Abstract: An attempt has been made to investigate the wall shear stress, resistance parameter and flow rate across mild stenosis situated symmetrically on steady blood flow through blood vessels with uniform or non-uniform cross-section by assuming the blood to be Non-Newtonian, incompressible and homogeneous fluid. An analytic solution for Power law fluid, Bingham Plastic fluid, Casson fluid has been obtained.

INTRODUCTION

Diseases in the blood vessels and in the heart, such as heart attack and stroke are the major causes of mortality world wide. The underlying cause for these events is the formation of lesions, known as atherosclerosis in the large and medium sized arteries in the human circulation. These lesions and plaques can grow and occlude the artery and hence prevent blood supply to the distal bed. Plaques with calcium in them can also rupture and initiate the formation of blood clots (thrombus). The clots can form as emboli and occlude the smaller vessels that can also result in interruption of blood supply to distal bed. Plaques formed in coronary arteries can lead to heart attacks and clots in the cerebral circulation can result in the stroke.

There are a number of risk factors for the presence of atherosclerotic lesions. The common sites for the formation and development of atherosclerosis include the coronary arteries, the branching of the subclavian and common carotids in the aortic arch, the bifurcation of the common carotid to internal and external carotids especially in the carotid sinus region distal to the bifurcation, the renal arterial branching in the descending aorta and in the ileo-femoral bifurcations of the descending aorta. The common feature in the location for the development of the lesion is the presence of curvature, branching and bifurcation present in these sites. The fluid dynamics at these sites can be anticipated to be vastly different from other segments of the arteries that are relatively straight and devoid of any branching segments. Hence, several investigators have attempted to link the fluid dynamically induced stress with the formation of atherosclerotic lesion in the human circulation (Guyton, 2006).

By assuming the artery to be circularly cylindrical in shape, Mishra (2003) discussed characteristics of blood in stenosed artery and the stenosis to be symmetric about the axis of artery. Mishra and Panda (2005) studied the flow of blood in stenosed artery for a power law fluid and Casson fluid. Chaturani and Sany (1985) investigated the various aspects of blood flow in stenosed artery assuming the blood to be Non-Newtonian. Young and Tsai (1973a, b) discussed the various characteristics of steady and unsteady flow of blood in stented arteries.

The initiation and development of atherosclerotic plaques is shown in Fig. 1 and 2. The blood vessels in Fig. 2 that we are talking about are the arteries. They are the structures that carry blood from the heart to all the organs and tissues of the body including brain, kidneys, gut, muscles and the heart itself. Below are a series of illustrations that will help us to understand the process of atherosclerosis (vascular disease) and the kinds of problems that can arise in this condition. In Fig. 2a, we are looking at the vessel sliced across. This vessel is normal like we all come equipped with at birth.

Fig. 1:	Blocked in right coronary artery

Fig. 2:	Development of atherosclerotic plaque

The white space marked lumen is where the blood from the heart would flow to the rest of the body. Figure 2b is showing the early stages of vascular disease with the formation of what is called a plaque generally made up of cholesterol and sometimes calcium. Figure 2c shows a very serious problem.

When we observe the 6 o'clock position (tip of the arrow), we see that the smooth layer or sleeve called the endothelium has ruptured. Up until now that layer has prevented any direct contact between the blood flowing by and the material within the plaque.

Once the material underneath the sleeve is exposed to the passing blood, the blood begins to clot. In this process, a narrowing that might have been 40 or 50% of the cross sectional area of the artery can become an 80, 90 or l00% narrowing within seconds or minutes. This is the process that causes most heart attacks and sudden death (Fig. 2d).

In the present study, blood is assumed to be Non-Newtonian, incompressible and homogeneous fluid; cylindrical polar co-ordinate is used with the axis of symmetry of artery taken as Z-axis. The stenoses are mild and the motion of the fluid is laminar and steady. The inertia term is neglected, as the motion is slow. No body force acts on the fluid and there is no slip at the wall.

MATERIALS AND METHODS

Basic equations: In the present analysis, it is assumed that that the stenosis develops in the arterial wall in an axially symmetric manner and depends upon the axial distance z and the height of its growth (Fig. 3). In such a case the radius of the artery, R(z) can be written as follows:

(1)

We assume one stenosis each in uniform and non-uniform portion of the artery (Fig. 3). To observe explicitly the effect of various parameters on resistance, wall shear stress and viscosity to the flow, the following function has been assumed for the artery radius, which is non-uniform.

