Effect of Periodic Body Acceleration in Blood Flow through Stenosed Arteries A Theoretical Model

Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 Effect of Periodic Body Acceleration in Blood Flow through Stenosed Arteries A Theoretical Model D. S. Sankar*, A. I. M. Ismail School of Mathematical Sciences, University Science Malaysia, Penang, Malaysia Abstract The pulsatile flow of blood through stenosed narrow arteries with body acceleration is analyzed by treating blood as Herschel-Bulkley fluid. Perturbation method is used to solve the resulting system of nonlinear partial differential equations with appropriate boundary conditions. The expressions for the shear stress, velocity, flow rate, wall shear stress, plug core radius and longitudinal impedance are obtained. The effects of different parameters on these flow quantities are discussed. It is found that the velocity and flow rate increase with the increase of the pressure gradient, body acceleration and pulsatile Reynolds number, whereas they decrease with the increase of the yield stress and depth of the stenosis. It is noted that the plug core radius and wall shear stress increase with the increase of the yield stress and depth of the stenosis, but they decrease with the increase of the body acceleration and pulsatile Reynolds number. It is also observed that the estimates of the increase in the longitudinal impedance rise with an increase of the stenosis depth and yield stress and an opposite trend is noticed with an increase of the body acceleration. Keywords: body acceleration, pulsatile blood flow, Herschel-Bulkley fluid, stenosed artery, perturbation method, longitudinal impedance 1. Introduction The human body is subject to body accelerations or vibrations in many situations, such as swinging of kids in a cradle, vibration therapy applied to a patient with heart disease, travel of passengers by surface, water or air transport, sudden movement of the body in sports activities and so on [1]. In certain situations, like a passenger sitting in a bus/train etc, our whole body may be subjected to vibrations. In some other situations, a specific part of our body might be subjected to vibrations as for example, while operating a jack hammer or lathe machine, driving a car etc []. Prolonged exposure to high level unintended external body accelerations may cause disturbance to the blood flow [3] and this leads to serious diseases which have symptoms like headache, abdominal pain, increase in pulse rate, venous pooling of blood in the extremities, *Corresponding Author: D. S. Sankar: sankar_ds@yahoo.co.in loss of vision, hemorrhage in the face, neck, eyesockets, lungs and brain [4]. The information about the magnitude, frequency and duration of vibration and their orientation with respect to the body of subject is very important in analyzing the effects of body accelerations on the human body [5]. Hence, the study of blood flow in arteries with body acceleration plays an important role in the diagnosis and therapeutic treatment of health problems [6]. Since the presence of body acceleration in blood flow causes major changes in the human physiology; several theoretical studies were performed to understand the influences of body acceleration on the physiologically important flow quantities [3-5, 7-9]. The development of arteriosclerosis in blood vessels is one of the serious arterial diseases, which is attributed to the accumulation of lipids on the arterial wall [10]. Arteries are narrowed by the development of atherosclerotic plaques that protrude into the lumen, resulting in stenosed arteries [11]. One

44 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries of the most serious consequences of the stenosis in the arteries is the increased resistance and the associated reduction of the blood flow to the particular vascular bed supplied by the artery [1]. Further, the continual flow of blood may lead to shearing of the superficial layer of the plaques, parts of which may be deposited in some other blood vessel forming thrombus [13]. Hence, the presence of a stenosis can lead to serious circulatory disorder. Several researchers discussed the blood flow characteristics due to the presence of a stenosis in the arterial lumen of blood vessels [14-17]. Long et al. [18] and Chakravarthy et al. [19] mention that the hydrodynamic factors play an important role in the formation of stenosis [18, 19] and hence, the study of the blood flow through stenosed blood vessels is very important. Many studies were performed to analyze the steady flow of blood, treating it as a Newtonian fluid [0-3]. It is well known that the blood flow through arteries is highly pulsatile, and thus more attempts were made to study the pulsatile flow of blood, treating it as a Newtonian fluid [16, 17, 4]. The Newtonian behavior of blood is acceptable only in larger arteries at high shear rates. But, blood, being a suspension of cells in plasma, exhibits remarkable non Newtonian behavior when it flows through narrow arteries at low shear rates, particularly, in diseased state; the actual flow is distinctly pulsatile [13, 5-7]. Several attempts have been made to study the pulsatile flow of blood through stenosed narrow arteries treating it as a non Newtonian fluid [1, 14, 8, 9]. Nagarani and Sarojamma [6] mathematically analyzed the pulsatile flow of Casson fluid for blood flow through stenosed narrow arteries with body acceleration, using perturbation method. Herschel-Bulkley fluid and Casson fluid are non-newtonian fluid models which are generally used in the studies of blood flow through narrow arteries [3, 30]. Tu and Deville [1] mention that blood obeys Casson s equation only for moderate shear rate, whereas, the Herschel-Bulkley equation can still be used at low shear rates and represents very closely what is occurring in blood [19]. Chaturani and Ponnalagar Samy [9] mention that for tube diameter 0.095mm blood behaves like Herschel Bulkley fluid rather than power law and Bingham fluids. Iida [31] reports as follows: The velocity profiles in the arterioles whose diameter less than 0.1mm, are generally explained by Casson and Herschel-Bulkley fluid models. However, velocity profiles in the arterioles whose diameters less than 0.065mm does not conform to the Casson fluid model, but, can still be explained by Herschel Bulkley fluid model. Also, the Casson fluid s constitutive equation has only one parameter namely the yield stress, whereas, the Herschel- Bulkley fluid s constitutive equation has one more parameter, namely the power law index n. Thus, one can get more detailed information about flow characteristics by using the Herschel-Bulkley fluid model rather than Casson fluid model. Hence, it is appropriate to model blood as Herschel Bulkley (H-B) fluid rather than Casson fluid when it flows through narrow arteries at low shear rates. So far no one has studied the pulsatile flow of H-B fluid for blood flow through stenosed narrow arteries with body acceleration. Hence, in the present study, we perform a theoretical analysis for pulsatile flow of blood through stenosed narrow arteries with body acceleration at low shear rates, treating blood as Herschel-Bulkley fluid model. The layout of the paper is as follows: Section formulates the problem as a mathematical model, while section 3 solves the problem using perturbation method. In section 4, the effects of pulsatility, stenosis, body acceleration and non-newtonian behavior of blood on various flow quantities are discussed through appropriate graphs. The results are summarized in the concluding section 5.. Mathematical Formulation Consider an axially symmetric, laminar, pulsatile and fully developed flow of blood (assumed to be incompressible) in the axial direction through a circular narrow artery with stenosis. The geometry of the stenosed artery is

