^{1}

^{1}

^{2}

The stenosis in the artery, which reduces the flow passage to blood, is a common cardiovascular disease that is responsible even for cardiac arrest sometimes. The hemodynaics reveals that the severe blockage in an artery due to stenosis generates pressure tangential stress that impacts adversely on the arterial wall downstream to stenosis and weakens the arterial wall. The site of weakened wall in the artery generates post stenotic dilatation. The objective of this paper is to study flow of blood, of non-Newtonian in nature described by Herschel-Bulkley model, in a diseased artery suffering with partly overlapped two stenoses and a dilatation distal to the stenoses. A mathematical model, describing the blood flow, has been derived using Navier-Stokes equations along with the prescribed geometry of the diseased artery. The expressions of velocity profile, resistive impedance to flow and wall shear stress (skin-friction) are derived. The effect of inclination of the vessel on the resistive impedance to flow is discussed along with the effect of rheological and geometrical parameters on the resistive impedance to flow and skin friction.

In general, the non-Newtonian effects in blood are depending on the magnitude of deformation rates; consequently non-Newtonian effects may exist or be enhanced at low shear rates flow regimes. The types of deformation, shear or elongation also influence the non-Newtonian effects of blood [

It is a common clinical finding that the dilatation in the arterial wall arises distal to the constriction [

In the present study blood is considered as non-Newtonian in nature described by the Herschel-Bulkley model that flowing in a rigid walled unhealthy artery suffering with two stenoses which are overlapped alongside and a dilatation distal to the stenotic region. The axis of the circular tube is taken along the z-axis and the stenoses and dilatation are assumed to be symmetrical about the z-axis. It is assumed that the tube is inclined at an angle α to the horizontal as shown in

where, 6L is the length of stenosed segment and 2L is the length of dilatation segment. The δ_{1} is maximum thickness of the stenosis and δ_{2} is maximum dilatation in the arterial wall. In the axisymmetric coordinates system

The function h(z) is plotted on MatLab to draw the

The equation of continuity for incompressible fluid is given by

Since, flow is axisymmetric, i.e.,

The governing equation of motion for steady developed flow of blood in the inclined vessel is given by

where, g is acceleration due to gravity and

The Herschel-Bulkley fluid is defined by the constitutive equation

where,

The boundary conditions are given by

Using the following non-dimensional parameters:

mensional form of these equations are given by (8) to (14), respectively

The equation of motion is

where,

Corresponding boundary conditions are reduced to

Equation (10) can be written as

where,

The solution of (15) under the boundary condition (13) gives

Using (16) in the constitutive Equation (11) for

Integrating Equation (17) w.r.t. r and applying boundary condition (14), the

The skin friction

Substituting the value of

when

From the Equation (22),

Using Equation (19) in Equation (23)

The pressure drop

In the absence of any constriction, the pressure drop

The resistance to flow

If there is no diseased portion then resistance to flow is given by

Therefore, the impedance-resistance to flow ratio may be expressed as

From Equation (27) and (28),

The values of

The effects of various parameters pertinent to the model on velocity profile, skin-fric- tion and impedance are computed and plotted to analyze them.

having peak at the location

as compared to that at

nificantly increases with the reduction of the flow passage. It is plausible that the flow velocity reduces from the velocity in the normal region of the vessel. In the dilatation region, the flow velocity reduces significantly in magnitude as compared to the other locations in the region of overlapped stenosis or normal region of vessel.

The flow velocity retarded with the increase in the maximum dilatation in the vessel wall. The results in the region of stenosis and effect of maximum thickness of stenosis on the flow profile pattern are in good agreement with results already published. The effect of non-Newtonian behaviour of blood as compare to blood of Newtonian nature is plausible as the profiles are not of parabolic shape.

In the healthy human blood, the value of yield stress ranges in between 0.01 dyne/s^{2} and 0.06 dyne/s^{2}, while in unhealthy human, suffering with myocardial infarction the yield stress may increases up to five times [

The measurement of yield stresses may used in the diagnosis of pathological conditions such as diabetes [^{2}. In the present study the values of yield stress are taken 0.05, 0.025 and 0.50. The last value correlates the diseased situation.

_{0} = 0.5, the plug flow radius increases irrespective of the locations whether it is falls within region of the stenosis or in the region of dilatation.

_{0} = 0.5 arterial wall bears more stresses which may responsible for the wall rupturing.

The typical values of the power index n for blood flow is taken to lie between 0.9 and 1.1 are used in [

in power law index overcomes the stress on wall of the vessel consequently on the endothelial cells.

Corresponding to a fixed thickness of stenosis, the increase in maximum expansion in dilatation of the vessel, the impedance to the flow is reduces as observed in

_{0} = 0.5, the resistance to flow ratio increases hence the reduction in the transportation of blood, results in the less oxygen and nutrients supply to the tissues. It is also evident from the

The impedance to the flow reduces with the increase of power law index as shown in

reduces with the increase of n since the pressure drop increases with the increasing value of n. Also, the impedance enhances on the advancement of thickness of stenotic region while it reduces with the increase in maximum expansion of the dilatation as observed in

The inclination α of the vessel affects significantly on the impedance to the flow,

value of α. At α = 0 the vessel is horizontal then the effect of gravitational field is uni-

form throughout. The case when α = π/6 and α = π/3 the body force term

play role to retard the flow.

§ In the dilatation region, the flow velocity reduces significantly in magnitude as com- pared to overlapped stenotic region or normal region of the vessel.

§ The effect of non-Newtonian behaviour on the flow profiles is plausible. The profiles are not of parabolic shape.

§ The increase in yield stress is due to unhealthy condition. The plug flow radius increases irrespective of the locations whether it falls within stenotic region or in the region of dilatation. On increasing power law index, the fluid gets impetus even in the plug flow region.

§ Near the extremities of the overlapped stenosis, the steepest enhancement in the skin-friction is observed.

§ With the increase in power law index, profound reduction in skin-friction is obtained. This suggests the importance of power law index of the blood during the theoretical studies.

§ In case of unhealthy situation, the yield stress increases from its normal value. Hence with the increase in yield stress, the resistance to flow ratio increases that results in reduction in the transportation of blood, oxygen and nutrients supply to the tissues.

§ The inclination α of the vessel affects significantly on the impedance to the flow.

§ The impedance to the flow reduces with the increase of power law index.

Sharma, M.K., Nasha, V. and Sharma, P.R. (2016) A Study for Analyzing the Effect of Overlapping Stenosis and Dilatation on Non-Newtonian Blood Flow in an Inclined Artery. J. Biomedical Science and Engineering, 9, 576- 596. http://dx.doi.org/10.4236/jbise.2016.912050

g Acceleration due to gravity.

R Radius of unstenosed artery.

h(z) Radius of stenosed artery defined in the Equation (1).

6L Length of stenosed portion shown in the

2L Length of dilatation portion shown in the

n, k Non-Newtonian parameters taken in the Equation (11).

Q Volumetric flow rate.

r Radial coordinate.

w Dimensionless axial velocity.

z Axial coordinate.

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