ff ect of Paired Stenosis on Blood Flow through Small Artery

In this paper the mathematical models have been developed to study the effect of paired stenosis on blood flow, where the blood flow is assumed to behave like a couple stress fluid, peripheral layer plasma (Newtonian fluid) and core layer of suspension of erythrocytes (NonNewtonian fluid). The study predicts that wall shear stress increases with the increase of the two heights of stenosis. Also wall shear stress has proportional direction relation with the separation factor, the maximum value of wall shear stress occurs at the peak separation of paired stenoses.


Introduction
The blood flow through small arteries or arteries with stenosis is considered non-Newtonian Chan et al 2005, Mann and Tarbe 1990, Shukla et al 1980and Nakamura and Sawada 1988.Non -Newtonian flow is the flow that does not obey the Newtonian relationship between shear stress and shear rate.In non Newtonian flow the slope of shear stress versus shear rate curve is not constant and the viscosity of fluid decreases with increasing shear rate.Wall shear rate of non Newtonian is considered less than 100 s-1 Tu and Delwille 1996, Misra et al 1993, Shukla et al 1990and Huang et al 1987.The simple example of a liquid exhibiting non-Newtonian flow is paints and printing inks.The simple constitutive equation is Cossan which is based on a structure model of the interactive behavior of solid and liquid phases of two-phase suspension.The model describes the flow of viscoplastic fluids that can be mathematically described as flow.
√ τ = √ τ y + k.γ , where τ is shear stress, γ is shear rate and k is a Cossan model constant.The abnormal and unnatural growth in the arterial wall thickness that develops at various location of the cardiovascular system under diseased conditions is called arteriosclerosis or stenosis, this can cause serious circulatory disorders by reducing of occluding the blood supply.For instance stenosis in the arteries supplying blood to brain can bring about cerebral strokes, likewise in coronary arteries it can cause myocardial infarction leading to the heart failure.The actual causes of the stenosis are not well known but it has been suggested that the deposits of the cholesterol on the arterial wall and the proliferation of connective tissues may be responsible.Ponalagusamy 2007 andSrivastava 2003 have assumed blood flow through stenosis to behave like a couple stress fluid, peripheral layer plasma Newtonian fluid which acceptable for high shear rate flow (γ > 100 sec-1) in layer arteries and with viscosity coefficients μ p , and a core region of suspension of erythrocytes non-Newtonian fluid which acceptable for low shear rate flow (γ <100 sec-1) through small diameter arteries and with viscosity coefficients μ c , Sanker and Ahmmed 2009 have assumed the non-Newtonian fluid in core layer as Casson fluid and indicated that Casson fluid model holds for blood flow in tubes of diameter 0.013 -0.13 cm also estimated the range of wss 1.677 -2.290 for hieght stenosis range 0.025 -0.125.
The initiation and localization of arteriosclerosis is closely related to local hemodynamic factor (such as wall shear stress, hydrostatic pressure and dynamic pressure).Wall shear stress is an important factor in the study of blood flow.Accurate predictions of the distribution of the wall shear stress are particularly useful for the understanding of the effect of blood flow on endothelial cells.Musad and Khan 2010 have developed mathematical models to study wall shear stress in larg and small artery with single stenosis.Thus in this paper we extend the models to study blood flow through artery with paired stenosis.Johnston and Kilpatrick 1990 have considered a mathematical model of paired stenosis.The two velocities of both layers have reported by Kapur 1985.

The Model Devloped
Consider an axially symmetric flow of blood which assumed to be incompressible in the two directions through a rigid walled artery with an axially symmetric paired stenosis.Also we consider the flow of a plasma layer and a core layer with viscosity coefficient μ p , and μ c respectively, we also assume that the plasma layer is of uniform thickness and δ 1 , δ 2 are the hight of stenosis 1 and stenosis 2 respectively The velocity of plasma layer (Newtonian fluid) and the geometry equation of the stenosis are given by 2z 2  1, Also the velocity of core layer (non-Newtonian fluid ) is given by While the geometry equation of core layer is R 1 (z) = βR(z), where (0 < β < 1) Since blood flow in peripheral is Newtonian, then wall shear stress is given by the equation Where τ is the shear stress, μ p is the viscosity of plasma, ∂v/∂r is the velocity gradient or rate of deformation (shear rate), v is the velocity, G is the pressure gradient and R(z) is the radius of artery in the stenosis part and R 0 is the radius of normal part of artery then Derivative(1), then From equations ( 4) and ( 5) we get By integrating(6) we get Equation ( 7) is the equation of wall shear stress of artery with paired stenoses in the case of blood is Newtonian fluid.
Similarly blood flow in core is non-Newtonian, then wall shear stress is given by the equation.
The velocity of blood flow in core region is given in equation ( 3), let Derivative(9), then from eq(8) and eq(10) we get .95 and substitute μ m in eq(13)where μ m =( μ c μ p − 1) then we get Equation ( 14) is the equation of wall shear stress of artery with paired stenoses in the case of blood is non-Newtonian fluid 3) when z 0 = z 1 + z 2 then

Results and Conclusion
The results based on the numerical solution of equation ( 14) by using mathematical software(Microsoft Mathematics 3.0) for z 0 = 0, z 0 = z 1 = z 2 and z 0 = z 1 + z 2 the values of (δ 1 /R 0 ) and (δ 2 /R 0 ) varied from 0.1 to 0.9 and values of (μ c /μ p ) varied from 1 to 2.5, observed that, wall shear stress has direct relation with the separation factor z 0 and wall shear stress is maximum at peak separation z 0 = z 1 + z 2 .Also wall shear stress increases with the increase of one or both heights of stenosis.Mathematically, we see the peak separation in the case of paired stenoses lead to peak separation of wall shear stress in plasma layer (Newtonian fluid), this mean the vessel of paired stenosis or multiple stenosis have wall shear Stress equals the sum of total stresses for each stenosis, while this does not happen in core layer (non-Newtonian fluid), the viscosity of the blood and the yield stress are not doubled by a doubling of stenosis, but change in a certain range.High stress leads to aggregate blood platelets, which in turn lead to the blocked of vessels.
The results listed in the (Tables 1 and 2) and curves (Figures 1 -3) show that the wall shear stress rang is (0.1 to 0.52 pa), pressure gradient rang is (200 to 300 pa /m), viscosity rang is (0.004 to 0,009 pa.-s) and yield stress rang is (0.01 to 0.03 pa.).