Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction

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1 Korea-Australia Rheology Journal, Vol.5, No.3, pp (August 03) DOI: 0.007/s Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal, * Department of Mathematics, Raiganj Surendranath College, Raiganj, W.B., India. Department of Mathematics, Visva-Bharati, Santiniketan 7335, W.B., India. (Received July 9, 0; final revision received April 5, 03; accepted April 6, 03) The present investigation deals with the effect of the shape of a stenosis on the flow characteristics of blood, having shear-thinning viscoelastic rheological properties by using a suitable mathematical model. Keeping the relevance of the physiological situation, the mathematical model is developed by treating blood as a non-newtonian shear-thinning viscoelastic fluid characterised by unsteady Oldroyd-3-constant model through an axisymmetric irregular arterial stenosis obtained from casting of a mildly stenosed artery (cf. Back et al., 984). Comparison with the well-known cosine-shaped stenosis, in order to estimate the effect of surface roughness on the flow characteristics of blood, has however not been ruled out from the present study. Numerical illustrations are presented for a physiological flow, as well as for an equivalent simple pulsatile flow with equal stroke volume to that of the physiological flow, and the differences in their flow behaviour are recorded and discussed. The Marker and Cell method is developed in cylindrical co-ordinate system in order to tackle the highly nonlinear governing equations of motion. The effects of the quantities of significance such as Reynolds number, Deborah number, blood viscoelasticity and flow pulsatility, as well on the velocity components, pressure drop, wall shear stress and patterns of streamlines are quantitatively investigated graphically. Comparison of the results reveals that although the behaviour of two different pulses are similar at the same instant of time, there exist some important deviations in the flow pattern, pressure drop and wall shear stress as well. The present results also predict that the excess pressure drop across the cosine stenosis compared with the irregular one is consistent with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration. Keywords: shear-thinning, viscoelastic, pulsatile, irregular stenosis, unsteady, MAC method.. Introduction *Corresponding author: pkmind0@yahoo.co.uk Atherosclerosis is an arterial disease causing major concerns to health in the form of heart attacks and strokes. It usually affects large and medium sized artery (cf. Ross, 993; Waters et al., 0). The role of blood flow dynamics on the formation of atherosclerosis, its subsequent progression to plaque rupture and thrombosis have been estimated experimentally by Brunette et al. (008). Mathematical models of blood flow constitute an alternative and useful tool for supporting experiments and detecting minor constriction phenomena including local features which are not always obvious by measurement. The linear theory of Navier-Stokes equations is frequently used to model the flow of blood in larger arteries, but the role of material nonlinearity and the influence of red cells on the blood viscosity becomes more important at low shear rates and in smaller vessels (cf. Chien et al., 984). Keeping this view in mind many successful studies have been carried out by treating blood as a non-newtonian fluid (cf. Mann and Tarbell, 990; Usha and Prema, 999; Pontrelli, 00; Khanafer et al., 006; Mandal et al., 007; Lukacova-Medvidova and Zauskova, 008; Sarifuddin et al., 008, 009; Ikbal et al., 00; Sankar and Lee, 00). Thurston (97) was among the earliest to recognise the viscoelastic nature of blood which is less prominent with increasing shear rate. As flow proceeds, the sliding of cells requires a continuous input of energy. This is dissipated through viscous friction (cf. Thurston, 979). These effects induce the blood to behave like a viscoelastic fluid. In addition, several pathologies are accompanied by significant changes in the mechanical properties of blood which result in alteration in blood viscosity and viscoelastic properties, as reported in the recent review articles by Robertson et al. (008, 009). As red blood cells (RBCs) form rouleaux, they tumble while flowing through vessels. Such tumbling disturbs the normal flow pattern and requires the consumption of energy resulting in an increase in blood viscosity at low shear. Other major factors in the viscoelasticity of streaming blood are primarily due to the elastic energy that is stored in the deformed red blood cells as the heart pumps the blood through the body. Red blood cells by themselves have been shown to exhibit viscoelastic properties. These properties are the largest contributing factors to the viscoelastic behaviour of blood (cf. Thurston, 989; Thurston and Henderson, 006). 03 The Korean Society of Rheology and Springer 63

