An unsteady analysis of non-newtonian blood flow through tapered arteries with a stenosis

Transcription

1 International Journal of Non-Linear Mechanics 4 (25) An unsteady analysis of non-newtonian blood flow through tapered arteries with a stenosis Prashanta Kumar Mandal Department of Mathematics, Krishnath College, Berhampore, Dt., Murshidabad 74211, WB, India Received 13 October 23; received in revised form24 June 24 Abstract The problemof non-newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non- Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of the time-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made with nonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier Stokes equations in cylindrical coordinates system governing flow assuming aial symmetry under laminar flow condition so that the problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rate of flow, the resistive impedance and the wall shear stress with minor significance of computational compleity by eploiting the appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle, wall deformation, severity of the stenosis within its fied length, steeper stenosis of the same severity, nonlinearity and non- Newtonian rheology of the flowing blood on the flow field. An etensive quantitative analysis is performed through numerical computations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present model. 24 Elsevier Ltd. All rights reserved. Keywords: Generalised power law; Tapered arteries; Steeper stenosis; Wall shear stress 1. Introduction There are considerable evidences that vascular fluid dynamics play important role in the development and progression of arterial stenosis, one of the most wide spread diseases in human beings leading to the malfunction of the cardiovascular system. Although the address: pkmind2@yahoo.co.uk (P.K. Mandal). eact mechanisms responsible for the initiation of this phenomena are not clearly known, it has been established that once a mild stenosis is developed, the resulting flow disorder further influences the development of the disease and arterial deformity, and change the regional blood rheology [1,2]. Understanding of stenotic flow has proceeded fromquite a good number of theoretical and computational and eperimental efforts. Steady flow through an aisymmetric stenosis /$ - see front matter 24 Elsevier Ltd. All rights reserved. doi:1.116/j.ijnonlinmec

2 152 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) has been investigated etensively by Smith [3] using an analytical approach indicating that the flow patterns strongly depend on the geometry of the stenosis and the upstreamreynolds number. Deshpande et al. [4] considered the steady flow through an aisymmetric stenosis using a finite difference technique. Realising the fact that the pulsatile nature of the blood flow cannot be neglected, many theoretical analyses and eperimental measurements on the flow through stenosis have been performed [5 14].Inmost of the studies mentioned above, the flowing blood is assumed to be Newtonian. The assumption of Newtonian behaviour of blood is acceptable for high shear rate flow, i.e. the case of flow through larger arteries. It is not, however, valid when the shear rate is low as is the flow in smaller arteries and in the downstreamof the stenosis. It has been pointed out that in some diseased conditions, blood ehibits remarkable non-newtonian properties. Hemorheological studies have documented three types of non-newtonian blood properties: Thiotropy, Viscoelasticity and Shear thinning. Thiotropy, a transient property of blood, is ehibited at low shear rates and has a fairly long time scale. This suggests that thiotropy is of secondary importance in physiological blood flow. Thurston [15,16] has shown conclusively that blood, being a suspension of enumerable number of cells, possesses significant viscoelastic properties in the frequency range of physiological importance. Studies pertaining to the viscoelasticity of blood are of great interest because of three main reasons. To medical scientists, an accurate knowledge of the mechanical properties of whole blood and the erythrocytes can suggest a new diagnostic tool. For specialists in fluid mechanics, detailed informations of the comple rheological behaviour of the system is of utmost importance in any attempt towards establishing the equations that govern the flow of blood in various parts