PERFORMANCE MODELING AND ANALYSIS OF BLOOD FLOW IN ELASTIC ARTERIES

Applied Mathematics and MechanicsPublished by Shanghai University,(English Edition) Vol. 26 No.3 Mar.2005Shanghai, ChinaCEditorial Conite of Appl. Math. Mech. ,ISSN 0253 4827Article ID: 0253-4827(2005)03-0345-10PERFORMANCE MODELING AND ANALYSIS OF BLOODFLOVIN ELASTIC ARTERIES *。Anil Kumar'，C.L. Varshney'，G. C. Sharma?(1. Department of Post- Graduate Studies and Research in Mathematics &Computer Science, S.Varshney College, Aligath-202001, India;2. Institute of Basic Science, Khandari, Agra-282002, India)(Communicated by zHOU Zhe-wei)Abstract: Two different non-Newtonian models for blood flow are considered, first a simplepower law model displaying shear thinning viscosity, and second a generalized Maxwellmodel displaying both shear thinning viscosity and oscillating flow viscous-elasticity .Thesemodels are used along with a Newtonian model to study sinusoidal flow of blood in rigid andelastic straight arteries in the presence of magnetic feld. The elasticity of blood does notappear to influence its flow behavior under physiological conditions in the large arteries,purely viscous shear thinning model should be quite realistic for simulating blood flow underthese conditions. On using the power law model with high shear rate for sinusoidal flowsimulation in elastic arteries, the mean and amplitude of the flow rate were found to be lowerfor a power law fluid compared to Newtonian fluid for the same pressure gradient. Thegoveming equations have been solved by Crank- Niclson scheme . The results are interpreted inthe context of blood in the elastic arteries keeping the magnetic effects in view. Forphysiological flow simulation in the aorta, an increase in mean wall shear stress, but areduction in peak wall shear stress were observed for power law model compared to aNewtonian fluid model for matched flow rate wave fomn. Blood flow in the presence oftransverse magnetic field in an elastic artery is investigated and the influence of factors suchas morphology and surface iregularity is evaluated.Key words: elastic artery model; Crank Niclson scheme; non-Newtonian fluid; wall shearstressChinese Library Classification; 0242.1; 0357.1; R318.01Document code: A2000 Mathematics Subject Classification: 92C10; 76A05Nomenclaturewall shear stressnon-Newtonian indexshear rateshear stress of the pth elementyield stressangular velocityCasson viscosity中国煤化工the consistency indexYHCNMHG* Received 0ct. 17 ,2003345346Anil Kumar, C.L. Varshney and G. C. SharmaMmagneticparameter ( Hartmannunsteadiness parameternumber)k,R mean parametersu,w velocity component in the r andTp relaxation time of the pth element若directions, respectivelydensityPpressureIntroductionThe importance to. atherogenesis of arterial flow phenomena such as flow separation,recirculation and stagnation secondary flow motion low and oscillatory wall shear stress and longparticle residence times, have become more and more evident during the past few decades. Thehemodynamics of flows through atherosclerotic vessels is of great interest, because these vesselspresent a substantial health risk and are major causes of mortality and morbidity in theindustrialized world. Some of the studies on the arterial flows are as narrated below. Thurston!11attempted to investigate all of the rheological properties of blood with a model including non-Newtonian viscosity, viscoelasticity and thixotropy. Liepsch and Moraveel21 investigated the flowof a shear thining blood, analog fluid in pulsatile flow through arterial branch model and observedlarge differences in velocity profles relative to these measured with Newtonian fluids having thehigh shear rate viscosity of the analog fluid. Rindt et al . L3J considered both experimentally andnumerically the two-dimensional steady and pulsatile flow. Nazemi et al .41 made importantcontributions to the identification of atherogenic sites. Rodkiewicz et al . [5] used several differentnon-Newtonian models for blood for simulation of blood flow in large arteries and they observedthat there is no effect of the yield stress of blood on either the velocity profiles or the wall shearstress. Boesiger et al.l6J used magnetic resonance imaging ( MRI ) to study arterialHemodynamics. Perktold et al.7J modeled the flow in stenotic vessels as that of anincompressible Newtonian fluid in the rigid vessels . Sharma and Kapoor-8J made a mathematicalanalysis of blood flow through arteries using finite element method. Dutta and Tarblll9] studiedthe two different rheological models of blood displaying shearing thining viscosity and oscillatoryflow visco- elasticity. Lee and Libby1o] made a study of vulnerable atherosclerotic plaquecontaining a large necrotic core, and covered by his fibrous cap.Korenage et al . l 1 considered biochemical factors such as gene expression and albumintransport in atherogenessis and in plaque rupture these have been shown to be activated byhemodynamic factors in wall shear stress. Rachev et al.12J have considered a model forgeometric and mechanical adaptation of arteries. Rees ane Thompson' 13] studied a simple modelderived from laminar boundary layer theory to investigate the flow of blood in arterial stenoses upto Reynolds oumbers of 1 000. Tang et al .[14]analysed triggering events are bemieved to beprimarily Hemo-dynamics including cap tension, bending of torsion of the artery. Zendehbudi andMoayary-15] made a comparison of physiological and simple pulsatile flows through stenosedarteries .Berger and Jou'e measured wall shear stress down stream of axi-symmetric stenoses in thepresence of hemodynamics forces acting on the plaque中国煤化工le for plaque!rupture. Botnar et al. l17] based on the correspondence|Y片C N M H Gsurements andnumerical simulations used two approaches to study in detail the role of diferent flow patterms forthe initiation and amplification of atherosclerotic plaque sedimentation. Stroud et al. [18] foundedPerformance Modeling and Analysis of Blood Flow in Elastic Arteries347the differences in flow fields and in quantities such as wall shear stress among stenotic vesselswith the same degree of stenosis. Sharma et al . [19) made a mathematical analysis of blood flowthrough arteries using finite element Galerkin approaches.In current study we are interested in the analysis of blood flow in elastic arteries in thepresence of transverse magnetic field. In the present paper we extend the local flow calculations toinclude the non- Newtonian rheology of blood in order to examine the effects of the shear thiningvisco elasticity of blood on flow phenomena in large elastic arteries in the presence of magneticfield using a suitable finite difference scheme.1 Constitutive Equations of the BloodThe constitutive equations proposed for the whole blood are as follows:.12 = η。γl1+r1/2,| τ|> Ty,(1)andY=0,|τ1