2 Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal Other factors contributing to the viscoelastic properties of blood are the plasma viscosity, plasma composition, temperature and the shear rate. Together, these factors make the streaming blood viscoelastic (cf. Thurston, 979). In this case, the viscosity µ cannot be considered merely as a constant, but as a decreasing function of the shear rate (shear-thinning fluid) (cf. Chmiel et al., 990; Mann and Tarbell, 990; Politsis et al., 99). The analytical function which offers the best fit for experimental data with a large range of flows has been found in the findings of Yeleswarapu (996) and we have made use of it in our present simulation. It is rather surprising to find that blood viscoelasticity has received much less attention in the literature than its Newtonian counterpart and/or other non-newtonian models even though interest has been growing in recent times due to its implications in clinical medicine and physiology. Etter and Schowalter (965) and Waters and King (97) studied the unsteady flow of Oldroyd-B fluid in a circular tube. Philips and Deutsch (975) applied a general state equation for human blood by using a four constant Oldroyd model. Experimental investigation highlighting the influence of viscoelasticity of human blood has been carried out by Gijsen et al. (999). Pontrelli (000) studied the unsteady flow of a Oldroyd-B fluid in a straight, long and rigid pipe driven by a suddenly imposed pressure gradient. The influence of the shear-thinning (Cross model) viscoelasticity on the shape of the flow domain has been successfully studied by Leuprecht and Perktold (000). A viscoelastic model of blood within a generalised thermodynamic framework is developed successfully by Anand and Rajagopal (004). Anand et al. (006) proposed the mechanics of a particular type of clot, formed from human plasma, within a thermodynamic framework by describing blood as a viscoelastic fluid. Arada and Sequeira (005) extensively studied the existence and uniqueness of such flow problems. Nadau and Sequeira (007) made an attempt to study the shear-dependent viscoelastic flow problem by using a combined finite element-finite volume method. The coupling of a generalised Newtonian fluid by taking into account the shear-thinning behaviour of blood, with an elastic structure describing the vessel wall has been successfully made to capture the pulse wave due to the interaction between blood and vessel wall, and to estimate the energy for the coupling (cf. Janela et al., 00). Very recently, Badnar et al. (00) concluded that the shear-thinning effects related to fluid velocity, pressure and wall shear stress are more pronounced than the viscoelastic ones. Most experimental and numerical studies of pulsatile flow through a stenosed artery are based on the assumption of simple periodic variation of the inflow with time in the form of single harmonic pulse (cf. O Brien and Ehrlich, 985; Stergiopulos et al., 996), but available data on canine and human arteries reveal that the arterial flux waves are different from a single harmonic pulse. For physiological flow, the waveform given by Daly (976) has been used in the present investigation. For the purpose of making comparison, a simple pulsatile flow having the same stroke volume as the physiological flow is considered. Zendehboodi and Moayeri (999) concluded that for better understanding of pulsatile flow behaviour in stenosed arteries, the actual physiological flow should be simulated. Most of the studies relating to stenotic flow have been performed with the basic assumption representing the stenosis by the cosine curve. However, arterial constriction contains many small valleys and ridges, analogous to a mountain range. For the purpose of deeper investigation into this problem one step closer to the real situation, the published clinical data (cf. Back et al., 984) were used to define the outline of the stenosis. Some attempts have already been made to investigate the flow characteristics through irregularly occluded vessels (cf. Johnston and Kilpatrick, 99; Andersson et al., 000; Yakhot et al., 005; Chakravarty et al., 005; Mustapha et al., 00) by treating blood as a Newtonian fluid with constant viscosity which perhaps ignores some important rheological aspects of physiological flow under stenotic