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 45 Fig. 1: Geometry of the stenosed artery. shown in Fig. 1. The segment of the artery under study is considered as long enough so that the entrance, end and special wall effects are neglected. Assume that the blood flow is subjected to the periodical body acceleration. The flowing blood is modeled as H-B fluid. Cylindrical polar coordinate system ( r, ψ, z) is used to analyze the flow. The wall of the stenosed artery is assumed as rigid. It is reported that the radial velocity is negligibly small and is neglected for low Reynolds number flow [6]. The momentum equations governing the flow reduce to u p 1 ρ = τ + t z r r ( r ) F( t ) (1) p 0 = () r where u is the axial component of the velocity, p is the pressure, ρ is the density, t is the time, τ = τ rz = τ rz is the shear stress, F ( t ) is the body acceleration. The constitutive equation of the H-B fluid (which represents the blood) is given by τ = u = 0 r 1 n μ H u r 1 n + τ H if τ τ H (3) if τ (4) τ H where, τ H is the yield stress and μ H is the coefficient of viscosity for Herschel Bulkley 1 fluid with dimension ( ML T ) n T. The geometry of the stenosis is given by R z πz R δ 1+ cos if z z = R0 otherwise 0 0 z0 (5) where δ denotes the half of the maximum projection of the stenosis from the wall of the artery to the lumen of the artery such that δ R 0 << 1, R( z ) is the radius of the artery in the stenosed region, R 0 is the radius of the normal artery. The boundary conditions are (i) τ is finite at r = 0 (6) (ii) u = 0 at r = R(z) (7) Since the pressure gradient is a function of z and t, we take p (z, t) = A 0 + A 1 cos ω p t z (8) where A 0 is the steady component of the pressure gradient, A 1 is the amplitude of the fluctuating component of the pressure gradient

46 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries and ω p = π fp, f p is the pulse frequency in Hz. A0 1 Both and A are functions of z [6]. The periodic body acceleration in the axial direction is given by = cos 0 ( + b ) F t a ω t φ (9) where a0 is the amplitude, ω b = π fb, f b is the frequency in Hz and is assumed to be small so that the wave effect can be neglected [4], φ is the lead angle of F ( t ) with respect to the heart action. Let us introduce the following non dimensional variables z = z R, Rz = R z 0 R, r = r 0 R, t = tω, 0 ω = b δ u τ ω, δ =, u =, τ =, ω p R 0 A0 R0 A 0 R 0 4μ0 τ H 0 θ =, = R ωρ 1 0 α, e= A, B= a A0R 0 μ0 A0 A0 (10) n 1 where μ0 = μ H, has dimension as R0 A0 that of Newtonian fluid s viscosity. α is the pulsatile Reynolds number or generalized Wormersly frequency parameter and when n = 1, we get the Womersly frequency parameter for Newtonian fluid. Substitution of Eq.(10) into Eqs. (1), (3) and (4) yields the following non-dimensional form of the momentum and constitutive equations. α u = 41 ( + ecost) + 4Bcos( t + ) ( r ) t ω φ r r τ (11) 1 n 1 u τ = + θ if τ θ (1) r u r = 0 if τ θ (13) The boundary conditions (in nondimensional form) are (i) τ is finite at r = 0 (14) (ii) u = 0 at r = R(z) (15) The geometry of the stenosis in dimensionless form is given by π z 1 δ 1+ cos if z z0 R( z) = z0 1 otherwise (16) The non-dimensional volume flow rate Q(t) is given by R ( z) Q z, t = 4 u( z, r, t) rdr (17) 0 4 where Qt = Qt π R 0 A 0 8μ, Q( z, t ) is the volumetric flow rate and μ is the viscosity of the Newtonian fluid. 3. Method of solution Since Eqs.(11) and (1) form a system of nonlinear partial differential equations, it is not possible to get an exact solution to this system of differential equations. Hence, perturbation method with pulsatile Reynolds number α as the small parameter of the series expansion is used to solve this system of nonlinear partial differential equations. As the present study deals with pulsatile flow of blood and the square of the pulsatile Reynolds number ( α ) occurs naturally when we non-dimensionalize the momentum equation (1), it is more appropriate to expand the Eqs. (11) and (1) in the perturbation series about α. Let us expand the velocity u in the perturbation series about the square of the pulsatile Reynolds number α as shown below (where α < < 1). 0 1 urzt (,, ) = u( rzt,, ) + α u( rzt,, ) +... (18) Similarly, one can expand the shear stress τ rzt,,, the plug core radius R ( zt,, the plug core velocity up (, ) shear stress ( zt, ) p p ) z t, and the plug core τ in terms of α.

