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A Computer Simulation of the Non-Newtonian Blood Flow at the Aortic Bifurcation

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A Computer Simulation of the Non-Newtonian Blood Flow at the Aortic Bifurcation Do21 9290/93 %S.W+.oO J. Biomechmics Vol. 26, NO. I, pp. 37-49. 1993. c#; 1992 Pergamon Press Ltd Pnnted m Great Britain A COMPUTER SIMULATION OF THE NON-NEWTONIAN BLOOD FLOW AT THE AORTIC BIFURCATION ZHENG Lou* and WEN-JEI YANG? *Transportation Research Institute, University of Michigan, Ann Arbor, MI 48109, U.S.A. and tMechanicil Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, U.S.A. Abstract-A two-dimensional numerical model was developed to determine the effect of the non-New- tonian behavior of blood on a pulsatile flow at the aortic bifurcation. The blood rheology was described by a weak-form Casson equation. The successive-over-relaxation (SOR) method was used to solve both the vorticity and Poisson equations numerically. It was disclosed that the non-Newtonian property of blood did not drastically change the flow patterns, but caused an appreciable increase in the shear stresses and a slightly higher resistance to both flow separations and the phase shifts between flow layers. NOMENCLATURE INTRODUCTION AR area ratio The curvatures, junctions and bifurcations of large D diameter and medium arteries are severely affected by athero- second invariant of the strain rate tensor Dll sclerosis. It has been suggested that fluid dynamics H half-width of a channel J, .i and J’ coordinate transformation coefficients may play a role in the genesis and progression of the I, reference length disease. The blood flows in these regions are disturbed Lb min minimum branch length and the arterial walls are exposed to either high or low parameter in the Papanastasiou’s weak form shear stresses. Many experimental and theoretical :Y Reynolds number studies have been conducted to investigate the role of Re,, peak Reynolds number in the trunk ri transition curve radius at the vertex fluid dynamics in atherogenesis. Comprehensive re- To transition curve radius at the hip views pertinent to the subject are available (Lou, .\ dimensionless displacement along a wall, with 1990; Nerem and Levesque, 1987). The involvement of the origin being at the vertex for the inner wall the fluid dynamics in atherogenesis is subtle; however, and at the hip for the outer wall i time various studies favor the association of a low shear V reference velocity region with atherosclerosis (Nerem and Levesque. c, Y spatial domain 1987). x Womersley number The current study is to reveal the effect of blood x, ? and a’ coordinate transformation coefficients bifurcation angle rheology on the fluid dynamics and the consequent i,$andB coordinate transformation coefficients implications in atherogenesis at the aortic bifurcation. ;‘, v and y’ coordinate transformation coefficients It is commonly believed that the non-Newtonian ef- ;’ shear rate fect is small in large vessels, where the shear rate is rl Casson viscosity or limiting apparent viscosity high. However, low shear spots also exist near bends u effective or apparent viscosity v kinematic viscosity and bifurcations. The current study uses the Casson :3 9 referential domain model, which is known to adequately represent non- P density Newtonian blood rheology (Charm and Kurland, 0, Crand u’ coordinate transformation coefficients 1965). Liepsch and coworkers (Liepsch, 1985. 1986; r. S and T’ coordinate transformation coefficients r shear stress Liepsch and McMillan, 1986; Liepsch and Moravec, ro time constant for a starting flow 1984; Moravec and Liepsch, 1983) studied flows in 3 yield shear stress renal and general T-shaped bifurcations with ?, and T; dimensionless yield shear stresses a power-law non-Newtonian fluid. Perktold et ul. dimensionless viscosity function (1989) studied a Y-shaped bifurcation with an ; stream function w cardiac frequency or vorticity aneurysm in the cerebrovascular system with the Cas- son model. More recently, Perktold et al. (1991a) Subscripts studied a carotid bifurcation with the Casson model. b branch The Casson model, like many other models, does not t trunk address a time-dependent thixotropic behavior that Y yield blood shows. There have been no studies using the non-Newtonian models on the aortic bifurcation, which is where the abdominal aorta is divided into Received in final form 9 June 1992. two iliac arteries. 37 38 2. Lou and W.