A Computational Analysis of a Two-Fluid non-linear Mathematical Model of Pulsatile Blood Flow through Constricted Artery

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1 A Comutational Analysis of a Two-Fluid non-linear Mathematial Model of Pulsatile Blood Flow through Constrited Artery S. U. Siddiqui, S. R. Shah, Geeta* Deartment of Mathematis, Harourt Butler Tehnologial Institute, Kanur- 8, India * geeta5_hbti@rediffmail.om Abstrat This aer deals with the ulsatile flow of blood through a stenosed artery with the effet of body aeleration. We have onsidered the axially non-symmetri mild stenosis & a two layered blood flow with a ore region of susension of all erythroytes assumed to be a Casson fluid and a eriheral layer of lasma free from ells as a ewtonian fluid. The non-linear differential equations governing the fluid flow are solved analytially and obtained the exressions for veloity, flow rate, wall shear stress, lug ore radius, effetive visosity. We have disussed the effet of body aeleration, ulsatality, eriheral stenosis height and non-ewtonian behavior of blood on above mentioned flow quantities. It is found that the inrease of stenosis size and yield stress inreases the lug ore radius, ressure dro, wall shear stress where as veloity and flow rate dereases. Body aeleration also lays a very imortant role in blood flow. The resent study is more useful for the urose of validation of the different models for blood flow in the different ases of stenosis. Keywrds: Atheroslerosis; Wall shear stress; Axial veloity; Body aeleration; Introdution Atheroslerosis is a major risk fator for many onditions involving the flow of blood. It is a otentially serious ondition where arteries beomes logged u by fatty substanes known as laques or atheroma. The laques ause affeted arteries to harden and narrow, whih an be dangerous as restrited blood flow an damage organs and sto them funtioning roerly. Aording to the exerts stenosis begins with the damage to the endothelium aused by the high blood ressure, smoking or high holesterol. The flow of blood through a stenosed artery an be reresented by different fluid models aording to the situation. In the literature it is found that some researhers reresented blood by ewtonian & non-ewtonian while some of the researhers onsidered single and two layered blood flow. Arteries are the blood vessels, arry blood from the heart throughout the body. Human heart ums the blood in the irulatory system and rodues the ressure gradient throughout the system. There are two omonents of the ressure gradient, one is onstant and other is flutuating or ulsatile. Body aeleration is a very imortant fator in blood flow modeling. Body aeleration disturbs the normal blood flow and auses many roblems suh as headahe, abdominal ain, inreases ulse rate and others. Rathod et. al. [] desribes the ulsatile flow of oule stress fluid through a orous medium with eriodi body aeleration and magneti field. Shaw et. al. [5] have shown the effet of body aeleration on the two dimensional flow of asson fluid through an artery with asymmetri stenosis where, the artery wall has been treated as an elasti ylindrial tube. htt://e-jst.teiath.gr 65

