Abstract: The present study deals with a mathematical model describing the resistance to flow across mild stenosis situated symmetrically on steady blood flow through arteries with uniform or non-uniform cross-section. This mathematical model involves the usual assumption that the blood is Non-Newtonian, incompressible and homogeneous fluid.

INTRODUCTION

Arteries throughout the body may be affected by hardening, which causes symptoms because hardened arteries cannot carry enough blood to the body. Narrowing or hardening of the arteries that feed the heart (the coronary arteries) can lead to a heart attack. Due to these serious consequences, attention has been given in studies of blood flow in stenotic region under different conditions.

By assuming the artery to be circularly cylindrical in shape, Mishra and Panda (2005c) discussed characteristics of blood in stenosed artery and the stenosis to be symmetric about the axis of artery. Mishra and Panda (2005a) studied the flow of blood in stenosed artery for a power law fluid.

Large number of researchers Smith et al. (2002), Tu and Deville (1996), Misra et al. (1993), Tu et al. (1992), Misra and Chakravarty (1986), Siouffi et al. (1984), Young and Tsai (1973a, b), Mishra (2003), Mishra and Panda (2005b) and Mishra et al. (2008) have contributed a lot in developing a mathematical model for blood flow in atherosclerosis.

By assuming that blood 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 (Fig. 1). 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.

Fig. 1:	Physical model and coordinate system

MATERIALS AND METHODS

Development of the model: For a Herschel Bulkley Fluid, the relationship between stress and strain is given by:

(1)

(2)

Where:

For the steady flow through circular artery,

(3)

Where, G = (dp/dz) is pressure gradient. From Eq. 1-3

(4)

Now,

(5)

(6)

From Eq. 4-6, we have,

or,

(7)

Now, Total flow flux

Where:

(8)

(9)

(10)

Now,

(11)

where, λ is resistance to flow at the wall for the flow of blood and

(12)

Where:

The resistance parameter λ' = λ/λ₀ is given by the expression:

(13)

We assume one stenosis each in uniform and non-uniform portion of the artery. The surface of stenosis as obtained by Young and Tsai (1973a, b) is:

(14)

To observe explicity the effect of various parameters resistance to the flow, the following function has been assumed for the artery radius for the portion of the artery which is non-uniform.

Where:

Using Eq. 14 in Eq. 13 and integrating, for n = 1/3, we get the expression of resistance parameter as:

Where:

RESULTS AND DISCUSSION

Physiological insight: In order to get the effect of stenosis on the resistance to flow for pulmonary artery, the following values are taken:

Length of the first stenosis, L₁ = 0.05 cm; Length of the second stenosis, L₂ = 0.05 cm; Height of the first stenosis δS₁ = 0.03 cm (Initially); Height of the second stenosis δS₁ = 0.03 cm (Initially); Length of the artery, l = 5 cm; Length of the uniform portion of the artery, l₁ = 4.0 cm; Radius of the uniform portion of the artery, R₁ = 1.5 cm; Position of second stenosis d₂ = 5 cm.

An investigation has been done for the resistance to flow across mild stenosis situated symmetrically on steady blood flow through arteries with uniform or non-uniform cross section by assuming the blood to be Non-Newtonian, incompressible and homogeneous fluid. We have obtained an analytical solution for Herschel-Bulkley fluid.

It can be observed from the Tables (1-3) that in the divergence of artery (K<0), uniform portion of the artery (K = 0) and the convergence of artery (K>0) the resistance parameter λ’ increases as the height of the stenosis in the uniform or non-uniform or both portions of the artery increases.

Table 1:	Variation of against (when n = 1/3) for various value of K

Table 2:	Variation of against (when n = 1/3) for various value of K

Table 3:	Variation of against and (when n = 1/3) for various value of K

CONCLUSION

The study reveals that as the height of the stenosis increases in the uniform or non-uniform or both portions of the artery the resistance to the flow also increases.