-J. YANG The present study employs a two-dimensional than during steady flow. There were other studies that model that is tractable computationally although the found the secondary flow with a quasi-steady flow related physiological situation is three-dimensional. rate, which has either a low frequency or a large ratio There have been some three-dimensional simulations of the mean flow over the peak flow (Walburn and for the T-shaped (Perktold and Peter, 1990), carotid Stein, 1982). The current model is similar to that of (Perktold et al., 1991a, b; Rindt et al., 1990) and aortic Walburn and Stein (1982) and will only cause small (Yung et al., 1990; Wille, 1984) bifurcations. Wille error, if any. In a recent study (Hearshen, 1991), the (1984) used a nonphysiological Reynolds number of results from the same two-dimensional model, but 10, while Yung et al. (1990) selected a nonphysiologi- with a Newtonian fluid, showed good agreement with cal area ratio of 2.0 to produce flow geometry for those from magnetic resonance measurements for pul- a convenient mesh generation. In addition, both Wille satile flows, and some deviations for steady flows. (1984) and Yung et al. (1990) applied steady flow. For the carotid bifurcation, a two-dimensional PHYSICAL MODEL model can give basic information on the complexity of the flow field, but it cannot take into account the Geometry important secondary flow that can be seen in a three- dimensional model (Perktold et al., 1991b). The devi- The flow is in channels of infinite depth (Fig. 1). ation induced by the secondary flow is found in the H, and Hb are the half-widths of the trunk and branch, more distal parts (van de Vosse et al., 1990; Perktold respectively, while B is the bifurcation angle. The et al., 1991b). In the case of the aortic bifurcation, vertex is the tip of the flow divider, while the hip is at secondary flows are not observed in the branches with the turning point of the outer wall. Let ri and r, be the the area ratios of 0.8 and 1.2 and a bifurcation angle of radii of the transition curves at the vertex and the hip, 70” during pulsatile flow, although they are observed respectively. The system geometry is symmetrical with during steady flow (Walburn and Stein, 1982). In respect to the trunk centerline and the flow rates in another study of the aortic bifurcation with a bifurca- the two branches are equal. Therefore, the analysis tion angle of 91” and an area ratio of 0.96, the second- needs to be performed on only one-half of the system. ary flow was found both in steady and pulsatile flow The actual aortic bifurcation may not be symmetric, (Fukushima et al., 1988). However, the region affected and the effect of asymmetry on the flow is also an by the secondary flow is smaller during pulsatile flow important subject (Friedman et al., 1983). Fig. 1. A schematic diagram of a two-dimensional aortic bifurcation. The channels are of infinite depth. Non-Newtonian blood flow 39 D w a=- ; for a circular pipe, 2 J where D is the pipe diameter and w is the frequency. The equation can also be expressed as (Lou. 1990) a* = 36.336 :, (4) where o. is the time constant for a starting flow in the pipe, and T the period of the oscillating flow. r. re- flects the influences from inertia and viscous forces. For a channel flow to be comparable with that in a circular pipe, the same equation should hold. With Fig. 2. Physiologically pulsatile flow rate over one cycle the time constant for a starting channel flow equal to proximal to the aortic bifurcation (adapted from Stettler H2/1.5712v, one gets et ai., 1981). The total cycle lasts for 1 s in a normal case. a= 1.531H w for a channel, (5) J V A peak Reynolds number is defined in the trunk, where H is H, for the current study. The trunk with the flow rate peaking at peak 1 (Fig. 2), as Womersley number is 8.8623 for a heart rate of 60 beats min _ I. Rep,= LVIv, (11 where L is the characteristic length, equal to the trunk hydrodynamic diameter (4H,), chosen to be 15 mm, The Casson model V the characteristic velocity, equal to the average velocity across a hydrodynamically equivalent circu- The Casson equation was originally proposed for lar pipe, and v the kinematic viscosity, equal to shear thinning fluids, such as printer ink (Casson, 4.5 x 10e6 m2 s-l. Re, is equal to 1300. The time- 1959), with its stress-strain-rate relation average Reynolds number over one cardiac cycle is ,l/2=p23~/2+7;/*, (6) 181, a laminar flow. The area ratio, AR, is defined as the ratio of the total area of two branches over the where ‘ty is the yield shear stress, 3 the shear rate, and trunk area, and its values are 0.75 and 1.1. tl the Casson viscosity or the asymptotic apparent Murray (1926) observed that the bifurcation angle viscosity, which is calculated from varies from 60 to 75”; Walburn et al. (1979) found this tl=pv, (7) angle to be between 37 and 80”; and Friedman et al. (1981) observed this angle to be approximately 51”. where p is the density, and v the kinematic viscosity in Feuerstein et al. (1976) adopted a bifurcation angle of an equivalent Newtonian fluid.
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