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A Casson Fluid Model for Multiple Stenosed Artery in the Presence of Magnetic Field

The flow of blood through a multistenosed artery under the influence of external applied magnetic field is studied. The artery is modeled as a circular tube. The effect of non-Newtonian nature of blood in small blood vessels has been taken into account by modeling blood as a Casson fluid. The effect of magnetic field, height of stenosis, parameter determining the shape of the stenosis on velocity field, volumetric flow rate in stenotic region and wall shear stress at surface of stenosis are obtained and shown graphically. Some important observations regarding the flow of blood in multi stenosed artery are obtained leading to medical interest.


Introduction
Stenosis narrowing of body passage tube or orifice [1] can cause series circulatory disorders by reducing or occluding the blood supply. Stenosis in the arteries supplying blood to the brain can cause cerebral strokes, and in coronary arteries, myocardial infarction, leading to the heart failure. The actual causes of the stenosis are not known, but it has been suggested that cholesterol deposition in arterial wall and profiliferation of connective tissues may be responsible [2], vascular fluid dynamics is reported to play a significant role in the development and progression of the pathological conditions [3].
Blood is suspension of cells in plasma. Due to the presence of hemoglobin (an iron compound) in red cells, blood can be regarded as a suspension of magnetic particle (red cells) in non-magnetic plasma. The effect of a magnetic field on blood flow has been analyzed theoretically by treating blood as an electrically conductive fluid [4]. The conductive flow in the presence of a magnetic field induces voltage and currents, resulting in a decrease in the flow. The importance of heat transfer on artery diseases and blood flow was mentioned by several researcher. Ugulu and Abby [5] claimed that, the heat transfer and a magnetic field have a significant effect on blood flow through constricted artery.
An analytical solution for the steady flow of a viscous fluid through an arbitrary shaped tube of variable crosssection has been presented by Manton [6] [6] for unsteady pulsatile flows. The steady and unsteady flow through channels and tubes of variable cross-section have been studied by Smith [9] and Duck [10]. Mathematical model for analyzing pulsatile flow in a single stenosed vessels have been proposed by Padmanabhan [11], Mehrotha et al. [12] and Mishra and Chakravorty [13].
The studies on the blood flow/unsteady blood flow through an artery with mild stenosis [14,15] ,effect of arterial dispensability on blood flow through model of mild axi-symmetric arterial stenosis [16], flow of micropolar fluid through a tube with a stenosis [17] non-Newtonian aspects of blood flow through stenosed arteries [18], flow of couple stress fluid through stenotic blood vessels [19], pulsatile flow of Casson's fluid through stenosed tube [20], oscillatory flow of blood in a stenosed artery [21] and in a single constrcted blood vessels [22], effect of erythrocytes on blood flow characteristics in an indented tube [23], effect of an externally applied uniform magnetic field on the blood flow in a single consitricted blood vessel [24] were also reported. In recent paper Manadal et al. [25] developed a two dimension mathematical model to study the effect of externally imposed periodic body acceleration on non-Newtonian blood flow through an elastic stenosed artery where the blood is characterized by the generalized power-law model.
In all the above studies none has applied magnetic field. But the application of magneto hydrodynamics principles in medicine and biology is of growing interest in the literature of bio-mathematics [26][27][28]. By Lenz's law, the Lorentz's force will oppose the motion of conducting fluid. Since blood is an electrically conducting fluid, The MHD principles may be used to deaccelerate the flow of blood in a human arterial system and thereby it is useful in the treatment of certain cardiovascular disorders and in the diseases which accelerate blood circulation like hemorrhages and hypertension etc. [29].
Our object in the present work is to study the effect of an externally applied uniform magnetic field on the multi-stenosed artery with core region. Blood is modeled as a Casson fluid by properly accounting for yield stress of blood in small blood vessel. The analytical expressions for the velocities (normal and core region), blood flow rate and wall shear stress are obtained. The effect of external magnetic field and other parameter has been shown graphically in these results.

