Suspension model for blood flow through a tapering catheterized inclined artery with asymmetric stenosis

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1 Available at Appl. Appl. Math. ISSN: Vol., Issue (June 25), pp Applications and Applied Mathematics: An International Journal (AAM) Suspension model or blood low through a tapering catheterized inclined artery with asymmetric stenosis * Corresponding author Devajyoti Biswas and Moumita Paul * Department o Mathematics Assam University Silchar: 788, India mpmaths@gmail.com Received: March 27, 23; Accepted: April 7, 25 Abstract We intend to study a particle luid suspension model or blood low through an axially asymmetric but radially symmetric mild stenosis in the annular region o an inclined tapered artery and a co-axial catheter in a suitable low geometry has been considered to investigate the inluence o velocity slip at the stenotic wall as well as hematocrit, shape parameter. The model also includes the tapering eect and inclination o the artery. Expressions or the low variables have been derived analytically and their variations with various low parameters are represented graphically. The results or the dierent values o the parameters involved show that the impedance to low increases with stenosis height, hematocrit and catheter radius. However, it decreases with the shape parameter, angle o inclination o artery and velocity slip at the stenotic wall. The present analysis is an extension o the work by Chakraborty et al. (2) and also includes several theoretical models o arterial stenosis in the uniorm, tapering and catheterized tubes, with the consideration o velocity slip or zero slip at the vessel wall. Finally, some biological implications o this theoretical modeling are included in brie. Keywords: Blood; hematocrit; catheter; impedance; shear stress; stenosis; slip; taper angle MSC 2 No.: 76Z5, 92C. Introduction Atherosclerosis (stenosis) is a wide spread Cardiovascular (CVS) disease. Majority o deaths in developed countries result rom CVS diseases and most o which are associated with abnormal low in arteries (Ku, 997). Stenosis is an abnormal and unnatural growth that develops at one or more locations o Cardiovascular System, under diseased conditions and causes serious circulatory disorders (Guyton, 97; Young, 968). There is no exact 474

2 AAM: Intern. J., Vol., Issue (June 25) 475 inormation regarding such unamiliar growth at an arterial wall. However, due to the deposits o atherosclerotic plaques, cholesterol, lipids, ats etc., at an innermost arterial wall, the kind o ormation may develop at the vessel wall. It is reported that circulatory disorders could be responsible or over seventy ive percent o all deaths (Srivastava et al. 2). These circulatory disorders may include (i) the narrowing in body passage or oriice, leading to the reduction in nutrient supply an impediment to blood low in constricted artery regions, (ii) blockage o artery, in turning the low irregular and causing abnormality o blood low and (iii) presence o stenosis at one or more major blood vessels, supplying blood to heart or brain could lead to various arterial and Cardiovascular diseases like, angina pectoris, myocardial inarction, cerebral accident, coronary thrombosis, heart attack, strokes, thrombosis etc. (Young, 968). In view o the above, blood low modeling in arterial system is a topic o recent interest to theoretical and clinical investigators. Blood is a suspension o dierent cells or corpuscles in plasma. Most o the models on blood low have dealt with a one phase model (Womersley, 955; Lightoot, 974; Sud and Sekhon, 985; Young, 968). As blood is a suspension, a two phase model seems to be more appropriate. At low shear rates and while lowing in narrow channels, blood behaves as a non- Newtonian luid (Merril and Pelletier, 967; Charm and Kurland, 974). The theoretical results o Haynes (96) indicate that blood cannot be considered as a single phase homogeneous viscous luid while lowing through narrow arteries (o diameter µm). Srivastava and Srivastava (983) have mentioned that blood can be suitably represented by a macroscopic two-phase model (i.e., a suspension o red cells in plasma) in small vessels (o diameter 24µm). Recently, Chakraborty et al. (2) have considered a two-phase model or blood low through a constricted artery. In most o the aorementioned works, blood vessels are considered horizontal. It is well known that many ducts in the human physiological systems are not horizontal, rather they have some inclination to the axis. The orce o gravity comes into the low ield due to the consideration as an inclined tube. Steady blood low through an inclined non-uniorm tube with multiple stenoses has been proposed by Maruti Prasad and Radhakrishnachandramacharya (28). Chakraborty et al. (2) have proposed a blood low model through an inclined tube with stenosis. Normally, in circulatory systems arteries are assumed to be clear pipes and blood can low easily without any hindrance and perorms speciic unctions like, transporting nutrients, maintaining metabolic processes and regulating body balance (Guyton, 97). The study o circulatory systems is pretty old and the quest or knowing the living world, especially the human physiological systems had its beginning in the long past (Fung, 98). In act, both Aristotle and Leonardo the Vinci were interested in blood low through human circulatory systems. Many investigators have proposed theoretical models on blood low rom various considerations (Fung, 98). Blood low through stenosed arteries with axially symmetric stenosis, have been proposed by many researchers (Young, 968; MacDonald, 979). Flow o blood in obstructed arteries with axially non-symmetrical stenosis is investigated by Mekheimer and Kothari (2), Chakraborty et al. (2). Pressure-low relationship alters the blood low in a stenosed artery. Sometimes, or some clinical purposes, catheters are inserted in arteries. The pressure-low relationship changes appreciably when a catheter is inserted in a stenosed artery. Blood low models through catheterized stenosed artery have been proposed by Mekheimer and Kothari (2), Chakraborty et al. (2). In human systems, there prevail dierent geometries in blood vessels such as, circular, branched, biurcated, tapered, inclined etc. (Guyton, 97). It could be important to