conditions in smaller arteries. Keeping all these in mind, an attempt is made in the present investigation to explore the influence of pulsatile as well as physiological flow through a rigid irregularly constricted artery in which the streaming blood is treated to be a generalised viscoelastic fluid which accommodates viscoelasticity (cf. Phan-Thien and Huilgel, 985) and shear-thinning (cf. Yeleswarapu, 996) characteristics. The assumption of wall rigidity may not seriously affect the flow since the development of atherosclerosis in arteries causes a significant reduction in vessel wall distensibility (cf. Nerem, 99). The governing equations of motion of the unsteady flow phenomena are successfully solved numerically by MAC method primarily introduced by Harlow and Welch (965) and achieve the desired degree of accuracy. The primary objective of the present study is to explore the effects of some essential issues like flow unsteadiness, different stenosis shapes, different inlet wave forms, Reynolds number and different viscoelastic parameters on the velocity profile of the blood stream and on the wall shear stresses quantitatively by using a relatively simple finite-difference scheme in rather complex geometries. The novelty of the present study is the treatment of shear-thinning viscoelastic fluid characterising blood rheology in general, and the choice of differently shaped stenosis models and different pulsatile flow wave forms which more closely resemble the physiological situation. 64 Korea-Australia Rheology J., Vol. 5, No. 3 (03)

3 Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction 3. Governing Equations The streaming fluid representing blood is generally considered as a viscoelastic fluid having shear-thinning characteristic (cf. Yeleswarapu, 996). The axisymmetric flow of blood through an axisymmetric irregular arterial stenosis can be regarded as two-dimensional by making use of cylindrical co-ordinate system under laminar flow conditions. The momentum and the continuity equations for an unsteady, incompressible, Oldroyd-3-constant model proposed by Phan-Thien and Huilgel (985) can be written as u u u () t = P + and u = 0 () where u = (u r, 0, u z ) is the velocity vector,, the density, and P, are the pressure and the extra stress respectively in which the extra stress consists of two parts: n from the Newtonian ( solvent stress) and s from the non-newtonian rheology of the flowing blood (particle stress) as = n + s Following Leuprecht and Perktold (00), the shearthinning behaviour of the viscoelastic fluid is realised by applying a shear rate dependent viscosity on the solvent fluid. Thus for the shear- thinning generalised Oldroyd-3- constant viscoelastic model, the fluid viscosity (µ) is no longer a constant rather it possesses a functional dependence of shear rate ( ). Hence we have n = u + u T, (3) and s + s = p D DD (4) D : DI where 0 +log e + = , =, -- D :D + with > 0, a material constant with the dimension of time representing degree of shear- thinning. 0 and ( 0 ) are two asymptotic values of viscosity at 0 and respectively. D is the strain rate tensor u+u T, I, the unit tensor,, the fluid relaxation time,, a dimensionless parameter p, the viscosity contribution from the particle part and s stands for the upper convective derivative defined by s = s+ u s s u u T s (5) We now introduce non-dimensional variables as r r --- r 0 z u, z ---, u r u r -----, u z tu z t 0 P , P , r 0 U 0 U 0 r 0 U 0 Fig.. Shear rate dependent viscosity function () with * = for four values of. U and ij ij r 0 where r 0 and U 0 are the unconstricted radius and cross-sectional average velocity over the inlet section respectively. The non-dimensionalised equations of motion for twodimensional unsteady flow are given by and u = 0 (7) in which n = u +u T, and Re (6) = D, (8) Re DeDD + -- DeD :DI with + log + e = , + =, (9) -- D : D U where the Reynolds number Re 0 r = , the Deborah p U number De o = and = The complex viscosity of blood is approximated here with a three-parameter model, where the apparent viscosity decreases dramatically as shear rate increases (cf. Fig. ). 4. Stenosis Model U 0 u u u = P + t n + s + De s r o s o Two differently shaped models of stenosis each shown in Fig. have been examined. The first profile of the stenosis considered here is the straight axisymmetric model of Back et al. (984) mimicking real surface irreg- Korea-Australia Rheology J., Vol. 5, No. 3 (03) 65