Mathematical Formulation
Let us consider the Casson fluid motion of blood through a multi-stenosed artery under the influence of an external applied uniform transverse magnetic fluid. The geometry of the stenosis is as shown in Figure 1. We have taken some assumption for solving the model. 1) Let us take the flow of blood as axially symmetric and fully developed (i.e. , flow in z-direction only). This is entirely reasonable and reinforces the fact that in steady-state incompressible flow in a circular tube of uniform cross-section. The velocity does not change in the direction of the flow, except near the entrance and exist regions.
2) Consider blood as a Casson fluid (non-Newtonian) and magnetic fluid. Since red cell is a major biomagnetic substance and blood flow may be influenced by the magnetic field.
3) Consider the transverse magnetic field. Since the biomagnetic fluid (blood) is subjected to a magnetic field, the action of magnetization will introduce a rotational motion to orient the magnetic fluid particle with the magnetic field).
The above assumptions for Navier-Stokes equation is given by (1) where and r  denote the radial and axial coordinates respectively, 0  magnetic permeability, M magnetization, H  magnetic field intensity, pressure and c P   the shear stress. For Casson fluid the relation between shear stress and shear rate is given by Fung [30], where 0   denotes yields stress and  the viscosity of blood.
The boundary conditions appropriate to the problem under study are 1) u is core velocity.

Solution of the Problem
Introducing the following non-dimensional scheme.
where 0 H is external transverse uniform constant magnetic field.
Using the non-dimensional scheme the governing equations from (1)-(3) are written as: The boundary conditions (3a), (3b), (3c) reduce 1)  is maximum height of stenosis where S (≥2) is the parameter for determining the shape of the stenosis.

Solution
On using analytical method in Equations (5)-(7) and using boundary conditions (7a), (7b), (7c) and (8) the expression for velocity u and core velocity are: The volumetric flow rate Q is given by: where c and 0 are the flow rate in core and annular region of stenotic tube.

Q Q
Using the Equations (11) and (12) in Equation (13) The wall shear stress w  is defined as: On differentiating Equation (11) with respect to r and substituting in Equation (14), then w  is given by:

Results and Discussions
In Figures 2(a) shows the axial velocity (u) with radial axis (r) for different values of induced magnetic field gradient (H = d d H z ), when the magnetic field gradient (H = d d H z ) increases then the curve shifts towards the origin. This is due to the fact that as magnetic field applied on the body, the Laurentz force oppose the flow of blood and hence reduces its velocity. This result compare with Das [29] and Ponalagusamy [31].
In Figure 2 Where high shearing velocity produced in order to attain uniform flow rate at given parameter, so the severity of the multi-stenosis affects the axial flow distribution significantly. This result agrees qualitively with Sanyal et al. [32] and Biswas et al. [33]. . This observation is in good agreement with those of Tzirtzilakis [34] although his studies were based on the Newtonian blood flow under the action of an applied magnetic field.
From Figure 4(a), it is clear that the ratio of the stenosis height and radius of the normal tube   0 R  increases the rate of flow diminishes appreciably for radial axis (r). The characterization of blood irrespective of the presence and absence of the magnetic field certainly ensures the importance of blood rheology in the flow phenomena. The flow rate diminishes as the artery gets narrowed gradually. It may be noted further that the flow rate drops sharply with increasing severity of the constriction in the absence of the magnetic field.
In Figure 4  the rate of blood flow increases at r = 0 and then diminishes for become the value of (0 < r < 1). The flow rate becomes higher in the absence of magnetic field ant it gradually diminishes with increasing magnetic field which is in good agreement with these of Haik et al. [35]. good agreement with that of Srivastava and Saxena [36], whose studies were based on a one-dimensional Casson model of the blood flow in rigid arteries under steadystate conditions in the absence of magnetic field. Figure 5( . The wall shear stress increases rapidly as the length of arterial increases z from (0 to 0.5) and gradually decreases for the length of arterial increases z from (0.5 to 1). This is due to large velocity gradient and therefore the severity of the stenosis significantly affects the wall shear stress characteristics. It is also clear from the figure that when magnetic gradient . Increases then the wall shear stress de-creases. However as the characteristic of the non-Newtonian fluid changes from shear-thinning to Newtonian. The wall shear stress is observed when the flowing blood is subjected to externally applied transverse magnetic field. The present stress distribution plays an important role in detecting the aggregation sites of platelets as mentioned by Fry [37].

Conclusion
From the above discussion, it is clear that the magnetic field, ratio of maximum height of stenosis and radius of normal tube and yield stress of the fluid are the strong parameters influencing the flow. It is observed that, in the presence of magnetic field the magnitude of velocity is decreased. The effect of yield stress and stenosis is to reduce the wall shear stress and flow rate in the presence of magnetic field. In view of these arguments, the present study may be more useful to control the blood flow in diseased state.

Acknowledgements
Authors gratefully acknowledge the financial assistance from the UGC major research Project No. 36-320/2008 (SR), for this work first.