2 Srivastava [] modeled a two layered blood flow through a narrow atheterized artery by onsidering blood as ewtonian, inomressible with variable blood visosity. Bugliarello and Sevilla [] and Hayden [] have exerimentally observed that when blood flows through narrow tubes there exists a ell free lasma layer near the wall. In view of their exeriments, it is referable to reresent the flow of blood through narrow tubes by a two layered model instead of single layered model. Pulsatile oule stress fluid model through stenosed artery with the orous effet for non-ewtonian blood has been onsidered by Singh and Rathee [7]. Singh and Singh [6] resented a aer that deals with the blood flow through a radially non-symmetri stenosed artery onsidering blood as a non-ewtonian asson fluid model. Few authors foused on the ulsatile nature of blood, magneti effet and body aeleration in stenoti artery with its aliation in different blood disease [, 4, 8]. Mathematially the solution an be done by using Hankel transformation aroah instead of either numerial or emirial aroah by Haghighi [8]. Sankar et. al. [] examine the effet of an external magneti field on the blood flow through a omosite stenosis by using a two layered blood flow model onsisting of a ell free eriheral layer and a ore region of erythroytes in lasma flowing through a omosite stenosis in the resene of an external transverse magneti field. Hazarika and Sharma [9] onsidered a two-layered mathematial model for blood flow through taering asymmetri stenosed artery with veloity sli at the interfae under the effet of transverse magneti field. Srivastava and saxena [8] investigated twolayered model of asson s fluid flow through stenoti blood vessels. Ponalgusamy [] investigated blood flow through an artery with mild stenosis: A two-layered model, different shaes of stenosis and sli at the wall. Ellahi et. al. [7] onsidered a study of non-ewtonian miroolar fluid in an arterial blood flow through omosite stenosis, sli veloity are taken into aount with ermeable wall effets. In the resent analysis an attemt has been made to rovide a model to examine the effet of an body aeleration on blood flow through an axially symmetri stenosis whih has not yet been examined in revious works. For this urose we have used a two-layered blood flow model onsisting of a ell-free eriheral layer and a ore region of erythroytes in lasma. This two-layered model for blood flow rovides a more realisti model for flow in small arteries sine we annot neglet the existene of the eriheral layer and the red blood ells in the lasma. The effet that the body aeleration has on the fluid's veloity, flow rate, wall shear stress and shear stress at the stenosis throat will be examined. Present model an lead to the imrovement of existing diagnosti tools for a more effetive treatment of atients suffering from aner, hyertension, myoardial infartion, stroke and aralysis. Mathematial formulation We onsider an axially symmetri, laminar, usatile and fully develoed flow of blood (assumed to be inomressible) through a irular tube with an axilly symmetri mild stenosis as shown in figure. It is assumed that the wall of the tube is rigid and the body fluid blood is reresented by a two-fluid model with a ore region of susension of all erythroytes as a osson fluid and a eriheral layer of lasma as a ewtonian fluid. The artery length is assumed to be large enough as omared to it s radius so that the entrane and the exit, seial wall effets an be negleted. (4), 5 66

3 The geometry of the stenosis in the eriheral region is given by R z R / P os z / z, For z z R, For z z The geometry of stenosis in the ore region is given by R z Where R / os /, z z For z z R, For z z R z is the radius of the stenosed artery with eriheral layer, R z R( z) radius of the artery in the stenosed ore region suh that R () () z is the, R and R are the radii of the normal artery and ore region of the normal artery resetively; P is the maximum height of the stenosis in the eriheral region, is the ratio of the entral ore radius to the normal artery radius, is the maximum height of the stenosis in the ore region suh that P and z is the half length of the stenosis. It has been reorted that the radial veloity is the negligible small for a low Reynolds s number flow in a tube with mild stenosis. The equation of motion governing the fluid flow are given by u r F( t) r R ( z) t z r r u r F( t) R ( z) r R( z) t z r r () (4) In the ore and eriheral regions resetively, where u and u are the fluid veloities in the ore region and eriheral regions resetively, and are the shear stresses htt://e-jst.teiath.gr 67

4 for the asson fluid and ewtonian fluid resetively, and are the densities for asson fluid and ewtonian fluid resetively, P is the ressure and Ft () is the body aeleration. The onseutive equations for asson fluid and ewtonian fluid are resetively given by / u ( ) y if y and R r R z r u if y and r R r u ( ) ( ) if R z r R z r (5) (6) Where R is the radius of the lug flow region. The eriodi body aeleration in the axial diretion is given by F( t) a os( t ) (7) b Where a is its amlitude, b f b, f b is its frequeny in Hz., is the lead angle of Ft () with reseted to the heart ation. The frequeny of body aeleration f b is assumed to be small so that wave effet an be negleted. The ressure gradient at any z and t may be reresented as follows ( z, t ) A A os( t ) z (8) Where A the steady omonent of the ressure gradient is, A is amlitude of the flutuating omonent and f where f is the ulse frequeny. Both A and A are funtion of z we introdue the following non-dimensional variables. z R( z) R ( z) r w z R z R z r t tw w R R R R w b, ( ), ( ),,,, u u,,,, R R A R / 4 r A R / 4r u u u A a u,,, e, B / / A A s s AR /4r A R A R (9) Where and are ulsatile Reynolds s number for asson fluid and ewtonian fluid resetively. Using non-dimensional variables, equation () and () beomes (4), 5 68