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 47 Substituting the perturbation series expansions of u and τ in Eq. (11) and then equating the constant term and α term, we get ( rτ 0 ) = r ( 1+ ecost) + Bcos( ωt + φ ) (19) r u0 = t r r ( r ) τ (0) 1 Using the binomial series approximation in Eq. (1) and applying the perturbation series expansions of u and τ in the resulting equation and then equating the constant term and α term, one can obtain u 0 n = τ 0 1 [ τ0 nθ ] (1) r u1 n = nτ 0 τ1[ τ0 ( n 1) θ ] () r Using the perturbation series expansions of u and τ in the boundary conditions (14) and (15), we obtain (i) τ 0 and τ 1 are finite at r = 0 (3) (ii) u0 = 0 and u1 = 0 at r= 0 (4) R 0 p Integrating Eq. (19) between 0 and, and then using the condition that τ 0 p is finite at r = 0, we obtain 0p τ = g t R (5) 0p where g( t) = ( 1+ ecost) + Bcos( ω t+φ ). R 0 p Integrating Eq. (19) between making use of Eq. (5), one can get and r and τ 0 = g ( t ) r (6) Substituting Eq. (6) into Eq.(1) and integrating between r and R(= R(z)) with the help of first of the boundary condition (4), we obtain n 1 n 1 + r q r u0 = g() t R R 1 1 ( n+ 1) R R R where q g( t) ( n (7) = θ ). The plug core velocity can be obtained from Eq. (7) as u 0 p n 1 n n 1 + R 0p q R0p u0p g() t R R = 1 1 ( n + 1) R R R (8) Neglecting the terms with α and higher powers of α in the perturbation series expansion of R and using Eq. (5), the expression for p R 0 p can be obtained as θ r = R 0p= 0p = =q τ θ g t () (9) Using Eq. (9) in Eq. (8), one can get n 1 n n 1 + q q q u0p g() t R R = 1 1 ( n + 1) R R R (30)

48 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries Similarly solving Eqs. (0) and () with the help of the boundary conditions (3) and (4) and Eqs. (5) (30), the expressions for τ1p, τ1,u1 and u 1 p can be obtained as shown below. ( n 1) τ1p = () ( + ) ( + ) n+ n n q q n q g t R DR n 1 R R n 1 R n n n + n + 3 r r τ 1 = g z R DR ( n + 1)( n + 3) R R ( + ) n+ 1 n+ 3 n 1 q n + r r 3n n q R n+ R R R ( n+ )( n + 3) R r (31) (3) n+ 1 n+ n 1 3 n r r u1 = n g() t R DR ( n ) ( n 3) + + + ( n 1) ( n 3) R R + + n n+ 1 n 1 q r r + ( n )( n 3)( n 1) n 1 n n 3 n 1 R + + + + + + + + R R n + 9n + 11n + 3 + n + 6n+ 3 3 r n n n 1 q r r + ( n + 1) ( n + ) + n n + R R R 3n + n n 1 q r + 1 ( n )( n+ ) ( n+ 3) R R n+ 4 n R n+ 1 n+ 3 n 1 3n + n q r + 1 ( n 1)( n + )( n + 3) R R (33) n 1 n n 1 3 n + + q q 1p = () ( + ) ( + ) + u n g t R DR n n 3 ( n + 1) ( n+ 3) R R ( + )( + )( + )( + ) n n+ 1 q n 1 q q + ( n + )( n + 3)( n + 1) + n 1 n n 3 n 1 R R R 3 q ( n + 9n + 11n+ 3) + ( n + 6n+ 3) R n ( n 1) q q n q + ( n + 1) ( n + ) + n ( n + ) R R R 3n + n q q + 1 n ( 1)( n+ )( n+ 3) R R n+ 3 n 1 ( )( + )( + ) n+ 1 3 n + n n 1 q q + 1 n n n 3 R R n+ 4 n (34)