of the circulatory systemin different states. To rheologists, blood (whose biochemical and cellular compositions are well-known in other respects) is an ecellent model for correlating its rheological behaviour with the underlying molecular or cellular structures. However, the viscoelasticity of blood diminishes very rapidly as shear rate rises and at physiological hematocrit values ( 45%) [17]. This suggests that viscoelasticity has a secondary impact on normal pulsatile blood flow at physiological hematocrit values. The shear thinning properties of blood, however, are not transient and are ehibited in normal blood at all shear rates upto about 1 s 1. The purely viscous shear thinning nature of blood is, therefore, the dominant non-newtonian effect. In an etensive study, Easthope and Brooks [18] found that Walburn and Schneck model [19] which reduces to familiar Power-law relationship, is highly effective in modelling blood flow. The Power-law fluid showed far more non-newtonian influence as evident fromthe eperimental findings of Perktold et al. [2]. A number of researchers have studied the flow of non-newtonian fluids [21 32] with various perspectives. In most of the investigations relevant to the domain under discussion, the flow is mainly considered in cylindrical pipes of uniformcross-section. But, it is well known that blood vessels bifurcate at frequent intervals and the diameter of the vessels varies with the distance as propounded by Whitemore [33]. Hence the concept of flow in a varying cross-section forms the prime basis of a large class of problems in understanding blood flow. Manton [34], Hall [35] and Porenta et al. [36] pointed out that most of the vessels could be considered as long and narrow, slowly tapering cones. Thus the effects of vessel tapering together with the non-newtonian behaviour of the streaming blood seemto be equally important and hence certainly deserve special attention. With the above motivation, an attempt is made in the present theoretical investigation to develop a mathematical model in order to study the notable characteristics of the non-newtonian blood flow through a fleible tapered arteries in the presence of stenosis subject to the pulsatile pressure gradient. The non-newtonian behaviour of the streaming blood is characterised by the generalised Power-law model. Malek et al. [37,38] etensively studied the eistence and uniqueness as well as the stability characteristics of such flow problems. Although the present paper does not deal with the eistence of the flows characterised by generalised Power-law fluids, the already cited references [37,38] bear the foundation to make an attempt safely to eplore the flow characteristics of the streaming blood. Although the general problemsuch as the present one is of major physiological significance, due attention is also paid to the effect of arterial wall motion on local fluid mechanics but not on the stresses and strains in the vessel wall. The consideration of a time-variant

3 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) geometry of the stenosis has not however been ruled out fromthe present analysis. An etensive quantitative analysis is carried out by performing large scale numerical computations of the desired quantities having more physiological significance to eplore the effects of vessel tapering, the severity of the stenosis, the wall deformability, the steeper stenosis of same severity, the nonlinearity and the non-newtonian behaviour of the flowing blood on the physiological flow phenomena which are etensively quantified through their graphical representations presented at the end of thepaper with appropriate scientific discussions. A few comparisons are also made with the other eisting results so as to substantiate the applicability of the present model under study. 