1. INTRODUCTION:

The fluid dynamical behaviour of blood through an arterial segment having stenosis play a vital role in cardiovascular disease. The narrowing in the artery, commonly referred to as stenosis, is a dangerous disease and is caused due to the abnormal growth in the lumen of the arterial wall. Stenosis may be formed at one or more locations of the result cardiovascular system. As a result of such undesirable formation at the endothelium of the vessel wall reduction of regular blood flow is likely to take place near the stenosis. If this disease takes a severe form, it may lead to stroke, heart attack and various cardiovascular disease. Many cardiovascular diseases such as due to the leading cause of death worldwide. The partial occlusion of the arteries due to stenotic obstruction not only restrict the regular blood flow but also characterizes the hardening and thickening of the arterial wall.

However, the main cause of the formation of stenosis is still unknown but it is well established that the fluid dynamical factors play an important role as to further development of stenosis.

The blood flow through an artery depends upon the pumping action of the heart gives rise to a pressure gradiant which produces an oscillatory flow in the blood vessel (Haldar, 1987). In fact blood is a suspension of cells in plasma. The plasma which is a solution of proteins, electrolyte and other substances, is an incompressible virtually Newtonian fluid. From biomechanical point of view, blood is considered as an intelligent fluid, probably the most one in the nature, capable of adapting itself in a great extent in order to provide nutrients to the organs. Human body experiences magnetic fields of moderate to high intensity in many situations of day to day life. In recent times, many medical diagnostic devices especially those used in diagnosing cardiovascular disease make use of magnetic fields. It is known from the magneto-hydrodynamics that when a stationary, transverse magnetic field is applied externally to a moving electrically conducting fluid, electrical currents are induced in the fluid. Since blood is electrically conducting fluid, its flow characteristics is influenced by the application of magnetic field. If a magnetic field is applied to a moving and electrically conducting fluid. it will induce electric as well as magnetic fields. The interaction of these fields produces a body force per unit volume known as Lorentz force, which has significant impact on the flow characteristics of blood. Such an analysis may be useful for the reduction of blood flow during surgery and Magnetic Resonance Imaging (MRI). Hematocrit is the most important determinant of whole blood viscosity. Therefore, blood viscosity and vascular resistance affect total peripheral resistance to blood flow, which is abnormally high in the primary stage of hypertension. Again hematocrit is a blood test that measures the percentage of red blood cells present in the whole blood of the body. The percentage of red blood cells blood cells may as in adult human body is approximately 40-45% [18]. Red blood cells may affect the viscosity of whole blood and thus the velocity distribution depends on the hematocrit. So blood cannot be considered as homogeneous fluid [2]. Due to the high shear rate near the arterial wall, the viscosity of blood is low and the concentration of red blood cells is high in the central core region. Therefore, blood may be treated as Newtonian fluid with variable viscosity particularly in the case of large blood vessels. The effect of magnetic field on blood flow has been analyzed theoretically and experimentally by many investigators[6-9)] under different situations. Shit and his co-investigators [10-14] explored variety of flow behavior of blood in arteries by treating Newtonian, non-Newtonian model in the presence of a uniform magnetic field. It is well known that blood behaves differently when flowing in large vessels, in which Newtonian behaviour is expected and in medium and small vessels where non-Newtonian effects appear [1]. Many researchers studied the pulsatile flow of blood in stenosed artery [4]. Misra and Shit studied the effect of magnetic field on blood flow through an artery in unsteady situation and observed the effect of magnetic parameter[15], unsteady parameter and the radius phase angle on the flow charecteristics. Sanyal et.al studied the effect of magnetic field on pulsatile blood flow through an inclined circular tube with periodic body accelerations and discussed the effect of magnetic field[16], gravitational parameter, inclined angle, body acceleration, time etc. on axial blood flow, flow rate and acceleration. Bhuyan and Hazarika studied the magnetic effect on flow through circular tube of non-uniform cross-section with permeable walls[17].