4 Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal Fig.. Comparison of two different stenosis models. ularities since the actual variation of the cross-sectional area of a left circumflex coronary artery casting from a human cadaver is retained. The second geometrical model of the stenosis encountered here is the most commonly used cosine curve z z Rz = + cos , r o z dzd+ z 0 0 and Rz =, elsewhere (0) where z 0 is the half-length,, the maximum width and z (= d + z 0 ), the centre of the stenosis with = 0.76r 0, r 0 being the unconstricted radius of the stenosed artery. 5. Boundary Conditions As the arterial wall is treated to be rigid, the velocity boundary conditions of the blood stream on the wall are the usual no-slip conditions given by u z rzt = 0 = u r rzt on r= Rz, () while zero transverse velocity gradient and zero cross flow on the axis of symmetry are taken as u z rzt = 0 = u on. () r r rzt r = 0 A pulsatile parabolic velocity profile at the inlet of the stenosed arterial lumen may be assumed as u z rzt = ---- r uz t at z = 0, R where u z t = uz + cost + 0 (3) with u z (0) = 0, is the angular frequency having the same stroke volume as the physiological flow. The physiological flow waveform [cf. Daly (976)] for the mean velocity u z (t) in the canine femoral artery is shown in Fig. 3. Using Fig. 3. Time variation of the mean velocity for the physiological flow (Daly,976) and for an equivalent simple pulsatile flow. uz the condition u z (0) = 0, the value of 0 = cos uz can be obtained from the expression of u z (t). The velocity gradients at the outlet of the arterial segment of finite length L may be taken to have the traction-fee conditions as u z rzt u r rzt = = z z at z = L for 0 rr z, (4) and finally rr = 0, rz = 0, zz = 0, = 0 r r r r for r= R z, and rr = 0, rz = 0, zz = 0, = 0 at r = 0. rr rz Also = 0, = 0, = 0, z z z for z = 0 and z= L (5) 6. Solution Procedure zz = 0 z In order to avoid interpolation error while discretising the governing equations, a suitable radial co-ordinate transformation transformation defined by x = r has been made use of. Rz The governing equations along with the set of initial and boundary conditions, duly transformed, are solved numerically by finite difference method. Control volume-based finite-difference discretisation of those equations is carried out in non-uniform staggered grids, usually known as MAC (Marker and Cell) method proposed initially by Harlow and Welch (965). In this type of grid alignment, the velocities, pressure, viscosity and stresses are calculated at different locations of the control volume as indicated in Fig. 4. The difference equations have been derived in three distinct cells corresponding to the continuity equation, the axial 66 Korea-Australia Rheology J., Vol. 5, No. 3 (03)

5 Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction Fig. 4. A typical MAC cell. Scheme. Flow chart for MAC algorithm. momentum and the radial momentum equations. The discretisation of the time derivative terms are based on the first order accurate two-level forward time differencing formula while those for the convective terms in the momentum equations are accorded with a hybrid formula consisting of central differencing and second order upwinding. The diffusive terms however, are discretised by second order accurate three-point central difference formula. The Poisson equation for pressure is derived from the discretised momentum and continuity equations. The advantage in using MAC cell is that the pressure boundary condition is not needed at boundaries where the velocity vector is specified, because the domain boundaries are chosen to fall on velocity nodes. The Poisson equation for pressure is solved iteratively by Successive-Over- Relaxation (S.O.R.) method with the chosen value of over relaxation parameter as., in order to get the intermediate pressure field using the velocity field. Subsequently the maximum cell divergence of the velocity field is calculated and checked for its tolerance of If the tolerance limit is not satisfied, then the pressure at each cell of the flow domain is corrected and the velocities at each cell are adjusted accordingly by repeating the process. No standard package for the finite difference scheme has been used in the present computations. However, the computational code based on the following algorithm has been successfully programmed using FORTRAN language. The MAC method consists of the following two stages: Stage : n n (i) Velocities u z and u r are initialized at each i -- + j ij + -- cell (i, j). This is done either from result of the previous cycle or from the prescribed initial conditions. (ii) Time step (t) is calculated from stability criteria. (iii) The Poisson equation for pressure is solved to get the intermediate pressure field p ij using velocities n n u and of the n th z u r time step. i -- + j