Journal of Fluids

Volume 2015, Article ID 546716, 22 pages

http://dx.doi.org/10.1155/2015/546716

## Rheological Behavior of Physiological Pulsatile Flow through a Model Arterial Stenosis with Moving Wall

Department of Mathematics & Physics, North South University, Dhaka 1229, Bangladesh

Received 30 December 2014; Revised 9 May 2015; Accepted 10 May 2015

Academic Editor: Chang Yi Kong

Copyright © 2015 Sumaia Parveen Shupti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents a numerical investigation of non-Newtonian modeling effects on unsteady periodic flows in a two-dimensional (2D) constricted channel with moving wall using finite volume method. The governing Navier-Stokes equations have been modified using the Cartesian curvilinear coordinates to handle complex geometries, such as, arterial stenosis. The physiological pulsatile flow has been used at the inlet position as an inlet velocity. The flow is characterized by the Reynolds numbers 300, 500, and 750 that are appropriate for large arteries. The investigations have been carried out to characterize four different non-Newtonian constitutive equations of blood, namely, the (i) Carreau, (ii) Cross, (iii) Modified-Casson, and (iv) Quemada. In these four models, blood viscosity is a nonlinear function of shear rates. The Newtonian model has been investigated to study the physics of fluid and the results are compared with the non-Newtonian viscosity models. The numerical results are presented in terms of streamwise velocity, wall shear stress, pressure distribution as well as the vorticity, streamlines, and vector plots indicating recirculation zones at the poststenotic region. Comparison has also been illustrated in terms of wall pressure and wall shear stress for the Cross model considering different amplitudes of wall oscillation.

#### 1. Introduction

Atherosclerosis is known as a major arterial disease. In atherosclerosis, localized deposits and accumulation of cholesterol and lipid compounds as well as proliferation of connective tissues originate a partial decline in the arterial cross-sectional area which, in particular, is called stenosis. Atherosclerotic lesions mainly occur in arterial segments with high curvature or bifurcations and junctions initiating notable alterations in flow structure and fluid loading on vessel walls [1]. Diabetes, smoking, inflammation, ischemia, and so forth are the main risk factors of stenosis development. Although the precise mechanisms responsible for the initiation of this phenomenon are not apparently known, it has been established that once a mild stenosis is developed, the resulting flow disorder plays a significant function in the further development of the disease that eventually changes the regional blood rheology as well [2, 3].

Non-Newtonian viscosity can be employed to characterize the rheological behavior of blood. The rheology and the fluid dynamical properties of blood flow can play a significant role in the fundamental understanding, diagnosis, and treatment of many cardiovascular and arterial diseases [4]. Therefore, many researchers have paid their attention to studying the hemodynamics for various viscous conditions. For example, Tu and Deville [5] implemented Galerkin finite-element method simulations for physiological pulsatile flow through a severe stenosis. They treated blood as non-Newtonian fluid employing a Herschel-Bulkley model which roughly behaves like blood. They illustrated results for steady and pulsatile flow conditions in terms of velocity profile, formation of vortex in separate regions, pressure drop across the stenosis, wall shear stress, and the vorticity contours. Non-Newtonian behavior of the blood is a crucial factor affecting the primary and secondary flow patterns near the junction between the bypass graft and the stenosed artery. Chen et al. [6] explored the non-Newtonian fluid flow in a stenosed coronary bypass numerically employing the Carreau-Yasuda model where they revealed significant differences in axial velocity profiles, secondary flow streamlines, and wall shear stress (WSS) between the non-Newtonian and Newtonian fluid flows. Moreover, the effects of both non-Newtonian behavior and the pulsation of blood flow on the distributions of luminal surface low-density lipoprotein (LDL) concentration and oxygen flux along the wall of the human aorta were numerically analyzed by Liu et al. [7] whereas Razavi et al. [8] performed a numerical analysis to study the viscous effects of blood. The power-law model demonstrated higher deviations in terms of velocity and wall shear stress compared to Newtonian and six other non-Newtonian viscosity models in their investigation. They also found that increasing stenosis intensity causes more disturbed flow patterns in the downstream of the stenosis and WSS develops remarkably at the stenosis throat. Sriram et al. [9] reveal the importance of incorporating non-Newtonian blood properties into estimates of WSS in microvessels.

Taking the physiological inlet condition of the blood flow into consideration, Long et al. [10] numerically simulated pulsatile blood flow in straight tube stenosis models with area reduction of 25%, 50%, and 75%. A measured human common carotid artery blood flow waveform was used as the upstream flow condition which has a mean Reynolds number of 300. The results indicated WSS oscillations (between positive and negative values) at various downstream locations in some models. On the other hand, comparison between numerical solution for simple pulsatile and physiologically pulsatile flow through a 61% stenosed artery was worked out by Zendehbudi and Moayeri [11]. They considered laminar flow, Newtonian and axisymmetric rigid tube for flow field computation.