2. Formulation of the problem The tapered blood vessel segment having a stenosis in its lumen is modelled as a thin elastic tube with a circular cross-section containing an incompressible non-newtonian fluid characterised by generalised Power-law model. Let (r, θ, z) be the coordinates of a material point in the cylindrical polar coordinates systemwhere the z-ais is taken along the ais of the artery while r, θ are taken along the radial and the circumferential directions, respectively. The geometry of the time-variant stenosed arterial segment (see Fig. 1) is constructed mathematically as R(z, t) [ (mz + a) τ m sec (z d) τ 2 m sin 2 l2 ] 4 = (1) {l (z d)} a 1 (t), d z d + l, (mz + a)a 1 (t), otherwise, where R(z, t) denotes the radius of the tapered arterial segment in the stenotic region, a the constant radius of the non-tapered artery in the non-stenotic region,, the angle of tapering, l, the length of the stenosis, d, the location of the stenosis and τ m sec is taken to be the critical height of the stenosis for the tapered artery appearing at z = d + l 2 + τ m sin and m(= tan ) represents the slope of the tapered vessel. The Fig. 1. Geometry of the stenosed tapered artery for different taper angle. time-variant parameter a 1 (t) is given by a 1 (t) = 1 b(cos ωt 1)e bωt, (2) in which ω represents the angular frequency where ω = 2πf p,f p being the pulse frequency and b is a constant. The arterial segment is taken to be of finite length L. One can eplore the possibility of different shapes of the artery viz., the converging tapering ( < ), non-tapered artery ( = ) and the diverging tapering ( > ) as shown in Fig. 1. Let us consider the stenotic blood flow in the tapered artery to be two-dimensional, unsteady, aisymmetric and fully developed, where the flowing blood is treated to be non-newtonian characterised by the generalised Power-law model. The governing equations for the z and r components of momentum together with the equation of continuity, in the cylindrical coordinate systemmay be written as t + u r + w = 1 ρ p 1 ρ [ 1 r u t + u u r + w u p r 1 ρ [ 1 r r (rτ rz) ] (τ zz), (3) r (rτ rr) + ] (τ rz) = 1 (4) ρ and u r + u r + =, (5) where the relationships between the shear stress and shear rate in case of two-dimensional motion are as

4 154 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) follows: [ ( u τ zz = 2 m r + ) 2 ( u + r ( u + ) ] 2 1/2 r ) ( ) n 1 ( τ rz = {...} r + u ) ( ), (6) (7) and ( ) u τ rr = 2{...}. (8) r Here w(r, z, t) and u(r,z,t)are the aial and the radial velocity components, respectively, p is the pressure and ρ, the density of blood. Since the lumen radius, R, is sufficiently smaller than the wavelength λ, of the pressure wave i.e. R/λ>1, the radial Navier Stokes equation simply reduces to p/ r = (cf. [39]) and hence Eq. (4) can be omitted. It is then reasonable and convenient to assume that the pressure is independent of radial coordinate [5,4] and eventually the pressure gradient p/ appearing in (3), the formof which has been taken following Burton [41] for human beings as p = A + A 1 cos ωt, t >, (9) where A is the constant amplitude of the pressure gradient, A 1 is the amplitude of the pulsatile component giving rise to systolic and diastolic pressure. 3. Boundary conditions On the symmetry ais, the normal component of the velocity, the aial velocity gradient and the shear stress vanish. These may be stated mathematically as (r, z, t) u(r,z,t)=, = and τ rz = r on r =. (1) The velocity boundary conditions on the arterial wall are taken as u(r,z,t)= R, w(r,z,t)= on r = R(z, t). (11) t It is further assumed that initially no flow takes place when the systemis at rest, that means u(r, z, ) = = w(r, z, ). (12) 4. Method of solution Let us introduce a radial coordinate transformation [4], given by = r R(z, t), (13) which has the effect of immobilizing the vessel wall in the transformed coordinate. Using this transformation, Eqs. (3), (5) (7) together with the prescribed conditions (1) (12) take the following form: { t = R R t w 1 ρ + R τ zz R 1 u R + u R + R [ ( τ zz = 2 m 1 R u R + } R R w 1 ρ { 1 R τ z + 1 τ z R p τ zz }, (14) R =, (15) ) u 2 ( u ) 2 + R ( + ) R 2 R + ( u R u R + 1 ) 2]1/2 R ( R R n 1 ), (16) ( u τ z = {...} R u R + 1 R with ) (17) (, z, t) u(,z,t)=, =, τ z = on = (18)

5 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) u(,z,t)= R, w(,z,t)= on = 1, (19) t and u(, z, ) = = w(, z, ). (2) Multiplying Eq. (15) by R and integrating with respect to fromthe limits to one finds, u(,z,t)= R w R d 2 R w d. (21) This equation takes the following formby making use of the boundary conditions (19) as 1 d 1 = [ 2 R R w + 1 R f() R t ] d. (22) Since the choice of f() is, of course, arbitrary, let f() be of the form f()= 4( 2 1) satisfying 1 f () d = 1. Taking the approimation of considering the equality between the integrals to integrands, we have from(22) = 2 R R w + 4 R (2 1) R t. (23) Introducing (23) into (21) one gets [ R u(,z,t)= w + R ] t (2 2 ). (24) 5. Finite difference approimations