NON-NEWTONIAN BLOOD FLOW MODEL WITH THE EFFECT OF DIFFERENT GEOMETRY OF STENOSIS

The objective of this paper is to present a non-Newtonian blood flow model with the effect of different geometry of stenosis on various flow quantities. The Power-law model is considered to explore the non-Newtonian property of blood. Two-point Gauss quadrature formula is applied to obtain the numerical expressions of dimensionless flow resistance, skin-friction and flow rate. The variation of dimensionless flow resistance, skinfriction and flow rate with degree of stenosis, axial distance and power-law index is shown graphically. Moreover, the power-law index is adjusted to explore the non-Newtonian characteristics of blood. The importance of the present work has been carried out by comparing the results with other theories both numerically and graphically. It has been found that resistance to flow becomes maximum with total blockage of artery for different shape of stenosis.


INTRODUCTION
Diseases in human blood vessels and in heart such as heart attack and brain stroke are the main cause of death worldwide. The major reason for such diseases is the constriction in artery or narrowing of the artery. In medical term, it is called stenosis. Stenosis is a pathological constriction of an artery, generally due to deposition of fat, cholesterol and unusual development of tissues.
The fatty substances reduce the cross-sectional area of blood vessels and hence each body part cannot receive an adequate quantity of blood. Therefore, the analytical study of blood flow through a stenosed artery is beneficial for a better understanding and diagnosis of the artery diseases [3,14].
It has been observed that some researchers work on blood by treating it as Newtonian fluid [5,12].
When blood flows at a high shear rate with a larger diameter arteries, then it is considered as Newtonian blood. While blood flows at a low shear rate with smaller diameter arteries, then it would be treated as non-Newtonian blood [9]. Ishikawa et al. [4] observed that non-Newtonian behavior of flow has high stability than Newtonian flow because non-Newtonian characteristics of blood affect the flow quantities. It reduces wall shear stress, vortex size and wall pressure. The Power-law [8,9], Herschel-Bulkley [7,11] and Casson model [10, 16, and 17] of non-Newtonian fluid are frequently used by many researcher. The Herschel-Bulkley model and Power-law model have more benefits than Casson model. The flow-behavior index of Herschel-Bulkley and Powerlaw model can be adjusted to a preferred value, while the flow-behavior index of Casson model is fixed. Furthermore, Easthope and Brooks [1] observed that power-law model has more significant role in modelling of non-Newtonian flow.
Misra and shit [7] studied the generalized model of blood flow by taking it as a Casson and Herschel-Bulkley fluid. Singh et al. [15] considered a blood flow model to analyze the impact of shape parameter and stenosis length on the wall shear stress and flow resistance with axially symmetric but radially non-symmetric stenosed artery. The impact of different parameters of stenosis shape on various flow quantities with slip conditions studied by Haldar [2], Singh [13] and Srivastava [6]. A non-Newtonian blood flow model with effect of stenosis shape and slip velocity at wall is considered by Bhatnagar et al. [18]. They found that axial velocity and flow rate 3 NON-NEWTONIAN BLOOD FLOW MODEL is increased with slip but it decreased with yield stress.
Kamanger et al. [19] investigated the blood flow behavior with severity of blockage area with different geometries like trapezium, triangular and elliptical shape. They conducted their study to different size of stenosis in terms of 70%, 80% and 90% area blockage of artery. They observed that wall shear stress is high for trapezoidal shape as compared to triangular and elliptical shape.
Sriyab [20] considered a non-Newtonian blood flow model with various geometry of shape of stenosis like bell and cosine shape. They observed that cosine shape displays higher pressure, flow resistance and flow rate than bell shape. Owasit and Sriyab [21] considered non-Newtonian blood flow model with various geometry of stenosis in stenosed artery. They studied power-law model of two-dimensional blood flow having vertically asymmetric and symmetric stenosis.
It has been observed that geometry of stenosis has a significant role in study of blood flow. In our present work, we presented a graphical and conceptual study of non-Newtonian blood flow model with effect of different geometry of stenosis, like rectangular, cosine and trapezoidal shape.
Furthermore, the impact of power-law index, depth of stenosis, and non-Newtonian behavior of blood on the various flow quantities for rectangular, cosine and trapezoidal shape are shown graphically.