Effects of compliance on diagnostic parameters have also been studied by Moayeri and Zendehbudi [2] where they numerically investigated pulsatile blood flow through stenotic arteries considering physiological flow in a dog femoral artery assuming Newtonian fluid. An isotropic elastic and incompressible material was assumed for the wall at each axial section but a nonuniform distribution of the shear modulus in axial direction to model the high stiffness of the wall at the stenotic location. Their results indicated that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Furthermore, flow field and stress field for different degrees of stenoses under physiological conditions were investigated by Li et al. [12] which suggests that severe stenoses cause considerably large pressure drop across the throat that inhibits wall motion resulting in higher blood velocities and higher peak wall shear stress and localization of hoop stress.

A different kind of enquiry was carried out to study the influence of arterial wall-stenosis compliance on the coronary diagnostic parameters by Konala et al. [13]. Three parameters were defined as fractional flow reserve (FFR), pressure drop coefficient (CDP), and lesion flow coefficient (LFC) to be assessed for varying degrees of epicardial stenoses. The paper concluded that the differences in diagnostic parameters with compliance at intermediate stenosis (78.7–82.7% area blockage) could lead to misinterpretation of the stenosis severity. Again, the critical rates of blood flow acceleration and deceleration at sites of artificially induced stenosis (vessel side-wall compression or ligation) are a function of tissue elasticity; this was found by Tovar-Lopez et al. [14] when they investigated the relationship between the local hydrodynamic strain-rates and the severity of arteriolar stenosis in the small bowel mesenteric vessels of mice. Vahidi and Fatouraee [15] presented a computational model using fluid structure interactions (FSI) to investigate the physical motion of a blood clot inside the human common carotid artery. They simulated transportation of a buoyant embolus in an unsteady flow within a finite length tube having stenosis. The maximum magnitude of arterial wall shear stress during embolism occurred at a short distance proximal to the throat of the stenosis.

Paul et al. [16, 17] and Molla et al. [18] investigated the simple sinusoidal pulsatile flow in a planer channel with a cosine shape single stenosis for maximum using the LES technique. They [19–21] also have studied the physiological pulsatile flows in a channel with a single stenosis for the Newtonian and non-Newtonian fluids using the large-eddy simulation technique. Again, rheological properties of blood have been studied for transition-to-turbulent condition using LES technique in [22].

From the physiological point of view, wall pressure and wall shear stress caused by physiological pulsatile flow in a stenosed artery play significant roles in hemodynamics. Fry [23] revealed that high wall shear stress initiated by atherosclerosis is a strong parameter for endothelial or inner side damage in an artery. It can also overstimulate platelet thrombosis causing blockage [24]. Therefore, it is important to study the hemodynamic factors to realize the basic scenario behind the physiology of arterial diseases. Moreover, from the above discussion it is apparent that many researchers performed various simulations to study the fluid flow pattern both physically and physiologically. In the present study, we also simulated the pulsatile flow through an axisymmetric artery considering Newtonian, Carreau [25], Cross [26], Modified Casson [27], and Quemada [28] models as molecular shear thinning viscosity models and studied different interesting flow phenomena varying the fluid viscosity, wall condition, flow velocity, and geometry. The objective of this study is to analyze the modeling consequences of various non-Newtonian fluids in terms of streamlines, wall pressure, and wall shear stress along with the Newtonian one. Efforts have been made to obtain a very good flow insight in a stenotic artery considering physiological inlet and moving arterial wall with a very small degree of oscillation.

#### 2. Hypothesis

The wall motion is generated using (A.8) that is given in the appendix. In this study, the wall moves sinusoidally along the height of the channel. Apparently, the wall oscillation is prescribed by the authors. As a result, it influences the flow pattern but the wall motion is not induced by the fluid flow. This is how our study differs from fluid structure interaction where both fluid and structure influence each other.

#### 3. Governing Equations

Physiological pulsatile flow is simulated for Reynolds number 300. The arterial wall is assumed moving and blood is modeled as both Newtonian and non-Newtonian fluids for the flow field computation. The governing momentum equations for non-Newtonian 2D flows are

The blood viscosity, , depends on the shear rate , and its magnitude is defined as . When blood is treated as a Newtonian fluid, its viscosity tends to become constant value which is denoted by Pa·s. Moreover, constitutive relations used for the apparent viscosity of the blood are presented in Section 6 for non-Newtonian models. The above governing equations have been modified using the general Cartesian curvilinear coordinate system which is described in the following section.

#### 4. Computational Geometry

The geometry is a two-dimensional (2D) channel with a cosine-shaped blockage or stenosis. Owing to the presence of the stenosis, the cross-sectional area of the channel, , is a variable in the streamwise direction [i.e., ]. Away from the stenosis, the height of the channel is constant and is represented here using (i.e., in the region either upstream or downstream of the stenosis). The stenosis is centered at downstream of the channel inlet (i.e., the inlet location is ) and upstream from the channel outlet. The stenosis is centered at and length of the stenosis is . The mathematical form of the stenosis chosen for this study iswhere is a parameter that controls the height of the stenosis. In the present study, is fixed to 1/2 that results in a 50% reduction of the cross-sectional area at the center of the stenosis. However, a schematic view of the model has been depicted in Figure 1(a) along with a subsequent illustration of the grid in Figure 1(b).