The finite difference scheme for solving Eq. (14) is based on the central difference approimations for all the spatial derivatives in the following manner: =(w)k i,j+1 (w)k i,j 1 = w f, 2Δ =(w)k i+1,j (w)k i 1,j 2Δz = w fz, (25) while the time derivative in (14) is approimated by t = wk+1 i,j wi,j k. (26) Δt Similar epressions can also be obtained for u, τ z and τ zz. Here w(, z, t) is discretised to w( j,z i,t k ) and in turn, to wi,j k where we define j = (j 1)Δ, (j = 1, 2,...N + 1) such that (N+1) = 1.,z i = (i 1)Δz, (i = 1, 2,...M+ 1) and t k = (k 1)Δt, (k = 1, 2,...) for the entire arterial segment under study with Δ,Δz are the increments in the radial and the aial directions, respectively, and Δt is the small time increment. Using (25) and (26), Eq. (14) may be transformed to the following difference equation: wi,j k+1 =wk i,j + Δt uk i,j R k i [ 1 ( ) { p k+1 j + ρ k + ( ) } j R k k wi,j k (w f ) k i,j i { w k i,j (w fz) k i,j 1 ρ 1 j R k i (τ z ) k i,j ( ) R k t i + 1 k [(τ z ) f ] k i,j [(τ zz) fz ] k i,j + ( ) }] j R k k [(τ zz ) f ] k i,j (27) i while Eqs. (16) and (17) have their discretised formas ( ) 2 ( ) (τ zz ) k i,j = 2 m 1 u k 2 k (u f ) k i,j i,j + j k ( + (w fz ) k i,j ( ) ) j R k 2 k (w f ) k i,j i ( + (u fz ) k i,j ( ) j R k k (u f ) k i,j i ) 1/2 2 n [ k (w f ) k i,j (w fz ) k i,j j R k i ( ) R k (w f ) k i,j i ], (28)

6 156 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) (τ z ) k i,j = {...} [(u fz ) k i,j j R k i ( ) R k (u f ) k i,j + 1 i k ] (w f ) k i,j. (29) Also the prescribed conditions (18) (2) have their finite difference representations, given by u k i,1 =, wk i,1 = wk i,2,(τ z) k i,1 =, (3) ( ) R k w i,n+1 =, u k i,n+1 =, t i (31) u 1 i,j = = w1 i,j. (32) The difference equation (27) is solved for w by making use of (28) (29) together with the prescribed conditions (3) (32) throughout the arterial segment under consideration. After having obtained the aial velocity of the flowing blood, the radial velocity can be calculated directly from(24). Now with the help of the aial and the radial velocity of the streaming blood, one can easily determine the volumetric flow rate (Q), the resistance to flow ( ) and the wall shear stress (τ w ) fromthe following relations, given by 1 Q k i = 2π(Rk i )2 j wi,j k d j, (33) L( p/) k k i = Q k, (34) i [ (τ w ) k i =μ 1 k (w f ) k i,j + (u fz) k i,j ] ( j R k (u f ) k i,j [ ( R cos arctan ) k i ) k i =1 ]. (35) The present analysis bears the potential to eplore several case studies by means of which the effects of wall distensibility, the non-newtonian rheology of the flowing blood and the nonlinearity on the flow pattern can be estimated both analytically and numerically through the following cases of reduction of the present systemto the particular consideration. Case I: When the wall motion is withdrawn from the present system, the analysis can be reduced to a formby considering R = R(z) only and consequently R/ t = so that the effect of wall distensibility can be estimated directly. Case II: The generalised Power-law model is quite different fromthe other formas it does not simply reduce to Newtonian model for n = 1 and hence to estimate the effects of non-newtonian rheology of the streaming blood, the following set of equations have been solved using the same methodology as in the case of non-newtonian model but for the shake of brevity, the detailed derivations are not presented here: t + u r + w [ 2 w r r r + w u + μ ρ u t + u u = 1 p ρ r + 2 w 2 = 1 ρ + μ [ 2 u ρ r r u and r + u r + =. p r ], u r u r u 2 ], Case III: When the convective acceleration terms viz. the nonlinear terms are disregarded from the governing equations mentioned in Case II, one finds the systemreducing to a linearized (Newtonian) one where the terms u(/ r), w(/), u( u/ r), w( u/) are totally absent. The effect of nonlinearity can therefore be measured through the direct comparison of the present system and its linearized (Newtonian) version. Case IV: Instead of treating the arterial wall as elastic (moving wall), if one clamps the nonlinear viscoelastic wall property so that the variation of the results can be quantified and the necessary comparison can also be made with those of the eisting result by making use of the following pressure radius relationship [1]: p(z, t) = μ w ( R/ t) + σ (e β(δ 1) 1 ) (r m /h)δ 2 2 1, where the input data for μ w, σ, β are taken to be similar to those of Ref. [1].