(2)

For the steady flow through circular artery, the wall shear stress is given by:

(3)

Where,

(4)

is the pressure gradient. The flow rate Q through the artery is the sum of the flow through the core region and that in the peripheral region, i.e.,

(5)

where the flow rate through the core and peripheral region, respectively is given by:

Fig. 3:	Physical model and coordinate system

(6)

(7)

The resistance to flow at the wall for the flow of blood can also be expressed as:

(8)

Development of the model:
Case 1: power law model: The constitutive relationship for the power fluid is given by the relationship:

(9)

The velocity of the fluid through the tube thus can be expressed in terms of:

(10)

Where C is constant of integration. Applying the boundary condition: u = 0; r = R, we have;

Thus the velocity of the fluid in the tube is given by Eq. 11;

(11)

The flow through the artery can be obtained from the basic equation

(12)

For n = ½, we get the expression of flow through the blood vessel as:

(13)

Also the expression of wall shear stress through the blood vessel is given in Eq. 14:

(14)

For n = 1/3, resistance to flow at the wall for the flow of blood is given by:

The resistance to flow at the wall for the flow of blood in uniform portion of blood vessel is:

Thus the resistance parameter for the flow of blood in the blood vessel is expressed as:

(15)

Case 2: Bingham plastic model: The constitutive relationship for the Bingham Plastic fluid is given by the relationship:

(16)

The velocity of the fluid thus can be expressed in terms of:

(17)

where C is constant of integration. Applying the boundary condition: u = 0; r = R, we have;

(18)

Thus we have the expression of velocity of the fluid given in Eq. 19;

(19)

Where R_c is radius of core region and

(20)

The velocity in the core region is thus given by:

(21)

The expression of flow in the core region can thus be expressed as:

(22)

Also the flow through the peripheral region can be expressed as:

(23)

Substituting the value of Q_core and Q_peripheral in Eq. 5, we get the expression for the flow through the blood vessel as:

(24)

We get the expression of flow through the artery as:

(25)

The expression of wall shear stress can be expressed as given in Eq. 26;

(26)

The resistance to flow at the wall for the flow of blood is given by:

The resistance to flow at the wall for the flow of blood in uniform portion of blood vessel:

Thus the resistance parameter for the flow of blood in the blood vessel is expressed as:

(27)

Case III: casson model: The constitutive relationship for the Casson fluid is given by the relationship:

(28)

The velocity of the fluid thus can be expressed in terms of:

(29)

where C is constant of integration. Applying the boundary condition: u = 0; r = R.

For r>R_c, the expression for the velocity profile is:

(30)

The core velocity is given by:

(31)

The flow through the core region thus can be expressed as:

(32)

The expression of flow through the peripheral region is given in Eq. 33:

(33)

Thus the expression of flow through the blood vessel is expressed as:

(34)

The expression of wall shear stress through the blood vessel is given in Eq. 35:

(35)

Resistance to flow at the wall for the flow of blood is given by:

The resistance to flow at the wall for the flow of blood in uniform portion of blood vessel is thus:

Thus the resistance parameter for the flow of blood in the blood vessel is expressed as:

(36)

Effect of various parameters on the flow of blood in stented blood vessels: In order to get a physiological insight into the effect of stenosis on the wall shear stress, flow rate and resistance parameter against oror both, for different values of wall exponent parameter K, i.e., K>0 (divergence of artery), K = 0 (uniform portion of capillary) and K<0 (convergence of veins), computations are made for Power law model, Bingham Plastic model and Casson model and are shown in the Table 1-11.

Effect of wall shear stress: Effect of wall shear stress is observed according to these three models as shown in Table 1-3.