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 49 where D= ( 1 g)( dg dt). The wall shear stress τ w is a physiologically important quantity which plays an important role in determining n 1 τw = ( τ0 + α τ 1) ( g() t R) α r= R =g() t R 1 ( n + )( n + 3) ( + ) ( ) ( + ) ( + ) R B n+ 3 q q n n n 1 n n 3 3 n n R R aggregate sites of platelets [11]. The expression for wall shear stress τ w is given by (35) From Eq. (17) and the expressions for velocity, one can derive the expression for the volumetric flow rate Q(z, t) as R0p R R0p R Q(z, t) = 4 r u 0 0p dr + r u R 0 dr + α r u1p dr + r u1 dr 0p 0 R 0p () n 3 n+ 3 g t R R q q ( n ) n( n 3) ( n n ) n 1 ndr = + + + + α g() t R n + n + 3 R R 4 3 n n 1 4n + 1n + 5 q n( n 1) ( n + 3) q ( n n 11n + 6) q n + + ( n + 1)( n + 3) R ( n + 1) R ( n + 1) R 3 5 4 3 ( n 1)( n n 11n + 6) ( 4n + 14n 8n 45n 3n + 18) n+ 3 q q n R n n 1 n 3 R n+ 4 n+ 4 (36) ( + )( + ) The correction to the plug core radius R 1p can be obtained by neglecting the terms with 4 α and higher powers ofα in the perturbation series expansion of R in the following manner. The shear stress τ = τ0 + α τ 1 at r = R p is given by p From the Taylor s series of τ 0 and τ 1 about R 0p and τ = θ, one can obtain R 0 r= R 0p g() = = τ t (38) 1p 1 r R 0p τ ατ = θ (37) 0 + 1 r= R p Substitution of Eqs. (6) and (38) in the perturbation series expansion of R p yields 3 n+ n 1 ndr q n 1 q q Rp = q + α g() t R ( n 1) + R n R R (39) The longitudinal impedance of the artery is given by Λ= g( t) Q( z,t) (40)

50 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries From Eq. (17) and the expressions for velocity, one can derive the expression for the volumetric flow rate Q(z, t) as 4. Numerical simulation of the results The aim of this study is to bring out the effects of the body acceleration, pulsatility of the flow, non-newtonian nature of the blood and pressure gradient on the plug core radius, plug flow velocity, velocity, wall shear stress, flow rate and longitudinal impedance to flow. The flowing blood is modeled as H-B fluid. The values of the power law index n for blood flow problems are generally taken to lie between 0.9 and 1.1 and in this analysis, we have used the value of n as 0.95 for n < 1 and 1.05 for n > 1 [3]. The values of the non-dimensional yield stress θ for the blood of the normal subject are between 0.01 and 0.04 and in the diseased state, it is quite high and in such a case, the value of the yield stress is taken to lie between 0.05 and 0. [11]. The origin of the coordinate system is fixed at the middle of the stenosis where the stenosis projection is maximum. The stenosis is assumed to lie between z= Z 0 and z= Z 0. The pressure gradient parameter e is taken in the range 0.5 0.7 [6]. The value of pulsatile Reynolds number is generally taken as 0. and only to pronounce its effect, we have taken the range 0. 0.7 [33]. To discuss the effects of the body acceleration parameter B on the various flow quantities, its value is taken in the range 0 [6]. Half of the maximum projection of the stenosis is denoted by δ and its value is taken in the range 0 0.3 [6]. All the flow quantities are computed at the maximum projection of the stenosis (at z = 0) to bring out the effect of the depth of stenosis on them. 4.1 Plug core radius The variation of the plug core radius in a time cycle for different values of B, e and θ with α = φ = 0., n = 0.95 and δ = 0.15 is depicted in Fig.. It is seen that the plug core radius increases very slowly as the time parameter t increases from 0± to 90± and then it increases rapidly as the time t increases from 90± to 10± and then it decreases as t increases from 10± to 180± and then it increases nonlinearly as t increases from 180± to 10± and then it decreases rapidly as the time t increases from 10± to 40± and then it decreases very slowly as time t increases further from 40± to 360±. The plug core radius is maximum at 10± in the first half of the time cycle and it is maximum at 10± in the second half of the cycle. It is observed that for a given set of values of B and e and increasing values of the yield stress θ, the plug core radius increases marginally when time t lies between 0± and 90± and also between 40± and 360± and increases significantly when time t lies between 90± and 40±. The plug core radius decreases with the increase of either body acceleration parameter B or the pressure gradient e when all the other parameters were held constant. The decrease in the plug core radius is very small when the body acceleration parameter increases and the decrease in the plug core radius is significant when the pressure gradient increases. Fig. 3 depicts the variation of plug core radius with yield stress for different values of n, δ, φ and α with t = 60±, B = 1, and e = 0.5. It is seen that the plug core radius increases linearly with the increase of the yield stress θ. For a given set of values of δ, φ and α, the plug core radius decreases slightly with the increase of the power law index n. It is also noted that the plug core radius increases significantly with the increase of either the stenosis depth δ or the pulsatile Reynolds number α when all the other parameters were held fixed. One can notice that for a given set of values of n, δ and α, the plug core radius decreases considerably with the increase of the lead angle φ of the body acceleration with respect to the heart

