Computational and Mathematical Methods in Medicine

Volume 2014, Article ID 479152, 10 pages

http://dx.doi.org/10.1155/2014/479152

## Mathematical Analysis of Non-Newtonian Blood Flow in Stenosis Narrow Arteries

Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand

Received 20 June 2014; Revised 18 November 2014; Accepted 18 November 2014; Published 17 December 2014

Academic Editor: Chung-Min Liao

Copyright © 2014 Somchai Sriyab. 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 flow of blood in narrow arteries with bell-shaped mild stenosis is investigated that treats blood as non-Newtonian fluid by using the K-L model. When skin friction and resistance of blood flow are normalized with respect to non-Newtonian blood in normal artery, the results present the effect of stenosis length. When skin friction and resistance of blood flow are normalized with respect to Newtonian blood in stenosis artery, the results present the effect of non-Newtonian blood. The effect of stenosis length and effect of non-Newtonian fluid on skin friction are consistent with the Casson model in which the skin friction increases with the increase of ither stenosis length or the yield stress but the skin friction decreases with the increase of plasma viscosity coefficient. The effect of stenosis length and effect of non-Newtonian fluid on resistance of blood flow are contradictory. The resistance of blood flow (when normalized by non-Newtonian blood in normal artery) increases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length. The resistance of blood flow (when normalized by Newtonian blood in stenosis artery) decreases when either the plasma viscosity coefficient or the yield stress increases, but it decreases with the increase of stenosis length.

#### 1. Introduction

Stenosis arteries are a narrowing or constriction of inner surface (lumen) of arteries. It is a main cause of well-known serious diseases such as atherosclerosis and cardiovascular disease to name a few (see [1, 2]). Therefore, the study of blood flow in a stenosis artery is useful for the understanding of circulatory disorders. Blood behaves as Newtonian fluid when blood flows through larger diameter arteries at high shear rate, but it exhibits a non-Newtonian fluid when it flows through small diameters arteries at low shear rate [3–5].

The Casson fluid model is a non-Newtonian fluid and widely used for blood flow in narrow arteries, for example, in [6–10]. Kuang and Luo have proposed the K-L model as an improvement of Casson model. This model is more effective in the describing of a non-Newtonian blood flow because it contains two parameters such as the yield stress and the plasma viscosity but Casson model is considered only yield stress. K-L model is more effective in describing the shear thinning behavior of blood within a wide shear rate [11]. Asharafizaadeh and Bakhshaei used the K-L model with the Lattice Boltzmann simulation [12]. Zhang and Kuang indicated that the K-L model is in good agreement with hemorheological characteristics of human [13]. Another model with two parameters, the yield stress and power law index, namely, H-B model, has also been proposed by Herschel-Bulkley; see [14, 15].

In our work, a mathematical model is developed to analyze the blood flow at low shear rate in narrow arteries with mild bell-shaped stenosis. We treated the blood as non-Newtonian by using the K-L model and discussed the effect of various parameters on the physiologically important flow quantities such as flow rate, skin friction, and resistance of blood flow.

#### 2. Mathematical Formulation

We consider an axially symmetric, laminar flow and non-Newtonian incompressible viscous blood in the -axial direction through a circular artery. The bell-shaped mild stenosis in arrow artery is studied and the artery wall is assumed to be rigid. Many researchers studied the non-Newtonian blood flow by the Casson model [6–10]. In our work, we consider the K-L model because it is more effective than the Casson model. The yield stress and the plasma viscosity are considered in the K-L model but only the yield stress is concerned in the Casson model [11]. The artery is assumed to be long enough so the entrance and the end effects can be neglected. A cylindrical polar coordinate is used to analyze the behavior of blood flow, where and are the radial and axial directions, respectively, and is the azimuthal angle.

Since the blood flow in narrow arteries is slow, the magnitude of the inertial forces is negligible, and the inertial terms in the momentum equations are neglected. The radial component of momentum equation is ignored because the considered flow is unidirectional. Therefore, the axial component of momentum equation is simplified to the following: where is the pressure and is the shear stress. The K-L model that is a relationship between shear and strain rate is defined as follows: where is the velocity of blood in the axial direction, is the yield stress, is plasma viscosity, and is a parameter constant in K-L model. In this work, the geometry of segment of the narrow artery with mild bell-shaped stenosis is shown in Figure 1 and defined as follows: where is the radius of artery in the stenosis region and is radius of normal artery. Note that is a nondimensional parameter of stenosis height, defined as , where is depth of stenosis. Parameter is a nondimensional parameter that is the length of the stenosis in the segment of narrow artery, defined as , where is the stenosis shape. When parameter is variable and is constant, the marginally increase along the -axis with decrease of (Figure 2). On the other hand, keep as constant and as variable (for different values of and fixed value of ); it is noticed that the width of the stenosis increases with increase in value of (Figure 3).