EFFECT OF EXTERNALLY APPLIED TRANSVERSE MAGNETIC FIELD ON BLOOD FLOW IN TAPERED STENOSED ARTERY: BY FROBENIUS METHOD

A mathematical model to observe the behaviour of blood flow in the presence of externally applied transverse magnetic field in a tapered stenosed artery is developed. Blood flow is assumed to follow Bingham plastic model. The effect of constant and variable viscosity on blood flow has been studied. To find expressions for important flow characteristics; volumetric flow rate, radial velocity and wall shear stress, the system of non-linear differential equations is solved by Frobenius method. Matlab 10.7.0 is used to analyze the behavior of blood flow. It is observed that these important flow characteristics change their behavior in stenosed tapered artery and externally applied transverse magnetic field helps in regulating these changes up to some extent.


INTRODUCTION
Cardiovascular diseases are one of the main causes of death in both developed and developing countries. Atherosclerosis is a type of arterial disease, which is due to cholesterol plaque in inner 1905 EFFECT OF TRANSVERSE MAGNETIC BY FROBENIUS METHOD wall of the artery. This narrows the artery passage and a loss of elasticity in the artery. Modeling and analysis of hemodynamic of human vascular system improve our understanding of vascular disease and provide valuable insights, which can help in the development of efficient treatment methods. Stenosis is the partial occlusion of blood vessels, which carries blood to heart. Because of this partial occlusion, the blood supply to the heart suffers; as a result, there will be changes in blood pressure, which causes heart diseases.
Magnetohydrodynamics is the study of magnetic properties of electrically conducting fluid. In 1840, an experimental study of flow in capillaries has been conducted by the French physician Poiseuille and is well-known as Poiseuille's flow. Later, in last two decades, lot of research has been carried out on blood flow dynamics under the influence of magnetic field. In particular, the study of flow of blood in stenosed arteries under the influence of magnetic field has been carried out with various perspectives by developing / modifying mathematical models. The first Blood Flow Dynamic (BFD) model was developed mathematically by Haik and is further extended by Tzirtzilakis and Kafoussias in 2001. Since then, many researchers have developed/modified many mathematical models [1][2][3][4]. It is reported that the normal flow changes to disturbed flow in presence of stenosis, resulting into atherosclerosis [5]. The effect of magnetization on blood flow in mild stenosed artery in the presence of erythrocytes through an intended tube is discussed by developing a mathematical model using Frobenius method [6,7]. The stress increases significantly with increase in magnetization. Increase in pressure gradient corresponds to the increase in both magnetic field and hematocrit but the increase in magnetic field is more substantial than that of hematocrit [8,9]. Around the turn of 20 th century, it has been observed that blood exhibits non-Newtonian behavior and because of complexity of problem, it is concluded that unsteady state non-Newtonian blood flow can be modeled only by higher order equations [10][11][12]. Externally applied transverse magnetic field of different intensities on time dependent blood flow model in non-tapered artery affects all the flow characteristics in multi stenosed artery. It is also observed that, wall shear stress increases, flow pattern changes in non-Newtonian rheology, as compared to the results of Newtonian rheology [13]. A pulsatile flow 3D model of human aorta at different stages of atherosclerosis is studied by considering 50% and 80% stenosis to the healthy aorta geometry, found that the wall shear stress distribution and magnitude are strongly affected by size of stenosis, its features and location [14,15]. This paper deals with pulsatile flow of blood in presence of externally applied transverse magnetic field through a tapered stenosed artery. Two cases are considered, a) blood viscosity is constant 1906 B. S. VEENA, ARUNDHATI WARKE and b) blood viscosity varying with shear stress (depends on hematocrit and position). The nonlinear differential equations governing the fluid flow are solved using Frobenius method and the expressions for velocity, flow rate and shear stress are obtained. This study is useful to regulate blood flow in tapered stenosed artery.

PROBLEM FORMULATION
The schematic diagram of tapered stenosed artery with tapering angle φ is as shown in Fig. 1. If φ 0  then the artery is converging, if φ 0  then diverging, and if φ 0 = then the artery is non-tapered. The mathematical model of tapered stenosed artery is given by the equation where φ is the tapering angle, m = tan φ , h = 4x0.4 Ro sec φ = height of stenosis, Ro is radius of non-stenosed part, R(z) is radius of stenosed part, l is length of stenosis d is starting point of stenosis and z is axial coordinate [16].
The problem is formulated by assuming the blood flow as unsteady, laminar and fully developed pulsatile flow, blood is incompressible and transverse magnetic field is applied externally with negligible electric field.
The governing equation satisfying the above conditions is ( )

EFFECT OF TRANSVERSE MAGNETIC BY FROBENIUS METHOD
where P is pressure, z is axial coordinate, r is radial coordinate, k is permeability, M is magnetization, τ is shear stress, ρ is density, u is axial velocity and T is time parameter.
Blood is assumed to be non-Newtonian, and it follows Bingham plastic fluid model, i.e.: 0 u r Here 0 τ is yield stress and μ is viscosity.
Boundary conditions are The following radial coordinate transformations are used:' where R0 is length and t0 is time scaling parameter. Then So using (7), equation (6) (11) where He is hematocrit value and n is the parameter determining the shape of flow.
As the flow is pulsatile, ( ) ( ) Using (13) and (14) in (12), The important force for blood to flow is local pressure gradient which depends on heart pressure pulse and is periodic in nature. Hence, it is assumed that where w =2πf, f is pulse frequency and c is amplitude of pulsatile flow. Hence (15) becomes

SOLUTION USING FROBENIUS METHOD
Frobenius method is used to solve equation (17) and for this, U must be bounded at y = 0. Therefore, the only admissible solution is 2?
where Ai and Bi are series constants, l is arbitrary constant to be determined using boundary conditions. Here Using no slip condition U = 0 at (20)

Solution of homogeneous part:
Using U, dU dy and 2 2 dU dy in LHS of (17), we get Comparing the coefficients of y i+1 , we get Hence, with 00 1 AB == and 0 mm AB −− == .