3 476 Devajyoti Biswas and Moumita Paul investigate blood low through an inclined tapered artery or Newtonian luid (Biswas and Paul, 22) and or the two phase low, in the present study. Recently, Chakraborty et al. (2) have used the slip condition in their two-phase stenosed, but to the authors knowledge, no theoretical or experimental work has been done to analyze the eects o velocity slip at the stenotic wall on macroscopic two-phase (plasma red-cell) blood low model, in an inclined constricted catheterized tapering artery. In the present analysis, an eort has been made to study the eects o velocity slip (at the stenotic vessel wall), hematocrit, tapering tube, catheterization and inclination o the artery on the low variables or annular blood low through an inclined, catheterized tapering artery with the ormation o an axially asymmetric mild stenosis, by considering blood to behave as a particle-luid suspension. 2. Mathematical Formulation: We consider a steady, laminar and ully developed low o blood (supposed to be incompressible) through the annular region o an inclined tapering circular tube and a co-axial rigid catheter, with the ormation o an axially asymmetric mild stenosis. The low geometry o the inclined catheterized tapering vessel and the non-axisymmetrical stenosis but radially symmetric growth, are presented in Figure and 2 respectively. The geometry o an arterial asymmetric stenosis, developed along a tapered wall, can be reproduced (Mekheimer and Kothari, 28) as n n R( z) b( z)[ A( L ( z d) ( z d) )], d z d L,

4 AAM: Intern. J., Vol., Issue (June 25) 477 b( z), otherwise, () with b z R z ( ) ( ), where bz ( ) represents the radius o the tapered arterial segment along the stenotic portion, ξ(=tanφ) is the tapering parameter, Φ the tapering angle, Rzis () the radius o the stenosed region, R is the radius o the artery in the non-stenotic region, R is the catheter radius, n( 2) is a parameter (treated as shape parameter), signiying the stenosis shape which includes the symmetric stenosis case when n=2, and, (r,z) are the radial and axial coordinates, the respective quantities L, d and L denote the stenosis length, its location and the total length o the obstructed artery. The parameter A is taken as n/( n) n A, n RL( n ) where is the maximum height o the stenosis at a distance L z d n. /( n ) (2) The body luid blood is assumed to behave like a two luid model that is a mixture o erythrocytes and plasma. The equations describing the steady low o a two phase model, comprising the luid phase and particle phase o blood may be expressed as (Srivastava and Srivastava, 983) as