ij +-- (iv) The momentum equations are solved to get intermediate velocities u z and u in an explicit i -- r + j ij + -- manner using the previously known velocities and pressure. Step : (v) The maximum cell divergence of the velocity field is calculated and checked for its limit. If satisfied, steadystate convergence is checked for whether to stop calculation. If the maximum divergence is not achieved with desirable value, it goes to step (vi). (vi) The pressure at each of the flow domain is corrected and subsequently the velocities at each cell are adjusted to get n+ n+ n u z, u r and. Then step (v) is again performed. i -- p ij + j ij + -- This completes the necessary calculations for advancing the flow field through one cycle in time. The process is to be repeated until steady-state convergence is achieved. The working procedure of the MAC methodology for determining the numerical solution of the problem under consideration is primarily based upon the flow chart [cf. Scheme ] presented herein. 6.. Numerical stability: Time-stepping procedure Amsden and Harlow (970) suggested that the number of calculation cycles and hence the running time could be reduced by the use of an adaptive time stepping routine Korea-Australia Rheology J., Vol. 5, No. 3 (03) 67

6 Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal which, at a given cycle, would automatically choose the time step most appropriate to the velocity field at that cycle. Welch et al. (966) discussed the stability and accuracy requirements for the MAC method. They suggested that two stability restrictions are required. The first is akin to the Courant condition which will only be appropriate for selected class of problems. The second stability restriction involves the Reynolds number: t Min Re z i x z i + x ij. (6) This stability condition is related to viscous effect (cf. Hirt, 968) which can be applied directly to select an appropriate time step. A more appropriate treatment used by Markham and Proctor (983), among others, is to require that no particle should cross more than one cell boundary in a given time interval i.e., Fig. 5a. Cross-sectional velocity profiles of the axial velocity for different grid size at the stenosis throat (48% stenosis, cosine geometry, Re = 00, De = 0.). t Min z i x u z u r ij. (7) We shall now discuss the implementation of this adaptive time stepping procedure. The time step to be used at a given point in the calculation will be t = a Mint t, (8) where 0 < a ; the reason for this extra added factor a in (8) leads to a considerable computational savings (cf. Markham and Proctor, 983) and our experience concurs with them. 7. Result and Discussions For the purpose of numerical computation of the desired quantities of major physiological significance, the following parameters have been ranged around some typical values in order to obtain results of physiological interest (cf. Pontrelli, 000; Anand and Rajagopal, 004): x = 0.05, L = 60.0, = 0.05, = 0.068, = 48.0, = The computational domain has been confined with a finite non-dimensional arterial length of 60.0 in which the upstream and downstream lengths have been selected to be 8 and 3.4 times the non-dimensional arterial radius respectively. For this computational domain, solutions are computed through the generation of staggered grid with a size of 8640 while the insertion of additional points whatsoever needed between any two consecutive original irregular stenosis data of Back et al. (984) by means of interpolation has been made for the purpose of generating finer mesh adequately. 7.. Model verification and validation The simulation concerning the grid independence study Fig. 5b. Centerline axial velocity profiles for different grid size (48% stenosis, cosine geometry, Re = 00, De = 0.). was performed for the purpose of examining the error associated with the grid sizes used and is presented in Figs. 5(a) and 5(b). One may notice from these figures that the profiles concerning three distinct grid sizes almost overlap one another for Re=00 and De=0.. Thus the grid independence study in the present context of numerical simulation has its own importance to establish the correctness of the results obtained. Fig. 6a displays the results of an axial velocity profile for an irregular stenosis at the throat as well as at the inlet and outlet of the stenosed arterial segment. All the curves of the present figure have a common feature that the maximum velocity at the axis of the artery diminishes at a rate depending upon the axial positions to become zero on the wall surface. It is noted that the usual parabolic profile is maintained at both the inlet and outlet of the arterial tube throughout the computation in conformity to the fully developed flow downstream of the stenosis. Moreover, at the throat of the stenosis, the streamwise velocity accelerates with increasing Deborah number and decelerates 68 Korea-Australia Rheology J., Vol. 5, No. 3 (03)