7 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Numerical results and discussion For the purpose of numerical computations of the desired quantities of major physiological significance, the following parameter values have been made use of [21,41 43]: a =.8mm,L = 5 mm, L = 16 mm, d = 2 mm, b =.1, m =.1735P, μ =.35P, n =.639 ρ = kg m 3, f p = 1.2Hz,A = 1 kg m 2 s 2, A 1 =.2A, τ m =.4a, Δ =.25, Δz =.1. The iterative method has been found to be quite effective in solving the equations numerically for different time periods. The results appeared to converge with an accuracy of the order 1 7 when the time step was chosen to be.1. The computed results obtained following the abovementioned method for various physical quantities of major physiological significance in order to have their quantitative measures are all ehibited through the Figs and discussed at length. Fig. 2 illustrates the results for the aial velocity profile of the flowing blood characterised by the generalised Power-law model (non-newtonian) at a specific location of z = 28 mm in the stenotic region of the tapered artery at an instant of t =.45 s comprising of si distinct curves for different perspectives with distinguishable marks. The curves are all featured to be analogous in the sense that they do decrease from their individual maima at the ais as one moves away fromit and finally drop to zero on the wall surface. Eamining the behaviour of the results of the present figure, one observes that the aial velocity profile assumes a flat shape in the presence of a converging tapering ( =.1 ) instead of a parabolic one for non-tapered ( = ) artery when both are treated under stenotic conditions. This observation can be interpreted physically that if the tube is tapered, then inertial forces associated with the convective accelerations manifest themselves in an amount of the same order as viscous forces while the former compel the aial velocity profile to attain a flat shape. This observation agrees qualitatively well with that of Belardinelli and Cavalcanti [9] though their observations were based on the Newtonian rheology of flowing blood past a taperedartery. Likewise, the effect of tapering without any arterial constriction can be visualised and quantified fromthe first and second curves fromthe top of the present figure where again the vessel tapering Fig. 2. Aial velocity profile at z=28 mm for t =.45 s. (τ m =.4a, d = 2 mm, l = 16 mm). diminishes the flow velocity significantly. Thus one may conclude that the aial flow velocity is reduced to some etent with vessel tapering irrespective of the presence of any arterial constriction. The corresponding results for a diverging tapering ( =.1 ) with arterial stenosis represented by the third curve fromthe top differ from all the remaining curves and become higher than those for a converging tapering as anticipated. The present figure also includes the result for a steeper stenosis (of same severity) for the same aial position for a non-tapered artery where the aial velocity profile gets reduced to a considerable etent in case of a steeper stenosis which can be quantified by comparing the relevant curves of the present figure. Thus, analysing the behaviour of all the curves of the present figure, one may conclude that the presence of stenosis, tapering and steeper stenosis affect the aial velocity of the streaming blood past a stenosed tapered artery significantly. Unlike the characteristics of the aial velocity profile, the results of the radial velocity component varying radially at the same critical location of z = 28 mm ehibited in Fig. 3 at the same instant of t =.45 s, are found to be negative. All the curves appear to decline fromzero on the ais as one moves fromaway from it and finally to increase towards the wall to attend some finite value on the wall surface which clearly

8 158 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Fig. 3. Radial velocity profile at z=28 mm for t=.45 s. (τ m =.4a, d = 2 mm, l = 16 mm) Fig. 4. Aial velocity profile for different times (z = 28 mm, τ m =.4a, d = 2 mm, l = 16 mm, =.1 ). reflects the presence of wall motion encountered in the present model. Analysing all the curves of the present figure, one can easily observe that the effect of tapering on the radial velocity profile is more prominent in case of stenosed artery than for a non-stenosed one, however, for non-stenosed artery, the maimum deviation occurs near the wall. The corresponding result for a steeper stenosis is almost unperturbed which can be visualised if one go through the relevant curves of the present figure. Even