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 51 action. Figs. and 3 show the effects of various parameters on the plug core radius of the blood flow through a stenosed narrow artery with body acceleration. Fig. : Variation of plug core radius in a time cycle for different values of B, e and θ with α = φ = 0., n = 0.95 and δ = 0.15. Fig. 3: Variation of plug core radius for different values of n, δ, φ and α with t = 60±, B = 1 and e = 0.5. 4. Plug flow velocity The variation of the plug flow velocity in a time cycle for different values of B, θ and α with n = 0.95, e = 0.5, φ = 0., δ = 0.1 and ω = 1 is illustrated in Fig. 4. It is clear that in the presence of the body acceleration, the plug flow velocity decreases rapidly with the increase of the time t from 0± to 10± and then it increases slowly when time t increases from 10± to 180± and then it decreases as time t increases from 180± to 10± and then it increases rapidly when time t increases from 10± to 360±. The plug flow velocity is maximum at 0± and 360± and minimum at 10± and 10±. In the absence of the body acceleration, the plug flow velocity decreases as time t increases from 0± to 180± and then it increases as time t increases from 180± to 360±, so that it is maximum at 0± and 360± and minimum at 180±. It is found that the plug flow velocity increases with the increase of the body acceleration when time t lies between 0± and 90±, 150± and 00±, and also between 70± and 360± and it decreases in the rest of the time cycle. It is seen that the plug flow velocity decreases marginally with the increase of the yield stress when all the other parameters where held constant. It is further noticed that for a given set of values of B and θ and increasing values of the pulsatile Reynolds number α, the plug flow velocity increases marginally when the time t lies between 0± and 180± and it decreases considerably when time t lies between 180± and 360±. Fig. 5 shows the variation of the plug flow velocity with stenosis depth δ for different values of n, φ and ω with B = 1, t = 60±, θ = δ = 0.1 and e = 0.5. It is clear that the plug flow velocity decreases nonlinearly (rapidly) when the depth of the stenosis increases from 0 to 0. and then it decreases very slowly from 0. to 0.45. It is to be noted that when the half of the depth of the stenosis δ is 0.45 (90% stenosis), the flow is very slow and its magnitude is almost zero. It is also observed that the plug flow velocity decreases with the increase of either the power law index n or the lead angle φ or the pulse frequency ω, but the decrease is marginal when the power law index n increases and is considerable when the lead angle φ or pulse frequency ω increases. Figs. 4 and 5 depict the effects of body acceleration, non-newtonian effects, depth of the stenosis and pulsatile Reynolds number on the plug flow velocity

5 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries when blood flows through a stenosed artery. Fig. 4: Variation of plug flow velocity in a time cycle for different values of B, θ and α with n = 0.95, e = 0.5, φ = 0., δ = 0.1 and ω = 1. Fig. 7. It is noticed that the velocity is maximum for the power law fluid with n = 0.95 and minimum for the H-B fluid model with n = 1.05. One can notice a flattened velocity profile for the H-B fluid models with n = 0.95 and n = 1.05, since H-B fluid model is a non-newtonian fluid model with non-zero yield stress. It is found that the velocity decreases marginally with the increase of the power law index and yield stress. It is of importance to note that the plot of the Newtonian fluid model is in good agreement with Fig. b of Nagarani and Sarojamma [6]. Fig. 5: Variation of plug flow velocity with depth of the stenosis for different values of n, φ and ω with B = 1, t = 60±, θ = δ = 0.1 and e = 0.5. 4.3 Velocity distribution Fig. 6 sketches the velocity distribution during a time cycle with n = 0.95, φ = α = 0., θ = δ = 0.1, B = ω = 1.0 and e = 0.5. Once can easily see the flattened velocity profile around the axis of the artery. It is observed that the velocity decreases when time t increases from 0± to 10± and then it increases as time t increases from10± to 180± and then it decreases as t increases from 180± to 40± and then it increases as t increases from 40± to 360±. It is found that the velocity is maximum at 0± and 360± and minimum at 10±. Velocity distribution for different fluid models with φ = α = 0., δ = 0.1, B = t = ω = 1.0 and e = 0.5 is shown in Fig. 6: Velocity distribution in a time cycle with n = 0.95, φ = α = 0., θ = δ = 0.1, B = ω = 1.0 and e = 0.5. Fig. 8 depicts the velocity distribution for different values of B, e, α, φ and δ with n = 0.95, θ = 0.1, t = 1 and ω = 1. It is seen that for a given set of values of the parameters B, e and α, the velocity decreases significantly with the increase of either the stenosis depth δ or lead angle φ of the body acceleration. It is found that the velocity increases marginally with the increase of the body acceleration B or the pulsatile Reynolds number α while all the other parameters are treated as constants. It is noticed that the velocity increases considerably with the increase of the pressure gradient e. Figs. 6, 7 and 8 show the effect of pressure gradient, body acceleration, non-newtonian effects and depth of the stenosis on the velocity of the