7 Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction Fig. 6a. Cross-sectional profiles of the axial velocity for irregular stenosis at the stenosis throat and at the inlet and outlet. Fig. 7a. Pressure drop across the stenosis for irregular model in different situations for pulsatile inlet. p Fig. 6b. Comparison of pressure drop [ ] for different Deborah U o number, different stenosis models and stenosis size. Fig. 7b. Pressure drop across the stenosis for different geometry for pulsatile inlet at De = 0.0. with increasing Reynolds number in the vicinity of the centerline whereas a reverse trend is observed near the wall. In order to validate the applicability of the model under consideration, we compared our numerical results of the normalized pressure drop obtained for an irregular stenosis with the experimental results of Young and Tsai (973) in Fig. 6b. The comparative study of this figure shows that the dimensionless pressure drop diminishes considerably owing to Newtonian rheology of the flowing blood together with the absence of surface roughness of the stenosis for 56% area occlusion. The present figure also depicts that as the Deborah number increases, the dimensionless pressure drop increases for all Reynolds numbers under consideration irrespective of the shape of the stenosis. Fig. 7c. Pressure drop across the stenosis for irregular model in different situations for physiological inlet. 7.. Pressure drop The dimensionless pressure drop, measured across a stenosis of 48% area reduction, for simple pulsatile and physiological flow are depicted in Figs. 7a-7e. The pressure drop decreases with increasing Reynolds number whereas it increases with increasing Deborah number for both simple pulsatile and physiological flow (cf. Figs. 7a and 7c) which may be justified in the sense that as Reynolds number increases, the fluid gets accelerated which, Korea-Australia Rheology J., Vol. 5, No. 3 (03) 69

8 Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal Fig. 7d. Pressure drop across the stenosis for different geometry for physiological inlet at De = 0.0. Fig. 8a. Variation of the wall shear stress through each of the stenosis geometries for pulsatile inlet at Re 00, De = 0.0. Fig. 7e. Pressure drop across the stenosis for irregular model in different situations for physiological inlet. in turn, reduces the resistance to flow and eventually the pressure drop as they are linearly proportional (cf. McDonald, 974). Moreover, as Deborah number increases, the fluid relaxation time increases and hence the pressure drop increases. Furthermore, the cosine-shaped stenosis model experiences excess pressure drop due to higher area covering than the irregular stenosis (cf. Figs. 7b, 7d). Although the general behaviour of the pressure drop as time progresses with the variation of parameters in terms of both pulsatile and physiological flow conditions is analogous, it is worthwhile to note that the velocity profiles corresponding to both pulsatile and physiological flow are appreciably reflected in the pressure drop causing significant deviations of the results and hence their influences on the pressure drop can be estimated. Moreover, it is worthwhile to record the influence of a viscoelastic fluid representation of blood on the pressure drop as evident from Fig. 7e, the viscoelastic fluid causes a larger pressure drop than that of the Newtonian counterpart. This may result in higher resistive impedances experienced by the viscoelastic fluid flow through the arterial tube under consideration. Fig. 8b. Variation of the wall shear stress of irregular stenosis geometry for different inlet at Re = 00, De = Wall shear stress The variations of the dimensionless wall shear stress (WSS) distributions over the entire arterial segment for the two different shapes of the present constricted artery under the influence of both the physiological and the simple pulsatile flow are recorded in Figs. 8a and 8b in the presence of both the peak forward and the peak backward stream. One may observe from the results of Fig. 8a that the irregular stenosis model experiences higher stress than that of the cosine model towards the throat of the stenosis while the deviation of the stress distributions becomes meager in the rest of the artery irrespective of the consideration of both the peak forward and the peak backward flows. Of the two inlet conditions, the pulsatile inlet influences higher wall shear stress distribution than that of its counterpart in the event of peak backward flow only as appeared in Fig. 8b. No significant deviations of the stress distribution between these two are observed for the peak forward flow situation. It may be further noted that all other critical locations corresponding to the peak back- 70 Korea-Australia Rheology J., Vol. 5, No. 3 (03)