5 478 Devajyoti Biswas and Moumita Paul p ( C) ( v u ) v ( C) r z r ( C) s( C)( ) v ( ) ( ) cos, CS v p v C g r z r r r p ( C) ( v u ) u ( C) r z z ( C) s( C)( ) u ( ) ( ) sin, CS u p u C g r z r r (3) (4) ( C) v ( C) u ( C) v. r z r (5) Particulate phase: p C p( v p u p ) v p C CS( v v p) C pg cos, r z r (6) p C p( v p u p ) u p C CS( u u p) C pg sin, (7) r z z ( Cv p) ( Cu p) ( Cv p), r z r (8) where ( rz, ) are (radial, axial) coordinates, ( u, u p) and ( v, v p) are the respective axial and radial velocities o luid and particle phases, (, ) are corresponding densities o two stages, ( ) is the suspension viscosity, C denotes the constant volume raction s s C density o the particles (called hematocrit), p is the pressure, S is the drag coeicient o interaction between these two (luid, particle) phases, g is the acceleration due to gravity and is the inclination o the tube to the horizontal. The expression or the drag coeicient o interaction S and empirical relation or the viscosity o suspension s, may be taken (Srivastava, 995; 22) as p C 9 [4 3(8C 3 C ) 3 C] S, 2 a (2 3 ) s, qc (9) () 7 K q.7 exp[2.49c exp(.69 C)], () T where a is the particle radius, is the plasma viscosity and T is measured in absolute temperature. We introduce the ollowing dimensionless quantities in the oregoing analysis:

6 AAM: Intern. J., Vol., Issue (June 25) 479 dp z d r R dp S z, d, r R,,, dz, S, R R R R R dz q R 2 / ( u, u ) A AR u u L L L L R F n p,(, p),(, ) (, ) /,, U (4( C) g) 2 q qr G, U, (4 C g) 4 p q (2) where U is the maximum velocity at the axis o a horizontal tube with radius R and q is the negative o the pressure gradient, in case o an unobstructed uniorm horizontal tube. Due to the non-linearity o the convective acceleration terms, integration o equations (3-8) is a diicult task. It is already reported by many investigators (Srivastava, 995, 22; Sankar and Lee, 29) that the radial velocity being very small or the case o a mild stenosis ( / R ), subject to additional conditions Re( n n / L) and n n R/ L~O(), can be neglected. In view o the above simpliied orm o above partial equations (3)-(8), governing the steady low o a particulate (particle- luid) suspension with the above parameters, can be expressed in the ollowing coupled equations d p s( C) u ( C) ( C) ( r ) CS( u p u ) ( C) g sin, d z r r r (3) dp C CS( u u p) C pg sin, dz (4) In integrating equations (3-4), boundary conditions employed are the ollowing: u us at r R, (5) u, up=inite at r R, (6) where u s is the slip velocity at the tapered constricted wall (Biswas and Chakraborty, 29). Using the non-dimensional quantities as included in equation (2), the ollowing equations have been obtained in dimensionless orm. The low geometry expressed in equation () can be written as R ( z)[ A( L ( z d) ( z d) )], d z d L, n n ( z), otherwise. (7)

7 48 Devajyoti Biswas and Moumita Paul The equations (3, 4) governing the luid low reduce to the orms dp s u 4( C) ( C) ( r ) CS( up u ) sin / F, dz r r r (8) dp 4 C CS( u u p) sin / G. (9) dz Boundary conditions inserted in equations (5, 6), or integrating the above (8-9) equations, reduce to the orms u u at r R, s u, u =inite at r R. p (2) (2) The non-dimensional low rate Q can be expressed as Q Q 4 r[( C) u Cu p] dr, ( R ) q /8 (22) 4 R R where R Q 2 r[( C) u Cu p] dr R is the volumetric low ratio. The non-dimensional shear stress at a radial distance r is given by qr ( C) u s / r. (23) 3. Integrals The expressions o velocity or luid and particle phases in non-dimensional orms are obtained by straight orward integration o equations (8, 9) as ollows: u log( R / r) u log( R / R) s 2 dp R log( R / r) r log( R / R) R log( r / R) { h }[ ], ( C) dz log( R / R) s (24) log( R / r) dp R log( R / r) r log( R / R) R log( r / R) up u { h}[ ] log( R / R) ( C) dz log( R / R) 2 s s

8 AAM: Intern. J., Vol., Issue (June 25) 48 4 dp [ k C ], CS dz (25) where sin sin h, k,. 4H 4G H F G The non-dimensional low rate rom equation (22), is given by 2 2R R R R R Q ( R R )[ ] u log( / ) s s ( R R ) dp ( R R ) 8 dp { h}[( R R ) ] ( R R )( k C ). ( C) dz log( R / R ) S dz (26) The expression or pressure gradient dp can be obtained directly rom equation (26) as dz dp ( C) si( z), (27) dz Where Q ( R R )[ Mus N ] I() z ( R R )( N ) And h k 8 C( C) R,,,, ( C) S S R R log( R / R ) s ( R R ) N ( R R ). log( / ) R R 2 8 s 2 M From equation (27), the expression or the pressure drop, obtained as p across the stenosis, may be p L dp ( ) dz dz L ( C) I( Z) dz. (28) s The resistance to low (impedance) is given by (using equation (28)) p ( C ) sj ( z ), (29) Q Q where