though the magnitudes of the radial velocity are smaller than those of the aial velocity, the noted feature is directly responsible for the local storage and is manifested in the convective acceleration of the nonlinear flow phenomena. Fig. 4 displays the variation of the aial velocity profile at the same critical location of z = 28 mm in a tapered artery for different times spread over a single cardiac period. The curves do shift towards the origin when the time is allowed to increase from.1 to.45 s and eventually at t =.7 s, the curve is found to shift away fromthe origin. Such behaviour is believed to be directly responsible for the pulsatile pressure gradient produced by the heart as it comes into play. It is interesting to note that the rate of decrease of the aial velocity in the systolic phase appears almost the same as the rate of increase in the diastolic phase over a single cardiac cycle, as anticipated. Moreover, the velocity is reduced to a considerable etent if the wall motion is totally withdrawn fromthe systemunder consideration as observed fromthe bottomtwo curves of the present figure. Here too, the observations that the results obtained by Newtonian model of the streaming blood are dramatically much higher than the non- Newtonian values. Thus the vascular wall deformability and the non-newtonian characteristics of the flowing blood affect the aial velocity profile which can be estimated by the relevant curves of the present figure. The results indicating the unsteady behaviour of the flowing blood over a single cardiac cycle, presented in the Fig. 5 at the same critical location are found to be quite interesting to note. The radial velocity profile assumes positive values with time advancement from.1 to.3 s in the systolic phase while with the passage of time from.45 to.7 s, the radial velocity profile assumes to continue with negative values only causing back flow. Such typical nature of the curves reflects very closely the radial motion of the arterial wall for a single cardiac cycle whose graphical presentation is not shown here for brevity. When the wall motion is totally disregarded, the radial velocity profile takes a complete symmetrical shape as recorded in the third curve fromthe top of the present figure. The present figure also displays the results for the flowing blood having Newtonian rheology and it turns out that Newtonian characteristics of the flowing blood affects the

9 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Fig. 5. Radial velocity profile for different times (z = 28 mm, τ m =.4a, d = 2 mm, l = 16 mm, =.1 ) Fig. 7. Radial velocity profile for different aial positions at t =.45 s. (d = 2 mm, τ m =.4a, l = 16 mm, =.1 ) Fig. 6. Aial velocity profile for different aial positions at t =.45 s. (d = 2 mm, τ m =.4a, l = 16 mm, =.1 ). radial velocity pattern less significantly than the aial velocity profile. In order to analyse the flow-field intensively along the arterial segment under study, Figs. 6 ehibits the aial velocity profiles at five distinct aial locations for =.1 at t =.45 s. The aial velocity profile is parabolic at the upstream (z = 15 mm) while a flattening trend is followed at the converging section (z = 24 mm) and subsequently it becomes much blunter at the specific location (z = 28 mm) than at the entry. The velocity appears to be enhanced at the diverging section (z = 34 mm) and finally at the offset of the stenosis, the aial flow velocity profile gets back again into the parabolic pattern. This result agrees qualitatively well with those of Tu et al. [7] though their studies were based on the stenotic blood flow in which the streaming blood was treated as Newtonian fluid. The results of the radial velocity component at t =.45 s ehibited in Fig. 7 are noted to be negative ecepting at the downstream. Most of the curves become concave near the wall ecepting that for the downstreamof the stenosis. It is interesting to record that at the downstream, back flow occurs near the wall where the direction of the velocity changes frompositive to negative and that, in turn, causes separation in the flow field. This observation is believed to be quite justified fromthe physiological point of view where the arterial tapering plays a key role in order to characterise the flow phenomena under study. If one disregards the non-newtonian rheology of the flowing blood and also ignores the convective acceleration terms in the momentum (case II) i.e. in case of linear Newtonian flow, a meagre deviation is observed in the radial velocity of the flowing blood in the present mathematical model.