RESULTS AND DISCUSSION
The list of parameters and their corresponding values used for simulation is given in the Table 1.
As the flow is pulsatile, the flow characteristics are in the form of imaginary numbers. So, to observe the profile of these important flow characteristics, only real part is considered.

Axial velocity
Substituting the value of l in (18), we get,     The table 2 gives the axial velocity profile for different tapering angles at 4 z = . It is observed that axial velocity decreases towards the wall of artery. As the artery is getting widened ( φ =0.02), axial velocity will be more as compared to axial velocity in less tapered artery. Also, axial velocity decreases faster as tapering angle decreases.  Table 4 presents the profile of axial velocity at different z in diverging artery keeping the remaining parameters constant. It also shows the radii of artery for different z at φ =0.01 It is found that axial velocity decreases as r increases irrespective of the position. It can also be observed that axial velocity decreases as the radius of artery increases.

Volumetric flow rate
The volumetric flow rate is  The following discussion will give the profiles of volumetric flow rate for a range of parameters of interest. The rate of decrease in the first half is less and rate of increase in the next half is more in diverging artery and rate of decrease is more in the first part and is less in the second half in case of converging artery. A comparison of flow rate profiles for different magnetic field gradients at 0.2 r = in diverging artery is shown in Fig. 3. It is observed that volumetric flow rate increases with increase in magnetic field gradient. Rate of change in volumetric flow rate is less in the beginning and is more in the later part of stenosis. There is no major change in the profile of volumetric flow rate in the middle of stenosis. Hence it can be concluded that by increasing magnetic field gradient, flow rate can be increased.  Volumetric flow rate at different Darcy numbers is shown in Fig. 4. It can be observed that as z increases from 2 to 5, flow rate decreases drastically and then as z increases from 5 to 7, flow

Wall shear stress
The wall shear stress is given by  Fig . 5 shows the influence of magnetic field intensity on wall shear stress for fixed value of other parameters. As z increases from 2 to 4, wall shear stress increases slowly and as z increases from 4 to 7, wall shear stress decreases drastically in diverging artery. As magnetic field intensity increases, wall shear stress decreases.
The variation in wall shear stress with z for different values of pressure gradient in converging artery is given in table 7. Wall shear increases drastically in the first part of stenosis and then decreases slowly. The same phenomena is observed for all values of pressure gradient, however the rate of change may be different. It is also observed that as pressure gradient increases, shear stress increases.
It is noticed from table 8 that as z increases from 2 to 5, wall shear stress increases and then it decreases in the converging artery in all cases of Womersley number. No considerable change in wall shear stress with respect to Womersley number is observed, however as Womersley number increases, wall shear stress decreases slightly at z = 2 and increases slightly for z >=3.

CONCLUSION
A pulsatile flow of blood is studied in tapered stenosed artery in the presence of externally applied transverse magnetic field with variable viscosity by assuming blood follow non-Newtonian path.
In this time dependent model, Bingham plastic model -a non-Newtonian model is used to relate shear stress and shear rate. Blood viscosity is allowed to vary with radial coordinate and hematocrit.
The expressions for wall shear stress, volumetric flow rate and axial velocity are obtained using Frobenius method and are found to be affected by externally magnetic field. Various combinations of parameters are used to analyze the profile of important flow characteristics. Based on the exhaustive analysis, it is found that tapering angle, viscosity and hematocrit are the important factors influencing the major characteristics of blood. Graphs are plotted to highlight the influence of different parameters on flow rate, shear stress and axial velocity. It is found that these important flow characteristics change their behavior in stenosed artery and the flow is mainly controlled by tapering angle and significantly influenced by the strength and gradient of magnetic field. It is found that axial velocity decreases faster as tapering angle decreases and the rise in yield stress will help in elevating axial velocity.The decrease in the magnitude of velocity and wall shear stress with the increased magnetic field intensity has been observed. The effect of stenosis and yield stress reduces wall shear stress and flow rate in the presence of magnetic field. Change in the flow pattern and increase in wall shear stress are observed by treating blood as Non-Newtonian.
It is also noticed that both Darcy number and Womersley number plays important role in assessing the effect of magnetic field on important flow characteristics. As the Womersley number increases, axial velocity decreases slowly. The rate of change in volumetric flow rate is less for higher values of Womersley number. Also, it is possible to increase the flow rate in the beginning and end of stenosis by increasing Darcy number, but no considerable change in wall shear stress with respect to Womersley number is observed.
It is observed from graphs and tables that the flow can be regulated up to some extent with the help of proper combination of fluid parameters in tapered arteries and application of external transverse magnetic field with suitable pressure gradient and hematocrit level.
This study has a potential to examine the complex flow behavior of blood, under the simultaneous influence of the shape/size of stenosis and strength of externally applied magnetic field. Hence it provides a powerful tool to probe biomechanical behavior of arteries which in turn would be useful in medical field and further research.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.