5 R R z z P / os z / z, For z z, For z z / os z / z, For z z, For z z () () The governing equation of motion given by equation () and (4) are reresented in the non dimensional form as u 4 f ( t) ( r) t r r u 4 f ( t) ( r) t r r () () where f ( t) ( eos t) B os( t ) (4) using non dimensional variables equation (5) & (6) redue to u r / / ( ) u r (5) (6) u r (7) The boundary onditions are u is finite and at r r u at r R (8) and u u at r R The boundary onditions in the dimensionless form are u is finite and at r r u at r R and u u at r R (9) The non-dimensional volumetri flow rate is given by htt://e-jst.teiath.gr 69

6 R( z) Q 4 u( z, r, t) rdr () Qt () Where Q( t) ; Q( t) is the volumetri flow rate. R A 4 8 Perturbation Method of Solution Sine it is not ossible to find an exat solution to the system of nonlinear equations ()-(7), the erturbation method is used to obtain the aroximate solution to the unknowns u, u, and. when we non-dimensionalize the momentum equations () and (4) and ours naturally and hene it is more aroriate to exand the equations ()-(7) about and. Let us exand the lug ore veloityu, the veloity in the ore region erturbation series of as below (where <<) u in the u ( z, t) u ( z, t) u ( z, t)... () u ( z, r, t) u ( z, r, t) u ( z, r, t)... () R ( z, t) R ( z, t) R ( z, t)... () u ( z, r, t) u ( z, r, t) u ( z, r, t)... (4) Substituting the erturbation series exansion in equation (), (5) and (6) and equating the ower of and onstant terms. u ( r ) t r r f ( t) r ( r ) r u r u r ( ) ( ) (5) (6) (7) (8) Similarly using the erturbation series exansion in equation () and (7) then equating onstant term and terms. (4), 5 7

7 ( r ) f ( t) r ( a) r u ( r ) ( b) t r r u ( ) r u ( d) r (9) ow substituting the erturbation series exansion in equation (9) and then equating the onstant term and,, we get u u and are finite,, at r r r,, u u, u u at r R u u at r R and are finite at r () On solving equation (5)-(9) for unknowns u, u, u, u, u, u,,,, using equation (), we an obtain, f () t r () f () t r () f () t R () u f ( t) R r R 8 u f ( t)( R r ) ( R r) f ( t) R r 8 u f ( t)( R r ) ( R r ) f ( t) R r (4) (5) (6) ( ) ( ) (7 4 ) 8 5 f t r R r k rr r (7) 7 k f ( t)( R r r ) f ( t) R (8) 8 7r 7 f ( t) r u R r r R kr ln R (9) htt://e-jst.teiath.gr 7

8 k r R r r R R R R u f ( t) k (7r R r 5 R ) R ( R 4R R r 4 R r ) R R R R kr ln R (4) k R R R R R R R R u f ( t) k (7R R R 5 R ) (4) R ( R 4R R R 4 R R ) R R R R kr ln R Where k f (t) egleting the terms of o( ) and higher ower of in equation (), the first aroximation lug ore radius an be obtained as R f () t k (4) Using equations (4) (5), (9), & (4) the exressions for axial veloities in the ore and eriheral regions are obtained as 7 f ( t) r u f ( t)( R r ) R r r R kr ln R (4) On relaing the value of u and u in equation (44) we get the value of u u u u (44) The exression for wall shear stress w an be obtained as w r R (4), 5 7

9 Axial Veloity 7 R k w f ( t) R f ( t) R 8 7R (45) From equation (), (4) and (44) the volumetri flow rate is given by R R R (46) Q 4 r u u dr 4 r u u dr 4 r u u dr R R 4. Results and Disussions In the resent model an attemt has been made to evaluate some of the imortant harateristis of blood flow ast an arterial stenosis with the effet of body aeleration and ulsatile ressure gradient. In order to oint out the biologial imortane and to examining the validity of the model, omuter odes are develoed to evaluate the analytial solution for flow rate, veloity rofile, wall shear stress, effetive visosity for different values of arameters involved in equations ()-(46). The value of womersley frequeny arameters are taken as ( B.5 ). The body aeleration arameter is taken in the range -, the ressure gradient arameter e is taken in the range.5, the ratio of the entral ore radius to the normal radius of the artery is taken as.95.we have onsidered the magnitude of the lead angle as., the range -.5 is taken for the eriheral stenosis height, yield stress is taken as, B B.5 B Radial Distane Fig. Variation of axial veloity with radial distane for L,.,.,.8, e htt://e-jst.teiath.gr 7