9 Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction Fig. 9a. Pattern of streamlines for pulsatile inlet and irregular model at which the velocity positive to instantaneous zero (Re = 00, De = 0.0). Fig. 8c. Variation of the wall shear stress of irregular stenosis geometry for physiological inlet at peak forward velocity. ward stream, where both flows occur in the reverse direction, experience a much larger magnitude of shear stress for simple pulsatile flow than for physiological flow. This is due to the fact that at the bottom most location, the magnitude of the flow rate corresponding to the simple pulsatile flow appears to have a three- fold enhancement over that of the physiological flow under consideration. All of these findings agree with the findings of Zendeboodi and Moayeri (999) who studied Newtonian flow through cosine-shaped arterial constriction. The rheological characteristics of the streaming blood on the distribution of the wall shear stress have however not been ruled out from the present investigation. Fig. 8c exhibits the results of the stress distribution over the entire arterial segment having an irregular shaped constriction in its lumen corresponding to three different rheological properties of blood at a specific Re=00 in the case of peak forward flow only. It appears that, in the case of Newtonian fluid, there is an appreciable increase in the peak stress value at the throat of the stenosis to the tune of more than ten times that of the viscoelastic one. The Newtonian fluid and the viscoelastic fluid having constant viscosity as well maintain the increasing trend of the wall shear stress distribution. Studying the behaviour of all these results of the present figure, one can conclude that the consideration of the shear-thinning viscoelastic fluid representing blood in the generated wall shear stress distribution arising from the constricted flow phenomena is quite significant and hence the inclusion of blood rheology in the model is justified Patterns of streamlines Figs. 9a 9f represent the results showing various patterns of the streamlines in the stenosed arterial segment corresponding to shear-thinning viscoelastic model idealisation of blood. It is interesting to observe that several flow lines are attracted towards the stenotic wall upstream with the formation of several recirculation zones. The Fig. 9b. Pattern of streamlines for physiological inlet and irregular model at which the velocity positive to instantaneous zero (Re = 00, De = 0.0). Fig. 9c. Pattern of streamlines for pulsatile inlet and cosine model at which the velocity positive to instantaneous zero (Re = 00, De = 0.0). Fig. 9d. Pattern of streamlines for physiological inlet and cosine model at which the velocity positive to instantaneous zero (Re = 00, De = 0.0). Fig. 9e. Pattern of streamlines for physiological inlet and irregular model at which the velocity positive to instantaneous zero in the case of Newtonian flow (Re = 00). notable features are that for both pulsatile and physiological flow past an irregular or a cosine-shaped stenosis, the onset of flow-detachment from a point upstream of the Korea-Australia Rheology J., Vol. 5, No. 3 (03) 7

10 Sarifuddin, Santabrata Chakravarty and Prashanta Kumar Mandal Fig. 9f. Pattern of streamlines for physiological inlet and irregular model at which the velocity positive to instantaneous zero for constant viscosity (Lamda =.0, Re = 00, De = 0.0). Fig. 0a. Pattern of streamlines for physiological inlet and irregular model at the end of the pulse rate. stenosis continues for the entire downstream of the irregular stenosis (cf. Figs. 9a and 9b) and a long recirculation zone is observed downstream of a cosine-shaped stenosis (cf. Figs. 9c and 9d) in contrast to multiple recirculation zones downstream of the irregular stenosis (cf. Figs. 9a and 9b). Moreover, for physiological flow corresponding to a Newtonian model, eddies, both upstream and downstream of the constriction, are visible but no downstream flow detachment takes place for an irregular stenosis model (cf. Fig. 9e). The formation of such eddies is caused primarily by the adverse pressure. The pattern of flow lines gets perturbed further with the formation of a