1. Introduction

The control of the blood pressure has been possible by using porous effect in cases of cholesterol and related diseases. For describing the mechanics of red blood cell motion in narrow capillaries, we distinguish two situations according to the convenience with which the cells fit into the vessels. Womersley ^{[1, 2]} considered the oscillatory flow in a cylindrical tube with uniform cross-section. Lee and Fung ^[3] studied the flow of blood through an artery with an axisymmetric stenosis taking blood as a Newtonian fluid. Bitoun Bellet ^[6] studied pulsatile flow of blood with reference to stenosis in microcirculation. Radhakrishnamacharya et al. ^[8] and Prasad et al. ^[9] studied the pulsatile flow of blood in circular tubes of varying cross-section with Suction/injection. But the non- Newtonian property is not taken into consideration in these studies. Nakayama and Sawada ^[13] studied the flow of a non- Newtonian fluid through an axisymmetric stenosis numerically. The pulsatile flow of a non- Newtonian biviscous fluid through a tube with varying cross-section and non- permeable walls in presence of external magnetic field has been analysed by Elnaby et al. ^[14]. Sanyal et al. ^[15] investigated the pulsatile flow of biviscous fluid through a tube of varying cross-section with suction/injection. But they considered no effect of slip velocity at the wall of the tube and so the effect of slip velocity has been neglected. Our main object in the present work is to study the pulsatile motion of blood in a circular tube of permeable wall and varying cross-section in presence of slip velocity at the tube wall. In this analysis, we assume that blood is a non- Newtonian biviscous fluid and the blood vessel is a straight, rigid circular tube of varying cross-section. Kumar at el. ^[18] founded computational technique for flow in blood vessels with porous effects and their using Galerkin finite element method. Gupta ^[19] investigated a performance and analysis of blood flow through carotid artery and their using finite element method. Gupta ^[20] made performance modeling and mechanical behaviour of blood vessel in the presence of magnetic effects and they are using finite difference method.

2. Mathematical Model

In the present communication of pulsatile motion for an incompressible non- Newtonian biviscous fluid in an axi-symmetric rigid circular tube of varying cross-section and permeable wall with slip flow is considered.

Then the radius of the tube is given by

(1)

The governing equation of the pulsatile flow of an incompressible non- Newtonian fluid obeying biviscosity model axisymmetric circular artery are given by:

(2)

(3)

(4)

The normal component of the fluid velocity at the tube wall is given by

(5)

where is the steady state suction/injection velocity, is the ratio of the amplitudes of the oscillatory and steady parts of the suction/injection velocity and is the frequency of the oscillation.

The non-dimensional form is given below:

(6)

(7)

(8)

(9)

where is the Reynolds number of entrance flow, is Womersley’s parameter, is the leakage parameter and and and .

3. Perturbation Technique

We assume that the pulsatile flow consists of the steady part and the oscillatory part of small amplitude of oscillation such that the terms of the order can be neglected (ie ) and the expression of vortices and stream function are given by

(10)

The shear stress is given by

Then using the boundary conditions at and equations (6) and (7), we obtain the dimensionless wall shear stress are given by:

(11)

The pressure drop are given by

(12)

where

4. Results and Discussion

The real part of dimensionless streamlines is plotted for different values of slip parameter, Reynolds number and upper limit of apparent viscosity in Figure 1 of the value of decreases in the converging region and increases in the diverging region of the tube. From Figure 2, Figure 3, it is seen that the similar results occur for a locally constricted tube. The effects of different parameters on the real part of dimensionless pressure drop are indicated graphically through Figure 2. And Figure 3 depicts that of the decreases with increase in and increases with increase in for both suction and injection velocities.

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Figure 1. ψ vs z for sinusiodal tube with injection at the permeable wall

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Figure 2. Δp vs z for sinusoidal tube with suction at the permeable wall

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Figure 3. Δp vs z for sinusoidal tube with injection at the permeable wall

5. Conclusions and Applications

This investigation helps us to note that the influence of permeable parameter in the pressure drop is much significant and decreases rapidly with increases in slip parameter. It is also to be noted that this presentation help us to draw the flow characteristic of blood and the wall shear stress on the inner permeable wall of capillaries and small blood vessels where suction, injection and slip velocities arises and Reynolds number is very low. So, this investigation may be helpful in various fields of medical science.

References

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