STUDY OF BLOOD FLOW THROUGH MODELLED VASCULAR STENOSIS

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1 STUDY OF BLOOD FLOW THROUGH MODELLED VASCULAR STENOSIS S.R. Verma Department of Mathematics D.A-V. (P.G.) College, Kanpur , India srverma303@gmail.com The effect of an axially symmetric mild stenosis on the steady flow of blood, when blood is represented by Newtonian model have been studied. Theoretical results for the velocity profile, pressure gradient, wall shearing stress and separation phenomena are obtained by solving the momentum integral equation. The results obtained are discussed graphically for mild stenosis for Reynolds number (Re) of Re = and the significance of the analysis are pointed out by comparing the results with similar studies. Our results are in good agreement with published experimental and theoretical results. 1 INTRODUCATION Vascular stenosis (atherosclerosis), localized narrowing of the blood vessels due to abnormal growth of tissues, is one of the disease in the cardiovascular system of man. This can cause serious circulatory disorders by reducing or occluding the blood supply. For instance, stenosis in the arteries supplying blood to brain can bring about cerebral strokes; like wise in coronary arteries, it can cause myocardinal infraction leading to heart failure. Many authors [1,2,3] have reported that the rheologic and fluid KEY WORDS : Vascular stenosis, myocardinal infraction, separation phenomena.

2 [2] dynamic properties of blood and its flow behaviour through non-uniform cross-section of the tube (stenosed tube, tapered etc.) could play an important role in the fundamental understanding, diagnosis and treatment of many cardiovascular diseases. Due to importance of the fluid dynamic factors in understanding of blood flow, in the present paper, it has been made to give a generalized model of blood flow and obtain important information about the blood flow. Several theoretical and experimental attempts have been made to study the blood flow characteristics due to the presence of stenosis in the lumen of the blood vessel by taking various model of blood (Young [4], Morgan and Young [5], Azuma and Fukushima [6], Young et al [7], MacDonald [8], Forrester and young [9], Misra and Kar [10], Srivastav [11], Perkkio and Keskinen [12], Chaturani and Ponnalagar Samy [13], Chaturani and Palanisamy [14]). Morgan and Young [5] applied momentum and integral-energy equations in integral form to obtain approximate solutions for velocity distribution, pressure drop, wall shearing stress and separation in axisymmetric constrictions. In recent years many workers have investigated the flow characteristics of blood through artery in the presence of stenosis (Srivastava [15, 16], Gupta and Gupta [17], Provenzano and Rutland [18], Ghalichi and Deng [19], Sapna Ratan Shah [20], Musad Mohammad Musad Saleh and Mear Yaseen Ali Khan [21], Bijendra Singh et al.[22], Sanjeev Kumar and Archana Dixit [23], and Verma [24]). Srivastava [15] present an analysis to study the effects of a mild stenosis on blood flow characteristics when blood is represented by a two-fluid model consisting of a core region of suspension of erythrocytes assumed to be a couple stress fluid and a pheripheral layer of plasma as a Newtonian fluid. Provenzano

3 and Rutland [18] have proposed a boundary layer model for wall shear stress in arterial stenosis. Wall shear stress is obtained by solving the momentum integral equation for Reynolds numbers (Re) of Re= The boundary layer model proposed can be easily implemented by clinical researchers. Wootton et al. [25] examined thrombogenic stenosis experimentally and computationally. Keeping in view of the above literature survey, we propose to study the effect of mild stenosis on the flow of blood, when blood is represented by a Newtonian fluid. By employing the momentum integral technique the equation of motion governing the flow are solved. Analytical expressions for velocity profile, pressure gradient, shearing stress and separation phenomena are derived. The solution is applicable for low Reynolds numbers. MATHEMATICAL FORMULATION AND ANALYSIS Consider the axisymmetric flow of blood in an uniform circular tube with an axisymmetric mild stenosis. The geometry of the stenosis as shown in Fig.1, is described as (Young [4]). RR (xx ) = RR 0 δδ CCCCCC 2ππ LL 0 xx (1 ) where R 0 is the radius of normal tube, RR (xx ) the radius of stenosed tube, δδ the maximum height of the stenosis, xx the axial co-ordinate and LL 0 the length of the stenosis. Some dimensional variables are designated by the notation ( ) to differentiate them from corresponding dimensionless variables. [3]

4 [4] R 0 rr xx RR (xx ) vv uu UU δδ Fig.1. Geometry of stenosis For a two-dimensional steady, incompressible flow, the governing Navier-Stokes equations may be taken in the form uu uu uu + vv xx rr = 1 pp + μμ ρρ xx ρρ ²uu + 1 uu rr ² rr rr in the axial direction and uu vv xx + vv vv rr in the radial direction. LL 0 + ²uu (2 ) xx ² = 1 pp + μμ ρρ rr ρρ ²vv + 1 vv + ²vv vv (3 ) rr ² rr rr xx ² rr ² The continuity equation can be written as uu xx + vv rr + vv rr = 0 (4 ) where uu represents the axial velocity, vv the radial velocity, ρ the density, pp the pressure, µ the viscosity coefficient of blood, r and xx the radial and axial co-ordinates respectively. Now we write above equation in dimensionless form by using the transformation rr = rr RR 0, xx = xx RR 0, RR = RR RR 0, uu = uu UU 0, vv = vv UU 0,