10 Axial veloity Variation of axial veloity with radial distane has been shown in Fig. and Fig.. From the Fig. it is learly observed that at r veloity is maximum and minimum value at the stenoti wall r R( z) for fixed values of o L,., t 45,.8, e and different values of body aeleration arameter ( B,, ).It an be seen from Fig. that axial veloity inreases with the inrease in body aeleration. Fig. shows the variation of axial veloity with radial distane for different values of time t. Time lays a very imortant role in blood flow modeling. It is learly seen from the Fig., as time inreases veloity dereases. When the blood is flowing to start i.e. when t veloity is maximum. ow as we inrease the time, veloity dereases. Veloity is less at (t=,.5) than veloity at t. Fig. 4 resents the flow rate distributions for the two fluid Casson models R.,.,.8 and t 45 at the throat of the stenosis (i.e.) when z. Like the axial veloity flow rate also inreases with the inrease in body aeleration. In the absene of yield stress, the urves are linear while the urves are nonlinear with the inrease in yield stress. It is also observed that when the yield stress inreases from = to =., flow rate dereases beause of inrease in width of the lug flow region..5.5 t.5 t.5 t Radial distane Fig. Variation of axial veloity with the radial distane for different values of time and fixed values of B,.5,.,. The variation of wall shear stress with time and eriheral stenosis height is desribed in figures (5,6 and 7) for different flow arameters (, eb,,, ). Wall shear stress is symmtrial about t 8. It is observed that the wall shear stress inreases linearly (4), 5 74

11 Wall Shear Stress Flow Rate with the inrease of the eriheral stenosis height from Fig. 5. Body aeleration also enhanes the wall shear stress B B B Pressure Gradient Fig 4 Variation of flow rate with ressure gradient e for R t z.,.,.8, 45, B B B Periheral Stenosis Height Fig. 5 Variation of wall shear stress with eriheral stenosis height for B,.5,.5,.95, t 45 htt://e-jst.teiath.gr 75

12 Wall Shear Stress Wall Shear Stress B B B Time Fig. 6 Variation of wall shear stress with time t for.4,.,.95,,. e Time Fig. 7 Variation of Wall shear stress with time for.4,.95, e,.,. Fig. 6 and Fig. 7 shows the variation of wall shear stress with time t, full sale of time t( t6 ) has been onsidered. It is found that in a stenosed artery wall shear stress is highly influened by body aeleration. Wall shear stress dereases with the inrease in body aeleration. It is dieted that attains it s minimum value at t 8 while, maximum at t and t 8. We an not ignore the effet of eriheral stenosis (4), 5 76