small separation zone downstream of the stenosis when the fluid is considered to be viscoelastic having constant viscosity as shown in Fig. 9f. The nature of the streamlines at the end of pulse rate for shear-thinning viscoelastic fluid for both physiological as well as pulsatile flow past both irregular and cosineshaped arterial stenosis is captured in the concluding Figs. 0a-0d. The general observations of the present flow lines are almost analogous to those of Figs. 9 so far as the flow behaviour is concerned. Thus, the irregular stenosis model bears the potential to measure the flow disturbances one step closer to the real situation compared to the usual cosine-shaped stenosis model. Moreover, the influence of the present rheological behaviour of the streaming blood on the constricted flow phenomena may be measured quantitatively through a direct comparison of the results exhibited by the corresponding Newtonian model. 8. Concluding Remarks At present, inclusion of the shear thinning viscoelastic characterisation of the streaming blood and both pulsatile and physiological flow waveforms in the arterial flow, under stenotic conditions, with both irregular and cosine geometry of the constriction, certainly contributes much to the domain of constricted flow phenomena in the arterial system. Unlike the previous investigations, the present updated model appears to have its own importance in the context of the quantitative estimation of the influences of blood rheology, the shape of the constriction and different flow waveforms as well on the unsteady constricted flow characteristics of Fig. 0b. Pattern of streamlines for pulsatile inlet and cosine model at the end of the pulse rate. Fig. 0c. Pattern of streamlines for physiological inlet and cosine model at the end of the pulse rate. Fig. 0d. Pattern of streamlines for physiological inlet and irregular model at the end of the pulse rate in the case of Newtonian flow. blood in terms of the pressure drop, the distribution of wall shear stress and the patterns of the streamlines. The present study makes an observation that, although the behaviour of both the pulsatile and the physiological flow are analogous at certain instances of time, they should always be treated as two completely different flows. This observation agrees well with those of Zendehboodi and Moayeri (999) and hence the real physiological flow needs to be modeled for the purpose of deeper understanding of the pulsatile flow through constricted arteries. The irregular outline of the stenosis is often found to be more realistic than that of the cosine geometrical shape where the flow experiences a greater number of separation zones both upstream and downstream of the constriction, 7 Korea-Australia Rheology J., Vol. 5, No. 3 (03)

11 Physiological flow of shear-thinning viscoelastic fluid past an irregular arterial constriction in the case of irregular stenosis, than that of the cosine stenosis irrespective of the use of both pulsatile and physiological flow wave forms. Moreover, apart from the considerable influence of blood viscoelasticity on the distribution of wall shear stresses, it has also been observed from the present quantitative analysis that the rheology of blood affects the peak pressure drop significantly at all times, irrespective of pulsatile and physiological flow wave forms at the inlet, over the entire physiological range of Reynolds numbers under consideration. The behaviour of the pressure drop with different Reynolds numbers appears to be closer to that of the experimental results of Young and Tsai (973) when the rheological parameter (De) value is continuously reduced. Such characteristics of the pressure drop based on the present rheology of the streaming blood influence the pattern of the velocity profile, the wall shear stress and the flow lines significantly and hence its incorporation appears to be quite useful in order to update the appropriate simulation and its validation in the realm of arterial biomechanics. Acknowledgements The final form of the paper owes much to the helpful suggestions of the referees, whose careful scrutiny we are pleased to acknowledge. The present work is part of the Special Assistance Programme (SAP-II) sponsored by the University Grants Commission (UGC), New Delhi, India. The authors are thankful to Dr. Martin Rowlands, University Division of Anaesthesia & Intensive Care, Queens Medical Centre Campus (Nottingham University Hospitals NHS Trust), University of Nottingham, Derby Road, Nottingham, NG7 UH, United Kingdom, for evaluating and correcting the English language of this paper. 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