In the present investigation, we consider the unsteady oscillatory flow of blood through a stenosed artery under the effect transverse magnetic field. The study pertains to a situation in which the variable viscosity of blood depending upon hematacrit is taken into consideration. It is assumed that the arterial segment is cylindrical tube with time dependent exponential pressure gradiant and governing equations are solved by using Frobenius method. The effect of Hartmann number, magnetic field and maximum Hematocrit at the center of the arterial segment on velocity profile, volumetric flow rate and wall shear stress are computed graphically.

2. FORMULATION OF THE PROBLEM:

2.1. Flow Geometry:

The Idealized geometry of stenosis is given by fig. 1

1 ; otherwise (1)

Where R(z) is the radius of the artery in the stenotic region, R₀is the radius of the normal artery, the length of stenosis, d length of non-stenosis and δ the maximum height of stenosis such that (fig 1)

Fig. 1 Geometry of a Stenosed artery

2.2. Flow analysis and coordinate system :

Let us consider the oscillatory flow of blood through an artery with mild constriction. The flow is assumed to be laminar, Newtonian, viscous, incompressible, unsteady and axially symmetric by assuming variable viscosity and density is constant.

We assumed that blood s compressible, suspension of erythrocytes in plasma and has uniform dense throughout but the viscosity µ(r) varies in the radial direction. According to Einstein’s formula for the variable viscosity of blood taken to be

(2)

where µ₀ is the coefficient of viscosity of plasma, β is a constant and h(r) stands for the hematocrit. The analysis will be carried out by using the flowing empirical formula for hematocrit.

(3)

In which R₀ represents the radius of a normal arterial segment, H is the maximum hematocrit at the center of the artery and (m≥2) a parameter that determines the exact shape of the velocity profile for blood. The shape of the hematocrit profile given by equation (3) is valid only for very dilute suspensions of erythrocytes, which are considered to be of spherical shape.

According to our considerations, the equation that governs the flow of blood under the action of an external magnetic field may be put as the governing equation of motion in axial direction is

(4)

The boundary conditions are

No slip condition : v=0 at r= R(z)

Symmetry condition : at r=0 (5)

3. METHOD OF SOLUTION

It is convenient to write these equations in dimensionless form by means of the following transformation variables.

(6)

Where and B₀ are the average velocity, radius in the unobstructed tube, pressure, density, viscosity of blood and the applied magnetic field in respectively.

Then the equation (5) reduces to the form, by dropping primes, we get

(7)

Where , and v=v(r,t) is the velocity in the axial direction.

The boundary corresponding conditions becomes

No slip condition : v=0 at r=R(z)

Symmetry condition : at r=0 (8)

Let the solutions for v and p be set in the forms

(9)

Sub. equation (9) in equation (7), we get

(10)

with the use of the transform defined in the equations (2) and (3), the governing equation (7) reduces to

(11)

With =1+, =βH.

The equation (11) can be solved subjected to the boundary conditions (8) using Frobenius method. For this, of course, v has to be bounded at r=0, then only admissible series solution of the equation (11) will exists and can put in the form

(12)

where C, and are arbitrary constants.

To find the arbitrary constant C, we use the no-slip boundary condition (8) and obtained as

(13)

Substituting the value of v from equation (12) into equation (11) on simplification, we get

(14)

Equating the coefficient of C and in equation (14) we have

(15)

And

(16)

Hence the constants are obtained by equating the coefficients of and from both side of equations (15) and (16) respectively, we get

(17)

(18)

With (19)

Substituting the expression of C in the equation (12), we have the velocity profile the arterial segment in the radial direction is

(20)

Where . The average velocity has the form

(21)

Where is the pressure gradient of the flow field in the normal artery in the absence of magnetic field. The non-dimensional expression for v is given by

(22)

The volumetric flow rate across the arterial segment is given by

(23)

Substituting v from Eq. (20) into Eq. (23) and then integrating with respect to r, we obtain

(24)

If be the volumetric flow rate in the normal portion of the artery, in the absence of magnetic field and porosity effect then

(25)

Therefore, the non-dimensional volumetric flow rate has the following form

(26)

The wall shear stress on the endothelial surface is given by

(27)