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 53 blood when it flows through a narrow stenosed artery. Fig. 7: Velocity distribution for different fluid models with θ = 0.1, δ = 0., φ = α = 0., ω = 1, t = 1, e = 0.5 and B = 1.0. stress is maximum at 0± and 360± and minimum at 10± and 10±. In the absence of the body acceleration, the wall shear stress decreases considerably as time t increases from 0± to 180± and then it increases as time t increases from 180± to 360±, so that it is maximum at 0± and 360± and minimum at 180±. It is found that for fixed values of e and α and the increasing values of the body acceleration parameter B, the wall shear stress increases when time t lies between 0± and 90±, 150± and 00±, and also between 70± and 360± and it decreases in the rest of the time cycle. For a given set of values of B and e and increasing values of the pulsatile Reynolds number α, the wall shear increases slightly when the time t lies between 0± and 180± and it decreases when the time t lies between 180± and 360±. It is observed that the wall shear stress increases with the increase of the pressure gradient e while all the other parameters are kept as constant. Fig. 8: Velocity distribution for different values of B, e, α, φ and δ with n = 0.95, θ = 0.1, t = 60±, and ω = 1.0. 4.4 Wall shear stress The variation of wall shear stress in a time cycle for different values of B, e and α with ω = 1, n = 0.95, φ = 0., θ = 0.05 and δ = 0.1 is depicted in Fig. 9. It is seen that in the presence of the body acceleration, the wall shear stress decreases rapidly with the increase of the time t from 0± to 10± and then it increases slowly when time t increases from 10± to 180± and then it decreases as time t increases from 180± to 10± and then it increases rapidly when time t increases from 10± to 360±. The wall shear Fig. 9: Variation of wall shear stress in a time cycle for different values of B, e and α with ω = 1, n = 0.95, φ = 0., θ = 0.05 and δ = 0.1. 4.5 Flow rate Fig. 10 depicts the variation of flow rate in a time cycle for different values of δ, α and φ with n = 0.95, B = ω = 1, θ = 0.05 and e = 0.5. It is clear that in a time cycle, the flow rate decreases rapidly as the time parameter t increases from 0± to 10± and then it increases as time t increases from 10± to 180± and then it decreases as the time t increases from 180± to 10± and then it increases rapidly as the time t increases further from 10± to 360±. It is found

54 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries that for a given set of values of α and φ, the flow rate decreases considerably with the increase of the stenosis depth δ. It is also noted that for fixed values of δ and φ and increasing values of α, the flow rate decreases slightly when the time t lies between 0± and 180± and increases slightly when the time parameter t lies between 180± and 360±. The same behavior is observed for the flow rate when the lead angle φ increases while all the other parameters were treated as constant, but the increase in this case is considerable. Fig. 11: Variation of flow rate with pressure gradient for different values of B, n and θ with t = 60±, δ = 0.1, α = φ = 0. and ω = 1. Fig. 10: Variation of flow rate in a time cycle for different values of θ, δ, α and φ with n = 0.95, B = ω = 1, θ = 0.05 and e = 0.5. The variation of flow rate with the pressure gradient for different values of B, n and θ with t = 60±, δ = 0.1, α = φ = 0. and ω = 1 is shown in Fig. 11. It is seen that the flow rate increases almost linearly with the increase of the pressure gradient e. It is also found that the flow rate decreases marginally when the yield stress θ increases and decreases considerably when the power law index n increases while all the other parameters were treated as invariables. One can also note that for the fixed values of n and θ, the flow rate increases significantly when the body acceleration parameter B increases. Figs. 10 and 11 bring out the effects of the body acceleration, pulsatility of the flow, pressure gradient, stenosis depth and non-newtonian nature of the blood on the flow rate of blood when it flows through stenosed narrow arteries. 4.6 Longitudinal impedance The variation of the longitudinal impedance with the stenosis depth for different values of α, B and θ with φ = 0., n = 0.95, ω = 1 and t = 60± is shown in Fig. 1. It is seen that the longitudinal impedance increases slowly with the increase of the stenosis depth δ from 0 to 0.15 and then it increases nonlinearly (rapidly) when the stenosis depth δ increases further from 0.15 to 0.5. One can observe that for the fixed value of the yield stress θ, the longitudinal impedance decreases with the increase of either the body acceleration parameter B or the pulsatile Reynolds number α, but the decrease in the longitudinal impedance is marginal when the pulsatile Reynolds number α increases and is considerable when the body acceleration parameter B increases. For a given set of values of B and α, the longitudinal impedance increases considerably when the yield stress θ increases. The increase in the longitudinal impedance is defined as the ratio between the longitudinal impedance of a fluid model for a given set of values of the parameters in the stenosed artery and the longitudinal impedance of the same fluid model for the same set of values of the parameters in the normal artery. The estimates of the increase in the longitudinal impedance due to the