5 [5] pp = pp ρρuu 0 2, UU = UU UU 0, δδ = δδ RR 0, LL 0 = LL 0 RR 0, where UU is the centerline velocity and UU 0 the average velocity in the unobstructed tube. form as The geometry of the stenosis can thus be written in dimensionless RR (xx) = 1 δδ 2 (1 + CCCCCC 2ππ LL 0 xx ) (1) Similarly, the Navier-Stokes equations are uu + vv = + 2 ²uu + 1 RR ee ² rr in the axial direction, vv + ²uu ² (2) + uu = + 2 ²vv + 1 vv + ²vv (3) RR ee ² rr rr² ² in the radial direction. The continuity equation becomes + + vv = 0 (4) rr It is assumed that the term due to viscous component of the normal stress in the axial direction ( ²u/²) are negligible. This assumption has been used in the analysis of non-uniform flow. Therefore equation (2) becomes uu + vv = RR ee rr rr (5) where RR ee = 2RR 0UU 0 ρρ is the Reynolds number. μμ

6 [6] An integral form of (5) is obtained by integration over the crosssection of the tube with no-slip boundary condition, that is u = v = 0 at the wall and using (4), the integral-momentum equation is RR rrrr²dddd 0 = RR² + 2RR 2 RR ee RR (6) The velocity-profile constraints for axisymmetric tube flow are u = U at r = 0 (7a) u = 0 at r = R (7b) = 0 at r = 0 (7c) ³uu ³ = 0 at r = 0 (7d) = 2 1 RR ee rr rr at r = R (7e) The first of these is the definition of the centreline velocity and the second is the no-slip condition at the wall. The third is regularity condition and is deduced by considering the forces on a cylindrical element as : If the pressure and inertial forces are to be finite, as the radius of the element tends to zero, the viscous force that is proportional to u/ r must tend to zero. The fourth constraint can be obtained by eliminating the pressure between equation (2) and (3) and considering the resulting equation as r tends to zero. The fifth constraint represents the validity of equation (5) at r = R. The assumed fourth-degree polynomial velocity profile is uu UU = AA + BB rr RR + CC rr RR 2 + DD rr RR 3 + EE rr RR 4 (8)

7 The coefficients A, B, C, D, E are evaluated from the velocityprofile constraints and using in (8), velocity profile is obtained as [7] uu UU = (4 λλ) rr RR (1 λλ) rr RR 4 (9) where λλ = RRRRRR2 8UU (10) From equation (13) it is noted that if λ=1 then it is parabolic profile corresponding to the Poiseuille profile uu UU = 1 rr RR 2 (11) The justification of equation (11) is in the Appendix. The flow flux Q is given by RR QQ = ππππ² UU aa = 2ππ rr uu dd rr 0 (12) where UU aa = UU /UU 0 is nondimensional average velocity at any cross-section. Substitution of (9) in (12) yields QQ = 1 ππππππ²(8 + λλ) (13) 18 Substituting the value of λλ from (10) in (13) we obtain the expression for centerline velocity as UU = 9 4ππ 1 RR 2 QQ + ππrr ee 144 RR4 (14) The parameter λ is determined from equation (6) after use of (9) and (10) as λλ = 1 + 3RR ee 8UU RR rrrr² dddd (15) 0

8 Now neglecting the terms higher than two in the velocity profile (9) we have only the Poiseuille profile uu = 2UU aa 1 rr RR 2 (16) where UU aa = RR eerr² 16 [8] (17) is the non-dimensional average velocity at any given cross-section and /<0. Using equations (10), (14) and (16) in (6) we obtain the expression for pressure gradient as = 32 9 QQ² 1 dddd 16 ππ² RR 5 dddd ππ QQ RR ee RR 4 (18) Further using (10) and (14) in (9), we have uu = RR eerr ππ QQ QQ 2 ππ 2 RR 5 dddd dddd 16 ππ QQ RR ee RR rr RR rr RR 4 RR rr RR 2 + rr RR 4 (19) The flow flux in normal (unobstructed blood vessel, QQ = ππrr 0 2 UU 0, this gives non-dimensional flow flux QQ 0 = QQ 0 /RR 0 2 UU 0 = ππ, which is same for obstructed blood vessel,. Therefore replacing Q by QQ 0 i.e. Q by π in (18) and (19) we obtain the expression for pressure gradient and velocity profile as = 32 9 and 1 dddd 16 1 RR 5 dddd RR ee RR 4 (20) uu = 2 RR rr RR RR ee dddd 3 12 RR³ dddd rr RR rr RR 4 (21) The wall shearing stress can be expressed as