9 482 Devajyoti Biswas and Moumita Paul L J ( z) I( z) dz d dl I( z) dz I( z) dz I( z) dz. R R d d L L The irst and the third integrals in the expression or J(z) are straight orward whereas the analytic evaluation o the second integral is a ormidable task. In view o this, one can obtain the inal expression or λ as ( C ) s Q 2 ( R ) Q ( R )[{ } u {( R ) }] log(/ ) log(/ ) ( LL ) 2 2 s R R R 2 2 ( R ) ( R ) log(/ R ) dl / Q( C) s I( z) dz. d (3) The expression or wall shear stress, may now be written rom equation (23) as R dp ( R R ) ( C) u ( ){ log( / ) }. (3) log( R / R ) dz 2R 2 Rlog( R / R ) s s h R R R The shear stress at the throat o the stenosis can be computed (at z d L/2, R, R s) rom equation (3) as dp (( ) R ) s { h }{( )log(( ) / R ) } log(( ) / R ) dz 2( ) ( C) s us, 2 ( )log(( ) / R ) where (32) dp ( ) ( C) sh( z) dz is the pressure gradient at the stenotic throat and Q (( ) R )[ Ous P ] H( z). (( ) )( P) R 2( ) (( ) R ) O, P (( ) R ). ( ) log(( ) / ) log(( ) / ) 2 R R R

10 AAM: Intern. J., Vol., Issue (June 25) Results and Discussions Analytic expressions or the important low variables are presented in an earlier section and their variations with several low parameters are considered here. For this purpose, computer codes have been developed to evaluate the eects o various low variables as included in equations (24) to (32). The variations o these low variables with dierent relevant parameters, like hematocrit C, maximum height attained by the stenosis δ, slip velocity u s in the stenotic wall, artery inclination α, tapering angle Φ, shape parameter n ( 2) and catheter radius R in the annular region, are presented graphically or better understanding o the problem. For this computation, we have considered some numerical measures like δ=-.5; u s =,.5; α= (horizontal tube), 3 (inclined vessel); C=-.6; R =.,.2,.3; ξ(=tanφ)= -.,,. or tapering cases Φ <,=, > ; n = 2, 6, ; a =4-6 m; T=25.5 C (Srivastava, 22); R =8-5 m; q =2, kgm -2 s -2 (Usha and Prema, 999); =25 kgm -3 ; =25 kgm -3 (Chakraborty et al., 2); Q= (Sankar and Lee, 29); z = d to d + L and R r R(z). In the oregoing analysis, an eort is taken to indicate the variations in low characteristics due to such parameters. The present analysis includes (i) the model o Chakraborty et al. (2) when R =, Φ = ; (ii) the analysis o Srivastava (22) or u s =, α = and R =, Φ = ; (iii) the Srivastava and Rastogi (2) model when u s = α, α= and Φ=; (iv) the tapered or non-tapering (Φ ) models with slip (u s > ) and zero slip (u s = ) or inclined (α > ) and horizontal (α = ) vessels, with or without stenosis (δ ); (v) the catheterized tapering or non-tapering (Φ ) models with slip or no-slip (u s ) cases or inclined or horizontal (α ) tube, with or without constriction (δ ), as its special cases. The velocity, in equations (24-25) is a unction o several low parameters and co-ordinates. The variations o axial velocity u versus the radial distance r or dierent values o R and α are shown in Figures (3-4). Figures (5-6) shows the variation o axial velocity u versus radial distance r or dierent values o slip velocity u s and hematocrit C. The variation o the pressure gradient, using equation (27), with the axial distance, or dierent values o catheter radius R, inclination α, slip velocity u s and hematocrit C are depicted in Figures (7-). p

11 484 Devajyoti Biswas and Moumita Paul In this unidirectional low, the proiles or the axial velocity versus the radial distance through the annular region (R r R), clearly indicate a deviation rom the usual parabolic proiles. As the radial co-ordinate r increases in the ull scale rom R to R. The velocity increases rapidly to a greater value, whererom, it gradually decreases to a lower value at or near the vessel wall. It is observed rom Figure 3 that as the catheter radius R increases, velocity

12 AAM: Intern. J., Vol., Issue (June 25) 485 decreases. As expected, the velocity decreases with the rise in catheter radius (Figure 3) but it is ound that the velocity increases, with the increase in inclination (α) o the artery (Figure 4).

