad Bhandari et al./ BIBECHANA 21 (2024) 300-310 309 lyze the Einstein viscosity in an axisymmetric di- rection, calculating model expressions for veloc- ity profile, volumetric flow rate, pressure, pressure drop, and shear stress in an artery. The velocity of blood is more influenced by Einstein’s viscos- ity than plasma viscosity. 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Introduction Methods Model Equation Geometry of Stenosis Velocity Profile Ratio of Shear Stress Results and Discussion Velocity of blood flow through a stenotic artery Volumetric flow rate in a stenotic artery Pressure gradient across the stenosis artery Shear Stress ratio acroos stenotic artery Conclusion