10 16 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Fig. 8. Aial velocity profile for different taper angles at t =.45 s. (d = 2 mm, z = 28 mm, τ m =.4a, l = 16 mm) Fig. 9. Radial velocity profile for different taper angles at t =.45 s. (d = 2 mm, z = 28 mm, τ m =.4a, l = 16 mm). Figs. 8 and 9 show how the constricted arterial tapering with varied taper angles influences the patterns of the flow-field at a particular instant of t =.45 s. It is observed that the curves representing both the aial and the radial flow velocity do shift towards the origin when the taper angle increases from.2 to.4 (converging tapering) while they shift away fromthe origin for a non-tapered ( = ) and positively ta Fig. 1. Flow dependence on the pressure gradient at t =.1s (z = 28 mm, d = 2 mm, l = 16 mm, τ m =.4a). pered ( =.2 ) artery. Eamining the behaviour of the results of the present figures, one can easily quantify the effect of arterial tapering on the flow-field. Moreover, by comparing the present results with our previous work [13], one may also take note that there is no significant change in behaviour of the flow-field patterns evaluated by Newtonian and non-newtonian model of the streaming blood. For the purpose of making a comparative study with the eisting results [9], several plots have been made in Fig. 1 just to characterize the behaviour of the flow rate with the pressure gradient for different taper angles at a particular instant of t=.1 s where the input pressure gradient data has been chosen to be similar to those of Ref. [9], at a particular location of z=28 mm. All the curves appear to be linear and they do shift towards the origin with increasing taper angle from to.1 or in other words, the flow rate diminishes as the artery gets narrowed gradually. The present figure also includes the results for a diverging tapering ( =.1 ) ehibiting an increasing trend as anticipated. In the absence of any constriction, the flow rate enhances to a considerable etent which can be quantified from the present figure. Finally, the effects of vascular wall deformability, the steeper arterial stenosis and of the non-newtonian rheology of the flowing blood with the noted range of the pressure gradient for a non-tapered artery ( = )are, however, not ruled out fromthe present investigation.

11 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Fig. 11. Aial velocity profile for z=37 mm at t=.1s.(d=2 mm, l = 16 mm, τ m =.5a). The representations of Fig. 11 record how the behaviour of the aial velocity profile is dramatically changed at the post-stenotic region (z = 37 mm) for a particular instant of t =.1 s when one takes into account the arterial wall to be isotropic, incompressible, visco-elastic material with circular cross-section [1]. For a converging tapered artery ( =.3 ), one observes that the aial velocity profile varying radially is always negative while for a diverging artery ( =.3 ), the aial flow velocity is always positive when the arterial wall is treated as elastic (moving wall). On the contrary, a vorte takes shape when the arterial wall deformation is treated to be similar to those of Cavalcanti [1] which may cause irregularities in the wall shear stress. This observation is in good agreement with those of Cavlcanti [1] though the later studied the flow phenomena in a mildly stenosed artery by considering the Newtonian rheology of the flowing blood. Perhaps the pressure radius relation considered by Cavalcanti [1] gives more realistic result, but it lacks tocompare the results with the rigid and straight circular arterial tube without any tapering and constriction. Fig. 12 includes more results showing the variation of flow rate at a specific location of z = 28 mm for certain distinct cases stretched over a period of nearly four cardiac cycles. The pulsatile nature of the flow rate has been found to be distributed for all the curves Fig. 12. Variation of the rate of flow with time at z = 28 mm (τ m =.4a, d = 2 mm, l = 16 mm). throughout the time scale considered here. In the absence of the constriction, the flow rate gets enhanced significantly for the entire time range. The deviation of the results thus obtained clearly estimates the effect of stenosis quantitatively on the flow rate in the tapered artery. The flow rate for a non-tapered artery ( = ) possesses relatively higher magnitudes