9 [9] ττ ww = μμ uu 1 + rr RR ddrr 2 (22) ddxx where ddrr /dddd is the slope of the wall. The dimensionless wall shearing stress is defined as ττ ww = ττ ww ττ ww 0 (23) where ττ ww0 is the corresponding shearing stress due to Poiseuille flow in the unobstructed tube. Thus in terms of dimensionless variables ττ ww = dddd RR dddd 2 (24) Using (21) in (24), the wall shearing stress is therefore obtain as ττ ww = RR ee RR 3 18 RR dddd dddd 1 + dddd dddd 2 (25) At points along the wall where ττ ww = 0 flow separation and reattachment will occur. From equation (25) the condition of zero wall shear stress given as dddd = 18RR dddd RR ee (26) when dddd > 18RR, ττ dddd RR ww is negative, indicating backflow and the separated ee region is clearly described. RESULTS AND DISCUSSION The flow characteristics of blood flow through narrow vessel with mild stenosis have been investigated by modelling of blood as a Newtonian fluid. The present investigation is concerned with the approximate solution for velocity profile, wall shearing stress, pressure gradient and separation phenomena for specific geometry of the stenosis. It involves integralmomentum method for the solution of the problem.

10 [10] The analytical expressions have been computed numerically for different Reynolds number. It has been suggested by the authors Forrester and Young [9] in the theoretical and experimental studies, the stenosis parameters RR 0, LL 0 and δδ are taken to be related as LL 0 = 8RR 0 = 24δδ The computation has been carried out for the location of stenosis defined by 4 xx 4 for δδ = 1/3 and LL 0 = 8 for different Reynolds number lies between Figures 2, 3 and 4 are the non-dimensional axial velocity profile of blood flow in the mild stenosed vessel segment (δδ = 1/3) under consideration for the Reynolds number 10, 50 and 100 respectively. It is observed that in conversing region near the throat of stenosis velocity decreases but increases away from the throat as Reynolds number increases for r/r upto 0.5 and then in reverse order for r/r > 0.5. While in diversing region all above observations are reversed as Reynolds number increases. The theoretical distribution of shearing stress along the wall for δδ = 1/3 is given in Figure 5. It is observed that for any Reynolds number, the shearing stress reaches a maximum value on the throat and then rapidly decreases in the diversing region. Absolute theoretical values of dimensionless pressure gradient / between x = ±6 for δ=1/3 are shown in Figure 6. The pressure gradient increases upto the throat of stenosis and then decreases in the diversing region. It is noted that calculated values of absolute pressure gradient at any location are decreases as Reynolds number increases in both conversing and diversing region of stenosis.

11 [11] The theoretical locations of zero shearing stress are shown in Figure 7 for δ=1/3. The values of separation and reattachment points were obtained from equation (26). At a given Reynolds number the upstream position corresponds to the separation point and the downstream position is the reattachment point. No separation occurs until a critical Reynolds number is reached. Separation and reattachment observations are in good agreement with the theoretical as well as experimental observations of Forrester and Young [9].

12 [12] 5.00 x= x= x= x=2.0 u 2.50 x= r/r Fig. 2. Velocity profile for RR ee = 10, δδ = 1/3

13 [13] x= x= x= x=2.0 u 2.50 x= r/r Fig. 3. Velocity profile for RR ee = 50 and δδ = 1/3

14 [14] x= x= x= x=2.0 u 2.50 x= r/r Fig. 4. Velocity profile for RR ee = 100 and δδ = 1/3

15 [15] Re=10 3 Re= Re=30 ττ ττw ww x Fig. 5. Wall shearing stress for δδ = 1/3

16 [16] Re=10 6 Re=20 5 Re=30 / ττw x Fig. 6. Absolute Pressure gradient for δδ = 1/3

17 [17] R e Critical Reynolds number x Fig. 7. Separation-reattachment data for δδ = 1/3

18 [18] APPENDIX Equation (10) is λλ = RRRRRR2 8UU For λλ = 1, UU = RRRRRR2 8 UU = 1 UU 2RR 0UU 0 ρρ 0 8 μμ RR 2 RR 0 pp RR 0 ρρuu 2 0 UU = 2 RR 2 pp 8μμ xx From equation (11) (i) uu = UU 1 rr UU 0 UU 0 RR 2 uu = UU 1 rr RR 2 (ii) Using (i) in (ii), we obtain uu = 2 RR ² 8μμ pp xx 1 rr RR 2 uu = 2UU 1 rr RR 2 (iii) where UU is the average velocity at any cross-section. Equation (iii) is the Poiseuille velocity profile for straight tube. Non dimensional form of (iii) is uu = 2UU aa 1 rr RR 2 (iv) where UU aa = UU /UU 0 is the average velocity at any cross-section. ACKNOWLEDGEMENT This research is supported by U.G.C. Grand No. F /2008 (S.R.)-MRP.

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