13 486 Devajyoti Biswas and Moumita Paul The velocity increases with slip employed at the stenotic wall. Its values are higher or the lows with slip (u s > ) than those with no slip (Figure 5). The proiles indicate that an increase in hematocrit C decreases the velocity o the blood (Figure 6). The magnitude o the velocity shows the lowest or converging tapering tube, higher in case o a non-tapered artery and highest or the diverging tapering vessel.

14 AAM: Intern. J., Vol., Issue (June 25) 487 It could be noticed that the pressure gradient increases rom a lower magnitude at one end o the stenosis, and gradually attains the maximum at the stenosis throat and there rom, it decreases to almost the same lower value. However pressure gradient attains the highest value at the throat o a stenosis and the lower value at the two ends o a stenosis.

15 488 Devajyoti Biswas and Moumita Paul As the catheter radius R increases the pressure gradient also increases (Figure 7), but the pressure gradient decreases with the increase in inclination (α) o the artery (Figure 8). It may be noted that in all such variations, pressure gradient decreases with the employment o a velocity slip (u s ) at the vessel wall (Figure 9) but increases with increase with the hematocrit C (Figure ). Also or a diverging tapering with taper angle Φ >, the pressure gradient is lower as compare to converging tapering (Φ < ) and without tapering (Φ = ). The variation o the resistance to the low λ versus the maximum height o the stenosis δ or dierent magnitudes o the parameters n, Φ, α and or dierent values o C, n and Φ, are shown in Figures -2. The variation o impedance λ with hematocrit parameter C or dierent values o n, u s and Φ, is presented in Figure 3 and, Figure 4 shows the proile or λ versus δ or variation in R, n and Φ. In Figure 5, the proile o the wall shear stress distribution τ R against the axial distance z or variation o n, R and Φ, is drawn. Figure 6 shows the variation o the shear stress at the stenotic throat versus stenosis height δ or dierent values o C and u s. In Figure 7 variation o τ s with R or dierent values o u s and δ is shown. In this catheterized tapering inclined artery region, it could be noticed that the resistance (impedance) to low λ increases with the maximum stenosis height δ and, decreases with the rise in both the shape parameter n and inclination α o the artery (Figure ), but the impedance λ increases as the hematocrit C increases (Figure 2). In Figure 3, it is clearly noticed that the resistance λ, experienced by the streaming luid over the whole arterial segment in the annular region increases with the hematocrit C but decreases with the slip velocity u s attained by the luid at the constricted wall and shape parameter n( 2) o the stenosis. It is seen in Figure 4 that as catheter radius R increases, impedance (λ) to low increases. However, resistance increases with stenosis height δ but decreases with the rise in

16 AAM: Intern. J., Vol., Issue (June 25) 489 shape parameter n. For any given value o stenosis height δ as the hematocrit increases rom to., resistance to low increases steeply, the increment is relatively slower rom C =. to.5 and again, it goes on increasing rapidly rom C =.5 to.6 (Figure 3). However, an

17 49 Devajyoti Biswas and Moumita Paul employment o velocity slip at vessel wall, decreases the resistance i the other parameters are kept constant. The behaviour o resistance in tapering region (Φ >,=, < ) o the constricted artery in this annular low is relected as, λ diverging tapering < λ non-tapering < λ converging tapering. The impedance (λ) to the low is inluenced by the shape parameter as its magnitude is seen to be lower in asymmetric stenosis (n > 2) than that in axi-symmetric stenosis (n = 2). As expected, resistance λ increases with the rise in catheter radius but it decreases, as we consider a horizontal artery (α = ) to an inclined tube (α > ). The consideration / R (or mild stenosis case) could make this analysis useul only in the ormative (early) stage o a stenosis. The range o the parameter δ is restricted upto.5 (i.e., 28% o area reduction), as beyond this quantity, low separation may happen or even very low magnitude Reynolds number (Young, 968). Numerical results urther reveal that the resistance assumes an asymptotic magnitude at about n=, that in turn implies that no signiicant change in the low would occur beyond this magnitude o shape parameter n. This behaviour in λ conorms to the results obtained by Srivastava (22), Srivastava and Rastogi (2), and, Chakraborty et al. (2).