than that of negatively tapered artery ( =.1 ). However, the magnitude of the flow rate for a diverging tapered artery is alltime higher than those of non-tapered and negatively tapered artery. Further, the corresponding Newtonian model yields an analogous behaviour with higher magnitudes. Hence the effect of taper angle and non-newtonian rheology of the flowing blood can be quantified if one analyses the relevant curves of the present figure. Moreover, when the wall motion is totally withdrawn fromthe present systemby disregarding the vessel wall distensibility, the flow rate declines keeping its pulsatile nature unaltered. One may also take note fromthe present figure that the flow rate diminishes to a considerable etent in the case of a steeper stenosis where stenosis has been made steeper just by reducing its length with thesame severity. Studying all the results referred to the present figure, one may conclude that the presence of stenosis, the taper angle, the steeper stenosis, the wall deformability and the non-newtonian rheology of the flowing

12 162 P.K. Mandal / International Journal of Non-Linear Mechanics 4 (25) Fig. 13. Variation of the resistance to flow with time at z = 28 mm (τ m =.4a, d = 2 mm, l = 16 mm). blood certainly bear the potential to influence the flow rate to a considerable etent in the realmof the physiological flow phenomena. Fig. 13 indicates how the resistances to flow are influenced by the unsteady flow behaviour of blood as well as by the vessel tapering, the vessel wall distensibility, the stenosis, the steeper stenosis and by the non-newtonian rheology of the streaming blood. The resistances to flow follow a reverse trend fromthose of Fig. 12 in a way that the streaming fluid eperiences higher resistance when the rate of flow in the constricted tapered artery are correspondingly lower and vice-versa. Unlike the characteristics of the flow rate, one may observe that the flowing blood eperiences much higher resistances to flow in the presence of arterial constriction, in the absence of vascular wall distensibility and in the presence of non-newtonian characteristics of the flowing blood. However, the effects of tapering and the steeper stenosis on the resistive impedances are, however, not ruled out from the present investigation. Finally, the variation of the time-dependent wall shear stress at a specific location of z = 28 mm corresponding to a constricted zone of a tapered artery has been portrayed in Fig. 14. The wall shear stresses represented by the curves of the concluding figure appear to be compressive in nature. It appears that the wall shear stress declines fromzero at the onset of the cardiac cycle and the rate of decline with negative val- Fig. 14. Variation of the wall shear stress with time at z = 28 mm (τ m =.4a, d = 2 mm, l = 16 mm). ues gradually diminishes for rest of the pulse cycles when the streaming blood is Newtonian past a tapered artery ( =.1 ) which has a remarkable deviation with the corresponding non-newtonian result if one goes through the relevant curves of the present figure and thereby the effect of non-newtonian rheology of the flowing blood on the wall shear stress can be well established. However, for the rest of the curves of the present figure, there is a remarkable variation of the stress characteristics almost immediately after the onset of the first cardiac cycle where small fluctuations with some fied amplitudes keep the stress steady with the advancement of time. The stress yields all time values higher for a diverging tapering than all other eisting results corresponding to converging tapering and without tapering so far as their magnitudes are concerned which is in good agreement with Sagayamary and Devanathan [44] who studied the steady flow behaviour of couple stress fluid in a stenosed tapered tube and disregarded walldeformability. The deviation in results for the rigid artery and for the constricted artery can be visualised and quantified their effects on the stresses fromthe relevant curves of the present figure. According to Glagov et al. [45], high tensile stresses are associated with vessel wall thickening and alterations in composition. One may also record fromthe present results that wall shear stress enhances to a considerable etent for steeper stenosis

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