Substituting v from Eq. (20) into Eq. (27), we obtain

(28)

If be the shear stress at the normal portion of the arterial wall, in the absence of magnetic field, the non-dimensional form of the wall shear stress is given by

(29)

4. RESULTS AND DISCUSSION

The theorectical study of oscillatory flow of blood in stenosed artery in the presence of Magnetic field with variable viscosity have been discussed. The study pertains to a situation in which the variable viscosity of blood depending upon hematocrit is taken into consideration. It is assumed that the arterial segment is in cylindrical tube with time dependent exponential pressure gradiant. The variable viscosity of blood depending on hematocrit is taken into account in order to improve resemblance to the real situation. It is assumed that the surface roughness is cosine shaped and the maximum height of roughness is very small compared with radius of the unconstructed tube. The analytical expression for velocity component, Volumetric flow rate and wall shear stress are obtained. The effect of magnetic field, Hartmann number and maximum Hematocrit at the center of the arterial segment on velocity, flow rate and stress are computed graphically.

In the previous section we have obtained analytical expressions for different flow characteristics of blood through the action of an external magnetic field. In this section we are to discuss the effect of various parameter on the flow characteristics graphically with the use of following numerical data which is applicable to blood.

d=0.25, H=0.2, m=2, M=2.5, β=2.5, l=0.5, Re =10.

Fig. 2 :Velocity distribution at z=2.5 with r for different values Hartmann number M, when hemotocirt H=0.2, frequency parameter β=2.5, Reynolds number Re=10.

Fig. 3 : Velocity distribution at z=2.5 with r for different values hematocrit H, when Hartmann number M=2.5, frequency parameter β=2.5, Reynolds number Re=10.

Fig. 4 : Variation of the rate of flow with for different Hematocriy H, Hartmann number M=2.5, frequency parameter β=2.5.

Fig. 5: Variation of the rate of flow with for different Hartmann number M, Hematocrit H=0.2, frequency parameter β=2.5.

Fig. 6 : Variation of the wall shear stress with time for different Hartmann number M, Hematocrit H=0.2, frequency parameter β=2.5.

Fig. 7 : Variation of the wall shear stress of flow with time for different Hematocrit H, Hartmann number M=2.5, frequency parameter β=2.5.

Figure 2 shows that the velocity profile gradually decreases at the centerline of the artery increase of Hartmann number. Figure 3 gives the distribution of axial velocity for different values of the hematocrit H, that the velocity decreases at the stenosed artery with the increase of hematocrit level H. This fact lies within the hematocrit as the blood viscosity is high in the stenosed artery due to the aggregation of blood cells rather than low viscosity in the plasma near the arterial wall. Figure 4 and 5 shows that the volume flow rate increases with the increase of Magnetic field strength B₀ and hematocrit H. Figure 6 and 7 gives the distribution of the wall shear stress for different values of the hematocrit H and Hartmann number M. We observe from figure 6 and 7 shows that wall shear stress increases as the hematocrit H and Hartmann number increases. One can note from this figures that the wall shear stress is low at the throat of the secondary stenosis as well as at downstream of the artery.

5. CONCLUSION

A theoretical study of blood flow through a stenosed artery in the presence magnetic field has been carried out. In this study the variable viscosity of blood depending on hematocrit. The problem is solved analytically by using the Frobenius method. The main findings of the present study may be listed as follows :

Ø The flow velocity at the central region decreases gradually with the increase of magnetic field strength.

Ø The hematocrit and the pressure has a linear relationship as reported.

Ø The lower range of hematocrit may leads to the further deposition of cholesterol at the endothelium of the vascular wall.

Ø Hematocrit contributes to the regulation of blood pressure.

Finally we can conclude that further potential improvement of the model are anticipated. Since the hematocrit positively affects blood pressure, further study should examine the other factors such as diet, tobacco, smoking, overweight etc. from a cardiovascular point of view. Moreover on the basis of the present results, it can be concluded that the flow of blood and pressure can be controlled by the application of an external magnetic field. All the flow characteristics are found to be affected by the influence of applied magnetic field with profile velocity stenosis.

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