18 AAM: Intern. J., Vol., Issue (June 25) 49 The wall shear stress τ R in this annular stenotic region (d z d+l ) increases rapidly rom its approached value (initiation point) at z=d in the upstream o the stenotic throat and attains its maximum magnitude at the throat, whererom it decreases rapidly to a lower magnitude at the termination (end point) o the constricted region at z = d+l (Figure 5). The wall shear stress τ R decreases with increasing shape parameter n in the upstream o the throat but this behaviour reverses in the downstream. As the catheter radius R increases in the constricted annular region (with other parameters keeping ixed), τ R decreases rom a higher value to a lower one. It is also observed that wall shear stress at any axial distance increases with tapering angle Φ (rom Φ >, Φ =, Φ < ) o the inclined tube as ollows τ R (divergent tapering tube) < τ R (non-tapering tube) < τ R (converging tapering tube). It could be observed rom Figures 6 and 7 that the wall shear stress at the peak o the stenosis τ s, increases with both stenosis height δ and hematocrit C, or any given value o the angle o inclination o the artery (α) and catheter radius (R ). However, τ s decreases with velocity slip u s, at the stenotic wall in the constricted annular region. As the catheter radius R increases or given magnitudes o C and α, τ s increases as δ increases but it decreases with velocity slip at the wall.

19 492 Devajyoti Biswas and Moumita Paul

20 AAM: Intern. J., Vol., Issue (June 25) Conclusion To account or the combined inluence o several low parameters like slip velocity, hematocrit, catheter radius, tapering geometry, shape parameter and inclination o the vessel a two phase annular model, on assuming that blood is represented by a suspension o erythrocytes in plasma, has been considered. The annular region is spaced within an inclined, tapering, constricted (-asymmetric stenosis) wall and a co-axial catheter. Analytical expressions o the low variables are obtained and the variations o the important low characteristics such as, velocity, pressure gradient, resistance (impedance) to low and wall shear stress, have been included. The low characteristics (viz., resistance to low, wall shear stress in the stenotic region and shear stress at the throat o the constriction) increase with hematocrit, stenosis height and catheter radius. It may be interesting to observe that the resistance to the low decreases with the inclination o the tapering vessel. However, all the three low characteristics decrease with velocity slip at the stenotic wall. Both the resistance to the low and the wall shear stress are seen to be higher in the case o axially symmetric stenosis than those in case o axially nonsymmetric ones. It is also exhibited that both these low variables attain the lower magnitude in a diverging tapering region than that in the non-tapering tube which value is lower than that obtained in a converging tapering vessel. The present analysis includes the models o Srivastava, Srivastava and Rastogi and Chakraborty et al., uniorm, tapering and catheterized models or inclined or horizontal vessels with velocity slip or zero slip at stenotic wall, as its special cases. It is already reported that arterial stenosis or atherosclerosis is a common and wide spread disease that may severally inluence human health in general and cardiovascular system in particular. In the present analysis, a mild stenosis ormation has been dealt with. However, its gradual growth rom mild to moderate stage and moderate to severe orms at certain locations eg., carotid, biurcations, coronary arteries, distal to abdominal aorta etc., could lead to serious complications inside the body and several health hazards, like reduction in blood supply, cut o in nutrition supply, stroke, thrombosis, renal problems, circulatory disorders etc. Theoretical models can throw some insight into the complicated situations and in turn, it could suggest some measures in regulating the normal blood low and nutrition supply to each and every body organ, tissue, cell etc. In this analysis, it is observed that a velocity slip condition employed at the constricted tapering vessel wall may accelerate the low o the one hand and, retard the resistance to low and wall shear stress on the other. Thus, by employing an appropriate velocity slip, damages to the diseased vessel wall could be reduced and bore o the blood vessel could be enlarged. It is thereore necessary to determine an appropriate velocity slip in accordance with the hematocrit, stenosis size, artery radius and other physiological situations. Theoretical analysis could be improved by considering a two-dimensional, pulsatile, two-layered annular model with employing slips in both radial and axial directions and, with permeability o the arterial wall. Such models could be used as a device in the initiation o atherosclerosis and also in the treatment modalities o cardiovascular complications, cardiac arrest, haematological, stroke, thrombosis, renal and sickle cell diseases and other arterial disorders.