Table 1:	Variation of t against for K = -0.001, 0.0.001 (Power law model)

Table 2:	Variation of τ against for K = -0.001, 0,0.001 (Bingham plastic model)

Table 3:	Variation of τ against for K = -0.001, 0, 0.001 (Casson model)

Analysis: In all the three models developed (Table 1-3), we observe that as the height of stenosis increases in the blood vessels, wall shear stress also steadily increases for different values of wall exponent parameter, i.e., K>0 (divergence of artery), K = 0 (uniform portion of capillary) and K<0 (convergence of veins). The Mean Arterial blood Pressure (MAP) in arteries is around 100 mm. Hg in the capillaries the MAP is 25 mm Hg and in the veins and venae cavae its mean pressure falls progressively to about 0 mm Hg in the systemic circulation. Similarly in the pulmonary circulation the MAP is 16 mm Hg, whereas, in the pulmonary capillary it is 7 mm Hg and in the pulmonary veins its mean pressure falls progressively to about 0 mm Hg like in systemic circulation. All the above three models depict the physiological conditions like K>0, K = 0, and K<0.

Effect of flow rate: Effect of flow rate on the three models is shown in Table 4-6.

Table 4:	Variation of Q against for K = -0.001, 0, 0.001 (Power law model)

Table 5:	Variation of Q against for K= -0.001, 0, 0.001 (Bingham plastic model)

Table 6:	Variation of Q against for K = -0.001, 0, 0.001 (Casson model)

Analysis: In all the three models developed, we observe that as the height of stenosis increases in the blood vessels, flow rate steadily decreases for different values of wall exponent parameter, i.e., K<0 (convergence of artery), K = 0 (uniform portion of artery) and K>0 (divergence of artery).

Effect of resistance parameter: Effect of resistance and parameter on the three models is analysed in Table 7-11.

In all the three models developed, we observe that as the height of stenosis increases in the blood vessels, resistance parameter steadily increases for different values of wall exponent parameter, i.e., K<0 (convergence of artery), K = 0 (uniform portion of artery) and K>0 (divergence of artery).

Table 7:	Variation of λ] against for K = -0.001, 0, 0.001 (Power law model)

Table 8:	Variation of λ] against for K = -0.001, 0, 0.001 (Bingham plastic model)

Table 9:	Variation of λ] against for K= -0.001, 0, 0.001 (Bingham plastc model)

Table 10:	Variation of λ] against and for K = -0.001, 0, 0.001 (Bingham plastic model)

Table 11:	Variation of λ] against for K= -0.001, 0, 0.001 (Casson model)

RESULTS AND DISCUSSION

Wall shear stress is an important factor in the study of blood flow. Accurate predictions of the distribution of the wall shear stress are particularly useful for the understanding of the effect of blood flow on endothelial cells.

However, the flow rate in the arteries is affected much compared to veins, as arteries are resistance vessels, whereas veins are capacitance vessels.

In hypertensive patients, the sustained increased pressure in arteries will lead to remodeling of the blood vessels and heart, especially in the resistance vessels where the pressure is very high.

Arteries tend to become less elastic and stiff. In the models discussed, the trends observed show that as the stenosis increases there is an increase in the MAP in the resistance vessels which may lead to remodeling of the arteries.

The remodeling is not prominent in capillaries and veins, where the resistance to flow is least compared to arteries.

In the models developed, we observe that Casson model and Power law fluid model well suits for the physiological data (Table 12).

Table 12:	Comparative analysis of various parameters on Non-Newtonian models for the flow of blood in blood vessels

Nomenclature:

ρ	=	Density of Blood
μ	=	Viscosity of Blood
P	=	Pressure
R1	=	Radius of uniform portion of tube
R (z)	=	Radius of obstructed portion of tube
Rsn(z)	=	Radius of obstructed portion due to the nth stenosis of tube
δSn	=	Amplitude of nth stenosis
Ln	=	Length of nth stenosis
dn	=	Location of nth stenosis
l1	=	Length of uniform portion of tube
l	=	Length of tube
K	=	Wall exponent parameter
τ	=	Wall shear stress
τ0	=	Measure of yield stress
e	=	strain rate = -
u	=	Velocity of fluid
Rc	=	Radius of the core region of the tube
λ	=	Resistance to flow at the wall for the flow of blood
λ0	=	Resistance to flow at the wall for the flow of blood in uniform portion of tube
	=	Resistance parameter

We assume the following non-dimensional quantities:

CONCLUSION

For the physiological insight of the problem various parameters of systemic and pulmonary artery are taken and the study reveals that as the height of the stenosis increases in blood vessels, the shear stress and resistance parameter steadily increases, whereas, flow rate decreases steadily.