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 55 presence of the stenosis, body acceleration and yield stress with α = φ = 0., ω = 1, e = 0.5 and t = 60± are computed in Table 1. It is observed that the estimates of the increase in the longitudinal impedance increases significantly when the stenosis depth δ increases and it increases marginally when the yield stress θ increases and decreases slightly when the body acceleration parameter B increases. Hence, the presence of stenosis in the artery, body acceleration in the flow and yield stress in the flowing fluid make considerable changes in the longitudinal impedance. n, lead angle φ and pulse frequency ω. It is observed that the velocity decreases marginally with the increase of the power law index n, yield stress θ, depth of the stenosis δ and lead angle φ of the body acceleration, and it increases with the increase of the body acceleration B, the pulsatile Reynolds number α, the pressure gradient e. It is also found that the wall shear stress increases with the increase of the pressure gradient e. It is of importance to mention that the flow rate decreases with the increase of the stenosis depth δ, yield stress θ and power law index n and, it increases with the increase of the pressure gradient e and body acceleration B. It is observed that the estimates of the increase in longitudinal impedance rise with the rise of the stenosis depth δ and yield stress θ, whereas it decreases with the increase of the body acceleration parameter B and the pulsatile Reynolds number α. Table 1. Estimates of the increase in the longitudinal impedance due to the increase in the stenosis depth and body acceleration with e = 0.5, α = φ = 0., ω = 1 and t = 60±. Fig. 1: Variation of longitudinal impedance with stenosis depth for different values of α, B and θ with φ = 0., n = 0.95, ω = 1.0 and t = 60±. 5. Conclusions This theoretical analysis brings out various interesting rheological properties of blood flow through narrow stenosed arteries with body acceleration, treating blood as Herschel-Bulkley fluid. It is found that the plug core radius decreases with the increase of the body acceleration parameter B and the pressure gradient e and, it increases with the increase of the yield stress θ, power law index n and depth of the stenosis δ. It is also noticed that the plug flow velocity decreases marginally with the increase of the yield stress θ, power law index δ θ = 0.1 B = 0 B = 1 B = θ=0 B=1.0 θ=0.0 5 θ=0.1 5 0.05 1.53 1.53 1.5 1.5 1.5 1.53 0.1.47.45.44.41.43.47 0.15 4.6 4.19 4.16 4.09 4.14 4.4 0. 8.01 7.80 7.71 7.5 7.65 7.96 0.5 17.03 16.33 16.05 15.45 15.87 16.85 0.3 43.3 40.58 39.54 37.9 38.83 4.6 It is hoped that the present theoretical study will be useful to physicians in predicting the severity of the stenosis and its consequences and will also enable them to take some crucial decisions regarding the treatment of patients, whether the disease can be treated with medicines or whether the patient should undergo surgery. Hence, it is concluded that the