13 height, sine it is a very imortant fator in two layered blood flow modeling. Inrease in eriheral stenosis height results an inrease in wall shear stress. Conlusion We have onsidered the two-layered blood flow through stenosed artery with all erythroytes in ore region as a Casson fluid and eriheral layer of lasma as a ewtonian fluid. From the resent results, it is lear that body aeleration is an imortant fator in blood flow modeling. In the resent aer we have studied the effet of body aeleration on various flow arameters. It is deited that flow rate and axial veloity inreases but wall shear stress shows both inreasing and dereasing trends with the inrease in body aeleration arameter B, aording to our imagination. This model is able to redit the some blood flow harateristis and may be useful in biomedial aliations. In the near future the resent study an be extended by onsidering magneti effet of blood and ermeability of wall. Referenes. Biswas, D. and Chakraborty, U. S. (9) Pulsatile flow of blood in a onstrited artery with body aeleration. Aliation and Alied Mathematis, 4: Bugliarello, G. and Sevilla, J. Veloity distribution & other harateristis of steady and ulsatile blood flow in a glass tubes, Biorheology, Vol. 7,. 85-7, (97).. Bugliarillo G., Hayden J. W., Detailed harateristi of flow of blood in vitro., Trans. So. Rheol., Vol , (96). 4. Chaturani P. and Palanisamy V., Casson fluid model for ulsatile flow of blood under eriodi body aeleration Biorheology, Vol. 7,. 69-6, (99). 5. Chaturani P. and Palanisamy V., Pulsatile flow of blood with eriodi body aeleration Int. J. Engg. Si., Vol. 9,. -, (99). 6. Chaturani P. and Palanisamy V., Pulsatile flow of Power law fluid model for blood flow under eriodi body aeleration Biorheology, Vol. 7, , (99). 7. Ellahi R.,Rahman, S. U., Gulzar, M. M., adeem, S., Vafai, K., A mathematial study of non-ewtonian miroolar fluid in arterial blood flow through omosite stenosis, Alied Mathematis and Information Sienes, Vol. 8(4), , (4). 8. Haghighi, A. R. and Asghari., A mathematial model for the effet of magneti, body aeleration and time deendene on blood flow in stenosed artery, Austrelian Journal of Basi and Alied Sienes, Vol. 6(),. 59-7, (). 9. Hazarika, G. C., & Sharma B., Two-layered mathematial model for blood flow through taering asymmetri stenosed artery with veloity sli at the interfae under the effet of transverse magneti field, Int. J. Comuter Aliations, Vol. 5(8),. 7-, (4).. Joshi, P., Pathak, A. and Joshi, B. K. Two-layered model of blood flow through omosite stenosed artery, Aliation and Alied Mathematis, Vol. 4(8), htt://e-jst.teiath.gr 77

14 . Ponalgusamy, R., Blood flow through an artery with mild stenosis: A two-layered model, different shaes of stenosis & sli at the wall, J. of Alied Siene, Vol. 7(7),. 7-77, (7).. Rathod V. P. and Tanveer S. Pulsatile flow of oule stress fluid through a orous medium with eriodi body aeleration and magneti field., Bulletin of the Malaysian Mathematial Sienes Soity,Vol., ,(9).. Sankar A. R., Gunakala, S. R. and Comissiong M. G., Two layered blood flow through a omosite stenosis in the resene of magneti field, IJAIEM, Vol. (),. -4, (). 4. Sankar, D. S., Lee,U., (9). Mathematial modeling of ulsatile flow of non- ewtonian fluid in stenosed arteries, Commun onlinear Si umer Simulat., Vol. 4, Shaw S., Murthy, P.V. S.. and Pradhan S. C., Effet of body aeleration on two dimensional flow of asson fluid through an artery with asymmetri stenosis The oen Transort Phenomena journal, Vol., , (). 6. Singh A. K. and Singh D. P., Blood flow obeying a asson fluid equation through a radially non-symmetri mild stenosis, Amerian Journal of Alied Mnathematis, Vol. (),. -4,(). 7. Singh J., Rathee R., Analysis of non-ewtoian blood flow through stenosed vessel in orous medium under the effet of magneti field, Int. J. of Physial Sienes, Vol. 6(), , (). 8. Srivastava, V. P. and Saxena, M., Two-layered model of asson s fluid flow through stenoti blood vessels: aliation to ardiovasular system, J. Biomeh., Vol. 7,. 9-98, (994). 9. Srivastava, V. P., Rastogi, R., Vishnoi, R, Two-layered susension blood flow through an overlaing stenosis, Comuter and Mehanis with Aliations, Vol. 6,. 4-44,().. Srivstava R. A two-layered blood flow through a narrow atheterized artery, e- Journal of Siene and Tehnology, Vol. (6),. -4, ().. Verma,. K., Siddiqui, S. U. and Guta, R. S., Mishra S. () A mathematial model for ulsatile flow of Hershel Bulkley fluid through stenosed arteries. Journal of siene and tehnology, 5(4) : (4), 5 78