STENOTIC ARTERY MODEL
We considered a mathematical model for different geometry of arterial stenosis, such as rectangular, cosines and trapezoidal stenosis shape. The geometrical representation of these shapes are shown in figure (1), (2) and (3) respectively.
The mathematical representation of the geometry of the rectangular shape stenosis is given as follows, Where 0 R indicates normal artery radius and ( ) Rz represents stenosed artery radius and S indicates degree of stenosis, We considered here that for rectangular shape, R is constant and equal to min R .

Fig.1. Geometry of axially symmetric rectangular shaped stenosed artery.
The mathematical representation for geometry of cosine shape stenosis is defined as follows, The mathematical representation for geometry of Trapezoidal shape stenosis is defined as

FORMULATION OF THE PROBLEM
In this mathematical model, we considered blood flow in stenosed artery is laminar, steady (independent of time), fully-developed, one-dimensional and blood is taken as non-Newtonian and incompressible fluid. Cylindrical coordinate ( ) ,, rz are taken to study this model, where r represents radial and z indicates axial direction. The governing equation of motion is given [19].
( ) Where, p represents pressure and  indicates shear stress. The power-law model defined the non-Newtonian characteristics of blood as Where, u represents blood velocity,  indicates index of power-law and  shows viscosity of plasma. The boundary conditions to solve eq. (4) and (5)

MATHEMATICAL ANALYSIS
To study non-Newtonian blood flow model for various geometry of stenosis, we have to find analytical expressions for various flow quantities like skin-friction, flow resistance, and flow rate.
In order to find the skin-friction, we integrate eq. (4) with respect to r and using boundary condition The skin-friction, The flow rate is defined by ( ) Using eq. (5) into (10), we get Integrating eq. (11) with respect to  , we get ( ) Now, from eq. (12), we get the expression of skin-friction ( ) The resistance of blood flow ( )  is defined as Where, 1 P represents input pressure and 2 P indicates output pressure of blood in the artery.
Using eq. (13) into eq. (9), we obtain, Now when the geometry is of cosines shape, flow resistance is given by When stenosis is not present in artery for / tan tan The non-dimensional blood skin friction (η) with the impact of Non-Newtonian behaviour of blood is considered as the ratio of the skin friction of Non-Newtonian blood in a stenosed artery to the skin friction of Newtonian blood in a normal artery. Hence, it can be stated as

NUMERICAL RESULT AND DISCUSSION
In this mathematical model, we studied the impact of different shape of stenosis and non -Newtonian behavior of blood on physiologically indispensable flow quantities, like flow rate, skin friction, and flow resistance. For graphical representation, we executed the dimensionless flow quantities from equations (21) to (26) and equations (29) to (34) by using MATLAB.

EFFECT OF DIFFERENT SHAPES OF STENOSIS ON FLOW RESISTANCE
The effect of dimensionless flow resistance (Λ , Λ , Λ ) along / 0 with different shape of stenosis is shown in fig. 4 to 6 for different value of α. Since every shape is depend on degree of stenosis S. As / 0 decreases, the degree of stenosis S increases. From fig.4 to 6, it is observed that when α is increasing, the dimensionless resistance of flow for rectangular, cosine and trapezoidal shape is decreasing and it is also decreasing with / 0 . It has been noticed that as value of / 0 reaches towards 0, the resistance of flow is attained maximum.
Therefore, retardation in flow is maximum, which represents the location of total blockage of artery and causes heart failure. Moreover, it is also pointed out that non-dimensional flow resistance in rectangular, cosine and trapezoidal shape is decreasing slightly when the blood takes Newtonian property i.e. = 1. A comparison of results are done with the result of karimi and it has been observed that results agree with the result of karimi.