56 D.S. Sankar: Effect of periodic body acceleration in blood flow through stenosed arteries present study can be treated as an improvement in the studies of blood flow through narrow stenosed arteries with body accelerations. This study could be extended further by introducing the permeability of the arterial wall and this will be done in the near future. Acknowledgement The present work was supported by the RU grant of Universiti Sains Malaysia, Malaysia (RU Grant Ref. No: 1001/PMATHS/816088). References [1] V. K. Sud, G. S. Sekhon, Arterial flow under periodic body acceleration, Bulletin of Mathematical Biology 47 (1985) 35 5. [] J. C. Misra, B. K. Sahu, Flow through blood vessels under the action of a periodic acceleration field: a mathematical analysis, Computers and Mathematics with Applications 16 (1988) 993 1016. [3] R. Usha, K. Prema, Pulsatile flow of particlefluid suspension model of blood under periodic body acceleration, ZAMP 50 (1999) 175 19. [4] P. Chaturani, V. Palanisami, Casson fluid model of pulsatile flow of blood flow under periodic body acceleration, Biorheology 7 (1990) 619 630. [5] P. Chaturani, V. Palanisami, Pulsatile flow of power law fluid model for blood flow under periodic body acceleration, Biorheology 7 (1990) 747 758. [6] P. Nagarani, G. Sarojamma, Effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery, Korea- Australia Rheology Journal 0 (008) 189 196. [7] P. Chaturani, S. A. Wassf Isaac, Blood flow with body acceleration forces, International Journal of Engineering Science 33 (1995) 1807 180. [8] G. Sarojamma, P. Nagarani, Pulsatile flow of Casson fluid in a homogenous porous medium subject to external acceleration, International journal of Non-Linear Differential Equations: Theory Methods and applications 7(00) 50 64. [9] P. K. Mandal, S. Chakravarthy, A. Mandal, N. Amin, Effect of body acceleration on unsteady pulsatile flow of non-newtonian fluid through a stenosed artery, applied Mathematics and Computation 189 (007) 766 779. [10] D. Liepsch, M. Singh, L. Martin, Experimental analysis of the influence of stenotic geometry on steady flow, Biorheology 9 (199) 419 431. [11] P. Chaturani, R. Ponnalagar Samy, Pulsatile flow of Casson s fluid through stenosed arteries with applications to blood flow, Biorheology 3 (1986) 499 511. [1] C. Tu, M. Deville, Pulsatile flow of non- Newtonian fluids through arterial stenosis, Journal of Biomechanics 9 (1996) 899 908. [13] D. S. Sankar, K. Hemalatha, Pulsatile flow of Herschel-Bulkey fluid through stenosed arteries-a mathematical model, International Journal of Non-Linear Mechanics 41 (006) 979-990. [14] P. K. Mandal, An unsteady analysis of non- Newtonian blood flow through tapered arteries with a stenosis, International Journal Non-Linear Mechanics 40 (005) 151 164. [15] I. Marshall, S. Zhao, P. Papathanasopoulou, P. Hoskins, XY. Xu, MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models, Journal of Biomechanics 37 (004) 679 687. [16] S. Chakravarthy, P. K. Mandal, Twodimensional blood flow through tapered arteries under stenotic conditions, International Journal of Non-Linear Mechanics 35 (000) 779 793. [17] G. T. Liu, X. J. Wang, B. Q. Ai, L. G. Liu, Numerical study of pulsating flow through a tapered artery with stenosis, Chinese Journal of Physics 4 (004) 401 409. [18] Q. Long, X. Y. Ku, K. V. Ramnarine, P. Hoskins, Numerical investigations of physiologically realistic pulsatile flow through arterial stenosis, Journal of Biomechanics 34 (001) 19 14. [19] S. Chakravarthy, A. Datta, P. K. Mandal, Analysis of nonlinear blood flow in a stenosed flexible artery, International Journal of Engineering Science 33 (1995) 181 1837.

ISSN: 1565-1339 International Journal of Nonlinear Sciences & Numerical Simulation 11(4): 43-57, 010 57 [0] M. Texon, The hemodynamic concept of atherosclerosis, Bulletin of the New York Academy of Medicine 36 (1960) 63 73. [1] D. F. Young, F. Y. Tsai, Flow characteristics in models of arterial stenosis I. Steady flow, Journal of Biomechanics 6 (1973) 395 411. [] B. E. Morgan, D. F. Young, An integral method for the analysis of flow in arterial stenosis, Bulletin of Mathamatical. Biolology 36 (1974) 39 53. [3] D. A. MacDonald, On steady flow through modeled vascular stenosis, Journal of Biomechanics 1 (1979) 13-0. [4] A. Sarkar, G. Jayaraman, Correction to flow rate-pressure drop relation in coronary angioplasty: steady streaming effect, Journal of Biomechanics 31 (1998) 781-791. [5] C. D. Han, B. Barnett, Measurement of rheological properties of biological fluids, In: Henry L. Gabelnick, Mitchel Litt (Eds.), Charles C. Thomas Publ., Illinois, U.S.A, 1973. [6] C. E. Huckaba, A. N. Hahn, A generalized approach to the modeling of arterial blood flow, Bulletin of.mathematical Biophysics 30 (1968) 645 66. [7] E. W. Merrill, Rheology of human blood and some speculations on its role in vascular homeostasis, In: Sawyer PN.(Eds), Biomechanical mechanisms in vascular homeostasis and intravascular thrombus. Appleton Century Crafts, New York 1965. [8] J. B. Shukla, R. S. Parihar, B. R. P. Rao, Effects of stenosis on non-newtonian flow of the blood in an artery, Bulletin of Mathematical Biology 4 (1980) 83 94. [9] P. Chaturani, R. Ponnalagar Samy, A study of non-newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases, Biorheology (1985) 51 531. [30] D. F. Young, Fluid mechanics of arterial stenosis, Journal of Biomechanical Engineering (Trans. ASME) 101 (1979) 157 175. [31] N. Iida, Influence of plasma layer on steady blood flow in micro vessels, Japanese Journal of Applied Physics 17 (1978) 03-14. [3] D. S. Sankar and Usik Lee, Two -fluid nonlinear mathematical model for pulsatile blood flow through catheterized arteries, Journal of Mechanical Science and Technology 3 (009) 1650 1669. [33] S. U. Siddiqui, N. K. Verma, Shailesh Mishra, R. S. Gupta, Mathematical modeling of pulsatile flow of Casson s fluid in arterial stenosis, Applied Mathematics and Computation 10 (009) 1-10.