21 494 Devajyoti Biswas and Moumita Paul Acknowledgement This research work is supported by the UGC, New Delhi, India (MRP Grant Reerence No. : AU:MS:UGC-MRPF: 2:). Authors are also grateul to the Editor-in-Chie Pro. A. M. Haghighi and the Reviewer, or their valuable comments. REFERENCES Biswas, D. and Chakraborty, U.S. (29). Pulsatile low o blood in a constricted artery with body acceleration, Appl. Appl. Math., Vol. 4, pp Biswas, D. and Paul, M. (22). Mathematical Modelling o Blood Flow Through Inclined Tapered Artery With Stenosis, AUJST, Vol. (II), pp -8. Charm, S.E. and Kurland, G.S. (974). Blood low and microcirculation, John Wiley, New York. Chakraborty, U.S., Biswas, D. and Paul, M. (2). Suspension model blood low through an inclined tube with an axially non-symmetrical stenosis, Korea Australia Rheology Journal, Vol. 23(), pp Fung, Y.C. (98). Biomechanics: Mechanical Properties o Living Tissues, Springer-Verlag, New York. Guyton, A.C. (97). Text Book o Medical Physiology, W.B. Saunders, Philadelphia. Haynes, R.H. (96). Physical basis on dependence o blood viscosity on tube radius, Am. J. Physiol., Vol. 98, pp Ku, D. (997). Blood low in arteries, Ann. Rev. Fluid Mech., Vol. 29, pp Lightoot, E.N. (974). Transport Phenomenon in Living Systems, New York, Wiley. Maruti Prasad, K. and Radhakrishnamcharya, G. (28). Flow o herschel-bulkley luid through an inclined tube o non-uniorm cross-section with multiple stenoses, Arch. Mech., Vol. 6(2), pp MacDonald, D.A. (979). On steady low through modelled vascular stenosis, J. Biomechanics, Vol. 2, pp 3-2. Mekheimer, K.S. and Kothari, M.A.E. (28). The micropolar luid model or blood low through a tapered artery with a stenosis, Acta Mech. Sin., Vol. 24, pp Mekheimer, K.S. and Kothari, M.A.E. (2). Suspension model or blood low through arterial catheterization, Chem. Eng. Comm., Vol. 97, pp Merril, E.W. and Pelletier, G.A. (967). Viscosity o human blood: transition rom Newtonian to non-newtonian, J. Appl. Physiol., Vol 23, pp Sankar, D.S. and Lee, U. (29). Mathematical modelling o pulsatile low o non-newtonian luid in stenosed arteries, Commun. Nonlinear Sci. Numer. Simulat., Vol. 4, pp Srivastava, V.P. (995). Particle-luid suspension model o blood low through stenotic vessels with applications, Int. J. Biomed. Compt., Vol. 38, pp Srivastava, V.P. (22). Particle suspension blood low through stenotic arteries: eects o hematocrit and stenosis shape, Indian J. Pure Appl. Math., Vol. 33, pp Srivastava, V.P. and Rastogi, R. (2). Blood low through a stenosed catheterized artery: Eects o hematocrit and stenosis shape, Comput. Math. Appl., Vol. 59, pp Srivastava, L.M. and Srivastava, V.P. (983). On Two-phase Model o Pulsatile Blood Flow with Entrance Eects, Biorheology, Vol. 2, pp Srivastava, V.P., Rastogi, R. and Vishnoi, R. (2). A two-layered suspension blood low through an overlapping stenosis, Comput. Math. Appl., Vol 6(3),

22 AAM: Intern. J., Vol., Issue (June 25) 495 Sud, V.K. and Sekhon, G.S. (985). Arterial low under Periodic body acceleration, Bull. Math. Biol., Vol 47(), pp Usha, R. and Prema, K. (999). Pulsatile low o particle-luid suspension model o blood under periodic body acceleration, Z. Angew. Math. Phys., Vol. 5, pp Womersley, J.R. (955). Method or the calculation o Velocity rate o Flow and Viscous Drag in Arteries when pressure gradient is known, J. Physiol., Vol. 27, pp Young, D.F. (968). Eects o Time-Dependent Stenosis on Flow Through a Tube, J. Eng. Ind. Trans. ASME., Vol. 9, pp